LIBRARY

CWWVERSITY OF CALIFORNIA
DAVIS

TH E O R I A

PHILOSOPHISE NATURALIS

REDACTA AD UNICAM LEGEM VIRIUM
IN NATURA EXISTENTIUM,

A V C T O

P^ROGERIO JOSEPHO BOSCOVICH

SOCIETATIS J £ S U,
NUNC AB IPSO PERPOLITA, ET AUCTA,

Ac a plurimis praeccclcntium edifionum
mendis expurgata.

EDITIO VENETA PR1MA

IPSO «fUCTORE PRJESENTE, ET CORRIGENTE.

V E N E T I I S,

MDCCLXIII.

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A THEORY OF

NATURAL PHILOSOPHY

PUT FORWARD AND EXPLAINED BY

ROGER JOSEPH BOSCOVICH, S.J.

LATIN— ENGLISH EDITION

FROM THE TEXT OF THE

FIRST VENETIAN EDITION

PUBLISHED UNDER THE PERSONAL

SUPERINTENDENCE OF THE AUTHOR

IN 1763

WITH

A SHORT LIFE OF BOSCOVICH

CHICAGO LONDON

OPEN COURT PUBLISHING COMPANY

1922

LIBRARY

UNIVERSITY OF CALIFORNIA
DAVIS

PRINTED IN GREAT BRITAIN

BY

BUTLER & TANNER, FROME, ENGLAND

Copyright

PREFACE

HE text presented in this volume is that of the Venetian edition of 1763.
This edition was chosen in preference to the first edition of 1758, published
at Vienna, because, as stated on the title-page, it was the first edition (revised
and enlarged) issued under the personal superintendence of the author.

In the English translation, an endeavour has been made to adhere as
closely as possible to a literal rendering of the Latin ; except that the some-
what lengthy and complicated sentences have been broken up. This has
made necessary slight changes of meaning in several of the connecting words. This will be
noted especially with regard to the word " adeoque ", which Boscovich uses with a variety
of shades of meaning, from " indeed ", " also " or " further ", through " thus ", to a decided
" therefore ", which would have been more correctly rendered by " ideoque ". There is
only one phrase in English that can also take these various shades of meaning, viz., " and so " ;
and this phrase, for the use of which there is some justification in the word " adeo " itself,
has been usually employed.

The punctuation of the Latin is that of the author. It is often misleading to a modern
reader and even irrational ; but to have recast it would have been an onerous task and
something characteristic of the author and his century would have been lost.

My translation has had the advantage of a revision by Mr. A. O. Prickard, M.A., Fellow
of New College, Oxford, whose task has been very onerous, for he has had to watch not
only for flaws in the translation, but also for misprints in the Latin. These were necessarily
many ; in the first place, there was only one original copy available, kindly loaned to me by
the authorities of the Cambridge University Library ; and, as this copy could not leave
my charge, a type-script had to be prepared from which the compositor worked, thus doub-
ling the chance of error. Secondly, there were a large number of misprints, and even
omissions of important words, in the original itself ; for this no discredit can be assigned to
Boscovich ; for, in the printer's preface, we read that four presses were working at the
same time in order to take advantage of the author's temporary presence in Venice. Further,
owing to almost insurmountable difficulties, there have been many delays in the production
of the present edition, causing breaks of continuity in the work of the translator and reviser ;
which have not conduced to success. We trust, however, that no really serious faults remain.
The short life of Boscovich, which follows next after this preface, has been written by
Dr. Branislav Petronievic, Professor of Philosophy at the University of Belgrade. It is to
be regretted that, owing to want of space requiring the omission of several addenda to the
text of the Theoria itself, a large amount of interesting material collected by Professor
Petronievic has had to be left out.

The financial support necessary for the production of such a costly edition as the present
has been met mainly by the Government of the Kingdom of Serbs, Croats and Slovenes ;
and the subsidiary expenses by some Jugo-Slavs interested in the publication.

After the " Life," there follows an " Introduction," in which I have discussed the ideas
of Boscovich, as far as they may be gathered from the text of the Tbeoria alone ; this
also has been cut down, those parts which are clearly presented to the reader in Boscovich's
own Synopsis having been omitted. It is a matter of profound regret to everyone that this
discussion comes from my pen instead of, as was originally arranged, from that of the late
Philip E. P. Jourdain, the well-known mathematical logician ; whose untimely death threw
into my far less capable hands the responsible duties of editorship.

I desire to thank the authorities of the Cambridge University Library, who time after
time over a period of five years have forwarded to me the original text of this work of
Boscovich. Great credit is also due to the staff of Messrs. Butler & Tanner, Frome,
for the care and skill with which they have carried out their share of the work ; and
my special thanks for the unfailing painstaking courtesy accorded to my demands, which were
frequently not in agreement with trade custom.

J. M. CHILD.
MANCHESTER UNIVERSITY,

December, 1921.

LIFE OF ROGER JOSEPH BOSCOVICH

By BRANISLAV PETRONIEVIC'

]HE Slav world, being still in its infancy, has, despite a considerable number
of scientific men, been unable to contribute as largely to general science
as the other great European nations. It has, nevertheless, demonstrated
its capacity of producing scientific works of the highest value. Above
all, as I have elsewhere indicated," it possesses Copernicus, Lobachevski,
Mendeljev, and Boscovich.

In the following article, I propose to describe briefly the life of the
Jugo-Slav, Boscovich, whose principal work is here published for the sixth time ; the first
edition having appeared in 1758, and others in 1759, 1763, 1764, and 1765. The present
text is from the edition of 1763, the first Venetian edition, revised and enlarged.

On his father's side, the family of Boscovich is of purely Serbian origin, his grandfather,
Bosko, having been an orthodox Serbian peasant of the village of Orakova in Herzegovina.
His father, Nikola, was first a merchant in Novi Pazar (Old Serbia), but later settled in
Dubrovnik (Ragusa, the famous republic in Southern Dalmatia), whither his father, Bosko,
soon followed him, and where Nikola became a Roman Catholic. Pavica, Boscovich's
mother, belonged to the Italian family of Betere, which for a century had been established
in Dubrovnik and had become Slavonicized — Bara Betere, Pavica's father, having been a
poet of some reputation in Ragusa.

Roger Joseph Boscovich (Rudjer Josif Boskovic', in Serbo-Croatian) was born at Ragusa
on September i8th, 1711, and was one of the younger members of a large family. He
received his primary and secondary education at the Jesuit College of his native town ;
in 1725 he became a member of the Jesuit order and was sent to Rome, where from 1728
to 1733 he studied philosophy, physics and mathematics in the Collegium Romanum.
From 1733 to 1738 he taught rhetoric and grammar in various Jesuit schools ; he became
Professor of mathematics in the Collegium Romanum, continuing at the same time his
studies in theology, until in 1744 he became a priest and a member of his order.

In 1736, Boscovich began his literary activity with the first fragment, " De Maculis
Solaribus," of a scientific poem, " De Solis ac Lunse Defectibus " ; and almost every
succeeding year he published at least one treatise upon some scientific or philosophic problem.
His reputation as a mathematician was already established when he was commissioned by
Pope Benedict XIV to examine with two other mathematicians the causes of the weakness
in the cupola of St. Peter's at Rome. Shortly after, the same Pope commissioned him to
consider various other problems, such as the drainage of the Pontine marshes, the regulariza-
tion of the Tiber, and so on. In 1756, he was sent by the republic of Lucca to Vienna
as arbiter in a dispute between Lucca and Tuscany. During this stay in Vienna, Boscovich
was commanded by the Empress Maria Theresa to examine the building of the Imperial
Library at Vienna and the cupola of the cathedral at Milan. But this stay in Vienna,
which lasted until 1758, had still more important consequences ; for Boscovich found
time there to finish his principal work, Theoria Philosophies Naturalis ; the publication
was entrusted to a Jesuit, Father Scherffer, Boscovich having to leave Vienna, and the
first edition appeared in 1758, followed by a second edition in the following year. With
both of these editions, Boscovich was to some extent dissatisfied (see the remarks made
by the printer who carried out the third edition at Venice, given in this volume on page 3) ;
so a third edition was issued at Venice, revised, enlarged and rearranged under the author's
personal superintendence in 1763. The revision was so extensive that as the printer
remarks, " it ought to be considered in some measure as a first and original edition " ;
and as such it has been taken as the basis of the translation now published. The fourth
and fifth editions followed in 1764 and 1765.

One of the most important tasks which Boscovich was commissioned to undertake
was that of measuring an arc of the meridian in the Papal States. Boscovich had designed
to take part in a Portuguese expedition to Brazil on a similar errand ; but he was per-

" Slav Achievements in Advanced Science, London, 1917.

vii

viii A THEORY OF NATURAL PHILOSOPHY

suaded by Pope Benedict XIV, in 1750, to conduct, in collaboration with an English Jesuit,
Christopher Maire, the measurements in Italy. The results of their work were published,
in 1755, by Boscovich, in a treatise, De Litter aria Expedition^ -per Pontificiam, &c. ; this
was translated into French under the title of Voyage astronomique et geograpbique dans
VEtat de VEglise, in 1770.

By the numerous scientific treatises and dissertations which he had published up to
1759, and by his principal work, Boscovich had acquired so high a reputation in Italy, nay
in Europe at large, that the membership of numerous academies and learned societies had
already been conferred upon him. In 1760, Boscovich, who hitherto had been bound to
Italy by his professorship at Rome, decided to leave that country. In this year we find
him at Paris, where he had gone as the travelling companion of the Marquis Romagnosi.
Although in the previous year the Jesuit order had been expelled from France, Boscovich
had been received on the strength of his great scientific reputation. Despite this, he did not
feel easy in Paris ; and the same year we find him in London, on a mission to vindicate
the character of his native place, the suspicions of the British Government, that Ragusa was
being used by France to fit out ships of war, having been aroused ; this mission he carried
out successfully. In London he was warmly welcomed, and was made a member of the
Royal Society. Here he published his work, De Solis ac Lunce defectibus, dedicating it to
the Royal Society. Later, he was commissioned by the Royal Society to proceed to Cali-
fornia to observe the transit of Venus ; but, as he was unwilling to go, the Society sent
him to Constantinople for the same purpose. He did not, however, arrive in time to
make the observation ; and, when he did arrive, he fell ill and was forced to remain at
Constantinople for seven months. He left that city in company with the English ambas-
sador, Porter, and, after a journey through Thrace, Bulgaria, and Moldavia, he arrived
finally at Warsaw, in Poland ; here he remained for a time as the guest of the family of
PoniatowsM. In 1762, he returned from Warsaw to Rome by way of Silesia and Austria.
The first part of this long journey has been described by Boscovich himself in his Giornale
di un viaggio da Constantinopoli in Polonia — the original of which was not published until
1784, although a French translation had appeared in 1772, and a German translation
in 1779.

Shortly after his return to Rome, Boscovich was appointed to a chair at the University
of Pavia ; but his stay there was not of long duration. Already, in 1764, the building
of the observatory of Brera had been begun at Milan according to the plans of Boscovich ;
and in 1770, Boscovich was appointed its director. Unfortunately, only two years later
he was deprived of office by the Austrian Government which, in a controversy between
Boscovich and another astronomer of the observatory, the Jesuit Lagrange, took the part
of his opponent. The position of Boscovich was still further complicated by the disbanding
of his company ; for, by the decree of Clement V, the Order of Jesus had been suppressed in
1773. In the same year Boscovich, now free for the second time, again visited Paris, where
he was cordially received in official circles. The French Government appointed him director
of " Optique Marine," with an annual salary of 8,000 francs ; and Boscovich became a
French subject. But, as an ex- Jesuit, he was not welcomed in all scientific circles. The
celebrated d'Alembert was his declared enemy ; on the other hand, the famous astronomer,
Lalande, was his devoted friend and admirer. Particularly, in his controversy with Rochon
on the priority of the discovery of the micrometer, and again in the dispute with Laplace
about priority in the invention of a method for determining the orbits of comets, did
the enmity felt in these scientific circles show itself. In Paris, in 1779, Boscovich
published a new edition of his poem on eclipses, translated into French and annotated,
under the title, Les Eclipses, dedicating the edition to the King, Louis XV.

During this second stay in Paris, Boscovich had prepared a whole series of new works,
which he hoped would have been published at the Royal Press. But, as the American
War of Independence was imminent, he was forced, in 1782, to take two years' leave of
absence, and return to Italy. He went to the house of his publisher at Bassano ; and here,
in 1 785^ were published five volumes of his optical and astronomical works, Opera pertinentia
ad opticam et astronomiam.

Boscovich had planned to return through Italy from Bassano to Paris ; indeed, he left
Bassano for Venice, Rome, Florence, and came to Milan. Here he was detained by illness
and he was obliged to ask the French Government to extend his leave, a request that was
willingly granted. His health, however, became worse ; and to it was added a melancholia.
He died on February I3th, 1787.

The great loss which Science sustained by his death has been fitly commemorated in
the eulogium by his friend Lalande in the French Academy, of which he was a member ;
and also in that of Francesco Ricca at Milan, and so on. But it is his native town, his
beloved Ragusa, which has most fitly celebrated the death of the greatest of her sons

A THEORY OF NATURAL PHILOSOPHY ix

in the eulogium of the poet, Bernardo Zamagna. " This magnificent tribute from his native
town was entirely deserved by Boscovich, both for his scientific works, and for his love and
work for his country.

Boscovich had left his native country when a boy, and returned to it only once after-
wards, when, in 1747, he passed the summer there, from June 20th to October 1st ; but
he often intended to return. In a letter, dated May 3rd, 1774, he seeks to secure a pension
as a member of the Jesuit College of Ragusa ; he writes : " I always hope at last to find
my true peace in my own country and, if God permit me, to pass my old age there in
quietness."

Although Boscovich has written nothing in his own language, he understood it per-
fectly ; as is shown by the correspondence with his sister, by certain passages in his Italian
letters, and also by his Giornale (p. 31 ; p. 59 of the French edition). In a dispute with
d'Alembert, who had called him an Italian, he said : " we will notice here in the first place
that our author is a Dalmatian, and from Ragusa, not Italian ; and that is the reason why
Marucelli, in a recent work on Italian authors, has made no mention of him." * That his
feeling of Slav nationality was strong is proved by the tributes he pays to his native town
and native land in his dedicatory epistle to Louis XV.

Boscovich was at once philosopher, astronomer, physicist, mathematician, historian,
engineer, architect, and poet. In addition, he was a diplomatist and a man of the world ;
and yet a good Catholic and a devoted member of the Jesuit order. His friend, Lalande,
has thus sketched his appearance and his character : " Father Boscovich was of great
stature ; he had a noble expression, and his disposition was obliging. He accommodated
himself with ease to the foibles of the great, with whom he came into frequent contact.
But his temper was a trifle hasty and irascible, even to his friends — at least his manner
gave that impression — but this solitary defect was compensated by all those qualities which
make up a great man. . . . He possessed so strong a constitution that it seemed likely that
he would have lived much longer than he actually did ; but his appetite was large, and his
belief in the strength of his constitution hindered him from paying sufficient attention
to the danger which always results from this." From other sources we learn that Boscovich
had only one meal daily, dejeuner.

Of his ability as a poet, Lalande says : " He was himself a poet like his brother, who was
also a Jesuit. . . . Boscovich wrote verse in Latin only, but he composed with extreme ease.
He hardly ever found himself in company without dashing off some impromptu verses to
well-known men or charming women. To the latter he paid no other attentions, for his
austerity was always exemplary. . . . With such talents, it is not to be wondered at that
he was everywhere appreciated and sought after. Ministers, princes and sovereigns all
received him with the greatest distinction. M. de Lalande witnessed this in every part
of Italy where Boscovich accompanied him in 1765."

Boscovich was acquainted with several languages — Latin, Italian, French, as well as
his native Serbo-Croatian, which, despite his long absence from his country, he did not
forget. Although he had studied in Italy and passed the greater part of his life there,
he had never penetrated to the spirit of the language, as his Italian biographer, Ricca, notices.
His command of French was even more defective ; but in spite of this fact, French men
of science urged him to write in French. English he did not understand, as he confessed
in a letter to Priestley ; although he had picked up some words of polite conversation
during his stay in London.

His correspondence was extensive. The greater part of it has been published in
the Memoirs de VAcademie Jougo-Slave of Zagrab, 1887 to 1912.

" Oratio in funere R. J. Boscovichii ... a Bernardo Zamagna.

* Voyage Astronomique, p. 750 ; also on pp. 707 seq.

• Journal des Sfavans, Fevrier, 1792, pp. 113-118.

INTRODUCTION

ALTHOUGH the title to this work to a very large extent correctly describes
the contents, yet the argument leans less towards the explanation of a
theory than it does towards the logical exposition of the results that must
follow from the acceptance of certain fundamental assumptions, more or
less generally admitted by natural philosophers of the time. The most
important of these assumptions is the doctrine of Continuity, as enunciated
by Leibniz. This doctrine may be shortly stated in the words : " Every-
thing takes place by degrees " ; or, in the phrase usually employed by Boscovich : " Nothing
happens -per saltum." The second assumption is the axiom of Impenetrability ; that is to
say, Boscovich admits as axiomatic that no two material points can occupy the same spatial,
or local, point simultaneously. Clerk Maxwell has characterized this assumption as " an
unwarrantable concession to the vulgar opinion." He considered that this axiom is a
prejudice, or prejudgment, founded on experience of bodies of sensible size. This opinion
of Maxwell cannot however be accepted without dissection into two main heads. The
criticism of the axiom itself would appear to carry greater weight against Boscovich than
against other philosophers ; but the assertion that it is a prejudice is hardly warranted.
For, Boscovich, in accepting the truth of the axiom, has no experience on which to found his
acceptance. His material points have absolutely no magnitude ; they are Euclidean points,
" having no parts." There is, therefore, no reason for assuming, by a sort of induction (and
Boscovich never makes an induction without expressing the reason why such induction can
be made), that two material points cannot occupy the same local point simultaneously ;
that is to say, there cannot have been a prejudice in favour of the acceptance of this axiom,
derived from experience of bodies of sensible size ; for, since the material points are non-
extended, they do not occupy space, and cannot therefore exclude another point from
occupying the same space. Perhaps, we should say the reason is not the same as that which
makes it impossible for bodies of sensible size. The acceptance of the axiom by Boscovich is
purely theoretical ; in fact, it constitutes practically the whole of the theory of Boscovich. On
the other hand, for this very reason, there are no readily apparent grounds for the acceptance
of the axiom ; and no serious arguments can be adduced in its favour ; Boscovich 's own
line of argument, founded on the idea that infinite improbability comes to the same thing
as impossibility, is given in Art. 361. Later, I will suggest the probable source from which
Boscovich derived his idea of impenetrability as applying to points of matter, as distinct
from impenetrability for bodies of sensible size.

Boscovich's own idea of the merit of his work seems to have been chiefly that it met the
requirements which, in the opinion of Newton, would constitute " a mighty advance in
philosophy." These requirements were the " derivation, from the phenomena of Nature,
of two or three general principles ; and the explanation of the manner in which the properties
and actions of all corporeal things follow from these principles, even if the causes of those
principles had not at the time been discovered." Boscovich claims in his preface to the
first edition (Vienna, 1758) that he has gone far beyond these requirements ; in that he has
reduced all the principles of Newton to a single principle — namely, that given by his Law
of Forces.

The occasion that led to the writing of this work was a request, made by Father Scherffer,
who eventually took charge of the first Vienna edition during the absence of Boscovich ; he
suggested to Boscovich the investigation of the centre of oscillation. Boscovich applied to
this investigation the principles which, as he himself states, " he lit upon so far back as the
year 1745." Of these principles he had already given some indication in the dissertations
De Viribus vivis (published in 1745), De Lege Firium in Natura existentium (1755), and
others. While engaged on the former dissertation, he investigated the production and
destruction of velocity in the case of impulsive action, such as occurs in direct collision.
In this, where it is to be noted that bodies of sensible size are under consideration, Boscovich
was led to the study of the distortion and recovery of shape which occurs on impact ; he
came to the conclusion that, owing to this distortion and recovery of shape, there was
produced by the impact a continuous retardation of the relative velocity during the whole
time of impact, which was finite ; in other words, the Law of Continuity, as enunciated by

XI

xii INTRODUCTION

Leibniz, was observed. It would appear that at this time (1745) Boscovich was concerned
mainly, if not solely, with the facts of the change of velocity, and not with the causes for
this change. The title of the dissertation, De Firibus vivis, shows however that a secondary
consideration, of almost equal importance in the development of the Theory of Boscovich,
also held the field. The natural philosophy of Leibniz postulated monads, without parts,
extension or figure. In these features the monads of Leibniz were similar to the material
points of Boscovich ; but Leibniz ascribed to his monads 1 perception and appetition in
addition to an equivalent of inertia. They are centres of force, and the force exerted is a
vis viva. Boscovich opposes this idea of a " living," or " lively " force ; and in this first
dissertation we may trace the first ideas of the formulation of his own material points.
Leibniz denies action at a distance ; with Boscovich it is the fundamental characteristic of
a material point.

The principles developed in the work on collisions of bodies were applied to the problem
of the centre of oscillation. During the latter investigation Boscovich was led to a theorem
on the mutual forces between the bodies forming a system of three ; and from this theorem
there followed the natural explanation of a whole sequence of phenomena, mostly connected
with the idea of a statical moment ; and his initial intention was to have published a
dissertation on this theorem and deductions from it, as a specimen of the use and advantage
of his principles. But all this time these principles had been developing in two directions,
mathematically and philosophically, and by this time included the fundamental notions
of the law of forces for material points. The essay on the centre of oscillation grew in length
as it proceeded ; until, finally, Boscovich added to it all that he had already published on
the subject of his principles and other matters which, as he says, " obtruded themselves on
his notice as he was writing." The whole of this material he rearranged into a more logical
(but unfortunately for a study of development of ideas, non-chronological) order before
publication.

As stated by Boscovich, in Art. 164, the whole of his Theory is contained in his statement
that : " Matter is composed of perfectly indivisible, non-extended, discrete points." To this
assertion is conjoined the axiom that no two material points can be in the same point of
space at the same time. As stated above, in opposition to Clerk Maxwell, this is no matter
of prejudice. Boscovich, in Art. 361, gives his own reasons for taking this axiom as part
of his theory. He lays it down that the number of material points is finite, whereas the
number of local points is an infinity of three dimensions ; hence it is infinitely improbable,
i.e., impossible, that two material points, without the action of a directive mind, should
ever encounter one another, and thus be in the same place at the same time. He even goes
further ; he asserts elsewhere that no material point ever returns to any point of space in
which it has ever been before, or in which any other material point has ever been. Whether
his arguments are sound or not, the matter does not rest on a prejudgment formed from
experience of bodies of sensible size ; Boscovich has convinced himself by such arguments
of the truth of the principle of Impenetrability, and lays it down as axiomatic ; and upon
this, as one of his foundations, builds his complete theory. The consequence of this axiom
is immediately evident ; there can be no such thing as contact between any two material
points ; two points cannot be contiguous or, as Boscovich states, no two points of matter
can be in mathematical contact. For, since material points have no
dimensions, if, to form an imagery of Boscovich's argument, we take
two little squares ABDC, CDFE to represent two points in mathema-
tical contact along the side CD, then CD must also coincide with AB,
and EF with CD ; that is the points which we have supposed to be
contiguous must also be coincident. This is contrary to the axiom of
Impenetrability ; and hence material points must be separated always O U Ir
by a finite interval, no matter how small. This finite interval however
has no minimum ; nor has it, on the other hand, on account of the infinity of space, any
maximum, except under certain hypothetical circumstances which may possibly exist.
Lastly, these points of matter float, so to speak, in an absolute void.

Every material point is exactly like every other material point ; each is postulated to
have an inherent propensity (determinatio) to remain in a state of rest or uniform motion in
a straight line, whichever of these is supposed to be its initial state, so long as the point is
not subject to some external influence. Thus it is endowed with an equivalent of inertia
as formulated by Newton ; but as we shall see, there does not enter the Newtonian idea
of inertia as a characteristic of mass. The propensity is akin to the characteristic ascribed
to the monad by Leibniz ; with this difference, that it is not a symptom of activity, as with
Leibniz, but one of inactivity.

1 See Bertrand Russell, Philosophy of Leibniz ; especially p. 91 for connection between Boscovich and Leibniz.

INTRODUCTION xiii

Further, according to Boscovich, there is a mutual vis between every pair of points,
the magnitude of which depends only on the distance between them. At first sight, there
would seem to be an incongruity in this supposition ; for, since a point has no magnitude,
it cannot have any mass, considered as " quantity of matter " ; and therefore, if the slightest
" force " (according to the ordinary acceptation of the term) existed between two points,
there would be an infinite acceleration or retardation of each point relative to the other.
If, on the other hand, we consider with Clerk Maxwell that each point of matter has a
definite small mass, this mass must be finite, no matter how small, and not infinitesimal.
For the mass of a point is the whole mass of a body, divided by the number of points of
matter composing that body, which are all exactly similar ; and this number Boscovich
asserts is finite. It follows immediately that the density of a material point must be infinite,
since the volume is an infinitesimal of the third order, if not of an infinite order, i.e., zero.
Now, infinite density, if not to all of us, to Boscovich at least is unimaginable. Clerk
Maxwell, in ascribing mass to a Boscovichian point of matter, seems to have been obsessed
by a prejudice, that very prejudice which obsesses most scientists of the present day, namely,
that there can be no force without mass. He understood that Boscovich ascribed to each
pair of points a mutual attraction or repulsion ; and, in consequence, prejudiced by Newton's
Laws of Motion, he ascribed mass to a material point of Boscovich.

This apparent incongruity, however, disappears when it is remembered that the word
vis, as used by the mathematicians of the period of Boscovich, had many different meanings ;
or rather that its meaning was given by the descriptive adjective that was associated with it.
Thus we have vis viva (later associated with energy), vis mortua (the antithesis of vis viva,
as understood by Leibniz), vis acceleratrix (acceleration), vis matrix (the real equivalent
of force, since it varied with the mass directly), vis descensiva (moment of a weight hung at
one end of a lever), and so on. Newton even, in enunciating his law of universal gravitation,
apparently asserted nothing more than the fact of gravitation — a propensity for approach —
according to the inverse square of the distance : and Boscovich imitates him in this. The
mutual vires, ascribed by Boscovich to his pairs of points, are really accelerations, i.e.
tendencies for mutual approach or recession of the two points, depending on the distance
between the points at the time under consideration. Boscovich's own words, as given in
Art. 9, are : " Censeo igitur bina quaecunque materise puncta determinari asque in aliis
distantiis ad mutuum accessum, in aliis ad recessum mutuum, quam ipsam determinationem
apello vim." The cause of this determination, or propensity, for approach or recession,
which in the case of bodies of sensible size is more correctly called " force " (vis matrix),
Boscovich does not seek to explain ; he merely postulates the propensities. The measures
of these propensities, i.e., the accelerations of the relative velocities, are the ordinates of
what is usually called his curve of forces. This is corroborated by the statement of Boscovich
that the areas under the arcs of his curve are proportional to squares of velocities ; which
is in accordance with the formula we should now use for the area under an " acceleration-
space " graph (Area = J f.ds = j-r-ds = I v.dv). See Note (f) to Art. 118, where it is

evident that the word vires, translated " forces," strictly means " accelerations ; " seejalso Art.64-
Thus it would appear that in the Theory of Boscovich we have something totally
different from the monads of Leibniz, which are truly centres of force. Again, although
there are some points of similarity with the ideas of Newton, more especially in the
postulation of an acceleration of the relative velocity of every pair of points of matter due
to and depending upon the relative distance between them, without any endeavour to
explain this acceleration or gravitation ; yet the Theory of Boscovich differs from that of
Newton in being purely kinematical. His material point is defined to be without parts,
i.e., it has no volume ; as such it can have no mass, and can exert no force, as we understand
such terms. The sole characteristic that has a finite measure is the relative acceleration
produced by the simultaneous existence of two points of matter ; and this acceleration
depends solely upon the distance between them. The Newtonian idea of mass is replaced
by something totally different ; it is a mere number, without " dimension " ; the " mass "
of a body is simply the number of points that are combined to " form " the body.

Each of these points, if sufficiently close together, will exert on another point of matter,
at a relatively much greater distance from every point of the body, the same acceleration
very approximately. Hence, if we have two small bodies A and B, situated at a distance s
from one another (the wording of this phrase postulates that the points of each body are
very close together as compared with the distance between the bodies) : and if the number
of points in A and B are respectively a and b, and / is the mutual acceleration between any
pair of material points at a distance s from one another ; then, each point of A will give to
each point of B an acceleration /. Hence, the body A will give to each point of B, and
therefore to the whole of B, an acceleration equal to a/. Similarly the body B will give to

xiv INTRODUCTION

the body A an acceleration equal to bf. Similarly, if we placed a third body, C, at a distance
j from A and B, the body A would give the body C an acceleration equal to af, and the body
B would give the body C an acceleration equal to bf. That is, the accelerations given to a
standard body C are proportional to the " number of points " in the bodies producing
these accelerations ; thus, numerically, the " mass " of Boscovich comes to the same thing
as the " mass " of Newton. Further, the acceleration given by C to the bodies A and B
is the same for either, namely, cf ; from which it follows that all bodies have their velocities
of fall towards the earth equally accelerated, apart from the resistance of the air ; and so on.
But the term " force," as the cause of acceleration is not applied by Boscovich to material
points ; nor is it used in the Newtonian sense at all. When Boscovich investigates the
attraction of " bodies," he introduces the idea of a cause, but then only more or less as a
convenient phrase. Although, as a philosopher, Boscovich denies that there is any possibility
of a fortuitous circumstance (and here indeed we may admit a prejudice derived from
experience ; for he states that what we call fortuitous is merely something for which we,
in our limited intelligence, can assign no cause), yet with him the existent thing is motion
and not force. The latter word is merely a convenient phrase to describe the " product " of
" mass " and " acceleration."

To sum up, it would seem that the curve of Boscovich is an acceleration-interval graph ;
and it is a mistake to refer to his cosmic system as a system of " force-centres." His material
points have zero volume, zero mass, and exert zero force. In fact, if one material point
alone existed outside the mind, and there were no material point forming part of the mind,
then this single external point could in no way be perceived. In other words, a single
point would give no sense-datum apart from another point ; and thus single points might
be considered as not perceptible in themselves, but as becoming so in relation to other
material points. This seems to be the logical deduction from the strict sense of the
definition given by Boscovich ; what Boscovich himself thought is given in the supplements
that follow the third part of the treatise. Nevertheless, the phraseology of " attraction "
and " repulsion " is so much more convenient than that of " acceleration of the velocity of
approach " and " acceleration of the velocity of recession," that it will be used in what
follows : as it has been used throughout the translation of the treatise.

There is still another point to be considered before we take up the study of the Boscovich
curve ; namely, whether we are to consider Boscovich as, consciously or unconsciously, an
atomist in the strict sense of the word. The practical test for this question would seem
to be simply whether the divisibility of matter was considered to be limited or unlimited.
Boscovich himself appears to be uncertain of his ground, hardly knowing which point of
view is the logical outcome of his definition of a material point. For, in Art. 394, he denies
infinite divisibility ; but he admits infinite componibility. The denial of infinite divisibility
is necessitated by his denial of " anything infinite in Nature, or in extension, or a self-
determined infinitely small." The admission of infinite componibility is necessitated by
his definition of the material point ; since it has no parts, a fresh point can always be placed
between any two points without being contiguous to either. Now, since he denies the
existence of the infinite and the infinitely small, the attraction or repulsion between two
points of matter (except at what he calls the limiting intervals) must be finite : hence, since
the attractions of masses are all by observation finite, it follows that the number of points
in a mass must be finite. To evade the difficulty thus raised, he appeals to the scale of
integers, in which there is no infinite number : but, as he says, the scale of integers is a
sequence of numbers increasing indefinitely, and having no last term. Thus, into any space,
however small, there may be crowded an indefinitely great number of material points ; this
number can be still further increased to any extent ; and yet the number of points finally
obtained is always finite. It would, again, seem that the system of Boscovich was not a
material system, but a system of relations ; if it were not for the fact that he asserts, in
Art. 7, that his view is that " the Universe does not consist of vacuum interspersed amongst
matter, but that matter is interspersed in a vacuum and floats in it." The whole question
is still further complicated by his remark, in Art. 393, that in the continual division of a
body, " as soon as we reach intervals less than the distance between two material points,
further sections will cut empty intervals and not matter " ; and yet he has postulated that
there is no minimum value to the interval between two material points. Leaving, however,
this question of the philosophical standpoint of Boscovich to be decided by the reader, after
a study of the supplements that follow the third part of the treatise, let us now consider the
curve of Boscovich.

Boscovich, from experimental data, gives to his curve, when the interval is large, a
branch asymptotic to the axis of intervals ; it approximates to the " hyperbola " x*y— c, in
which x represents the interval between two points, and y the vis corresponding to that
interval, which we have agreed to call an attraction, meaning thereby, not a force, but an

INTRODUCTION xv

acceleration of the velocity of approach. For small intervals he has as yet no knowledge
of the quality or quantity of his ordinates. In Supplement IV, he gives some very ingenious
arguments against forces that are attractive at very small distances and increase indefinitely,
such as would be the case where the law of forces was represented by an inverse power of
the interval, or even where the force varied inversely as the interval. For the inverse fourth
or higher power, he shows that the attraction of a sphere upon a point on its surface would
be less than the attraction of a part of itself on this point ; for the inverse third power, he con-
siders orbital motion, which in this case is an equiangular spiral motion, and deduces that
after a finite time the particle must be nowhere at all. Euler, considering this case, asserted
that on approaching the centre of force the particle must be annihilated ; Boscovich, with
more justice, argues that this law of force must be impossible. For the inverse square law,
the limiting case of an elliptic orbit, when the transverse velocity at the end of the major
axis is decreased indefinitely, is taken ; this leads to rectilinear motion of the particle to the
centre of force and a return from it ; which does not agree with the otherwise proved
oscillation through the centre of force to an equal distance on either side.

Now it is to be observed that this supplement is quoted from his dissertation De Lege
Firium in Natura existentium, which was published in 1755 ; also that in 1743 he had
published a dissertation of which the full title is : De Motu Corporis attracti in centrum
immobile viribus decrescentibus in ratione distantiarum reciproca duplicata in spatiis non
resistentibus. Hence it is not too much to suppose that somewhere between 1741 and 1755
he had tried to find a means of overcoming this discrepancy ; and he was thus led to suppose
that, in the case of rectilinear motion under an inverse square law, there was a departure
from the law on near approach to the centre of force ; that the attraction was replaced by a
repulsion increasing indefinitely as the distance decreased ; for this obviously would lead to
an oscillation to the centre and back, and so come into agreement with the limiting case of
the elliptic orbit. I therefore suggest that it was this consideration that led Boscovich to
the doctrine of Impenetrability. However, in the treatise itself, Boscovich postulates the
axiom of Impenetrability as applying in general, and thence argues that the force at infinitely
small distances must be repulsive and increasing indefinitely. Hence the ordinate to the
curve near the origin must be drawn in the opposite direction to that of the ordinates for
sensible distances, and the area under this branch of the curve must be indefinitely great.
That is to say, the branch must be asymptotic to the axis of ordinates ; Boscovich however
considers that this does not involve an infinite ordinate at the origin, because the interval
between two material points is never zero ; or, vice versa, since the repulsion increases
indefinitely for very small intervals, the velocity of relative approach, no matter how great,
of two material points is always destroyed before actual contact ; which necessitates a finite
interval between two material points, and the impossibility of encounter under any circum-
stances : the interval however, since a velocity of mutual approach may be supposed to be
of any magnitude, can have no minimum. Two points are said to be in physical contact,
in opposition to mathematical contact, when they are so close together that this great mutual
repulsion is sufficiently increased to prevent nearer approach.

Since Boscovich has these two asymptotic branches, and he postulates Continuity,
there must be a continuous curve, with a one-valued ordinate for any interval, to represent
the " force " at all other distances ; hence the curve must cut the axis at some point in
between, or the ordinate must become infinite. He does not lose sight of this latter possi-
bility, but apparently discards it for certain mechanical and physical reasons. Now, it is
known that as the degree of a curve rises, the number of curves of that degree increases very
rapidly ; there is only one of the first degree, the conic sections of the second degree, while
Newton had found over three-score curves with equations of the third degree, and nobody
had tried to find all the curves of the fourth degree. Since his curve is not one of the known
curves, Boscovich concludes that the degree of its equation is very high, even if it is not
transcendent. But the higher the degree of a curve, the greater the number of possible
intersections with a given straight line ; that is to say, it is highly probable that there are a
great many intersections of the curve with the axis ; i.e., points giving zero action for
material points situated "at the corresponding distance from one another. Lastly, since the
ordinate is one-valued, the equation of the curve, as stated in Supplement III, must be of
the form P-Qy = o, where P and Q are functions of x alone. Thus we have a curve winding
about the axis for intervals that are very small and developing finally into the hyperbola of
the third degree for sensible intervals. This final branch, however, cannot be exactly this
hyperbola ; for, Boscovich argues, if any finite arc of the curve ever coincided exactly with
the hyperbola of the third degree, it would be a breach of continuity if it ever departed from
it. Hence he concludes that the inverse square law is observed approximately only, even
at large distances.

As stated above, the possibility of other asymptotes, parallel to the asymptote at the

INTRODUCTION

origin, is not lost sight of. The consequence of one occurring at a very small distance from
the origin is discussed in full. Boscovich, however, takes great pains to show that all the
phenomena discussed can be explained on the assumption of a number of points of inter-
section of his curve with the axis, combined with different characteristics of the arcs that lie
between these points of intersection. There is, however, one suggestion that is very
interesting, especially in relation to recent statements of Einstein and Weyl. Suppose that
beyond the distances of the solar system, for which the inverse square law obtains approxi-
mately at least, the curve of forces, after touching the axis (as it may do, since it does not
coincide exactly with the hyperbola of the third degree), goes off to infinity in the positive
direction ; or suppose that, after cutting the axis (as again it may do, for the reason given
above), it once more begins to wind round the axis and finally has an asymptotic attractive
branch. Then it is evident that the universe in which we live is a self-contained cosmic
system ; for no point within it can ever get beyond the distance of this further asymptote.
If in addition, beyond this further asymptote, the curve had an asymptotic repulsive branch
and went on as a sort of replica of the curve already obtained, then no point outside our
universe could ever enter within it. Thus there is a possibility of infinite space being
filled with a succession of cosmic systems, each of which would never interfere with any
other ; indeed, a mind existing in any one of these universes could never perceive the
existence of any other universe except that in which it existed. Thus space might be in
reality infinite, and yet never could be perceived except as finite.

The use Boscovich makes of his curve, the ingenuity of his explanations and their logic,
the strength or weakness of his attacks on the theories of other philosophers, are left to the
consideration of the reader of the text. It may, however, be useful to point out certain
matters which seem more than usually interesting. Boscovich points out that no philosopher
has attempted to prove the existence of a centre of gravity. It would appear especially that
he is, somehow or other, aware of the mistake made by Leibniz in his early days (a mistake
corrected by Huygens according to the statement of Leibniz), and of the use Leibniz later
made of the principle of moments ; Boscovich has apparently considered the work of Pascal
and others, especially Guldinus ;, it looks almost as if (again, somehow or other) he had seen
some description of " The Method " of Archimedes. For he proceeds to define the centre
of gravity geometrically, and to prove that there is always a centre of gravity, or rather a
geometrical centroid ; whereas, even for a triangle, there is no centre of magnitude, with
which Leibniz seems to have confused a centroid before his conversation with Huygens.
This existence proof, and the deductions from it, are necessary foundations for the centro-
baryc analysis of Leibniz. The argument is shortly as follows : Take a plane outside, say
to the right of, all the points of all the bodies under consideration ; find the sum of all the
distances of all the points from this plane ; divide this sum by the number of points ; draw
a plane to the left of and parallel to the chosen plane, at a distance from it equal to the
quotient just found. Then, observing algebraic sign, this is a plane such that the sum of
the distances of all the points from it is zero ; i.e., the sum of the distances of all the points
on one side of this plane is equal arithmetically to the sum of the distances of all the points on
the other side. Find a similar plane of equal distances in another direction ; this intersects
the first plane in a straight line. A third similar plane cuts this straight line in a point ;
this point is the centroid ; it has the unique property that all planes through it are planes
of equal distances. If some of the points are conglomerated to form a particle, the sum
of the distances for each of the points is equal to the distance of the particle multiplied by
the number of points in the particle, i.e., by the mass of the particle. Hence follows the
theorem for the statical moment for lines and planes or other surfaces, as well as for solids
that have weight.

Another interesting point, in relation to recent work, is the subject-matter of Art. 230-
236 ; where it is shown that, due solely to the mutual forces exerted on a third point by
two points separated by a proper interval, there is a series of orbits, approximately confocal
ellipses, in which the third point is in a state of steady motion ; these orbits are alternately
stable and stable. If the steady motion in a stable orbit is disturbed, by a sufficiently great
difference of the velocity being induced by the action of a fourth point passing sufficiently
near the third point, this third point will leave its orbit and immediately take up another
stable orbit, after some initial oscillation about it. This elegant little theorem does not
depend in any way on the exact form of the curve of forces, so long as there are •portions of the
curve winding about the axis for very small intervals between the points.

It is sufficient, for the next point, to draw the reader's attention to Art. 266-278, on
collision, and to the articles which follow on the agreement between .resolution and com-
position of forces as a working hypothesis. From what Boscovich says, it would appear that
philosophers of his time were much perturbed over the idea that, when a force was resolved
into two forces at a sufficiently obtuse angle, the force itself might be less than either of

INTRODUCTION xvii

the resolutes. Boscovich points out that, in his Theory, there is no resolution, only com-
position ; and therefore the difficulty does not arise. In this connection he adds that there
are no signs in Nature of anything approaching the vires viva of Leibniz.

In Art. 294 we have Boscovich's contribution to the controversy over the correct
measure of the " quantity of motion " ; but, as there is no attempt made to follow out the
change in either the velocity or the square of the velocity, it cannot be said to lead to any-
thing conclusive. As a matter of fact, Boscovich uses the result to prove the non-existence
of vires vivce.

In Art. 298-306 we have a mechanical exposition of reflection and refraction of light.
This comes under the section on Mechanics, because with Boscovich light is matter moving
with a very high velocity, and therefore reflection is a case of impact, in that it depends
upon the destruction of the whole of the perpendicular velocity upon entering the " surface "
of a denser medium, the surface being that part of space in front of the physical surface of
the medium in which the particles of light are near enough to the denser medium to feel the
influence of the last repulsive asymptotic branch of the curve of forces. If this perpendicular
velocity is not all destroyed, the particle enters the medium, and is refracted ; in which
case, the existence of a sine law is demonstrated. It is to be noted that the " fits " of
alternate attraction and repulsion, postulated by Newton, follow as a natural consequence
of the winding portion of the curve of Boscovich.

In Art. 328-346 we have a discussion of the centre of oscillation, and the centre of
percussion is investigated as well for masses in a plane perpendicular to the axis of rotation,
and masses lying in a straight line, where each mass is connected with the different centres.
Boscovich deduces from his theory the theorems, amongst others, that the centres of suspen-
sion and oscillation are interchangeable, and that the distance between them is equal to the
distance of the centre of percussion from the axis of rotation ; he also gives a rule for finding
the simple equivalent pendulum. The work is completed in a letter to Fr. Scherffer, which
is appended at the end of this volume.

In the third section, which deals with the application of the Theory to Physics, we
naturally do not look for much that is of value. But, in Art. 505, Boscovich evidently has
the correct notion that sound is a longitudinal vibration of the air or some other medium ;
and he is able to give an explanation of the propagation of the disturbance purely by means
of the mutual forces between the particles of the medium. In Art. 507 he certainly states
that the cause of heat is a " vigorous internal motion " ; but this motion is that of the
" particles of fire," if it is a motion ; an alternative reason is however given, namely, that it
may be a " fermentation of a sulphurous substance with particles of light." " Cold is
a lack of this substance, or of a motion of it." No attention will be called to this part
of the work, beyond an expression of admiration for the great ingenuity of a large part
of it.

There is a metaphysical appendix on the seat of the mind, and its nature, and on the
existence and attributes of GOD. This is followed by two short discussions of a philosophical
nature on Space and Time. Boscovich does not look on either of these as being in themselves
existent ; his entities are modes of existence, temporal and local. These three sections are
full of interest for the modern philosophical reader.

Supplement V is a theoretical proof, purely derived from the theory of mutual actions
between points of matter, of the law of the lever ; this is well worth study.

There are two points of historical interest beyond the study of the work of Boscovich
that can be gathered from this volume. The first is that at this time it would appear that
the nature of negative numbers and quantities was not yet fully understood. Boscovich, to
make his curve more symmetrical, continues it to the left of the origin as a reflection in the
axis of ordinates. It is obvious, however, that, if distances to the left of the origin stand for
intervals measured in the opposite direction to the ordinary (remembering that of the two
points under consideration one is supposed to be at the origin), then the force just the other
side of the axis of ordinates must be repulsive ; but the repulsion is in the opposite direction
to the ordinary way of measuring it, and therefore should appear on the curve represented
by an ordinate of attraction. Thus, the curve of Boscovich, if completed, should have point
symmetry about the origin, and not line symmetry about the axis of ordinates. Boscovich,
however, avoids this difficulty, intentionally or unintentionally, when showing how the
equation to the curve may be obtained, by taking z = x* as his variable, and P and Q as
functions of z, in the equation P-Qy = o, referred to above. Note. — In this connection
(p. 410, Art. 25, 1. 5), Boscovich has apparently made a slip over the negative sign : as the
intention is clear, no attempt has been made to amend the Latin.

The second point is that Boscovich does not seem to have any idea of integrating between
limits. He has to find the area, in Fig. I on p. 134, bounded by the axes, the curve and the
ordinate ag ; this he does by the use of the calculus in Note (1) on p. 141. He assumes that

xviii INTRODUCTION

gt

the equation of the curve is xmyn = I, and obtains the integral - - xy + A, where A is the

n—m

constant of integration. He states that, if n is greater than m, A = o, being the initial area
at the origin. He is then faced with the necessity of making the area infinite when n = m,
and still more infinite when n<jn. He says : " The area is infinite, when n = m, because
this makes the divisor zero ; and thus the area becomes still more infinite if n<^m." Put

into symbols, the argument is : Since «-OT<O, >- > oo . The historically interesting

n— m o

point about this is that it represents the persistance of an error originally made by Wallis
in his Ariihmetica Infinitorum (it was Wallis who invented the sign oc to stand for " simple
infinity," the value of i/o, and hence of «/o). Wallis had justification for his error, if
indeed it was an error in his case ; for his exponents were characteristics of certain infinite
series, and he could make his own laws about these so that they suited the geometrical
problems to which they were applied ; it was not necessary that they should obey the laws
of inequality that were true for ordinary numbers. Boscovich's mistake is, of course, that
of assuming that the constant is zero in every case ; and in this he is probably deceived by

using the formula xy -f- A, instead of ^B/("-*l) -}- A, for the area. From the latter

n—m n — m

it is easily seen that since the initial area is zero, we must have A = ow/("~m). If n is

m— n

equal to or greater than m, the constant A is indeed zero ; but if n is less than m, the constant
is infinite. The persistence of this error for so long a time, from 1655 to 175%> during which
we have the writings of Newton, Leibniz, the Bernoullis and others on the calculus, seems
to lend corroboration to a doubt as to whether the integral sign was properly understood as
a summation between limits, and that this sum could be expressed as the difference of two
values of the same function of those limits. It appears to me that this point is one of
very great importance in the history of the development of mathematical thought.

Some idea of how prolific Boscovich was as an author may be gathered from the catalogue
of his writings appended at the end of this volume. This catalogue has been taken from the
end of the original first Venetian edition, and brings the list up to the date of its publication,
1763. It was felt to be an impossible task to make this list complete up to the time of the
death of Boscovich ; and an incomplete continuation did not seem desirable. Mention
must however be made of one other work of Boscovich at least ; namely, a work in five
quarto volumes, published in 1785, under the title of Opera pertinentia ad Opticam et
Astronomiam.

Finally, in order to bring out the versatility of the genius of Boscovich, we may mention
just a few of his discoveries in science, which seem to call for special attention. In astro-
nomical science, he speaks of the use of a telescope filled with liquid for the purpose of
measuring the aberration of light ; he invented a prismatic micrometer contemporaneously
with Rochon and Maskelyne. He gave methods for determining the orbit of a comet from
three observations, and for the equator of the sun from three observations of a " spot " ;
he carried out some investigations on the orbit of Uranus, and considered the rings of Saturn.
In what was then the subsidiary science of optics, he invented a prism with a variable angle
for measuring the refraction and dispersion of different kinds of glass ; and put forward a
theory of achromatism for the objectives and oculars of the telescope. In mechanics and
geodesy, he was apparently the first to solve the problem of the " body of greatest attraction " ;
he successfully attacked the question of the earth's density ; and perfected the apparatus
and advanced the theory of the measurement of the meridian. In mathematical theory,
he seems to have recognized, before Lobachevski and Bolyai, the impossibility of a proof of
Euclid's " parallel postulate " ; and considered the theory of the logarithms of negative
numbers.

J. M. C.

N.B. — The page numbers on the left-hand pages of the index are the pages of the
original Latin Edition of 1763 ; they correspond with the clarendon numbers inserted
throughout the Latin text of this edition.

CORRIGENDA

Attention is called to the following important corrections, omissions, and alternative renderings ; misprints
involving a single letter or syllable only are given at the end of the volume.

p. 27, 1. 8, for in one plane read in the same direction

p. 47, 1. 62, literally on which ... is exerted

p. 49, 1. 33, for just as ... is read so that . . . may be

P- S3> 1- 9> after a line add but not parts of the line itself

p. 61, Art. 47, Alternative rendering: These instances make good the same point as water making its way through

the pores of a sponge did for impenetrability ;

p. 67, 1. 5, for it is allowable for me read I am disposed ; unless in the original libet is taken to be a misprint for licet
p. 73, 1. 26, after nothing add in the strict meaning of the term
p. 85, 1. 27, after conjunction add of the same point of space

p. 91, 1. 25, Alternative rendering : and these properties might distinguish the points even in the view of the followers
of Leibniz

1. 5 from bottom, Alternative rendering : Not to speak of the actual form of the leaves present in the seed
p. 115, 1. 25, after the left add but that the two outer elements do not touch each other

1. 28, for two little spheres read one little sphere
p. 117, 1. 41, for precisely read abstractly
p. 125, 1. 29, for ignored read urged in reply
p. 126, 1. 6 from bottom, it is -possible that acquirere is intended for acquiescere, with a corresponding change in the

translation

p. 129, Art. 162, marg. note, for on what they may be founded read in what it consists,
p. 167, Art. 214, 1. 2 of marg. note, transpose by and on

footnote, 1. I, for be at read bisect it at
p. 199, 1. 24, for so that read just as
p. 233, 1. 4 from bottom, for base to the angle read base to the sine of the angle

last line, after vary insert inversely

p. 307, 1. 5 from end, for motion, as (with fluids) takes place read motion from taking place
p. 323, 1. 39, for the agitation will read the fluidity will
P- 345» 1- 32> for described read destroyed
p. 357, 1. 44, for others read some, others of others

1. 5 from end, for fire read a fiery and insert a comma before substance

XIX

THEORIA
PHILOSOPHIC NATURALIS

TYPOGRAPHUS

VENETUS

LECTORI

PUS, quod tibi offero, jam ab annis quinque Viennse editum, quo plausu
exceptum sit per Europam, noveris sane, si Diaria publica perlegeris, inter
quse si, ut omittam caetera, consulas ea, quae in Bernensi pertinent ad
initium anni 1761 ; videbis sane quo id loco haberi debeat. Systema
continet Naturalis Philosophise omnino novum, quod jam ab ipso Auctore
suo vulgo Boscovichianum appellant. Id quidem in pluribus Academiis
jam passim publice traditur, nee tantum in annuis thesibus, vel disserta-
tionibus impressis, ac propugnatis exponitur, sed & in pluribus elementaribus libris pro
juventute instituenda editis adhibetur, exponitur, & a pluribus habetur pro archetype.
Verum qui omnem systematis compagem, arctissimum partium nexum mutuum, fcecun-
ditatem summam, ac usum amplissimum ac omnem, quam late patet, Naturam ex unica
simplici lege virium derivandam intimius velit conspicere, ac contemplari, hoc Opus
consulat, necesse est.

Haec omnia me permoverant jam ab initio, ut novam Operis editionem curarem :
accedebat illud, quod Viennensia exemplaria non ita facile extra Germaniam itura videbam,
& quidem nunc etiam in reliquis omnibus Europse partibus, utut expetita, aut nuspiam
venalia prostant, aut vix uspiam : systema vero in Italia natum, ac ab Auctore suo pluribus
hie apud nos jam dissertationibus adumbratum, & casu quodam Viennae, quo se ad breve
tempus contulerat, digestum, ac editum, Italicis potissimum typis, censebam, per univer-
sam Europam disseminandum. Et quidem editionem ipsam e Viennensi exemplari jam
turn inchoaveram ; cum illud mihi constitit, Viennensem editionem ipsi Auctori, post cujus
discessum suscepta ibi fuerat, summopere displicere : innumera obrepsisse typorum menda :
esse autem multa, inprimis ea, quas Algebraicas formulas continent, admodum inordinata,
& corrupta : ipsum eorum omnium correctionem meditari, cum nonnullis mutationibus,
quibus Opus perpolitum redderetur magis, & vero etiam additamentis.

Illud ergo summopere desideravi, ut exemplar acquirerem ab ipso correctum, & auctum
ac ipsum edition! praesentem haberem, & curantem omnia per sese. At id quidem per
hosce annos obtinere non licuit, eo universam fere Europam peragrante ; donee demum
ex tarn longa peregrinatione redux hue nuper se contulit, & toto adstitit editionis tempore,
ac praeter correctores nostros omnem ipse etiam in corrigendo diligentiam adhibuit ;
quanquam is ipse haud quidem sibi ita fidit, ut nihil omnino effugisse censeat, cum ea sit
humanas mentis conditio, ut in eadem re diu satis intente defigi non possit.

Haec idcirco ut prima quaedam, atque originaria editio haberi debet, quam qui cum
Viennensi contulerit, videbit sane discrimen. E minoribus mutatiunculis multae pertinent
ad expolienda, & declaranda plura loca ; sunt tamen etiam nonnulla potissimum in pagin-
arum fine exigua additamenta, vel mutatiunculas exiguae factae post typographicam
constructionem idcirco tantummodo, ut lacunulae implerentur quae aliquando idcirco
supererant, quod plures ph'ylirae a diversis compositoribus simul adornabantur, & quatuor
simul praela sudabant; quod quidem ipso praesente fieri facile potuit, sine ulla pertur-
batione sententiarum, & ordinis.

THE PRINTER AT VENICE

TO

THE READER

! OU will be well aware, if you have read the public journals, with what applause
the work which I now offer to you has been received throughout Europe
since its publication at Vienna five years ago. Not to mention others, if
you refer to the numbers of the Berne Journal for the early part of the
year 1761, you will not fail to see how highly it has been esteemed. It
contains an entirely new system of Natural Philosophy, which is already
commonly known as the Boscovichian theory, from the name of its author,
As a matter of fact, it is even now a subject of public instruction in several Universities in
different parts ; it is expounded not only in yearly theses or dissertations, both printed &
debated ; but also in several elementary books issued for the instruction of the young it is
introduced, explained, & by many considered as their original. Any one, however, who
wishes to obtain more detailed insight into the whole structure of the theory, the close
relation that its several parts bear to one another, or its great fertility & wide scope for
the purpose of deriving the whole of Nature, in her widest range, from a single simple law
of forces ; any one who wishes to make a deeper study of it must perforce study the work
here offered.

All these considerations had from the first moved me to undertake a new edition of
the work ; in addition, there was the fact that I perceived that it would be a matter of some
difficulty for copies of the Vienna edition to pass beyond the confines of Germany — indeed,
at the present time, no matter how diligently they are inquired for, they are to be found
on sale nowhere, or scarcely anywhere, in the rest of Europe. The system had its birth in
Italy, & its outlines had already been sketched by the author in several dissertations pub-
lished here in our own land ; though, as luck would have it, the system itself was finally
put into shape and published at Vienna, whither he had gone for a short time. I therefore
thought it right that it should be disseminated throughout the whole of Europe, & that
preferably as the product of an Italian press. I had in fact already commenced an edition
founded on a copy of the Vienna edition, when it came to my knowledge that the author
was greatly dissatisfied with the Vienna edition, taken in hand there after his departure ;
that innumerable printer's errors had crept in ; that many passages, especially those that
contain Algebraical formulae, were ill-arranged and erroneous ; lastly, that the author
himself had in mind a complete revision, including certain alterations, to give a better
finish to the work, together with certain additional matter.

That being the case, I was greatly desirous of obtaining a copy, revised & enlarged
by himself ; I also wanted to have him at hand whilst the edition was in progress, & that
he should superintend the whole thing for himself. This, however, I was unable to procure
during the last few years, in which he has been travelling through nearly the whole of
Europe ; until at last he came here, a little while ago, as he returned home from his lengthy
wanderings, & stayed here to assist me during the whole time that the edition was in
hand. He, in addition to our regular proof-readers, himself also used every care in cor-
recting the proof ; even then, however, he has not sufficient confidence in himself as to
imagine that not the slightest thing has escaped him. For it is a characteristic of the human
mind that it cannot concentrate long on the same subject with sufficient attention.

It follows that this ought to be considered in some measure as a first & original
edition ; any one who compares it with that issued at Vienna will soon see the difference
between them. Many of the minor alterations are made for the purpose of rendering
certain passages more elegant & clear ; there are, however, especially at the foot of a
page, slight additions also, or slight changes made after the type was set up, merely for
the purpose of filling up gaps that were left here & there — these gaps being due to the
fact that several sheets were being set at the same time by different compositors, and four
presses were kept hard at work together. As he was at hand, this could easily be done
without causing any disturbance of the sentences or the pagination.

4 TYPOGRAPHUS VENETUS LECTORI

Inter mutationes occurret ordo numerorum mutatus in paragraphis : nam numerus 82
de novo accessit totus : deinde is, qui fuerat 261 discerptus est in 5 ; demum in Appendice
post num. 534 factse sunt & mutatiunculse nonnullae, & additamenta plura in iis, quse
pertinent ad sedem animse.

Supplementorum ordo mutatus est itidem ; quse enim fuerant 3, & 4, jam sunt I, &
2 : nam eorum usus in ipso Opere ante alia occurrit. Illi autem, quod prius fuerat primum,
nunc autem est tertium, accessit in fine Scholium tertium, quod pluribus numeris complec-
titur dissertatiunculam integram de argumento, quod ante aliquot annos in Parisiensi
Academia controversise occasionem exhibuit in Encyclopedico etiam dictionario attactum,
in qua dissertatiuncula demonstrat Auctor non esse, cur ad vim exprimendam potentia
quaepiam distantiae adhibeatur potius, quam functio.

Accesserunt per totum Opus notulae marginales, in quibus eorum, quae pertractantur
argumenta exponuntur brevissima, quorum ope unico obtutu videri possint omnia, & in
memoriam facile revocari.

Postremo loco ad calcem Operis additus est fusior catalogus eorum omnium, quse hue
usque ab ipso Auctore sunt edita, quorum collectionem omnem expolitam, & correctam,
ac eorum, quse nondum absoluta sunt, continuationem meditatur, aggressurus illico post
suum regressum in Urbem Romam, quo properat. Hie catalogus impressus fuit Venetisis
ante hosce duos annos in reimpressione ejus poematis de Solis ac Lunae defectibus.
Porro earn, omnium suorum Operum Collectionem, ubi ipse adornaverit, typis ego meis
excudendam suscipiam, quam magnificentissime potero.

Haec erant, quae te monendum censui ; tu laboribus nostris fruere, & vive felix.

THE PRINTER AT VENICE TO THE READER 5

Among the more Important alterations will be found a change in the order of numbering
the paragraphs. Thus, Art. 82 is additional matter that is entirely new ; that which was
formerly Art. 261 is now broken up into five parts ; &, in the Appendix, following Art.
534, both some slight changes and also several additions have been made in the passages
that relate to the Seat of the Soul.

The order of the Supplements has been altered also : those that were formerly num-
bered III and IV are now I and II respectively. This was done because they are required
for use in this work before the others. To that which was formerly numbered I, but is
now III, there has been added a third scholium, consisting of several articles that between
them give a short but complete dissertation on that point which, several years ago caused
a controversy in the University of Paris, the same point being also discussed in the
Dictionnaire Encydopedique. In this dissertation the author shows that there is no reason
why any one power of the distance should be employed to express the force, in preference
to a function.

Short marginal summaries have been inserted throughout the work, in which the
arguments dealt with are given in brief ; by the help of these, the whole matter may be
taken in at a glance and recalled to mind with ease.

Lastly, at the end of the work, a somewhat full catalogue of the whole of the author's
publications up to the present time has been added. Of these publications the author
intends to make a full collection, revised and corrected, together with a continuation of
those that are not yet finished ; this he proposes to do after his return to Rome, for which
city he is preparing to set out. This catalogue was printed in Venice a couple of years ago
in connection with a reprint of his essay in verse on the eclipses of the Sun and Moon.
Later, when his revision of them is complete, I propose to undertake the printing of this
complete collection of his works from my own type, with all the sumptuousness at my
command.

Such were the matters that I thought ought to be brought to your notice. May you
enjoy the fruit of our labours, & live in happiness.

EPISTOLA AUCTORIS DEDICATORIA

EDITIONIS VIENNENSIS

AD CELSISSIMUM TUNC PRINCIPEM ARCHIEPISCOPUM

VIENNENSEM, NUNC PR^TEREA ET CARDINALEM

EMINENTISSIMUM, ET EPISCOPUM VACCIENSEM

CHRISTOPHORUM E COMITATIBUS DE MIGAZZI

IA.BIS veniam, Princeps Celsissime, si forte inter assiduas sacri regirninis curas
importunus interpellator advenio, & libellum Tibi offero mole tenuem, nee
arcana Religionis mysteria, quam in isto tanto constitutus fastigio adminis-
tras, sed Naturalis Philosophise principia continentem. Novi ego quidem,
quam totus in eo sis, ut, quam geris, personam sustineas, ac vigilantissimi
sacrorum Antistitis partes agas. Videt utique Imperialis haec Aula, videt
universa Regalis Urbs, & ingenti admiratione defixa obstupescit, qua dili-
gentia, quo labore tanti Sacerdotii munus obire pergas. Vetus nimirum illud celeberrimum
age, quod agis, quod ab ipsa Tibi juventute, cum primum, ut Te Romas dantem operam
studiis cognoscerem, mihi fors obtigit, altissime jam insederat animo, id in omni
reliquo amplissimorum munerum Tibi commissorum cursu haesit firmissime, atque idipsum
inprimis adjectum tarn multis & dotibus, quas a Natura uberrime congestas habes, &
virtutibus, quas tute diuturna Tibi exercitatione, atque assiduo labore comparasti, sanc-
tissime observatum inter tarn varias forenses, Aulicas, Sacerdotales occupationes, istos Tibi
tarn celeres dignitatum gradus quodammodo veluti coacervavit, & omnium una tarn
populorum, quam Principum admirationem excitavit ubique, conciliavit amorem ; unde
illud est factum, ut ab aliis alia Te, sublimiora semper, atque honorificentiora munera
quodammodo velut avulsum, atque abstractum rapuerint. Dum Romse in celeberrimo illo,
quod Auditorum Rotae appellant, collegio toti Christiano orbi jus diceres, accesserat
Hetrusca Imperialis Legatio apud Romanum Pontificem exercenda ; cum repente Mech-
liniensi Archiepiscopo in amplissima ilia administranda Ecclesia Adjutor datus, & destinatus
Successor, possessione prsestantissimi muneris vixdum capta, ad Hispanicum Regem ab
Augustissima Romanorum Imperatrice ad gravissima tractanda negotia Legatus es missus,
in quibus cum summa utriusque Aulae approbatione versatum per annos quinque ditissima
Vacciensis Ecclesia adepta est ; atque ibi dum post tantos Aularum strepitus ea, qua
Christianum Antistitem decet, & animi moderatione, & demissione quadam, atque in omne
hominum genus charitate, & singular! cura, ac diligentia Religionem administras, & sacrorum
exceres curam ; non ea tantum urbs, atque ditio, sed universum Hungariae Regnum,
quanquam exterum hominem, non ut civem suum tantummodo, sed ut Parentem aman-
tissimum habuit, quern adhuc ereptum sibi dolet, & angitur ; dum scilicet minore, quam
unius anni intervallo ab Ipsa Augustissima Imperatrice ad Regalem hanc Urbem, tot
Imperatorum sedem, ac Austriacae Dominationis caput, dignum tantis dotibus explicandis
theatrum, eocatum videt, atque in hac Celsissima Archiepiscopali Sede, accedente Romani
Pontificis Auctoritate collocatum ; in qua Tu quidem personam itidem, quam agis, diligen-
tissime sustinens, totus es in gravissimis Sacerdotii Tui expediendis negotiis, in iis omnibus,
quae ad sacra pertinent, curandis vel per Te ipsum usque adeo, ut saepe, raro admodum per

AUTHOR'S EPISTLE DEDICATING

THE FIRST VIENNA EDITION

TO

CHRISTOPHER, COUNT DE MIGAZZI, THEN HIS HIGHNESS
THE PRINCE ARCHBISHOP OF VIENNA, AND NOW ALSO
IN ADDITION HIS EMINENCE THE CARDINAL,

BISHOP OF VACZ

OU will pardon me, Most Noble Prince, if perchance I come to disturb at an
inopportune moment the unremitting cares of your Holy Office, & offer
you a volume so inconsiderable in size ; one too that contains none of the
inner mysteries of Religion, such as you administer from the highly exalted
position to which you are ordained ; one that merely deals with the prin-
ciples of Natural Philosophy. I know full well how entirely your time is
taken up with sustaining the reputation that you bear, & in performing
the duties of a highly conscientious Prelate. This Imperial Court sees, nay, the whole of
this Royal City sees, with what care, what toil, you exert yourself to carry out the duties of
so great a sacred office, & stands wrapt with an overwhelming admiration. Of a truth,
that well-known old saying, " What you do, DO," which from your earliest youth, when
chance first allowed me to make your acquaintance while you were studying in Rome, had
already fixed itself deeply in your mind, has remained firmly implanted there during the
whole of the remainder of a career in which duties of the highest importance have been
committed to your care. Your strict observance of this maxim in particular, joined with
those numerous talents so lavishly showered upon you by Nature, & those virtues which
you have acquired for yourself by daily practice & unremitting toil, throughout your
whole career, forensic, courtly, & sacerdotal, has so to speak heaped upon your shoulders
those unusually rapid advances in dignity that have been your lot. It has aroused the
admiration of all, both peoples & princes alike, in every land ; & at the same time it has
earned for you their deep affection. The consequence was that one office after another,
each ever more exalted & honourable than the preceding, has in a sense seized upon you
& borne you away a captive. Whilst you were in Rome, giving judicial decisions to the
whole Christian world in that famous College, the Rota of Auditors, there was added the
duty of acting on the Tuscan Imperial Legation at the Court of the Roman Pontiff. Sud-
denly you were appointed coadjutor to the Archbishop of Malines in the administration of
that great church, & his future successor. Hardly had you entered upon the duties of
that most distinguished appointment, than you were despatched by the August Empress of
the Romans as Legate on a mission of the greatest importance. You occupied yourself on
this mission for the space of five years, to the entire approbation of both Courts, & then
the wealthy church of Vacz obtained your services. Whilst there, the great distractions of
a life at Court being left behind, you administer the offices of religion & discharge the
sacred rights with that moderation of spirit & humility that befits a Christian prelate, in
charity towards the whole race of mankind, with a singularly attentive care. So that not
only that city & the district in its see, but the whole realm of Hungary as well, has looked
upon you, though of foreign race, as one of her own citizens ; nay, rather as a well beloved
father, whom she still mourns & sorrows for, now that you have been taken from her.
For, after less than a year had passed, she sees you recalled by the August Empress herself to
this Imperial City, the seat of a long line of Emperors, & the capital of the Dominions of
Austria, a worthy stage for the display of your great talents ; she sees you appointed, under
the auspices of the authority of the Roman Pontiff, to this exalted Archiepiscopal see.
Here too, sustaining with the utmost diligence the part you play so well, you throw your-
self heart and soul into the business of discharging the weighty duties of your priesthood,
or in attending to all those things that deal with the sacred rites with your own hands : so
much so that we often see you officiating, & even administering the Sacraments, in our

8 EPISTOLA AUCTORIS DEDICATORIA PRI1VLE EDITIONIS VIENNENSIS

haec nostra tempora exemplo, & publico operatum, ac ipsa etiam Sacramenta administrantem
videamus in templis, & Tua ipsius voce populos, e superiore loco docentum audiamus, atque
ad omne virtutum genus inflammantem.

Novi ego quidem haec omnia ; novi hanc indolem, hanc animi constitutionem ; nee
sum tamen inde absterritus, ne, inter gravissimas istas Tuas Sacerdotales curas, Philosophicas
hasce meditationes meas, Tibi sisterem, ac tantulae libellum molis homini ad tantum culmen
evecto porrigerem, ac Tuo vellem Nomine insignitum. Quod enim ad primum pertinet
caput, non Theologicas tantum, sed Philosophicas etiam perquisitiones Christiano Antistite
ego quidem dignissimas esse censeo, & universam Naturae contemplationem omnino
arbitror cum Sacerdotii sanctitate penitus consentire. Mirum enim, quam belle ab ipsa
consideratione Naturae ad caslestium rerum contemplationem disponitur animus, & ad
ipsum Divinum tantae molis Conditorem assurgit, infinitam ejus Potentiam Sapientiam,
Providentiam admiratus, quae erumpunt undique, & utique se produnt.

Est autem & illud, quod ad supremi sacrorum Moderatoris curam pertinet providere,
ne in prima ingenuae juventutis institutione, quae semper a naturalibus studiis exordium
ducit, prava teneris mentibus irrepant, ac perniciosa principia, quae sensim Religionem
corrumpant, & vero etiam evertant penitus, ac eruant a fundamentis ; quod quidem jam
dudum tristi quodam Europae fato passim evenire cernimus, gliscente in dies malo, ut fucatis
quibusdam, profecto perniciosissimis, imbuti principiis juvenes, turn demum sibi sapere
videantur, cum & omnem animo religionem, & Deum ipsum sapientissimum Mundi
Fabricatorem, atque Moderatorem sibi mente excusserint. Quamobrem qui veluti ad
tribunal tanti Sacerdotum Principis Universae Physicae Theoriam, & novam potissimum
Theoriam sistat, rem is quidem praestet sequissimam, nee alienum quidpiam ab ejus munere
Sacerdotali offerat, sed cum eodem apprime consentiens.

Nee vero exigua libelli moles deterrere me debuit, ne cum eo ad tantum Principem
accederem. Est ille quidem satis tenuis libellus, at non & tenuem quoque rem continet.
Argumentum pertractat sublime admodum, & nobile, in quo illustrando omnem ego quidem
industriam coUocavi, ubi si quid praestitero, si minus infiliclter me gessero, nemo sane me
impudentiae arguat, quasi vilem aliquam, & tanto indignam fastigio rem offeram. Habetur
in eo novum quoddam Universae Naturalis Philosophiae genus a receptis hue usque, usi-
tatisque plurimam discrepans, quanquam etiam ex iis, quae maxime omnium per haec tempora
celebrantur, casu quodam praecipua quasque mirum sane in modum compacta, atque inter
se veluti coagmentata conjunguntur ibidem, uti sunt simplicia atque inextensa Leibnitian-
orum elementa, cum Newtoni viribus inducentibus in aliis distantiis accessum mutuum, in
aliis mutuum recessum, quas vulgo attractiones, & repulsiones appellant : casu, inquam :
neque enim ego conciliandi studio hinc, & inde decerpsi quaedam ad arbitrium selecta, quae
utcumque inter se componerem, atque compaginarem : sed omni praejudicio seposito, a
principiis exorsus inconcussis, & vero etiam receptis communiter, legitima ratiocinatione
usus, & continue conclusionum nexu deveni ad legem virium in Natura existentium unicam,
simplicem , continuam, quae mihi & constitutionem elementorum materiae, & Mechanicae
leges, & generales materiae ipsius proprietates, & praecipua corporum discrimina, sua
applicatione ita exhibuit, ut eadem in iis omnibus ubique se prodat uniformis agendi ratio,
non ex arbitrariis hypothesibus, & fictitiis commentationibus, sed ex sola continua ratio-
cinatione deducta. Ejusmodi autem est omnis, ut eas ubique vel definiat, vel adumbret
combinationes elementorum, quae ad diversa prasstanda phaenomena sunt adhibendas, ad
quas combinationes Conditoris Supremi consilium, & immensa Mentis Divinae vis ubique
requiritur, quae infinites casus perspiciat, & ad rem aptissimos seligat, ac in Naturam
inducat.

Id mihi quidem argumentum est operis, in quo Theoriam meam expono, comprobo,
vindico : turn ad Mechanicam primum, deinde ad Physicam applico, & uberrimos usus
expono, ubi brevi quidem libello, sed admodum diuturnas annorum jam tredecim medita-
tiones complector meas, eo plerumque tantummodo rem deducens, ubi demum cum

AUTHOR'S EPISTLE DEDICATING THE FIRST VIENNA EDITION 9

churches (a somewhat unusual thing at the present time), and also hear you with your own
voice exhorting the people from your episcopal throne, & inciting them to virtue of
every kind.

I am well aware of all this ; I know full well the extent of your genius, & your con-
stitution of mind ; & yet I am not afraid on that account of putting into your hands,
amongst all those weighty duties of your priestly office, these philosophical meditations of
mine ; nor of offering a volume so inconsiderable in bulk to one who has attained to such
heights of eminence ; nor of desiring that it should bear the hall-mark of your name. With
regard to the first of these heads, I think that not only theological but also philosophical
investigations are quite suitable matters for consideration by a Christian prelate ; & in
my opinion, a contemplation of all the works of Nature is in complete accord with the
sanctity of the priesthood. For it is marvellous how exceedingly prone the mind becomes
to pass from a contemplation of Nature herself to the contemplation of celestial, things, &
to give honour to the Divine Founder of such a mighty structure, lost in astonishment at
His infinite Power & Wisdom & Providence, which break forth & disclose themselves
in all directions & in all things.

There is also this further point, that it is part of the duty of a religious superior to take
care that, in the earliest training of ingenuous youth, which always takes its start from the
study of the wonders of Nature, improper ideas do not insinuate themselves into tender
minds ; or such pernicious principles as may gradually corrupt the belief in things Divine,
nay, even destroy it altogether, & uproot it from its very foundations. This is what we
have seen for a long time taking place, by some unhappy decree of adverse fate, all over
Europe ; and, as the canker spreads at an ever increasing rate, young men, who have been
made to imbibe principles that counterfeit the truth but are actually most pernicious doc-
trines, do not think that they have attained to wisdom until they have banished from their
minds all thoughts of religion and of God, the All- wise Founder and Supreme Head of the
Universe. Hence, one who so to speak sets before the judgment-seat of such a prince of
the priesthood as yourself a theory of general Physical Science, & more especially one that
is new, is doing nothing but what is absolutely correct. Nor would he be offering him
anything inconsistent with his priestly office, but on the contrary one that is in complete
harmony with it.

Nor, secondly, should the inconsiderable size of my little book deter me from approach-
ing with it so great a prince. It is true that the volume of the book is not very great, but
the matter that it contains is not unimportant as well. The theory it develops is a strik-
ingly sublime and noble idea ; & I have done my very best to explain it properly. If in
this I have somewhat succeeded, if I have not failed altogether, let no one accuse me of
presumption, as if I were offering some worthless thing, something unworthy of such dis-
tinguished honour. In it is contained a new kind of Universal Natural Philosophy, one that
differs widely from any that are generally accepted & practised at the present time ;
although it so happens that the principal points of all the most distinguished theories of the
present day, interlocking and as it were cemented together in a truly marvellous way, are
combined in it ; so too are the simple unextended elements of the followers of Leibniz,
as well as the Newtonian forces producing mutual approach at 'some distances & mutual
separation at others, usually called attractions and repulsions. I use the words " it so
happens " because I have not, in eagerness to make the whole consistent, selected one thing
here and another there, just as it suited me for the purpose of making them agree & form
a connected whole. On the contrary, I put on one side all prejudice, & started from
fundamental principles that are incontestable, & indeed are those commonly accepted ; I
used perfectly sound arguments, & by a continuous chain of deduction I arrived at a
single, simple, continuous law for the forces that exist in Nature. The application of this
law explained to me the constitution of the elements of matter, the laws of Mechanics, the
general properties of matter itself, & the chief characteristics of bodies, in such a manner
that the same uniform method of action in all things disclosed itself at all points ; being
deduced, not from arbitrary hypotheses, and fictitibus explanations, but from a single con-
tinuous chain of reasoning. Moreover it is in all its parts of such a kind as defines, or
suggests, in every case, the combinations of the elements that must be employed to produce
different phenomena. For these combinations the wisdom of the Supreme Founder of the
Universe, & the mighty power of a Divine Mind are absolutely necessary ; naught but
one that could survey the countless cases, select those most suitable for the purpose, and
introduce them into the scheme of Nature.

This then is the argument of my work, in which I explain, prove & defend my theory ;
then I apply it, in the first instance to Mechanics, & afterwards to Physics, & set forth
the many advantages to be derived from it. Here, although the book is but small, I yet
include the well-nigh daily meditations of the last thirteen years, carrying on my conclu-

io EPISTOLA AUCTORIS DEDICATORIA PRIM.£ EDITIONIS VIENNENSIS

communibus Philosophorum consentio placitis, & ubi ea, quae habemus jam pro compertis,
ex meis etiam deductionibus sponte fluunt, quod usque adeo voluminis molem contraxit.
Dederam ego quidem dispersa dissertatiunculis variis Theorise meae qusedam velut specimina,
quae inde & in Italia Professores publicos nonnullos adstipulatores est nacta, & jam ad
exteras quoque gentes pervasit ; sed ea nunc primum tota in unum compacta, & vero etiam
plusquam duplo aucta, prodit in publicum, quern laborem postremo hoc mense, molestiori-
bus negotiis, quae me Viennam adduxerant, & curis omnibus exsolutus suscepi, dum in
Italiam rediturus opportunam itineri tempus inter assiduas nives opperior, sed omnem in
eodem adornando, & ad communem mediocrum etiam Philosophorum captum accommo-
dando diligentiam adhibui.

Inde vero jam facile intelliges, cur ipsum laborem meum ad Te deferre, & Tuo
nuncupare Nomini non dubitaverim. Ratio ex iis, quae proposui, est duplex : primo quidem
ipsum argumenti genus, quod Christianum Antistitem non modo non dedecet, sed etiam
apprime decet : turn ipsius argumenti vis, atque dignitas, quae nimirum confirmat, & erigit
nimium fortasse impares, sed quantum fieri per me potuit, intentos conatus meos ; nam
quidquid eo in genere meditando assequi possum, totum ibidem adhibui, ut idcirco nihil
arbitrer a mea tenuitate proferri posse te minus indignum, cui ut aliquem offerrem laborum
meorum fructum quantumcunque, exposcebat sane, ac ingenti clamore quodam efnagitabat
tanta erga me humanitas Tua, qua jam olim immerentem complexus Romae, hie etiam
fovere pergis, nee in tanto dedignatus fastigio, omni benevolentiae significatione prosequeris.
Accedit autem & illud, quod in hisce terris vix adhuc nota, vel etiam ignota penitus Theoria
mea Patrocinio indiget, quod, si Tuo Nomine insignata prodeat in publicum, obtinebit sane
validissimum, & secura vagabitur : Tu enim illam, parente velut hie orbatam suo, in dies
nimirum discessuro, & quodammodo veluti posthumam post ipsum ejus discessum typis
impressam, & in publicum prodeuntem tueberis, fovebisque.

Haec sunt, quae meum Tibi consilium probent, Princeps Celsissime : Tu, qua soles
humanitate auctorem excipere, opus excipe, & si forte adhuc consilium ipsum Tibi visum
fuerit improbandum ; animum saltern aequus respice obsequentissimum Tibi, ac devinct-
issimum. Vale.

Dabam Viennce in Collegia Academico Soc. JESU
Idibus Febr. MDCCLFIIL

AUTHOR'S EPISTLE DEDICATING THE FIRST VIENNA EDITION 11

sions for the most part only up to the point where I finally agreed with the opinions com-
monly held amongst philosophers, or where theories, now accepted as established, are the
natural results of my deductions also ; & this has in some measure helped to diminish the
size of the volume. I had already published some instances, so to speak, of my general
theory in several short dissertations issued at odd times ; & on that account the theory
has found some supporters amongst the university professors in Italy, & has already made
its way into foreign countries. But now for the first time is it published as a whole in a
single volume, the matter being indeed more than doubled in amount. This work I have
carried out during the last month, being quit of the troublesome business that brought me
to Vienna, and of all other cares ; whilst I wait for seasonable time for my return journey
through the everlasting snow to Italy. I have however used my utmost endeavours in
preparing it, and adapting it to the ordinary intelligence of philosophers of only moderate
attainments.

From this you will readily understand why I have not hesitated to bestow this book
of mine upon you, & to dedicate it to you. My reason, as can be seen from what I have
said, was twofold ; in the first place, the nature of my theme is one that is not only not
unsuitable, but is suitable in a high degree, for the consideration of a Christian priest ;
secondly, the power & dignity of the theme itself, which doubtless gives strength &
vigour to my efforts — perchance rather feeble, but, as far as in me lay, earnest. What-
ever in that respect I could gain by the exercise of thought, I have applied the whole of it
to this matter ; & consequently I think that nothing less unworthy of you can be pro-
duced by my poor ability ; & that I should offer to you some such fruit of my labours
was surely required of me, & as it were clamorously demanded by your great kindness
to me ; long ago in Rome you had enfolded my unworthy self in it, & here now you
continue to be my patron, & do not disdain, from your exalted position, to honour me
with every mark of your goodwill. There is still a further consideration, namely, that my
Theory is as yet almost, if not quite, unknown in these parts, & therefore needs a patron's
support ; & this it will obtain most effectually, & will go on its way in security if it
comes before the public franked with your name. For you will protect & cherish it,
on its publication here, bereaved as it were of that parent whose departure in truth draws
nearer every day ; nay rather posthumous, since it will be seen in print only after he has
gone.

Such are my grounds for hoping that you will approve my idea, most High Prince.
I beg you to receive the work with the same kindness as you used to show to its author ;
&, if perchance the idea itself should fail to meet with your approval, at least regard
favourably the intentions of your most humble & devoted servant. Farewell.

University College of the Society of Jesus,
VIENNA,

February i$th, 1758.

AD LECTOREM

EX EDITIONS VIENNENSI

amice Lector, Philosophic Naturalis Theoriam ex unica lege virium
deductam, quam & ubi jam olim adumbraverim, vel etiam ex parte explica-
verim, y qua occasione nunc uberius pertractandum, atque augendam etiam,
susceperim, invenies in ipso -primes •partis exordia. Libuit autem hoc opus
dividere in partes tres, quarum prima continet explicationem Theories ipsius,
ac ejus analyticam deductionem, & vindicationem : secunda applicationem-
satis uberem ad Mechanicam ; tertia applicationem ad Physicam.

Porro illud inprimis curandum duxi, ut omnia, quam liceret, dilucide exponerentur, nee
sublimiore Geometria, aut Calculo indigerent. Et quidem in prima, ac tertia parte non tantum
nullcs analyticee, sed nee geometries demonstrations occurrunt, paucissimis qiiibusdam, quibus
indigeo, rejectis in adnotatiunculas, quas in fine paginarum quarundam invenies. Queedam
autem admodum pauca, quce majorem Algebra, & Geometries cognitionem requirebant, vel erant
complicatiora aliquando, & alibi a me jam edita, in fine operis apposui, quce Supplementorum
appellavi nomine, ubi W ea addidi, quce sentio de spatio, ac tempore, Theories mece consentanea,
ac edita itidem jam alibi. In secunda parte, ubi ad Mechanicam applicatur Theoria,a geome-
tricis, W aliquando etiam ab algebraicis demonstrationibus abstinere omnino non potui ; sed
ece ejusmodi sunt, ut vix unquam requirant aliud, quam Euclideam Geometriam, & primas
Trigonometries notiones maxime simplices, ac simplicem algorithmum.

In prima quidem parte occurrunt Figures geometricce complures, quce prima fronte vide-
buntur etiam complicate? rem ipsam intimius non perspectanti ; verum ece nihil aliud exhibent,
nisi imaginem quandam rerum, quce ipsis oculis per ejusmodi figuras sistuntur contemplandce.
Ejusmodi est ipsa ilia curva, quce legem virium exhibet. Invenio ego quidem inter omnia
materice puncta vim quandam mutuam, quce a distantiis pendet, £5" mutatis distantiis mutatur
ita, ut in aliis attractiva sit, in aliis repulsiva, sed certa quadam, y continua lege. Leges
ejusmodi variationis binarum quantitatum a se invicem pendentium, uti Jiic sunt distantia,
y vis, exprimi possunt vel per analyticam formulam, vel per geometricam curvam ; sed ilia
prior expressio & multo plures cognitiones requirit ad Algebram pertinentes, & imaginationem
non ita adjuvat, ut heec posterior, qua idcirco sum usus in ipsa prima operis parte, rejecta in
Supplementa formula analytica, quce y curvam, & legem virium ab ilia expressam exhibeat.

Porro hue res omnis reducitur. Habetur in recta indefinita, quce axis dicitur, punctum
quoddam, a quo abscissa ipsius rectce segmenta referunt distantias. Curva linea protenditur
secundum rectam ipsam, circa quam etiam serpit, y eandem in pluribus secat punctis : rectce
a fine segmentorum erectce perpendiculariter usque ad curvam, exprimunt vires, quce majores
sunt, vel minores, prout ejusmodi rectce sunt itidem majores, vel minores ; ac eesdem ex attrac-
tivis migrant in repulsivis, vel vice versa, ubi illce ipsce perpendiculares rectce directionem
mutant, curva ab alter a axis indefiniti plaga migrante ad alter am. Id quidem nullas requirit
geometricas demonstrations, sed meram cognitionem vocum quarundam, quce vel ad prima per-
tinent Geometries elementa, y notissimce sunt, vel ibi explicantur, ubi adhibentur. Notissima
autem etiam est significatio vocis Asymptotus, unde & crus asymptoticum curvce appellatur ;
dicitur nimirum recta asymptotus cruris cujuspiam curvce, cum ipsa recta in infinitum producta,
ita ad curvilineum arcum productum itidem in infinitum semper accedit magis, ut distantia
minuatur in infinitum, sed nusquam penitus evanescat, illis idcirco nunquam invicem con-
venientibus.

Consider atio porro attenta curvce propositce in Fig. I, &rationis, qua per illam exprimitur

12

THE PREFACE TO THE READER

THAT APPEARED IN THE VIENNA EDITION

EAR Reader, you have before you a Theory of Natural Philosophy deduced
from a single law of Forces. You will find in the opening paragraphs of
the first section a statement as to where the Theory has been already
published in outline, & to a certain extent explained ; & also the occasion
that led me to undertake a more detailed treatment & enlargement of it.
For I have thought fit to divide the work into three parts ; the first of
these contains the exposition of the Theory itself, its analytical deduction
& its demonstration ; the second a fairly full application to Mechanics ; & the third an
application to Physics.

The most important point, I decided, was for me to take the greatest care that every-
thing, as far as was possible, should be clearly explained, & that there should be no need for
higher geometry or for the calculus. Thus, in the first part, as well as in the third, there
are no proofs by analysis ; nor are there any by geometry, with the exception of a very few
that are absolutely necessary, & even these you will find relegated to brief notes set at the
foot of a page. I have also added some very few proofs, that required a knowledge of
higher algebra & geometry, or were of a rather more complicated nature, all of which have
been already published elsewhere, at the end of the work ; I have collected these under
the heading Supplements ; & in them I have included my views on Space & Time, which
are in accord with my main Theory, & also have been already published elsewhere. In
the second part, where the Theory is applied to Mechanics, I have not been able to do
without geometrical proofs altogether ; & even in some cases I have had to give algebraical
proofs. But these are of such a simple kind that they scarcely ever require anything more
than Euclidean geometry, the first and most elementary ideas of trigonometry, and easy
analytical calculations.

It is true that in the first part there are to be found a good many geometrical diagrams,
which at first sight, before the text is considered more closely, will appear to be rather
complicated. But these present nothing else but a kind of image of the subjects treated,
which by means of these diagrams are set before the eyes for contemplation. The very
curve that represents the law of forces is an instance of this. I find that between all points
of matter there is a mutual force depending on the distance between them, & changing as
this distance changes ; so that it is sometimes attractive, & sometimes repulsive, but always
follows a definite continuous law. Laws of variation of this kind between two quantities
depending upon one another, as distance & force do in this instance, may be represented
either by an analytical formula or by a geometrical curve ; but the former method of
representation requires far more knowledge of algebraical processes, & does not assist the
imagination in the way that the latter does. Hence I have employed the latter method in
the first part of the work, & relegated to the Supplements the analytical formula which
represents the curve, & the law of forces which the curve exhibits.

The whole matter reduces to this. In a straight line of indefinite length, which is
called the axis, a fixed point is taken ; & segments of the straight line cut off from this
point represent the distances. A curve is drawn following the general direction of this
straight line, & winding about it, so as to cut it in several places. Then perpendiculars that
are drawn from the ends of the segments to meet the curve represent the forces ; these
forces are greater or less, according as such perpendiculars are greater or less ; & they pass
from attractive forces to repulsive, and vice versa, whenever these perpendiculars change
their direction, as the curve passes from one side of the axis of indefinite length to the other
side of it. Now this requires no geometrical proof, but only a knowledge of certain terms,
which either belong to the first elementary principles of "geometry, & are thoroughly well
known, or are such as can be defined when they are used. The term Asymptote is well
known, and from the same idea we speak of the branch of a curve as being asymptotic ;
thus a straight line is said to be the asymptote to any branch of a curve when, if the straight
line is indefinitely produced, it approaches nearer and nearer to the curvilinear arc which
is also prolonged indefinitely in such manner that the distance between them becomes
indefinitely diminished, but never altogether vanishes, so that the straight line & the curve
never really meet.

A careful consideration of the curve given in Fig. I, & of the way in which the relation

14 AD LECTOREM EX EDITIONE VIENNENSI

nexus inter vires, y distantias, est utique admodum necessaria ad intelligendam Theoriam ipsam,
cujus ea est prcecipua qucedam veluti clavis, sine qua omnino incassum tentarentur cetera ; sed
y ejusmodi est, ut tironum, & sane etiam mediocrium, immo etiam longe infra mediocritatem
collocatorum, captum non excedat, potissimum si viva accedat Professoris vox mediocriter etiam
versati in Mechanica, cujus ope, pro certo habeo, rem ita patentem omnibus reddi posse, ut
ii etiam, qui Geometric penitus ignari sunt, paucorum admodum explicatione vocabulorum
accidente, earn ipsis oculis intueantur omnino perspicuam.

In tertia parte supponuntur utique nonnulla, quce demonstrantur in secunda ; sed ea ipsa
sunt admodum pauca, & Us, qui geometricas demonstrationes fastidiunt, facile admodum exponi
possunt res ipsce ita, ut penitus etiam sine ullo Geometries adjumento percipiantur, quanquam
sine Us ipsa demonstratio baberi non poterit ; ut idcirco in eo differre debeat is, qui secundam
partem attente legerit, & Geometriam calleat, ab eo, qui earn omittat, quod ille primus veritates
in tertia parte adhibitis, ac ex secunda erutas, ad, explicationem Physicce, intuebitur per evi-
dentiam ex ipsis demonstrationibus haustam, hie secundus easdem quodammodo per fidem Geo-
metris adhibitam credet. Hujusmodi inprimis est illud, particulam compositam ex punctis
etiam homogeneis, prceditis lege virium proposita, posse per solam diversam ipsorum punctorum
dispositionem aliam particulam per certum intervallum vel perpetuo attrahere, vel perpetuo
repellere, vel nihil in earn agere, atque id ipsum viribus admodum diversis, y quce respectu diver-
sarum particularum diver see sint, & diver see respectu partium diver sarum ejusdem particulce,
ac aliam particulam alicubi etiam urgeant in latus, unde plurium phcenomenorum explicatio in
Physica sponte fluit.

Verum qui omnem Theories, y deductionum compagem aliquanto altius inspexerit, ac
diligentius perpenderit, videbit, ut spero, me in hoc perquisitionis genere multo ulterius
progressum esse, quam olim Newtonus ipse desideravit. Is enim in postremo Opticce questione
prolatis Us, quce per vim attractivam, & vim repulsivam, mutata distantia ipsi attractive suc-
cedentem, explicari poterant, hcec addidit : " Atque hcec quidem omnia si ita sint, jam Natura
universa valde erit simplex, y consimilis sui, perficiens nimirum magnos omnes corporum
ccelestium motus attractione gravitatis, quce est mutua inter corpora ilia omnia, & minores fere
omnes particularum suarum motus alia aliqua vi attrahente, & repellente, qua est inter particulas
illas mutua" Aliquanto autem inferius de primigeniis particulis agens sic habet : " Porro
videntur mihi hce particulce primigenice non modo in se vim inertice habere, motusque leges passivas
illas, quce ex vi ista necessario oriuntur ; verum etiam motum perpetuo accipere a certis principiis
actuosis, qualia nimirum sunt gravitas, £ff causa fermentationis, & cohcerentia corporum. Atque
hcec quidem principia considero non ut occultas qualitates, quce ex specificis rerum formis oriri
fingantur, sed ut universales Naturce leges, quibus res ipsce sunt formatce. Nam principia
quidem talia revera existere ostendunt phenomena Naturce, licet ipsorum causce quce sint,
nondum fuerit explicatum. Affirmare, singulas rerum species specificis prceditas esse qualita-
tibus occultis, per quas eae vim certam in agenda habent, hoc utique est nihil dicere : at ex
phcenomenis Naturce duo, vel tria derivare generalia motus principia, & deinde explicare,
quemadmodum proprietates, & actiones rerum corporearum omnium ex istis principiis conse-
quantur, id vero magnus esset factus in Philosophia progressus, etiamsi principiorum istorum
causce nondum essent cognitce. Quare motus principia supradicta proponere non dubito, cum
per Naturam universam latissime pateant"

Hcec ibi Newtonus, ubi is quidem magnos in Philosophia progressus facturum arbitratus
est eum, qui ad duo, vel tria generalia motus principia ex Naturce phcenomenis derivata pheeno-
menorum explicationem reduxerit, & sua principia protulit, ex quibus inter se diversis eorum
aliqua tantummodo explicari posse censuit. Quid igitur, ubi tf? ea ipsa tria, & alia prcecipua
quceque, ut ipsa etiam impenetrabilitas, y impulsio reducantur ad principium unicum legitima
ratiocinatione deductum ? At id -per meam unicam, & simplicem virium legemprcestari, patebit
sane consideranti operis totius Synopsim quandam, quam hie subjicio ; sed multo magis opus
ipsum diligentius pervolventi.

THE PRINTER AT VENICE

TO

THE READER

\ OU will be well aware, if you have read the public journals, with what applause
the work which I now offer to you has been received throughout Europe
since its publication at Vienna five years ago. Not to mention others, if
you refer to the numbers of the Berne Journal for the early part of the
year 1761, you will not fail to see how highly it has been esteemed. It
contains an entirely new system of Natural Philosophy, which is already
commonly known as the Boscovichian theory, from the name of its author,
As a matter of fact, it is even now a subject of public instruction in several Universities in
different parts ; it is expounded not only in yearly theses or dissertations, both printed &
debated ; but also in several elementary books issued for the instruction of the young it is
introduced, explained, & by many considered as their original. Any one, however, who
wishes to obtain more detailed insight into the whole structure of the theory, the close
relation that its several parts bear to one another, or its great fertility & wide scope for
the purpose of deriving the whole of Nature, in her widest range, from a single simple law
of forces ; any one who wishes to make a deeper study of it must perforce study the work
here offered.

All these considerations had from the first moved me to undertake a new edition of
the work ; in addition, there was the fact that I perceived that it would be a matter of some
difficulty for copies of the Vienna edition to pass beyond the confines of Germany — indeed,
at the present time, no matter how diligently they are inquired for, they are to be found
on sale nowhere, or scarcely anywhere, in the rest of Europe. The system had its birth in
Italy, & its outlines had already been sketched by the author in several dissertations pub-
lished here in our own land ; though, as luck would have it, the system itself was finally
put into shape and published at Vienna, whither he had gone for a short time. I therefore
thought it right that it should be disseminated throughout the whole of Europe, & that
preferably as the product of an Italian press. I had in fact already commenced an edition
founded on a copy of the Vienna edition, when it came to my knowledge that the author
was greatly dissatisfied with the Vienna edition, taken in hand there after his departure ;
that innumerable printer's errors had crept in ; that many passages, especially those that
contain Algebraical formulae, were ill-arranged and erroneous ; lastly, that the author
himself had in mind a complete revision, including certain alterations, to give a better
finish to the work, together with certain additional matter.

That being the case, I was greatly desirous of obtaining a copy, revised & enlarged
by himself ; I also wanted to have him at hand whilst the edition was in progress, & that
he should superintend the whole thing for himself. This, however, I was unable to procure
during the last few years, in which he has been travelling through nearly the whole of
Europe ; until at last he came here, a little while ago, as he returned home from his lengthy
wanderings, & stayed here to assist me during the whole time that the edition was in
hand. He, in addition to our regular proof-readers, himself also used every care in cor-
recting the proof ; even then, however, he has not sufficient confidence in himself as to
imagine that not the slightest thing has escaped him. For it is a characteristic of the human
mind that it cannot concentrate long on the same subject with sufficient attention.

It follows that this ought to be considered in some measure as a first & original
edition ; any one who compares it with that issued at Vienna will soon see the difference
between them. Many of the minor alterations are made for the purpose of rendering
certain passages more elegant & clear ; there are, however, especially at the foot of a
page, slight additions also, or slight changes made after the type was set up, merely for
the purpose of filling up gaps that were left here & there — these gaps being due to the
fact that several sheets were being set at the same time by different compositors, and four
presses were kept hard at work together. As he was at hand, this could easily be done
without causing any disturbance of the sentences or the pagination.

14 AD LECTOREM EX EDITIONE VIENNENSI

nexus inter vires, & distantias, est utique admodum necessaria ad intelligendam Theoriam ipsam,
cujus ea est prcecipua queedam veluti clavis, sine qua omnino incassum tentarentur cetera ; sed
y ejusmodi est, ut tironum, & sane etiam mediocrium, immo etiam longe infra mediocritatem
collocatorum, captum non excedat, potissimum si viva accedat Professoris vox mediocriter etiam
versati in Mechanics, cujus ope, pro certo habeo, rem ita patentem omnibus reddi posse, ut
ii etiam, qui Geometric? penitus ignari sunt, paucorum admodum explicatione vocabulorum
accidente, earn ipsis oculis intueantur omnino perspicuam,

In tertia parte supponuntur utique nonnulla, que? demonstrantur in secunda ; sed ea ipsa
sunt admodum pauca, & Us, qui geometricas demonstrationes fastidiunt, facile admodum exponi
possunt res ipsee ita, ut penitus etiam sine ullo Geometric adjumento percipiantur, quanquam
sine Us ipsa demonstratio haberi non poterit ; ut idcirco in eo differre debeat is, qui secundam
partem attente legerit, y Geometriam calleat, ab eo, qui earn omittat, quod ille primus veritates
in tertia parte adhibitis, ac ex secunda erutas, ad explicationem Physics, intuebitur per evi-
dentiam ex ipsis demonstrationibus baustam, hie secundus easdem quodammodo per fidem Geo-
metris adhibitam credet. Hujusmodi inprimis est illud, particulam compositam ex punctis
etiam bomogeneis, preeditis lege virium proposita, posse per solam diversam ipsorum punctorum
dispositionem aliam particulam per cerium intervallum vel perpetuo attrahere, vel perpetuo
repellere, vel nihil in earn agere, atque id ipsum viribus admodum diversis, y que? respectu diver-
sarum particularum diver see sint, y diverse? respectu partium diver sarum ejusdem particulce,
ac aliam particulam alicubi etiam urgeant in latus, unde plurium pheenomenorum explicatio in
Physica sponte ftuit.

Ferum qui omnem Theorie?, y deductionum compagem aliquanto altius inspexerit, ac
diligentius perpenderit, videbit, ut spero, me in hoc perquisitionis genere multo ulterius
progressum esse, quam olim Newtonus ipse desideravit. Is enim in postremo Opticce questione
prolatis Us, qua per vim attractivam, y vim repulsivam, mutata distantia ipsi attractive? suc-
cedentem, explicari poterant, he?c addidit : " Atque he?c quidem omnia si ita sint, jam Natura
universa valde erit simplex, y consimilis sui, perficiens nimirum magnos omnes corporum
ccelestium motus attractione gravitatis, quee est mutua inter corpora ilia omnia, y minores fere
omnes particularum suarum motus alia aliqua vi attrabente, y repellente, quiz est inter particulas
illas mutua." Aliquanto autem inferius de primigeniis particulis agens sic habet : " Porro
videntur mihi he? particule? primigeniee non modo in se vim inertice habere, motusque leges passivas
illas, que? ex vi ista necessario oriuntur ; verum etiam motum perpetuo accipere a certis principiis
actuosis, qualia nimirum sunt gravitas, y causa fermentationis, y cohcerentia corporum. Atque
heec quidem principia considero non ut occultas qualitates, que? ex specificis rerum formis oriri
fingantur, sed ut universales Nature? leges, quibus res ipse? sunt formates. Nam principia
quidem talia revera existere ostendunt phenomena Nature?, licet ipsorum cause? que? sint,
nondum fuerit explicatum. Affirmare, singulas rerum species specificis preeditas esse qualita-
tibus occultis, per quas eae vim certam in agenda habent, hoc utique est nihil dicere : at ex
phcenomenis Nature? duo, vel tria derivare generalia motus principia, y deinde explicare,
quemadmodum proprietates, y actiones rerum corporearum omnium ex istis principiis conse-
quantur, id vero magnus esset factus in Philosophia progressus, etiamsi principiorum istorum
cause? nondum essent cognite?. Quare motus principia supradicta proponere non dubito, cum
per Naturam universam latissime pateant"

Hc?c ibi Newtonus, ubi is quidem magnos in Philosophia progressus facturum arbitratus
est eum, qui ad duo, vel tria generalia motus principia ex Nature? pheenomenis derivata phe?no-
menorum explicationem reduxerit, y sua principia protulit, ex quibus inter se diversis eorum
aliqua tantummodo explicari posse censuit. Quid igitur, ubi y ea ipsa tria, y alia preecipua
quczque, ut ipsa etiam impenetrabilitas, y impulsio reducantur ad principium unicum legitima
ratiocinatione deductum ? At id per meam unicam, y simplicem virium legempr<zstari,patebit
sane consideranti operis totius Synopsim quandam, quam hie subjicio ; sed multo magis opus
ipsum diligentius pervolventi.

THE PRINTER AT VENICE

TO

THE READER

|JOU will be well aware, if you have read the public journals, with what applause
the work which I now offer to you has been received throughout Europe
since its publication at Vienna five years ago. Not to mention others, if
you refer to the numbers of the Berne Journal for the early part of the
year 1761, you will not fail to see how highly it has been esteemed. It
contains an entirely new system of Natural Philosophy, which is already
commonly known as the Boscovicbian theory, from the name of its author,
As a matter of fact, it is even now a subject of public instruction in several Universities in
different parts ; it is expounded not only in yearly theses or dissertations, both printed &
debated ; but also in several elementary books issued for the instruction of the young it is
introduced, explained, & by many considered as their original. Any one, however, who
wishes to obtain more detailed insight into the whole structure of the theory, the close
relation that its several parts bear to one another, or its great fertility & wide scope for
the purpose of deriving the whole of Nature, in her widest range, from a single simple law
of forces ; any one who wishes to make a deeper study of it must perforce study the work
here offered.

All these considerations had from the first moved me to undertake a new edition of
the work ; in addition, there was the fact that I perceived that it would be a matter of some
difficulty for copies of the Vienna edition to pass beyond the confines of Germany — indeed,
at the present time, no matter how diligently they are inquired for, they are to be found
on sale nowhere, or scarcely anywhere, in the rest of Europe. The system had its birth in
Italy, & its outlines had already been sketched by the author in several dissertations pub-
lished here in our own land ; though, as luck would have it, the system itself was finally
put into shape and published at Vienna, whither he had gone for a short time. I therefore
thought it right that it should be disseminated throughout the whole of Europe, & that
preferably as the product of an Italian press. I had in fact already commenced an edition
founded on a copy of the Vienna edition, when it came to my knowledge that the author
was greatly dissatisfied with the Vienna edition, taken in hand there after his departure ;
that innumerable printer's errors had crept in ; that many passages, especially those that
contain Algebraical formulae, were ill-arranged and erroneous ; lastly, that the author
himself had in mind a complete revision, including certain alterations, to give a better
finish to the work, together with certain additional matter.

That being the case, I was greatly desirous of obtaining a copy, revised & enlarged
by himself ; I also wanted to have him at hand whilst the edition was in progress, & that
he should superintend the whole thing for himself. This, however, I was unable to procure
during the last few years, in which he has been travelling through nearly the whole of
Europe ; until at last he came here, a little while ago, as he returned home from his lengthy
wanderings, & stayed here to assist me during the whole time that the edition was in
hand. He, in addition to our regular proof-readers, himself also used every care in cor-
recting the proof ; even then, however, he has not sufficient confidence in himself as to
imagine that not the slightest thing has escaped him. For it is a characteristic of the human
mind that it cannot concentrate long on the same subject with sufficient attention.

It follows that this ought to be considered in some measure as a first & original
edition ; any one who compares it with that issued at Vienna will soon see the difference
between them. Many of the minor alterations are made for the purpose of rendering
certain passages more elegant & clear ; there are, however, especially at the foot of a
page, slight additions also, or slight changes made after the type was set up, merely for
the purpose of filling up gaps that were left here & there — these gaps being due to the
fact that several sheets were being set at the same time by different compositors, and four
presses were kept hard at work together. As he was at hand, this could easily be done
without causing any disturbance of the sentences or the pagination.

4 TYPOGRAPHUS VENETUS LECTORI

Inter mutationes occurret ordo numerorum mutatus in paragraphis : nam numerus 82
de novo accessit totus : deinde is, qui fuerat 261 discerptus est in 5 ; demum in Appendice
post num. 534 factae sunt & mutatiunculae nonnullae, & additamenta plura in iis, quae
pertinent ad sedem animse.

Supplementorum ordo mutatus est itidem ; quae enim fuerant 3, & 4, jam sunt i, &
2 : nam eorum usus in ipso Opere ante alia occurrit. UK autem, quod prius fuerat primum,
nunc autem est tertium, accessit in fine Scholium tertium, quod pluribus numeris complec-
titur dissertatiunculam integrant de argumento, quod ante aliquot annos in Parisiensi
Academia controversiae occasionem exhibuit in Encyclopedico etiam dictionario attactum,
in qua dissertatiuncula demonstrat Auctor non esse, cur ad vim exprimendam potentia
quaepiam distantice adhibeatur potius, quam functio.

Accesserunt per totum Opus notulae marginales, in quibus eorum, quae pertractantur
argumenta exponuntur brevissima, quorum ope unico obtutu videri possint omnia, & in
memoriam facile revocari.

Postremo loco ad calcem Operis additus est fusior catalogus eorum omnium, quae hue
usque ab ipso Auctore sunt edita, quorum collectionem omnem expolitam, & correctam,
ac eorum, quse nondum absoluta sunt, continuationem meditatur, aggressurus illico post
suum regressum in Urbem Romam, quo properat. Hie catalogus impressus fuit Venetisis
ante hosce duos annos in reimpressione ejus poematis de Solis ac Lunae defectibus.
Porro earn omnium suorum Operum Collectionem, ubi ipse adornaverit, typis ego meis
excudendam suscipiam, quam magnificentissime potero.

Haec erant, quae te monendum censui ; tu laboribus nostris fruere, & vive felix.

THE PREFACE TO THE READER

THAT APPEARED IN THE VIENNA EDITION

Reader, you have before you a Theory of Natural Philosophy deduced
from a single law of Forces. You will find in the opening paragraphs of
the first section a statement as to where the Theory has been already
published in outline, & to a certain extent explained ; & also the occasion
that led me to undertake a more detailed treatment & enlargement of it.
For I have thought fit to divide the work into three parts ; the first of
these contains the exposition of the Theory itself, its analytical deduction
& its demonstration ; the second a fairly full application to Mechanics ; & the third an
application to Physics.

The most important point, I decided, was for me to take the greatest care that every-
thing, as far as was possible, should be clearly explained, & that there should be no need for
higher geometry or for the calculus. Thus, in the first part, as well as in the third, there
are no proofs by analysis ; nor are there any by geometry, with the exception of a very few
that are absolutely necessary, & even these you will find relegated to brief notes set at the
foot of a page. I have also added some very few proofs, that required a knowledge of
higher algebra & geometry, or were of a rather more complicated nature, all of which have
been already published elsewhere, at the end of the work ; I have collected these under
the heading Supplements ; & in them I have included my views on Space & Time, which
are in accord with my main Theory, & also have been already published elsewhere. In
the second part, where the Theory is applied to Mechanics, I have not been able to do
without geometrical proofs altogether ; & even in some cases I have had to give algebraical
proofs. But these are of such a simple kind that they scarcely ever require anything more
than Euclidean geometry, the first and most elementary ideas of trigonometry, and easy
analytical calculations.

It is true that in the first part there are to be found a good many geometrical diagrams,
which at first sight, before the text is considered more closely, will appear to be rather
complicated. But these present nothing else but a kind of image of the subjects treated,
which by means of these diagrams are set before the eyes for contemplation. The very
curve that represents the law of forces is an instance of this. I find that between all points
of matter there is a mutual force depending on the distance between them, & changing as
this distance changes ; so that it is sometimes attractive, & sometimes repulsive, but always
follows a definite continuous law. Laws of variation of this kind between two quantities
depending upon one another, as distance & force do in this instance, may be represented
either by an analytical formula or by a geometrical curve ; but the former method of
representation requires far more knowledge of algebraical processes, & does not assist the
imagination in the way that the latter does. Hence I have employed the latter method in
the first part of the work, & relegated to the Supplements the analytical formula which
represents the curve, & the law of forces which the curve exhibits.

The whole matter reduces to this. In a straight line of indefinite length, which is
called the axis, a fixed point is taken ; & segments of the straight line cut off from this
point represent the distances. A curve is drawn following the general direction of this
straight line, & winding about it, so as to cut it in several places. Then perpendiculars that
are drawn from the ends of the segments to meet the curve represent the forces ; these
forces are greater or less, according as such perpendiculars are greater or less ; & they pass
from attractive forces to repulsive, and vice versa, whenever these perpendiculars change
their direction, as the curve passes from one side of the axis of indefinite length to the other
side of it. Now this requires no geometrical proof, but only a knowledge of certain terms,
which either belong to the first elementary principles of geometry, & are thoroughly well
known, or are such as can be defined when they are used. The term Asymptote is well
known, and from the same idea we speak of the branch of a curve as being asymptotic ;
thus a straight line is said to be the asymptote to any branch of a curve when, if the straight
line is indefinitely produced, it approaches nearer and nearer to the curvilinear arc which
is also prolonged indefinitely in such manner that the distance between them becomes
indefinitely diminished, but never altogether vanishes, so that the straight line & the curve
never really meet.

A careful consideration of the curve given in Fig. I, & of the way in which the relation

13

i4 AD LECTOREM EX EDITIONE VIENNENSI

nexus inter vires, & distantias, est utique ad.rn.odum necessaria ad intelligendam Theoriam ipsam,
cujus ea est prescipua qucsdam veluti clavis, sine qua omnino incassum tentarentur cetera ; sea
y ejusmodi est, ut tironum, & sane etiam mediocrium, immo etiam longe infra mediocritatem
collocatorum, captum non excedat, potissimum si viva accedat Professoris vox mediocriter etiam
versati in Mechanica, cujus ope, pro certo habeo, rem ita patentem omnibus reddi posse, ut
ii etiam, qui Geometries penitus ignari sunt, paucorum admodum explicatione vocabulorum
accidente, earn ipsis oculis intueantur omnino perspicuam.

In tertia parte supponuntur utique nonnulla, ques demonstrantur in secunda ; sed ea ipsa
sunt admodum pauca, & Us, qui geometricas demonstrationes fastidiunt, facile admodum exponi
possunt res ipsez ita, ut penitus etiam sine ullo Geometries adjumento percipiantur, quanquam
sine Us ipsa demonstratio haberi non poterit ; ut idcirco in eo differre debeat is, qui secundam
partem attente legerit, y Geometriam calleat, ab eo, qui earn omittat, quod ille primus veritates
in tertia parte adbibitis, ac ex secunda erutas, ad explicationem Physices, intuebitur per evi-
dentiam ex ipsis demonstrationibus haustam, hie secundus easdem quodammodo per fidem Geo-
metris adhibitam credet. Hujusmodi inprimis est illud, particulam compositam ex punctis
etiam homogeneis, presditis lege virium proposita, posse per solam diversam ipsorum punctorum
dispositionem aliam particulam per certum intervallum vel perpetuo attrahere, vel perpetuo
repellere, vel nihil in earn agere, atque id ipsum viribus admodum diversis, y qua respectu diver-
sarum particularum diver see sint, & diver see respectu partium diver sarum ejusdem particules,
ac aliam particulam alicubi etiam urgeant in latus, unde plurium phesnomenorum explicatio in
Physica sponte ftuit.

Verum qui omnem Theories, y deductionum compagem aliquanto altius inspexerit, ac
diligentius perpenderit, videbit, ut spero, me in hoc perquisitionis genere multo ulterius
progressum esse, quam olim Newtonus ipse desideravit. Is enim in postremo Optices questione
prolatis Us, ques per vim attractivam, & vim repulsivam, mutata distantia ipsi attractives suc-
cedentem, explicari poterant, hesc addidit : " Atque h<sc quidem omnia si ita sint, jam Natura
universa valde erit simplex, y consimilis sui, perficiens nimirum magnos omnes corporum
ceslestium motus attractione gravitatis, qucs est mutua inter corpora ilia omnia, & minores fere
omnes particularum suarum motus alia aliqua vi attrahente, y repellente, ques est inter particulas
illas mutua." Aliquanto autem inferius de primigeniis particulis agens sic habet : " Porro
videntur mihi hce particules primigenics non modo in se vim inerties habere, motusque leges passivas
illas, ques ex vi ista necessario oriuntur ; verum etiam motum perpetuo accipere a certis principiis
actuosis, qualia nimirum sunt gravitas, y causa fermentationis, y cohesrentia corporum. Atque
hesc quidem principia considero non ut occultas qualitates, ques ex specificis rerum formis oriri
fingantur, sed ut universales Natures leges, quibus res ipscs sunt formates. Nam principia
quidem talia revera existere ostendunt phesnomena Natures, licet ipsorum causes ques sint,
nondum fuerit explicatum. Affirmare, singulas rerum species specificis presditas esse qualita-
tibus occultis, per quas eae vim certam in agenda habent, hoc utique est nihil dicere : at ex
phesnomenis Natures duo, vel tria derivare generalia motus principia, y deinde explicare,
quemadmodum proprietates, y actiones rerum corporearum omnium ex istis principiis conse-
quantur, id vero magnus esset factus in Philosophia progressus, etiamsi principiorum istorum
causes nondum essent cognites. Quare motus principia supradicta proponere non dubito, cum
per Naturam universam latissime pateant."

Hcsc ibi Newtonus, ubi is quidem magnos in Philosophia progressus facturum arbitratus
est eum, qui ad duo, vel tria generalia motus principia ex Natures phesnomenis derivata phesno-
menorum explicationem reduxerit, y sua principia protulit, ex quibus inter se diversis eorum
aliqua tantummodo explicari posse censuit. Quid igitur, ubi y ea ipsa tria, y alia prcscipua
quesque, ut ipsa etiam impenetrabilitas, y impulsio reducantur ad principium unicum legitima
ratiocinatione deductum ? At id per meam unicam, y simplicem virium legem presstari, patebit
sane consideranti operis totius Synopsim quandam, quam hie subjicio ; sed multo magis opus
ipsum diligentius pervolventi.

PREFACE TO READER THAT APPEARED IN THE VIENNA EDITION 15

between the forces & the distances is represented by it, is absolutely necessary for the under-
standing of the Theory itself, to which it is as it were the chief key, without which it would
be quite useless to try to pass on to the rest. But it is of such a nature that it does not go
beyond the capacity of beginners, not even of those of very moderate ability, or of classes
even far below the level of mediocrity ; especially if they have the additional assistance of
a teacher's voice, even though he is only moderately familiar with Mechanics. By his help,
I am sure, the subject can be made clear to every one, so that those of them that are quite
ignorant of geometry, given the explanation of but a few terms, may get a perfectly good
idea of the subject by ocular demonstration.

In the third part, some of the theorems that have been proved in the second part are
certainly assumed, but there are very few such ; &, for those who do not care for geo-
metrical proofs, the facts in question can be quite easily stated in such a manner that they
can be completely understood without any assistance from geometry, although no real
demonstration is possible without them. There is thus bound to be a difference between
the reader who has gone carefully through the second part, & who is well versed in geo-
metry, & him who omits the second part ; in that the former will regard the facts, that
have been proved in the second part, & are now employed in the third part for the ex-
planation of Physics, through the evidence derived from the demonstrations of these facts,
whilst the second will credit these same facts through the mere faith that he has in geome-
tricians. A specially good instance of this is the fact, that a particle composed of points
quite homogeneous, subject to a law of forces as stated, may, merely by altering the arrange-
ment of those points, either continually attract, or continually repel, or have no effect at
all upon, another particle situated at a known distance from it ; & this too, with forces that
differ widely, both in respect of different particles & in respect of different parts of the same
particle ; & may even urge another particle in a direction at right angles to the line join-
ing the two, a fact that readily gives a perfectly natural explanation of many physical
phenomena.

Anyone who shall have studied somewhat closely the whole system of my Theory, &
what I deduce from it, will see, I hope, that I have advanced in this kind of investigation
much further than Newton himself even thought open to his desires. For he, in the last
of his " Questions " in his Opticks, after stating the facts that could be explained by means
of an attractive force, & a repulsive force that takes the place of the attractive force when
the distance is altered, has added these words : — " Now if all these things are as stated, then
the whole of Nature must be exceedingly simple in design, & similar in all its parts, accom-
plishing all the mighty motions of the heavenly bodies, as it does, by the attraction of
gravity, which is a mutual force between any two bodies of the whole system ; and Nature
accomplishes nearly all the smaller motions of their particles by some other force of attrac-
tion or repulsion, which is mutual between any two of those particles." Farther on, when
he is speaking about elementary particles, he says : — " Moreover, it appears to me that these
elementary particles not only possess an essential property of inertia, & laws of motion,
though only passive, which are the necessary consequences of this property ; but they also
constantly acquire motion from the influence of certain active principles such as, for
instance, gravity, the cause of fermentation, & the cohesion of solids. I do not consider these
principles to be certain mysterious qualities feigned as arising from characteristic forms of
things, but as universal laws of Nature, by the influence of which these very things have
been created. For the phenomena of Nature show that these principles do indeed exist,
although their nature has not yet been elucidated. To assert that each & every species is
endowed with a mysterious property characteristic to it, due to which it has a definite mode
in action, is really equivalent to saying nothing at all. On the other hand, to derive from
the phenomena of Nature two or three general principles, & then to explain how the pro-
perties & actions of all corporate things follow from those principles, this would indeed be
a mighty advance in philosophy, even if the causes of those principles had not at the time
been discovered. For these reasons I do not hesitate in bringing forward the principles of
motion given above, since they are clearly to be perceived throughout the whole range of
Nature."

These are the words of Newton, & therein he states his opinion that he indeed will
have made great strides in philosophy who shall have reduced the explanation of phenomena
to two or three general principles derived from the phenomena of Nature ; & he
brought forward his own principles, themselves differing from one another, by which he
thought that some only of the phenomena could be explained. What then if not only the
three he mentions, but also other important principles, such as impenetrability & impul-
sive force, be reduced to a single principle, deduced by a process of rigorous argument ! It
will be quite clear that this is exactly what is done by my single simple law of forces, to
anyone who studies a kind of synopsis of the whole work, which I add below ; but it will be
iar more clear to him who studies the whole work with some earnestness,

SYNOPSIS TOTIUS OPERIS

EX EDITIONE VIENNENSI

PARS I

sex numeris exhibeo, quando, & qua occasione Theoriam meam
invenerim, ac ubi hucusque de ea egerim in dissertationibus jam editis, quid
ea commune habeat cum Leibnitiana, quid cum Newtoniana Theoria, in
quo ab utraque discrepet, & vero etiam utrique praestet : addo, quid
alibi promiserim pertinens ad aequilibrium, & oscillationis centrum, &
quemadmodum iis nunc inventis, ac ex unico simplicissimo, ac elegant-
issimo theoremate profluentibus omnino sponte, cum dissertatiunculam

brevem meditarer, jam eo consilio rem aggressus ; repente mihi in opus integrum justse

molis evaserit tractatio.

7 Turn usque ad num. II expono Theoriam ipsam : materiam constantem punctis

prorsus simplicibus, indivisibilibus, & inextensis, ac a se invicem distantibus, quae puncta
habeant singula vim inertiae, & praeterea vim activam mutuam pendentem a distantiis, ut
nimirum, data distantia, detur & magnitude, & directio vis ipsius, mutata autem distantia,
mutetur vis ipsa, quae, imminuta distantia in infinitum, sit repulsiva, & quidem
excrescens in infinitum : aucta autem distantia, minuatur, evanescat, mutetur in attrac-
tivam crescentem primo, turn decrescentem, evanescentem, abeuntem iterum in repul-
sivam, idque per multas vices, donee demum in majoribus distantiis abeat in attractivam
decrescentem ad sensum in ratione reciproca duplicata distantiarum ; quern nexum virium
cum distantiis, & vero etiam earum transitum a positivis ad negativas, sive a repulsivis ad
attractivas, vel vice versa, oculis ipsis propono in vi, qua binae elastri cuspides conantur ad
es invicem accedere, vel a se invicem recedere, prout sunt plus justo distractae, vel con-
tractae.

II Inde ad num. 16 ostendo, quo pacto id non sit aggregatum quoddam virium temere

coalescentium, sed per unicam curvam continuam exponatur ope abscissarum exprimentium
distantias, & ordinatarum exprimentium vires, cujus curvae ductum, & naturam expono,
ac ostendo, in quo differat ab hyperbola ilia gradus tertii, quae Newtonianum gravitatem
exprimit : ac demum ibidem & argumentum, & divisionem propono operis totius.

1 6 Hisce expositis gradum facio ad exponendam totam illam analysim, qua ego ad ejusmodi

Theoriam deveni, & ex qua ipsam arbitror directa, & solidissima ratiocinatione deduci
totam. Contendo nimirum usque ad numerum 19 illud, in collisione corporum debere vel
haberi compenetrationem, vel violari legem continuitatis, velocitate mutata per saltum, si
cum inaequalibus velocitatibus deveniant ad immediatum contactum, quae continuitatis lex
cum (ut evinco) debeat omnino observari, illud infero, antequam ad contactum deveniant
corpora, debere mutari eorum velocitates per vim quandam, quae sit par extinguendse
velocitati, vel velocitatum differentiae, cuivis utcunque magnae.

19 A num. 19 ad 28 expendo effugium, quo ad eludendam argumenti mei vim utuntur ii,

qui negant corpora dura, qua quidem responsione uti non possunt Newtoniani, & Corpus-
culares generaliter, qui elementares corporum particulas assumunt prorsus duras : qui autem
omnes utcunque parvas corporum particulas molles admittunt, vel elasticas, difficultatem
non effugiunt, sed transferunt ad primas superficies, vel puncta, in quibus committeretur
omnino saltus, & lex continuitatis violaretur : ibidem quendam verborum lusum evolvo,
frustra adhibitum ad eludendam argumenti mei vim.

* Series numerorum, quibus tractari incipiunt, quae sunt in textu,

16

SYNOPSIS OF THE WHOLE WORK

(FROM THE VIENNA EDITION)

PART I

N the first six articles, I state the time at which I evolved my Theory, what i *
led me to it, & where I have discussed it hitherto in essays already pub-
lished : also what it has in common with the theories of Leibniz and
Newton ; in what it differs from either of these, & in what it is really
superior to them both. In addition I state what I have published else-
where about equilibrium & the centre of oscillation ; & how, having found
out that these matters followed quite easily from a single theorem of the
most simple & elegant kind, I proposed to write a short essay thereon ; but when I set to
work to deduce the matter from this principle, the discussion, quite unexpectedly to me,
developed into a whole work of considerable magnitude.

From this .until Art. II, I explain the Theory itself : that matter is unchangeable, 7
and consists of points that are perfectly simple, indivisible, of no extent, & separated from
one another ; that each of these points has a property of inertia, & in addition a mutual
active force depending on the distance in such a way that, if the distance is given, both the
magnitude & the direction of this force are given ; but if the distance is altered, so also is
the force altered ; & if the distance is diminished indefinitely, the force is repulsive, & in
fact also increases indefinitely ; whilst if the distance is increased, the force will be dimin-
ished, vanish, be changed to an attractive force that first of all increases, then decreases,
vanishes, is again turned into a repulsive force, & so on many times over ; until at greater
distances it finally becomes an attractive force that decreases approximately in the inverse
ratio of the squares of the distances. This connection between the forces & the distances,
& their passing from positive to negative, or from repulsive to attractive, & conversely, I
illustrate by the force with which the two ends of a spring strive to approach towards, or
recede from, one another, according as they are pulled apart, or drawn together, by more
than the natural amount.

From here on to Art. 1 6 I show that it is not merely an aggregate of forces combined n
haphazard, but that it is represented by a single continuous curve, by means of abscissse
representing the distances & ordinates representing the forces. I expound the construction
& nature of this curve ; & I show how it differs from the hyperbola of the third degree
which represents Newtonian gravitation. Finally, here too I set forth the scope of the
whole work & the nature of the parts into which it is divided.

These statements having been made, I start to expound the whole of the analysis, by 16
which I came upon a Theory of this kind, & from which I believe I have deduced the whole
of it by a straightforward & perfectly rigorous chain of reasoning. I contend indeed, from
here on until Art. 19, that, in the collision of solid bodies, either there must be compene-
tration, or the Law of Continuity must be violated by a sudden change of velocity, if
the bodies come into immediate contact with unequal velocities. Now since the Law of
Continuity must (as I prove that it must) be observed in every case, I infer that, before
the bodies reach the point of actual contact, their velocities must be altered by some force
which is capable of destroying the velocity, or the difference of the velocities, no matter how
great that may be.

From Art. 19 to Art. 28 I consider the artifice, adopted for the purpose of evading the 19
strength of my argument by those who deny the existence of hard bodies ; as a matter of
fact this cannot be used as an argument against me by the Newtonians, or the Corpuscular-
ians in general, for they assume that the elementary particles of solids are perfectly hard.
Moreover, those who admit that all the particles of solids, however small they may be, are
soft or elastic, yet do not escape the difficulty, but transfer it to prime surfaces, or points ;
& here a sudden change would be made & the Law of Continuity violated. In the same
connection I consider a certain verbal quibble, used in a vain attempt to foil the force of
my reasoning.

* These numbers are the numbers of the articles, in which the matters given in the text are first discussed.

17 C

1 8 SYNOPSIS TOTIUS OPERIS

28 Sequentibus num. 28 & 29 binas alias responsiones rejicio aliorum, quarum altera, ut

mei argument! vis elidatur, affirmat quispiam, prima materiae elementa compenetrari, alter
dicuntur materiae puncta adhuc moveri ad se invicem, ubi localiter omnino quiescunt, &
contra primum effugium evinco impenetrabilitatem ex inductione ; contra secundum
expono aequivocationem quandam in significatione vocis motus, cui aequivocationi totum
innititur.

30 Hinc num. 30, & 31 ostendo, in quo a Mac-Laurino dissentiam, qui considerata eadem,

quam ego contemplatus sum, collisione corporum, conclusit, continuitatis legem violari,
cum ego eandem illaesam esse debere ratus ad totam devenerim Theoriam meam.

32 Hie igitur, ut meae deductionis vim exponam, in ipsam continuitatis legem inquire, ac

a num. 32 ad 38 expono, quid ipsa sit, quid mutatio continua per gradus omnes intermedios,
quae nimirum excludat omnem saltum ab una magnitudine ad aliam sine transitu per

39 intermedias, ac Geometriam etiam ad explicationem rei in subsidium advoco : turn earn
probo primum ex inductione, ac in ipsum inductionis principium inquirens usque ad num.
44, exhibeo, unde habeatur ejusdem principii vis, ac ubi id adhiberi possit, rem ipsam
illustrans exemplo impenetrabilitatis erutae passim per inductionem, donee demum ejus vim

45 applicem ad legem continuitatis demonstrandam : ac sequentibus numeris casus evolvo
quosdam binarum classium, in quibus continuitatis lex videtur laedi nee tamen laeditur.

48 Post probationem principii continuitatis petitam ab inductione, aliam num. 48 ejus

probationem aggredior metaphysicam quandam, ex necessitate utriusque limitis in quanti-
tatibus realibus, vel seriebus quantitatum realium finitis, quae nimirum nee suo principio,
nee suo fine carere possunt. Ejus rationis vim ostendo in motu locali, & in Geometria

52 sequentibus duobus numeris : turn num. 52 expono difficultatem quandam, quas petitur
ex eo, quod in momento temporis, in quo transitur a non esse ad esse, videatur juxta ejusmodi
Theoriam debere simul haberi ipsum esse, & non esse, quorum alterum ad finem praecedentis
seriei statuum pertinet, alterum ad sequentis initium, ac solutionem ipsius fuse evolvo,
Geometria etiam ad rem oculo ipsi sistendam vocata in auxilium.

63 Num. 63, post epilogum eorum omnium, quae de lege continuitatis sunt dicta, id

principium applico ad excludendum saltum immediatum ab una velocitate ad aliam, sine
transitu per intermedias, quod & inductionem laederet pro continuitate amplissimam, &
induceret pro ipso momento temporis, in quo fieret saltus, binas velocitates, ultimam
nimirum seriei praecedentis, & primam novas, cum tamen duas simul velocitates idem mobile
habere omnino non possit. Id autem ut illustrem, & evincam, usque ad num. 72 considero
velocitatem ipsam, ubi potentialem quandam, ut appello, velocitatem ab actuali secerno,
& multa, quae ad ipsarum naturam, ac mutationes pertinent, diligenter evolvo, nonnullis
etiam, quae inde contra meae Theoriae probationem objici possunt, dissolutis.

His expositis conclude jam illud ex ipsa continuitate, ubi corpus quodpiam velocius
movetur post aliud lentius, ad contactum immediatum cum ilia velocitatum inaequalitate
deveniri non posse, in quo scilicet contactu primo mutaretur vel utriusque velocitas, vel
alterius, per saltum, sed debere mutationem velocitatis incipere ante contactum ipsum.

73 Hinc num. 73 infero, debere haberi mutationis causam, quae appelletur vis : turn num. 74

74 hanc vim debere esse mutuam, & agere in partes contrarias, quod per inductionem evinco,

75 & inde infero num. 75, appellari posse repulsivam ejusmodi vim mutuam, ac ejus legem
exquirendam propono. In ejusmodi autem perquisitione usque ad num. 80 invenio illud,
debere vim ipsam imminutis distantiis crescere in infinitum ita ut par sit extinguendae
velocitati utcunque magnse ; turn & illud, imminutis in infinitum etiam distantiis, debere
in infinitum augeri, in maximis autem debere esse e contrario attractivam, uti est gravitas :
inde vero colligo limitem inter attractionem, & repulsionem : turn sensim plures, ac etiam
plurimos ejusmodi limites invenio, sive transitus ab attractione ad repulsionem, & vice
versa, ac formam totius curvae per ordinatas suas exprimentis virium legem determino.

SYNOPSIS OF THE WHOLE WORK 19

In the next articles, 28 & 29, I refute a further pair of arguments advanced by others ; 28
in the first of these, in order to evade my reasoning, someone states that there is compene-
tration of the primary elements of matter ; in the second, the points of matter are said to
be moved with regard to one another, even when they are absolutely at rest as regards
position. In reply to the first artifice, I prove the principle of impenetrability by induc-
tion ; & in reply to the second, I expose an equivocation in the meaning of the term motion,
an equivocation upon which the whole thing depends.

Then, in Art. 30, 31, I show in what respect I differ from Maclaurin, who, having 30
considered the same point as myself, came to the conclusion that in the collision of bodies
the Law of Continuity was violated ; whereas I obtained the whole of my Theory from the
assumption that this law must be unassailable.

At this point therefore, in order that the strength of my deductive reasoning might 32
be shown, I investigate the Law of Continuity ; and from Art. 32 to Art. 38, I set forth its
nature, & what is meant by a continuous change through all intermediate stages, such as
to exclude any sudden change from any one magnitude to another except by a passage
through intermediate stages ; & I call in geometry as well to help my explanation of the
matter. Then I investigate its truth first of all by induction ; &, investigating the prin- 39
ciple of induction itself, as far as Art. 44, 1 show whence the force of this principle is derived,
& where it can be used. I give by way of illustration an example in which impenetrability
is derived entirely by induction ; & lastly I apply the force of the principle to demonstrate
the Law of Continuity. In the articles that follow I consider certain cases of two kinds, 45
in which the Law of Continuity appears to be violated, but is not however really violated.

After this proof of the principle of continuity procured through induction, in Art. 48, 48
I undertake another proof of a metaphysical kind, depending upon the necessity of a limit
on either side for either real quantities or for a finite series of real quantities ; & indeed it
is impossible that these limits should be lacking, either at the beginning or the end. I
demonstrate the force of this reasoning in the case of local motion, & also in geometry, in the
next two articles. Then in Art. 52 I explain a certain difficulty, which is derived from the S2
fact that, at the instant at which there is a passage from non-existence to existence, it appears
according to a theory of this kind that we must have at the same time both existence and
non-existence. For one of these belongs to the end of the antecedent series of states, & the
other to the beginning of the consequent series. I consider fairly fully' the solution of this
problem ; and I call in geometry as well to assist in giving a visual representation of the
matter.

In Art. 63, after summing up all that has been said about the Law of Continuity, I 63
apply the principle to exclude the possibility of any sudden change from one velocity to
another, except by passing through intermediate velocities ; this would be contrary to the
very full proof that I give for continuity, as it would lead to our having two velocities at
the instant at which the change occurred. That is to say, there would be the final velocity
of the antecedent series, & the initial velocity of the consequent series ; in spite of the fact
that it is quite impossible for a moving body to have two different velocities at the same
time. Moreover, in order to illustrate & prove the point, from here on to Art. 72, I
consider velocity itself ; and I distinguish between a potential velocity, as I call it, & an
actual velocity ; I also investigate carefully many matters that relate to the nature of these
velocities & to their changes. Further, I settle several difficulties that can be brought
up in opposition to the proof of my Theory, in consequence.

This done, I then conclude from the principle of continuity that, when one body with
a greater velocity follows after another body having a less velocity, it is impossible that
there should ever be absolute contact with such an inequality of velocities ; that is to say,
a case of the velocity of each, or of one or the other, of them being changed suddenly at
the instant of contact. I assert on the other hand that the change in the velocities must
begin before contact. Hence, in Art. 73, I infer that there must be a cause for this change : 73
which is to be called " force." Then, in Art. 74, I prove that this force is a mutual one, & 74
that it acts in opposite directions ; the proof is by induction. From this, in Art. 75, I 75
infer that such a mutual force may be said to be repulsive ; & I undertake the investigation
of the law that governs it. Carrying on this investigation as far as Art. 80, I find that this
force must increase indefinitely as the distance is diminished, in order that it may be capable
of destroying any velocity, however great that velocity may be. Moreover, I find that,
whilst the force must be indefinitely increased as the distance is indefinitely decreased, it
must be on the contrary attractive at very great distances, as is the case for gravitation.
Hence I infer that there must be a limit-point forming a boundary between attraction &
repulsion ; & then by degrees I find more, indeed very many more, of such limit-points,
or points of transition from attraction to repulsion, & from repulsion to attraction ; & I
determine the form of the entire curve, that expresses by its ordinates the law of these forces.

20 SYNOPSIS TOTIUS OPERIS

8 1 Eo usque virium legem deduce, ac definio ; turn num. 81 eruo ex ipsa lege consti-

tutionem elementorum materiae, quae debent esse simplicia, ob repulsionem in minimis
distantiis in immensum auctam ; nam ea, si forte ipsa elementa partibus constarent, nexum
omnem dissolveret. Usque ad num. 88 inquire in illud, an hasc elementa, ut simplicia esse
debent, ita etiam inextensa esse debeant, ac exposita ilia, quam virtualem extensionem
appellant, eandem exclude inductionis principio, & difficultatem evolvo turn earn, quae peti
possit ab exemplo ejus generis extensionis, quam in anima indivisibili, & simplice per aliquam
corporis partem divisibilem, & extensam passim admittunt : vel omnipraesentiae Dei : turn
earn, quae peti possit ab analogia cum quiete, in qua nimirum conjungi debeat unicum
spatii punctum cum serie continua momentorum temporis, uti in extensione virtuali unicum
momentum temporis cum serie continua punctorum spatii conjungeretur, ubi ostendo, nee
quietem omnimodam in Natura haberi usquam, nee adesse semper omnimodam inter

88 tempus, & spatium analogiam. Hie autem ingentem colligo ejusmodi determinationis
fructum, ostendens usque ad num. 91, quantum prosit simplicitas, indivisibilitas, inextensio
elementorum materiae, ob summotum transitum a vacuo continue per saltum ad materiam
continuam, ac ob sublatum limitem densitatis, quae in ejusmodi Theoria ut minui in
infinitum potest, ita potest in infinitum etiam augeri, dum in communi, ubi ad contactum
deventum est, augeri ultra densitas nequaquam potest, potissimum vero ob sublatum omne
continuum coexistens, quo sublato & gravissimae difficultates plurimse evanescunt, &
infinitum actu existens habetur nullum, sed in possibilibus tantummodo remanet series
finitorum in infinitum producta.

91 His definitis, inquire usque ad num. 99 in illud, an ejusmodi elementa sint censenda

homogenea, an heterogenea : ac primo quidem argumentum pro homogeneitate saltern in
eo, quod pertinet ad totam virium legem, invenio in homogenietate tanta primi cruris
repulsivi in minimis distantiis, ex quo pendet impenetrabilitas, & postremi attractivi, quo
gravitas exhibetur, in quibus omnis materia est penitus homogenea. Ostendo autem, nihil
contra ejusmodi homogenietatem evinci ex principio Leibnitiano indiscernibilium, nihil ex
inductione, & ostendo, unde tantum proveniat discrimen in compositis massulis, ut in
frondibus, & foliis ; ac per inductionem, & analogiam demonstro, naturam nos ad homo-
geneitatem elementorum, non ad heterogeneitatem deducere.

100 Ea ad probationem Theoriae pertinent ; qua absoluta, antequam inde fructus colli-
gantur multiplices, gradum hie facio ad evolvendas difficultates, quae vel objectae jam sunt,
vel objici posse videntur mihi, primo quidem contra vires in genere, turn contra meam
hanc expositam, comprobatamque virium legem, ac demum contra puncta ilia indivisibilia,
& inextensa, quae ex ipsa ejusmodi virium lege deducuntur.

101 Primo quidem, ut iis etiam faciam satis, qui inani vocabulorum quorundam sono
perturbantur, a num. 101 ad 104 ostendo, vires hasce non esse quoddam occultarum
qualitatum genus, sed patentem sane Mechanismum, cum & idea earum sit admodum
distincta, & existentia, ac lex positive comprobata ; ad Mechanicam vero pertineat omnis

104 tractatio de Motibus, qui a datis viribus etiam sine immediate impulsu oriuntur. A num.
104 ad 106 ostendo, nullum committi saltum in transitu a repulsionibus ad attractiones,

1 06 & vice versa, cum nimirum per omnes inter medias quantitates is transitus fiat. Inde vero
ad objectiones gradum facio, quae totam curvas formam impetunt. Ostendo nimirum usque
ad num. 116, non posse omnes repulsiones a minore attractione desumi ; repulsiones ejusdem
esse seriei cum attractionibus, a quibus differant tantummodo ut minus a majore, sive ut
negativum a positivo ; ex ipsa curvarum natura, quae, quo altioris sunt gradus, eo in
pluribus punctis rectam secare possunt, & eo in immensum plures sunt numero ; haberi
potius, ubi curva quaeritur, quae vires exprimat, indicium pro curva ejus naturae, ut rectam
in plurimis punctis secet, adeoque plurimos secum afferat virium transitus a repulsivis ad
attractivas, quam pro curva, quae nusquam axem secans attractiones solas, vel solas pro
distantiis omnibus repulsiones exhibeat : sed vires repulsivas, & multiplicitatem transituum
esse positive probatam, & deductam totam curvas formam, quam itidem ostendo, non esse
ex arcubus natura diversis temere coalescentem, sed omnino simplicem, atque earn ipsam

SYNOPSIS OF THE WHOLE WORK 21

So far I have been occupied in deducing and settling the law of these forces. Next,
in Art. 8r, I derive from this law the constitution of the elements of matter. These must be 81
quite simple, on account of the repulsion at very small distances being immensely great ;
for if by chance those elements were made up of parts, the repulsion would destroy all
connections between them. Then, as far as Art. 88, I consider the point, as to whether
these elements, as they must be simple, must therefore be also of no extent ; &, having ex-
plained what is called " virtual extension," I reject it by the principle of induction. I
then consider the difficulty which may be brought forward from an example of this kind of
extension ; such as is generally admitted in the case of the indivisible and one-fold soul
pervading a divisible & extended portion of the body, or in the case of the omnipresence
of GOD. Next I consider the difficulty that may be brought forward from an analogy with
rest ; for here in truth one point of space must be connected with a continuous series of
instants of time, just as in virtual extension a single instant of time would be connected with
a continuous series of points of space. I show that there can neither be perfect rest any- gg
where in Nature, nor can there be at all times a perfect analogy between time and space.
In this connection, I also gather a large harvest from such a conclusion as this ; showing,
as far as Art. 91, the great advantage of simplicity, indivisibility, & non-extension in the
elements of matter. For they do away with the idea of a passage from a continuous vacuum
to continuous matter through a sudden change. Also they render unnecessary any limit
to density : this, in a Theory like mine, can be just as well increased to an indefinite extent,
as it can be indefinitely decreased : whilst in the ordinary theory, as soon as contact takes
place, the density cannot in any way be further increased. But, most especially, they do
away with the idea of everything continuous coexisting ; & when this is done away with,
the majority of the greatest difficulties vanish. Further, nothing infinite is found actually
existing ; the only thing possible that remains is a series of finite things produced inde-
finitely.

These things being settled, I investigate, as far as Art. 99, the point as to whether QJ
elements of this kind are to be considered as being homogeneous or heterogeneous. I find
my first evidence in favour of homogeneity — at least as far as the complete law of forces
is concerned — in the equally great homogeneity of the first repulsive branch of my curve
of forces for very small distances, upon which depends impenetrability, & of the last attrac-
tive branch, by which gravity is represented. Moreover I show that there is nothing that
can be proved in opposition to homogeneity such as this, that can be derived from either
the Leibnizian principle of " indiscernibles," or by induction. I also show whence arise
those differences, that are so great amongst small composite bodies, such as we see in boughs
& leaves ; & I prove, by induction & analogy, that the very nature of things leads us to
homogeneity, & not to heterogeneity, for the elements of matter.

These matters are all connected with the proof of my Theory. Having accomplished IOo
this, before I start to gather the manifold fruits to be derived from it, I proceed to consider
the objections to my theory, such as either have been already raised or seem to me capable
of being raised ; first against forces in general, secondly against the law of forces that I
have enunciated & proved, & finally against those indivisible, non-extended points that
are deduced from a law of forces of this kind.

First of all then, in order that I may satisfy even those who are confused over the 101
empty sound of certain terms, I show, in Art. 101 to 104, that these forces are not some
sort of mysterious qualities ; but that they form a readily intelligible mechanism, since
both the idea of them is perfectly distinct, as well as their existence, & in addition the law
that governs them is demonstrated in a direct manner. To Mechanics belongs every dis-
cussion concerning motions that arise from given forces without any direct impulse. In
Art. 104 to 106, I show that no sudden change takes place in passing from repulsions to 104
attractions or vice versa ; for this transition is made through every intermediate quantity.
Then I pass on to consider the objections that are made against the whole form of my 106
curve. I show indeed, from here on to Art. 116, that all repulsions cannot be taken to
come from a decreased attraction ; that repulsions belong to the self-same series as attrac-
tions, differing from them only as less does from more, or negative from positive. From
the very nature of the curves (for which, the higher the degree, the more points there are
in which they can intersect a right line, & vastly more such curves there are), I deduce
that there is more reason for assuming a curve of the nature of mine (so that it may cut a
right line in a large number of points, & thus give a large number of transitions of the forces
from repulsions to attractions), than for assuming a curve that, since it does not cut
the axis anywhere, will represent attractions alone, or repulsions alone, at all distances.
Further, I point out that repulsive forces, and a multiplicity of transitions are directly
demonstrated, & the whole form of the curve is a matter of deduction ; & I also show that
it is not formed of a number of arcs differing in nature connected together haphazard ;

22 SYNOPSIS TOTIUS OPERIS

simplicitatem in Supplementis cvidentissime demonstro, exhibens methodum, qua deveniri
possit ad aequationem ejusmodi curvse simplicem, & uniformem ; licet, ut hie ostendo, ipsa
ilia lex virium possit mente resolvi in plures, quae per plures curvas exponantur, a quibus
tamen omnibus ilia reapse unica lex, per unicam illam continuant, & in se simplicem curvam
componatur.

121 A num. 121 refello, quae objici possunt a lege gravitatis decrescentis in ratione reciproca

duplicata distantiarum, quae nimirum in minimis distantiis attractionem requirit crescentem
in infinitum. Ostendo autem, ipsam non esse uspiam accurate in ejusmodi ratione, nisi
imaginarias resolutiones exhibeamus ; nee vero ex Astronomia deduci ejusmodi legem
prorsus accurate servatam in ipsis Planetarum, & Cometarum distantiis, sed ad summum ita

124 proxime, ut differentia ab ea lege sit perquam exigua : ac a num. 124 expendo argumentum,
quod pro ejusmodi lege desumi possit ex eo, quod cuipiam visa sit omnium optima, &
idcirco electa ab Auctore Naturae, ubi ipsum Optimismi principium ad trutinam revoco, ac
exclude, & vero illud etiam evinco, non esse, cur omnium optima ejusmodi lex censeatur :
in Supplementis vero ostendo, ad qua; potius absurda deducet ejusmodi lex, & vero etiam
aliae plures attractionis, quae imminutis in infinitum distantiis excrescat in infinitum.

131 Num. 131 a viribus transeo ad elementa, & primum ostendo, cur punctorum inexten-

sorum ideam non habeamus, quod nimirum earn haurire non possumus per sensus, quos
solae massae, & quidem grandiores, afficiunt, atque idcirco eandem nos ipsi debemus per
reflexionem efformare, quod quidem facile possumus. Ceterum illud ostendo, me non
inducere primum in Physicam puncta indivisibilia, & inextensa, cum eo etiam Leibnitianae
monades recidant, sed sublata extensione continua difficultatem auferre illam omnem, quae
jam olim contra Zenonicos objecta, nunquam est satis soluta, qua fit, ut extensio continua
ab inextensis effici omnino non possit.

140 Num. 140 ostendo, inductionis principium contra ipsa nullam habere vim, ipsorum

autem existentiam vel inde probari, quod continuitas se se ipsam destruat, & ex ea assumpta
probetur argumentis a me institutis hoc ipsum, prima elementa esse indivisibilia, & inextensa,

143 nee ullum haberi extensum continuum. A num. 143 ostendo, ubi continuitatem admittam,
nimirum in solis motibus ; ac illud explico, quid mihi sit spatium, quid tempus, quorum
naturam in Supplementis multo uberius expono. Porro continuitatem ipsam ostendo a
natura in solis motibus obtineri accurate, in reliquis affectari quodammodo ; ubi & exempla
quaedam evolvo continuitatis primo aspectu violatae, in quibusdam proprietatibus luminis,
ac in aliis quibusdam casibus, in quibus quaedam crescunt per additionem partium, non (ut
ajunt) per intussumptionem.

\

153 A num. 153 ostendo, quantum haec mea puncta a spiritibus differant ; ac illud etiam

evolvo, unde fiat, ut in ipsa idea corporis videatur includi extensio continua, ubi in ipsam
idearum nostrarum originem inquire, & quae inde praejudicia profluant, expono. Postremo

165 autem loco num. 165 innuo, qui fieri possit, ut puncta inextensa, & a se invicem distantia,
in massam coalescant, quantum libet, cohaerentem, & iis proprietatibus praeditam, quas in
corporibus experimur, quod tamen ad tertiam partem pertinet, ibi multo uberius pertrac-
tandum ; ac ibi quidem primam hanc partem absolve.

PARS II

166 Num. 166 hujus partis argumentum propono ; sequenti vero 167, quae potissimum in

curva virium consideranda sint, enuncio. Eorum considerationem aggressus, primo quidem

1 68 usque ad num. 172 in ipsos arcus inquire, quorum alii attractivi, alii repulsivi, alii asym-
ptotici, ubi casuum occurrit mira multitudo, & in quibusdam consectaria notatu digna, ut
& illud, cum ejus formae curva plurium asymptotorum esse possit, Mundorum prorsus
similium seriem posse oriri, quorum alter respectu alterius vices agat unius, & indissolubilis

SYNOPSIS OF THE WHOLE WORK 23

but that it is absolutely one-fold. This one-fold character I demonstrate in the Supple-
ments in a very evident manner, giving a method by which a simple and uniform equation
may be obtained for a curve of this kind. Although, as I there point out, this law of forces
may be mentally resolved into several, and these may be represented by several correspond-
ing curves, yet that law, actually unique, may be compounded from all of these together
by means of the unique, continuous & one-fold curve that I give.

In Art. 121, I start to give a refutation of those objections that may be raised from I2i
a consideration of the fact that the law of gravitation, decreasing in the inverse duplicate
ratio of the distances, demands that there should be an attraction at very small distances,
& that it should increase indefinitely. However, I show that the law is nowhere exactly in
conformity with a ratio of this sort, unless we add explanations that are merely imaginative ;
nor, I assert, can a law of this kind be deduced from astronomy, that is followed with per-
fect accuracy even at the distances of the planets & the comets, but one merely that is at
most so very nearly correct, that the difference from the law of inverse squares is very
slight. From Art. 124 onwards,! examine the value of the argument that can be drawn 124
in favour of a law of this sort from the view that, as some have thought, it is the best of
all, & that on that account it was selected by the Founder of Nature. In connection with
this I examine the principle of Optimism, & I reject it ; moreover I prove conclusively
that there is no reason why this sort of law should be supposed to be the best of all. Fur-
ther in the Supplements, I show to what absurdities a law of this sort is more likely to lead ;
& the same thing for other laws of an attraction that increases indefinitely as the distance
is diminished indefinitely.

In Art. 131 I pass from forces to elements. I first of all show the reason why we may 1*1
not appreciate the idea of non-extended points ; it is because we are unable to perceive
them by means of the senses, which are only affected by masses, & these too must be of
considerable size. Consequently we have to build up the idea by a process of reasoning ;
& this we can do without any difficulty. In addition, I point out that I am not the first
to introduce indivisible & non-extended points into physical science ; for the " monads "
of Leibniz practically come to the same thing. But I show that, by rejecting the idea of
continuous extension, I remove the whole of the difficulty, which was raised against the
disciples of Zeno in years gone by, & has never been answered satisfactorily ; namely, the
difficulty arising from the fact that by no possible means can continuous extension be
made up from things of no extent.

In Art. 140 I show that the principle of induction yields no argument against these 140
indivisibles ; rather their existence is demonstrated by that principle, for continuity is
self-contradictory. On this assumption it may be proved, by arguments originated by
myself, that the primary elements are indivisible & non-extended, & that there does not
exist anything possessing the property of continuous extension. From Art. 143 onwards, j .,
I point out the only connection in which I shall admit continuity, & that is in motion.
I state the idea that I have with regard to space, & also time : the nature of these I explain
much more fully in the Supplements. Further, I show that continuity itself is really a
property of motions only, & that in all other things it is more or less a false assumption.
Here I also consider some examples in which continuity at first sight appears to be
violated, such as in some of the properties of light, & in certain other cases where things
increase by addition of parts, and not by intussumption, as it is termed.

From Art. 153 onwards, I show how greatly these points of mine differ from object- 153
souls. I consider how it comes about that continuous extension seems to be included
in the very idea of a body ; & in this connection, I investigate the origin of our ideas
& I explain the prejudgments that arise therefrom. Finally, in Art. 165, I lightly 165
sketch what might happen to enable points that are of no extent, & at a distance from
one another, to coalesce into a coherent mass of any size, endowed with those properties
that we experience in bodies. This, however, belongs to the third part ; & there it will be
much more fully developed. This finishes the first part.

PART II

In Art. 1 66 I state the theme of this second part ; and in Art. 167 I declare what 166
matters are to be considered more especially in connection with the curve of forces. Com-
ing to the consideration of these matters, I first of all, as far as Art. 172, investigate the 168
arcs of the curve, some of which are attractive, some repulsive and some asymptotic. Here
a marvellous number of different cases present themselves, & to some of them there are
noteworthy corollaries ; such as that, since a curve of this kind is capable of possessing a
considerable number of asymptotes, there can arise a series of perfectly similar cosmi, each
of which will act upon all the others as a single inviolate elementary system. From Art. 172

24 SYNOPSIS TOTIUS OPERIS

172 element!. Ad. num. 179 areas contemplor arcubus clausas, quae respondentes segmento axis
cuicunque, esse possunt magnitudine utcunque magnae, vel parvae, sunt autem mensura

179 incrementi, vel decrement! quadrat! velocitatum. Ad num. 189 inquire in appulsus curvse
ad axem, sive is ibi secetur ab eadem (quo casu habentur transitus vel a repulsione ad
attractionem, vel ab attractione ad repulsionem, quos dico limites, & quorum maximus est
in tota mea Theoria usus), sive tangatur, & curva retro redeat, ubi etiam pro appulsibus
considero recessus in infinitum per arcus asymptoticos, & qui transitus, sive limites, oriantur
inde, vel in Natura admitti possint, evolvo.

189 Num. 189 a consideratione curvae ad punctorum combinationem gradum facio, ac

primo quidem usque ad num. 204 ago de systemate duorum punctorum, ea pertractans,
quas pertinent ad eorum vires mutuas, & motus, sive sibi relinquantur, sive projiciantur
utcunque, ubi & conjunctione ipsorum exposita in distantiis limitum, & oscillationibus
variis, sive nullam externam punctorum aliorum actionem sentiant, sive perturbentur ab
eadem, illud innuo in antecessum, quanto id usui futurum sit in parte tertia ad exponenda
cohaesionis varia genera, fermentationes, conflagrationes, emissiones vaporum, proprietates
luminis, elasticitatem, mollitiem.

204 Succedit a Num. 204 ad 239 multo uberior consideratio trium punctorum, quorum

vires generaliter facile definiuntur data ipsorum positione quacunque : verum utcunque
data positione, & celeritate nondum a Geometris inventi sunt motus ita, ut generaliter pro
casibus omnibus absolvi calculus possit. Vires igitur, & variationem ingentem, quam
diversae pariunt combinationes punctorum, utut tantummodo numero trium, persequor

209 usque ad num. 209. Hinc usque ad num. 214 quaedam evolvo, quae pertinent ad vires
ortas in singulis ex actione composita reliquorum duorum, & quae tertium punctum non ad
accessum urgeant, vel recessum tantummodo respectu eorundem, sed & in latus, ubi &
soliditatis imago prodit, & ingens sane discrimen in distantiis particularum perquam exiguis
ac summa in maximis, in quibus gravitas agit, conformitas, quod quanto itidem ad Naturae

214 explicationem futurum sit usui, significo. Usque ad num. 221 ipsis etiam oculis contem-
plandum propono ingens discrimen in legibus virium, quibus bina puncta agunt in tertium,
sive id jaceat in recta, qua junguntur, sive in recta ipsi perpendiculari, & eorum intervallum
secante bifariam, constructis ex data primigenia curva curvis vires compositas exhibentibus :

221 turn sequentibus binis numeris casum evolvo notatu dignissimum, in quo mutata sola
positione binorum punctorum, punctum tertium per idem quoddam intervallum, situm in
eadem distantia a medio eorum intervallo, vel perpetuo attrahitur, vel perpetuo repellitur,
vel nee attrahitur, nee repellitur ; cujusmodi discrimen cum in massis haberi debeat multo

222 majus, illud indico, num. 222, quantus inde itidem in Physicam usus proveniat.

223 Hie jam num. 223 a viribus binorum punctorum transeo ad considerandum totum

ipsorum systema, & usque ad num. 228 contemplor tria puncta in directum sita, ex quorum
mutuis viribus relationes quaedam exurgunt, quas multo generaliores redduntur inferius, ubi
in tribus etiam punctis tantummodo adumbrantur, quae pertinent ad virgas rigidas, flexiles,
elasticas, ac ad vectem, & ad alia plura, quae itidem inferius, ubi de massis, multo generaliora

228 fiunt. Demum usque ad num. 238 contemplor tria puncta posita non in directum, sive in
aequilibrio sint, sive in perimetro ellipsium quarundam, vel curvarum aliarum ; in quibus
mira occurrit analogia limitum quorundam cum limitibus, quos habent bina puncta in axe
curvae primigeniae ad se invicem, atque ibidem multo major varietas casuum indicatur pro
massis, & specimen applicationis exhibetur ad soliditatem, & liquationem per celerem

238 intestinum motum punctis impressum. Sequentibus autem binis numeris generalia quaedam
expono de systemate punctorum quatuor cum applicatione ad virgas solidas, rigidas, flexiles,
ac ordines particularum varies exhibeo per pyramides, quarum infimae ex punctis quatuor,
superiores ex quatuor pyramidibus singulae coalescant.

24° A num. 240 ad massas gradu facto usque a num. 264 considero, quae ad centrum gravi-

tatis pertinent, ac demonstro generaliter, in quavis massa esse aliquod, & esse unicum :
ostendo, quo pacto determinari generaliter possit, & quid in methodo, quae communiter
adhibetur, desit ad habendam demonstrationis vim, luculenter expono, & suppleo, ac

SYNOPSIS OF THE WHOLE WORK 25

to Art. 179, I consider the areas included by the arcs; these, corresponding to different 172
segments of the axis, may be of any magnitude whatever, either great or small ; moreover
they measure the increment or decrement in the squares of the velocities. Then, on as 179
far as Art. 189, 1 investigate the approach of the curve to the axis ; both when the former
is cut by the latter, in which case there are transitions from repulsion to attraction and
from attraction to repulsion, which I call ' limits,' & use very largely in every part of my
Theory ; & also when the former is touched by the latter, & the curve once again recedes
from the axis. I consider, too, as a case of approach, recession to infinity along an asymp-
totic arc ; and I investigate what transitions, or limits, may arise from such a case, &
whether such are admissible in Nature.

In Art. 189, I pass on from the consideration of the curve to combinations of points. l%9
First, as far as Art. 204, I deal with a system of two points. I work out those things that
concern their mutual forces, and motions, whether they are left to themselves or pro-
jected in any manner whatever. Here also, having explained the connection between
these motions & the distances of the limits, & different cases of oscillations, whether they
are affected by external action of other points, or are not so disturbed, I make an antici-
patory note of the great use to which this will be put in the third part, for the purpose
of explaining various kinds of cohesion, fermentations, conflagrations, emissions of vapours,
the properties of light, elasticity and flexibility.

There follows, from Art. 204 to Art. 239, the much more fruitful consideration of a 204
system of three points. The forces connected with them can in general be easily deter-
mined for any given positions of the points ; but, when any position & velocity are given,
the motions have not yet been obtained by geometricians in such a form that the general
calculation can be performed for every possible case. So I proceed to consider the forces,
& the huge variation that different combinations of the points beget, although they are
only three in number, as far as Art. 209. From that, on to Art. 214, I consider certain 209
things that have to do with the forces that arise from the action, on each of the points, of
the other two together, & how these urge the third point not only to approach, or recede
from, themselves, but also in a direction at right angles ; in this connection there comes
forth an analogy with solidity, & a truly immense difference between the several cases when
the distances are very small, & the greatest conformity possible at very great distances
such as those at which gravity acts ; & I point out what great use will be made of this also
in explaining the constitution of Nature. Then up to Art. 221, I give ocular demonstra- 214
tions of the huge differences that there are in the laws of forces with which two points act
upon a third, whether it lies in the right line joining them, or in the right line that is the
perpendicular which bisects the interval between them ; this I do by constructing, from
the primary curve, curves representing the composite forces. Then in the two articles 221
that follow, I consider the case, a really important one, in which, by merely changing the
position of the two points, the third point, at any and the same definite interval situated
at the same distance from the middle point of the interval between the two points, will
be either continually attracted, or continually repelled, or neither attracted nor repelled ;
& since a difference of this kind should hold to a much greater degree in masses, I point
out, in Art. 222, the great use that will be made of this also in Physics. 222

At this point then, in Art. 223, I pass from the forces derived from two points to the 223
consideration of a whole system of them ; and, as far as Art. 228, I study three points
situated in a right line, from the mutual forces of which there arise certain relations, which
I return to later in much greater generality ; in this connection also are outlined, for three
points only, matters that have to do with rods, either rigid, flexible or elastic, and with
the lever, as well as many other things ; these, too, are treated much more generally later
on, when I consider masses. Then right on to Art. 238, I consider three points that do
not lie in a right line, whether they are in equilibrium, or moving in the perimeters of
certain ellipses or other curves. Here we come across a marvellous analogy between certain
limits and the limits which two points lying on the axis of the primary curve have with
respect to each other ; & here also a much greater variety of cases for masses is shown,
& an example is given of the application to solidity, & liquefaction, on account of a quick
internal motion being impressed on the points of the body. Moreover, in the two articles
that then follow, I state some general propositions with regard to a system of four points,
together with their application to solid rods, both rigid and flexible ; I also give an illus-
tration of various classes of particles by means of pyramids, each of which is formed of four
points in the most simple case, & of four of such pyramids in the more complicated cases.

From Art. 240 as far as Art. 264, I pass on to masses & consider matters pertaining to 24°
the centre of gravity ; & I prove that in general there is one, & only one, in any given mass.
I show how it can in general be determined, & I set forth in clear terms the point that is
lacking in the usual method, when it comes to a question of rigorous proof ; this deficiency

26 SYNOPSIS TOTIUS OPERIS

exemplum profero quoddam ejusdem generis, quod ad numerorum pertinet multiplica-
tionem, & ad virium compositionem per parallelogramma, quam alia methodo generaliore
exhibeo analoga illi ipsi, qua generaliter in centrum gravitatis inquire : turn vero ejusdem
ope demonstro admodum expedite, & accuratissime celebre illud Newtoni theorema de
statu centri gravitatis per mutuas internas vires numquam turbato.

264 Ejus tractionis fructus colligo plures : conservationem ejusdem quantitatis motuum in

265 Mundo in eandem plagam num. 264, sequalitatem actionis, & reactionis in massis num. 265,

266 collisionem corporum, & communicationem motus in congressibus directis cum eorum
276 legibus, inde num. 276 congressus obliques, quorum Theoriam a resolutione motuum reduce

277, 278 ad compositionem num. 277, quod sequent! numero 278 transfero ad incursum etiam in
270 planum immobile ; ac a num. 279 ad 289 ostendo nullam haberi in Natura veram virium,
aut motuum resolutionem, sed imaginariam tantummodo, ubi omnia evolvo, & explico
casuum genera, quae prima fronte virium resolutionem requirere videntur.

289 A num. 289 ad 297 leges expono compositionis virium, & resolutionis, ubi & illud

notissimum, quo pacto in compositione decrescat vis, in resolutione crescat, sed in ilia priore
conspirantium summa semper maneat, contrariis elisis ; in hac posteriore concipiantur
tantummodo binae vires contrarise adjectas, quse consideratio nihil turbet phenomena ;
unde fiat, ut nihil inde pro virium vivarum Theoria deduci possit, cum sine iis explicentur
omnia, ubi plura itidem explico ex iis phsenomenis, quse pro ipsis viribus vivis afferri solent.

2Q7 A num. 297 occasione inde arrepta aggredior qusedam, quae ad legem continuitatis

pertinent, ubique in motibus sancte servatam, ac ostendo illud, idcirco in collisionibus
corporum, ac in motu reflexo, leges vulgo definitas, non nisi proxime tantummodo observari,
& usque ad num. 307 relationes varias persequor angulorum incidentisa, & reflexionis, sive
vires constanter in accessu attrahant, vel repellant constanter, sive jam attrahant, jam
repellant : ubi & illud considero, quid accidat, si scabrities superficiei agentis exigua sit,
quid, si ingens, ac elementa profero, quae ad luminis reflexionem, & refractionem explican-
dam, definiendamque ex Mechanica requiritur, relationem itidem vis absolutae ad relativam
in obliquo gravium descensu, & nonnulla, quae ad oscillationum accuratiorem Theoriam
necessaria sunt, prorsus elementaria, diligenter expono.

307 A num. 307 inquire in trium massarum systema, ubi usque ad num. 313 theoremata

evolvo plura, quae pertinent ad directionem virium in singulis compositarum e binis
reliquarum actionibus, ut illud, eas directiones vel esse inter se parallelas, vel, si utrinque

313 indefinite producantur, per quoddam commune punctum transire omnes : turn usque ad
321 theoremata alia plura, quae pertinent ad earumdem compositarum virium rationem ad
se invicem, ut illud & simplex, & elegans, binarum massarum vires acceleratrices esse semper
in ratione composita ex tribus reciprocis rationibus, distantise ipsarum a massa tertia, sinus
anguli, quern singularum directio continet cum sua ejusmodi distantia, & massae ipsius earn
habentis compositam vim, ad distantiam, sinum, massam alteram ; vires autem motrices
habere tantummodo priores rationes duas elisa tertia.

321 Eorum theorematum fructum colligo deducens inde usque ad num. 328, quae ad

aequilibrium pertinent divergentium utcumque virium, & ipsius aequilibrii centrum, ac
nisum centri in fulcrum, & quae ad prseponderantiam, Theoriam extendens ad casum etiam,
quo massae non in se invicem agant mutuo immediate, sed per intermedias alias, quse nexum
concilient, & virgarum nectentium suppleant vices, ac ad massas etiam quotcunque, quarum
singulas cum centro conversionis, & alia quavis assumpta massa connexas concipio, unde
principium momenti deduce pro machinis omnibus : turn omnium vectium genera evolvo,
ut & illud, facta suspensione per centrum gravitatis haberi aequilibrium, sed in ipso centro
debere sentiri vim a fulcro, vel sustinente puncto, sequalem summae ponderum totius
systematis, unde demum pateat ejus ratio, quod passim sine demonstratione assumitur,
nimirum systemate quiescente, & impedito omni partium motu per aequilibrium, totam
massam concipi posse ut in centro gravitatis collectam.

SYNOPSIS OF THE WHOLE WORK 27

I supply, & I bring forward a certain example of the same sort, that deals with the multi-
plication of numbers, & to the composition of forces by the parallelogram law ; the latter
I prove by another more general method, analogous to that which I use in the general
investigation for the centre of gravity. Then by its help I prove very expeditiously &
with extreme rigour that well-known theorem of Newton, in which he affirmed that the
state of the centre of gravity is in no way altered by the internal mutual forces.

I gather several good results from this method of treatment. In Art. 264, the con- 264
servation of the same quantity of motion in the Universe in one plane ; in Art. 265 the 265
equality of action and reaction amongst masses ; then the collision of solid bodies, and the 266
communication of motions in direct impacts & the laws that govern them, & from that, 276
in Art. 276, oblique impacts ; in Art. 277 I reduce the theory of these from resolution of 277
motions to compositions, & in the article that follows, Art. 278, I pass to impact on to a 278
fixed plane; from Art. 279 to Art. 289 I show that there can be no real resolution of forces 279
or of motions in Nature, but only a hypothetical one ; & in this connection I consider &
explain all sorts of cases, in which at first sight it would seem that there must be resolution.

From Art. 289 to Art. 297, 1 state the laws for the composition & resolution of forces ; 289
here also I give the explanation of that well-known fact, that force decreases in composition,
increases in resolution, but always remains equal to the sum of the parts acting in the same
direction as itself in the first, the rest being equal & opposite cancel one another ; whilst
in the second, all that is done is to suppose that two equal & opposite forces are added on,
which supposition has no effect on the phenomena. Thus it comes about that nothing
can be deduced from this in favour of the Theory of living forces, since everything can be
explained without them ; in the same connection, I explain also many of the phenomena,
which are usually brought forward as evidence in favour of these ' living forces.'

In Art. 297, I seize the opportunity offered by the results just mentioned to attack 207
certain matters that relate to the law of continuity, which in all cases of motion is strictly
observed ; & I show that, in the collision of solid bodies, & in reflected motion, the laws,
as usually stated, are therefore only approximately followed. From this, as far as Art. 307,
I make out the various relations between the angles of incidence & reflection, whether the
forces, as the bodies approach one another, continually attract, or continually repel, or
attract at one time & repel at another. I also consider what will happen if the roughness
of the acting surface is very slight, & what if it is very great. I also state the first principles,
derived from mechanics, that are required for the explanation & determination of the
reflection & refraction of light ; also the relation of the absolute to the relative force in
the oblique descent of heavy bodies ; & some theorems that are requisite for the more
accurate theory of oscillations ; these, though quite elementary, I explain with great care.

From Art. 307 onwards, I investigate the system of three bodies ; in this connection,
as far as Art. 313, I evolve several theorems dealing with the direction of the forces on each
one of the three compounded from the combined actions of the other two ; such as the
theorem, that these directions are either all parallel to one another, or all pass through
some one common point, when they are produced indefinitely on both sides. Then, as ^j,
far as Art. 321, I make out several other theorems dealing with the ratios of these same
resultant forces to one another ; such as the following very simple & elegant theorem, that
the accelerating forces of two of the masses will always be in a ratio compounded of three
reciprocal ratios ; namely, that of the distance of either one of them from the third mass,
that of the sine of the angle which the direction of each force makes with the corresponding
distance of this kind, & that of the mass itself on which the force is acting, to the corre-
sponding distance, sine and mass for the other : also that the motive forces only have the
first two ratios, that of the masses being omitted.

I then collect the results to be derived from these theorems, deriving from them, as far ,2I
as Art. 328, theorems relating to the equilibrium of forces diverging in any manner, & the
centre of equilibrium, & the pressure of the centre on a fulcrum. I extend the theorem
relating to preponderance to the case also, in which the masses do not mutually act upon
one another in a direct manner, but through others intermediate between them, which
connect them together, & supply the place of rods joining them ; and also to any number of
masses, each of which I suppose to be connected with the centre of rotation & some other
assumed mass, & from this I derive the principles of moments for all machines. Then I
consider all the different kinds of levers ; one of the theorems that I obtain is, that, if a
lever is suspended from the centre of gravity, then there is equilibrium ; but a force should
be felt in this centre from the fulcrum or sustaining point, equal to the sum of the weights
of the whole system ; from which there follows most clearly the reason, which is every-
where assumed without proof, why the whole mass can be supposed to be collected at its
centre of gravity, so long as the system is in a state of rest & all motions of its parts are pro-
hibited by equilibrium.

28 SYNOPSIS TOTIUS OPERIS

328 A num. 328 ad 347 deduce ex iisdem theorematis, quae pertinent ad centrum oscilla-

tionis quotcunque massarum, sive sint in eadem recta, sive in piano perpendiculari ad axem
rotationis ubicunque, quse Theoria per systema quatuor massarum, excolendum aliquanto
diligentius, uberius promoveri deberet & extendi ad generalem habendum solidorum nexum,

344 qua re indicata, centrum itidem percussionis inde evolve, & ejus analogiam cum centre
oscillationis exhibeo.

347 Collecto ejusmodi fructu ex theorematis pertinentibus ad massas tres, innuo num. 347,
quae mihi communia sint cum ceteris omnibus, & cum Newtonianis potissimum, pertinentia
ad summas virium, quas habet punctum, vel massa attracta, vel repulsa a punctis singulis

348 alterius massae ; turn a num. 348 ad finem hujus partis, sive ad num. 358, expono quasdam,
quae pertinent ad fluidorum Theoriam, & primo quidem ad pressionem, ubi illud innuo
demonstratum a Newtono, si compressio fluidi sit proportionalis vi comprimenti, vires
repulsivas punctorum esse in ratione reciproca distantiarum, ac vice versa : ostendo autem
illud, si eadem vis sit insensibilis, rem, praeter alias curvas, exponi posse per Logisticam,
& in fluidis gravitate nostra terrestri prseditis pressiones haberi debere ut altitudines ;
deinde vero attingo ilia etiam, quae pertinent ad velocitatem fluidi erumpentis e vase, &
expono, quid requiratur, ut ea sit sequalis velocitati, quae acquiretur cadendo per altitudinem
ipsam, quemadmodum videtur res obtingere in aquae efHuxu : quibus partim expositis,
partim indicatis, hanc secundam partem conclude.

PARS III

358 Num. 358 propono argumentum hujus tertise partis, in qua omnes e Theoria mea

360 generales materis proprietates deduce, & particulares plerasque : turn usque ad num. 371
ago aliquanto fusius de impenetrabilitate, quam duplicis generis agnosco in meis punctorum
inextensorum massis, ubi etiam de ea apparenti quadam compenetratione ago, ac de luminis
trarlsitu per substantias intimas sine vera compenetratione, & mira quaedam phenomena

371 hue pertinentia explico admodum expedite. Inde ad num. 375 de extensione ago, quae
mihi quidem in materia, & corporibus non est continua, sed adhuc eadem praebet phaeno-
menae sensibus, ac in communi sententia ; ubi etiam de Geometria ago, quae vim suam in

375 mea Theoria retinet omnem : turn ad num. 383 figurabilitatem perseqUor, ac molem,
massam, densitatem singillatim, in quibus omnibus sunt quaedam Theoriae meae propria

383 scitu non indigna. De Mobilitate, & Motuum Continuitate, usque ad num. 388 notatu

388 digna continentur : turn usque ad num. 391 ago de aequalitate actionis, & reactionis, cujus
consectaria vires ipsas, quibus Theoria mea innititur, mirum in modum conformant.
Succedit usque ad num. 398 divisibilitas, quam ego ita admitto, ut quaevis massa existens
numerum punctorum realium habeat finitum tantummodo, sed qui in data quavis mole
possit esse utcunque magnus ; quamobrem divisibilitati in infinitum vulgo admissae sub-
stituo componibilitatem in infinitum, ipsi, quod ad Naturae phenomena explicanda

398 pertinet, prorsus aequivalentem. His evolutis addo num. 398 immutabilitatem primorum
materiae elementorum, quse cum mihi sint simplicia prorsus, & inextensa, sunt utique
immutabilia, & ad exhibendam perennem phasnomenorum seriem aptissima.

399 A num. 399 ad 406 gravitatem deduco ex mea virium Theoria, tanquam ramum

quendam e communi trunco, ubi & illud expono, qui fieri possit, ut fixae in unicam massam

406 non coalescant, quod gravitas generalis requirere videretur. Inde ad num. 419 ago de
cohaesione, qui est itidem veluti alter quidam ramus, quam ostendo, nee in quiete con-
sistere, nee in motu conspirante, nee in pressione fluidi cujuspiam, nee in attractione
maxima in contactu, sed in limitibus inter repulsionem, & attractionem ; ubi & problema
generale propono quoddam hue pertinens, & illud explico, cur massa fracta non iterum
coalescat, cur fibrae ante fractionem distendantur, vel contrahantur, & innuo, quae ad
cohaesionem pertinentia mihi cum reliquis Philosophis communia sint.

419 A cohacsione gradum facio num. 419 ad particulas, quae ex punctis cohaerentibus

efformantur, de quibus .ago usque ad num. 426. & varia persequor earum discrimina :

SYNOPSIS OF THE WHOLE WORK 29

From Art. 328 to Art. 347, I deduce from these same theorems, others that relate to 328
the centre of oscillation of any number of masses, whether they are in the same right line,
or anywhere in a plane perpendicular to the axis of rotation ; this theory wants to be worked
somewhat more carefully with a system of four bodies, to be gone into more fully, & to
be extended so as to include the general case of a system of solid bodies ; having stated
this, I evolve from it the centre of percussion, & I show the analogy between it & the centre 344
of oscillation.

I obtain all such results from theorems relating to three masses. After that, in Art. 347
347, I intimate the matters in which I agree with all others, & especially with the followers
of Newton, concerning sums of forces, acting on a point, or an attracted or repelled mass,
due to the separate points of another mass. Then, from Art. 348 to the end of this part, 348
i.e., as far as Art. 359, I expound certain theorems that belong to the theory of fluids ; &
first of all, theorems with regard to pressure, in connection with which I mention that one
which was proved by Newton, namely, that, if the compression of a fluid is proportional to
the compressing force, then the repulsive forces between the points are in the reciprocal
ratio of the distances, & conversely. Moreover, I show that, if the same force is insen-
sible, then the matter can be represented by the logistic & other curves ; also that in fluids
subject to our terrestrial gravity pressures should be found proportional to the depths.
After that, I touch upon those things that relate to the velocity of a fluid issuing from a
vessel ; & I show what is necessary in order that this should be equal to the velocity which
would be acquired by falling through the depth itself, just as it is seen to happen in the
case of an efflux of water. These things in some part being explained, & in some part
merely indicated, I bring this second part to an end.

PART III

In Art. 358, I state the theme of this third part ; in it I derive all the general & most 358
of the special, properties of matter from my Theory. Then, as far as Art. 371, I deal some- 360
what more at length with the subject of impenetrability, which I remark is of a twofold
kind in my masses of non-extended points ; in this connection also, I deal with a certain
apparent case of compenetrability, & the passage of light through the innermost parts of
bodies without real compenetration ; I also explain in a very summary manner several
striking phenomena relating to the above. From here on to Art. 375, I deal with exten- 371
sion ; this in my opinion is not continuous either in matter or in solid bodies, & yet it
yields the same phenomena to the senses as does the usually accepted idea of it ; here I
also deal with geometry, which conserves all its power under my Theory. Then, as far 375
as Art. 383, I discuss figurability, volume, mass & density, each in turn ; in all of these
subjects there are certain special points of my Theory that are not unworthy of investi-
gation. Important theorems on mobility & continuity of motions are to be found from
here on to Art. 388 ; then, as far as Art. 391, I deal with the equality of action & reaction,
& my conclusions with regard to the subject corroborate in a wonderful way the hypothesis
of those forces, upon which my Theory depends. Then follows divisibility, as far as Art. 39 1
398 ; this principle I admit only to the extent that any existing mass may be made up of
a number of real points that are finite only, although in any given mass this finite number
may be as great as you please. Hence for infinite divisibility, as commonly accepted, I
substitute infinite multiplicity ; which comes to exactly the same thing, as far as it is
concerned with the explanation of the phenomena of Nature. Having considered these
subjects I add, in Art. 398, that of the immutability of the primary elements of matter ; 398
according to my idea, these are quite simple in composition, of no extent, they are every-
where unchangeable, & hence are splendidly adapted for explaining a continually recurring
set of phenomena.

From Art. 399 to Art. 406, 1 derive gravity from my Theory of forces, as if it were a 399
particular branch on a common trunk ; in this connection also I explain how it can happen
that the fixed stars do not all coalesce into one mass, as would seem to be required under 406
universal gravitation. Then, as far as Art. 419, I deal with cohesion, which is also as it
were another branch ; I show that this is not dependent upon quiescence, nor on motion
that is the same for all parts, nor on the pressure of some fluid, nor on the idea that the
attraction is greatest at actual contact, but on the limits between repulsion and attraction.
I propose, & solve, a general problem relating to this, namely, why masses, once broken,
do not again stick together, why the fibres are stretched or contracted before fracture
takes place ; & I intimate which of my ideas relative to cohesion are the same as those
held by other philosophers.

In Art. 419, 1 pass on from cohesion to particles which are formed from a number of 4J9
cohering points ; & I consider these as far as Art. 426, & investigate the various distinctions

30 SYNOPSIS TOTIUS OPERIS

ostendo nimirum, quo pacto varias induere possint figuras quascunque, quarum tenacissime
sint ; possint autem data quavis figura discrepare plurimum in numero, & distributione
punctorum, unde & oriantur admodum inter se diversae vires unius particulae in aliam, ac
itidem diversae in diversis partibus ejusdem particulae respectu diversarum partium, vel
etiam respectu ejusdem partis particulse alterius, cum a solo numero, & distributione
punctorum pendeat illud, ut data particula datam aliam in datis earum distantiis, &
superficierum locis, vel attrahat, vel repellat, vel respectu ipsius sit prorsus iners : turn illud
addo, particulas eo dimcilius dissolubiles esse, quo minores sint ; debere autem in gravitate
esse penitus uniformes, quaecunque punctorum dispositio habeatur, & in aliis proprietatibus
plerisque debere esse admodum (uti observamus) diversas, quae diversitas multo major in
majoribus massis esse debeat.

426 A num. 426 ad 446 de solidis, & fluidis, quod discrimen itidem pertinet ad varia

cohaesionum genera ; & discrimen inter solida, & fluida diligenter expono, horum naturam
potissimum repetens ex motu faciliori particularum in gyrum circa alias, atque id ipsum ex
viribus circumquaque aequalibus ; illorum vero ex inaequalitate virium, & viribus quibusdam
in latus, quibus certam positionem ad se invicem servare debeant. Varia autem distinguo
fluidorum genera, & discrimen profero inter virgas rigidas, flexiles, elasticas, fragiles, ut &
de viscositate, & humiditate ago, ac de organicis, & ad certas figuras determinatis corporibus,
quorum efformatio nullam habet difficultatem, ubi una particula unam aliam possit in
certis tantummodo superficiei partibus attrahere, & proinde cogere ad certam quandam
positionem acquirendam respectu ipsius, & retinendam. Demonstro autem & illud, posse
admodum facile ex certis particularum figuris, quarum ipsae tenacissimae sint, totum etiam
Atomistarum, & Corpuscularium systema a mea Theoria repeti ita, ut id nihil sit aliud,
nisi unicus itidem hujus veluti trunci foecundissimi ramus e diversa cohaesionis ratione
prorumpens. Demum ostendo, cur non quaevis massa, utut constans ex homogeneis
punctis, & circa se maxime in gyrum mobilibus, fluida sit ; & fluidorum resistentiam quoque
attingo, in ejus leges inquirens.

446 A num. 446 ad 450 ago de iis, quae itidem ad diversa pertinent soliditatis genera, nimirum

de elasticis, & mollibus, ilia repetens a magna inter limites proximos distantia, qua fiat, ut
puncta longe dimota a locis suis, idem ubique genus virium sentiant, & proinde se ad
priorem restituant locum ; hasc a limitum frequentia, atque ingenti vicinia, qua fiat, ut ex
uno ad alium delata limitem puncta, ibi quiescant itidem respective, ut prius. Turn vero
de ductilibus, & malleabilibus ago, ostendens, in quo a fragilibus discrepent : ostendo autem,
haec omnia discrimina a densitate nullo modo pendere, ut nimirum corpus, quod
multo sit altero densius, possit tarn multo majorem, quam multo minorem soliditatem, &
cohaesionem habere, & quaevis ex proprietatibus expositis aeque possit cum quavis vel majore,
vel minore densitate componi.

450 Num. 450 inquire in vulgaria quatuor elementa ; turn a num. 451 ad num. 467 persequor

452 chemicas operationes ; num. 452 explicans dissolutionem, 453 praecipitationem, 454, & 455
commixtionem plurium substantiarum in unam : turn num. 456, & 457 liquationem binis
methodis, 458 volatilizationem, & effervescentiam, 461 emissionem efHuviorum, quae e massa
constanti debeat esse ad sensum constans, 462 ebullitionem cum variis evaporationum
generibus ; 463 deflagrationem, & generationem aeris ; 464 crystallizationem cum certis
figuris ; ac demum ostendo illud num. 465, quo pacto possit fermentatio desinere ; & num.
466, quo pacto non omnia fermentescant cum omnibus.

467 A fermentatione num. 467 gradum facio ad ignem, qui mihi est fermentatio quaedam

substantiae lucis cum sulphurea quadam substantia, ac plura inde consectaria deduce usque

471 ad num. 471 ; turn ab igne ad lumen ibidem transeo, cujus proprietates praecipuas, ex

472 quibus omnia lucis phaenomena oriuntur, propono num. 472, ac singulas a Theoria mea
deduce, & fuse explico usque ad num. 503, nimirum emissionem num. 473, celeritatem 474,
propagationem rectilineam per media homogenea, & apparentem tantummodo compene-
trationem a num. 475 ad 483, pellucidatem, & opacitatem num. 483, reflexionem ad angulos
aequales inde ad 484, refractionem ad 487, tenuitatem num. 487, calorem, & ingentes
intestines motus allapsu tenuissimae lucis genitos, num. 488, actionem majorem corporum
eleosorum, & sulphurosorum in lumen num. 489 : turn num. 490 ostendo, nullam resist-

SYNOPSIS OF THE WHOLE WORK 31

between them. I show how it is possible for various shapes of all sorts to be assumed,
which offer great resistance to rupture ; & how in a given shape they may differ very greatly
in the number & disposition of the points forming them. Also that from this fact there
arise very different forces for the action of one particle upon another, & also for the action
of different parts of this particle upon other different parts of it, or on the same part of
another particle. For that depends solely on the number & distribution of the points,
so that one given particle either attracts, or repels, or is perfectly inert with regard to
another given particle, the distances between them and the positions of their surfaces being
also given. Then I state in addition that the smaller the particles, the greater is the diffi-
culty in dissociating them ; moreover, that they ought to be quite uniform as regards
gravitation, no matter what the disposition of the points may be ; but in most other
properties they should be quite different from one another (which we observe to be the
case) ; & that this difference ought to be much greater in larger masses.

From Art. 426 to Art. 446, 1 consider solids & fluids, the difference between which is 426
also a matter of different kinds of cohesion. I explain with great care the difference
between solids & fluids ; deriving the nature of the latter from the greater freedom of motion
of the particles in the matter of rotation about one another, this being due to the forces
being nearly equal ; & that of the former from the inequality of the forces, and from certain
lateral forces which help them to keep a definite position with regard to one another. I
distinguish between various kinds of fluids also, & I cite the distinction between rigid,
flexible, elastic & fragile rods, when I deal with viscosity & humidity ; & also in dealing with
organic bodies & those solids bounded by certain fixed figures, of which the formation
presents no difficulty ; in these one particle can only attract another particle in certain
parts of the surface, & thus urge it to take up some definite position with regard to itself,
& keep it there. I also show that the whole system of the Atomists, & also of the Corpus-
cularians, can be quite easily derived by my Theory, from the idea of particles of definite
shape, offering a high resistance to deformation ; so that it comes to nothing else than
another single branch of this so to speak most fertile trunk, breaking forth from it
on account of a different manner of cohesion. Lastly, I show the reason why it is that
not every mass, in spite of its being constantly made up of homogeneous points, & even
these in a high degree capable of rotary motion about one another, is a fluid. I also touch
upon the resistance of fluids, & investigate the laws that govern it.

From Art. 446 to Art. 450, I deal with those things that relate to the different kinds 446
of solidity, that is to say, with elastic bodies, & those that are soft. I attribute the nature
of the former to the existence of a large interval between the consecutive limits, on account
of which it comes about that points that are far removed from their natural positions still
feel the effects of the same kind of forces, & therefore return to their natural positions ;
& that of the latter to the frequency & great closeness of the limits, on account of which it
comes about that points that have been moved from one limit to another, remain there
in relative rest as they were to start with. Then I deal with ductile and malleable solids,
pointing out how they differ from fragile solids. Moreover I show that all these differ-
ences are in no way dependent on density ; so that, for instance, a body that is much more
dense than another body may have either a much greater or a much less solidity and
cohesion than another ; in fact, any of the properties set forth may just as well be combined
with any density either greater or less.

In Art. 450 I consider what are commonly called the " four elements " ; then from 450
Art. 451 to Art. 467, I treat of chemical operations ; I explain solution in Art. 452, preci- 452
pitation in Art. 453, the mixture of several substances to form a single mass in Art. 454,
455, liquefaction by two methods in Art. 456, 457, volatilization & effervescence in Art.
458, emission of effluvia (which from a constant mass ought to be approximately constant)
in. Art. 461, ebullition & various kinds of evaporation in Art. 462, deflagration & generation
of gas in Art. 463, crystallization with definite forms of crystals in Art. 464 ; & lastly, I show,
in Art. 465, how it is possible for fermentation to cease, & in Art. 466, how it is that any
one thing does not ferment when mixed with any other thing.

From fermentation I pass on, in Art. 467, to fire, which I look upon as a fermentation 467
of some substance in light with some sulphureal substance ; & from this I deduce several
propositions, up to Art. 471. There I pass on from fire to light, the chief properties of 471
which, from which all the phenomena of light arise, I set forth in Art. 472 ; & I deduce 472
& fully explain each of them in turn as far as Art. 503. Thus, emission in Art. 473, velo-
city in Art. 474, rectilinear propagation in homogeneous media, & a compenetration that
is merely apparent, from Art. 475 on to Art. 483, pellucidity & opacity in Art. 483, reflec-
tion at equal angles to Art. 484, & refraction to Art. 487, tenuity in Art. 487, heat & the
great internal motions arising from the smooth passage of the extremely tenuous light in
Art. 488, the greater action of oleose & sulphurous bodies on light in Art. 489. Then I

32 SYNOPSIS TOTIUS OPERIS

entiam veram pati, ac num. 491 explico, unde sint phosphora, num. 492 cur lumen cum
majo e obliquitate incidens reflectatur magis, num. 493 & 494 unde diversa refrangibilitas
ortum ducat, ac num. 495, & 496 deduce duas diversas dispositiones ad asqualia redeuntes
intervalla, unde num. 497 vices illas a Newtono detectas facilioris reflexionis, & facilioris
transmissus eruo, & num. 498 illud, radios alios debere reflecti, alios transmitti in appulsu
ad novum medium, & eo plures reflecti, quo obliquitas incidentise sit major, ac num.
499 & 500 expono, unde discrimen in intervallis vicium, ex quo uno omnis naturalium
colorum pendet Newtoniana Theoria. Demum num. 501 miram attingo crystalli
Islandicse proprietatem, & ejusdem causam, ac num. 502 diffractionem expono, quse est
quaedam inchoata refractio, sive reflexio.

503 Post lucem ex igne derivatam, quse ad oculos pertinet, ago brevissime num. 503 de

504 sapore, & odore, ac sequentibus tribus numeris de sono : turn aliis quator de tactu, ubi
507 etiam de frigore, & calore : deinde vero usque ad num. 514 de electricitate, ubi totam
511 Franklinianam Theoriam ex meis principiis explico, eandem ad bina tantummodo reducens

principia, quse ex mea generali virium Theoria eodem fere pacto deducuntur, quo prsecipi-
514 tationes, atque dissolutiones. Demum num. 514, ac 515 magnetismum persequor, tam
directionem explicans, quam attractionem magneticam.

516 Hisce expositis, quas ad particulares .etiam proprietates pertinent, iterum a num. 516

ad finem usque generalem corporum complector naturam, & quid materia sit, quid forma,
quse censeri debeant essentialia, quse accidentialia attributa, adeoque quid transformatio
sit, quid alteratio, singillatim persequor, & partem hanc tertiam Theorise mesa absolve.

De Appendice ad Metaphysicam pertinente innuam hie illud tantummodo, me ibi
exponere de anima illud inprimis, quantum spiritus a materia differat, quern nexum anima
habeat cum corpore, & quomodo in ipsum agat : turn de DEO, ipsius & existentiam me
pluribus evincere, quae nexum habeant cum ipsa Theoria mea, & Sapientiam inprimis, ac
Providentiam, ex qua gradum ad revelationem faciendum innuo tantummodo. Sed hsec
in antecessum veluti delibasse sit satis.

SYNOPSIS OF THE WHOLE WORK 33

show, in Art. 490, that it suffers no real resistance, & in Art. 491 I explain the origin of
bodies emitting light, in Art. 492 the reason why light that falls with greater obliquity
is reflected more strongly, in Art. 493, 494 the origin of different degrees of refrangibility,
& in Art. 495, 496 I deduce that there are two different dispositions recurring at equal
intervals ; hence, in Art. 497, I bring out those alternations, discovered by Newton, of
easier reflection & easier transmission, & in Art. 498 I deduce that some rays should be
reflected & others transmitted in the passage to a fresh medium, & that the greater the obli-
quity of incidence, the greater the number of reflected rays. In Art. 499, 500 I state the
origin of the difference between the lengths of the intervals of the alternations ; upon this
alone depends the whole of the Newtonian theory of natural colours. Finally, in Art. 501,
I touch upon the wonderful property of Iceland spar & its cause, & in Art. 502 I explain
diffraction, which is a kind of imperfect refraction or reflection.

After light derived from fire, which has to do with vision, I very briefly deal with
taste & smell in Art. 503, £ of sound in the three articles that follow next. Then, in the S°3
next four articles, I consider touch, & in connection with it, cold & heat also. After that, 5°4
as far as Art. 514, I deal with electricity ; here I explain the whole of the Franklin theory 5°7
by means of my principles ; I reduce this theory to two principles only, & these are 5 1 1
derived from my general Theory of forces in almost the same manner as I have already derived
precipitations & solutions. Finally, in Art. 514, 515, I investigate magnetism, explaining 5H
both magnetic direction £ attraction.

These things being expounded, all of which relate to special properties, I once more
consider, in the articles from 516 to the end, the general nature of bodies, what matter is, 516
its form, what things ought to be considered as essential, & what as accidental, attributes ;
and also the nature of transformation and alteration are investigated, each in turn ; &
thus I bring to a close the third part of my Theory.

I will mention here but this one thing with regard to the appendix on Metaphysics ;
namely, that I there expound more especially how greatly different is the soul from matter,
the connection between the soul & the body, & the manner of its action upon it. Then
with regard to GOD, I prove that He must exist by many arguments that have a close con-
nection with this Theory of mine ; I especially mention, though but slightly, His Wisdom
and Providence, from which there is but a step to be made towards revelation. But I think
that I have, so to speak, given my preliminary foretaste quite sufficiently.

[I] PHILOSOPHIC NATURALIS THEORIA

In quo conveniat
cum systemate
Newtoniano, &
Leibnitiano.

Cujusmodi systema>
Theoria exhibeat.

PARS I

Theorice expositio, analytica deductio^ & vindicatio.

lRIUM mutuarum Theoria, in quam incidi jam ab Anno 1745, dum e
notissimis principiis alia ex aliis consectaria eruerem, & ex qua ipsam
simplicium materise elementorum constitutionem deduxi, systema
exhibet medium inter Leibnitianum, & Newtonianum, quod nimirum
& ex utroque habet plurimum, & ab utroque plurimum dissidet ; at
utroque in immensum simplicius, proprietatibus corporum generalibus
sane omnibus, & [2] peculiaribus quibusque praecipuis per accuratissimas
demonstrationes deducendis est profecto mirum in modum idoneum.

2. Habet id quidem ex Leibnitii Theoria elementa prima simplicia, ac prorsus inex-
tensa : habet ex Newtoniano systemate vires mutuas, quae pro aliis punctorum distantiis a
se invicem aliae sint ; & quidem ex ipso itidem Newtono non ejusmodi vires tantummodo,
quse ipsa puncta determinent ad accessum, quas vulgo attractiones nominant ; sed etiam
ejusmodi, quae determinent ad recessum, & appellantur repulsiones : atque id ipsum ita,
ut, ubi attractio desinat, ibi, mutata distantia, incipiat repulsio, & vice versa, quod nimirum
Newtonus idem in postrema Opticse Quaestione proposuit, ac exemplo transitus a positivis
ad negativa, qui habetur in algebraicis formulis, illustravit. Illud autem utrique systemati
commune est cum hoc meo, quod quaevis particula materiae cum aliis quibusvis, utcunque
remotis, ita connectitur, ut ad mutationem utcunque exiguam in positione unius cujusvis,
determinationes ad motum in omnibus reliquis immutentur, & nisi forte elidantur omnes
oppositas, qui casus est infinities improbabilis, motus in iis omnibus aliquis inde ortus
habeatur.

In quo differat a
Leibnitiano & ipsi
praestet.

3. Distat autem a Leibnitiana Theoria longissime, turn quia nullam extensionem
continuam admittit, quae ex contiguis, & se contingentibus inextensis oriatur : in quo
quidem dirficultas jam olim contra Zenonem proposita, & nunquam sane aut soluta satis,
aut solvenda, de compenetratione omnimoda inextensorum contiguorum, eandem vim
adhuc habet contra Leibnitianum systema : turn quia homogeneitatem admittit in elementis,
omni massarum discrimine a sola dispositione, & diversa combinatione derivato, ad quam
homogeneitatem in elementis, & discriminis rationem in massis, ipsa nos Naturae analogia
ducit, ac chemicae resolutiones inprimis, in quibus cum ad adeo pauciora numero, & adeo
minus inter se diversa principiorum genera, in compositorum corporum analysi deveniatur,
id ipsum indicio est, quo ulterius promoveri possit analysis, eo ad majorem simplicitatem,
& homogeneitatem devenire debere, adeoque in ultima demum resolutione ad homogenei-
tatem, & simplicitatem summam, contra quam quidem indiscernibilium principium, &
principium rationis sufficients usque adeo a Leibnitianis depraedicata, meo quidem judicio,
nihil omnino possunt.

in quo differat a A Distat itidem a Newtoniano systemate quamplunmum, turn in eo, quod ea, quae

Newtoniano & ipsi XT . . r\ • r\ • r

praestet. Newtonus in ipsa postremo (Juaestione (Jpticae conatus est expncare per tna pnncipia,

gravitatis, cohsesionis, fermentationis, immo & reliqua quamplurima, quae ab iis tribus
principiis omnino non pendent, per unicam explicat legem virium, expressam unica, & ex
pluribus inter se commixtis non composita algebraica formula, vel unica continua geometrica
curva : turn in eo, quod in mi-[3]-nimis distantiis vires admittat non positivas, sive
attractivas, uti Newtonus, sed negativas, sive repulsivas, quamvis itidem eo majores in

34

A THEORY OF NATURAL PHILOSOPHY

PART I

Exposition ^ ^Analytical Derivation & Proof of the Theory

I. ' ^i ^^ HE following Theory of mutual forces, which I lit upon as far back as the year The kind of sys-
1745, whilst I was studying various propositions arising from other very p^ents.6
well-known principles, & from which I have derived the very constitu-
tion of the simple elements of matter, presents a system that is midway
between that of Leibniz & that of Newton ; it has very much in common
with both, & differs very much from either ; &, as it is immensely more
simple than either, it is undoubtedly suitable in a marvellous degree for

deriving all the general properties of bodies, & certain of the special properties also, by

means of the most rigorous demonstrations.

2. It indeed holds to those simple & perfectly non-extended primary elements upon what there is in
which is founded the theory of Leibniz ; & also to the mutual forces, which vary as the * s£^"0"{ to$^
distances of the points from one another vary, the characteristic of the theory of Newton ; ton *& Leibniz.

in addition, it deals not only with the kind of forces, employed by Newton, which oblige
the points to approach one another, & are commonly called attractions ; but also it
considers forces of a kind that engender recession, & are called repulsions. Further, the
idea is introduced in such a manner that, where attraction ends, there, with a change of
distance, repulsion begins ; this idea, as a matter of fact, was suggested by Newton in the
last of his ' Questions on Optics ', & he illustrated it by the example of the passage from
positive to negative, as used in algebraical formulas. Moreover there is this common point
between either of the theories of Newton & Leibniz & my own ; namely, that any particle
of matter is connected with every other particle, no matter how great is the distance
between them, in such a way that, in accordance with a change in the position, no matter
how slight, of any one of them, the factors that determine the motions of all the rest are
altered ; &, unless it happens that they all cancel one another (& this is infinitely impro-
bable), some motion, due to the change of position in question, will take place in every one
of them.

3. But my Theory differs in a marked degree from that of Leibniz. For one thing, How it differs from,
because it does not admit the continuous extension that arises from the idea of consecutive,
non-extended points touching one another ; here, the difficulty raised in times gone by in

opposition to Zeno, & never really or satisfactorily answered (nor can it be answered), with
regard to compenetration of all kinds with non-extended consecutive points, still holds the
same force against the system of Leibniz. For another thing, it admits homogeneity
amongst the elements, all distinction between masses depending on relative position only,
& different combinations of the elements ; for this homogeneity amongst the elements, &
the reason for the difference amongst masses, Nature herself provides us with the analogy.
Chemical operations especially do so ; for, since the result of the analysis of compound
substances leads to classes of elementary substances that are so comparatively few in num-
ber, & still less different from one another in nature ; it strongly suggests that, the further
analysis can be pushed, the greater the simplicity, & homogeneity, that ought to be attained ;
thus, at length, we should have, as the result of a final decomposition, homogeneity &
simplicity of the highest degree. Against this homogeneity & simplicity, the principle of
indiscernibles, & the doctrine of sufficient reason, so long & strongly advocated by the
followers of Leibniz, can, in my opinion at least, avail in not the slightest degree.

4. My Theory also differs as widely as possible from that of Newton. For one thing, HOW it differs from,
because it explains by means of a single law of forces all those things that Newton himself, * surpasses, the

i i i i. . X • f-\ • , i • i theory of Newton.

in the last of his Questions on Uptics , endeavoured to explain by the three principles
of gravity, cohesion & fermentation ; nay, & very many other things as well, which do not
altogether follow from those three principles. Further, this law is expressed by a single
algebraical formula, & not by one composed of several formulae compounded together ; or
by a single continuous geometrical curve. For another thing, it admits forces that at very
small distances are not positive or attractive, as Newton supposed, but negative or repul-

35

missum.

36 PHILOSOPHIC NATURALIS THEORIA

infinitum, quo distantise in infinitum decrescant. Unde illud necessario consequitur, ut nee
cohaesio a contactu immediate oriatur, quam ego quidem longe aliunde desumo ; nee ullus
immediatus, &, ut ilium appellare soleo, mathematicus materiae contactus habeatur, quod
simplicitatem, & inextensionem inducit elementorum, quae ipse variarum figurarum voluit,
& partibus a se invicem distinctis composita, quamvis ita cohasrentia, ut nulla Naturae vi
dissolvi possit compages, & adhaesio labefactari, quas adhaesio ipsi, respectu virium nobis
cognitarum, est absolute infinita.

Ubi de ipsa ctum 5. Quae ad ejusmodi Theoriam pertinentia hucusque sunt edita, continentur disserta-

ante ; & quid pro- tionibus meis, De viribus vivis, edita Anno 1741;, De Lumine A. 1748, De Leee Continuitatis

ml«<mm " . T r^ ... . • . rj . ...

A. 1754, De Lege virium in natura existentium A. 1755, De divisibihtate materite, C5 principiis
corporum A. 1757, ac in meis Supplementis Stayanae Philosophiae versibus traditae, cujus primus
Tomus prodiit A. 1755 : eadem autem satis dilucide proposuit, & amplissimum ipsius per
omnem Physicam demonstravit usum vir e nostra Societate doctissimus Carolus Benvenutus
in sua Physics Generalis Synopsi edita Anno 1754. In ea Synopsi proposuit idem & meam
deductionem aequilibrii binarum massarum, viribus parallelis animatarum, quas ex ipsa mea
Theoria per notissimam legem compositionis virium, & aequalitatis inter actionem, & reac-
tionem, fere sponte consequitur, cujus quidem in supplementis illis § 4. ad lib. 3. mentionem
feci, ubi & quae in dissertatione De centra Gravitatis edideram, paucis proposui ; & de centre
oscillationis agens, protuli aliorum methodos praecipuas quasque, quae ipsius determinationem
a subsidiariis tantummodo principiis quibusdam repetunt. Ibidem autem de sequilibrii
centre agens illud affirmavi : In Natura nullce sunt rigidce virgce, infiexiles, & omni gravitate,
ac inertia carentes, adeoque nee revera ullce leges pro Us conditcz ; & si ad genuina, & simpli-
cissima natures principia, res exigatur, invenietur, omnia pendere a compositione virium, quibus in
se invicem agunt particula materice ; a quibus nimirum viribus omnia Natures pb&nomena
proficiscuntur. Ibidem autem exhibitis aliorum methodis ad centrum oscillationis perti-
nentibus, promisi, me in quarto ejusdem Philosophiae tomo ex genuinis principiis investiga-
turum, ut aequilibrii, sic itidem oscillationis centrum.

Qua occasione hoc 6. Porro cum nuper occasio se mihi praebuisset inquirendi in ipsum oscillationis centrum
turn 'opus." Cnp ex meis principiis, urgente Scherffero nostro viro doctissimo, qui in eodem hoc Academico
Societatis Collegio nostros Mathesim docet ; casu incidi in theorema simplicisimum sane, &
admodum elegans, quo trium massarum in se mutuo agentium comparantur vires, [4] quod
quidem ipsa fortasse tanta sua simplicitate effugit hucusque Mechanicorum oculos ; nisi
forte ne effugerit quidem, sed alicubi jam ab alio quopiam inventum, & editum, me, quod
admodum facile fieri potest, adhuc latuerit, ex quo theoremate & asquilibrium, ac omne
vectium genus, & momentorum mensura pro machinis, & oscillationis centrum etiam pro
casu, quo oscillatio fit in latus in piano ad axem oscillationis perpendiculari, & centrum
percussionis sponte fluunt, & quod ad sublimiores alias perquisitiones viam aperit admodum
patentem. Cogitaveram ego quidem initio brevi dissertatiuncula hoc theorema tantummodo
edere cum consectariis, ac breve Theoriae meae specimen quoddam exponere ; sed paullatim
excrevit opusculum, ut demum & Theoriam omnem exposuerim ordine suo, & vindicarim,
& ad Mechanicam prius, turn ad Physicam fere universam applicaverim, ubi & quae maxima
notatu digna erant, in memoratis dissertationibus ordine suo digessi omnia, & alia adjeci
quamplurima, quae vel olim animo conceperam, vel modo sese obtulerunt scribenti, & omnem
hanc rerum farraginem animo pervolventi.

eiementa in- 7. Prima elementa materiae mihi sunt puncta prorsus indivisibilia, & inextensa, quae in

imrftenso vacuo ita dispersa sunt, ut bina quaevis a se invicem distent per aliquod intervallum,
quod quidem indefinite augeri potest, & minui, sed penitus evanescere non potest, sine
conpenetratione ipsorum punctorum : eorum enim contiguitatem nullam admitto possi-
bilem ; sed illud arbitror omnino certum, si distantia duorum materiae punctorum sit nulla,
idem prorsus spatii vulgo concept! punctum indivisibile occupari ab utroque debere, &

A THEORY OF NATURAL PHILOSOPHY 37

sive ; although these also become greater & greater indefinitely, as the distances decrease
indefinitely. From this it follows of necessity that cohesion is not a consequence of imme-
diate contact, as I indeed deduce from totally different considerations ; nor is it possible
to get any immediate or, as I usually term it, mathematical contact between the parts of
matter. This idea naturally leads to simplicity & non-extension of the elements, such as
Newton himself postulated for various figures ; & to bodies composed of parts perfectly
distinct from one another, although bound together so closely that the ties could not be
broken or the adherence weakened by any force in Nature ; this adherence, as far as the
forces known to us are concerned, is in his opinion unlimited.

5. What has already been published relating to this kind of Theory is contained in my when & where I
dissertations, De Viribus vivis, issued in 1745, De Lumine, 1748, De Lege Continuitatis, *£££ th^theory'*
1754, De Lege virium in natura existentium, 1755, De divisibilitate materia, y principiis & a promise that i
corporum, 1757, & in my Supplements to the philosophy of Benedictus Stay, issued in verse, made>

of which the first volume was published in 1755. The same theory was set forth with
considerable lucidity, & its extremely wide utility in the matter of the whole of Physics
was demonstrated, by a learned member of our Society, Carolus Benvenutus, in his Physics
Generalis Synopsis published in 1754. In this synopsis he also at the same time gave my
deduction of the equilibrium of a pair of masses actuated by parallel forces, which follows
quite naturally from my Theory by the well-known law for the composition of forces, &
the equality between action & reaction ; this I mentioned in those Supplements, section
4 of book 3, & there also I set forth briefly what I had published in my dissertation De
centra Gravitatis. Further, dealing with the centre of oscillation, I stated the most note-
worthy methods of others who sought to derive the determination of this centre from
merely subsidiary principles. Here also, dealing with the centre of equilibrium, I asserted : —
" In Nature there are no rods that are rigid, inflexible, totally devoid of weight & inertia ;
y so, neither are there really any laws founded on them. If the matter is worked back to the
genuine W simplest natural principles, it will be found that everything depends on the com-
position of the forces with which the particles of matter act upon one another ; y from these
very forces, as a matter of fact, all phenomena of Nature take their origin." Moreover, here
too, having stated the methods of others for the determination of the centre of oscillation,
I promised that, in the fourth volume of the Philosophy, I would investigate by means of
genuine principles, such as I had used for the centre of equilibrium, the centre of
oscillation as well.

6. Now, lately I had occasion to investigate this centre of oscillation, deriving it from The occasion that
my own principles, at the request of Father Scherffer, a man of much learning, who teaches |^
mathematics in this College of the Society. Whilst doing this, I happened to hit upon a matter.

really most simple & truly elegant theorem, from which the forces with which three
masses mutually act upon one another are easily to be found ; this theorem, perchance
owing to its extreme simplicity, has escaped the notice of mechanicians up till now (unless
indeed perhaps it has not escaped notice, but has at some time previously been discovered
& published by some other person, though, as may very easily have happened, it may not
have come to my notice). From this theorem there come, as the natural consequences,
the equilibrium & all the different kinds of levers, the measurement of moments for
machines, the centre of oscillation for the case in which the oscillation takes place sideways
in a plane perpendicular to the axis of oscillation, & also the centre of percussion ; it opens
up also a beautifully clear road to other and more sublime investigations. Initially, my
idea was to publish in a short esssay merely this theorem & some deductions from it, & thus
to give some sort of brief specimen of my Theory. But little by little the essay grew in
length, until it ended in my setting forth in an orderly manner the whole of the theory,
giving a demonstration of its truth, & showing its application to Mechanics in the first place,
and then to almost the whole of Physics. To it I also added not only those matters that
seemed to me to be more especially worth mention, which had all been already set forth
in an orderly manner in the dissertations mentioned above, but also a large number of other
things, some of which had entered my mind previously, whilst others in some sort pb truded
themselves on my notice as I was writing & turning over in my mind all this conglomer-
ation of material.

7. The primary elements of matter are in my opinion perfectly indivisible & non- The primary eie-
extended points ; they are so scattered in an immense vacuum that every two of them are ^biVnon^xtended
separated from one another by a definite interval ; this interval can be indefinitely & they are not
increased or diminished, but can never vanish altogether without compenetration of the c

points themselves ; for I do not admit as possible any immediate contact between them.
On the contrary I consider that it is a certainty that, if the distance between two points
of matter should become absolutely nothing, then the very same indivisible point of space,
according to the usual idea of it, must be occupied by both together, & we have true

38 PHILOSOPHIC NATURALIS THEORIA

haberi veram, ac omnimodam conpenetrationem. Quamobrem non vacuum ego quidem
admitto disseminatum in materia, sed materiam in vacuo disseminatam, atque innatantem.

Eorum inertias vis g jn n;sce punctis admitto determinationem perseverandi in eodem statu quietis, vel

cujusmodi. . r . r. ,. , . . , . J . . • i * XT '

motus umiormis in directum l«) m quo semel sint posita, si seorsum smgula in JNatura
existant ; vel si alia alibi extant puncta, componendi per notam, & communem metho-
dum compositionis virium, & motuum, parallelogrammorum ope, praecedentem motum
cum mo-[5]-tu quern determinant vires mutuae, quas inter bina quaevis puncta agnosco
a distantiis pendentes, & iis mutatis mutatas, juxta generalem quandam omnibus com-
munem legem. In ea determinatione stat ilia, quam dicimus, inertiae vis, quae, an a
libera pendeat Supremi Conditoris lege, an ab ipsa punctorum natura, an ab aliquo iis
adjecto, quodcunque, istud sit, ego quidem non quaere ; nee vero, si velim quasrere, in-
veniendi spem habeo ; quod idem sane censeo de ea virium lege, ad quam gradum jam facio.

Eorundem vires g Censeo igitur bina quaecunque materiae puncta determinari asque in aliis distantiis

mutuae in alus , y •,.. , -1 . . .

distantiis attrac- ad mutuum accessum, in alns ad recessum mutuum, quam ipsam determinationem appello
tivae, in aliis re- vim, in priore casu attractivam, in posteriore repulsivam, eo nomine non agendi modum, sed

pulsivae : v i n u m . ,r . . . , '. . .

ejusmodi exempia. ipsam determinationem expnmens, undecunque provemat, cujus vero magnitude mutatis
distantiis mutetur & ipsa secundum certam legem quandam, quae per geometricam lineam
curvam, vel algebraicam formulam exponi possit, & oculis ipsis, uti moris est apud Mechanicos
repraesentari. Vis mutuae a distantia pendentis, & ea variata itidem variatae, atque ad omnes
in immensum & magnas, & parvas distantias pertinentis, habemus exemplum in ipsa
Newtoniana generali gravitate mutata in ratione reciproca duplicata distantiarum, qua;
idcirco numquam e positiva in negativam migrare potest, adeoque ab attractiva ad repul-
sivam, sive a determinatione ad accessum ad determinationem ad recessum nusquam migrat.
Verum in elastris inflexis habemus etiam imaginem ejusmodi vis mutuae variatae secundum
distantias, & a determinatione ad recessum migrantis in determinationem ad accessum, &
vice versa. Ibi enim si duae cuspides, compresso elastro, ad se invicem accedant, acquirunt
determinationem ad recessum, eo majorem, quo magis, compresso elastro, distantia
decrescit ; aucta distantia cuspidum, vis ad recessum minuitur, donee in quadam distantia
evanescat, & fiat prorsus nulla ; turn distantia adhuc aucta, incipit determinatio ad accessum,
quae perpetuo eo magis crescit, quo magis cuspides a se invicem recedunt : ac si e contrario
cuspidum distantia minuatur perpetuo ; determinatio ad accessum itidem minuetur,
evanescet, & in determinationem ad recessum mutabitur. Ea determinatio oritur utique
non ab immediata cuspidum actione in se invicem, sed a natura, & forma totius intermediae
laminae plicatae ; sed hie physicam rei causam non merer, & solum persequor exemplum
determinationis ad accessum, & recessum, quae determinatio in aliis distantiis alium habeat
nisum, & migret etiam ab altera in alteram.

virium earundero 10. Lex autem virium est ejusmodi, ut in minimis distantiis sint repulsivae, atque eo

majores in infmitum, quo distantiae ipsae minuuntur in infinitum, ita, ut pares sint extinguen-
[6]-dae cuivis velocitati utcunque magnae, cum qua punctum alterum ad alterum possit
accedere, antequam eorum distantia evanescat ; distantiis vero auctis minuuntur ita, ut in
quadam distantia perquam exigua evadat vis nulla : turn adhuc, aucta distantia, mutentur in
attractivas, prime quidem crescentes, turn decrescentes, evanescentes, abeuntes in repulsivas,
eodem pacto crescentes, deinde decrescentes, evanescentes, migrantes iterum in attractivas,
atque id per vices in distantiis plurimis, sed adhuc perquam exiguis, donee, ubi ad aliquanto
majores distantias ventum sit, incipiant esse perpetuo attractivae, & ad sensum reciproce

(a) Id quidem respectu ejus spatii, in quo continemur nos, W omnia quis nostris observari sensibus possunt, corpora ;
quod quiddam spatium si quiescat, nihil ego in ea re a reliquis differo ; si forte moveatur motu quopiam, quern motum
ex hujusmodi determinatione sequi debeant ipsa materia puncta ; turn bcec mea erit quiedam non absoluta, sed respectiva
inertia: vis, quam ego quidem exposui W in dissertatione De Maris aestu fcf in Supplementis Stayanis Lib. I. § 13 ;
ubi etiam illud occurrit, quam oh causam ejusmodi respectivam inertiam excogitarim, & quibus rationihus evinci putem,
absolutam omnino demonstrari non posse ; sed ea hue non pertinent.

A THEORY OF NATURAL PHILOSOPHY 39

compenetration in every way. Therefore indeed I do not admit the idea of vacuum
interspersed amongst matter, but I consider that matter is interspersed in a vacuum &
floats in it.

8. As an attribute of these points I admit an inherent propensity to remain in the The nat.ure ?f the
same state of rest, or of uniform motion in a straight line, («) in which they are initially the" possess.1*
set, if each exists by itself in Nature. But if there are also other points anywhere, there

is an inherent propensity to compound (according to the usual well-known composition of
forces & motions by the parallelogram law), the preceding motion with the motion which
is determined by the mutual forces that I admit to act between any two of them, depending
on the distances & changing, as the distances change, according to a certain law common
to them all. This propensity is the origin of what we call the ' force of inertia ' ; whether
this is dependent upon an arbitrary law of the Supreme Architect, or on the nature of points
itself, or on some attribute of them, whatever it may be, I do not seek to know ; even if I
did wish to do so, I see no hope of finding the answer ; and I truly think that this also
applies to the law of forces, to which I now pass on.

9. I therefore consider that any two points of matter are subject to a determination The mutual forces
to approach one another at some distances, & in an equal degree recede from one another at Stw^*^™!*
other distances. This determination I call ' force ' ; in the first case ' attractive ', in the distances & repui-
second case ' repulsive ' ; this term does not denote the mode of action, but the propen- ^mpies

sity itself, whatever its origin, of which the magnitude changes as the distances change ; this kind,
this is in accordance with a certain definite law, which can be represented by a geometrical
curve or by an algebraical formula, & visualized in the manner customary with Mechanicians.
We have an example of a force dependent on distance, & varying with varying distance, &
pertaining to all distances either great or small, throughout the vastness of space, in the
Newtonian idea of general gravitation that changes according to the inverse squares of the
distances : this, on account of the law governing it, can never pass from positive to nega-
tive ; & thus on no occasion does it pass from being attractive to being repulsive, i.e., from
a propensity to approach to a propensity to recession. Further, in bent springs we have
an illustration of that kind of mutual force that varies according as the distance varies, &
passes from a propensity to recession to a propensity to approach, and vice versa. For
here, if the two ends of the spring approach one another on compressing the spring, they
acquire a propensity for recession that is the greater, the more the distance diminishes
between them as the spring is compressed. But, if the distance between the ends is
increased, the force of recession is diminished, until at a certain distance it vanishes and
becomes absolutely nothing. Then, if the distance is still further increased, there begins a
propensity to approach, which increases more & more as the ends recede further & further
away from one another. If now, on the contrary, the distance between the ends is con-
tinually diminished, the propensity to approach also diminishes, vanishes, & becomes changed
into a propensity to recession. This propensity certainly does not arise from the imme-
diate action of the ends upon one another, but from the nature & form of the whole of the
folded plate of metal intervening. But I do not delay over the physical cause of the thing
at this juncture ; I only describe it as an example of a propensity to approach & recession,
this propensity being characterized by one endeavour at some distances & another at other
distances, & changing from one propensity to another.

10. Now the law of forces is of this kind ; the forces are repulsive at very small dis- The Iaw .of forces
tances, & become indefinitely greater & greater, as the distances are diminished indefinitely, for the pomts-

in such a manner that they are capable of destroying any velocity, no matter how large it
may be, with which one point may approach another, before ever the distance between
them vanishes. When the distance between them is increased, they are diminished in such
a way that at a certain distance, which is extremely small, the force becomes nothing.
Then as the distance is still further increased, the forces are change-d to attractive forces ;
these at first increase, then diminish, vanish, & become repulsive forces, which in the same
way first increase, then diminish, vanish, & become once more attractive ; & so on, in turn,
for a very great number of distances, which1 are all still very^ minute : until, finally, when
we get to comparatively great distances, they begin to be continually attractive & approxi-

(a) This indeed holds true for that space in which we, and all bodies that can be observed by our senses, are
contained. Now, if this space is at rest, I do not differ from other philosophers with regard to the matter in question ;
but if perchance space itself moves in some way or other, what motion ought these points of matter to comply with owing
to this kind of propensity ? In that case Ms force of inertia that I postulate is not absolute, but relative ; as indeed
I explained both in the dissertation De Maris Aestu, and also in the Supplements to Stay's Philosophy, book I, section
13. Here also will be found the conclusions at which I arrived with regard to relative inertia of this sort, and the
arguments by which I think it is proved that it is impossible to show that it is generally abxlute. But these things do
not concern us at present.

4°

PHILOSOPHI/E NATURALIS THEORIA

proportionales quadratis distantiarum, atque id vel utcunque augeantur distantiae etiam in
infinitum, vel saltern donee ad distantias deveniatur omnibus Planetarum, & Cometarum
distantiis longe majores.

Leg is simpiicitas ii. Hujusmodi lex primo aspectu videtur admodum complicata, & ex diversis legibus

exprimibihs per temere jnter se coagmentatis coalescens ; at simplicissima, & prorsus incomposita esse potest,

COIlLlIlUtllTl CUf VclIUi • t i • • • 1*1* A 1 1 " J" 1

expressa videlicet per unicam contmuam curvam, vel simphcem Algebraicam iormulam, uti
innui superius. Hujusmodi curva linea est admodum apta ad sistendam oculis ipsis ejusmodi
legem, nee requirit Geometram, ut id praestare possit : satis est, ut quis earn intueatur
tantummodo, & in ipsa ut in imagine quadam solemus intueri depictas res qualescunque,
virium illarum indolem contempletur. In ejusmodi curva eae, quas Geometrae abscissas
dicunt, & sunt segmenta axis, ad quern ipsa refertur curva, exprimunt distantias binorum
punctorum a se invicem : illae vero, quae dicuntur ordinatae, ac sunt perpendiculares lineee
ab axe ad curvam ductae, referunt vires : quae quidem, ubi ad alteram jacent axis partem,
exhibent vires attractivas ; ubi jacent ad alteram, rcpulsivas, & prout curva accedit ad axem,
vel recedit, minuuntur ipsae etiam, vel augentur : ubi curva axem secat, & ab altera ejus
parte transit ad alteram, mutantibus directionem ordinatis, abeunt ex positivis in negativas,
vel vice versa : ubi autem arcus curvae aliquis ad rectam quampiam axi perpendicularem
in infinitum productam semper magis accedit ita ultra quoscumque limites, ut nunquam in
earn recidat, quern arcum asymptoticum appellant Geometrae, ibi vires ipsae in infinitum
excrescunt.

Forma curvae ips-
ius.

12. Ejusmodi curvam exhibui, & exposui in dissertationibus De viribus vivis a Num. 51,
De Lumine Num. 5, De Lege virium in Naturam existentium a Num. 68, & in sua Synopsi
Physics Generalis P. Benvenutus eandem protulit a Num. 108. En brevem quandemejus
ideam. In Fig. i, Axis C'AC habet in puncto A asymptotum curvae rectilineam AB
indefinitam, circa quam habentur bini curvae rami hinc, & inde aequales, prorsus inter se, &
similes, quorum alter DEFGHIKLMNOPQRSTV habet inprimis arcum ED [7] asympto-
ticum, qui nimirum ad partes BD, si indefinite producatur ultra quoscunque limites, semper
magis accedit ad rectam AB productam ultra quoscunque limites, quin unquam ad eandem
deveniat ; hinc vero versus DE perpetuo recidit ab eadam recta, immo etiam perpetuo
versus V ab eadem recedunt arcus reliqui omnes, quin uspiam recessus mutetur in accessum.
Ad axem C'C perpetuo primum accedit, donee ad ipsum deveniat alicubi in E ; turn eodem
ibi secto progreditur, & ab ipso perpetuo recedit usque ad quandam distantiam F, postquam
recessum in accessum mutat, & iterum ipsum axem secat in G, ac flexibus continuis contor-
quetur circa ipsum, quern pariter secat in punctis quamplurimis, sed paucas admodum
ejusmodi sectiones figura exhibet, uti I, L, N, P, R. Demum is arcus desinit in alterum
crus TpsV, jacens ex parte opposita axis respectu primi cruris, quod alterum crus ipsum
habet axem pro asymptoto, & ad ipsum accedit ad sensum ita, ut distantiae ab ipso sint in
ratione reciproca duplicata distantiarum a recta BA.

Abscissae exprimen-

d!nateStaexprimen-
tes vires.

13. Si ex quovis axis puncto a, b, d, erigatur usque ad curvam recta ipsi perpendicularis
aS> ^r' ^h , segmentum axis Aa, Ab, Ad, dicitur abscissa, & refert distantiam duorum materiae
punctorum quorumcunque a se invicem ; perpendicularis ag, br, db , dicitur ordinata, &
exhibet vim repulsivam, vel attractivam, prout jacet respectu axis ad partes D, vel oppositas.

Mutationes ordina-
tarum, & virium iis
expressarum.

14. Patet autem, in ea curvae forma ordinatam ag augeri ultra quoscunque limites, si
abscissa Aa, minuatur pariter ultra quoscunque limites ; quae si augeatur, ut abeat in Ab,
ordinata minuetur, & abibit in br, perpetuo imminutam in accessu b ad E, ubi evanescet :
turn aucta abscissa in Ad, mutabit ordinata directionem in dh , ac ex parte opposita augebitur
prius usque ad F, turn decrescet per il usque ad G, ubi evanescet, & iterum mutabit
directionem regressa in mn ad illam priorem, donee post evanescentiam, & directionis
mutationem factam in omnibus sectionibus I, L, N, P, R, fiant ordinatas op, vs, directionis
constantis, & decrescentes ad sensum in ratione reciproca duplicata abscissarum Ao, Av.
Quamobrem illud est manifestum, per ejusmodi curvam exprimi eas ipsas vires, initio

A THEORY OF NATURAL PHILOSOPHY

0

PHILOSOPHIC NATURALIS THEORIA

o

A THEORY OF NATURAL PHILOSOPHY 43

mately inversely proportional to the squares of the distances. This holds good as the
distances are increased indefinitely to any extent, or at any rate until we get to distances
that are far greater than all the distances of the planets & comets.

11. A law of this kind will seem at first sight to be very complicated, & to be the result The simplicity of

of combining together several different laws in a haphazard sort of way ; but it can be of the law can ^ re~
^.t. • i 1 i • j o v j • i v i • 1 i r presented by means

the simplest kind & not complicated in the slightest degree ; it can be represented for of a continuous
instance by a single continuous curve, or by an algebraical formula, as I intimated above. curve-
A curve of this sort is perfectly adapted to the .graphical representation of this sort of law,
& it does not require a knowledge of geometry to set it forth. It is sufficient for anyone
merely to glance at it, & in it, just as in a picture we are accustomed to view all manner of
things depicted, so will he perceive the nature of these forces. In a curve of this kind,
those lines, that geometricians call abscissae, namely, segments of the axis to which the
curve is referred, represent the distances of two points from one another ; & those, which
we called ordinates, namely, lines drawn perpendicular to the axis to meet the curve, repre-
sent forces. These, when they lie on one side of the axis represent attractive forces, and,
when they lie on the other side, repulsive forces ; & according as the curve approaches the
axis or recedes from it, they too are diminished or increased. When the curve cuts the
axis & passes from one side of it to the other, the direction of the ordinates being changed
in consequence, the forces pass from positive to negative or vice versa. When any arc of
the curve approaches ever more closely to some straight line perpendicular to the axis and
indefinitely produced, in such a manner that, even if this goes on beyond all limits, yet
the curve never quite reaches the line (such an arc is called asymptotic by geometricians),
then the forces themselves will increase indefinitely.

12. I set forth and explained a curve of this sort in my dissertations De Firibus vivis The form of the
(Art. 51), De Lumine (Art. 5), De lege virium in Natura existentium (Art. 68) ; and Father curve-
Benvenutus published the same thing in his Synopsis Physicce Generalis (Art. 108). This

will give you some idea of its nature in a few words.

In Fig. i the axis C'AC has at the point A a straight line AB perpendicular to itself,
which is an asymptote to the curve ; there are two branches of the curve, one on each side
of AB, which are equal & similar to one another in every way. Of these, one, namely
DEFGHIKLMNOPQRSTV, has first of all an asymptotic arc ED ; this indeed, if it is
produced ever so far in the direction ED, will approach nearer & nearer to the straight line
AB when it also is produced indefinitely, but will never reach it ; then, in the direction
DE, it will continually recede from this straight line, & so indeed will all the rest of the arcs
continually recede from this straight line towards V. The first arc continually approaches
the axis C'C, until it meets it in some point E ; then it cuts it at this point & passes on,
continually receding from the axis until it arrives at a certain distance given by the point
F ; after that the recession changes to an approach, & it cuts the axis once more in G ; &
so on, with successive changes of curvature, the curve winds about the axis, & at the same
time cuts it in a number of points that is really large, although only a very few of the
intersections of this kind, as I, L, N, P, R, are shown in the diagram. Finally the arc of the
curve ends up with the other branch TpsV, lying on the opposite side of the axis with
respect to the first branch ; and this second branch has the axis itself as its asymptote,
& approaches it approximately in such a manner that the distances from the axis are in
the inverse ratio of the squares of the distances from the straight line AB.

13. If from any point of the axis, such as a, b, or d, there is erected a straight line per- The abscissae re-
pendicular to it to meet the curve, such as ag, br, or db then the segment of the axis, Aa, £res^Jg

Ab, or Ad, is called the abscissa, & represents the distance of any two points of matter from forces,
one another ; the perpendicular, ag, br, or dh, is called the ordinate, & this represents the
force, which is repulsive or attractive, according as the ordinate lies with regard to the
axis on the side towards D, or on the opposite side.

14. Now it is clear that, in a curve of this form, the ordinate ag will be increased Change in the or-
beyond all bounds, if the abscissa Aa is in the same way diminished beyond all bounds ; & fbat tlfey reprSent!
if the latter is increased and becomes Ab, the ordinate will be diminished, & it will become

br, which will continually diminish as b approaches to E, at which point it will vanish.
Then the abscissa being increased until it becomes Ad, the ordinate will change its direction
as it becomes db, & will be increased in the opposite direction at first, until the point F is
reached, when it will be decreased through the value il until the point G is attained, at
which point it vanishes ; at the point G, the ordinate will once more change its direction
as it returns to the position mn on the same side of the axis as at the start. Finally, after
vanishing & changing direction at all points of intersection with the axis, such as I, L, N,
P, R, the ordinates take the several positions indicated by op, vs : here the direction remains
unchanged, & the ordinates decrease approximately in the inverse ratio of the squares of
the abscissae Ao, Av. Hence it is perfectly evident that, by a curve of this kind, we can

44

PHILOSOPHIC NATURALIS THEORIA

Discrimen hu us
legis virium a
gravitate N e w-
toniana : ejus usus
in Physica : ordo
pertractandorum.

Occasio inveniendae
Theories ex consid-
eraticine impulsus.

V

repulsivas, & imminutis in infinitum distantiis auctas in infinitum, auctis imminutas, turn
evanescentes, abeuntes, mutata directione, in attractivas, ac iterum evenescentes, mutatasque
per vices : donee demum in satis magna distantia evadant attractive ad sensum in ratione
reciproca duplicata distantiarum.

15. Haec virium lex a Newtoniana gravitate differt in ductu, & progressu curvae earn
exprimentis quse nimirum, ut in fig. 2, apud Newtonum est hyperbola DV gradus tertii,
jacens tota citra axem, quern nuspiam

secat, jacentibus omni-[8]-bus ordinatis
vs, op, bt, ag ex parte attractiva, ut
idcirco nulla habeatur mutatio e positivo
ad negativum, ex attractione in repulsi-
onem, vel vice versa ; caeterum utraque
per ductum exponitur curvae continue
habentis duo crura infinita asymptotica
in ramis singulis utrinque in infinitum
productis. Ex hujusmodi autem virium
lege, & ex solis principiis Mechanicis
notissimis, nimirum quod ex pluribus
viribus, vel motibus componatur vis, vel
motus quidam ope parallelogrammorum,
quorum latera exprimant vires, vel mo-
tus componentes, & quod vires ejusmodi

in punctis singulis, tempusculis singulis aequalibus, inducant velocitates, vel motus proportion-
ales sibi, omnes mihi profluunt generales, & praecipuae quacque particulars proprietates cor-
porum,uti etiam superius innui, nee ad singulares proprietates derivandas in genere afHrmo, eas
haberi per diversam combinationem, sed combinationes ipsas evolvo, & geometrice demon-
stro, quae e quibus combinationibus phasnomena, & corporum species oriri debeant. Verum
antequam ea evolvo in parte secunda, & tertia, ostendam in hac prima, qua via, & quibus
positivis rationibus ad earn virium legem devenerim, & qua ratione illam elementorum
materiae simplicitatem eruerim, turn quas difHcultatem aliquam videantur habere posse,
dissolvam.

1 6. Cum anno 1745 De Viribus vivis dissertationem conscriberem, & omnia, quse a
viribus vivis repetunt, qui Leibnitianam tuentur sententiam, & vero etiam plerique ex iis,
qui per solam velocitatem vires vivas metiuntur, repeterem immediate a sola velocitate
genita per potentiarum vires, quae juxta communem omnium Mechanicorum sententiam
velocitates vel generant, vel utcunque inducunt proportionales sibi, & tempusculis, quibus
agunt, uti est gravitas, elasticitas, atque aliae vires ejusmodi ; ccepi aliquant: o diligentius
inquirere in earn productionem velocitatis, quae per impulsum censetur fieri, ubi tota
velocitas momento temporis produci creditur ab iis, qui idcirco percussionis vim infinities
majorem esse censent viribus omnibus, quae pressionem solam momentis singulis exercent.
Statim illud mihi sese obtulit, alias pro percussionibus ejusmodi, quee nimirum momento
temporis finitam velocitatem inducant, actionum leges haberi debere.

FIG

origo ejusdem ex 17. Verum re altius considerata, mihi illud incidit, si recta utamur ratiocinandi methodo,

susTmrnedUatTalin eum agendi modum submovendum esse a Natura, quae nimirum eandem ubique virium

lege Continuitatis. legem, ac eandem agendi rationem adhibeat : impulsum nimirum immediatum alterius

corporis in alterum, & immediatam percussionem haberi non posse sine ilia productione

finitse velocitatis facta momento temporis indivisibili, & hanc sine saltu quodam, & Isesione

illius, quam legem Continuitatis appellant, quam quidem legem in Natura existere, & quidem

satis [9] valida ratione evinci posse existimabam. En autem ratiocinationem ipsam, qua

turn quidem primo sum usus, ac deinde novis aliis, atque aliis meditationibus illustravi, ac

confirmavi.

minus velox.

Laesio legis Continu- 18. Concipiantur duo corpora aequalia, quae moveantur in directum versus eandem

cOTpus^efocruTim- plagam> & id, quod praecedit, habeat gradus velocitatis 6, id vero, quod ipsum persequitur
mediate incurrat in gradus 12. Si hoc posterius cum sua ilia velocitate illaesa deveniat ad immediatum contactum
cum illo priore ; oportebit utique, ut ipso momento temporis, quo ad contactum devenerint,
illud posterius minuat velocitatem suam, & illud primus suam augeat, utrumque per saltum,
abeunte hoc a 12 ad 9, illo a 6 ad 9, sine ullo transitu per intermedios gradus n, & 7 ; 10, &
8 ; 9^, & 8i, &c. Neque enim fieri potest, ut per aliquam utcunque exiguam continui

A THEORY OF NATURAL PHILOSOPHY 45

represent the forces in question, which are initially repulsive & increase indefinitely as the
distances are diminished indefinitely, but which, as the distances increase, are first of all
diminished, then vanish, then become changed in direction & so attractive, again vanish,
& change their direction, & so on alternately ; until at length, at a distance comparatively
great they finally become attractive & are sensibly proportional to the inverse squares of
the distance.

ic. This law of forces differs from the law of gravitation enunciated by Newton in Difference between

. J -nii r i i • i i • • this 'aw °f forces

the construction & development or the curve that represents it ; thus, the curve given in & Newton's law of
Fie. 2, which is that according to Newton, is DV, a hyperbola of the third degree, lying gravitation ; i t s

ii • i r i • i • i • i • nil'6 use ln Physics ;

altogether on one side of the axis, which it does not cut at any point ; all the ordmates, the order in which
such as vs, op, bt, ag lie on the side of the axis representing attractive forces, & there- ^ets^ects are to
fore there is no change from positive to negative, i.e., from attraction to repulsion, or
vice versa. On the other hand, each of the laws is represented by the construction of a
continuous curve possessing two infinite asymptotic branches in each of its members, if
produced to infinity on both sides. Now, from a law of forces of this kind, & with the
help of well-known mechanical principles only, such as that a force or motion can be com-
pounded from several forces or motions by the help of parallelograms whose sides represent
the component forces or motions, or that the forces of this kind, acting on single points
for single small equal intervals of time, produce in them velocities that are proportional to
themselves ; from these alone, I say, there have burst forth on me in a regular flood all
the general & some of the most important particular properties of bodies, as I intimated
above. Nor, indeed, for the purpose of deriving special properties, do I assert that they
ought to be obtained owing to some special combination of points ; on the contrary I
consider the combinations themselves, & prove geometrically what phenomena, or what
species of bodies, ought to arise from this or that combination. Of course, before I
come to consider, both in the second part and in the third, all the matters mentioned
above, I will show in this first part in what way, & by what direct reasoning, I have arrived
at this law of forces, & by what argument I have made out the simplicity of the elements
of matter ; then I will give an explanation of every point that may seem to present any
possible difficulty.

16. In the year 1745, I was putting together my dissertation De Firibus vivis, & had The occasion that
derived everything that they who adhere to the idea of Leibniz, & the greater number of o^my^L^Trom
those who measure ' living forces ' by means of velocity only, derive from these ' living the consideration
forces ' ; as, I say I had derived everything directly & solely from the velocity generated by of imPulsive action,
the forces of those influences, which, according to the generally accepted view taken by

all Mechanicians, either generate, or in some way induce, velocities that are proportional
to themselves & the intervals of time during which they act ; take, for instance, gravity,
elasticity, & other forces of the same kind. I then began to investigate somewhat more
carefully that production of velocity which is thought to arise through impulsive action,
in which the whole of the velocity is credited with being produced in an instant of time by
those, who think, because of that, that the force of percussion is infinitely greater than all
forces which merely exercise pressure for single instants. It immediately forced itself upon
me that, for percussions of this kind, which really induce a finite velocity in an instant of
time, laws for their actions must be obtained different from the rest.

17. However, when I considered the matter more thoroughly, it struck me that, if The cause of
we employ a straightforward method of argument, such a mode of action must be with- w^s the^pposftion
drawn from Nature, which in every case adheres to one & the same law of forces, & the raised to the Law
same mode of action. I came to the conclusion that really immediate impulsive action of °he idea'

one body on another, & immediate percussion, could not be obtained, without the pro- impulse,
duction of a finite velocity taking place in an indivisible instant of time, & this would have
to be accomplished without any sudden change or violation of what is called the Law of
Continuity ; this law indeed I considered as existing in Nature, & that this could be shown
to be so by a sufficiently valid argument. The following is the line of argument that I
employed initially ; afterwards I made it clearer & confirmed it by further arguments &
fresh reflection.

1 8. Suppose there are two equal bodies, moving in the same straight line & in the violation of the
same direction ; & let the one that is in front have a degree of velocity represented by ^ tod^movrng1
6, & the one behind a degree represented by 12. If the latter, i.e., the body that was be- more swiftly comes
hind, should ever reach with its velocity undiminished, & come into absolute contact with, J"*° with^another
the former body which was in front, then in every case it would be necessary that, at the body moving more
very instant of time at which this contact happened, the hindermost body should diminish slowlv-

its velocity, & the foremost body increase its velocity, in each case by a sudden change :
one of them would pass from 12 to 9, the other from 6 to 9, without any passage through
the intermediate degrees, n & 7, 10 & 8, 9$ & 8f, & so on. For it cannot possibly happen

46 PHILOSOPHIC NATURALIS THEORIA

temporis particulam ejusmodi mutatio fiat per intermedios gradus, durante contactu. Si
enim aliquando alterum corpus jam habuit 7 gradus velocitatis, & alterum adhuc retinet
1 1 ; toto illo tempusculo, quod effluxit ab initio contactus, quando velocitates erant 12, & 6,
ad id tempus, quo sunt n, & 7, corpus secundum debuit moveri cum velocitate majore,
quam primum, adeoque plus percurrere spatii, quam illud, £ proinde anterior ejus superficies
debuit transcurrere ultra illius posteriorem superficiem, & idcirco pars aliqua corporis
sequentis cum aliqua antecedentis corporis parte compenetrari debuit, quod cum ob
impenetrabilitatem, quam in materia agnoscunt passim omnes Physici, & quam ipsi tri-
buendam omnino esse, facile evincitur, fieri omnino non possit ; oportuit sane, in ipso
primo initio contactus, in ipso indivisibili momento temporis, quod inter tempus continuum
praecedens contactum, & subsequens, est indivisibilis limes, ut punctum apud Geometras
est limes indivisibilis inter duo continue lineae segmenta, mutatio velocitatum facta fuerit
per saltum sine transitu per intermedias, laesa penitus ilia continuitatis lege, quae itum ab
una magnitudine ad aliam sine transitu per intermedias omnino vetat. Quod autem in
corporibus aequalibus diximus de transitu immediato utriusque ad 9 gradus velocitatis,
recurrit utique in iisdem, vel in utcunque inaequalibus de quovis alio transitu ad numeros
quosvis. Nimirum ille posterioris corporis excessus graduum 6 momento temporis auferri
debet, sive imminuta velocitate in ipso, sive aucta in priore, vel in altero imminuta utcunque,
& aucta in altero, quod utique sine saltu, qui omissis infinitis intermediis velocitatibus
habeatur, obtineri omnino non poterit.

Objectio petita a ig. Sunt, qui difficultatem omnem submoveri posse censeant, dicendo, id quidem ita se

cofporum.dl ' habere debere, si corpora dura habeantur, quae nimirum nullam compressionem sentiant,
nullam mutationem figurae ; & quoniam hsec a multis excluduntur penitus a Natura ; dum
se duo globi contingunt, introcessione, [10] & compressione partium fieri posse, ut in ipsis
corporibus velocitas immutetur per omnes intermedios gradus transitu facto, & omnis
argumenti vis eludatur.

Ea uti non posse, 2O fa mprjmis ea responsione uti non possunt, quicunque cum Newtono, & vero etiam

qui admittunt ele- _, \ . . r . j • o

menta soiida, & cum plerisquc veterum Pnilosopnorum pnma elementa matenae omnino dura admittunt, &

dura- soiida, cum adhaesione infinita, & impossibilitate absoluta mutationis figurae. Nam in primis

elementis illis solidis, & duris, quae in anteriore adsunt sequentis corporis parte, & in praece-

dentis posteriore, quae nimirum se mutuo immediate contingunt, redit omnis argumenti vis

prorsus illaesa.

Extensionem con- 2i. Deinde vero illud omnino intelligi sane non potest, quo pacto corpora omnia partes

primoT pores,1™*! aliquas postremas circa superficiem non habeant penitus solidas, quae idcirco comprimi
parietes soiidos, ac ornnino non possint. In materia quidem, si continua sit, divisibilitas in infinitum haberi
potest, & vero etiam debet ; at actualis divisio in infinitum difficultates secum trahit sane
inextricablies ; qua tamen divisione in infinitum ii indigent, qui nullam in corporibus
admittunt particulam utcunque exiguam compressionis omnis expertem penitus, atque
incapacem. Ii enim debent admittere, particulam quamcunque actu interpositis poris
distinctam, divisamque in plures pororum ipsorum velut parietes, poris tamen ipsis iterum
distinctos. Illud sane intelligi non potest, qui fiat, ut, ubi e vacuo spatio transitur ad corpus,
non aliquis continuus haberi debeat alicujus in se determinatae crassitudinis paries usque ad
primum porum, poris utique carens ; vel quomodo, quod eodem recidit, nullus sit extimus,
& superficiei externae omnium proximus porus, qui nimirum si sit aliquis, parietem habeat
utique poris expertem, & compressionis incapacem, in quo omnis argumenti superioris vis
redit prorsus illaesa.

legis Con- 22. At ea etiam, utcunque penitus inintelligibili, sententia admissa, redit omnis eadem

iprimis su^r™ argument! vis in ipsa prima, & ultima corporum se immediate contingentium superficie, vel
debus, vel punctis. s{ nullae continuae superficies congruant, in lineis, vel punctis. Quidquid enim sit id, in quo
contactus fiat, debet utique esse aliquid, quod nimirum impenetrabilitati occasionem
praestet, & cogat motum in sequente corpore minui, in prascedente augeri ; id, quidquid est,
in quo exeritur impenetratibilitatis vis, quo fit immediatus contactus, id sane velocitatem
mutare debet per saltum, sine transitu per intermedia, & in eo continuitatis lex abrumpi

A THEORY OF NATURAL PHILOSOPHY 47

that this kind of change is made by intermediate stages in some finite part, however small,
of continuous time, whilst the bodies remain in contact. For if at any time the one
body then had 7 degrees of velocity, the other would still retain 1 1 degrees ; thus, during
the whole time that has passed since the beginning of contact, when the velocities were
respectively 12 Si 6, until the time at which they are 1 1 & 7, the second body must be moved
with a greater velocity than the first ; hence it must traverse a greater distance in space
than the other. It follows that the front surface of the second body must have passed
beyond the back surface of the first body ; & therefore some part of the body that follows
behind must be penetrated by some part of the body that goes in front. Now, on account
of impenetrability, which all Physicists in all quarters recognize in matter, & which can be
easily proved to be rightly attributed to it, this cannot possibly happen. There really
must be, in the commencement of contact, in that indivisible instant of time which is an
indivisible limit between the continuous time that preceded the contact & that subsequent
to it (just in the same way as a point in geometry is an indivisible limit between two seg-
ments of a continuous line), a change of velocity taking place suddenly, without any passage
through intermediate stages ; & this violates the Law of Continuity, which absolutely
denies the possibility of a passage from one magnitude to another without passing through
intermediate stages. Now what has been said in the case of equal bodies concerning the
direct passing of both to 9 degrees of velocity, in every case holds good for such equal bodies,
or for bodies that are unequal in any way, concerning any other passage to any numbers.
In fact, the excess of velocity in the hindmost body, amounting to 6 degrees, has to be got
rid of in an instant of time, whether by diminishing the velocity of this body, or by increasing
the velocity of the other, or by diminishing somehow the velocity of the one & increasing
that of the other ; & this cannot possibly be done in any case, without the sudden change
that is obtained by omitting the infinite number of intermediate velocities.

19. There are some people, who think that the whole difficulty can be removed by An objection de-
saying that this is just as it should be, if hard bodies, such as indeed experience no com- ^edexr^ncenyilo1
pression or alteration of shape, are dealt with ; whereas by many philosophers hard bodies hard bodies.

are altogether excluded from Nature ; & therefore, so long as two spheres touch one
another, it is possible, by introcession & compression of their parts, for it to happen that in
these bodies the velocity is changed, the passage being made through all intermediate stages ;
& thus the whole force of the argument will be evaded.

20. Now in the first place, this reply can not be used by anyone who, following New- This reP'y cannot
ton, & indeed many of the ancient philosophers as well, admit the primary elements of ^"admit^oiid0*
matter to be absolutely hard & solid, possessing infinite adhesion & a definite shape that it hard elements.

is perfectly impossible to alter. For the whole force of my argument then applies quite
unimpaired to those solid and hard primary elements that are in the anterior part of the
body that is behind, & in the hindmost part of the body that is in front ; & certainly these
parts touch one another immediately.

21. Next it is truly impossible to understand in the slightest degree how all bodies do Continuous exten-
not have some of their last parts just near to the surface perfectly solid, & on that account mary ^resT* walls
altogether incapable of being compressed. If matter is continuous, it may & must be sub- bounding them,
ject to infinite divisibility ; but actual division carried on indefinitely brings in its train

difficulties that are truly inextricable ; however, this infinite division is required by those
who do not admit that there are any particles, no matter how small, in bodies that are
perfectly free from, & incapable of, compression. For they must admit the idea that every
particle is marked off & divided up, by the action of interspersed pores, into many boundary
walls, so to speak, for these pores ; & these walls again are distinct from the pores them-
selves. It is quite impossible to understand why it comes about that, in passing from
empty vacuum to solid matter, we are not then bound to encounter some continuous wall of
some definite inherent thickness from the surface to the first pore, this wall being everywhere
devoid of pores ; nor why, which comes to the same thing in the end, there does not exist
a pore that is the last & nearest to the external surface ; this pore at least, if there were one,
certainly has a wall that is free from pores & incapable of compression ; & here then the
whole force of the argument used above applies perfectly unimpaired.

22. Moreover, even if this idea is admitted, although it may be quite unintelligible, Violation of the
then the whole force of the same argument applies to the first or last surface of the bodies ta^s'piace^any
that are in immediate contact with one another ; or, if there are no continuous surfaces rate, in prime sur-
congruent, then to the lines or points. For, whatever the manner may be in which contact

takes place, there must be something in every case that certainly affords occasion for
impenetrability, & causes the motion of the body that follows to be diminished, & that of
the one in front to be increased. This, whatever it may be, from which the force of impene-
trability is derived, at the instant at which immediate contact is obtained, must certainly
change the velocity suddenly, & without any passage through intermediate stages ; & by

48

PHILOSOPHIC NATURALIS THEORIA

debet, atque labefactari, si ad ipsum immediatum contactum illo velocitatum discrimine
deveniatur. Id vero est sane aliquid in quacunque e sententiis omnibus continuam
extensionem tribuentibus materise. Est nimirum realis affectio qusedam corporis, videlicet
ejus limes ultimus realis, superficies, realis superficiei limes linea, realis lineae limes punctum,
qua affectiones utcunque in iis sententiis sint prorsus inseparabiles [n] ab ipso corpore,
sunt tamen non utique intellectu confictae, sed reales, quas nimirum reales dimensiones
aliquas habent, ut superficies binas, linea unam, ac realem motum, & translationem cum ipso
corpore, cujus idcirco in iis sententiis debent, esse affectiones quaedam, vel modi.

Objectio petita a 27. Est, qui dicat, nullum in iis committi saltum idcirco, quod censendum sit, nullum

vucemassa, &,.J r . .. , ,, i\/r x

motns. quae super- habere motum, superficiem, Imeam, punctum, quae massam habeant nullam. Motus, mquit,

ficiebus, & punctis a Mechanicis habet pro mensura massam in velocitatem ductam : massa autem est super-
non convemant. _.. , . • • «• • i • j • • • • /-^

ficies baseos ducta in crassitudmem, sive altitudmem, ex. gr. m pnsmatis. Quo minor est
ejusmodi crassitude, eo minor est massa, & motus, ac ipsa crassitudine evanescente, evanescat
oportet & massa, & motus.

Kesponsionis ini- 24. Verum qui sic ratiocinatur, inprimis ludit in ipsis vocibus. Massam vulgo appellant

tacam.^punctmn! quantitatem materiae, & motum corporum metiuntur per massam ejusmodi, ac velocitatem.

posita extensione At quemadmodum in ipsa geometrica quantitate tria genera sunt quantitatum, corpus, vel

contmua, e - ^11^^ qUO(J trinam dimensionem habet, superficies quae binas, linae, quae unicam, quibus

accedit linese limes punctum, omni dimensione, & extensione carens ; sic etiam in Physica

habetur in communi corpus tribus extensionis speciebus praeditum ; superficies realis extimus

corporis limes, praedita binis ; linea, limes realis superficiei, habens unicam; & ejusdem

lineae indivisibilis limes punctum. Utrobique alterum alterius est limes, non pars, & quatuor

diversa genera constituunt. Superficies est nihil corporeum, sed non & nihil superficial,

quin immo partes habet, & augeri potest, & minui ; & eodem pacto linea in ratione quidem

superficiei est nihil, sed aliquid in ratione linese ; ac ipsum demum punctum est aliquid in

suo genere, licet in ratione lineae sit nihil.

QUO pacto nomen 25. Hinc autem in iis ipsis massa quaedam considerari potest duarum dimensionum, vel

motus 'debeat8 con- unius, vel etiam nullius continuae dimensionis, sed numeri punctorum tantummodo, uti

venire superficie- quantitas ejus genere designetur ; quod si pro iis etiam usurpetur nomen massae generaliter,

bus, imeis, punctis. motus quantitas definiri poterit per productum ex velocitate, & massa ; si vero massae nomen

tribuendum sit soli corpori, turn motus quidem corporis mensura erit massa in velocitatem

ducta ; superficiei, lineae, punctorum quotcunque motus pro mensura habebit quantitatem

superficiei, vel lineae, vel numerum punctorum in velocitatem ducta ; sed motus utique iis

omnibus speciebus tribuendus erit, eruntque quatuor motuum genera, ut quatuor sunt

quantitatum, solidi, superficiei, lineae, punctorum ; ac ut altera harum erit nihil in alterius

ratione, non in sua ; ita alterius motus erit nihil in ratione alterius sed erit sane aliquid in

ratione sui, non purum nihil.

Fore, ut ea laedatur
saltern in velocitate
punctorum.

Motum passim rI2i 2Q- gt quidem jpSj Mechanici vulgo motum tribuunt & superficiebus & lineis, &

tnbui punctis; ,'•..* , . '. -m • • j

fore, lit in eo ixda- punctis, ac centri gravitatis motum ubique nommant rhysici, quod centrum utique punctum
i^r Continuitatis est aliquod, non corpus trina praeditum dimensione, quam iste ad motus rationem, &
appellationem requirit, ludendo, ut ajebam, in verbis. Porro in ejusmodi motibus exti-
marum saltern superficierum, vel linearum, vel punctorum, saltus omnino committi debet,
si ea ad contactum immediatum deveniant cum illo velocitatum discrimine, & continuitatis
lex violari.

27. Verum hac omni disquisitione omissa de notione motus, & massae, si factum ex
velocitate, & massa, evanescente una e tribus dimensionibus, evanescit ; remanet utique
velocitas reliquarum dimensionum, quae remanet, si eae reapse remanent, uti quidem omnino
remanent in superficie, & ejus velocitatis mutatio haberi deberet per saltum, ac in ea violari
continuitatis lex jam toties memorata.

-, ti°exin?P<!ne- 28. Haec quidem ita evidentia sunt, ut omnino dubitari non possit, quin continuitatis

trabilitate admissa ,.,..,/ .-KT • » • • i « i • • j- • • j ••

in minimis parti- lex infnngi debeat, & saltus m Naturam induci, ubi cum velocitatis discrimine ad se invicem

cuiis. & ejus confu- accedant corpora, & ad immediatum contactum deveniant, si modo impenetrabilitas

corporibus tribuenda sit, uti revera est. Earn quidem non in integris tantummodo corpori-

bus, sed in minimis etiam quibusque corporum particulis, atque elementis agnoverunt

Physici universi. Fuit sane, qui post meam editam Theoriam, ut ipsam vim mei argument}

A THEORY OF NATURAL PHILOSOPHY

49

that the Law of Continuity must be broken & destroyed, if immediate contact is arrived
at with such a difference of velocity. Moreover, there is in truth always something of this
sort in every one of the ideas that attribute continuous extension to matter. There is some
real condition of the body, namely, its last real boundary, or its surface, a real boundary of
a surface, a line, & a real boundary of a line, a point ; & these conditions, however insepar-
able they may be in these theories from the body itself, are nevertheless certainly not
fictions of the brain, but real things, having indeed certain real dimensions (for instance, a
surface has two dimensions, & a line one) ; they also have real motion & movement of trans-
lation along with the body itself ; hence in these theories they must be certain conditions
or modes of it.

23. Someone may say that there is no sudden change made, because it must be con- Objection derived
sidered that a surface, a line or a point, having no mass, cannot have any motion. He may 1™™ mo/io^w^idi
say that motion has, according to Mechanicians, as its measure, the mass multiplied by the do not accord with
velocity ; also mass is the surface of the base multiplied by the thickness or the altitude, surfaces & P°mis-
as for instance in prisms. Hence the less the thickness, the less the mass & the motion ;

thus, if the thickness vanishes, then both the mass & therefore the motion must vanish
as well.

24. Now the man who reasons in this manner is first of all merely playing with words. Commencement of
Mass is commonly called quantity of matter, & the motion of bodies is measured by mass the answer to tl?ls :

. i • * • i « . , •*• —- . * , . c • -_* cl SUrlclCC, OF ii 11116,

of this kind & the velocity. But, just as in a geometrical quantity there are three kinds of or a point, is some-
quantities, namely, a body or a solid having three dimensions, a surface with two, & a line \
with one : to which is added the boundary of a line, a point, lacking dimensions altogether, is supposed to ex-
& of no extension. So also in Physics, a body is considered to be endowed with three lst'
species of extension ; a surface, the last real boundary of a body, to be endowed with two ;
a line, the real boundary of a surface, with one ; & the indivisible boundary of the line, to
be a point. In both subjects, the one is a boundary of the other, & not a part of it ; &
they form four different kinds. There is nothing solid about a surface ; but that does not
mean that there is also nothing superficial about it ; nay, it certainly has parts & can be
increased or diminished. In the same way a line is nothing indeed when compared with
a surface, but a definite something when compared with a line ; & lastly a point is a definite
something in its own class, although nothing in comparison with a line.

25. Hence also in these matters, a mass can be considered to be of two dimensions, or The manner in
of one, or even of no continuous dimension, but only numbers of points, just as quantity of wn'^ma^and^the
this kind is indicated. Now, if for these also, the term mass is employed in a generalized term motus is bound
sense, we shall be able to define the quantity of motion by the product of the velocity & !:°;^p^'to?ujfaces'

1 Ii • ' • 1 1 1 • • • 1 1 * 1 1 1 i HOPS, <X pOintS.

the mass. But if the term mass is only to be used in connection with a solid body, then
indeed the motion of a solid body will be measured by the mass multiplied by the velocity ;
but the motion of a surface, or a line, or any number of points will have as their measure
the quantity of the surface, or line, or the number- of the points, multiplied by the velocity.
Motion at any rate will be ascribed in all these cases, & there will be four kinds of motion,
as there are four kinds of quantity, namely, for a solid, a surface, a line, or for points ; and, as
each class of the latter will be as nothing compared with the class before it, but something
in its own class, so the motion of the one will be as nothing compared with the motion
of the other, but yet really something, & not entirely nothing, compared with those of
its own class.

26. Indeed, Mechanicians themselves commonly ascribe motion to surfaces, lines & Motion is ascribed
points, & Physicists universally speak of the motion of the centre of gravity ; this centre is minateiy3 the'i^w
undoubtedly some point, & not a body endowed with three dimensions, which the objector of Continuity is vio-
demands for the idea & name of motion, by playing with words, as I said above. On the '

other hand, in this kind of motions of ultimate surfaces, or lines, or points, a sudden change
must certainly be made, if they arrive at immediate contact with a difference of velocity
as above, & the Law of Continuity must be violated.

27. But, omitting all debate about the notions of motion & mass, if the product of it is at least a fact
the velocity & the mass vanishes when one of the three dimensions vanish, there will still fated^tf^the^idea
remain the velocity of the remaining dimensions ; & this will persist so long as the dimen- of the velocity of
sions persist, as they do persist undoubtedly in the case of a surface. Hence the change P°mts-

in its velocity must have been made suddenly, & thereby the Law of Continuity, which I
have already mentioned so many times, is violated.

28. These things are so evident that it is absolutely impossible to doubt that the Law objection derived

/./-!•••• r- i „ i j j i . . j j . »T iv from the admission

of Continuity is infringed, & that a sudden change is introduced into Nature, when bodies Of impenetrability
approach one another with a difference of velocity & come into immediate contact, if only in verv small Par-
we are to ascribe impenetrability to bodies, as we really should. And this property too, tion. '
not in whole bodies only, but in any of the smallest particles of bodies, & in the elements as
well, is recognized by Physicists universally. There was one, I must confess, who, after I

50

PHILOSOPHIC NATURALIS THEORIA

infringeret, affirmavit, minimas corporum particulas post contactum superficierum com-
penetrari non nihil, & post ipsam compenetrationem mutari velocitates per gradus. At id
ipsum facile demonstrari potest contrarium illi inductioni, & analogiae, quam unam habemus
in Physica investigandis generalibus naturae legibus idoneam, cujus inductionis vis quae sit,
& quibus in locis usum habeat, quorum locorum unus est hie ipse impenetrabilitatis ad
minimas quasque particulas extendendae, inferius exponam.

Objectio a voce

motus assumpta

pro mutatione;
confutatio ex
reahtate motus

2Q. Fuit itidem e Leibnitianorum familia, qui post evulgatam Theoriam meam cen-

. ' ,./>- •, • j- • j« j j -i •

suerit, dimcultatem ejusmodi amoveri posse dicendo, duas monades sibi etiam mvicem

occurrentes cum velocitatibus quibuscunque oppositis aequalibus, post ipsum contactum
..... . i, . . .' r .....

pergere moven sine locali progressione. Ham progressionem, ajebat, revera omnmo nihil
esse, si a spatio percurso sestimetur, cum spatium sit nihil ; motum utique perseverare, &
extingui per gradus, quia per gradus extinguatur energia ilia, qua in se mutuo agunt, sese
premendo invicem. Is itidem ludit in voce motus, quam adhibet pro mutatione quacunque,
& actione, vel actionis modo. Motus locaiis, & velocitas motus ipsius, sunt ea, quse ego
quidem adhibeo, & quae ibi abrumpuntur per saltum. Ea, ut evidentissime constat, erant
aliqua ante contactum, & post contactum mo-[i3]-mento temporis in eo casu abrumpuntur ;
nee vero sunt nihil ; licet spatium pure imaginarium sit nihil. Sunt realis affectio rei
mobilis fundata in ipsis modis localiter existendi, qui modi etiam relationes inducunt dis-
tantiarum reales utique. Quod duo corpora magis a se ipsis invicem distent, vel minus ;
quod localiter celerius moveantur, vel lentius ; est aliquid non imaginarie tantummodo, sed
realiter diversum ; in eo vero per immediatum contactum saltus utique induceretur in eo
casu, quo ego superius sum usus.

Qui Continuitatu, 30. Et sane summus nostri aevi Geometra, & Philosophus Mac-Laurinus, cum etiam ipse

jegem summover- conisjonem corporum contemplatus vidisset, nihil esse, quod continuitatis legem in collisione
corporum facta per immediatum contactum conservare, ac tueri posset, ipsam continuitatis
legem deferendam censuit, quam in eo casu omnino violari affirmavit in eo opere, quod de
Newtoni Compertis inscripsit, lib. I, cap. 4. Et sane sunt alii nonnulli, qui ipsam con-
tinuitatis legem nequaquam admiserint, quos inter Maupertuisius, vir celeberrimus, ac de
Republica Litteraria optime meritus, absurdam etiam censuit, & quodammodo inexplica-
bilem. Eodem nimirum in nostris de corporum collisione contemplationibus devenimus
Mac-Laurinus, & ego, ut viderimus in ipsa immediatum contactum, atque impulsionem cum
continuitatis lege conciliari non posse. At quoniam de impulsione, & immediate corporum
contactu ille ne dubitari quidem posse arbitrabatur, (nee vero scio, an alius quisquam omnem
omnium corporum immediatum contactum subducere sit ausus antea, utcunque aliqui aeris
velum, corporis nimirum alterius, in collisione intermedium retinuerint) continuitatis
legem deseruit, atque infregit.

Theorise exortus,
t^'t Uf fien

31. Ast ego cum ipsam continuitatis legem aliquanto diligentius considerarim, &
, quibus ea innititur, perpenderim, arbitratus sum, ipsam omnino e Natura
submoveri non posse, qua proinde retenta contactum ipsum immediatum submovendum
censui in collisionibus corporum, ac ea consectaria persecutus, quae ex ipsa continuitate
servata sponte profluebant, directa ratiocinatione delatus sum ad earn, quam superius
exposui, virium mutuarum legem, quae consectaria suo quaeque ordine proferam, ubi ipsa,
quae ad continuitatis legem retinendam argumenta me movent, attigero.

Lex Continuitatis 32. Continuitatis lex, de qua hie agimus, in eo sita est, uti superius innui, ut quaevis

quid sit : discn- • j i • i- T i- • • r

men inter status, quantitas, dum ab una magmtudme ad aliam migrat, debeat transire per omnes intermedias
& incrementa. ejusdem generis magnitudines. Solet etiam idem exprimi nominandi transitum per gradus
intermedios, quos quidem gradus Maupertuisius ita accepit, quasi vero quaedam exiguae
accessiones fierent momento temporis, in quo quidem is censuit violari jam necessario legem
ipsam, quae utcunque exiguo saltu utique violatur nihilo minus, quam maximo ; cum
nimi-[l4]-rum magnum, & parvum sint tantummodo respectiva ; & jure quidem id censuit ;
si nomine graduum incrementa magnitudinis cujuscunque momentanea intelligerentur.

A THEORY OF NATURAL PHILOSOPHY 51

had published my Theory, endeavoured to overcome the force of the argument I had used
by asserting that the minute particles of the bodies after contact of the surfaces were
subject to compenetration in some measure, & that after compenetration the velocities
were changed gradually. But it can be easily proved that this is contrary to that induction
& analogy, such as we have in Physics, one peculiarly adapted for the investigation of the
general laws of Nature. What the power of this induction is, & where it can be used (one
of the cases is this very matter of extending impenetrability to the minute particles of a
body), I will set forth later.

29. There was also one of the followers of Leibniz who, after I had published my Objection to the
Theory, expressed his opinion that this kind of difficulty could be removed by saying that used for°a"change^
two monads colliding with one another with any velocities that were equal & opposite refutation from the

,,,., .. . .-I , , • TT reality of local mo-

would, alter they came into contact, go on moving without any local progression, rle tion.
added that that progression would indeed be absolutely nothing, if it were estimated by the
space passed over, since the space was nothing ; but the motion would go on & be destroyed
by degrees, because the energy with which they act upon one another, by mutual pressure,
would be gradually destroyed. He also is playing with the meaning of the term motus,
which he uses both for any change, & for action & mode of action. Local motion, & the
velocity of that motion are what I am dealing with, & these are here broken off suddenly.
These, it is perfectly evident, were something definite before contact, & after contact in
an instant of time in this case they are broken off. Not that they are nothing ; although
purely imaginary space is nothing. They are real conditions of the movable thing
depending on its modes of extension as regards position ; & these modes induce relations
between the distances that are certainly real. To account for the fact that two bodies
stand at a greater distance from one another, or at a less ; or for the fact that they are
moved in position more quickly, or more slowly ; to account for this there must be some-
thing that is not altogether imaginary, but real & diverse. In this something there would
be induced, in the question under consideration, a sudden change through immediate
contact.

30. Indeed the finest geometrician & philosopher of our times, Maclaurin, after he too There are some who
had considered the collision of solid bodies & observed that there is nothing which could i^doi continuity5
maintain & preserve the Law of Continuity in the collision of bodies accomplished by

immediate contact, thought that the Law of Continuity ought to be abandoned. He
asserted that, in general in the case of collision, the law was violated, publishing his idea in
the work that he wrote on the discoveries of Newton, bk. i, chap. 4. True, there are some
others too, who would not admit the Law of Continuity at all ; & amongst these, Mauper-
tuis, a man of great reputation & the highest merit in the world of letters, thought it was
senseless, & in a measure inexplicable. Thus, Maclaurin came to the same conclusion as
myself with regard to our investigations on the collision of bodies ; for we both saw that, in
collision, immediate contact & impulsive action could not be reconciled with the Law of
Continuity. But, whereas he came to the conclusion that there could be no doubt about
the fact of impulsive action & immediate contact between the bodies, he impeached &
abrogated the Law of Continuity. Nor indeed do I know of anyone else before me, who
has had the courage to deny the existence of all immediate contact for any bodies whatever,
although there are some who would retain a thin layer of air, (that is to say, of another body),
in between the two in collision.

31. But I, after considering the Law of Continuity somewhat more carefully, & The origin of my
pondering over the fundamental ideas on which it depends, came to the conclusion that this°Law, as'shouid
it certainly could not be withdrawn altogether out of Nature. Hence, since it had to be be done,
retained, I came to the conclusion that immediate contact in the collision of solid bodies

must be got rid of ; &, investigating the deductions that naturally sprang from the
conservation of continuity, I was led by straightforward reasoning to the law that I have set
forth above, namely, the law of mutual forces. These deductions, each set out in order,
I will bring forward when I come to touch upon those arguments that persuade me to
retain the Law of Continuity.

32. The Law of Continuity, as we here deal with it, consists in the idea that, as I Jhe nature of the

. j , ..''... . , , Law of Continuity ;

intimated above, any quantity, in passing from one magnitude to another, must pass through distinction between
all intermediate magnitudes of the same class. The same notion is also commonly expressed stat<~s & incre-

, , ° .,,. ,. -11 ments.

by saying that the passage is made by intermediate stages or steps ; these steps indeed
Maupertuis accepted, but considered that they were very small additions made in an
instant of time. In this he thought that the Law of Continuity was already of necessity
violated, the law being indeed violated by any sudden change, no matter how small, in no
less a degree than by a very great one. For, of a truth, large & small are only relative terms ;
& he rightly thought as he did, if by the name of steps we are to understand momentaneous

PHILOSOPHIC NATURALIS THEORIA

Geometriae usus ad
earn exponendam :
momenta punctis,
tempera continua
lineis expressa.

Fluxus ordinatae
transeuntis per
m agnit u d i nes
omnes intermedias.

Idem in quantitate
variabili expressa :
aequivocatio in
voce gradus.

FKMH K' M' D'

FIG. 3.

Verum id ita intelligendum est ; ut singulis momentis singuli status respondeant ; incre-
menta, vel decrementa non nisi continuis tempusculis.

33. Id sane admodum facile concipitur ope Geometriae. Sit recta quaedam AB in
fig. 3, ad quam referatur quaedam alia linea CDE. Exprimat prior ex iis tempus, uti solet
utique in ipsis horologiis circularis peripheria

ab indicis cuspide denotata tempus definire.
Quemadmodum in Geometria in lineis
puncta sunt indivisibiles limites continuarum
lineas partium, non vero partes linese ipsius ;
ita in tempore distinguenda; erunt partes
continui temporis respondentes ipsis lines
partibus, continue itidem & ipsas, a mo-
mentis, quae sunt indivisibiles earum partium
limites, & punctis respondent ; nee inpos-
terum alio sensu agens de tempore momenti
nomen adhibebo, quam eo indivisibilis
limitis ; particulam vero temporis utcunque
exiguam, & habitam etiam pro infinitesima,
tempusculum appellabo.

34. Si jam a quovis puncto rectae AB, ut F, H, erigatur ordinata perpendicularis FG,
HI, usque ad lineam CD ; ea poterit repraesentare quantitatem quampiam continuo
variabilem. Cuicunque momento temporis F, H, respondebit sua ejus quantitatis magnitudo
FG, HI ; momentis autem intermediis aliis K, M, aliae magnitudines, KL, MN, respondebunt ;
ac si a puncto G ad I continua, & finita abeat pars linese CDE, facile patet & accurate de-
monstrari potest, utcunque eadem contorqueatur, nullum fore punctum K intermedium,
cui aliqua ordinata KL non respondeat ; & e converse nullam fore ordinatam magnitu-
dinis intermediae inter FG, HI, quae alicui puncto inter F, H intermedio non respondeat.

35. Quantitas ilia variabilis per hanc variabilem ordinatam expressa mutatur juxta
continuitatis legem, quia a magnitudine FG, quam habet momento temporis F, ad magni-
tudinem HI, quae respondet momento temporis H, transit per omnes intermedias magnitu-
dines KL, MN, respondentes intermediis momentis K, M, & momento cuivis respondet
determinata magnitudo. Quod si assumatur tempusculum quoddam continuum KM
utcunque exiguum ita, ut inter puncta L, N arcus ipse LN non mutet recessum a recta AB
in accessum ; ducta LO ipsi parallela, habebitur quantitas NO, quas in schemate exhibito
est incrementum magnitudinis ejus quantitatis continuo variatae. Quo minor est ibi
temporis particula KM, eo minus est id incrementum NO, & ilia evanescente, ubi congruant
momenta K, M, hoc etiam evanescit. Potest quaevis magnitudo KL, MN appellari status
quidam variabilis illius quantitatis, & gradus nomine deberet potius in-[i5]-telligi illud
incrementum NO, quanquam aliquando etiam ille status, ilia magnitudo KL nomine gradus
intelligi solet, ubi illud dicitur, quod ab una magnitudine ad aliam per omnes intermedios
gradus transeatur ; quod quidem aequivocationibus omnibus occasionem exhibuit.

status singuios 36. Sed omissis aequivocationibus ipsis, illud, quod ad rem facit, est accessio incremen-

menta^vero'utcun" torum facta non momento temporis, sed tempusculo continuo, quod est particula continui

que parva tem- temporis. Utcunque exiguum sit incrementum ON, ipsi semper respondet tempusculum

respondereC°ntinuis q.u°ddam KM continuum. Nullum est in linea punctum M ita proximum puncto K, ut sit

primum post ipsum ; sed vel congruunt, vel intercipiunt lineolam continua bisectione per

alia intermedia puncta perpetuo divisibilem in infinitum. Eodem pacto nullum est in

tempore momentum ita proximum alteri praecedenti momento, ut sit primum post ipsum,

sed vel idem momentum sunt, vel inter jacet inter ipsa tempusculum continuum per alia

intermedia momenta divisibile in infinitum ; ac nullus itidem est quantitatis continuo

variabilis status ita proximus praecedenti statui, ut sit primus post ipsum accessu aliquo

momentaneo facto : sed differentia, quae inter ejusmodi status est, debetur intermedio

continuo tempusculo ; ac data lege variationis, sive natura lineae ipsam exprimentis, &

quacunque utcunque exigua accessione, inveniri potest tempusculum continuum, quo ea

accessio advenerit.

Transitus sine sal- 37- Atque sic quidem intelligitur, quo pacto fieri possit transitus per intermedias

tu, etiamapositivis magnitudines omnes, per intermedios status, per gradus intermedios, quin ullus habeatur

ad negativa perm- ,° . r -, . ... , ' "

hiium, quod tamen saltus utcunque exiguus momento temporis factus. Notari mud potest tantummodo,
m°" eSstedVereu'ida1m mutati°nem neri alicubi per incrementa, ut ubi KL abit, in MN per NO ; alicubi per
reaiis status,1"0 ' decrementa, ut ubi K'L' abeat in N'M' per O'N' ; quin immo si linea CDE, quse legem

A THEORY OF NATURAL PHILOSOPHY 53

increments of any magnitude whatever. But the idea should be interpreted as follows :
single states correspond to single instants of time, but increments or decrements only to
small intervals of continuous time.

33. The idea can be very easily assimilated by the help of geometry. Explanation by the
Let AB be any straight line (Fig. 3), to which as axis let any other line CDE be referred. "nsseta°tfs ^eTes^

Let the first of them represent the time, in the same manner as it is customary to specify ted by points, con-
the time in the case of circular clocks by marking off the periphery with the end of a pointer. 1™°^ "^s*** °f
Now, just as in geometry, points are the indivisible boundaries of the continuous parts of
a line, so, in time, distinction must be made between parts of continuous time, which cor-
respond to these parts of a line, themselves also continuous, & instants of time, which are
the indivisible boundaries of those parts of time, & correspond to points. In future I shall
not use the term instant in any other sense, when dealing with time, than that of the
indivisible boundary ; & a small part of time, no matter how small, even though it is
considered to be infinitesimal, I shall term a tempuscule, or small interval of time.

34. If now from any points F,H on the straight line AB there are erected at right angles T.he flux °.f the or~
to it ordinates FG, HI, to meet the line CD ; any of these ordinates can be taken to repre- through^ ail *interS
sent a quantity that is continuously varying. To any instant of time F, or H, there will mediate values,
correspond its own magnitude of the quantity FG, or HI ; & to other intermediate instants

K, M, other magnitudes KL, MN will correspond. Now, if from the point G, there pro-
ceeds a continuous & finite part of the line CDE, it is very evident, & it can be rigorously
proved, that, no matter how the curve twists & turns, there is no intermediate point K,
to which some ordinate KL does not correspond ; &, conversely, there is no ordinate of
magnitude intermediate between FG & HI, to which there does not correspond a point
intermediate between F & H.

35. The variable quantity that is represented by this variable ordinate is altered in The same holds
accordance with the Law of Continuity ; for, from the magnitude FG, which it has at able1 quantity w
the instant of time F, to the magnitude HI, which corresponds to the instant H, it passes represented ; equi-
through all intermediate magnitudes KL, MN, which correspond to the intermediate oUhe1(term Itep^
instants K, M ; & to every instant there corresponds a definite magnitude. But if we take

a definite small interval of continuous time KM, no matter how small, so that between the
points L & N the arc LN does not alter from recession from the line AB to approach, &
draw LO parallel to AB, we shall obtain the quantity NO that in the figure as drawn is the
increment of the magnitude of the continuously varying quantity. Now the smaller the
interval of time KM, the smaller is this increment NO ; & as that vanishes when the
instants of time K, M coincide, the increment NO also vanishes. Any magnitude KL, MN
can be called a state of the variable quantity, & by the name step we ought rather to under-
stand the increment NO ; although sometimes also the state, or the magnitude KL is
accustomed to be called by the name step. For instance, when it is said that from one
magnitude to another there is a passage through all intermediate stages or steps ; but this
indeed affords opportunity for equivocations of all sorts.

36. But, omitting all equivocation of this kind, the point is this : that addition of single states cor-

.' 1-11 • • <• • i • 11 . respond to instants,

increments is accomplished, not m an instant 01 time, but in a small interval of con- but increments
tinuous time, which is a part of continuous time. However small the increment ON may however sma11 to

i i i Tru if mi •»•> intervals of con-

DC, there always corresponds to it some continuous interval KM. 1 here is no point M tinuous time.

in the straight line AB so very close to the point K, that it is the next after it ; but either
the points coincide, or they intercept between them a short length of line that is divisible
again & again indefinitely by repeated bisection at other points that are in between M &
K. In the same way, there is no instant of time that is so near to another instant that has
gone before it, that it is the next after it ; but either they are the same instant, or there
lies between them a continuous interval that can be divided indefinitely at other inter-
mediate instants. Similarly, there is no state of a continuously varying quantity so very
near to a preceding state that it is the next state to it, some momentary addition having
been made ; any difference that exists between two states of the same kind is due to a
continuous interval of time that has passed in the meanwhile. Hence, being given the
law of variation, or the nature of the line that represents it, & any increment, no matter
how small, it is possible to find a small interval of continuous time in which the increment
took place.

37. In this manner we can understand how it is possible for a passage to take place Passages without
through all intermediate magnitudes, through intermediate states, or through intermediate from^positive1 8 to
stages, without any sudden change being made, no matter how small, in an instant of time, negative through

T' 11 1111 • i 111- /i zero : zero how-

It can merely be remarked that change in some places takes place by increments (as when ever ;s not realiy

KL becomes MN by the addition of NO), in other places by decrements (as when K'L' nothing, but acer-

' tain real state.

54 PHILOSOPHIC NATURALIS THEORIA

variationis exhibit, alicubi secet rectam, temporis AB, potest ibidem evanescere magnitude,
ut ordinata M'N', puncto M' allapso ad D evanesceret, & deinde mutari in negativam PQ,
RS, habentem videlicet directionem contrariam, quae, quo magis ex oppositae parte crescit,
eo minor censetur in ratione priore, quemadmodum in ratione possessionis, vel divitiarum,
pergit perpetuo se habere pejus, qui iis omnibus, quae habebat, absumptis, aes alienum
contrahit perpetuo majus. Et in Geometria quidem habetur a positivo ad negativa
transitus, uti etiam in Algebraicis formulis, tarn transeundo per nihilum, quam per innnitum,
quos ego transitus persecutus sum partim in dissertatione adjecta meis Sectionibus Conicis,
partim in Algebra § 14, & utrumque simul in dissertatione De Lege Continuitatis ; sed in
Physica, ubi nulla quantitas in innnitum excrescit, is casus locum non habet, & non, nisi
transeundo per nihilum, transitus fit a positi-[i6]-vis ad negativa, ac vice versa ; quanquam,
uti inferius innuam, id ipsum sit non nihilum revera in se ipso, sed realis quidem status, &
habeatur pro nihilo in consideration quadam tantummodo, in qua negativa etiam, qui sunt
veri status, in se positivi, ut ut ad priorem seriem pertinentes negative quodam modo,
negativa appellentur.

Proponitur pro- ,§_ Exposita hoc pacto, & vindicata continuitatis lege, earn in Natura existere plerique

banda existentia _, ., J . . r . . .... ... P . . ,-, r.

legis Continuitat.s. Philosophi arbitrantur, contradicentibus nonnullis, uti supra mnui. Ego, cum in earn
primo inquirerem, censui, eandem omitti omnino non posse ; si earn, quam habemus unicam,
Naturae analogiam, & inductionis vim consulamus, ope cujus inductionis earn demonstrare
conatus sum in pluribus e memoratis dissertationibus, ac eandem probationem adhibet
Benvenutus in sua Synopsi Num. 119; in quibus etiam locis, prout diversis occasionibus
conscripta sunt, repetuntur non nulla.

Ejus probatio ab ,g Longum hie esset singula inde excerpere in ordinem redacta : satis erit exscribere

mductione satis ,. Jy . °_ , ~ . P . r ,-, -n • i • • j

ampia. dissertatioms De lege Continuitatis numerum 138. Post mductionem petitam praecedente

numero a Geometria, quae nullum uspiam habet saltum, atque a motu locali, in quo nunquam
ab uno loco ad alium devenitur, nisi ductu continue aliquo, unde consequitur illud, dis-
tantiam a dato loco nunquam mutari in aliam, neque densitatem, quae utique a distantiis
pendet particularum in aliam, nisi transeundo per intermedias ; fit gradus in eo numero ad
motuum velocitates, & ductus, quas magis hie ad rem faciunt, nimirum ubi de velocitate
agimus non mutanda per saltum in corporum collisionibus. Sic autem habetur : " Quin
immo in motibus ipsis continuitas servatur etiam in eo, quod motus omnes in lineis continuis
fiunt nusquam abruptis. Plurimos ejusmodi motus videmus. Planetae, & cometse in lineis
continuis cursum peragunt suum, & omnes retrogradationes fiunt paullatim, ac in stationibus
semper exiguus quidem motus, sed tamen habetur semper, atque hinc etiam dies paullatim
per auroram venit, per vespertinum crepusculum abit, Solis diameter non per saltum, sed
continuo motu supra horizontem ascendit, vel descendit. Gravia itidem oblique projecta
in lineis itidem pariter continuis motus exercent suos, nimirum in parabolis, seclusa ^aeris
resistentia, vel, ea considerata, in orbibus ad hyperbolas potius accedentibus, & quidem
semper cum aliqua exigua obliquitate projiciuntur, cum infinities infinitam improbabilitatem
habeat motus accurate verticalis inter infinities infinitas inclinationes, licet exiguas, & sub
sensum non cadentes, fortuito obvenienfe, qui quidem motus in hypothesi Telluris^motae a
parabolicis plurimum distant, & curvam continuam exhibent etiam pro casu projectionis
accurate verticalis, quo, quiescente penitus Tellure, & nulla ventorum vi deflectente motum,
haberetur [17] ascensus rectilineus, vel descensus. Immo omnes alii motus a gravitate
pendentes, omnes ab elasticitate, a vi magnetica, continuitatem itidem servant ; cum earn
servent vires illse ipsae, quibus gignuntur. Nam gravitas, cum decrescat in ratione reciproca
duplicata distantiarum, & distantise per saltum mutari non possint, mutatur per omnes
intermedias magnitudines. Videmus pariter, vim magneticam a distantiis pendere lege
continua ; vim elasticam ab inflexione, uti in laminis, vel a distantia, ut in particulis aeris
compressi. In iis, & omnibus ejusmodi viribus, & motibus, quos gignunt, continuitas habetur
semper, tarn in lineis quae describuntur, quam in velocitatibus, quae pariter per omnes
intermedias magnitudines mutantur, ut videre est in pendulis, in ascensu corporum gravium,

A THEORY OF NATURAL PHILOSOPHY 55

becomes N'M' by the subtraction of O'N') ; moreover, if the line CDE, which represents
the law of variation, cuts the straight AB, which is the axis of time, in any point, then the
magnitude can vanish at that point (just as the ordinate M'N' would vanish when the
point M' coincided with D), & be changed into a negative magnitude PQ, or RS, that is
to say one having an opposite direction ; & this, the more it increases in the opposite sense,
the less it is to be considered in the former sense (just as in the idea of property or riches,
a man goes on continuously getting worse off, when, after everything he had has been
taken away from him, he continues to get deeper & deeper into debt). In Geometry too
we have this passage from positive to negative, & also in algebraical formulae, the passage
being made not only through nothing, but also through infinity ; such I have discussed,
the one in a dissertation added to my Conic Sections, the other in my Algebra (§ 14), & both
of them together in my essay De Lege Continuitatis ; but in Physics, where no quantity
ever increases to an infinite extent, the second case has no place ; hence, unless the passage
is made through the value nothing, there is no passage from positive to negative, or vice
versa. Although, as I point out below, this nothing is not really nothing in itself, but a
certain real state ; & it may be considered as nothing only in a certain sense. In the same
sense, too, negatives, which are true states, are positive in themselves, although, as they
belong to the first set in a certain negative way, they are called negative.

38. Thus explained & defended, the Law of Continuity is considered by most philoso- I propose to prove
phers to exist in Nature, though there are some who deny it, as I mentioned above. I, LaVof^Continuity6
when first I investigated the matter, considered that it was absolutely impossible that it
should be left out of account, if we have regard to the unparalleled analogy that there is
with Nature & to the power of induction ; & by the help of this induction I endeavoured
to prove the law in several of the dissertations that I have mentioned, & Benvenutus also
used the same form of proof in his Synopsis (Art. 119). In these too, as they were written
on several different occasions, there are some repetitions.

39. It would take too long to extract & arrange in order here each of the passages in Proof by induction
these essays ; it will be sufficient if I give Art. 138 of the dissertation De Lege Continuitatis. s~^^ for the
After induction derived in the preceding article from geometry, in which there is no sudden
change anywhere, & from local motion, in which passage from one position to another
never takes place unless by some continuous progress (the consequence of which is that a
distance from any given position can never be changed into another distance, nor the
density, which depends altogether on the distances between the particles, into another density,
except by passing through intermediate stages), the step is made in that article to the
velocities of motions, & deductions, which have more to do with the matter now in hand,
namely, where we are dealing with the idea that the velocity is not changed suddenly in the
collision of solid bodies. These are the words : " Moreover in motions themselves
continuity is preserved also in the fact that all motions take place in continuous lines that
are not broken anywhere. We see a great number of motions of this kind. The planets &
the comets pursue their courses, each in its own continuous line, & all retrogradations are
gradual ; & in stationary positions the motion is always slight indeed, but yet there is
always some ; hence also daylight comes gradually through the dawn, & goes through the
evening twilight, as the diameter of the sun ascends above the horizon, not suddenly, but
by a continuous motion, & in the same manner descends. Again heavy bodies projected
obliquely follow their courses in lines also that are just as continuous ; namely, in para-
bolae, if we neglect the resistance of the air, but if that is taken into account, then in orbits
that are more nearly hyperbolae. Now, they are always projected with some slight obli-
quity, since there is an infinitely infinite probability against accurate vertical motion, from
out of the infinitely infinite number of inclinations (although slight & not capable of being
observed), happening fortuitously. These motions are indeed very far from being para-
bolae, if the hypothesis that the Earth is in motion is adopted. They give a continuous
curve also for the case of accurate vertical projection, in which, if the Earth were at rest,
& no wind-force deflected the motion, rectilinear ascent & descent would be obtained.
All other motions that depend on gravity, all that depend upon elasticity, or magnetic
force, also preserve continuity ; for the forces themselves, from which the motions arise,
preserve it. For gravity, since it diminishes in the inverse ratio of the squares of the dis-
tances, & the distances cannot be changed suddenly, is itself changed through every inter-
mediate stage. Similarly we see that magnetic force depends on the distances according
to a continuous law ; that elastic force depends on the amount of bending as in plates, or
according to distance as in particles of compressed air. In these, & all other forces of the
sort, & in the motions that arise from them, we always get continuity, both as regards the
lines which they describe & also in the velocities which are changed in similar manner
through all intermediate magnitudes ; as is seen in pendulums, in the ascent of heavy

56 PHILOSOPHISE NATURALIS THEORIA

& in aliis mille ejusmodi, in quibus mutationes velocitatis fiunt gradatim, nee retro cursus
reflectitur, nisi imminuta velocitate per omnes gradus. Ea diligentissime continuitatem
servat omnia. Hinc nee ulli in naturalibus motibus habentur anguli, sed semper mutatio
directionis fit paullatim, nee vero anguli exacti habentur in corporibus ipsis, in quibus
utcunque videatur tennis acies, vel cuspis, microscopii saltern ope videri solet curvatura,
quam etiam habent alvei fluviorum semper, habent arborum folia, & frondes, ac rami, habent
lapides quicunque, nisi forte alicubi cuspides continuae occurrant, vel primi generis, quas
Natura videtur affectare in spinis, vel secundi generis, quas videtur affectare in avium
unguibus, & rostro, in quibus tamen manente in ipsa cuspide unica tangente continuitatem
servari videbimus infra. Infinitum esset singula persequi, in quibus continuitas in Natura
observatur. Satius est generaliter provocare ad exhibendum casum in Natura, in quo
eontinuitas non servetur, qui omnino exhiberi non poterit."

Duplex inductionis 40. Inductio amplissima turn ex hisce motibus, ac velocitatibus, turn ex aliis pluribus

vimhabeatittductio exemPn's> <lU3e habemus in Natura, in quibus ea ubique, quantum observando licet depre-
incompieta. hendere, continuitatem vel observat accurate, vel affcctat, debet omnino id efficere, ut ab

ea ne in ipsa quidem corporum collisione recedamus. Sed de inductionis natura, & vi, ac
ejusdem usu in Physica, libet itidem hie inserere partem numeri 134, & totum 135, disserta-
tionis De Lege Continuitatis. Sic autem habent ibidem : " Inprimis ubi generales Naturae
leges investigantur, inductio vim habet maximam, & ad earum inventionem vix alia ulla
superest via. Ejus ope extensionem, figurabilitem, mobilitatem, impenetrabilitatem
corporibus omnibus tribuerunt semper Philosophi etiam veteres, quibus eodem argumento
inertiam, & generalem gravitatem plerique e recentioribus addunt. Inductio, ut demon-
strationis vim habeat, debet omnes singulares casus, quicunque haberi possunt percurrere.
Ea in Natu-[i8]-rae legibus stabiliendis locum habere non potest. Habet locum laxior
qusedam inductio, quae, ut adhiberi possit, debet esse ejusmodi, ut inprimis in omnibus iis
casibus, qui ad trutinam ita revocari possunt, ut deprehendi debeat, an ea lex observetur,
eadem in iis omnibus inveniatur, & ii non exiguo numero sint ; in reliquis vero, si quse prima
fronte contraria videantur, re accuratius perspecta, cum ilia lege possint omnia conciliari ;
licet, an eo potissimum pacto concilientur, immediate innotescere, nequaquam possit. Si
eae conditiones habeantur ; inductio ad legem stabiliendam censeri debet idonea. Sic quia
videmus corpora tarn multa, quae habemus prae manibus, aliis corporibus resistere, ne in
eorum locum adveniant, & loco cedere, si resistendo sint imparia, potius, quam eodem
perstare simul ; impenetrabilitatem corporum admittimus ; nee obest, quod qusedam
corpora videamus intra alia, licet durissima, insinuari, ut oleum in marmora, lumen in
crystalla, & gemmas. Videmus enim hoc phsenomenum facile conciliari cum ipsa impene-
trabilitate, dicendo, per vacuos corporum poros ea corpora permeare. (Num. 135).
Praeterea, qusecunque proprietates absolutae, nimirum quae relationem non habent ad
nostros sensus, deteguntur generaliter in massis sensibilibus corporum, easdem ad quascunque
utcunque exiguas particulas debemus transferre ; nisi positiva aliqua ratio obstet, & nisi sint
ejusmodi, quae pendeant a ratione totius, seu multitudinis, contradistincta a ratione partis.
Primum evincitur ex eo, quod magna, & parva sunt respectiva, ac insensibilia dicuntur ea,
quse respectu nostrae molis, & nostrorum sensuum sunt exigua. Quare ubi agitur de
proprietatibus absolutis non respectivis, quaecunque communia videmus in iis, quse intra
limites continentur nobis sensibiles, ea debemus censere communia etiam infra eos limites :
nam ii limites respectu rerum, ut sunt in se, sunt accidentales, adeoque siqua fuisset analogise
Isesio, poterat ilia multo facilius cadere intra limites nobis sensibiles, qui tanto laxiores sunt,
quam infra eos, adeo nimirum propinquos nihilo. Quod nulla ceciderit, indicio est, nullam
esse. Id indicium non est evidens, sed ad investigationis principia pertinet, quae si juxta

A THEORY OF NATURAL PHILOSOPHY 57

bodies, & in a thousand other things of the same kind, where the changes of velocity occur
gradually, & the path is not retraced before the velocity has been diminished through all
degrees. All these things most strictly preserve continuity. Hence it follows that no
sharp angles are met with in natural motions, but in every case a change of direction occurs
gradually ; neither do perfect angles occur in bodies themselves, for, however fine an edge
or point in them may seem, one can usually detect curvature by the help of the microscope
if nothing else. We have this gradual change of direction also in the beds of rivers, in the
leaves, boughs & branches of trees, & stones of all kinds ; unless, in some cases perchance,
there may be continuous pointed ends, either of the first kind, which Nature is seen to
affect in thorns, or of the second kind, which she is seen to do in the claws & the beak of
birds ; in these, however, we shall see below that continuity is still preserved, since we are
left with a single tangent at the extreme end. It would take far too long to mention every
single thing in which Nature preserves the Law of Continuity ; it is more than sufficient
to make a general statement challenging the production of a single case in Nature, in which
continuity is not preserved ; for it is absolutely impossible for any such case to be brought
forward."

40. The effect of the very complete induction from such motions as these & velocities, induction of a two-
as well as from a large number of other examples, such as we have in Nature, where Nature *old , kil\d '• when

e c ,...-..& why incomplete

in every case, as far as can be gathered from direct observation, maintains continuity or induction has vaii-

tries to do so, should certainly be that of keeping us from neglecting it even in the case

of collision of bodies. As regards the nature & validity of induction, & its use in Physics,

I may here quote part of Art. 134 & the wjiole of Art. 135 from my dissertation De Lege

Continuitatis, The passage runs thus : " Especially when we investigate the general laws

of Nature, induction has very great power ; & there is scarcely any other method beside

it for the discovery of these laws. By its assistance, even the ancient philosophers attributed

to all bodies extension, figurability, mobility, & impenetrability ; & to these properties,

by the use of the same method of reasoning, most of the later philosophers add inertia &

universal gravitation. Now, induction should take account of every single case that can

possibly happen, before it can have the force of demonstration ; such induction as this has no

place in establishing the laws of Nature. But use is made of an induction of a less rigorous

type ; in order that this kind of induction may be employed, it must be of such a nature

that in all those cases particularly, which can be examined in a manner that is bound to

lead to a definite conclusion as to whether or no the law in question is followed, in all of

them the same result is arrived at ; & that these cases are not merely a few. Moreover,

in the other cases, if those which at first sight appeared to be contradictory, on further &

more accurate investigation, can all of them be made to agree with the law ; although,

whether they can be made to agree in this way better than in any other whatever, it is

impossible to know directly anyhow. If such conditions obtain, then it must be considered

that the induction is adapted to establishing the law. Thus, as we see that so many of

the bodies around us try to prevent other bodies from occupying the position which they

themselves occupy, or give way to them if they are not capable of resisting them, rather

than that both should occupy the same place at the same time, therefore we admit the

impenetrability of bodies. Nor is there anything against the idea in the fact that we see

certain bodies penetrating into the innermost parts of others, although the latter are very

hard bodies ; such as oil into marble, & light into crystals & gems. For we see that this

phenomenon can very easily be reconciled with the idea of impenetrability, by supposing

that the former bodies enter and pass through empty pores in the latter bodies (Art.

135). In addition, whatever absolute properties, for instance those that bear no relation

to our senses, are generally found to exist in sensible masses of bodies, we are bound to

attribute these same properties also to all small parts whatsoever, no matter how small

they may be. That is to say, unless some positive reason prevents this ; such as that they

are of such a nature that they depend on argument having to do with a body as a whole,

or with a group of particles, in contradistinction to an argument dealing with a part only.

The proof comes in the first place from the fact that great & small are relative terms, &

those things are called insensible which are very small with respect to our own size & with

regard to our senses. Therefore, when we consider absolute, & not relative, properties,

whatever we perceive to be common to those contained within the limits that are sensible

to us, we should consider these things to be still common to those beyond those limits.

For these limits, with regard to such matters as are self-contained, are accidental ; & thus,

if there should be any violation of the analogy, this would be far more likely to happen

between the limits sensible to us, which are more open, than beyond them, where indeed

they are so nearly nothing. Because then none did happen thus, it is a sign that there is

none. This sign is not evident, but belongs to the principles of investigation, which

generally proves successful if it is carried out in accordance with certain definite wisely

5 8 PHILOSOPHIC NATURALIS THEORIA

quasdam prudentes regulas fiat, successum habere solet. Cum id indicium fallere possit ;
fieri potest, ut committatur error, sed contra ipsum errorem habebitur praesumptio, ut
etiam in jure appellant, donee positiva ratione evincatur oppositum. Hinc addendum fuit,
nisi ratio positiva obstet. Sic contra hasce regulas peccaret, qui diceret, corpora quidem
magna compenetrari, ac replicari, & inertia carere non posse, compenetrari tamen posse, vel
replicari, vel sine inertia esse exiguas eorum partes. At si proprietas sit respectiva, respectu
nostrorum sensuum, ex [19] eo, quod habeatur in majoribus massis, non debemus inferre,
earn haberi in particulis minoribus, ut est hoc ipsum, esse sensibile, ut est, esse coloratas,
quod ipsis majoribus massis competit, minoribus non competit ; cum ejusmodi magnitudinis
discrimen, accidentale respectu materiae, non sit accidentale respectu ejus denominationis
sensibile, coloratum. Sic etiam siqua proprietas ita pendet a ratione aggregati, vel totius, ut
ab ea separari non possit ; nee ea, ob rationem nimirum eandem, a toto, vel aggregate debet
transferri ad partes. Est de ratione totius, ut partes habeat, nee totum sine partibus haberi
potest. Est de ratione figurabilis, & extensi, ut habeat aliquid, quod ab alio distet, adeoque,
ut habeat partes ; hinc eae proprietates, licet in quovis aggregate particularum materiae,
sive in quavis sensibili massa, inveniantur, non debent inductionis vi transferri ad particulas
quascunque."

Et impenetrabili- 41. Ex his patet, & impenetrabilitatem, & continuitatis legem per ejusmodi inductionis

ultatem tvtad""pCT genus abunde probari, atque evinci, & illam quidem ad quascunque utcunque exiguas

inductionem : "* ad particulas corporum, hanc ad gradus utcunque exiguos momento temporis adjectos debere

ipsam quid requu-a- exten(jj< Requiritur autem ad hujusmodi inductionem primo, ut ilia proprietas, ad quam

probandam ea adhibetur, in plurimis casibus observetur, aliter enim probabilitas esset exigua ;

& ut nullus sit casus observatus, in quo evinci possit, earn violari. Non est necessarium illud,

ut in iis casibus, in quibus primo aspectu timeri possit defectus proprietatis ipsius, positive

demonstretur, earn non deficere ; satis est, si pro iis casibus haberi possit ratio aliqua

conciliandi observationem cum ipsa proprietate, & id multo magis, si in aliis casibus habeatur

ejus conciliationis exemplum, & positive ostendi possit, eo ipso modo fieri aliquando

conciliationem.

Ejus appiicatio ad 42. Id ipsum fit, ubi per inductionem impenetrabilitas corporum accipitur pro generali

impenetrab;htatem. jege ]sjaturaEi Nam impenetrabilitatem ipsam magnorum corporum observamus in exemplis
sane innumeris tot corporum, quae pertractamus. Habentur quidem & casus, in quibus earn
violari quis credent, ut ubi oleum per ligna, & marmora penetrat, atque insinuatur, & ubi
lux per vitra, & gemmas traducitur. At praesto est conciliatio phasnomeni cum impenetra-
bilitate, petita ab eo, quod ilia corpora, in quse se ejusmodi substantiae insinuant, poros
habeant, quos 633 permeent. Et quidem haec conciliatio exemplum habet manifestissimum
in spongia, quae per poros ingentes aqua immissa imbuitur. Poros marmorum illorum, &
multo magis vitrorum, non videmus, ac multo minus videre possumus illud, non insinuari
eas substantias nisi per poros. Hoc satis est reliquae inductionis vi, ut dicere debeamus, eo
potissimum pacto se rem habere, & ne ibi quidem violari generalem utique impenetrabilitatis
legem.

simiiisad continu- [2O] 43- Eodem igitur pacto in lege ipsa continuitatis agendum est. Ilia tarn ampla

itatem : duo cas- inductio, quam habemus, debet nos movere ad illam generaliter admittendam etiam pro iis

quibus ean<videatur casibus, in quibus determinare immediate per observations non possumus, an eadem

lacdi- habeatur, uti est collisio corporum ; ac si sunt casus nonnulli, in quibus eadem prima fronte

violari videatur ; ineunda est ratio aliqua, qua ipsum phsenomenum cum ea lege conciliari

possit, uti revera potest. Nonnullos ejusmodi casus protuli in memoratis dissertationibus,

quorum alii ad geometricam continuitatem pertinent, alii ad physicam. In illis prioribus

non immorabor ; neque enim geometrica continuitas necessaria est ad hanc physicam

propugnandam, sed earn ut exemplum quoddam ad confirmationem quandam inductionis

majoris adhibui. Posterior, ut saepe & ilia prior, ad duas classes reducitur ; altera est eorum

casuum, in quibus saltus videtur committi idcirco, quia nos per saltum omittimus intermedias

quantitates : rem exemplo geometrico illustro, cui physicum adjicio.

A THEORY OF NATURAL PHILOSOPHY 59

chosen rules. Now, since the indication may possibly be fallacious, it may happen that an
error may be made ; but there is presumption against such an error, as they call it in law,
until direct evidence to the contrary can be brought forward. Hence we should add :
unless some positive argument is against it. Thus, it would be offending against these rules
to say that large bodies indeed could not suffer compenetration, or enfolding, or be deficient
in inertia, but yet very small parts of them could suffer penetration, or enfolding, or be
without inertia. On the other hand, if a property is relative with respect to our senses,
then, from a result obtained for the larger masses we cannot infer that the same is to be
obtained in its smaller particles ; for instance, that it is the same thing to be sensible, as
it is to be coloured, which is true in the case of large masses, but not in the case of small
particles ; since a distinction of this kind, accidental with respect to matter, is not accidental
with respect to the term sensible or coloured. So also if any property depends on an argu-
ment referring to an aggregate, or a whole, in such a way that it cannot be considered
apart from the whole, or the aggregate ; then, neither must it (that is to say, by that same
argument), be transferred from the whole, or the aggregate, to parts of it. It is on account
of its being a whole that it has parts ; nor can there be a whole without parts. It is on
account of its being figurable & extended that it has some thing that is apart from some
other thing, & therefore that it has parts. Hence those properties, although they are
found in any aggregate of particles of matter, or in any sensible mass, must not however be
transferred by the power of induction to each & every particle."

41. From what has been said it is quite evident that both impenetrability & the Law Both impenetra-
of Continuity can be proved by a kind of induction of this type ; & the former must be c^bf dTm""^
extended to all particles of bodies, no matter how small, & the latter to all additional steps, strated by indue-
however small, made in an instant of time. Now, in the first place, to use this kind of quired for thisSpur-
induction, it is required that the property, for the proof of which it is to be used, must be pose.
observed in a very large number of cases ; for otherwise the probability would be very

small. Also it is required that no case should be observed, in which it can be proved that
it is violated. It is not necessary that, in those cases in which at first sight it is feared that
there may be a failure of the property, that it should be directly proved that there is no
failure. It is sufficient if in those cases some reason can be obtained which will make the
observation agree with the property ; & all the more so, if in other cases an example of
reconciliation can be obtained, & it can be positively proved that sometimes reconciliation
can be obtained in that way.

42. This is just what does happen, when the impenetrability of solid bodies is accepted Application of in-
as a law of Nature through inductive reasoning. For we observe this impenetrability of tr'abiuty.*0 lmpene"
large bodies in innumerable examples of the many bodies that we consider. There are

indeed also cases, in which one would think that it was violated, such as when oil penetrates
wood and marble, & works its way through them, or when light passes through glasses &
gems. But we have ready a means of making these phenomena agree with impenetrability,
derived from the fact that those bodies, into which substances of this kind work their way,
possess pores which they can permeate. There is a very evident example of this recon-
ciliation in a sponge, which is saturated with water introduced into it by means of huge
pores. We do not see the pores of the marble, still less those of glass ; & far less can we see
that these substances do not penetrate except by pores. It satisfies the general force of
induction if we can say that the matter can be explained in this way better than in any
other, & that in this case there is absolutely no contradiction of the general law of impene-
trability.

43. In the same way, then, we must deal with the Law of Continuity. The full Similar application

"

induction that we possess should lead us to admit in general this law even in those cases in ^sisses "oT cases

which it is impossible for us to determine directly by observation whether the same law which there seems

holds good, as for instance in the collision of bodies. Also, if there are some cases in which *

the law at first sight seems to be violated, some method must be followed, through which

each phenomenon can be reconciled with the law, as is in every case possible. I brought

forward several cases of this kind in the dissertations I have mentioned, some of which

pertained to geometrical continuity, & others to physical continuity. I will not delay over

the first of these : for geometrical continuity is not necessary for the defence of the physical

variety ; I used it as an example in confirmation of a wider induction. The latter, as well

as very frequently the former, reduces to two classes ; & the first of these classes is that class

in which a sudden change seems to have been made on account of our having omitted the

intermediate quantities with a jump. I give a geometrical illustration, and then add one

in physics.

6o

PHILOSOPHISE NATURALIS THEORIA

Exemplum geome-
tricum primi gene-
ris, ubi nos inter-
mcdias magnitu-
dines omittimus.

Quando id accidat
exempla physica
dierum, & oscilla-
tionum consequen-
tium.

44. In axe curvae cujusdam in fig. 4. sumantur segmenta AC, CE, EG aequalia, &
erigantur ordinatae AB, CD, EF, GH. Area; BACD, DCEF, FEGH videntur continue
cujusdam seriei termini ita, ut ab ilia BACD acl DCEF, & inde ad FEGH immediate
transeatur, & tamen secunda a prima, ut

& tertia a secunda, differunt per quanti-
tates finitas : si enim capiantur CI, EK
sequales BA, DC, & arcus BD transferatur
in IK ; area DIKF erit incrementum se-
cundae supra primam, quod videtur imme-
diate advenire totum absque eo, quod
unquam habitum sit ejus dimidium, vel
quaevis alia pars incrementi ipsius ; ut idcirco
a prima ad secundam magnitudinem areae
itum sit sine transitu per intermedias. At
ibi omittuntur a nobis termini intermedii,

qui continuitatem servant ; si enim ac aequalis FIG. 4.

AC motu continue feratur ita, ut incipiendo

ab AC desinat in CE ; magnitude areae BACD per omnes intermedias bacd abit in magnitu-
dinem DCEF sine ullo saltu, & sine ulla violatione continuitatis.

45. Id sane ubique accidit, ubi initium secundae magnitudinis aliquo intervallo distal
ab initio primas ; sive statim veniat post ejus finem, sive qua vis alia lege ab ea disjungatur.
Sic in pliysicis, si diem concipiamus intervallum temporis ab occasu ad occasum, vel etiam
ab ortu ad occasum, dies praecedens a sequent! quibusdam anni temporibus differt per plura
secunda, ubi videtur fieri saltus sine ullo intermedio die, qui minus differat. At seriem
quidem continuam ii dies nequaquam constituunt. Concipiatur parallelus integer Telluris,
in quo sunt continuo ductu disposita loca omnia, quae eandem latitudinem geographicam
habent ; ea singula loca suam habent durationem diei, & omnium ejusmodi dierum initia,
ac fines continenter fluunt ; donee ad eundem redeatur locum, cujus pr£e-[2i]-cedens dies
est in continua ilia serie primus, & sequens postremus. Illorum omnium dierum magni-
tudines continenter fluunt sine ullo saltu : nos, intermediis omissis, saltum committimus
non Natura. Atque huic similis responsio est ad omnes reliquos casus ejusmodi, in quibus
initia, & fines continenter non fluunt, sed a nobis per saltum accipiuntur. Sic ubi pendulum
oscillat in acre ; sequens oscillatio per finitam magnitudinem distat a praecedente ; sed &
initium & finis ejus finite intervallo temporis distat a prascedentis initio, & fine, ac intermedii
termini continua serie fluente a prima oscillatione ad secundam essent ii, qui haberentur, si
primae, & secundae oscillationis arcu in aequalem partium numerum diviso, assumeretur via
confecta, vel tempus in ea impensum, inter jacens inter fines partium omnium proportion-
alium, ut inter trientem, vel quadrantem prioris arcus, & trientem,vel quadrantem posterioris,
quod ad omnes ejus generis casus facile transferri potest, in quibus semper immediate etiam
demonstrari potest illud, continuitatem nequaquam violari.

Exempla secundi 46. Secunda classis casuum est ea, in qua videtur aliquid momento temporis peragi,

atne«iOTimeUtSsed & tamen peragitur tempore successive, sed perbrevi. Sunt, qui objiciant pro violatione
non momento' tem- continuitatis casum, quo quisquam manu lapidem tenens, ipsi statim det velocitatem
quandam finitam : alius objicit aquae e vase effluentis, foramine constitute aliquanto infra
superficiem ipsius aquae, velocitatem oriri momento temporis finitam. At in priore casu
admodum evidens est, momento temporis velocitatem finitam nequaquam produci. Tempore
opus est, utcunque brevissimo, ad excursum spirituum per nervos, & musculos, ad fibrarum
tensionem, & alia ejusmodi : ac idcirco ut velocitatem aliquam sensibilem demus lapidi,
manum retrahimus, & ipsum aliquandiu, perpetuo accelerantes, retinemus. Sic etiam, ubi
tormentum bellicum exploditur, videtur momento temporis emitti globus, ac totam
celeritatem acquirere ; at id successive fieri, patet vel inde, quod debeat inflammari tota
massa pulveris pyrii, & dilatari aer, ut elasticitate sua globum acceleret, quod quidem fit
omnino per omnes gradus. Successionem multo etiam melius videmus in globe, qui ab
elastro sibi relicto propellatur : quo elasticitas est major, eo citius, sed nunquam momento
temporis velocitas in globum inducitur.

AppUcatio ipsorum 47. Hsec exempla illud praestant, quod aqua per pores spongiae ingressa respectu

ad emuxum1IaquK impenetrabilitatis, ut ea responsione uti possimus in aliis casibus omnibus, in quibus accessio
e vase. aliqua magnitudinis videtur fieri tota momento temporis ; ut nimirum dicamus fieri tempore

A THEORY OF NATURAL PHILOSOPHY 61

44. In the axis of any curve (Fig. 4) let there be taken the segments AC, CE, EG equal Geometrical ex-
to one another ; & let the ordinates AB, CD, EF, GH be erected. The areas BACD, DCEF, kind where6 Ivl
FEGH seem to be terms of some continuous series such that we can pass directly from BACD omit . intermediate
to DCEF and then on to FEGH, & yet the second differs from the first, & also the third from

the second, by a finite quantity. For if CI, EK are taken equal to BA, DC, & the arc BD
is transferred to the position IK ; then the area DIKF will be the increment of the second
area beyond the first ; & this seems to be directly arrived at as a whole without that which
at any one time is considered to be the half of it, or indeed any other part of the increment
itself : so that, in consequence, we go from the first to the second magnitude of area without
passing through intermediate magnitudes. But in this case we omit intermediate terms
which maintain the continuity ; for if ac is equal to AC, & this is carried by a continuous
motion in such a way that, starting from the position AC it ends up at the position CE,
then the magnitude of the area BACD will pass through all intermediate values such as
bacd until it reaches the magnitude of the area DCEF without any sudden change, & hence
without any breach of continuity.

45. Indeed this always happens when the beginning of the second magnitude is distant when this will
by a definite interval from the beginning of the first ; whether it comes immediately after happen = physical
the end of the first or is disconnected from it by some other law. Thus in physics, if we casTof Consecutive
look upon the day as the interval of time between sunset & sunset, or even between sunrise day^ OI. consecutive
& sunset, the preceding day differs from that which follows it at certain times of the year

by several seconds ; in which case we see that there is a sudden change made, without there
being any intermediate day for which the change is less. But the fact is that these days do
not constitute a continuous series. Let us consider a complete parallel of latitude on the
Earth, along which in a continuous sequence are situated all those places that have the same
geographical latitude. Each of these places has its own duration of the day, & the begin-
nings & ends of days of this kind change uninterruptedly ; until we get back again to the
same place, where the preceding day is the first of that continuous series, & the day that fol-
lows is the last of the series. The magnitudes of all these days continuously alter without there
being any sudden change : it was we who, by omitting the intermediates, made the sudden
change, & not Nature. Similar to this is the answer to all the rest of the cases of the same
kind, in which the beginnings & the ends do not change uninterruptedly, but are observed by
us discontinuously. Similarly, when a pendulum oscillates in air, the oscillation that follows
differs from the oscillation that has gone before by a finite magnitude. But both the begin-
ning & the end of the second differs from the beginning & the end of the first by a finite inter-
val of time ; & the intermediate terms in a continuously varying series from the first oscillation
to the second would be those that would be obtained, if the arcs of the first & second oscilla-
tions were each divided into the same number of equal parts, & the path traversed (or the
time spent in traversing the path) is taken between the ends of all these proportional paths ;
such as that between the third or fourth part of the first arc & the third or fourth part
of the second arc. This argument can be easily transferred so as to apply to all cases of this
kind ; & in such cases it can always be directly proved that there is no breach of continuity.

46. The second class of cases is that in which something seems to have been done in an Examples of the
instant of time, but still it is really done in a continuous, but very short, interval of time. ^iS? the^chan'e
There are some who bring forward, as an objection in favour of a breach of continuity, the is very rapid, but
case in which a man, holding a stone in his hand, gives to it a definite velocity all at once ; f^an^nstant^of
another raises an objection that favours a breach of continuity, in the case of water flowing time.

from a vessel, where, if an opening is made below the level of the surface of the water, a
finite velocity is produced in an instant of time. But in the first case it is perfectly clear
that a finite velocity is in no wise produced in an instant of time. For there is need of
time, although this is exceedingly short, for the passage of cerebral impulses through
the nerves and muscles, for the tension of the fibres, and other things of that sort ; and
therefore, in order to give a definite sensible velocity to the stone, we draw back the hand,
and then retain the stone in it for some time as we continually increase its velocity forwards.
So too when an engine of war is exploded, the ball seems to be driven forth and to acquire
the whole of its speed in an instant of time. But that it is done continuously is clear, if
only from the fact that the whole mass of the gunpowder has to be inflamed and the gas
has to be expanded in order that it may accelerate the ball by its elasticity ; and this latter
certainly takes place by degrees. The continuous nature of this is far better seen in the
case of a ball propelled by releasing a spring ; here the stronger the elasticity, the greater
the speed ; but in no case is the speed imparted to the ball in an instant of time.

47. These examples are superior to that of water entering through the pores of a sponge, Application of
which we employed in the matter of impenetrability ; so that we can make use of this reply *£.gss.e particularly
in all other cases in which some addition to a magnitude seems to have taken place entirely in to the flow of water
an instant of time. Thus, without doubt we may say that it takes place in an exceedingly from a vesse1'

62

PHILOSOPHIC NATURALIS THEORIA

brevissimo, utique per omnes intermedias magnitudines, ac illsesa penitus lege continuitatis.
Hinc & in aquae effluentis exemplo res eodem redit, ut non unico momento, sed successive
aliquo tempore, & per [22] omnes intermedias magnitudines progignatur velocitas, quod
quidem ita se habere optimi quique Physici affirmant. Et ibi quidem, qui momento
temporis omnem illam velocitatem progigni, contra me affirmet, principium utique, ut
ajunt, petat, necesse est. Neque enim aqua, nisi foramen aperiatur, operculo dimoto,
effluet ; remotio vero operculi, sive manu fiat, sive percussione aliqua, non potest fieri
momento temporis, sed debet velocitatem suam acquirere per omnes gradus ; nisi illud
ipsum, quod quaerimus, supponatur jam definitum, nimirum an in collisione corporum
communicatio motus fiat momento temporis, an per omnes intermedios gradus, & magni-
tudines. Verum eo omisso, si etiam concipiamus momento temporis impedimentum
auferri, non idcirco momento itidem temporis omnis ilia velocitas produceretur ; ilia enim
non a percussione aliqua, sed a pressione superincumbentis aquae orta, oriri utique non
potest, nisi per accessiones continuas tempusculo admodum parvo, sed non omnino nullo :
nam pressio tempore indiget, ut velocitatem progignat, in communi omnium sententia.

Transitus ad meta-

continuis

ut in Geometria.

48. Illaesa igitur esse debet continuitatis lex, nee ad earn evertendam contra inductionem,
tam uberem quidquam poterunt casus allati hucusque, vel iis similes. At ejusdem con-
umcus, tinuitatis aliam metaphysicam rationem adinveni, & proposui in dissertatione De Lege
Continuitatis, petitam ab ipsa continuitatis natura, in qua quod Aristoteles ipse olim
notaverat, communis esse debet limes, qui praecedentia cum consequentibus conjungit, qui
idcirco etiam indivisibilis est in ea ratione, in qua est limes. Sic superficies duo solida
dirimens & crassitudine caret, & est unica, in qua immediatus ab una parte fit transitus ad
aliam ; linea dirimens binas superficiei continuae partes latitudine caret ; punctum continuae
lineae segmenta discriminans, dimensione omni : nee duo sunt puncta contigua, quorum
alterum sit finis prioris segmenti, alterum initium sequentis, cum duo contigua indivisibilia,
& inextensa haberi non possint sine compenetratione, & coalescentia quadam in unum.

idem in tempore 49. Eodem autem pacto idem debet accidere etiam in tempore, ut nimirum inter tempus

«> ti ua^'evide"6 contmuum praecedens, & continuo subsequens unicum habeatur momentum, quod sit
tius in quibusdam. indivisibilis terminus utriusque ; nee duo momenta, uti supra innuimus, contigua esse
possint, sed inter quodvis momentum, & aliud momentum debeat intercedere semper
continuum aliquod tempus divisibile in infinitum. Et eodem pacto in quavis quantitate,
quae continuo tempore duret, haberi debet series quasdam magnitudinum ejusmodi, ut
momento temporis cuivis respondeat sua, quae praecedentem cum consequente conjungat,
& ab ilia per aliquam determinatam magnitudinem differat. Quin immo in illo quantitatum
genere, in quo [23] binae magnitudines simul haberi non possunt, id ipsum multo evidentius
conficitur, nempe nullum haberi posse saltum immediatum ab una ad alteram. Nam illo
momento temporis, quo deberet saltus fieri, & abrumpi series accessu aliquo momentaneo,
deberent haberi duae magnitudines, postrema seriei praecedentis, & prima seriei sequentis.
Id ipsum vero adhuc multo evidentius habetur in illis rerum statibus, in quibus ex una
parte quovis momento haberi debet aliquis status ita, ut nunquam sine aliquo ejus generis
statu res esse possit ; & ex alia duos simul ejusmodi status habere non potest.

inde cur motus ip- ro> \& quidem satis patebit in ipso locali motu, in quo habetur phsenomenum omnibus

calls non fiat, nisi •> . . * , . r r. ...... \ ,. . , . .

per Hneam contin- sane notissimum, sed cujus ratio non ita facile ahunde redditur, inde autem patentissima est,
Corpus a quovis loco ad alium quemvis devenire utique potest motu continuo per lineas
quascunque utcunque contortas, & in immensum productas quaquaversum, quae numero
infinities infinitae sunt : sed omnino debet per continuam aliquam abire, & nullibi inter-
ruptam. En inde rationem ejus rei admodum manifestam. Si alicubi linea motus abrum-
peretur ; vel momentum temporis, quo esset in primo puncto posterioris lineae, esset
posterius eo momento, quo esset in puncto postremo anterioris, vel esset idem, vel anterius ?
In primo, & tertio casu inter ea momenta intercederet tempus aliquod continuum divisibile
in infinitum per alia momenta intermedia, cum bina momenta temporis, in eo sensu accepta,
in quo ego hie ea accipio, contigua esse non possint, uti superiusexposui. Quamobrem in

A THEORY OF NATURAL PHILOSOPHY 63

short interval of time, and certainly passes through every intermediate magnitude, and that
the Law of Continuity is not violated. Hence also in the case of water flowing from a
vessel it reduces to the same example : so that the velocity is generated, not in a single
instant, but in some continuous interval of time, and passes through all intermediate magni-
tudes ; and indeed all the most noted physicists assert that this is what really happens.
Also in this matter, should anyone assert in opposition to me that the whole of the speed
is produced in an instant of time, then he must use a •petitio principii, as they call it. For
the water can-not flow out, unless the hole is opened, & the lid removed ; & the removal of
the lid, whether done by hand or by a blow, cannot be effected in an instant of time, but
must acquire its own velocity by degrees ; unless we suppose that the matter under investi-
gation is already decided, that is to say, whether in collision of bodies communication of
motion takes place in an instant of time or through all intermediate degrees and magnitudes.
But even if that is left out of account, & if also we assume that the barrier is removed
in an instant of time, none the more on that account would the whole of the velocity
also be produced in an instant of time ; for it is impossible that such velocity can arise,
not from some blow, but from a pressure arising from the superincumbent water, except by
continuous additions in a very short interval of time, which is however not absolutely
nothing ; for pressure requires time to produce velocity, according to the general opinion
of everybody.

48. The Law of Continuity ought then to be subject to no breach, nor will the cases Passing to a meta-
hitherto brought forward, nor others like them, have any power at all to controvert this haveT'smrie'iinUt
law in opposition to induction so copious. Moreover I discovered another argument, a in the case of con-
metaphysical one, in favour of this continuity, & published it in my dissertation De Lege g'^n^iy1"11^' &S "*
Continuitatis, having derived it from the very nature of continuity ; as Aristotle himself long

ago remarked, there must be a common boundary which joins the things that precede to
those that follow ; & this must therefore be indivisible for the very reason that it is a
boundary. In the same way, a surface of separation of two solids is also without thickness
& is single, & in it there is immediate passage from one side to the other ; the line of
separation of two parts of a continuous surface lacks any breadth ; a point determining
segments of a continuous line has no dimension at all ; nor are there two contiguous points,
one of which is the end of the first segment, & the other the beginning of the next ; for
two contiguous indivisibles, of no extent, cannot possibly be considered to exist, unless
there is compenetration & a coalescence into one.

49. In the same way, this should also happen with regard to time, namely, that between similarly for time
a preceding continuous time & the next following there should be a single instant, which ^£y. mor^evi-
is the indivisible boundary of either. There cannot be two instants, as we intimated above, dent in some than
contiguous to one another ; but between one instant & another there must always intervene m others-

some interval of continuous time divisible indefinitely. In the same way, in any quantity
which lasts for a continuous interval of time, there must be obtained a series of magnitudes
of such a kind that to each instant of time there is its corresponding magnitude ; & this
magnitude connects the one that precedes with the one that follows it, & differs from the
former by some definite magnitude. Nay even in that class of quantities, in which we
cannot have two magnitudes at the same time, this very point can be deduced far more
clearly, namely, that there cannot be any sudden change from one to another. For at that
instant, when the sudden change should take place, & the series be broken by some momen-
tary definite addition, two magnitudes would necessarily be obtained, namely, the last of
the first series & the first of the next. Now this very point is still more clearly seen in those
states of things, in which on the one hand there must be at any instant some state so that
at no time can the thing be without some state of the kind, whilst on the other hand it can
never have two states of the kind simultaneously.

50. The above will be sufficiently clear in the case of local motion, in regard to which Hence the reason
the phenomenon is perfectly well known to all ; the reason for it, however, is not so easily ^Jj^ Recurs™;:!10"
derived from any other source, whilst it follows most clearly from this idea. A body can continuous line,
get from any one position to any other position in any case by a continuous motion along

any line whatever, no matter how contorted, or produced ever so far in any direction ;
these lines being infinitely infinite in number. But it is bound to travel by some continuous
line, with no break in it at any point. Here then is the reason of this phenomenon quite
clearly explained. If the motion in the line should be broken at any point, either the
instant of time, at which it was at the first point of the second part of the line, would be
after the instant, at which it was at the last point of the first part of the line, or it would
be the same instant, or before it. In the first & third cases, there would intervene between
the two instants some definite interval of continuous time divisible indefinitely at other
intermediate instants ; for two instants of time, considered in the sense in which I have

PHILOSOPHIC NATURALIS THEORIA

primo casu in omnibus iis infinitis intermediis momentis nullibi esset id corpus, in secundo
casu idem esset eodem illo memento in binis locis, adeoque replicaretur ; in terio haberetur
replicatio non tantum respectu eorum binorum momentorum, sed omnium etiam inter-
mediorum, in quibus nimirum omnibus id corpus esset in binis locis. Cum igitur corpus
existens nee nullibi esse possit, nee simul in locis pluribus ; ilia vias mutatio, & ille saltus
haberi omnino non possunt.

51. Idem ope Geometric magis adhuc oculis ipsis subjicitur. Exponantur per rectam
AB tempora, ac per ordinatas ad lineas CD, EF, abruptas alicubi, diversi status rei cujuspiam.
e metaphysica, Ductis ordinatis DG, EH, vel punctum H iaceret post G, ut in Fie. c : vel cum ipso

ibus exemphs . • /• i • ij . r T . o J • r

congrueret, ut in 6 ; vel ipsum prsccederet, ut in 7. In pnmo casu nulla responderet
ordinata omnibus punctis rectae GH ; in secundo binae responderent GD, & HE eidem puncto
G ; in tertio vero binae HI, & HE puncto H, binas GD, GK puncto G, & binae LM, LN

Illustratio ejus
i ex Geo-
ratiocina-

tione

pluribus exempl

D E.

D

G H

FIG. 5.

B A

GH

FIG. 6.

H L G

FIG. 7.

puncto cuivis intermedio L ; nam ordinata est relatio quaedam distantly, quam habet
punctum curvae cum puncto axis sibi respondente, adeoque ubi jacent in recta eadem
perpendiculari axi bina curvarum puncta, habentur binae ordinatae respondentes eidem
puncto axis. Quamobrem si nee o-[24]-mni statu carere res possit, nee haberi possint
status simul bini ; necessario consequitur, saltum ilium committi non posse. Saltus ipse, si
deberet accidere, uti vulgo fieri concipitur, accideret binis momentis G, & H, quae sibi in
fig. 6 immediate succederent sine ullo immediato hiatu, quod utique fieri non potest ex
ipsa limitis ratione, qui in continuis debet esse idem, & antecedentibus, & consequentibus
communis, uti diximus. Atque idem in quavis reali serie accidit ; ut hie linea finita sine
puncto primo, & postremo, quod sit ejus limes, & superficies sine linea esse non potest ; unde
fit, ut in casu figurae 6 binae ordinatae necessario respondere debeant eidem puncto : ita in
quavis finita reali serie statuum primus terminus, & postremus haberi necessario debent ;
adeoque si saltus fit, uti supra de loco diximus ; debet eo momento, quo saltus confici
dicitur, haberi simul status duplex ; qui cum haberi non possit : saltus itidem ille haberi
omnino non potest. Sic, ut aliis utamur exemplis, distantia unius corporis ab alio mutari
per saltum non potest, nee densitas, quia dux simul haberentur distantiae, vel duae densitates,
quod utique sine replicatione haberi non potest ; caloris itidem, & frigoris mutatio in
thermometris, ponderis atmosphaerae mutatio in barometris, non fit per saltum, quia binae
simul altitudines mercurii in instrumento haberi deberent eodem momento temporis, quod
fieri utique non potest ; cum quovis momento determinate unica altitude haberi debeat,
ac unicus determinatus caloris gradus, vel frigoris ; quae quidem theoria innumeris casibus
pariter aptari potest.

52. Contra hoc argumentum videtur primo aspectu adesse aliquid, quod ipsum pforsus
non esse conjun- evertat, & tamen ipsi illustrando idoneum est maxime. Videtur nimirum inde erui,

gend s in creatione • •« M • • • • o • • £»• • • J

& annihiiatione, ac impossibilem esse & creationem rei cujuspiam, Scintentum. 01 enim conjungendus est
ejus soiutio. postremus terminus praecedentis seriei cum primo sequentis ;" in ipso transitu a non esse ad

esse, vel vice versa, debebit utrumque conjungi, ac idem simul erit, & non erit, quod est
absurdum. Responsio in promptu est. Seriei finita; realis, & existentis, reales itidem, &
existentes termini esse debent ; non vero nihili, quod nullas proprietates habet, quas exigat,
Hinc si realium statuum seriei altera series realium itidem statuum succedat, quae non
sit communi termino conjuncta ; bini eodem momento debebuntur status, qui nimirum
sint bini limites earundem. At quoniam non esse est merum nihilum ; ejusmodi series
limitem nullum extremum requirit, sed per ipsum esse immediate, & directe excluditur.
Quamobrem primo, & postremo momento temporis ejus continui, quo res est, erit utique,
nee cum hoc esse suum non esse conjunget simul ; at si densitas certa per horam duret, turn
momento temporis in aliam mutetur duplam, duraturam itidem per alteram sequentem
horam ; momento temporis, [25] quod horas dirimit, binae debebunt esse densitates simul,
nimirum & simplex, & dupla, quae sunt reales binarum realium serierum termini.

Objectio ab esse, &

A THEORY OF NATURAL PHILOSOPHY 65

considered them, cannot be contiguous, as I explained above. Wherefore in the first case,
at all those infinite intermediate instants the body would be nowhere at all ; in the second
case, it would be at the same instant in two different places & so there would be replication.
In the third case, there would not only occur replication in respect of these two instants
but for all those intermediate to them as well, in all of which the body would forsooth be
in two places at the same time. Since then a body that exists can never be nowhere, nor
in several places at one & the same time, there can certainly be no alteration of path & no
sudden change.

51. The same thing can be visualized better with the aid of Geometry. illustration of this

Let times be represented by the straight line AB, & diverse states of any thing by SSyT^STS
ordinates drawn to meet the lines CD, EF, which are discontinuous at some point. If the reasoning being
ordmates DG, EH are drawn, either the point H will fall after the point G, as in Fig. 5 ;
or it will coincide with it, as in Fig. 6 ; or it will fall before it, as in Fig. 7. In the first
case, no ordinate will correspond to any one of the points of the straight line GH ; in the
second case, GD and HE would correspond to the same point G ; in the third case, two
ordinates, HI, HE, would correspond to the same point H, two, GD, GK, to the same
point G, and two, LM, LN, to any intermediate point L. Now the ordinate is some relation
as regards distance, which a point on the curve bears to the point on the axis that corresponds
with it ; & thus, when two points of the curve lie in the same straight line perpendicular
to the axis, we have two ordinates corresponding to the same point of the axis. Wherefore,
if the thing in question can neither be without some state at each instant, nor is it possible
that there should be two states at the same time, then it necessarily follows that the sudden
change cannot be made. For this sudden change, if it is bound to happen, would take place
at the two instants G & H, which immediately succeed the one the other without any direct
gap between them ; this is quite impossible, from the very nature of a limit, which should
be the same for,& common to, both the antecedents & the consequents in a continuous set,
as has been said. The same thing happens in any series of real things ; as in this case there
cannot be a finite line without a first & last point, each to be a boundary to it, neither can
there be a surface without a line. Hence it comes about that in the case of Fig. 6 two
ordinates must necessarily correspond to the same point. Thus, in any finite real series of
states, there must of necessity be a first term & a last ; & so if a sudden change is made, as
we said above with regard to position, there must be at the instant, at which the sudden
change is said to be accomplished, a twofold state at one & the same time. Now since this
can never happen, it follows that this sudden change is also quite impossible. Similarly, to
make use of other illustrations, the distance of one body from another can never be altered
suddenly, no more can its density ; for there would be at one & the same time two distances,
or two densities, a thing which is quite impossible without replication. Again, the change
of heat, or cold, in thermometers, the change in the weight of the air in barometers, does
not happen suddenly ; for then there would necessarily be at one & the same time two
different heights for the mercury in the instrument ; & this could not possibly be the case.
For at any given instant there must be but one height, & but one definite degree of heat,
& but one definite degree of cold ; & this argument can be applied just as well to innu-
merable other cases.

52. Against this argument it would seem at first sight that there is something ready to
hand which overthrows it altogether ; whilst as a matter of fact it is peculiarly fitted to together of existence
exemplify it. It seems that from this argument it follows that both the creation of any * non-existence a.t

•> • „ • i • • -11 rf >r T i <• -i i • the time of creation

thing, & its destruction, are impossible, r or, it the last term of a series that precedes is to Or annihilation ; &

be connected with the first term of the series that follows, then in the passage from a state its solution.

of existence to one of non-existence, or vice versa, it will be necessary that the two are

connected together ; & then at one & the same time the same thing will both exist & not

exist, which is absurd. The answer to this is immediate. For the ends of a finite series

that is real & existent must themselves be real & existent, not such as end up in absolute

nothing, which has no properties. Hence, if to one series of real states there succeeds

another series of real states also, which is not connected with it by a common term, then

indeed there must be two states at the same instant, namely those which are their two

limits. But since non-existence is mere nothing, a series of this kind requires no last limiting

term, but is immediately & directly cut off by fact of existence. Wherefore, at the first &

at the last instant of that continuous interval of time, during which the matter exists, it will

certainly exist ; & its non-existence will not be connected with its existence simultaneously.

On the other hand if a given density persists for an hour, & then is changed in an instant

of time into another twice as great, which will last for another hour ; then in that instant

of time which separates the two hours, there would have to be two densities at one & the

same time, the simple & the double, & these are real terms of two real series.

66

PHILOSOPHIC NATURALIS THEORIA

Unde hue transfer-
enda solutio ipsa.

Solutio petita ex
geometrico exem-
plo.

Solutio
physica
atione.

ex meta-
consider-

Illustratio ulterior
geometrica.

Applicatio ad crea-
tionem, & annihi-
lationem.

D

F

i

\

F

D

f

m m*

\

G

G'

P

L

5

\

MJVI,

'

A

B

C E H H'E'C7

FIG. 8.

53. Id ipsum in dissertatione De lege virium in Natura existentium satis, ni fallor,
luculenter exposui, ac geometricis figuris illustravi, adjectis nonnullis, quae eodem recidunt,
& quae in applicatione ad rem, de qua agimus, & in cujus gratiam haec omnia ad legem con-
tinuitatis pertinentia allata sunt, proderunt infra ; libet autem novem ejus dissertationis
numeros hue transferre integros, incipiendo ab octavo, sed numeros ipsos, ut & schematum
numeros mutabo hie, ut cum superioribus consentiant.

54. " Sit in fig. 8 circulus GMM'wz, qui referatur ad datam rectam AB per ordinatas
HM ipsi rectae perpendiculares ; uti itidem perpendiculares sint binae tangentes EGF,
E'G'F'. Concipiantur igitur recta quaedam indefinita ipsi rectse AB perpendicularis, motu
quodam continuo delata ab A ad B. Ubi ea habuerit, positionem quamcumque GD, quae
praecedat tangentem EF, vel C'D', quae consequatur tangentem E'F' ; ordinata ad circulum
nulla erit, sive erit impossibilis, & ut Geometrae

loquuntur, imaginaria. Ubicunque autem ea sit

inter binas tangentes EGF, E'G'F', in HI, HT,

occurret circulo in binis punctis M, m, vel M', m',

& habebitur valor ordinate HM, HOT, vel H'M',

H'm'. Ordinata quidem ipsa respondet soli inter-

vallo EE' : & si ipsa linea AB referat tempus ;

momentum E est limes inter tempus praecedens

continuum AE, quo ordinata non est, & tempus

continuum EE' subsequens, quo ordinata est ; punc-

tum E' est limes inter tempus praecedens EE', quo

ordinata est, & subsequens E'B, quo non est. Vita

igitur quaedam ordinatae est tempus EE' ; ortus

habetur in E, interitus in E'. Quid autem in

ipso ortu, & interitu ? Habetur-ne quoddam esse

ordinatas, an non esse ? Habetur utique esse, nimi-

rum EG, vel E'G', non autem non esse. Oritur

tota finitae magnitudinis ordinata EG, interit tota finite magnitudinis E'G', nee tamen

ibi conjungit esse, & non esse, nee ullum absurdum secum trahit. Habetur momento E

primus terminus seriei sequentis sine ultimo seriei praecedentis, & habetur momento E'

ultimus terminus seriei praecedentis sine primo termino seriei sequentis."

55. " Quare autem id ipsum accidat, si metaphysica consideratione rem perpendimus,
statim patebit. Nimirum veri nihili nullae sunt verae proprietates : entis realis verae, &
reales proprietates sunt. Quaevis realis series initium reale debet, & finem, sive primum, &
ultimum terminum. Id, quod non est, nullam habet veram proprietatem, nee proinde sui
generis ultimum terminum, aut primum exigit. Series praecedens ordinatae nullius, ultimum
terminum non [26] habet, series consequens non habet primum : series realis contenta
intervallo EE', & primum habere debet, & ultimum. Hujus reales termini terminum ilium
nihili per se se excludunt, cum ipsum esse per se excludat non esse."

56. " Atque id quidem manifestum fit magis : si consideremus seriem aliquam
praecedentem realem, quam exprimant ordinatae ad lineam continuam PLg, quae respondeat
toti tempori AE ita, ut cuivis momento C ejus temporis respondeat ordinata CL. Turn
vero si momento E debeat fieri saltus ab ordinata Eg ad ordinatam EG : necessario ipsi
momento E debent respondere binae ordinatae EG, Eg. Nam in tota linea PLg non potest
deesse solum ultimum punctum g ; cum ipso sublato debeat adhuc ilia linea terminum
habere suum, qui terminus esset itidem punctum : id vero punctum idcirco fuisset ante
contiguum puncto g, quod est absurdum, ut in eadem dissertatione De Lege Continuitatis
demonstravimus. Nam inter quodvis punctum, & aliud punctum linea aliqua interjacere
debet ; quae si non inter jaceat ; jam ilia puncta in unicum coalescunt. Quare non potest
deesse nisi lineola aliqua gL ita, ut terminus seriei praecedentis sit in aliquo momento C
praecedente momentum E, & disjuncto ab eo per tempus quoddam continuum, in cujus
temporis momentis omnibus ordi'nata sit nulla."

57. " Patet igitur discrimen inter transitum a vero nihilo, nimirum a quantitate
imaginaria, ad esse, & transitum ab una magnitudine ad aliam. In primo casu terminus
nihili non habetur ; habetur terminus uterque seriei veram habentis existentiam, & potest
quantitas, cujus ea est series, oriri, vel occidere quantitate finita, ac per se excludere non esse.
In secundo casu necessario haberi debet utriusque seriei terminus, alterius nimirum postre-
mus, alterius primus. Quamobrem etiam in creatione, & in annihilatione potest quantitas
oriri, vel interire magnitudine finita, & primum, ac ultimum esse erit quoddam esse, quod
secum non conjunget una non esse. Contra vero ubi magnitude realis ab una quantitate ad

A THEORY OF NATURAL PHILOSOPHY 67

c*. I explained this very point clearly enough, if I mistake not, in my dissertation The s0"166 from

n i • • • JIT- . • ' . • a T -11 j v i ... • i A 'IT ^ which the solution

D,? lege vmum in Natura existentium, & 1 illustrated it by geometrical figures ; also I made u to be borrowed.

some additions that reduced to the same thing. These will appear below, as an application

to the matter in question ; for the sake of which all these things relating to the Law of

Continuity have been adduced. It is allowable for me to quote in this connection the

whole of nine articles from that dissertation, beginning with Art. 8 ; but I will here

change the numbering of the articles, & of the diagrams as well, so that they may agree

with those already given.

54. " In Fig. 8, let GMM'm be a circle, referred to a given straight line AB as axis, by Sotoion derived
means of ordinates HM drawn perpendicular to that straight line ; also let the two tan- exampief"
gents EGF, E'G'F' be perpendiculars to the axis. Now suppose that an unlimited straight

line perpendicular to the axis AB is carried with a continuous motion from A to B. When
it reaches some such position as CD preceding the tangent EF, or as C'D' subsequent to
the tangent E'F', there will be no ordinate to the circle, or it will be impossible &, as the
geometricians call it, imaginary. Also, wherever it falls between the two tangents EGF,
E'G'F', as at HI or HT, it will meet the circle in two points, M, m or M', m' ; & for the
value of the ordinate there will be obtained HM & Hm, or H'M' & H'm'. Such an ordinate
will correspond to the interval EE' only ; & if the line AB represents time, the instant E
is the boundary between the preceding continuous time AE, in which the ordinate does
not exist, £ the subsequent continuous time EE', in which the ordinate does exist. The
point E' is the boundary between the preceding time EE', in which the ordinate does exist,
& the subsequent time E'B, in which it does not ; the lifetime, as it were, of the ordinate,
is EE' ; its production is at E & its destruction at E'. But what happens at this production
& destruction ? Is it an existence of the ordinate, or a non-existence I Of a truth there
is an existence, represented by EG & E'G', & not a non-existence. The whole ordinate EG
of finite magnitude is produced, & the whole ordinate E'G' of finite magnitude is destroyed;
& yet there is no connecting together of the states of existence & non-existence, nor does it
bring in anything absurd in its train. At the instant E we get the first term of the sub-
sequent series without the last term of the preceding series ; & at the instant E' we have
the last term of the preceding series without the first term of the subsequent series."

55. " The reason why this should happen is immediately evident, if we consider the Sol«tion from a
matter metaphysically. Thus, to absolute nothing there belong no real properties ; but Sderatwn!*

the properties of a real absolute entity are also real. Any real series must have a real
beginning & end, or a first term & a last. That which does not exist can have no true
property ; & on that account does not require a last term of its kind, or a first. The
preceding series, in which there is no ordinate, does not have a last term ; & the subsequent
series has likewise no first term ; whilst the real series contained within the interval EE'
must have both a first term & a last term. The real terms of this series of themselves
exclude the term of no value, since the fact of existence of itself excludes non-existence"

56. " This indeed will be still more evident, if we consider some preceding series of Further illustration

i • • 11 i • i i i • T.T r „ i i • by geometry.

real quantities, expressed by the ordinates to the curved line PLg ; & let this curve
correspond to the whole time AE in such a way that to every instant C of the time there
corresponds an ordinate CL. Then, if at the instant E there is bound to be a sudden
change from the ordinate Eg to the ordinate EG, to that instant E there must of necessity
correspond both the ordinates EG, Eg. For it is impossible that in the whole line PLg
the last point alone should be missing ; because, if that point is taken away, yet the line
is Bound to have an end to it, & that end must also be a point ; hence that point would be
before & contiguous to the point g ; & this is absurd, as we have shown in the same
dissertation De Lege Continuitatis. For between any one point & any other point there
must lie some line ; & if such a line does not intervene, then those points must coalesce
into one. Hence nothing can be absent, except it be a short length of line gL, so that
the end of the series that precedes occurs at some instant, C, preceding the instant E, &
separated from it by an interval of continuous time, at all instants of which there is no
ordinate."

157. "Evidently, then, there is a distinction between passing from absolute nothing, Application to crea-

•f' . '' . ... °. , . Y tion& annihilation.

i.e., from an imaginary quantity, to a state of existence, & passing from one magnitude
to another. In the first case the term which is naught is not reckoned in ; the term at
either end of a series which has real existence is given, & the quantity, of which it is the
series, can be produced or destroyed, finite in amount ; & of itself it will exclude non-
existence. In the second case, there must of necessity be an end to either series, namely
the last of the one series & the first of the other. Hence, in creation & annihilation,
a quantity can be produced or destroyed, finite in magnitude ; & the first & last
state of existence will be a state of existence of some kind ; & this will not associate with
itself a state of non-existence. But, on the other hand, where a real magnitude is bound

68

PHILOSOPHIC NATURALIS THEORIA

Aliquando videri
nihtium id, quod
est aliquid.

Ordinatam nullam,
ut & distantiam
nullam existentium
esse compenetra-
tionem.

Ad idem pertinere
seriei realis genus
earn distan t i a m
nullam, & aliquam.

Alia, quje videntur
nihil, & sunt ali-
quid : discrimen
inter radicem ima-
ginariam, & zero.

aliam transire debet per saltum ; momento temporis, quo saltus committitur, uterque
terminus haberi deberet. Manet igitur illaesum argumentum nostrum metaphysicum pro
exclusione saltus a creatione & annihilatione, sive ortu, & interitu."

58. "At hie illud etiam notandum est ; quoniam ad ortum, & interitum considerandum
geometricas contemplationes assumpsimus, videri quidem prima fronte, aliquando etiam
realis seriei terminum postremum esse nihilum ; sed re altius considerata, non erit vere
nihilum ; sed status quidam itidem realis, & ejusdem generis cum prsecedentibus, licet alio
nomine insignitus."

[27] 59. " Sit in Fig. 9. Linea AB, ut prius, ad quam linea qusedam PL deveniat in G
(pertinet punctum G ad lineam PL, E ad AB continuatas, & sibi occurrentes ibidem), & sive
pergat ultra ipsam in GM', sive retro resiliat per GM'. Recta CD habebit ordinatam CL,
quas evanescet, ubi puncto C abeunte in E, ipsa CD abibit in EF, turn in positione ulteriori
rectse perpendicularis HI, vel abibit in nega-
tivam HM, vel retro positiva regredietur
in HM'. Ubi linea altera cum altera coit,
& punctum E alterius cum alterius puncto
G congreditur, ordinata CL videtur abire in
nihilum ita, ut nihilum, quemadmodum &
supra innuimus, sit limes quidam inter seriem
ordinatarum positivarum CL, & negativarum
HM ; vel positivarum CL, & iterum posi-
tivarum HM'. Sed, si res altius considere-
tur ad metaphysicum conceptum reducta,
in situ EF non habetur verum nihilum.
In situ CD, HI habetur distantia quaedam
punctorum C, L ; H, M : in situ EF
habetur eorundem punctorum compene-

tratio. Distantia est relatio quaedam FJG

binorum modorum, quibus bina puncta

existunt ; compenetratio itidem est relatio binorum modorum, quibus ea existunt,
quae compenetratio est aliquid reale ejusdem prorsus generis, cujus est distantia, constituta
nimirum per binos reales existendi modos."

60. " Totum discrimen est in vocabulis, quae nos imposuimus. Bini locales existendi
modi infinitas numero relationes possunt constituere, alii alias. Hae omnes inter se &
differunt, & tamen simul etiam plurimum conveniunt ; nam reales sunt, & in quodam genere
congruunt, quod nimirum sint relationes ortae a binis localibus existendi modis. Diversa
vero habent nomina ad arbitrarium instituta, cum alise ex ejusmodi relationibus, ut CL,
dicantur distantiae positivae, relatio EG dicatur compenetratio, relationes HM dicantur
distantiae negativse. Sed quoniam, ut a decem palmis distantiae demptis 5, relinquuntur 5,
ita demptis aliis 5, habetur nihil (non quidem verum nihil, sed nihil in ratione distantiae a
nobis ita appellatae, cum remaneat compenetratio) ; ablatis autem aliis quinque, remanent
quinque palmi distantiae negativae ; ista omnia realia sunt, & ad idem genus pertinent ; cum
eodem prorsus modo inter se differant distantia palmorum 10 a distantia palmorum 5, haec
a distantia nulla, sed reali, quas compenetrationem importat, & haec a distantia negativa
palmorum 5. Nam ex prima ilia quantitate eodem modo devenitur ad hasce posteriores per
continuam ablationem palmorum 5. Eodem autem pacto infinitas ellipses, ab infinitis
hyperbolis unica interjecta parabola discriminat, quae quidem unica nomen peculiare sortita
est, cum illas numero infinitas, & a se invicem admodum discrepantes unico vocabulo com-
plectamur ; licet altera magis oblonga ab altera minus oblonga plurimum itidem diversa sit."

[28] 61. " Et quidem eodem pacto status quidam realis est quies, sive perseverantia in
eodem modo locali existendi ; status quidam realis est velocitas nulla puncti existentis.
nimirum determinatio perseverandi in eodem loco ; status quidam realis puncti existentis
est vis nulla, nimirum determinatio retinendi praecedentem velocitatem, & ita porro ;
plurimum haec discrepant a vero non esse. Casus ordinatae respondentis lineae EF in fig. 9,
differt plurimum a casu ordinatae circuli respondentis lineae CD figurae 8 : in prima existunt
puncta, sed compenetrata, in secunda alterum punctum impossible est. Ubi in solutione
problematum devenitur ad quantitatem primi generis, problema determinationem peculiarem
accipit ; ubi devenitur ad quantitatem secundi generis, problema evadit impossible ; usque
adeo in hoc secundo casu habetur verum nihilum, omni reali proprietate carens ; in illo
primo habetur aliquid realibus proprietatibus praeditum, quod ipsis etiam solutionibus
problematum, & constructionibus veras sufficit, & reales determinationes ; cum realis, non
imaginaria sit radix equationis cujuspiam, quae sit = o, sive nihilo aequalis."

A THEORY OF NATURAL PHILOSOPHY 69

to pass suddenly from one quantity to another, then at the instant in which the sudden
change is accomplished, both terms must be obtained. Hence, our argument on
metaphysical grounds in favour of the exclusion of a sudden change from creation or
annihilation, or production & destruction, remains quite unimpaired."

58. " In this connection the following point must be noted. As we have used geometrical Sometimes what is
ideas for the consideration of production & destruction, it seems also that sometimes reallysomethingap-
the last term of a real series is nothing. But if we go deeper into the matter, we find

that it is not in reality nothing, but some state that is also real and of the same kind as
those that precede it, though designated by another name."

59. " In Fig. 9, let AB be a line, as before, which some line PL reaches at G (where the When the ordinate
point G belongs to the line PL, & E to the line AB, both being produced to meet one whe^thT'dlst^n'13
another at this point) ; & suppose that PL either goes on beyond the point as GM, or between two exis-
recoils along GM'. Then the straight line CD will contain the ordinate CL, which will ^ tJ1™gs .u no"

_ & „, . . . ' . thing, there is com-

vanish when, as the point L, gets to H, L-D attains the position r,r ; & after that, in the penetration.

further position of the perpendicular straight line HI, will either pass on to the negative

ordinate HM or return, once more positive, to HM'. Now when the one line meets the

other, & the point E of the one coincides with the point G of the other, the ordinate

CL seems to run off into nothing in such a manner that nothing, as we remarked above,

is a certain boundary between the series of positive ordinates CL & the negative ordinates

HM, or between the positive ordinates CL & the ordinates HM' which are also positive.

But if the matter is more deeply considered & reduced to a metaphysical concept, there

is not an absolute nothing in the position EF. In the position CD, or HI, we have given

a certain distance between the points C,L, or H,M ; in the position EF, there is

compenetration of these points. Now distance is a relation between the modes of existence

of two points ; also compenetration is a relation between two modes of existence ; &

this compenetration is something real of the very same nature as distance, founded as it is

on two real modes of existence."

60. " The whole difference lies in the words that we have given to the things in question. ™s ' no ' distance
Two local modes of existence can constitute an infinite number of relations, some of one kmdT^f °series "of
sort & some of another. All of these differ from one another, & yet agree with one real quantities as

•i • i • i j r ia • • j • i • • j j ' some ' distance.

another in a high degree ; ior they are real & to a certain extent identical, since indeed
they are all relations arising from a pair of local modes of existence. But they have different
names assigned to them arbitrarily, so that some of the relations of this kind, as CL, are
called positive distances, the relation EG is called compenetration, & relations like HM
are called negative distances. But, just as when five palms of distance are taken away
from ten palms, there are left five palms, so when five more are taken away, there is nothing
left (& yet not really nothing, but nothing in comparison with what we usually call
distance ; for compenetration is left). Again, if we take away another five, there remain
five palms of negative distance. All of these are real & belong to the same class ; for
they differ amongst themselves in exactly the same way, namely, the distance of ten palms
from the distance of five palms, the latter from ' no ' distance (which however is something
real that denotes compenetration), & this again from a negative distance of five palms.
For starting with the first quantity, the others that follow are obtained in the same manner,
by a continual subtraction of five palms. In a similar manner a single intermediate
parabola discriminates between an infinite number of ellipses & an infinite number of
hyperbolas ; & this single curve receives a special name, whilst under the one term we include
an infinite number of them that to a certain extent are all different from one another,
although one that is considerably elongated may be very different from another that is
less elongated."

61. "In the same way, rest, i.e., a perseverance in the same mode of local existence, other things that
is some real state ; so is ' no ' velocity a real state of an existent point, namely, a propensity ^ndVet^re^eaJi^
to remain in the same place ; so also is ' no ' force a real state of an existent point, namely, something ; d i s-
a propensity to retain the velocity that it has already; & so on. All these differ from a'~"

a state of non-existence in the highest degree. The case of the ordinate corresponding & zero/
to the line EF in Fig. 9 differs altogether from the case of the ordinate of the circle
corresponding to the line CD in Fig. 8. In the first there exist two points, but there is
compenetration of these points ; in the other case, the second point cannot possibly exist.
When, in the solution of problems, we arrive at a quantity of the first kind, the problem
receives a special sort of solution ; but when the result is a quantity of the second kind,
the problem turns out to be incapable of solution. So much indeed that, in this second case,
there is obtained a true nothing that lacks every real property ; in the first case, we get
something endowed with real properties, which also supplies true & real values to the
solutions & constructions of the problems. For the root of any equation that = o, or is
equal to nothing, is something that is real, & is not an imaginary thing."

70 PHILOSOPHIC NATURALIS THEORIA

Conciusip prosolu- fa. " Firmum igitur manebit semper. & stabile, seriem realem quamcunque. quas

tione ejus objec- . ~ °. , , , v » „ ... a i • r

contmuo tempore finito duret, debere habere £ primum prmcipium, & ultimum nnem
realem, sine ullo absurdo, & sine conjunctione sui esse cum non esse, si forte duret eo solo
tempore : dum si prascedenti etiam exstitit tempore, habere debet & ultimum terminum
seriei praecedentis, & primum sequentis, qui debent esse unicus indivisibilis communis limes,
ut momentum est unicus indivisibilis limes inter tempus continuum praecedens, & subsequens.
Sed haec de ortu, & interitu jam satis."

Apphcatio leg is ft- ijt igitur contrahamus iam vela, continuitatis lex & inductione, & metaphysico

contmuitatis ad J , ° , . . • i • • • • • . .. . . r ' .

coiiisionem corpo- argumento abunde nititur, quas idcirco etiam in velocitatis commumcatione retmeri omnmo
rum- debet, ut nimirum ab una velocitate ad aliam numquam transeatur, nisi per intermedias

velocitates omnes sine saltu. Et quidem in ipsis motibus, & velocitatibus inductionem
habuimus num. 39, ac difficultates solvimus num. 46, & 47 pertinentes ad velocitates, quae
videri possent mutatse per saltum. Quod autem pertinet ad metaphysicum argumentum, si
toto tempore ante contactum subsequentis corporis superficies antecedens habuit 12 gradus
velocitatis, & sequenti 9, saltu facto momentaneo ipso initio contactus ; in ipso momento ea
tempora dirimente debuisset habere & 12, & 9 simul, quod est absurdum. Duas enim
velocitates simul habere corpus non potest, quod ipsum aliquanto diligentius demonstrabo.

DUO velocitatum g, Velocitatis nomen, uti passim usurpatur a Mechanicis, asquivocum est; potest

genera, potentials, T r r . T. . r

& actuaiis. enim sigmncare velocitatem actuaiem, quas nimirum est relatio quaedam in motu asquabm

spatii percursi divisi per tempus, quo percurritur ; & potest significare [29] quandam, quam
apto Scholiasticorum vocabulo potentialem appello, quae nimirum est determinatio, ad
actuaiem, sive determinatio, quam habet mobile, si nulla vis mutationem inducat, percur-
rendi motu asquabili determinatum quoddam spatium quovis determinato tempore, quas
quidem duo & in dissertatione De Viribus Fivis, & in Stayanis Supplements distinxi,
distinctione utique .necessaria ad aequivocationes evitandas. Prima haberi non potest
momento temporis, sed requirit tempus continuum, quo motus fiat, & quidem etiam motum
aequabilem requirit ad accuratam sui mensuram ; secunda habetur etiam momento quovis
determinata ; & hanc alteram intelligunt utique Mechanici, cum scalas geometricas effor-
mant pro motibus quibuscunque difformibus, sive abscissa exprimente tempus, & ordinata
velocitatem, utcunque etiam variatam, area exprimat spatium : sive abscissa exprimente
itidem tempus, & ordinata vim, area exprimat velocitatem jam genitam, quod itidem in aliis
ejusmodi scalis, & formulis algebraicis fit passim, hac potentiali velocitate usurpata, quas sit
tantummodo determinatio ad actuaiem, quam quidem ipsam intelligo, ubi in collisione
corporum earn nego mutari posse per saltum ex hoc posteriore argumento.

^5' Jam vero velocitates actuales non posse simul esse duas in eodem mobili, satis patet ;
potentials 'simul quia oporteret, id mobile, quod initio dati cujusdam temporis fuerit in dato spatii puncto,
ne^eturn<vei exf<*a- ^n omnibus sequentibus occupare duo puncta ejusdem spatii, ut nimirum spatium percursum
tur compenetratfo. sit duplex, alterum pro altera velocitate determinanda, adeoque requireretur actuaiis
replicatio, quam non haberi uspiam, ex principio inductionis colligere sane possumus
admodum facile. Cum nimirum nunquam videamus idem mobile simul ex eodem loco
discedere in partes duas, & esse simul in duobis locis ita, ut constet nobis, utrobique esse illud
idem. At nee potentiales velocitates duas simul esse posse, facile demonstratur. Nam
velocitas potentialis est determinatio ad existendum post datum tempus continuum quodvis
in dato quodam puncto spatii habente datam distantiam a puncto spatii, in quo mobile est
eo temporis momento, quo dicitur habere illam potentialem velocitatem determinatam.
Quamobrem habere simul illas duas potentiales velocitates est esse determinatum ad occu-
panda eodem momento temporis duo puncta spatii, quorum singula habeant suam diversam
distantiam ab eo puncto spatii, in quo turn est mobile, quod est esse determinatum ad
replicationem habendam momentis omnibus sequentis temporis. Dicitur utique idem
mobile a diversis causis acquirere simul diversas velocitates, sed eae componuntur in unicam
ita, ut singulas constituant statum mobilis, qui status respectu dispositionum, quas eo
momento, in quo turn est, habet ipsum mobile, complectentium omnes circumstantias
praeteritas, & praesentes, est tantummodo conditionatus, non absolutus ; nimirum ut con-
tineant determi-[3o]-nationem, quam ex omnibus praeteritis, & praesentibus circumstantiis
haberet ad occupandum illud determinatum spatii punctum determinato illo momento

A THEORY OF NATURAL PHILOSOPHY 71

62. "Hence in all cases it must remain a firm &stable conclusion that any real series, Conclusion in

,.,, , c. . . .1 i, c i • • r- i favour of a solution

which lasts for some finite continuous time, is bound to have a first beginning & a final Of this difficulty.

end, without any absurdity coming in, & without any linking up of its existence with

a state of non-existence, if perchance it lasts for that interval of time only. But if it existed

at a previous time as well, it must have both a last term of the preceding series & a first

term of the subsequent series ; just as an instant is a single indivisible boundary between

the continuous time that precedes & that which follows. But what I have said about

production & destruction is already quite enough."

63. But, to come back at last to our point, the Law of Continuity is solidly founded Application of the
both on induction & on metaphysical reasoning ; & on that account it should be retained ^The* co5ision"af
in every case of communication of velocity. So that indeed there can never be any passing solid bodies.
from one velocity to another except through all intermediate velocities, & then without

any sudden change. We have employed induction for actual motions & velocities in
Art. 39 & solved difficulties with regard to velocities in Art. 46, 47, in cases in which they
might seem to be subject to sudden changes. As regards metaphysical argument, if in the
whole time before contact the anterior surface of the body that follows had 12 degrees of
velocity & in the subsequent time had 9, a sudden change being made at the instant of first
contact ; then at the instant that separates the two times, the body would be bound to have
12 degrees of velocity, & 9, at one & the same time. This is absurd ; for a body cannot at
the same time have two velocities, as I will now demonstrate somewhat more carefully.

64. The term velocity, as it is used in general by Mechanicians is equivocal. For it Two kinds of veio-
may mean actual velocity, that is to say, a certain relation in uniform motion given by Clty< P°tentlal &
the space passed over divided by the time taken to traverse it. It may mean also something

which, adopting a term used by the Scholastics, I call potential velocity. The latter is
a propensity for actual velocity, or a propensity possessed by the movable body (should
no force cause an alteration) for traversing with uniform motion some definite space in
any definite time. I made the distinction between these two meanings, both in the
dissertation De Firibus Fivis & in the Supplements to Stay's Philosophy ; the distinction
being very necessary to avoid equivocations. The former cannot be obtained in an instant
of time, but requires continuous time for the motion to take place ; it also requires uniform
motion in order to measure it accurately. The latter can be determined at any given
instant ; & it is this kind that is everywhere intended by Mechanicians, when they make
geometrical measured diagrams for any non-uniform velocities whatever. In which, if
the abscissa represents time & the ordinate velocity, no matter how it is varied, then
the area will express the distance passed over ; or again, if the abscissa represents time
& the ordinate force, then the area will represent the velocity already produced. This
is always the case, for other scales of the same kind, whenever algebraical formulae &
this potential velocity are employed ; the latter being taken to be but the propensity for
actual velocity, such indeed as I understand it to be, when in collision of bodies I deny
from the foregoing argument that there can be any sudden change.

65. Now it is quite clear that there cannot be two actual velocities at one & the same I4 is impossible
time in the same moving body. For, then it would be necessary that the moving body, have two velocities"
which at the beginning of a certain time occupied a certain given point of space, should at either actual or
all times afterwards occupy two points of that space ; so that the space traversed would be ^given) or we are
twofold, the one space being determined by the one velocity & the other by the other, forced to admit,
Thus an actual replication would be required ; & this we can clearly prove in a perfectly penetration 'S
simple way from the principle of induction. Because, for instance, we never see the same

movable body departing from the same place in two directions, nor being in two places at
the same time in such a way that it is clear to us that it is in both. Again, it can be easily
proved that it is also impossible that there should be two potential velocities at the same
time. For potential velocity is the propensity that the body has, at the end of any given
continuous time, for existing at a certain given point of space that has a given distance
from that point of space, which the moving body occupied at the instant of time in which
it is said to have the prescribed potential velocity. Wherefore to have at one & the same
time two potential velocities is the same thing as being prescribed to occupy at the same
instant of time two points of space ; each of which has its own distinct distance from that
point of space that the body occupied at the start ; & this is the same thing as prescribing
that there should be replication at all subsequent instants of time. It is commonly said
that a movable body acquires from different causes several velocities simultaneously ; but
these velocities are compounded into one in such a way that each produces a state of the
moving body ; & this state, with regard to the dispositions that it has at that instant (these
include all circumstances both past & present), is only conditional, not absolute. That is
to say, each involves the propensity which the body, on account of all past & present
circumstances, would have for occupying that prescribed point of space at that particular

72 PHILOSOPHISE NATURALIS THEORIA

temporis ; nisi aliunde ejusmodi determinatio per conjunctionem alterius causae, quae turn
agat, vel jam egerit, mutaretur, & loco ipsius alia, quae composita dicitur, succederet. Sed
status absolutus resultans ex omnibus eo momento praasentibus, & prseteritis circumstantiis
ipsius mobilis, est unica determinatio ad existendum pro quovis determinato momento
temporis sequentis in quodam determinato puncto spatii, qui quidem status pro circum-
stantiis omnibus praeteritis, & prsesentibus est absolutus, licet sit itidem conditionatus pro
futuris : si nimirum esedem, vel alias causa; agentes sequentibus momentis non mutent
determinationem, & punctum illud loci, ad quod revera deveniri deinde debet dato illo
momento temporis, & actu devenitur ; si ipsae nihil aliud agant. Porro patet ejusmodi
status ex omnibus prseteritis, & praesentibus circumstantiis absolutes non posse eodem
momento temporis esse duos sine determinatione ad replicationem, quam ille conditionatus
status resultans e singulis componentibus velocitatibus non inducit ob id ipsum, quod
conditionatus est. Jam vero si haberetur saltus a velocitate ex omnibus prsteritis, &
praesentibus circumstantiis exigente, ex. gr. post unum minutum, punctum spatii distans
per palmos 6 ad exigentem punctum distans per palmos 9 ; deberet eo momento temporis,
quo fieret saltus, haberi simul utraque determinatio absoluta respectu circumstantiarum
omnium ejus momenti, & omnium praeteritarum ; nam toto prsecedenti tempore habita
fuisset realis series statuum cum ilia priore, & toto sequenti deberet haberi cum ilia
posteriore, adeoque eo momento, simul utraque, cum neutra series realis sine reali suo
termino stare possit.

Quovis momento 66. Praeterea corporis, vel puncti existentis potest utique nulla esse velocitas actualis,

denbeUre hTbeTe saltern accurate talis ; si nimirum difformem habeat motum, quod ipsum etiam semper in

statum reaiem ex Natura accidit, ut demonstrari posse arbitror, sed hue non pertinet ; at semper utique

potentialis'6 li!itlS haberi debet aliqua velocitas potentialis, vel saltern aliquis status, qui licet alio vocabulo

appellari soleat, & dici velocitas nulla, est tamen non nihilum quoddam, sed realis status,

nimirum determinatio ad quietem, quanquam hanc ipsam, ut & quietem, ego quidem

arbitrer in Natura reapse haberi nullam, argumentis, quae in Stayanis Supplementis exposui

in binis paragraphis de spatio, ac tempore, quos hie addam in fine inter nonnulla, quae hie

etiam supplementa appellabo, & occurrent primo, ac secundo loco. Sed id ipsum itidem

nequaquam hue pertinet. lis etiam penitus praetermissis, eruitur e reliquis, quae diximus,

admisso etiam ut existente, vel possibili in Natura motu uniformi, & quiete, utramque

velocitatem habere conditiones necessarias ad [31] hoc, ut secundum argumentum pro

continuitatis lege superius allatum vim habeat suam, nee ab una velocitate ad alteram abiri

possit sine transitu per intermedias.

ento te^oris'trari" ^7' Patet autenij nmc illud evinci, nee interire momento temporis posse, nee oriri

sin ab una veioci- velocitatem totam corporis, vel puncti non simul intereuntis, vel orientis, nee hue transferri
demonstratliai& Posse» quod de creatione, & morte diximus ; cum nimirum ipsa velocitas nulla corporis, vel
vindicatur. puncti existentis, sit non purum nihil, ut monui, sed realis quidam status, qui simul cum

alio reali statu determinatae illius intereuntis, vel orientis velocitatis deberet conjungi ; unde
etiam fit, ut nullum effugium haberi possit contra superiora argumenta, dicendo, quando a
12 gradibus velocitatis transitur ad 9, durare utique priores 9, & interire reliquos tres, in
quo nullum absurdum sit, cum nee in illorum duratione habeatur saltus, nee in saltu per
interitum habeatur absurdi quidpiam, ejus exemplo, quod superius dictum fuit, ubi ostensum
est, non conjungi non esse simul, & esse. Nam in primis 12 gradus velocitatis non sunt quid
compositum e duodecim rebus inter se distinctis, atque disjunctis, quarum 9 manere possint,
3 interire, sed sunt unica determinatio ad existendum in punctis spatii distantibus certo
intervallo, ut palmorumi2, elapsis datis quibusdam temporibus aequalibus quibusvis. Sic
etiam in ordinatis GD, HE, quae exprimunt velocitates in fig. 6, revera, in mea potissimuim
Theoria, ordinata GD non est quaedam pars ordinatae HE communis ipsi usque ad D, sed
sunt duae ordinatae, quarum prima constitit in relatione distantiaa, puncti curvae D a puncto
axis G, secunda in relatione puncti curvae E a puncto axis H, quod estibi idem, ac punctum G.

A THEORY OF NATURAL PHILOSOPHY 73

instant of time ; were it not for the fact that that particular propensity is for other reasons
altered by the conjunction of another cause, which acts at the time, or has already done so ;
& then another propensity, which is termed compound, will take the place of the former.
But the absolute propensity, which arises from the combination of all the past & present
circumstances of the moving body for that instant, is but a single propensity for existing at
any prescribed instant of subsequent time in a certain prescribed point of space ; & this
state is absolute for all past & present circumstances, although it may be conditional for
future circumstances. That is to say, if the same or other causes, acting during subsequent
instants, do not change that propensity, & the point of space to which it ought to get
thereafter at the given instant of time, & which it actually does reach if these causes have
no other effect. Further, it is clear that we cannot have two such absolute states, arising
from all past & present circumstances, at the same time without prescribing replication ;
& this the conditional state arising from each of the component velocities does not induce
because of the very fact that it is conditional. If now there should be a jump from the
velocity, arising out of all the past & present circumstances, which, after one minute for
example, compels a point of space to move through 6 palms, to a velocity that compels the
point to move through 9 palms ; then, at the instant of time, in which the sudden change
takes place, there would be each of two absolute propensities in respect of all the circum-
stances of that instant & all that had gone before, existing simultaneously. For in the
whole of the preceding time there would have been a real series of states having the former
velocity as a term, & in the whole of the subsequent time there must be one having the
latter velocity as a term ; hence at that particular instant each of them must occur at one
& the same time, since neither real series can stand good without each having its own
real end term.

66. Again, it is at least possible that the actual velocity of a body, or of an existing At any instant an
point, may be nothing ; that is to say, if the motion is non-uniform. Now, this always ^^l *?££ ""**
is the case in Nature ; as I think can be proved, but it does not concern us at present. But, arising from a kind
at any rate, it is bound to have some potential velocity, or at least some state, which, °£yP°tentlal vel°-
although usually referred to by another name, & the velocity stated to be nothing, yet is

not definitely nothing, but is a real state, namely, a propensity for rest. I have come to
the conclusion, however, that in Nature there is not really such a thing as this state, or
absolute rest, from arguments that I gave in the Supplements to Stay's Philosophy in
two paragraphs concerning space & time ; & these I will add at the end of the work, amongst
some matters, that I will call by the name of supplements in this work as well ; they will
be placed first & second amongst them. But that idea also does not concern us at present.
Now, putting on one side these considerations altogether, it follows from the rest of what
I have said that, if we admit both uniform motion & rest as existing in Nature, or even
possible, then each velocity must have conditions that necessarily lead to the conclusion
that according to the argument given above in support of the Law of Continuity it has its
own corresponding force, & that no passage from one velocity to another can be made
except through intermediate stages.

67. Further, it is quite clear that from this it can be rigorously proved that the whole Rigorous proof that

e i . i . . . , 9 J r, . it is impossible to

velocity of a body cannot perish or arise in an instant of time, nor for a point that does pass from one veio-
not perish or arise along with it ; nor can our arguments with regard to production & city to a™*11?1 in

1-1 i r i • T-i • i.*«««°»«*» an instant of time.

destruction be made to refer to this. For, since that no velocity of a body, or of an
existing point, is not absolutely nothing, as I remarked, but is some real state ; & this real
state is bound to be connected with that other real state, namely, that of the prescribed
velocity that is being created or destroyed. Hence it comes about that there can be no
escape from the arguments I have given above, by saying that when the change from twelve
degrees of velocity is made to nine degrees, the first nine at least endure, whilst the
remaining three are destroyed ; & then by asserting that there is nothing absurd in this,
since neither in the duration of the former has there been any sudden change, nor is there
anything absurd in the jump caused by the destruction of the latter, according to the instance
of it given above, where it was shown that non-existence & existence must be disconnected.
For in the first place those twelve degrees of velocity are not something compounded of
twelve things distinct from, & unconnected with, one another, of which nine can endure
& three can be destroyed ; but are a single propensity for existing, after the lapse of any
given number of equal times of any given length, in points of space at a certain interval,
say twelve palms, away from the original position. So also, with regard to the ordinates
GD, HE, which in Fig. 6. express velocities, it is the fact that (most especially in my Theory)
the ordinate GD is not some part of the ordinate HE, common with it as far as the point
D ; but there are two ordinates, of which the first depends upon the relation of the distance
of the point D of the curve from the point G on the axis, & the second upon the relation
of the distance of point E on the curve from the point H on the axis, which is here the

74

PHILOSOPHIC NATURALIS THEORIA

Relationem distantiae punctorum D, & G constituunt duo reales modi existendi ipsorum,
relationem distantias punctorum D. & E duo reales modi existendi ipsorum, & relationem
distantiae punctorum H, & E duo reales modi existendi ipsorum. Haec ultima relatio
constat duobus modis realibus tantummodo pertinentibus ad puncta E, & H, vel G, &
summa priorum constat modis realibus omnium trium, E, D, G. Sed nos indefinite con-
cipimus possibilitatem omnium modorum realium intermediorum, ut infra dicemus, in qua
praecisiva, & indefinita idea stat mini idea spatii continui ; & intermedii modi possibles inter
G, & D sunt pars intermediorum inter E, & H. Praeterea omissis etiam hisce omnibus ipse
ille saltus a velocitate finita ad nullam, vel a nulla ad finitam, haberi non potest.

Cur adhibita col- 68. Atque hinc ego quidem potuissem etiam adhibere duos globos asquales, qui sibi

eaiuicm^aKanTpro mv*cem occurrant cum velocitatibus sequalibus, quae nimirum in ipso contactu deberent

Thcoria deducenda. momento temporis intcrirc ; sed ut hasce ipsas considerationes evitarem de transitu a statu

reali ad statum itidem realem, ubi a velocitate aliqua transitur ad velocitatem nullam ;

adhibui potius [32] in omnibus dissertationibus meis globum, qui cum 12 velocitatis gradibus

assequatur alterum praecedentem cum 6 ; ut nimirum abeundo ad velocitatem aliam

quamcunque haberetur saltus ab una velocitate ad aliam, in quo evidentius esset absurdum.

Quo pacto mutata
velocitate poten-
tial! per saltum,
non mutetur per
saltum actualis.

69. Jam vero in hisce casibus utique haberi deberet saltus quidam, & violatio legis
continuitatis, non quidem in velocitate actuali, sed in potentiali, si ad contactum deveniretur
cum velocitatum discrimine aliquo determinato quocunque. In velocitate actuali, si earn
metiamur spatio, quod conficitur, diviso per tempus, transitus utique fieret per omnes
intermedias, quod sic facile ostenditur ope Geometriae. In fig. 10 designent AB, BC bina
tempora ante & post contactum, & momento quolibet H sit velocitas potentialis ilia major
HI, quae aequetur velocitati primae AD : quovis autem momento Q posterioris temporis sit

velocitas potentialis minor QR, quae aequetur
velocitati cuidam data: CG. Assumpto quovis
tempore HK determinatae magnitudinis, area
IHKL divisa per tempus HK, sive recta HI,
exhibebit velocitatem actualem. Moveatur
tempus HK versus B, & donee K adveniat ad
B, semper eadem habebitur velocitatis men-
sura ; eo autem progressoin O ultra B, sed adhuc
H existente in M citra B, spatium illi tem-
pori respondens componetur ex binis MNEB,
BFPO, quorum summa si dividatur per MO ;
jam nee erit MN aequalis priori AD, nee BF,
ipsa minor per datam quantitatem FE ; sed
facile demonstrari potest (&), capta VE asquali

D! ~ L V N E Y

Irrcgularitas alia
in cxpressione act-
ualis velocitatis.

\

"1

1

X

\

p; R T G

1

1

1

1

AH K

M B OQ S C

FIG. 10.

IL, vel HK, sive MO, & ducta recta VF, quae secet MN in X, quotum ex illo divisione
prodeuntem fore MX, donee, abeunte toto illo tempore ultra B in QS, jam area QRTS
divisa per tempus QS exhibeat velocitatem constantem QR.

70. Patet igitur in ea consideratione a velocitate actuali praecedente HI ad sequentem
QR transiri per omnes intermedias MX, quas continua recta VF definiet ; quanquam ibi
etiam irregulare quid oritur inde, quod velocitas actualis XM diversa obvenire debeat pro
diversa magnitudine temporis assumpti HK, quo nimirum assumpto majore, vel minore
removetur magis, vel minus V ab E, & decrescit, vel crescit XM. Id tamen accidit in
motibus omnibus, in quibus velocitas non manet eadem toto tempore, ut nimirum turn
etiam, si velocitas aliqua actualis debeat agnosci, & determinari spatio diviso per tempus ;
pro aliis, atque aliis temporibus assumptis pro mensura alias, atque alias velocitatis actualis
mensuras ob-[33]-veniant, secus ac accidit in motu semper aequabili, quam ipsam ob causam,
velocitatis actualis in motu difformi nulla est revera mensura accurata, quod supra innui
sed ejus idea praecisa, ac distincta aequabilitatem motus requirit, & idcirco Mechanic! in
difformibus motibus ad actualem velocitatem determinandam adhibere solent spatiolum
infinitesimo tempusculo percursum, in quo ipso motum habent pro aequabili.

(b) Si enim producatur OP usque ad NE in T, erit ET = VN, ob VE = MO =NT. Est autem

VE : VN : : EF : NX ; quart VN X EF = VE X NX, sive posito ET pro VN, W MO pro VE, erit
ET XEF =MO X NX. Totum MNTO est MO X MN, pars FETP est = EY X EF. Quafe residuus
gnomon NMOPFE est MOx(MN-NX), sive est MO X MX, quo diviso per MO babetur MX.

A THEORY OF NATURAL PHILOSOPHY 75

same as the point G. The relation of the distance between the points D & G is determined
by the two real modes of existence peculiar to them, the relation of the distance between
the points D & E by the two real modes of existence peculiar to them, & the relation of
the distance between the points H & E by the two real modes of existence peculiar to them.
The last of these relations depends upon the two real modes of existence that pertain to the
points E & H (or G), & upon these alone ; the sum of the first & second depends upon all
three of the modes of the points E, D, & G. But we have some sort of ill-defined conception
of the possibility of all intermediate real modes of existence, as I will remark later ; & on
this disconnected & ill-defined idea is founded my conception of continuous space ; also
the possible intermediate modes between G & D form part of those intermediate between
E & H. Besides, omitting all considerations of this sort, -that sudden change from a finite
velocity to none at all, or from none to a finite, cannot happen.

68. Hence I might just as well have employed two equal balls, colliding with one why the collision
another with equal velocities, which in truth at the moment of contact would have to be thebsameTirecfion
destroyed in an instant of time. But, in order to avoid the very considerations just stated is employed for the
with regard to the passage from a real state to another real state (when we pass from a In "
definite velocity to none), I have preferred to employ in all my dissertations a ball having

1 2 degrees of velocity, which follows another ball going in front of it with 6 degrees ;
so that, by passing to some other velocity, there would be a sudden change from one
velocity to another ; & by this means the absurdity of the idea would be made more
evident.

69. Now, at least in such cases as these, there is bound to be some sudden change &

a breach of the Law of Continuity, not indeed in the actual velocity, but in the potential sudden change in
velocity, if the collision occurs with any given difference of velocities whatever. In the ^ ^^T^ might
actual velocity, measured by the space traversed divided by the time, the change will at any not 'be a sudden
rate be through all intermediate stages ; & this can easily be shown to be 50 by the aid of ^^veioclty16 ***"
Geometry.

In Fig. 10 let AB, BC represent two intervals of time, respectively before & after
contact ; & at any instant let the potential velocity be the greater velocity HI, equal to the .
first velocity AD ; & at any instant Q of the time subsequent to contact let the potential
velocity be the less velocity QR, equal to some given velocity CG. If any prescribed interval
of time HK be taken, the area IHKL divided by the time HK, i.e., the straight line HI,
will represent the actual velocity. Let the time HK be moved towards B ; then until
K comes to B, the measure of the velocity will always be the same. If then, K goes on
beyond B to O, whilst H still remains on the other side of B at M ; then the space corre-
sponding to that time will be composed of the two spaces MNEB, BFPO. Now, if the
sum of these is divided by MO, the result will not be equal to either MN (which is equal
to the first AD), or BF (which is less than MN by the given quantity FE). But it can
easily be proved ( ) that, if VE is taken equal to IL, or HK, or MO, & the straight line
VF is drawn to cut MN in X ; then the quotient obtained by the division will be MX.
This holds until, when the whole of the interval of time has passed beyond B into the
position QS, the area QRTS divided by the time QS now represents a constant velocity
equal to QR.

70. From the foregoing reasoning it is therefore clear that the change from the A further irregu-
preceding actual velocity HI to the subsequent velocity QR is made through all intermediate larity m the repre-

r , . . TV/TTT- i • i MI i i • i i i ••>•>• sentation of actual

velocities such as MX, which will be determined by the continuous straight line VF. There velocity,
is, however, some irregularity arising from the fact that the actual velocity XM must turn
out to be different for different magnitudes of the assumed interval of time HK. For,
according as this is taken to be greater or less, so the point V is removed to a greater or
less distance from E ; & thereby XM will be decreased or increased correspondingly. This
is the case, however, for all motions in which the velocity does not remain the same during
the whole interval ; as for instance in the case where, if any actual velocity has to be found
& determined by the quotient of the space traversed divided by the time taken, far other
& different measures of the actual velocities will arise to correspond with the different
intervals of time assumed for their measurement ; which is not the case for motions that
are always uniform. For this reason there is no really accurate measure of the actual
velocity in non-uniform motion, as I remarked above ; but a precise & distinct idea of it
requires uniformity of motion. Therefore Mechanicians in non-uniform motions, as a
means to the determination of actual velocity, usually employ the small space traversed in
an infinitesimal interval of time, & for this interval they consider that the motion is uniform.

(b) For if OP be produced to meet NE in T, then EY = VN ; for VE = MO = NT. Moreover

VE : VN=EF : NX ; and therefore VN.EF=VE.NX. Hence, replacing VN hy EY, and. VE hy MO, we have
EYEF=MO.NX. Now, the whole MNYO = MO.MN, and the part FEYP= ET.EF. Hence the remainder
(the gnomon NMOPFE) = MO.(MN — NX) = MO.MX .- and this, on division by MO, will give MX.

76 PHILOSOPHIC NATURALIS THEORIA

" \mmc yi- At velocitas potcntialis, quas singulis momentis temporis respondet sua, mutaretur

citatum non posse utique per saltum ipso momento B, quo deberet haberi & ultima velocitatum praecedentium
entianivciodtatumr" ^' ^ P"ma sequentium BF, quod cum haberi nequeat, uti demonstratum est, fieri non
potest per secundum ex argumentis, quae adhibuimus pro lege continuitatis, ut cum ilia
velocitatum inasqualitate deveniatur ad immediatum contactum ; atque id ipsum excludit
etiam inductio, quam pro lege continuitatis in ipsis quoque velocitatibus, atque motibus
primo loco proposui.

Prpmovenda ana- 72. Atque hoc demum pacto illud constitit evidenter, non licere continuitatis legem

deserere in collisione corporum, & illud admittere, ut ad contactum immediatum deveniatur
cum illaesis binorum corporum velocitatibus integris. Videndum igitur, quid necessario
consequi debeat, ubi id non admittatur, & haec analysis ulterius promovenda.

ifaberimu- 73' Quoniam a^ immediatum contactum devenire ea corpora non possunt cum praece-

tationem veiocita- dentibus velocitatibus ; oportet, ante contactum ipsum immediatum incipiant mutari
auk mutat Ue Vlm> velocitates ipsae, & vel ea consequentis corporis minui, vel ea antecedentis augeri, vel
utrumque simul. Quidquid accidat, habebitur ibi aliqua mutatio status, vel in altero
corpore, vel in utroque, in ordine ad motum, vel quietem, adeoque habebitur aliqua
mutationis causa, quaecunque ilia sit. Causa vero mutans statum corporis in ordine ad
motum, vel quietem, dicitur vis ; habebitur igitur vis aliqua, quae effectum gignat, etiam
ubi ilia duo corpora nondum ad contactum devenerint.

Earn vim debere 74. Ad impediendam violationem continuitatis satis esset, si ejusmodi vis ageret in

. iSf-SSi & alterum tantummodo e binis corporibus, reducendo praecedentis velocitatem ad gradus 12,

agere m panes op- . r. . ' .

positas. vel sequentis ad 6. Videndum igitur aliunde, an agere debeat in alterum tantummodo, an

in utrumque simul, & quomodo. Id determinabitur per aliam Naturae legem, quam nobis
inductio satis ampla ostendit, qua nimirum evincitur, omnes vires nobis cognitas agere
utrinque & aequaliter, & in partes oppositas, unde provenit principium, quod appellant
actionis, & reactionis aequalium ; est autem fortasse quaedam actio duplex semper aequaliter
agens in partes oppositas. Ferrum, & magnes aeque se mutuo trahunt ; elastrum binis
globis asqualibus interjectum aeque utrumque urget, & aequalibus velocitatibus propellit ;
gravitatem ipsam generalem mutuam esse osten-[34]-dunt errores Jovis, ac Saturni potissi-
mum, ubi ad se invicem accedunt, uti & curvatura orbitae lunaris orta ex ejus gravitate in
terram comparata cum aestu maris orto ex inaequali partium globi terraquei gravitate in
Lunam. Ipsas nostrae vires, quas nervorum ope exerimus, semper in partes oppositas agunt,
nee satis valide aliquid propellimus, nisi pede humum, vel etiam, ut efficacius agamus,
oppositum parietem simul repellamus. En igitur inductionem, quam utique ampliorem
etiam habere possumus, ex qua illud pro eo quoque casu debemus inferre, earn ibi vim in
utrumque corpus agere, quae actio ad aequalitatem non reducet inaequales illas velocitates,
nisi augeat praecedentis, minuat consequentis corporis velocitatem ; nimirum nisi in iis
producat velocitates quasdam contrarias, quibus, si solae essent, deberent a se invicem
recedere : sed quia eae componuntur cum praecedentibus ; hasc utique non recedunt, sed
tantummodo minus ad se invicem accedunt, quam accederent.

Hinc dicendam 75. Invenimus igitur vim ibi debere esse mutuam, quae ad partes oppositas agat, & quae

esse

sua natura determinet per sese ilia corpora ad recessum mutuum a se invicem. Hujusmodi

quaerendam ejus . . . . . ./„ . . 11 • • i • /~» j • i

legem. igitur vis ex nomims denmtione appellari potest vis repulsiva. Uuaerendum jam ulterius,

qua lege progredi debeat, an imminutis in immensum distantiis ad datam quandam mensuram
deveniat, an in infinitum excrescat ?

Ea vi debere totum 76. Ut in illo casu evitetur saltus ; satis est in allato exemplo ; si vis repulsiva, ad quam

crimenateHdi ante delati sumus, extinguat velocitatum differentiam illam 6 graduum, antequam ad contactum

contactum. immediatum corpora devenirent : quamobrem possent utique devenire ad eum contactum

eodem illo momento, quo ad aequalitatem velocitatum deveniunt. At si in alio quopiam

casu corpus sequens impellatur cum velocitatis gradibus 20, corpore praecedente cum suis 6 ;

A THEORY OF NATURAL PHILOSOPHY 77

71. The potential velocity, each corresponding to its own separate instant of time, The conclusion is

ij • f i j jj i ^.t i. • t • n a i • • tnat immediate

would certainly be changed suddenly at that instant ot time r> ; & at this point we are contact with a dif-

bound to have both the last of the preceding velocities, BE, & the first of the subsequent ference of velocities

velocities, BF. Now, since (as has been already proved) this is impossible, it follows from

the second of the arguments that I used to prove the Law of Continuity, that it cannot

come about that the bodies come into immediate contact with the inequality of velocities

in question. This is also excluded by induction, such as I gave in the first place for the

Law of Continuity, in the case also of these velocities & motions.

72. In this manner it is at length clearly established that it is not right to neglect the immediate contact
Law of Continuity in the collision of bodies, & admit the idea that they can come into ^Sysis^tobe ca^
immediate contact with the whole velocities of both bodies unaltered. Hence, we must ried further,
now investigate the consequences that necessarily follow when this idea is not admitted ;

& the analysis must be carried further.

73. Since the bodies cannot come into immediate contact with the velocities they had There must be then,
at first, it is necessary that those velocities should commence to change before that immediate change in the v'eioa
contact ; & either that of the body that follows should be diminished, or that of the one city '• & therefore
going in front should be increased, or that both these changes should take place together, causes the change!1
Whatever happens, there will be some change of state at the time, in one or other of the

bodies, or in both, with regard to motion or rest ; & so there must be some cause for this
change, whatever it is. But a cause that changes the state of a body as regards motion or
rest is called force. Hence there must be some force, which gives the effect, & that too
whilst the two bodies have not as yet come into contact.

74. It would be enough, to avoid a breach of the Law of Continuity, if a force of The f°rce omust V6

i • i.ii 11 r -I IT i i • i i • <• i i i • mutual, & act m

this kind should act on one of the two bodies only, altering the velocity of the body in opposite directions,
front to 12 degrees, or that of the one behind to 6 degrees. Hence we must find out,
from other considerations, whether it should act on one of the two bodies only, or on both
of them at the same time, & how. This point will be settled by another law of Nature,
which sufficiently copious induction brings before us ; that is, the law in which it is estab-
lished that all forces that are known to us act on both bodies, equally, and in opposite
directions. From this comes the principle that is called ' the principle of equal action
& reaction ' ; perchance this may be a sort of twofold action that always produces its
effect equally in opposite directions. Iron & a loadstone attract one another with the
same strength ; a spring introduced between two balls exerts an equal action on either
ball, & generates equal velocities in them. That universal gravity itself is mutual is proved
by the aberrations of Jupiter & of Saturn especially (not to mention anything else) ; that
is to say, the way in which they err from their orbits & approach one another mutually.
So also, when the curvature of the lunar orbit arising from its gravitation towards the
Earth is compared with the flow of the tides caused by the unequal gravitation towards
the Moon of different parts of the land & water that make up the Earth. Our own bodily
forces, which produce their effect by the help of our muscles, always act in opposite direc-
tions ; nor have we any power to set anything in motion, unless at the same time we press
upon the earth with our feet or, in order to get a better purchase, upon something that
will resist them, such as a wall opposite. Here then we have an induction, that can be
made indeed more ample still ; & from it we are bound in this case also to infer that the
force acts on each of the two bodies. This action will not reduce to equality those two
unequal velocities, unless it increases that of the body which is in front & diminishes that
of the one which follows. That is to say, unless it produces in them velocities that are
opposite in direction ; & with these velocities, if they alone existed, the bodies would
move away from one another. But, as they are compounded with those they had to start
with, the bodies do not indeed recede from one another, but only approach one another
less quickly than they otherwise would have done.

75. We have then found that the force must be a mutual force which acts in opposite Hence the force
directions ; one which from its very nature imparts to those bodies a natural propensity ™pu*sive*r ^"1^
for mutual recession from one another. Hence a force of this kind, from the very meaning governing it is now
of the term, may be called a repulsive force. We have now to go further & find the law to ^ found-
that it follows, & whether, when the distances are indefinitely diminished, it attains any

given measure, or whether it increases indefinitely.

76. In this case, in order that any sudden change may be avoided, it is sufficient, in The whole differ-
the example under consideration, if the repulsive force, to which our arguments have led veiocitiesWmust *be
us, should destroy that difference of 6 degrees in the velocities before the bodies should destroyed by the
have come into immediate contact. Hence they might possibly at least come into contact t°^e

at the instant in which they attained equality between the velocities. But if in another
case, say, the body that was behind were moving with 20 degrees of velocity, whilst the

78 PHILOSOPHL/E NATURALIS THEORIA

turn vero ad contactum deveniretur cum differentia velocitatum majore, quam graduum 8.
Nam illud itidem amplissima inductione evincitur, vires omnes nobis cognitas, quas aliquo
tempore agunt, ut velocitatem producant, agere in ratione temporis, quo agunt, & sui
ipsius. Rem in gravibus oblique descendentibus experimenta confirmant ; eadem & in
elastris institui facile possunt, ut rem comprobent ; ac id ipsum est fundamentum totius
Mechanicae, quae inde motuum leges eruit, quas experimenta in pendulis, in projectis
gravibus, in aliis pluribus comprobant, & Astronomia confirmat in caelestibus motibus.
Quamobrem ilia vis repulsiva, quae in priore casu extinxit 6 tantummodo gradus discriminis,
si agat breviore tempore in secundo casu, non poterit extinguere nisi pauciores, minore
nimirum velocitate producta utrinque ad partes contrarias. At breviore utique tempore
aget : nam cum majore velocitatum discrimine velocitas respectiva est major, ac proinde
accessus celerior. [35] Extingueret igitur in secundo casu ilia vis minus, quam 6 discriminis
gradus, si in primo usque ad contactum extinxit tantummodo 6. Superessent igitur plures,
quam 8 ; nam inter 20 & 6 erant 14, ubi ad ipsum deveniretur contactum, & ibi per saltum
deberent velocitates mutari, ne compenetratio haberetur, ac proinde lex continuitatis
violari. Cum igitur id accidere non possit ; oportet, Natura incommodo caverit per
ejusmodi vim, quae in priore casu aliquanto ante contactum extinxerit velocitatis discrimen,
ut nimirum imminutis in secundo casu adhuc magis distantiis, vis ulterior illud omne
discrimen auferat, elisis omnibus illis 14 gradibus discriminis, qui habebantur.

Earn vim debere
augeri in infinitum,
imminutis, & qui-
dem in infinitum,
distantiis : habente
virium curva ali-
quam asymptotum
in origine abscissa-
rum.

77. Quando autem hue jam delati sumus, facile est ulterius progredi, & illud con-
siderare, quod in secundo casu accidit respectu primi, idem accidere aucta semper velocitate
consequentis corporis in tertio aliquo respectu secundi, & ita porro. Debebit igitur ad
omnem pro omni casu evitandum saltum Natura cavisse per ejusmodi vim, quae imminutis
distantiis crescat in infinitum, atque ita crescat, ut par sit extinguendas cuicunque velocitati,
utcunque magnae. Devenimus igitur ad vires repulsivas imminutis distantiis crescentes
in infinitum, nimirum ad arcum ilium asymptoticum ED curae virium in fig. i propositum.
Illud quidem ratiocinatione hactenus instituta immediate non deducitur, hujusmodi
incrementa virium auctarum in infinitum respondere distantiis in infinitum imminutis.
Posset pro hisce corporibus, quae habemus prae manibus, quasdam data distantia quascunque
esse ultimus limes virium in infinitum excrescentium, quo casu asymptotus AB non transiret
per initium distantiae binorum corporum, sed tanto intervallo post ipsum, quantus esset
ille omnium distantiarum, quas remotiores particulse possint acquirere a se invicem, limes
minimus ; sed aliquem demum esse debere extremum etiam asymptoticum arcum curvas
habentem pro asymptote rectam transeuntem per ipsum initium distantiae, sic evincitur ;
si nullus ejusmodi haberetur arcus ; particulae materiae minores, & primo collocatae in
distantia minore, quam esset ille ultimus limes, sive ilia distantia asymptoti ab initio
distantias binorum punctorum materiae, in mutuis incursibus velocitatem deberent posse
mutare per saltum, quod cum fieri nequeat, debet utique aliquis esse ultimus asymptoticus
arcus, qui asymptotum habeat transeuntem per distantiarum initium, & vires inducat
imminutis in infinitum distantiis crescentes in infinitum ita, ut sint pares velocitati extin-
guendae cuivis, utcunque magnae. Ad summum in curva virium haberi possent plures
asymptotici arcus, alii post alios, habentes ad exigua intervalla asymptotes inter se parallelas,
qui casus itidem uberrimum aperit contemplationibus fcecundissimis campum, de quo
aliquid inferius ; sed aliquis arcus asympto-[36]-ticus postremus, cujusmodi est is, quern
in figura i proposui, haberi omnino debet. Verum ea perquisitione hie omissa, pergendum
est in consideratione legis virium, & curvae earn exprimentis, quae habentur auctis distantiis.

vim in majoribus
tractfvam, ^

78. In primis gravitas omnium corporum in Terram, quam quotidie experimur, satis
, evmcit> repulsionem illam, quam pro minimis distantiis invenimus, non extendi ad distantias

secante axem in quascunque, sed in magnis jam distantiis haberi determinationem ad accessum, quam vim
aliquo hmite. attractivam nominavimus. Quin immo Keplerianae leges in Astronomia tarn feliciter a

Newtono adhibitae ad legem gravitatis generalis deducendam, & ad cometas etiam traductas,

A THEORY OF NATURAL PHILOSOPHY

79

I?
3

0

O

8o

PHILOSOPHIC NATURALIS THEORIA

o

A THEORY OF NATURAL PHILOSOPHY 81

body in front still had its' original 6 degrees ; then they would come into contact with
a difference of velocity greater than 8 degrees. For, it can also be proved by the fullest
possible induction that all forces known to us, which act for any intervals of time so as to
produce velocity, give effects that are proportional to the times for which they act, & also
to the magnitudes of the forces themselves. This is confirmed by experiments with heavy
bodies descending obliquely ; the same things can be easily established in the case of springs
so as to afford corroboration. Moreover it is the fundamental theorem of the whole of
Mechanics, & from it are derived the laws of motion ; these are confirmed by experiments
with pendulums, projected weights, & many other things ; they are corroborated also by
astronomy in the matter of the motions of the heavenly bodies. Hence the repulsive force,
which in the first case destroyed only 6 degrees difference of velocity, if it acts for a shorter
time in the second case, will not be able to destroy aught but a less number of degrees, as
the velocity produced in the two bodies in opposite directions is less. Now it certainly
will act for a shorter time ; for, owing to the greater difference of velocities, the relative
velocity is greater & therefore the approach is faster. Hence, in the second case the force
would destroy less than 6 degrees of the difference, if in the first case it had, just at contact,
destroyed 6 degrees only. There would therefore be more than 8 degrees left over (for,
between 20 & 6 there are 14) when contact happened, & then the velocities would have
to be changed suddenly unless there was compenetration ; & thereby the Law of Continuity
would be violated. Since, then, this cannot be the case, Nature would be sure to guard
against this trouble by a force of such a kind as that which, in the former case, extinguished
the difference of velocity some time before contact ; that is to say, so that, when the
distances are still further diminished in the second case, a further force eliminates all
that difference, all of the 14 degrees of difference that there were originally being
destroyed.

77. Now, after that we have been led so far, it is easy to go on further still & to consider 'nie fon:e mus* "*•
that, what happens in the second case when compared with the first, will happen also in SThe distances Ire
a third case, in which the velocity of the body that follows is once more increased, when diminished, also
compared with the second case ; & so on, & so on. Hence, in order to guard against any Sn-ve"^*6 forces has

sudden change at all in every case whatever, Nature will necessarily have taken measures an asymptote at the

for this purpose by means of a force of such a kind that, as the distances are diminished the ongm

force increases indefinitely, & in such a manner that it is capable of destroying any velocity,

however great it may be. We have arrived therefore at repulsive forces that increase as

the distances diminish, & increase indefinitely ; that is to say, to the asymptotic arc, ED,

of the curve of forces exhibited in Fig. i . It is indeed true that by the reasoning given so

far it is not immediately deduced that increments of the forces when increased to infinity

correspond with the distances diminished to infinity. There may be for these bodies,

such as we have in consideration, some fixed distance that acts as a boundary limit to forces

that increase indefinitely ; in this case the asymptote AB will not pass through the

beginning of the distance between the two bodies, but at an interval after it as great as the

least limit of all distances that particles, originally more remote, might acquire from one

another. But, that there is some final asymptotic arc of the curve having for its asymptote

the straight line passing through the very beginning of the distance, is proved as follows.

If there were no arc of this kind, then the smaller particles of matter, originally collected

at a distance less than this final limit would be, i.e., less than the distance of the asymptote

from the beginning of the distance between the two points of matter, must be capable of

having- their velocities, on collision with one another, suddenly changed. Now, as this is

impossible, then at any rate there must be some asymptotic arc, which has an asymptote

passing through the very beginning of the distances ; & this leads us to forces that, as the

distances are indefinitely diminished, increase indefinitely in such a way that they are

capable of destroying any velocity, no matter how large it may be. In general, in a curve

of forces there may be several asymptotic arcs, one after the other, having at short intervals

asymptotes parallel to one another ; & this case also opens up a very rich field for fruitful

investigations, about which I will say something later. But there must certainly be some

one final asymptotic arc of the kind that I have given in Fig. i. However, putting

this investigation on one side, we must get on with the consideration of the law

of forces, & the curve that represents them, which are obtained when the distances

are increased.

78. First of all, the gravitation of all bodies towards the Earth, which is an everyday The force at greater
experience, proves sufficiently that the repulsion that we found for very small distances fv^he^curve^cut-
does not extend to all distances ; but that at distances that are now great there is a ting the axis at
propensity for approach, which we have called an attractive force. Moreover the Keplerian s
Laws in astronomy, so skilfully employed by Newton to deduce the law of universal
gravitation, & applied even to the comets, show perfectly well that gravitation extends,

82 PHILOSOPHIC NATURALIS THEORIA

satis ostendunt, gravitatem vel in infinitum, vel saltern per totum planetarium, & come-
tarium systema extendi in ratione reciproca duplicata distantiarum. Quamobrem virium
curva arcum habet aliquem jacentem ad partes axis oppositas, qui accedat, quantum sensu
percipi possit, ad earn tertii gradus hyperbolam, cujus ordinatae sunt in ratione reciproca
duplicata distantiarum, qui nimirum est ille arcus STV figuras I. Ac illud etiam hinc
patet, esse aliquem locum E, in quo curva ejusmodi axem secet, qui sit limes attractionum,
& repulsionum, in quo ab una ad alteram ex iis viribus transitus fiat.

Plures esse debere, 79. Duos alios nobis indicat limites ejusmodi, sive alias duas intersectiones, ut G & I,

linStes3 Pn3enomenum vaporum, qui oriuntur ex aqua, & aeris, qui a fixis corporibus gignitur ;
cum in iis ante nulla particularum repulsio fuerit, quin immo fuerit attractio, ob
cohaerentiam, qua, una parte retracta, altera ipsam consequebatur, & in ilia tanta expansione,
& elasticitatis vi satis se manifesto prodat repulsio, ut idcirco a repulsione in minimis distantiis
ad attractionem alicubi sit itum, turn inde iterum ad repulsionem, & iterum inde ad generalis
gravitatis attractiones. Effervescentiae, & fermentationes adeo diversae, in quibus cum
adeo diversis velocitatibus eunt, ac redeunt, & jam ad se invicem accedunt, jam recedunt
a se invicem particulae, indicant utique ejusmodi limites, atque transitus multo plures ;
sed illos prorsus evincunt substantise molles, ut cera, in quibus compressiones plurimse
acquiruntur cum distantiis admodum adversis, in quibus, tamen omnibus limites haberi
debent ; nam, anteriore parte ad se attracta, posteriores earn sequuntur, eadem propulsa,
illae recedunt, distantiis ad sensum non mutatis, quod ob illas repulsiones in minimis
distantiis, quae contiguitatem impediunt, fieri alio modo non potest, nisi si limites ibidem
habeantur in iis omnibus distantiis inter attractiones, & repulsiones, quae nimirum requi-
runtur ad hoc, ut pars altera alteram consequatur retractam, vel prsecedat propulsam.

Hinc tota curvae 80. Habentur igitur plurimi limites, & plurimi flexus curvse hinc, & inde ab axe prseter

a°yroptotLm& bTu- ^uos arcus> quorum prior ED in infinitum protenditur, & asymptoticus est, alter STV,
ribus flexibus, ac [37] si gravitas generalis in infinitum protenditur, est asymptoticus itidem, & ita accedit
ad crus illud hyperbolae gradus tertii, ut discrimen sensu percipi nequeat : nam cum ipso
penitus congruere omnino non potest ; non enim posset ab eodem deinde discedere, cum
duarum curvarum, quarum diversa natura est, nulli arcus continui, utcunque exigui, possint
penitus congruere, sed se tantummodo secare, contingere, osculari possint in punctis
quotcunque, & ad se invicem accedere utcumque. Hinc habetur jam tota forma curvae
virium, qualem initio proposui, directa ratiocinatione a Naturae phsenomenis, & genuinis
principiis deducta. Remanet jam determinanda constitutio primorum elementorum
materiae ab iis viribus deducta, quo facto omnis ilia Theoria, quam initio proposui, patebit,
nee erit arbitraria quaedam hypothesis, ac licebit progredi ad amovendas apparentes quasdam
difHcultates, & ad uberrimam applicationem ad omnem late Physicam qua exponendam,
qua tantummodo, ne hoc opus plus aequo excrescat, indicandam.

Hinc elementorum 81. Quoniam, imminutis in infinitum distantiis, vis repulsiva augetur in infinitum ;

m facile patet, nullam partem materias posse esse contiguam alteri parti : vis enim ilia repulsiva

carens

partibus. protinus alteram ab altera removeret. Quamobrem necessario inde consequitur, prima

materiae elementa esse omnino simplicia, & a nullis contiguis partibus composita. Id
quidem immediate, & necessario fluit ex ilia constitutione virium, quae in minimis distantiis
sunt repulsivae, & in infinitum excrescunt.

Soiutio objectionis 82. Objicit hie fortasse quispiam illud, fieri posse, ut particulae primigenias materias

petitaeex eo quod sjnt compOsitae quidem, sed nulla Naturae vi divisibiles a se invicem, quarum altera tota

vires repulsivas r. . ^ . .... . . . ,. .. i • i

habere possent non respectu altenus totius habeat vires illas in minimis distantiis repulsivas, vel quarum pars
puncta smguia, se qu3evis respectu reliquarum partium eiusdem particulae non solum nullam habeat repulsivam

particulae primi- T-. 1,1 '-.,, J.r ,. ,.,. r . .

geniae. vim, sed habeat maximam illam attractivam, qua; ad ejusmodi cohaesionem requintur :

eo pacto evitari debere quemvis immediatum impulsum, adeoque omnem saltum, & con-
tinuitatis laesionem. At in primis id esset contra homogeneitatem materiae, de qua agemus
infra : nam eadem materiae pars in iisdem distantiis respectu quarundam paucissimarum
partium, cum quibus particulam suam componit, haberet vim repulsivam, respectu autem

A THEORY OF NATURAL PHILOSOPHY 83

either to infinity or at least to the limits of the system including all the planets & comets,
in the inverse ratio of the squares of the distances. Hence the curve will have an arc
lying on the opposite side of the axis, which, as far as can be perceived by our senses,
approximates to that hyperbola of the third degree, of which the ordinates are in the inverse
ratio of the squares of the distances ; & this indeed is the arc STV in Fig. i. Now from
this it is evident that there is some point E, in which a curve of this kind cuts the axis ;
and this is a limit-point for attractions and repulsions, at which the passage from one to
the other of these forces is made.

79. The phenomenon of vapour arising from water, & that of gas produced from There are bound to
fixed bodies lead us to admit two more of these limit-points, i.e., two other intersections, ^Syof'tiSep^
say, at G & I. Since in these there would be initially no repulsion, nay rather there sages, with corre-
would be an attraction due to cohesion, by which, when one part is retracted, another 1"6 hmit
generally followed it : & since in the former, repulsion is clearly evidenced by the

greatness of the expansion, & by the force of its elasticity ; it therefore follows that
there is, somewhere or other, a passage from repulsion at very small distances to attraction,
then back again to repulsion, & from that back once more to the attractions of universal
gravitation. Effervescences & fermentations of many different kinds, in which the
particles go & return with as many different velocities, & now approach towards &
now recede from one another, certainly indicate many more of these limit-points &
transitions. But the existence of these limit-points is perfectly proved by the case of
soft substances like wax ; for in these substances a large number of compressions are acquired
with very different distances, yet in all of these there must be limit-points. For, if the
front part is drawn out, the part behind will follow ; or if the former is pushed inwards,
the latter will recede from it, the distances remaining approximately unchanged. This, on
account of the repulsions existing at very small distances, which prevent contiguity, can-
not take place in any way, unless there are limit-points there in all those distances between
attractions & repulsions ; namely, those that are requisite to account for the fact that one
part will follow the other when the latter is drawn out, & will recede in front of the
latter when that is pushed in.

80. Therefore there are a large number of limit-points, & a large number of flexures Hence we get the
on the curve, first on one side & then on the other side of the axis, in addition to two whole for™hof t^e
arcs, one of which, ED, is continued to infinity & is asymptotic, & the other, STV, is asymptotes, many
asymptotic also, provided that universal gravitation extends to infinity. It approximates flexures & many

J ir j- r i i r i i • i i -11 111 i intersections with

to the form of the hyperbola of the third degree mentioned above so closely that the the axis,
difference from it is imperceptible ; but it cannot altogether coincide with it, because, in
that case it would never depart from it. For, of two curves of different nature, there
cannot be any continuous arcs, no matter how short, that absolutely coincide ; they can
only cut, or touch, or osculate one another in an indefinitely great number of points, &
approximate to one another indefinitely closely. Thus we now have the whole form of
the curve of forces, of the nature that I gave at the commencement, derived by a straight-
forward chain of reasoning from natural phenomena, & sound principles. It only remains
for us now to determine the constitution of the primary elements of matter, derived from
these forces ; £: in this manner the whole of the Theory that I enunciated at the start
will become quite clear, & it will not appear to be a mere arbitrary hypothesis. We
can proceed to remove certain apparent difficulties, & to apply it with great profit to
the whole of Physics in general, explaining some things fully &, to prevent the work
from growing to an unreasonable size, merely mentioning others.

81. Now, because the repulsive force is indefinitely increased when the distances are The simplicity of
indefinitely diminished, it is quite easy to see clearly that no part of matter can be contiguous ments^oT^att^r "
to any other part ; for the repulsive force would at once separate one from the other, they are altogether
Therefore it necessarily follows that the primary elements of matter are perfectly simple, w^110"* Parts.

& that they are not composed of any parts contiguous to one another. This is an
immediate & necessary deduction from the constitution of the forces, which are repulsive
at very small distances & increase indefinitely.

82. Perhaps someone will here raise the objection that it may be that the primary Solution of the ob-
particles of matter are composite, but that they cannot be disintegrated by any force in jnetlo^SSertiond that
Nature; that one whole with regard to another whole may possibly have those forces single points can-
that are repulsive at very small distances, whilst any one part with regard to any other part ?OTces,a™u7Pt h'at
of the same particle may not only have no repulsive force, but indeed may have a very primary particles
great attractive force such as is required for cohesion of this sort ; that, in this way, we can have them-
are bound to avoid all immediate impulse, & so any sudden change or breach of continuity.

But, in the first place, this would be in opposition to the homogeneity of matter, which
we will consider later ; for the same part of matter, at the same distances with regard to
those very few parts, along with which it makes up the particle, would have a repulsive

84

PHILOSOPHISE NATURALIS THEORIA

aliarum omnium attractivam in iisdem distantiis, quod analogic adversatur. Deinde si a
Deo agente supra vires Naturae sejungerentur illas partes a se invicem, turn ipsius Naturae
vi in se invicem incurrerent ; haberetur in earum collisione saltus naturalis, utut praesup-
ponens aliquid factum vi agente supra Naturam. Demum duo turn cohaesionum genera
deberent haberi in Natura admodum diversa, alterum per attractionem in minimis distantiis,
alterum vero longe alio pacto in elementarium particularum massis, nimirum per limites
cohaesionis ; adeoque multo minus simplex, & minus uniformis evaderet Theoria.

An elementa sint [38]
extensa : argumen-
ta pro virtual! eor-
um extensione.

83. Simplicitate & incompositione elementorum defmita, dubitari potest, an ea
sint etiam inextensa, an aliquam, utut simplicia, extensionem habeant ejus generis, quam
virtualem extensionem appellant Scholastici. Fuerunt enim potissimum inter Peripateticos,
qui admiserint elementa simplicia, & carentia partibus, atque ex ipsa natura sua prorsus
indivisibilia, sed tamen extensa per spatium divisibile ita, ut alia aliis ma jus etiam occupent
spatium, ac eo loco, quo unum stet, possint, eo remote, stare simul duo, vel etiam plura ;
ac sunt etiamnum, qui ita sentiant. Sic etiam animam rationalem hominis utique prorsus
indivisibilem censuerunt alii per totum corpus diffusam : alii minori quidem corporis parti,
sed utique parti divisibili cuipiam, & extensae, praesentem toti etiamnum arbitrantur.
Deum autem ipsum praesentem ubique credimus per totum utique divisibile spatium,
quod omnia corpora occupant, licet ipse simplicissimus sit, nee ullam prorsus compositionem
admittat. Videtur autem sententia eadem inniti cuidam etiam analogiae loci, ac temporis.
Ut enim quies est conjunctio ejusdem puncti loci cum serie continua omnium moment-
orum ejus temporis, quo quies durat : sic etiam ilia virtualis extensio est conjunctio unius
momenti temporis cum serie continua omnium punctorum spatii, per quod simplex illud
ens virtualiter extenditur ; ut idcirco sicut ilia quies haberi creditur in Natura, ita & haec
virtualis extensio debeat admitti, qua admissa poterunt utique ilia primse materiae elementa
esse simplicia, & tamen non penitus inextensa.

Exciuditur virtu-

rite appiicato.

84. At ego quidem arbitror, hanc itidem sententiam everti penitus eodem inductionis
principio, ex quo alia tarn multa hucusque, quibus usi sumus, deduximus. Videmus enim
in his corporibus omnibus, quae observare possumus, quidquid distinctum occupat locum,
distinctum esse itidem ita, ut etiam satis magnis viribus adhibitis separari possint, quae
diversas occupant spatii partes, nee ullum casum deprehendimus, in quo magna haec corpora
partem aliquam habeant, quae eodem tempore diversas spatii partes occupet, & eadem
sit. Porro haec proprietas ex natura sua ejus generis est, ut aeque cadere possit in
magnitudines, quas per sensum deprehendimus, ac in magnitudines, quae infra sensuum
nostrorum limites sunt ; res nimirum pendet tantummodo a magnitudine spatii, per quod
haberetur virtualis extensio, quae magnitudo si esset satis ampla, sub sensus caderet. Cum
igitur nunquam id comperiamus in magnitudinibus sub sensum cadentibus, immo in
casibus innumeris deprehendamus oppositum : debet utique res transferri ex inductionis
principio supra exposito ad minimas etiam quasque materiae particulas, ut ne illae quidem
ejusmodi habeant virtualem extensionem.

Responsioadexem- [39] 85. Exempla, quae adduntur, petita ab anima rational}, & ab omnipraesentia
plum anima & Dei. j)ej} nj^ positive evincunt, cum ex alio entium genere petita sint ; praeterquam quod nee
illud demonstrari posse censeo, animam rationalem non esse unico tantummodo, simplici,
& inextenso corporis puncto ita praesentem, ut eundem locum obtineat, exerendo inde
vires quasdam in reliqua corporis puncta rite disposita, in quibus viribus partim necessariis,
& partim liberis, stet ipsum animae commercium cum corpore. Dei autem praesentia
cujusmodi sit, ignoramus omnino ; quem sane extensum per spatium divisibile nequaquam
dicimus, nee ab iis modis omnem excedentibus humanum captum, quibus ille existit,
cogitat, vult, agit, ad humanos, ad materiales existendi, agendique modos, ulla esse potest
analogia, & deductio.

itidem ad analo- 86. Quod autem pertinet ad analogiam cum quiete, sunt sane satis valida argumenta,

giam cum quiete. quibus, ut supra innui, ego censeam, in Natura quietem nullam existere. Ipsam nee posse

A THEORY OF NATURAL PHILOSOPHY 85

»

force ; but it would have an attractive force with regard to all others, at the very same
distances ; & this is in opposition to analogy. Secondly, if, due to the action of GOD
surpassing the forces of Nature, those parts are separated from one another, then urged
by the forces of Nature they would rush towards one another ; & we should have, from
their collision, a sudden change appertaining to Nature, although conveying a presumption
that something was done by the action of a supernatural force. Lastly, with this idea,
there would have to be two kinds of cohesion in Nature that were altogether different in
constitution ; one due to attraction at very small distances, & the other coming about
in a far different way in the case of masses of elementary particles, that is to say, due to
the limit-points of cohesion. Thus a theory would result that is far less simple & less
uniform than mine.

83. Taking it for granted, then, that the elements are simple & non-composite, whether the ele-
there can be no doubt as to whether they are also non-extended or whether, although ments are extended;

, , , , .' , 1-1 • i V certain arguments

simple, they have an extension of the kind that is termed virtual extension by the m favour of virtual

Scholastics. For there were some, especially among the Peripatetics, who admitted elements extension.

that were simple, lacking in all parts, & from their very nature perfectly indivisible ;

but, for all that, so extended through divisible space that some occupied more room than

others ; & such that in the position once occupied by one of them, if that one were

removed, two or even more others might be placed at the same time ; & even now there

are some who are of the same opinion. So also some thought that the rational soul in

man, which certainly is altogether indivisible, was diffused throughout the whole of the

body ; whilst others still consider that it is present throughout the whole of, indeed, a

smaller part of the body, but yet a part that is at any rate divisible & extended.

Further we believe that GOD Himself is present everywhere throughout the whole of the

undoubtedly divisible space that all bodies occupy ; & yet He is onefold in the highest

degree & admits not of any composite nature whatever. Moreover, the same idea seems

to depend on an analogy between space & time. For, just as rest is a conjunction with

a continuous series of all the instants in the interval of time during which the rest endures ;

so also this virtual extension is a conjunction of one instant of time with a continuous series

of all the points of space throughout which this one-fold entity extends virtually. Hence,

just as rest is believed to exist in Nature, so also are we bound to admit virtual extension ;

& if this is admitted, then it will be possible for the primary elements of matter to be

simple, & yet not absolutely non-extended.

84. But I have come to the conclusion that this idea is quite overthrown by that same virtual extension
principle of induction, by which we have hitherto deduced so many results which we have isr .excluded^ by the
employed. For we see, in all those bodies that we can bring under observation, that auction6 correctly
whatever occupies a distinct position is itself also a distinct thing ; so that those that occupy aPPlied-
different parts of space can be separated by using a sufficiently large force ; nor can we

detect a case in which these larger bodies have any part that occupies different parts of
space at one & the same time, & yet is the same part. Further, this property by its very
nature is of the sort for which it is equally probable that it happens in magnitudes that we can
detect by the senses & in magnitudes which are below the limits of our senses. In fact,
the matter depends only upon the size of the space, throughout which the virtual extension
is supposed to exist ; & this size, if it were sufficiently ample, would become sensible
to us. Since then we never find this virtual extension in magnitudes that fall within the
range of our senses, nay rather, in innumerable cases we perceive the contrary ; the matter
certainly ought to be transferred by the principle of induction, as explained above, to
any of the smallest particles of matter as well ; so that not even they are admitted to have
such virtual extension.

85. The illustrations that are added, derived from a consideration of the rational Reply to the
soul & the omnipresence of GOD, prove nothing positively ; for they are derived from s^uf&'cot)6 '
another class of entities, except that, I do not think that it can even be proved that the

rational soul does not exist in merely a single, simple, & non-extended point of the body ;
so that it maintains the same position, & thence it puts forth some sort of force into the
remaining points of the body duly disposed about it ; & the intercommunication between
the soul & the body consists of these forces, some of which are involuntary whilst others
are voluntary. Further, we are absolutely ignorant of the nature of the presence of GOD ;
& in no wise do we say that He is really extended throughout divisible space ; nor from
those modes, surpassing all human intelligence, by which HE exists, thinks, wills & acts,
can any analogy or deduction be made which will apply to human or material modes of
existence & action.

86. Again, as regards the analogy with rest, we have arguments that are sufficiently Again with regard

IT T i j i i i_ • t ^v • vr .. ' to the analogy with

strong to lead us to believe, as I remarked above, that there is no such thing m Nature rest.
as absolute rest. Indeed, I proved that such a thing could not be, by a direct argument

86 PHILOSOPHISE NATURALIS THEORIA

existere, argumento quodam positive ex numero combinationum possibilium infinite
contra alium finitum, demonstravi in Stayanis Supplementis, ubi de spatio, & tempore
quae juxta num. 66 occurrent infra Supplementorum § i, & § 2 ; numquam vero earn
existere in Natura, patet sane in ipsa Newtoniana sententia de gravitate generali, in qua in
planetario systemate ex mutuis actionibus quiescit tantummodo centrum commune gravi-
tatis, punctum utique imaginarium, circa quod omnia planetarum, cometarumque corpora
moventur, ut & ipse Sol ; ac idem accidit fixis omnibus circa suorum systematum gravitatis
centra ; quin immo ex actione unius systematis in aliud utcunque distans, in ipsa gravitatis
centra motus aliquis inducetur ; & generalius, dum movetur quaecunque materiae particula,
uti luminis particula qusecunque ; reliquae omnes utcunque remotae, quas inde positionem
ab ilia mutant, mutant & gravitatem, ac proinde moventur motu aliquo exiguo, sed sane
motu. In ipsa Telluris quiescentis sententia, quiescit quidem Tellus ad sensum, nee tota
ab uno in alium transfertur locum ; at ad quamcunque crispationem maris, rivuli decursum,
muscae volatum, asquilibrio dempto, trepidatio oritur, perquam exigua ilia quidem, sed
ejusmodi, ut veram quietem omnino impediat. Quamobrem analogia inde petita evertit
potius virtualem ejusmodi simplicium elementorum extensionem positam in conjunctione
ejusdem momenti temporis cum serie continua punctorum loci, quam comprobet.

in quo deficiat ana- 87. Sed nee ea ipsa analogia, si adesset, rem satis evinceret ; cum analogiam inter tempus,

logia loci, & tem- £ locum videamus in aliis etiam violari : nam in iis itidem paragraphis Supplementorum
demonstravi, nullum materiae punctum unquam redire ad punctum spatii quodcunque,
in quo semel fuerit aliud materiae punctum, ut idcirco duo puncta materiae nunquam
conjungant idem [40] punctum spatii ne cum binis quidem punctis temporis, dum quam-
plurima binaria punctorum materiae conjungunt idem punctum temporis cum duobus
punctis loci ; nam utique coexistunt : ac praeterea tempus quidem unicam dimensionem
habet diuturnitatis, spatium vero habet triplicem, in longum, latum, atque profundum.

inextensio utilis 88. Quamobrem illud jam tuto inferri potest, haec primigenia materiae elementa, non

ad exciudendum soium esse simplicia, ac indivisibilia, sed etiam inextensa. Et quidem haec ipsa simplicitas,

transitum momen- , t ; . i • «i_ j ji_

taneum a densitate & inextensio elementorum praestabit commoda sane plunma, quibus eadem adnuc magis
nuiia ad summam. fuicitur, ac comprobatur. Si enim prima elementa materiae sint quaedam partes solidse,
ex partibus compositae, vel etiam tantummodo extensae virtualiter, dum a vacuo spatio
motu continue pergitur per unam ejusmodi particulam, fit saltus quidam momentaneus
a densitate nulla, quae habetur in vacuo, ad densitatem summam, quae habetur, ubi ea
particula spatium occupat totum. Is vero saltus non habetur, si elementa simplicia sint,
& inextensa, ac a se invicem distantia. Turn enim omne continuum est vacuum tantum-
modo, & in motu continue per punctum simplex fit transitus a vacuo continue ad vacuum
continuum. Punctum illud materiae occupat unicum spatii punctum, quod punctum
spatii est indivisibilis limes inter spatium praecedens, & consequens. Per ipsum non
immoratur mobile continue motu delatum, nee ad ipsum transit ab ullo ipsi immediate
proximo spatii puncto, cum punctum puncto proximum, uti supra diximus, nullum sit ;
sed a vacuo continue ad vacuum continuum transitur per ipsum spatii punctum a materiae
puncto occupatum.

itidem ad hoc, ut go,. Accedit, quod in sententia solidorum, extensorumque elementorum habetur illud,

possit, ut p"test densitatem corporis minui posse in infinitum, augeri autem non posse, nisi ad certum limitem
minui in infinitum. in quo increment! lex necessario abrumpi debeat. Primum constat ex eo, quod eadem
particula continua dividi possit in particulas minores quotcunque, quae idcirco per spatium
utcunque magnum diffundi potest ita, ut nulla earum sit, quae aliquam aliam non habeat
utcunque libuerit parum a se distantem. Atque eo pacto aucta mole, per quam
eadem ilia massa diffusa sit, eaque aucta in ratione quacunque minuetur utique
densitas in ratione itidem utcunque magna. Patet & alterum : ubi enim omnes
particulae ad contactum devenerint ; densitas ultra augeri non poterit. Quoniam
autem determinata quaedam erit utique ratio spatii vacui ad plenum, nonnisi in ea ratione
augeri poterit densitas, cujus augmentum, ubi ad contactum deventum fuerit, adrumpetur.
At si elementa sint puncta penitus indivisibilia, & inextensa ; uti augeri eorum distantia
poterit in infinitum, ita utique poterit etiam minui pariter in ratione quacunque ; cum

A THEORY OF NATURAL PHILOSOPHY 87

founded upon the infiniteness of a number of possible combinations as against the finiteness
of another number, in the Supplements to Stay's Philosophy, in connection with space
& time ; these will be found later immediately after Art. 14 of the Supplements, §§ I
and II. That it never does exist in Nature is really clear in the Newtonian theory of
universal gravitation ; according to this theory, in the planetary system the common centre
of gravity alone is at rest under the action of the mutual forces ; & this is an altogether
imaginary point, about which all the bodies of the planets & comets move, as also does
the sun itself. Moreover the same thing happens in the case of all the fixed stars with regard
to the centres of gravity of their systems ; & from the action of one system on another
at any distance whatever from it, some motion will be imparted to these very centres of
gravity. More generally, so long as any particle of matter, so long as any particle of light,
is in motion, all other particles, no matter how distant, which on account of this motion
have their distance from the first particle altered, must also have their gravitation altered,
& consequently must move with some very slight motion, but yet a true motion. In
the idea of a quiescent Earth, the Earth is at rest approximately, nor is it as a whole translated
from place to place ; but, due to any tremulous motion of the sea, the downward course
of rivers, even to the fly's flight, equilibrium is destroyed & some agitation is produced,
although in truth it is very slight ; yet it is quite enough to prevent true rest altogether.
Hence an analogy deduced from rest contradicts rather than corroborates virtual extension
of the simple elements of Nature, on the hypothesis of a conjunction of the same instant
of time with a continuous series of points of space.

87. But even if the foregoing analogy held good, it would not prove the matter Where the analogy
satisfactorily ; since we see that in other ways the analogy between space & time is impaired. 2^pace and tlme
For I proved, also in those paragraphs of the Supplements that I have mentioned, that

no point of matter ever returned to any point of space, in which there had once been any
other point of matter ; so that two points of matter never connected the same point of
space with two instants of time, let alone with more ; whereas a huge number of pairs of
points connect the same instant of time with two points of space, since they certainly coexist.
Besides, time has but one dimension, duration ; whilst space has three, length, breadth
& depth.

88. Therefore it can now be safely accepted that these primary elements of matter Non-extension use-
are not only simple & indivisible, but also that they are non-extended. Indeed this aun \nstanTaneous
very simplicity & non-extension of the elements will prove useful in a really large number passage from • no •
of cases for still further strengthening & corroborating the results already obtained. J^-one.0 a Very
For if the primary elements were certain solid parts, themselves composed of parts or even

virtually extended only, then, whilst we pass by a continuous motion from empty space
through one particle of this kind, there would be a sudden change from a density that is
nothing when the space is empty, to a density that is very great when the particle occupies
the whole of the space. But there is not this sudden change if we assume that the elements
are simple, non-extended & non-adjacent. For then the whole of space is merely a
continuous vacuum, &, in the continuous motion by a simple point, the passage is made
from continuous vacuum to continuous vacuum. The one point of matter occupies but
one point of space ; & this point of space is the indivisible boundary between the space
that precedes & the space that follows. There is nothing to prevent the moving point
from being carried through it by a continuous motion, nor from passing to it from any
point of space that is in immediate proximity to it : for, as I remarked above, there
is no point that is the next point to a given point. But from continuous vacuum
to continuous vacuum the passage is made through that point of space which is occupied
by the point of matter.

89. There is also the point, that arises in the theory of solid extended elements, namely Also for the idea
that the density of a body can be diminished indefinitely, but cannot be increased except j^^a'^ ^can
up to a certain fixed limit, at which the law of increase must be discontinuous. The first be decreased,
comes from the fact that this same continuous particle can be divided into any number mdefinltely-

of smaller particles ; these can be diffused through space of any size in such a way that
there is not one of them that does not have some other one at some little (as little as you
will) distance from itself. In this way the volume through which the same mass is diffused
is increased ; & when that is increased in any ratio whatever, then indeed the density
will be diminished in the same ratio, no matter how great the ratio may be. The second
thing is also evident ; for when the particles have come into contact, the density cannot
be increased any further. Moreover, since there will undoubtedly be a certain determinate
ratio for the amount of space that is empty compared with the amount of space that is
full, the density can only be increased in that ratio ; & the regular increase of density
will be arrested when contact is attained. But if the elements are points that are perfectly
indivisible & non-extended, then, just as their distances can be increased indefinitely,

88 PHILOSOPHIC NATURALIS THEORIA

in [41] ratione quacunque lineola quaecunque secari sane possit : adeoque uli nullus est
limes raritatis auctae, ita etiam nullus erit auctae densitatis.

Et ad excludendum 9°- Sed & illud commodum accidet, quod ita omne continuum coexistens eliminabitur

continuum extcn- e Natura, in quo explicando usque adeo dcsudarunt, & fere incassum, Philosophi, ncc idcirco

sum, & in infinitum j« • • « **• • • • r • j • • i • i •

in existentibus. divisio ulla realis entis in innmtum produci potent, nee naerebitur, ubi quaeratur, an numerus
partium actu distinctarum, & separabilium, sit finitus, an infinitus ; nee alia ejusmodi
sane innumera, quae in continui compositione usque adeo negotium facessunt Philosophis,
jam habebuntur. Si enim prima materiae elementa sint puncta penitus inextensa, &
indivisibilia, a se invicem aliquo intervallo disjuncta ; jam erit finitus punctorum numerus
in quavis massa : nam distantiae omnes finitae erunt ; infinitesimas enim quantitates in se
determinatas nullas esse, satis ego quidem, ut arbitror, luculcnter demonstravi & in disser-
tatione De Natura, t$ Usu infinitorum, ac infinite parvorum, & in dissertatione DC Lege
Continuitatis, & alibi. Intervallum quodcunque finitum erit, & divisibile utique in
infinitum per interpositionem aliorum, atque aliorum punctorum, quae tamen singula,
ubi fuerint posita, finita itidem erunt, & aliis pluribus, finitis tamen itidem, ubi extiterint,
locum reliquent, ut infinitum sit tantummodo in possibilibus, non autem in existentibus,
in quibus possibilibus ipsis omnem possibilium seriem idcirco ego appellare soleo constantem
terminis finitis in infinitum, quod quaecunque, quae existant, finita esse debeant, sed nullus
sit existentium finitus numerus ita ingens, ut alii, & alii majores, sed itidem finiti, haberi
non possint, atque id sine ullo limite, qui nequeat praeteriri. Hoc autcm pacto, sublato
ex existentibus omni actuali infinite, innumerae sane difficultates auferentur.

inextensionem 91. Cum igitur & positive argumento. a lege virium positive demonstrata desumpto,

qua'rend^m^e simplicitas, & inextensio primorum materiae elementorum deducatur, £ tam multis aliis

homogeneitate. vel indiciis fulciatur, vel emolumentis inde derivatis confirmetur ; ipsa itidem admitti

jam debet, ac supererit quaerendum illud tantummodo, utrum haec elementa homogenca

censeri debeant, & inter se prorsus similia, ut ea initio assumpsimus, an vero heterogenea,

ac dissimilia.

Homogeneitatem 92. Pro homogeneitate primorum materiae elementorum illud est quoddani veluti

genefta1teaprim°i!n(& Prmcipium, quod in simplicitate, & inextensione conveniant, ac etiam vires quasdam habeant
uitimi asymptotici utique omnia. Deinde curvam ipsam virium eandem esse omnino in omnibus illud indicat,
omnibus'0 P"' S ve^ etiani evincit, quod primum crus repulsivum impenetrabilitatem secum trahens, &
postremum attractivum gravitatem definiens, omnino communia in omnibus sint : nam
corpora omnia aeque impenetrabilia sunt, & vero etiam aeque gravia pro quantitate materiae
suae, uti satis [42] evincit aequalis velocitas auri, & plumse cadentis in Boyliano recipiente
Si reliquus curvae arcus intermedius esset difformis in diversis materiae punctis ; infinities
probabilius esset, difformitatem extendi etiam ad crus primum, & ultimum, cum infinities
plures sint curvae, quae, cum in reliquis differant partibus, differant plurimum etiam in
hisce extremis, quam quae in hisce extremis tantum modo tam arete consentiant. Et hoc
quidem argumento illud etiam colligitur, curvam virium in quavis directione ab eodem
primo materiae elemento, nimirum ab eodem materiae puncto eandem esse, cum & primum
impenetrabilitatis, & postremum gravitatis crus pro omnibus directionibus sit ad sensum
idem. Cum primum in dissertatione De Firibus Vivis hanc Theoriam protuli, suspicabar
diversitatem legis ' virium respondentis diversis directionibus ; sed hoc argumento adi
majorem simplicitatem, & uniformitatem deinde adductus sum. Diversitas autem legum
virium pro diversis particulis, & pro diversis respectu ejusdem particulae directionibus,
habetur utique ex diverso numero, & positione punctorum earn componentium, qua de
re inferius aliquid.

i contra deduci 93- Nee vero huic homogeneitati opponitur inductionis principium, quo ipsam
ex principio indis- Leibnitiani oppugnare solent, nee principium rationis sufficients, atque indiscernibilium,

cermbUium, & rati- . • ° . T £ • TV- • • /"• j- • -j

onis sufficients. quod supenus innui numero 3. Innmtam Divini v_onditons mentem, ego quidem omnino.
arbitror, quod & tam multi Philosophi censuerunt, ejusmodi perspicacitatem habere, atque
intuitionem quandam, ut ipsam etiam, quam individuationem appellant, omnino similium
individuorum cognoscat, atque ilia inter se omnino discernat. Rationis autem sufficientis

A THEORY OF NATURAL PHILOSOPHY 89

so also can they just as well be diminished in any ratio whatever. For it is certainly possible
that a short line can be divided into parts in any ratio whatever ; & thus, just as there
is no limit to increase of rarity, so also there is none to increase of density.

qo. The theory of non-extension is also convenient for eliminating from Nature all ^lso-/or excludms

7 / 1 • 1 • 1 1 •! 1 1 Ml 11 1 6 *"ea ° a C011"

idea of a coexistent continuum — to explain which philosophers have up till now laboured tinuum in existing

so very hard & generally in vain. Assuming non-extension, no division of a real entity thmRs- that

can be carried on indefinitely ; we shall not be brought to a standstill when we seek to
find out whether the number of parts that are actually distinct & separable is finite or
infinite ; nor with it will there come in any of those other truly innumerable difficulties
that, with the idea of continuous composition, have given so much trouble to philosophers.
For if the primary elements of matter are perfectly non-extended & indivisible points
separated from one another by some definite interval, then the number of points in any
given mass must be finite ; because all the distances are finite. I proved clearly enough,
I think, in the dissertation De Natura, & Usu infinitorum ac infinite parvorum, & in the
dissertation De Lege Continuitatis, & in other places, that there are no infinitesimal
quantities determinate in themselves. Any interval whatever will be finite, & at least
divisible indefinitely by the interpolation of other points, & still others ; each such set
however, when they have been interpolated, will be also finite in number, & leave room
for still more ; & these too, when they existed, will also be finite in number. So that
there is only an infinity of possible points, but not of existing points ; & with regard
to these possible points, I usually term the whole series of possibles a series that ends at
finite limits at infinity. This for the reason that any of them that exist must be finite
in number ; but there is no finite number of things that exist so great that other numbers,
greater & greater still, but yet all finite, cannot be obtained ; & that too without any
limit, which cannot be surpassed. Further, in this way, by doing away with all idea of
an actual infinity in existing things, truly countless difficulties are got rid of.

91. Since therefore, by a direct argument derived from a law of forces that has been Non-extension
directly proved, we have both deduced the simplicity & non-extension of the primary w"5 havea now e to
elements of matter, & also we have strengthened the theory by evidence pointing towards investigate homo-
it, or corroborated it by referring to the advantages to be derived from it ; this theory gen

ought now to be accepted as true. There only remains the investigation as to whether
these elements ought to be considered to be homogeneous & perfectly similar to one
another, as we assumed at the start, or whether they are really heterogeneous & dissimilar.

92. In favour of the homogeneity of the primary elements of matter we have so to Homogeneity for
speak some foundation derived from the fact that all of them agree in simplicity & non- Voca!tedStf0romaa
extension, & also that they are all endowed with forces of some sort. Now, that this consideration of
curve of forces is exactly the same for all of them is indicated or even proved by the fact Of 6the °fir?t86™ last
that the first repulsive branch necessitating impenetrability, & the last attractive branch a s y"m p t o t i c
determining gravitation, are exactly the same in all respects. For all bodies are equally c^l S forces*
impenetrable ; & also all are equally heavy in proportion to the amount of matter

contained in them, as is sufficiently proved by the equal velocity of the piece of gold &
the feather when falling in Boyle's experiment. If the remaining intermediate arc of the
curve were non-uniform for different points of matter, it would be infinitely more probable
that the non-uniformity would extend also to the first & last branches also ; for there
are infinitely more curves which, when they differ in the remaining parts, also differ to
the greatest extent in the extremes, than there are curves, which agree so closely only in
these extremes. Also from this argument we can deduce that the curve of forces is indeed
exactly the same from the same point of matter, in any direction whatever from the same
primary element of matter ; for both the first branch of impenetrability & the last branch
of gravitation are the same, so far as we can perceive, for all directions. When I first
published this Theory in my dissertation De Firibus Fivis, I was inclined to believe that
there was a diversity in the law of forces corresponding to diversity of direction ; but I
was led by the argument given above to the greater simplicity & the greater uniformity
derived therefrom. Further, diversity of the laws of forces for diverse particles, & for
different directions with the same particle, is certainly to be obtained from the diverse
number & position of the points composing it ; about which I shall have something
to say later.

93. Nor indeed is there anything opposed to this idea of homogeneity to be derived Notl?ins t? b«

r i • • i r • i J i o rr t o /_ . brought against

from the principle of induction, by means of which the followers of Leibniz usually raise this from the doc-
an objection to it ; nor from the principle of sufficient reason, & of indiscernibles, that fc™es°f .indj.scern:

J . , . . . -rr • i i • -TO i_ ibles & sufficient

1 mentioned above in Art. 3. I am indeed quite convinced, & a great many other reason.1
philosophers too have thought, that the Infinite Will of the Divine Founder has a
perspicacity & an intuition of such a nature that it takes cognizance of that which is
called individuation amongst individuals that are perfectly similar, & absolutely

90

PHILOSOPHIC NATURALIS THEORIA

principium falsum omnino esse censeo, ac ejusmodi, ut omnem verse libertatis ideam omnino
tollat ; nisi pro ratione, ubi agitur de voluntatis determinatione, ipsum liberum arbitrium,
ipsa libera determinatio assumatur, quod nisi fiat in voluntate divina, quaccunque existunt,
necessario existunt, & qusecunque non existunt, ne possibilia quidem erunt, vera aliqua
possibilitate, uti facile admodum demonstratur ; quod tamen si semel admittatur, mirum
sane, quam prona demum ad fatalem necessitatem patebit via. Quamobrem potest divina
voluntas determinari ex toto solo arbitrio suo ad creandum hoc individuum potius, quam
illud ex omnibus omnino similibus, & ad ponendum quodlibet ex iis potius eo loco, quo
ponit, quam loco alterius. Sed de rationis sufficientis principio haec ipsa fusius pertractavi
turn in aliis locis pluribus, turn in Stayanis Supplementis, ubi etiam illud ostendi, id prin-
cipium nullum habere usum posse in iis ipsis casibus, in quibus adhibetur, & praedicari solet
tantopere, atque id idcirco, quod nobis non innotescant rationes omnes, quas tamen
oporteret utique omnes nosse ad hoc, ut eo principio uti possemus, amrmando, nullam
esse rationem sufncientem pro hoc potius, quam pro illo [43] alio : sane in exemplo illo
ipso, quod adhiberi solet, Archimedis hoc principio aequilibrium determinantis, ibidem
ostendi, ex ignoratione causarum, sive rationum, quse postea detectae sunt, ipsum in suae
investigationis progressu errasse plurimum, deducendo per abusum ejus principii sphsericam
figuram marium, ac Telluris.

combinatiombus.

Posse etiam puncta 94. Accedit & illud, quod ilia puncta materiae, licet essent prorsus similia in simplicitate,

dlfierrTin aiiis 11S> & extensione, ac mensura virium, pendentium a distantia, possent alias habere proprietates

metaphysicas diversas inter se, nobis ignotas, quae ipsa etiam apud ipsos Leibnitianos

discriminarent.

Non vaierehicprin- 95. Quod autem attinet ad inductionem, quam Leibnitiani desumunt a dissimilitudine,

a^ma^sis^eas^de! quam observamus in rebus omnibus, cum nimirum nusquam ex. gr. in amplissima silva reperire
ferre ex diversis sit duo folia prorsus similia ; ea sane me nihil movet ; cum nimirum illud discrimen sit
prOprietas relativa ad rationem aggregati, & nostros sensus, quos singula materiae elementa
non afficiunt vi sufficiente ad excitandam in animo ideam, nisi multa sint simul, & in molem
majorem excrescant. Porro scimus utique combinationes ejusdem numeri terminorum
in immensum excrescere, si ille ipse numerus sit aliquanto major. Solis 24 litterulis
Alphabet! diversimodo combinatis formantur voces omnes, quibus hue usque usa sunt
omnia idiomata, quae extiterunt, & quibus omnia ilia, quae possunt existere, uti possunt.
Quid si numerus earum existeret tanto major, quanto major est numerus puuctorum
materiae in quavis massa sensibili ? Quod ibi diversus est litterarum diversarum ordo, id
in punctis etiam prorsus homogeneis sunt positiones, & distantia, quibus variatis, variatur
utique forma, & vis, qua sensus afficitur in aggregatis. Quanto major est numerus
combinationum diversarum possibilium in massis sensibilibus, quam earum massarum, quas
possumus observare, & inter se conferre (qui quidem ob distantias, & directiones in infinitum
variabiles praescindendo ab aequilibrio virium, est infinitus, cum ipso aequilibrio est immen-
sus) ; tanto major est improbabilitas duarum massarum omnino similium, quam omnium
aliquantisper saltern inter se dissimilium.

Physica ratio dis- 96. Et quidem accedit illud etiam, quod alicujus dissimilitudinis in aggregatis physicam

massU1ut1in1foriiuUS I1100!116 rationem cernimus in iis etiam casibus, in quibus maxime inter se similia esse
deberent. Cum enim mutuae vires ad distantias quascunque pertineant ; status uniuscu-
jusque puncti pendebit saltern aliquantisper a statu omnium aliorum punctorum, quae
sunt in Mundo. Porro utcunque puncta quaedam sint parum a se invicem remota, uti
sunt duo folia in eadem silva, & multo magis in eodem ramo ; adhuc tamen non eandem
prorsus relationem distantiae, & virium habent ad reliqua omnia materiae puncta, quae
[44] sunt in Mundo, cum non eundem prorsus locum obtineant ; & inde jam in aggregate
discrimen aliquod oriri debet, quod perfectam similitudinem omnino impediat. Sed illud
earn inducit magis, quod quae maxime conferunt ad ejusmodi dispositionem, necessario
respectu diversarum frondium diversa non nihil esse debeant. Omissa ipsa earum forma
in semine, solares radii, humoris ad nutritionem necessarii quantitas, distantia, a qua debet
is progredi, ut ad locum suum deveniat, aura ipsa, & agitatio inde orta, non sunt omnino
similia, sed diversitatem aliquam habent, ex qua diversitas in massas inde efformatas
redundat.

A THEORY OF NATURAL PHILOSOPHY 91

distinguishes them one from the other. Moreover, I consider that the principle of sufficient
reason is altogether false, & one that is calculated to take away all idea of true freewill.
Unless free choice or free determination is assumed as the basis of argument, in discussing
the determination of will, unless this is the case with the Divine Will, then, whatever
things exist, exist because they must do so, & whatever things do not exist will not even
be possible, i.e., with any real possibility, as is very easily proved. Nevertheless, once this
idea is accepted, it is truly wonderful how it tends to point the way finally to fatalistic
necessity. Hence the Divine Will is able, of its own pleasure alone, to be determined
to the creation of one individual rather than another out of a whole set of exactly similar
things, & to the setting of any one of these in the place in which it puts it rather than in
the place of another. But I have discussed these very matters more at length, besides several
other places, in the Supplements to Stay's Philosophy ; where I have shown that the
principle cannot be employed in those instances in which it is used & generally so strongly
asserted. The reason being that all possible reasons are not known to us ; & yet they
should certainly be known, to enable us to employ the principle by stating that there is
no sufficient reason in favour of this rather than that other. In truth, in that very example
of the principle generally given, namely, that of Archimedes' determination of equilibrium
by means of it, I showed also that Archimedes himself had made a very big mistake in following
out his investigation because of his lack of knowledge of causes or reasons that were discovered
in later days, when he deduced a spherical figure for the seas & the Earth by an abuse
of this principle.

94. There is also this, that these points of matter, although they might be perfectly it is possible for
similar as regards simplicity & extension, & in having the measure of their forces depen- ^"^ese^ro erties
dent on their distances, might still have other metaphysical properties different from one but to disagree in
another, & unknown to us ; & these distinctions also are made by the followers of others-
Leibniz.

95. As regards the induction which the followers of Leibniz make from the lack of The principle does
similitude that we see in all things, (for instance such as that there never can be found in n°t.hold g°°d here

T_ i j i i vi \ i • i • . , °* induction from

the largest wood two leaves exactly alike), their argument does not impress me in the masses; they differ

slightest degree. For that distinction is a property that is concerned with reasoning for °.n account of

an aggregate, & also with our senses ; & these senses single elements of matter cannot tionsof their parta.

influence with sufficient force to excite an idea in the mind, except when there are many

of them together at a time, & they develop into a mass of considerable size. Further

it is well known that combinations of the same number of terms increase enormously, if

that number itself increase a little. From the 24 letters of the alphabet alone, grouped

together in different ways, are formed all the words that have hitherto been used in all

expressions that have existed, or can possibly come into existence. What then if their

number were increased to equal the number of points of matter in any sensible mass ?

Corresponding to the different order of the several letters in the one, we have in perfectly

homogeneous points also different positions & distances ; & if these are altered at least

the form & the force, which affect our senses in the groups, are altered as well. How

much greater is the number of different combinations that are possible in sensible masses

than the number of those masses that we can observe & compare with one another (&

this number, on account of the infinitely variable distances & directions of the forces,

when equilibrium is precluded, is infinite, since including equilibrium it is very great) ;

just so much greater is the improbability of two masses being exactly similar than of

their being all at least slightly different from one another.

96. There is also this point in addition ; we discern a physical reason as well for some Physical reason for
dissimilarity in groups for those cases too, in which they ought to be especially similar to the difference in

.1 -n • i f • 11 -11 T r i ' several masses, as

one another, ror since mutual forces pertain to all possible distances, the state of any in leaves.
one point will depend upon, at least in some slight degree, the state of all other points
that are in the universe. Further, however short the distance between certain points may
be, as of two leaves in the same wood, much more so on the same branch, still for all
that they do not have quite the same relation as regards distance & forces as all the rest
of the points of matter that are in the universe, because they do not occupy quite the
same place. Hence in a group some distinction is bound to arise which will entirely prevent
perfect similarity. Moreover this tendency is all the stronger, because those things which
especially conduce to this sort of disposition must necessarily be somewhat different with
regard to different leaves. For the form itself being absent in the seed, the rays of the
sun, the quantity of moisture necessary for nutrition, the distance from which it has to
proceed to arrive at the place it occupies, the air itself & the continual motion derived
from this, these are not exactly similar, but have some diversity ; & from this diversity
there proceeds a diversity in the masses thus formed.

92 PHILOSOPHIC NATURALIS THEORIA

simiiitudine quaii- 97. Patet igitur, varietatem illam a numero pendere combinationum possibilium in

^ numero punctorum necessario ad sensationem, & circumstantiarum, quae ad formationem

geneitatem, quam massze sunt neccssariae, adeoque ejusmodi inductionem extend! ad elementa non posse.

* '

Quin immo ilia tanta similitude, quae cum exigua dissimilitudine commixta invenitur in
tarn multis corporibus, indicat potius similitudinem ingentem in elementis. Nam ob
tantum possibilium combinationum numerum, massae elementorum etiam penitus homo-
geneorum debent a se invicem differre plurimum, adeoque si elementa heterogenea sint,
in immensum majorem debent habere dissimilitudinem, quam ipsa prima elementa, ex
quibus idcirco nullae massas, ne tantillum quidem, similes provenire deberent. Cum
elementa multo minus dissimilia esse debeant, quam aggregata elementorum, multo
magis ad elementorum homogeneitatem valere debet ilia quaecunque similitudo, quam
in corporibus observamus, potissimum in tarn multis, quae ad eandem pertinent speciem,
quam ad homogeneitatem eorundem tarn exiguum illud discrimen, quod in aliis tarn
multis observatur. Rem autem penitus conficit ilia tanta similitudo, qua superius usi
sumus, in primo crure exhibente impenetrabilitatem, & in postremo exhibente gravitatem
generalem, quae crura cum ob hasce proprietates corporibus omnibus adeo generales, adeo
inter se in omnibus similia sint, etiam reliqui arcus curvae exprimentis vires omnimodam
similitudinem indicant pro corporibus itidem omnibus.

Homogeneitatem 98. Superest, quod ad hanc rem pertinet, illud unum iterum hie monendum, quod

insinuarr' ^xem* ipsum etiam initio hujus Operis innui, ipsam Naturam, & ipsum analyseos ordinem nos

plum a libris, lit- ducere ad simplicitatem & homogeneitatem elementorum, cum nimirum, quo analysis

ns> pul promovetur magis, eo ad pauciora, & inter se minus discrepantia principia deveniatur, uti

patet in resolutionibus Chemicis. Quam quidem rem ipsum litterarum, & vocum exemplum

multo melius animo sistet. Fieri utique possent nigricantes litteras, non ductu atramenti

continue, sed punctulis rotundis nigricantibus, & ita parum a se invicem remotis, ut inter-

valla non nisi ope microscopii discerni possent, & quidem ipsae litterarum formae pro typis

fieri pos-[45]-sent ex ejusmodi rotundis sibi proximis cuspidibus constantes. Concipiatur

ingens quaedam bibliotheca, cujus omnes libri constent litteris impressis, ac sit incredibilis

in ea multitude librorum conscriptorum linguis variis, in quibus omnibus forma charac-

terum sit eadem. Si quis scripturae ejusmodi, & linguarum ignarus circa ejusmodi libros,

quos omnes a se invicem discrepantes intueretur, observationem institueret cum diligenti

contemplatione ; primo quidem inveniret vocum farraginem quandam, quae voces in

quibusdam libris occurrerent saepe, cum eaedem in aliis nusquam apparent, & inde lexica

posset quaedam componere totidem numero, quot idiomata sunt, in quibus singulis omnes

ejusdem idiomatis voces reperirentur, quae quidem numero admodum pauca essent, discri-

mine illo ingenti tot, tarn variorum librorum redacto ad illud usque adeo minus discrimen,

quod contineretur lexicis illis, & haberetur in vocibus ipsa lexica constituentibus. At

inquisitione promota, facile adverteret, omnes illas tarn varias voces constare ex 24

tantummodo diversis litteris, discrimen aliquod inter se habentibus in ductu linearum,

quibus formantur, quarum combinatio diversa pareret omnes illas voces tarn varias, ut

earum combinatio libros efformaret usque adeo magis a se invicem discrepantes. Et ille

quidem si aliud quodcunque sine microscopic examen institueret, nullum aliud inveniret

magis adhuc simile elementorum genus, ex quibus diversa ratione combinatis orirentur

ipsae litterse ; at microscopic arrepto, intueretur utique illam ipsam litterarum composi-

tionem e punctis illis rotundis prorsus homogeneis, quorum sola diversa positio, ac

distributio litteras exhiberet.

Appiicatio exempli 99. Haec mihi quaedam imago videtur esse eorum, quae cernimus in Natura. Tarn

a<^ Naturae analy- mu\t{} tam var;j fift ijbrj corpora sunt, & quae ad diversa pertinent regna, sunt tanquam
diversis conscripta linguis. Horum omnium Chemica analysis principia quaedam invenit
minus inter se difrormia, quam sint libri, nimirum voces. Hae tamen ipsae inter se habent
discrimen aliquod, ut tam multas oleorum, terrarum, salium species eruit Chemica analysis
e diversis corporibus. Ulterior analysis harum, veluti vocum, litteras minus adhuc inter
se difformes inveniret, & ultima juxta Theoriam meam deveniret ad homogenea punctula,
quae ut illi circuli nigri litteras, ita ipsa diversas diversorum corporum particulas per solam
dispositionem diversam efformarent : usque adeo analogia ex ipsa Naturae consideratione

A THEORY OF NATURAL PHILOSOPHY 93

97. It is clear then that this variety depends on the number of possible combinations Homogeneity is to
to be found for the number of points that are necessary to make the mass sensible, & ^m d<^° ™ort * ot
of the circumstances that arenecessary for the formation of the mass ; & so it is not similitude in some
possible that the induction should be extended to the elements. Nay rather, the great heterogeneity from"
similarity that is found accompanied by some very slight dissimilarity in so many bodies dissimilarity.
points more strongly to the greatest possible similarity of the elements. For on account

of the great number of the possible combinations, even masses of elements that are perfectly
homogeneous must be greatly different from one another ; & thus if the elements are
heterogeneous, the masses must have an immensely greater dissimilarity than the primary
elements themselves ; & therefore no masses formed from these ought to come out similar,
not even in the very slightest degree. Since the elements are bound to be much less
dissimilar than aggregates formed from these elements, homogeneity of the elements must
be indicated by that certain similarity that we observe in bodies, especially in so many
of those that belong to the same species, far more strongly than heterogeneity of the elements
is indicated by the slight differences that are observed in so many others. The whole
discussion is made perfectly complete by that great similarity, which we made use of above,
that exists in the first branch representing impenetrability, & in the last branch representing
universal gravitation ; for since these branches, on account of properties that are so general
to all bodies, are so similar to one another in all cases, they indicate complete similarity
of the remaining arc of the curve expressing the forces for all bodies as well.

98. Naught that concerns this subject remains but for me to once more mention in Homogeneity is
this connection that one thing, which I have already remarked at the beginning of this anftysis of Nature"
work, namely, that Nature itself & the method of analysis lead us towards simplicity & example taken
homogeneity of the elements ; since in truth the farther the analysis is pushed, the fewer ancj dots° '

the fundamental substances we arrive at & the less they differ from one another ; as is
to be seen in chemical experiments. This will be presented to the mind far more clearly
by an illustration derived from letters & words. Suppose we have made black letters,
not by drawing a continuous line with ink, but by means of little black dots which are at
such small distances from one another that the intervals cannot be perceived except with
the aid of a microscope — & indeed such forms of letters may be made as types from round
points of this sort set close to one another. Now imagine that we have a huge library,
all the books in it consisting of printed letters, & let there be an incredible multitude
of books printed in various languages, in all which the form of the characters is the same.
If anyone, who was ignorant of such compositions or languages, started on a careful study
of books of this kind, all of which he would perceive differed from one another ; then first
of all he would find a medley of words, some of which occurred frequently in certain books
whilst they never appeared at all in others. Hence he could compose lexicons, as many
in number as there are languages ; in each of these all words of the same language would
be found, & these would indeed be very few in number ; for the immense multiplicity
of words in this numerous collection of books of so many kinds is now reduced to what
is still a multiplicity, but smaller, than is contained in the lexicons & the words forming
these lexicons. Now if he continued his investigation, he would easily perceive that the
whole of these words of so many different kinds were formed from 24 letters only ; that
these differed in some sort from one another in the manner in which the lines forming
them were drawn ; that the different combinations of these would produce the whole of
that great variety of words, & that combinations of these words would form books differing
from one another still more widely. Now if he made yet another examination without the
aid of a microscope, he would not find any other kind of elements that were more similar
to one another than these letters, from a combination of which in different ways the letters
themselves could be produced. But if he took a microscope, then indeed would he see
the mode of formation of the letters from the perfectly homogeneous round points, by
the different position & distribution of which the letters were depicted.

99. This seems to me to be a sort of picture of what we perceive in Nature. Those Application of the

i,-7-7 . ,..,. r., i T n i 1-111 illustration to the

books, so many m number & so different in character are bodies, & those which belong analysis of Nature.

to the different kingdoms are written as it were in different tongues. Of all of these,

chemical analysis finds out certain fundamental constituents that are less unlike one another

than the books ; these are the words. Yet these constituent substances have some sort

of difference amongst themselves, & thus chemical analysis produces a large number of

species of oils, earths & salts from different bodies. Further analysis of these, like that

of the words, would disclose the letters that are still less unlike one another ; & finally,

according to my Theory, the little homogeneous points would be obtained. These, just

as the little black circles formed the letters, would form the diverse particles of diverse

bodies through diverse arrangement alone. So far then the analogy derived from such a

94 PHILOSOPHIC NATURALIS THEORIA

derivata non ad difformitatem, sed ad conformitatem elementorum nos ducit.

Transitus a pro- ioo. Atque hoc demum pacto ex principiis certis & vulgo receptis, per legitimam,

ad consectariorum seriem devenimus ad omnem illam, quam initio proposui, Theoriam,
nimirum ad legem virium mutuarum, & ad constitutionem primorum materiae elementorum
ex ilia ipsa virium lege derivatorum. [46] Videndum jam superest, quam uberes inde
fructus per universam late Physicam colligantur, explicatis per earn unam praecipuis cor-
porum proprietatibus, & Naturae phaenomenis. Sed antequam id aggredior, praecipuas
quasdam e difficultatibus, quae contra Theoriam ipsam vel objectae jam sunt, vel in oculos
etiam sponte incurrunt, dissolvam, uti promisi.

Legem virium non ioi. Contra vires mutuas illud sclent objicere, illas esse occultas quasdam qualitates,

in distans,anec0esse ve^ etiam actionem in distans inducere. His satisfit notione virium exhibita numero 8,
occuitam quaiita- & 9. Illud unum praeterea hie addo, admodum manifestas eas esse, quarum idea admodum
facile efformatur, quarum existentia positive argumento evincitur, quarum effectus multi-
plices continue oculis observantur. Sunt autem ejusmodi hae vires. Determinationis
ad accessum, vel recessum idea efformatur admodum facile. Constat omnibus, quid sit
accedere, quid recedere ; constat, quid sit esse indifferens, quid determinatum ; adeoque
& determinationis ad accessum, vel recessum habetur idea admodum sane distincta.
Argumenta itidem positiva, quae ipsius ejusmodi determinationis existentiam probant,
superius prolata sunt. Demum etiam motus varii, qui ab ejusmodi viribus oriuntur, ut
ubi corpus quoddam incurrit in aliud corpus, ubi partem solidi arreptam pars alia sequitur,
ubi vaporum, vel elastrorum particulae se invicem repellunt, ubi gravia descendunt, hi
motus, inquam, quotidie incurrant in oculos. Patet itidem saltern in genere forma curvae
ejusmodi vires exprimentis. Haec omnia non occuitam, sed patentem reddunt ejusmodi
virium legem.

Quid adhuc lateat : IO2. Sunt quidem adhuc quaedam, quae ad earn pertinent, prorsus incognita, uti est

admittendam om- numerus, & distantia intersectionum curvae cum axe, forma arcuum intermediorum, atque

nino : quo pacto .. . ,. -11 i -11 i i i •

evitetur hie actio alia ejusmodi, quae quidem longe superant humanum captum, & quas me solus habuit
in distans. omnia simul prae oculis, qui Mundum condidit ; sed id omnino nil officit. Nee sane

id ipsum in causa esse debet, ut non admittatur illud, cujus existentiam novimus, & cujus
proprietates plures, & effectus deprehendimus ; licet alia multa nobis incognita eodem
pertinentia supersint. Sic aurum incognitam, occultamque substantiam nemo appellant,
& multo minus ejusdem existentiam negabit idcirco, quod admodum probabile sit, plures
alias latere ipsius proprietates, olim forte detegendas, uti'aliae tarn multae subinde detectae
sunt, & quia non patet oculis, qui sit particularum ipsum componentium textus, quid, &
qua ratione Natura ad ejus compositionem adhibeat. Quod autem pertinet ad actionem
in distans, id abunde ibidem praevenimus, cum inde pateat fieri posse, ut punctum quodvis
in se ipsum agat, & ad actionis directionem, ac energiam determinetur ab altero puncto,
vel ut Deus juxta liberam sibi legem a se in Natura condenda stabilitam motum progignat
in utroque pun-[47]-cto. Illud sane mihi est evidens, nihilo magis occuitam esse, vel explicatu,
& captu difficilem productionem. motus per hasce vires pendentes a certis distantiis, quam
sit productio motus vulgo concepta per immediatum impulsum, ubi ad motum determinat
impenetrabilitas, quae itidem vel a corporum natura, vel a libera conditoris lege repeti
debet.

sine impuisione 103. Et quidem hoc potius pacto, quam per impulsionem, in motuum causas, & leges

Mst'hucus^'u^N™ inquirendum esse, illud etiam satis indicat, quod ubi hue usque, impuisione omissa, vires

turam, & menus ex- adhibitae sunt a distantiis pendentes, ibi sane tantummodo accurate definita sunt omnia,

phcajidam. impost- atque determinata, & ad calculum redacta cum phaenomenis congruunt ultra, quam sperare

liceret, accuratissime. Ego quidem ejusmodi in explicando, ac determinando felicitatem

nusquam alibi video in universa Physica, nisi tantummodo in Astronomia mechanica, quae

abjectis vorticibus, atque omni impuisione submota, per gravitatem generalem absolvit

omnia, ac in Theoria luminis, & colorum, in quibus per vires in aliqua distantia agentes,

& reflexionem, & refractionem, & diffractionem Newtonus exposuit, ac priorum duarum,

potissimum leges omnes per calculum, & Geometriam determinavit, & ubi ilia etiam, quae

ad diversas vices facilioris transmissus, & facilioris reflexionis, quas Physici passim relinquunt

A THEORY OF NATURAL PHILOSOPHY 95

consideration of Nature leads us not to non-uniformity but to uniformity of the
elements.

100. Thus at length, from known principles that are commonly accepted, by a Pa^g ,,fro™ the

... , , , ° .' . .r r , i i % i n-<i i T • i proof of the Theory

legitimate series of deductions, we have arrived at the whole of the I heory that I enunciated to the considera-
at the start ; that is to say. at a law of mutual forces & the constitution of the primary tion °f. objections

, ' i • i ,- i re XT • i r i ' against it.

elements of matter derived from that law of forces. Now it remains to be seen what a
bountiful harvest is to be gathered throughout the wide field of general physics ; for from
this one theory we obtain explanations of all the chief properties of bodies, & of the
phenomena of Nature. But before I go on to that, I will give solutions of a few of the
principal difficulties that have been raised against the Theory itself, as well as some that
naturally meet the eye, according to the promise I made.

10 1. The objection is frequently brought forward against mutual forces that they The law o{ forces

,J . * «. . « , . . j. mi • does not necessi-

are some sort of mysterious qualities or that they necessitate action at a distance. This tate action at a

is answered by the idea of forces outlined in Art. 8, & 9. In addition, I will make just distance, nor is it

one remark, namely, that it is quite evident that these forces exist, that an idea of them quTuty. "

can be easily formed, that their existence is demonstrated by direct reasoning, & that

the manifold results that arise from them are a matter of continual ocular observation.

Moreover these forces are of the following nature. The idea of a propensity to approach

or of a propensity to recede is easily formed. For everybody knows what approach means,

and what recession is ; everybody knows what it means to be indifferent, & what having

a propensity means ; & thus the idea of a propensity to approach, or to recede, is perfectly

distinctly obtained. Direct arguments, that prove the existence of this kind of propensity,

have been given above. Lastly also, the various motions that arise from forces of this

kind, such as when one body collides with another body, when one part of a solid is seized

& another part follows it, when the particles of gases, & of springs, repel one another,

when heavy bodies descend, these motions, I say, are of everyday occurrence before our

eyes. It is evident also, at least in a general way, that the form of the curve represents

forces of this kind. In all of these there is nothing mysterious ; on the contrary they all

tend to make the law of forces of this kind perfectly plain.

102. There are indeed certain things that relate to the law of forces of which we are What is so far un
altogether ignorant, such as the number & distances of the intersections of the curve J^0^ idmitted°m
with the axis, the shape of the intervening arcs, & other things of that sort ; these indeed ail detail ; the way
far surpass human understanding, & He alone, Who founded the universe, had the whole ^ action1 atha to*
before His eyes. But truly there is no reason on that account, why a thing, whose existence tance is eliminated,
we fully recognize, & many of the properties & results of which are readily understood,

should not be accepted ; although certainly there do remain many other things pertaining
to it that are unknown to us. For instance, nobody would call gold an unknown &
mysterious substance, & still less would deny its existence, simply because it is quite
probable that many of its properties are unknown to us, to be discovered perhaps in the
future, as so many others have been already discovered from time to time, or because it
is not visually apparent what is the texture of the particles composing it, or why & in
what way Nature adopts that particular composition. Again, as regards action at a distance,
we amply guard against this by the same means ; for, if this is admitted, then it would
be possible for any point to act upon itself, & to be determined as to its direction of action
& energy apart from another point, or that God should produce in either point a motion
according to some arbitrary law fixed by Him when founding the universe. To my mind
indeed it is clear that motions produced by these forces depending on the distances are
not a whit more mysterious, involved or difficult of understanding than the production
of motion by immediate impulse as it is usually accepted ; in which impenetrability
determines the motion, & the latter has to be derived just the same either from the nature
of solid bodies, or from an arbitrary law of the founder of the universe.

103. Now, that the investigation of the causes & laws of motion are better made As far as we have
by my method, than through the idea of impulse, is sufficiently indicated by the fact that, f^' m0reUIciearJy
where hitherto we have omitted impulse & employed forces depending on the distances, explained without
only in this way has everything been accurately defined & determined, & when reduced g^1 what

to calculation everything agrees with the phenomena with far more accuracy than we will be so too.
could possibly have expected. Indeed I do not see anywhere such felicity in explaining
& determining the matters of general physics, except only in celestial mechanics ; in
which indeed, rejecting the idea of vortices, & doing away with that of impulse entirely,
Newton gave a solution of everything by means of universal gravitation ; & in the theory
of light & colours, where by means of forces acting at some distance he explained reflection,
refraction & diffraction ; &, especially in the two first mentioned, he determined all
the laws by calculus & Geometry. Here also those things depending on alternate fits
of easier transmission & easier reflection, which physicists everywhere leave almost

96

PHILOSOPHISE NATURALIS THEORIA

fere intactas, ac alia multa admodum feliciter determinantur, explicanturque, quod & ego
praestiti in dissertatione De Lumine, & praestabo hie in tertia parte ; cum in ceteris Physicae
partibus plerumque explicationes habeantur subsidariis quibusdam principiis innixae &
vagas admodum. Unde jam illud conjectare licet, si ab impulsione immediata penitus
recedatur, & sibi constans ubique adhibeatur in Natura agendi ratio a distantiis pendens,
multo sane facilius, & certius explicatum iri cetera ; quod quidem mihi omnino successit,
ut patebit inferius, ubi Theoriam ipsam applicavero ad Naturam.

Non fieri saltum in

tracfiva ad repui-
sivam.

104. Solent & illud objicere, in hac potissimo Theoria virium committi saltum ilium,
a(^ quern evitandum ea inprimis admittitur ; fieri enim transitum ab attractionibus ad
repulsiones per saltum, ubi nimirum a minima ultima repulsione ad minimam primam
attractionem transitur. At isti continuitatis naturam, quam supra exposuimus, nequaquam
intelligunt. Saltus, cui evitando Theoria inducitur, in eo consistit, quod ab una magnitudine
ad aliam eatur sine transitu per intermedias. Id quidem non accidit in casu exposito.
Assumatur quaecunque vis repulsiva utcunque parva ; turn quaecunque vis attractiva.
Inter eas intercedunt omnes vires repulsivae minores usque ad zero, in quo habetur deter-
minatio ad conservandum praecedentum statum quietis, vel motus uniformis in directum :
turn omnes vires attractivae a z^-[48]-ro usque ad earn determinatam vim, & omnino nullus
erit ex hisce omnibus intermediis statibus, quern aliquando non sint habitura puncta, quae
a repulsione abeunt ad attractionem. Id ipsum facile erit contemplari in fig. i, in qua a
vi repulsiva br ad attractionem dh itur utique continue motu puncti b ad d transeundo
per omnes intermedias, & per ipsum zero in E sine ullo saltu ; cum ordinata in eo motu
habitura sit omnes magnitudines minores priore br usque ad zero in E ; turn omnes oppositas
majores usque ad posteriorem dh. Qui in ea veluti imagine mentis oculos defigat, is omnem
apparentem difficultatem videbit plane sibi penitus evanescere.

Nuiium esse post- JOP Quod autem additur de postremo repulsionis gradu, & primo attractionis nihil

remum attractions, -1. ... r,....r, .,

A: primum repuisio- sane probaret, quando etiam essent aliqui n gradus postrerm, & primi ; nam ab altero
ms gradum, qm si eorum transiretur ad alterum per intermedium illud zero, & ex eo ipso, quod illi essent

essent, adhuc tran- .....*. . ,. . .

sire per omnes in- postremus, ac primus, mhil omitteretur mtermeaium, quae tamen sola intermedn omissio
termedios. continuitatis legem evertit, & saltum inducit. Sed nee habetur ullus gradus postremus,

aut primus, sicut nulla ibi est ordinata postrema, aut prima, nulla lineola omnium minima.
Data quacunque lineola utcunque exigua, aliae ilia breviores habentur minores, ac minores
ad infinitum sine ulla ultima, in quo ipso stat, uti supra etiam monuimus, continuitatis
natura. Quamobrem qui primum, aut ultimum sibi confingit in lineola, in vi, in celeritatis
gradu, in tempusculo, is naturam continuitatis ignorat, quam supra hie innui, & quam ego
idcirco initio meae dissertationis De Lege Continuitatis abunde exposui.

potest cuipiam saltern illud, ejusmodi legem virium, & curvam, quam in
curvae, & duobus fig. I protuli, esse nimium complicatam, compositam, & irregularem, quae nimirum coalescat
virium genenbus. ex }ngenti numero arcuum jam attractivorum, jam repulsivorum, qui inter se nullo pacto
cohaereant ; rem eo redire, ubi erat olim, cum apud Peripateticos pro singulis proprietatibus
corporum singulae qualitates distinctae, & pro diversis speciebus diversae formae substantiales
confingebaritur ad arbitrium. Sunt autem, qui & illud addant, repulsionem, & attractionem
esse virium genera inter se diversa ; satius esse, alteram tantummodo adhibere, & repulsionem
explicare tantummodo per attractionem minorem.

repuisivam positive I07- Inprimis quod ad hoc postremum pertinet, satis patet, per positivam meae

demonstrari prater Theoriae probationem immediate evinci repulsionem ita, ut a minore attractione repeti
omnino non possit ; nam duae materiae particulae si etiam solae in Mundo essent, & ad se
invicem cum aliqua velocitatum inaequalitate accederent, deberent utique ante contactum
ad sequalitatem devenire vi, quse a nulla attractione pendere posset.

A THEORY OF NATURAL PHILOSOPHY 97

untouched, & many other matters were most felicitously determined & explained by
him ; & also that which I enunciated in the dissertation De Lumine, & will repeat in
the third part of this work. For in other parts of physics most of the explanations are
independent of, & disconnected from, one another, being based on several subsidiary
principles. Hence we may now conclude that if, relinquishing all idea of immediate
impulses, we employ a reason for the action of Nature that is everywhere the same &
depends on the distances, the remainder will be explained with far greater ease & certainty ;
& indeed it is altogether successful in my hands, as will be evident later, when I come
to apply the Theory to Nature.

104. It is very frequently objected that, in this Theory more especially, a sudden change There is no sudden
is made in the forces, whilst the theory is to be accepted for the very purpose of avoiding sitfwPfrom aiTat-
such a thing. For it is said that the transition from attractions to repulsions is made tractive to a repui-
suddenly, namely, when we pass from the last extremely minute repulsive force to the
first extremely minute attractive force. But those who raise these objections in no wise
understand the nature of continuity, as it has been explained above. The sudden change,
to avoid which the Theory has been brought forward, consists in the fact that a passage
is made from one magnitude to another without going through the intermediate stages.
Now this kind of thing does not take place in the case under consideration. Take any
repulsive force, however small, & then any attractive force. Between these two there
lie all the repulsive forces that are less than the former right down to zero, in which there
is the propensity for preserving the original state of rest or of uniform motion in a straight
line ; & also all the attractive forces from zero up to the prescribed attractive force,
& there will be absolutely no one of all these intermediate states, which will not be possessed
at some time or other by the points as they pass from repulsion to attraction. This can
be readily understood from a study of Fig. I, where indeed the passage is made from the
repulsive force br to the attractive force dh by the continuous motion of a point from b to
d ; the passage is made through every intermediate stage, & through zero at E, without
any sudden change. For in this motion there will be obtained as ordinates all magnitudes,
less than the first one br, down to zero at E, & after that all magnitudes of opposite sign
greater than zero as far as the last ordinate dh. Anyone, who will fix his intellectual vision
on this as on a sort of pictorial illustration cannot fail to perceive for himself that all the
apparent difficulty vanishes completely.

i OS. Further, as regards what is said in addition about the last stage of repulsion & T^1"6 & no !ast

, r . . 11 11 TI 1111 stage of attraction,

the first stage of attraction, it would really not matter, even if there were these so called and no first for re-
last & first stages ; for, from one of them to the other the passage would be made through puisjon ; and even

6 ,.' . . , i c° if there were, the

the one intermediate stage, namely zero ; since it passes zero, & because they are the nrst passage would be

& last, therefore no intermediate stage is omitted. Nevertheless the omission of this pade through ail

intermediate alone would upset the law of continuity, & introduce a sudden change.

But, as a matter of fact, there cannot possibly be a last stage or a first ; just as there cannot

be a last ordinate or a first in the curve, that is to say, a short line that is the least of

them all. Given any short line, no matter how short, there will be others shorter than

it, less & less in infinite succession without any limit whatever ; & in this, as we remarked

also above, there lies the nature of continuity. Hence anyone who brings forward the

idea of a first or a last in the case of a line, or a force, or a degree of velocity, or an

interval of time, must be ignorant of continuity ; this I have mentioned before in this

work, & also for this very reason I explained it very fully at the beginning of my

dissertation De Lege Continuitatis.

1 06. It may seem to some that at least a law of forces of this nature, & the curve °gb^cstt10?herappard
expressing it, which I gave in Fig. I, is very complicated, composite & irregular, being ent composite cha-
indeed made up of an immense number of arcs that are alternately attractive & repulsive, ^te[h°f t

& that these are joined together according to no definite plan ; & that it reduces to Of forces,
the same thing as obtained amongst the ancients, since with the Peripatetics separate
distinct qualities were invented for the several properties of bodies, & different substantial
forms for different species. Moreover there are some who add that repulsion & attraction
are kinds of forces that differ from one another ; & that it would be quite enough to
use only the latter, & to explain repulsion merely as a smaller attraction.

107. First of all, as regards the last objection, it is clear enough from what has been p^ssibie^o prove
directly proved in my Theory that the existence of repulsion has been rigorously demonstrated directly the exist-
in such a way that it cannot possibly be derived from the idea of a smaller attraction. For f^ce °f apart PUfrom
two particles of matter, if they were also the only particles in the universe, & approached attraction.

one another with some difference of velocity, would be bound to attain to an equality of
velocity on account of a force which could not possibly be derived from an attraction of

any kind,

H

PHILOSOPHIC NATURALIS THEORIA

tiva, & negativa.

Hinc nihu pbstare, Iogi Deinde vefo quod pertinet ad duas diversas species attractionis, & repulsionis ;

si diversi suit gene- ., . , ,. . ,^. r.r-ii-i i

ris; sed esse ejus- id quidem licet ita se haberet, m-[49j-hil sane obesset, cum positive argumento evmcatur
dem uti sunt posi- & repulsio. & attractio, uti vidimus; at id ipsum est omnino falsum. Utraque vis ad

. f . . . . ^ . . .

eandem pertinet speciem, cum altera respectu alterms negativa sit, & negativa a positivis
specie non differant. Alteram negativam esse respectu alterius, patet inde, quod tantum-
modo differant in directione, quae in altera est prorsus opposita direction! alterius ; in
altera enim habetur determinatio ad accessum, in altera ad recessum, & uti recessus, &
accessus sunt positivum, ac negativum ; ita sunt pariter & determinationes ad ipsos. Quod
autem negativum, & positivum ad eandem pertineant speciem, id sane patet vel ex eo
principio : magis, W minus non differunt specie. Nam a positive per continuam subtrac-
tionem, nimirum diminutionem, habentur prius minora positiva, turn zero, ac demum
negativa, continuando subtractionem eandem.

Probatio hujus a
progressu, & re-
gressu, in fluvio.

109. Id facile patet exemplis solitis. Eat aliquis contra fluvii directionem versus locum
aliquem superiori alveo proximum, & singulis minutis perficiat remis, vel vento too hexapedas,
dum a cursu fluvii retroagitur per hexapedas 40 ; is habet progressum hexapedarum 60
singulis minutis. Crescat autem continue impetus fluvii ita, ut retroagatur per 50, turn per
60, 70, 80, 90, 100, no, 120, &c. Is progredietur per 50, 40, 30, 20, 10, nihil ; turn
regredietur per 10, 20, quae erunt negativa priorum ; nam erat prius 100 — 50, 100 — 60,
100—70,100 — 80,100 — 90, turn 100 — 100=0,100 — no, = — 10, 100 — 120 = —20, et ita
porro. Continua imminutione, sive subtractione itum est a positivis in negativa, a
progressu ad regressum, in quibus idcirco eadem species mansit, non duae diversae.

Probatio ex Alge-
bra, & Geometria :
applicatio ad omnes
quantitates varia-
biles.

An habeatur trans-
itus e positivis in
negativa ; investi-
gatio ex sola curv-
arum natura.

B

FHN

MAC

FIG. ii.

i to. Idem autem & algebraicis formulis, & geometricis lineis satis manifeste ostenditur.
Sit formula 10— x, & pro x ponantur valores 6, 7, 8, 9, 10, n, 12, &c. ; valor formulae
exhibebit 4, 3, 2, I, o, — I, — 2, &c., quod eodem redit, ubi erat superius in progressu, &
regressu, qui exprimerentur simulper formulam 10— x. Eadem ilia formula per continuam
mutationem valoris x migrat e valore positive in negativum, qui aeque ad eandem formulam
pertinent. Eodem pacto in Geometria in fig.
u,siduae lineae MN, OP referantur invicem
per ordinatas AB, CD, &c. parallelas inter se,
secent autem se in E ; continue motu ipsius
ordinatae a positive abitur in negativum, mutata
directione AB, CD, quae hie habentur pro
positivis, in FG, HI, post evanescentiam in E.
Ad eandem lineam continuam OEP aeque
pertinet omnis ea ordinatarum series, nee est
altera linea, alter locus geometricus OE, ubi
ordinatae sunt positivae, ac EP, ubi sunt nega-
tivae. Jam vero variabilis quantitatis cujusvis
natura, & lex plerumque per formulam aliquam analyticam, semper per ordinatas ad lineam
aliquam exprimi potest ; si [50] enim singulis ejus statibus ducatur perpendicularis
respondens ; vertices omnium ejusmodi perpendicularium erunt utique ad lineam quandam
continuam. Si ea linea nusquam ad alteram abeat axis partem, si ea formula nullum valorem
negativum habeat ; ilia etiam quantitas semper positiva manebit. Sed si mutet latus linea,
vel formula valoris signum ; ipsa ilia quantitatis debebit itidem ejusmodi mutationem
habere. Ut autem a formulae, vel lineae exprimentis natura, & positione respectu axis
mutatio pendet ; ita mutatio eadem a natura quantitatis illius pendebit ; & ut nee duas
formulae, nee duae lineae speciei diversae sunt, quae positiva exhibent, & negativa ; ita nee in
ea quantitate duae erunt naturae, duae species, quarum altera exhibeat positiva, altera
negativa, ut altera progressus, altera regressus ; altera accessus, altera recessus ; & hie altera
attractiones, altera repulsiones exhibeat ; sed eadem erit, unica, & ad eandem pertinens
quantitatis speciem tota.

in. Quin immo hie locum habet argumentum quoddam, quo usus sum in dissertatione
De Lege Continuitatis, quo nimirum Theoria virium attractivarum, & repulsivarum pro
diversis distantiis, multo magis rationi consentanea evincitur, quam Theoria ^ virium
tantummodo attractivarum, vel tantummodo repulsivarum. Fingamus illud, nos ignorare
penitus, quodnam virium genus in Natura existat, an tantummodo attractivarum, vel
repulsivarum tantummodo, an utrumque simul : hac sane ratiocinatione ad earn perquisi-
tionem uti liceret. Erit utique aliqua linea continua, quae per suas ordinatas ad axem
exprimentem distantias, vires ipsas determinabit, & prout ipsa axem secuerit, vel non

A THEORY OF NATURAL PHILOSOPHY

99

108. Next, as regards attraction & repulsion being of different species, even if it Hence it does not
were a fact that they were so, it would not matter in the slightest degree, since by rigorous Satdifferenthkmds!
argument the existence of both attraction & repulsion is proved, as we have seen ; but but as a matter of
really the supposition is untrue. Both kinds of force belong to the same species ; for one same^kmdnusVas
is negative with regard to the other, & a negative does not differ in species from positives. a positive and a
That the one is negative with regard to the other is evident from the fact that they only negatlve are so-
differ in direction, the direction of one being exactly the opposite of the direction of the

other ; for in the one there is a propensity to approach, in the other a propensity to recede ;
& just as approach & recession are positive & negative, so also are the propensities
for these equally so. Further, that such a negative & a positive belong to the same species,
is quite evident from the principle the greater & the less are not different in kind. For
from a positive by continual subtraction, or diminution, we first obtain less positives, then
zero, & finally negatives, the same subtraction being continued throughout.

109. The matter is easily made clear by the usual illustrations. Suppose a man Demonstration by
to go against the current of a river to some place on the bank up-stream; & suppose "veTndretrogS
that he succeeds in doing, either by rowing or sailing, 100 fathoms a minute, whilst he motion on a river.
is carried back by the current of the river through 40 fathoms ; then he will get forward

a distance of 60 fathoms a minute. Now suppose that the strength of the current continually
increases in such a way that he is carried back first 50, then 60, 70, 80, 90, ipo, no, 120,
&c. fathoms per minute. His forward motion will be successively 50, 40, 30, 20, 10 fathoms
per minute, then nothing, & then he will be carried backward through 10, 20, &c. fathoms
a minute ; & these latter motions are the negatives of the former. For first of all we
had 100 — 50, 100 — 60, 100 — 70, 100 — 80, 100 — 90, then 100 — 100 (which = o),
then 100 — no (which = — 10), 100 — 120 (which = — 20), and so on. By a continual
diminution or subtraction we have passed from positives to negatives, from a progressive
to a retrograde motion ; & therefore in these there was a continuance of the same species,
and there were not two different species.

no. Further, the same thing is shown plainly enough by algebraical formulae, & Proof from algebra
by lines in geometry. Consider the formula 10 — x, & for x substitute the values, 6, pucatfon™^ Oy : a n
7, 8, 9, 10, n, 12, &c. ; then the value of the formula will give in succession 4, 3, 2, variable quantities.
I, o, — i, — 2, &c. ; & this comes to the same thing as we had above in the case of the
progressive & retrograde motion, which may be expressed by the formula 10 — x, all
together. This same formula passes, by a continuous change in the value of x, from a
positive value to a negative, which equally belong to the same formula. In the same
manner in geometry, in Fig. 1 1, if two lines MN, OP are referred to one another by ordinates
AB, CD, & also cut one another in E ; then by a continuous motion of the ordinate
itself it passes from positive to negative, the direction of AB, CD, which are here taken
to be positive, being changed to that of FG, HI, after evanescence at E. To the same
continuous line OEP belongs equally the whole of this series of ordinates ; & OE, where
the ordinates are positive, is not a different line, or geometrical locus from EP, where the
ordinates are negative. Now the nature of any variable quantity, & very frequently
also the law, can be expressed by an algebraical formula, & can always be expressed by
some line ; for if a perpendicular be drawn to correspond to each separate state of the
quantity, the vertices of all perpendiculars so drawn will undoubtedly form some continuous
line. If the line never passes over to the other side of the axis, if the formula has no negative
value, then also the quantity will always remain positive. But if the line changes side,
or the formula the sign of its value, then the quantity itself must also have a change of the
same kind. Further, as the change depends on the nature of the formula & the line
expressing it, & its position with respect to the axis ; so also the same change will depend
on the nature of the quantity ; & just as there are not two formulas, or two lines of
different species to represent the positives & the negatives, so also there will not be in the
quantity two natures, or two species, of which the one yields positives & the other negatives,
as the one a progressive & the other a retrograde motion, the one approach & the other
recession, & in the matter under consideration the one will give attractions & the other
repulsions. But it will be one & the same nature & wholly belonging to the same
spec es of quantity.

in. Lastly, this is the proper place for me to bring forward an argument that I used whether there can

i i . • T\ T /-. -T.r. ,..,!.. , .be a transition

in the dissertation De Lege Continmtatis ; by it indeed it is proved that a theory of attractive {rom positive to
& repulsive forces for different distances is far more reasonable than one of attractive negative ; mves-
forces only, or of repulsive forces only. Let us imagine that we are quite ignorant of the of8the°nature of the
kind of forces that exist in Nature, whether they are only attractive or only repulsive, or curve only,
both ; it would be allowable to use the following reasoning to help us to investigate the
matter. Without doubt there will be some continuous line which, by means of ordinates
drawn from it to an axis representing distances, will determine the forces ; & according

ioo

PHILOSOPHIC NATURALIS THEORIA

cent.qi

secuerit ; vires erunt alibi attractive, alibi repulsivae ; vel ubique attractive tantum, aut
repulsive tantum. Videndum igitur, an sit ration! consentaneum magis, lineam ejus
naturae, & positionis censere, ut axem alicubi secet, an ut non secet.

Transitum deduci U2. Inter rectas axem rectilineum unica parallela ducta per quod vis datum punctum

sint 0>curvse, Pquas non secatj omnes alie numero infinitae secant alicubi. Curvarum nulla est, quam infinitae
recte secent, quam numero rectae secare non possint ; & licet aliquae curvae ejus naturae sint, ut eas aliquae rectae
non secent ; tamen & eas ipsas aliae infinite numero recte secant, & infinite numero curve,
quod Geometrie sublimioris peritis est notissimum, sunt ejus nature, ut nulla prorsus sit
recta linea, a qua possint non secari. Hujusmodi ex. gr. est parabola ilia, cujus ordinate
sunt in ratione triplicata abscissarum. Quare infinite numero curve sunt, & infinite
numero rectae, que sectionem necessario habeant, pro quavis recta, que non habeat, & nulla
est curva, que sectionem cum axe habere non possit. Ergo inter casus possibles multo
plures sunt ii, qui sectionem admittunt, quam qui ea careant ; adeoque seclusis rationibus
aliis omnibus, & sola casuum probabilitate, & rei [51] natura abstracte considerata, multo
magis rationi consentaneum est, censere lineam illam, que vires exprimat, esse unam ex iis,
que axem secant, quam ex iis, que non secant, adeoque & ejusmodi esse virium legem, ut
attractiones, & repulsiones exhibeat simul pro diversis distantiis, quam ut alteras tantummodo
referat ; usque adeo rei natura considerata non solam attractionem, vel solam repulsionem,
sed utramque nobis objicit simul.

punctis

a recta.

secabiles

Ulterior perqui- u* ged eodem argumento licet ultenus quoque progredi, & primum etiam difficultatis

sitio: curvarum J , ° o -j • • • • • i •

genera : quo aiti- caput amovere, quod a sectionum, & idcirco etiam arcuum jam attractivorum, jam repulsi-
ores, eo in piuribus vorum multiplicitate desumitur. Curvas lineas Geometre in quasdam classes dividunt

uni •, .........

°Pe anaiyseos, que earum naturam expnmit per mas, quas Analyste appellant, equationes,
& que ad varies gradus ascendunt. Aequationes primi gradus exprimunt rectas ; equati-
ones secundi gradus curvas primi generis ; equationes tertii gradus curvas secundi generis,
atque ita porro ; & sunt curve, que omnes gradus transcendunt finite Algebre, & que
idcirco dicuntur transcendentes. Porro illud demonstrant Geometre in Analysi ad
Geometriam applicata, lineas, que exprimuntur per equationem primi gradus, posse
secari a recta in unico puncto ; que equationem habent gradus secundi, tertii, & ita porro,
secari posse a recta in punctis duobus, tribus, & ita porro : unde fit, ut curva noni, vel
nonagesimi noni generis secari possit a recta in punctis decem, vel centum.

itidem

sum plures in eo-

Jam vero curvae primi generis sunt tantummodo tres conice sectiones, ellipis,
parabola, hyperbola, adnumerato ellipsibus etiam circulo, que quidem veteribus quoque
Geometris innotuerunt. Curvas secundi generis enumeravit Newtonus omnium primus,
& sunt circiter octoginta ; curvarum generis tertii nemo adhuc numerum exhibuit accura-
tum, & mirum sane, quantus sit is ipse illarum numerus. Sed quo altius assurgit curve
genus, eo plures in eo genere sunt curve, progressione ita in immensum crescente, ut ubi
aliquanto altius ascenderit genus ipsum, numerus curvarum omnem superet humane
imaginationis vim. Idem nimirum ibi accidit, quod in combinationibus terminorum, de
quibus supra mentionem fecimus, ubi diximus a 24 litterulis omnes exhiberi voces linguarum
omnium, & que fuerunt, aut sunt, & que esse possunt.

Deductio inde piu- jjr Inde iam pronum est argumentationem hujusmodi instituere. Numerus

rimarum mtersec- .. J .. . ,..J..

tionum, axis, & linearum, que axem secare possint in punctis quamplunmis, est in immensum major earum
curvae exprimentis numero, quae non possint, nisi in paucis, vel unico : igitur ubi agitur de linea exprimente
legem virium, ei, qui nihil aliunde sciat, in immensum probabilius erit, ejusmodi lineam
esse ex prio-[52]-rum genere unam, quam ex genere posteriorum, adeoque ipsam virium
naturam plurimos requirere transitus ab attractionibus ad repulsiones, & vice versa, quam
paucos, vel nullum.

- Sed omissa ista conjecturali argumentatione quadam, formam curve exprimentis

simpiicem: in quo vires positive argumento a phenomenis Nature deducto nos supra determinavimus cum
plurimis intersectionibus, que transitus ejusmodi quamplurimos exhibeant. Nee ejusmodi
curva debet esse e piuribus arcubus temere compaginata, & compacta : diximus enim,

11 *

A THEORY OF NATURAL PHILOSOPHY 101

as it will cut the axis, or will not, the forces will be either partly attractive & partly
repulsive, or everywhere only attractive or only repulsive. Accordingly it is to be seen •
if it is more reasonable to suppose that a line of this nature & position cuts the axis anywhere,
or does not.

112. Amongst straight lines there is only one, drawn parallel to the rectilinear axis, intersection is to
through any given point that does not cut the axis; all the rest (infinite in number) will the factThat tfhere
cut it somewhere. There is no curve that an infinite number of straight lines cannot cut ; are more lines that
& although there are some curves of such a nature that some straight lines do not cut them, thL^es^hat^o
yet there are an infinite number of other straight lines that do cut these curves ; & there not.

are an infinite number of curves, as is well-known to those versed in higher geometry, of
such a nature that there is absolutely not a single straight line by which they cannot be
cut. An example of this kind of curve is that parabola, in which the ordinates are in the
triplicate ratio of the abscissae. Hence there are an infinite number of curves & an
infinite number of straight lines which necessarily have intersection, corresponding to any
straight line that has not ; & there is no curve that cannot have intersection with an
axis. Therefore amongst the cases that are possible, there are far more curves that admit
intersection than those that are free from it ; hence, putting all other reasons on one side,
& considering only the probability of the cases & the nature of the matter on its own
merits, it is far more reasonable to suppose that the line representing the forces is one of
those, which cut the axis, than one of those that do not cut it. Thus the law of forces
is such that it yields both attractions & repulsions (for different distances), rather than
such that it deals with either alone. Thus far the nature of the matter has been considered,
with the result that it presents to us, not attraction alone, nor repulsion alone, but both of
these together.

113. But we can also proceed still further adopting the same line of argument, & Further investiga-
first of all remove the chief point of the difficulty, that is derived from the multiplicity S^L.^JILhi

ri* */i i i p i i i • curves , nit, iijgiicr

of the intersections, & consequently also of the arcs alternately attractive & repulsive, their order, the

Geometricians divide curves into certain classes by the help of analysis, which expresses wWcV^a ^teaight

their nature by what the analysts call equations ; these equations rise to various degrees, line can cut them.

Equations of the first degree represent straight lines, equations of the second degree represent

curves of the first class, equations of the third degree curves of the second class, & so on.

There are also curves which transcend all degrees of finite algebra, & on that account

these are called transcendental curves. Further, geometricians prove, in analysis applied

to geometry, that lines that are expressed by equations of the first degree can be cut by a

straight line in one point only ; those that have equations of the second, third, & higher

degrees can be cut by a straight line in two, three, & more points respectively. Hence

it comes about that a curve of the ninth, or the ninety-ninth class can be cut by a straight

line in ten, or in a hundred, points.

114. Now there are only three curves of the first class, namely the conic sections, the As the class gets
parabola, the ellipse & the hyperbola; the circle is included under the name of ellipse; gh "

of that
& these three curves were known to the ancient geometricians also. Newton was the class becomes im-

first of all persons to enumerate the curves of the second class, & there are about eighty mensely greater.

of them. Nobody hitherto has stated an exact number for the curves of the third class ;

& it is really wonderful how great is the number of these curves. Moreover, the higher

the class of the curve becomes, the more curves there are in that class, according to a

progression that increases in such immensity that, when the class has risen but a little higher,

the number of curves will altogether surpass the fullest power of the human imagination.

Indeed the same thing happens in this case as in combinations of terms ; we mentioned

the latter above, when we said that by means of 24 little letters there can be

expressed all the words of all languages that ever have been, or are, or can be in

the future.

115. From what has been said above we are led to set up the following line of argument. Hence we deduce
The number of lines that can cut the axis in very many points is immensely greater than that there. are ^

, , , .... ' , ' r. . . ' f> many intersections

the number of those that can cut it in a few points only, or in a single point. Hence, when Of the axis and the
the line representing the law of forces is in question, it will appear to one. who otherwise ?urve representing

i i • i • i • • • i r i 111 , forces.

knows nothing about its nature, that it is immensely more probable that the curve is of
the first kind than that it is of the second kind ; & therefore that the nature of the forces
must be such as requires a very large number of transitions from attractions to repulsions
& back again, rather than a small number or none at all.

116. But, omitting this somewhat conjectural line of reasoning, we have already it may be that the
determined, by what has been said above, the form of the curve representing forces by a j|£™? I^SSlJ8

. ' rxr /iii simple , tnecuarac-

ngorous argument derived trom the phenomena of Nature, & that there are very many teristic of simplicity
intersections which represent just as many of these transitions. Further, a curve of this mcurves-

102

PHILOSOPHIC NATURALIS THEORIA

notum esse Geometris, infinita esse curvarum genera, quae ex ipsa natura sua debeant axem
in plurimis secare punctis, adeoque & circa ipsum sinuari ; sed praeter hanc generalem
responsionem desumptam a generali curvarum natura, in dissertatione De Lege Firium in
Natura existentium ego quidem directe demonstravi, curvam illius ipsius formae, cujusmodi
ea est, quam in fig. i exhibui, simplicem esse posse, non ex arcubus diversarum curvarum
compositam. Simplicem autem ejusmodi curvam affirmavi esse posse : earn enim simplicem
appello, quae tota est uniformis naturae, quae in Analysi exponi possit per aequationem non
resolubilem in plures, e quarum multiplicatione eadem componatur cujuscunque demum
ea curva sit generis, quotcunque habeat flexus, & contorsiones. Nobis quidem altiorum
generum curvae videntur minus simplices ; quh nimirum nostrae humanae menti, uti pluribus
ostendi in dissertatione De Maris Aestu, & in Stayanis Supplementis, recta linea videtur
omnium simplicissima, cujus congruentiam in superpositione intuemur mentis oculis
evidentissime, & ex qua una omnem nos homines nostram derivamus Geometriam ; ac
idcirco, quae lineae a recta recedunt magis, & discrepant, illas habemus pro compositis, &
magis ab ea simplicitate, quam nobis confinximus, recedentibus. At vero lineae continuae,
& uniformis naturae omnes in se ipsis sunt aeque simplices ; & aliud mentium genus, quod
cujuspiam ex ipsis proprietatem aliquam aeque evidenter intueretur, ac nos intuemur
congruentiam rectarum, illas maxime simplices esse crederet curvas lineas, ex ilia earum
proprietate longe alterius Geometrise sibi elementa conficeret, & ad illam ceteras referret
lineas, ut nos ad rectam referimus ; quas quidem mentes si aliquam ex. gr. parabolae pro-
prietatem intime perspicerent, atque intuerentur, non illud quaarerent, quod nostri
Geometrae quaerunt, ut parabolam rectificarent, sed, si ita loqui fas est, ut rectam
parabolarent.

Problema continens 1 1 7. Et quidem analyseos ipsius profundiorem cognitionem requirit ipsa investigatio

naturam curvaeana- aequationis, qua possit exprimi curva ems formae, quae meam exhibet virium legem.

lytice expnmendam. „/! j- • • 11 ji -i

Quamobrem hie tantummodo exponam conditiones, quas ipsa curva habere debet, & quibus
aequatio ibi inventa satis facere [53] debeat. (c) Continetur autem id ipsum num. 75,
illius dissertationis, ubi habetur hujusmodi Problema : Invenire naturam curvce, cujus
abscissis exprimentibus distantias, ordinal exprimant vires, mutatis distantiis utcunque
mutatas, y in datis quotcunque limitibus transeuntes e repulsivis in attractivas, ac ex attractivis
in repulsivas, in minimis autem distantiis repulsivas, W ita crescentes, ut sint pares extinguendce
cuicunque velocitati utcunque magnce. Proposito problemate illud addo : quoniam posuimus
mutatis distantiis utcunque mutatas, complectitur propositio etiam rationem quee ad rationem
reciprocam duplicatam distantiarum accedat, quantum libuerit, in quibusdam satis magnis
distantiis.

Conditiones ejus
problematis.

1 18. His propositis numero illo 75, sequenti numero propono sequentes sex conditiones,
quae requirantur, & sufficiant ad habendam curvam, quse quaeritur. Primo : ut sit regularis,
ac simplex, & non composita ex aggregate arcuum diversarum curvarum. Secundo : ut secet
axem C'AC figures i. tantum in punctis quibusdam datis ad binas distantias AE', AE ; AG',
AG ; y ita porro cequales (d) bine, y inde. Tertio : ut singulis abscissis respondeant singulcs
ordinatcf. (e) Quarto : ut sumptis abscissis cequalibus hinc, y inde ab A, respondeant ordinal*

(c) Qui velit ipsam rei determinationem videre, poterit hie in fine, ubi Supphmentorum, § 3. exhibebitur solutio
problematis, qua in memorata dissertatione continetur a num. 77. ad no. Sed W numerorum ordo, & figurarum
mutabitur, ut cum reliquis hujusce operis cohtereat.

Addetur prieterea eidem §. postremum scholium pertinens ad qu<sstionem agitatam ante has aliquot annos Parisiis ;
an vis mutua inter materite particulas debeat omnino exprimi per solam aliquam distantiee potenttam, an possit per
aliquam ejus functionem ; W constabit, posse utique per junctionem, ut hie ego presto, qute uti superiore numero de curvts
est dictum, est in se eeque simplex etiam, ubi nobis potentias ad ejus expressionem adhibentibus videatur admodum
composita.

(d) Id, ut y quarta conditio, requiritur, ut curva utrinque sit sui similis, quod ipsam magis uniformem reddit ;
quanquam de illo crure, quod est citra asymptotum AB, nihil est, quod soliciti simus ; cum ob vim repulsivam imminutis
distantiis ita in infinitum excrescentem, non possit abscissa distantiam exprimens unquam evadere zero, W abire in
negativam.

(e) Nam singulis distantiis singulte vires respondent.

A THEORY -OF NATURAL PHILOSOPHY 103

kind is not bound to be built up by connecting together a number of independent arcs.
For, as I said, it is well known to Geometricians that there are an infinite number of classes
of curves that, from their very nature, must cut the axis in a very large number of points,
& therefore also wind themselves about it. Moreover, in addition to this general answer
to the objector, derived from the general nature of curves, in my dissertation De Lege
Firium in Natura existentium, I indeed proved in a straightforward manner that a curve,
of the form that I have given in Fig. i, might be simple & not built up of arcs of several
different curves. Further, I asserted that a simple curve of this kind was perfectly feasible ;
for I call a curve simple, when the whole of it is of one uniform nature. In analysis, this
can be expressed by an equation that is not capable of being resolved into several other
equations, such that the former is formed from the latter by multiplication ; & that too,
no matter of what class the curve may be, or how many flexures or windings it may have.
It is true that the curves of higher classes seem to us to be less simple ; this is so because,
as I have shown in several places in the dissertation De Marts Aestu, & the supplements
to Stay's Philosophy, a straight line seems to our human mind to be the simplest of all
lines ; for we get a real clear mental perception of the congruence on superposition in the
case of a straight line, & from this we human beings form the whole of our geometry.
On this account, the more that lines depart from straightness & the more they differ,
the more we consider them to be composite & to depart from that simplicity that we have
set up as our standard. But really all lines that are continuous & of uniform nature
are just as simple as one another. Another kind of mind, which might form an equally
clear mental perception of some property of any one of these curves, as we do the congruence
of straight lines, might believe these curves to be the simplest of all & from that property
of these curves build up the elements of a far different geometry, referring all other curves
to that one, just as we compare them with a straight line. Indeed, these minds, if they
noticed & formed an extremely clear perception of some property of, say, the parabola,
would not seek, as our geometricians do, to rectify the parabola ; they would endeavour,
if one may use the words, to parabolify a straight line.

1 17. The investigation of the equation, by which a curve of the form that will represent pT°bl!Jn .

' i ~ • j' i i j f i • • ir 1T71. r "*™ the analytical

my law of forces can be expressed, requires a deeper knowledge 01 analysis itselt. Wnereiore expression of the

I will here do no more than set out the necessary requirements that the curve must fulfil nature of the curve.

& those that the equation thereby discovered must satisfy. (c) It is the subject of Art. 75

of the dissertation De Lege Firium, where the following problem is proposed. Required

to find the nature of the curve, whose abscissa represent distances & whose ordinates represent

forces that are changed as the distances are changed in any manner, y pass from attractive

forces to repulsive, & from repulsive to attractive, at any given number of limit-points ; further,

the forces are repulsive at extremely small distances and increase in such a manner that they

are capable of destroying any velocity, however great it may be. To the problem as there

proposed I now add the following : — As we have used the words are changed as the distances

are changed in any manner, the proposition includes also the ratio that approaches as nearly

as you please to the reciprocal ratio of the squares of the distances, whenever the distances are

sufficiently great.

1 1 8. In addition to what is proposed in this Art. 75, I set forth in the article that The 0* of
follows it the following six conditions ; these are the necessary and sufficient conditions

for determining the curve that is required.

(i) The curve is regular & simple, & not compounded of a number of arcs of different curves.

(ii) It shall cut the axis C'AC of Fig. I, only in certain given points, whose distances,
AE',AE, AG', AG, and so on, are equal (<t) in pairs on each side of A [see p. 80].

(iii) To each abscissa there shall correspond one ordinate y one only, (f)

(iv) To equal abscisses, taken one on each side of A, there shall correspond equal ordinates.

(c) Anyone who desires to see the solution of the -problem will be able to do seat the end of this work; it will be
found in § 3 of the Supplements ; it is the solution of the problem, as it was given in the dissertation mentioned above,
from Art. 77 to no. But here both the numbering of the articles W of the diagrams have been changed, so as to
agree with the rest of the work. In addition, at the end of this section, there will be found a final note dealing
with a question that was discussed some years ago in Paris. Namely, whether the mutual force between particles of mat-
ter is bound to be expressible by some one power of the distance only, or by some function of the distance. It will be
evident that at any rate it may be expressible by a function as I here assert ; y that function, as has been stated in the
article above, is perfectly simple in itself also ; whereas, if we adhere to an expression by means of powers, the curve will
seem to be altogether complex.

(d) This, y the fourth condition too, is required to make the curve symmetrical, thus giving it greater uniformity ;
although we are not concerned with the branch on the other side of the asymptote AB at all. For, on account of the
repulsive force at very small distances increasing indefinitely in such a manner as postulated, it is impossible that the
abscissa that represents the distance should ever become zero y then become negative.

(e) For to each distance one force, & and only one, corresponds.

104

PHILOSOPHIC NATURALIS THEORIA

czquales. Quinto : ut babeant rectam AB pro asymptoto, area asymptotica BAED existente (£)
infinita. Sexto : ut arcus binis quibuscunque intersectionibus terminati possint variari, ut
libuerit, fcf? ad quascunque distantias recedere ab axe C'AC, ac accedcre ad quoscunque quarum-
cunque curvarum arcus, quantum libuerit, eos secanda, vel tangendo, vel osculando ubicunque,
£ff quomodocunque libuerit.

soiutio IrTUattrac IS4] IT9- Verum quod ad multiplicitatem virium pertinet, quas diversis jam Physici
tionem gravitatis nominibus appellant, illud hie etiam notari potest, si quis singulas seorsim considerare
velit, licere illud etiam, hanc curvam in se unicam per resolutionem virium cogitatione
nostra, atque fictione quadam, dividere in plures. Si ex. gr. quis velit considerare in materia
gravitatem generalem accurate reciprocam distantiarum quadratis ; poterit sane is describere
ex parte attractiva hyperbolam illam, quae habeat accurate ordinatas in ratione reciproca
duplicata distantiarum, quse quidem erit quaedam velut continuatio cruris VTS, turn singulis
ordinatis ag, dh curvae virium expressae in fig. I. adjungere ordinatas hujus novae hyperbolae
ad partes AB incipiendo a punctis curvae g, b, & eo pacto orietur nova quaedam curva, quae
versus partes pV coincidet ad sensum cum axe oC, in reliquis locis ab eo distabit, & contor-
quebitur etiam circa ipsum, si vertices F, K, O distiterint ab axe magis, quam distet ibidem
hyperbola ilia. Turn poterit dici, puncta omnia materiae habere gravitatem decrescentem
accurate in ratione reciproca duplicata distantiarum, & simul habere vim aliam expressam
ab ilia nova curva : nam idem erit, concipere simul hasce binas leges virium, ac illam
praecedentem unicam, & iidem effectus orientur.

Hujus posterioris
vis resolutio in alias
plures.

1 20. Eodem pacto haec nova curva potest dividi in alias duas, vel plures, concipiendo
aliam quamcunque vim, ut ut accurate servantem quasdam determinatas leges, sed simul
mutando curvam jam genitam, translatis ejus punctis per intervalla aequalia ordinatis
respondentibus novae legi ass.umptae. Hoc pacto habebuntur plures etiam vires diversae,
quod aliquando, ut in resolutione virium accidere diximus, inserviet ad faciliorem deter-
minationem effectuum, & ea erit itidem vera virium resolutio quaedam ; sed id omne erit
nostrae mentis partus quidam ; nam reipsa unica lex virium habebitur, quam in fig. I .
exposui, & quae ex omnibus ejusmodi legibus componetur.

Non obesse theo-
r i a m gravitatis ;

distantiis locum
non habet.

121. Quoniam autem hie mentio injecta est gravitatis decrescentis accurate in ratione
cujusiex1naminimis reciproca duplicata distantiarum ; cavendum, ne cui difficultatem aliquam pariat illud,
'"""m quod apud Physicos, & potissimum apud Astronomiae mechanicae cultores, habetur pro
comperto, gravitatem decrescere in ratione reciproca duplicata distantiarum accurate,
cum in hac mea Theoria lex virium discedat plurimum ab ipsa ratione reciproca duplicata
distantiarum. Inprimis in minoribus distantiis vis integra, quam in se mutuo exercent
particulae, omnino plurimum discrepat a gravitate, quae sit in ratione reciproca duplicata
distantiarum. Nam & vapores, qui tantam exercent vim ad se expandendos, repulsionem
habent utique in illis minimis distantiis a se invicem, non attractionem ; & ipsa attractio,
quae in cohaesione se prodit, est ilia quidem in immensum major, quam quae ex generali
gravitate consequitur ; cum ex ipsis Newtoni compertis attractio gravitati respondens [55]
in globes homogeneos diversarum diametrorum sit in eadem ratione, in qua sunt globorum
diametri, adeoque vis ejusmodi in exiguam particulam est eo minor gravitate corporum in
Terram, quo minor est diameter particulae diametro totius Terrae, adeoque penitus insen-
sibilis. Et idcirco Newtonus aliam admisit vim pro cohaesione, quae decrescat in ratione
majore, quam sit reciproca duplicata distantiarum ; & multi ex Newtonianis admiserunt

vim respondentem huic formulae -3 + -v cujus prior pars respectu posterioris sit in

immensum minor, ubi x sit in immensum major unitate assumpta ; sit vero major, ubi x
sit in immensum minor, ut idcirco in satis magnis distantiis evanescente ad sensum prima
parte, vis remaneat quam proxime in ratione reciproca duplicata distantiarum x, in minimis
vero distantiis sit quam proxime in ratione reciproca triplicata : usque adeo ne apud
Newtonianos quidem servatur omnino accurate ratio duplicata distantiarum.

EX

pianetarum
™

I22. Demonstravit quidem Newtonus, in ellipsibus planetariis, earn, quam Astronomi
q^ampro™ lineam apsidum nominant, & est axis ellipseos, habituram ingentem motum, si ratio virium
ime, non accurate. a reciproca duplicata distantiarum aliquanto magis aberret, cumque ad sensum quiescant

(f) Id requiritur, quia in Mecbanica demonstrator, aream curves, cujus abscissa fxprimant distantias, 13 ordinatx
vires, exprimere incrementum, vel decrementum quadrati velocitatis : quare ut illte vires sint pares extinguendte veloci-
tati cuivis utcunque magna, debet ilia area esse omni finita major.

A THEORY OF NATURAL PHILOSOPHY 105

(v) The straight line AB shall be an asymptote, and the asymptotic area BAED shall be
infinite. (f)

(vi) The arcs lying between any two intersections may vary to any extent, may recede to any
distances whatever from the axis C AC, and approximate to any arcs of any curves to any degree
of closeness, cutting them, or touching them, or osculating them, at any points and in any manner.

119. Now, as regards the multiplicity of forces which at the present time physicists call Resolution of the
by different names, it can also here be observed that, if anyone wants to consider one of these £Uj^f of N^wtonUn
separately, the curve though it is of itself quite one-fold can yet be divided into several attraction of
parts by a sort of mental & fictitious resolution of the forces. Thus, for instance, if f^fother1 force*1 d
anyone wishes to consider universal gravitation of matter exactly reciprocal to the squares
of the distances ; he can indeed describe on the attractive side the hyperbola which has
its ordinates accurately in the inverse ratio of the squares of the distances, & this will be
as it were a continuation of the branch VTS. Then he can add on to every ordinate, such
as ag, dh, the ordinates of this new hyperbola, in the direction of AB, starting in each case
from points on the curve, as g,h ; & in this way there will be obtained a fresh curve, which
for the part pV will approximately coincide with the axis 0C, & for the remainder will
recede from it & wind itself about it, if the vertices F,K,O are more distant from the
axis than the corresponding point on the hyperbola. Then it can be stated that all points
of matter have gravitation accurately decreasing in the inverse square of the distance,
together with another force represented by this new curve. For it comes to the same
thing to think of these two laws of forces acting together as of the single law already
given ; & the results that arise will be the same also.

1 20. In the same manner this new curve can be divided into two others, or several The resolution of
others, by considering some other force, in some way or other accurately obeying certain ^o several other
fixed laws, & at the same time altering the curve just obtained by translating the points of it forces.
through intervals equal to the ordinates corresponding to the new law that has been taken.
In this manner several different forces will be obtained ; & this will be sometimes useful,
as we mentioned that it would be in resolution of forces, for determining their effects more
readily ; & will be a sort of true resolution of forces. But all this will be as it were only
a conception of our mind ; for, in reality, there is a single law of forces, & that is the one
which I gave in Fig. i, & it will be the compounded resultant of all such forces as the above.

121. Moreover, since I here make mention of gravitation decreasing accurately in the The.. t.heor.y °*

, r11. .. 111 1111 gravitation is not

inverse ratio ot the squares ot the distances, it is to be remarked that no one should make in opposition ; this

any difficulty over the fact that, amongst physicists & more especially those who deal with l!J^0'Hd°tesv not holn

celestial mechanics, it is considered as an established fact that gravitation decreases accurately distances.

in the inverse ratio of the squares of the distances, whilst in my Theory the law of forces

is very different from this ratio. Especially, in the case of extremely small distances, the

whole force, which the particles exert upon one another, will differ very much in every

case from the force of gravity, if that is supposed to be inversely proportional to the

squares of these distances. For, in the case of gases, which exercise such a mighty

force of self-expansion, there is certainly repulsion at those very small distances from one

another, & not attraction ; again, the attraction that arises in cohesion is immensely

greater than it ought to be according to the law of universal gravitation. Now, from the

results obtained by Newton, the attraction corresponding to gravitation in homogeneous

spheres of different diameters varies as the diameters of the spheres ; & therefore this

kind of force for the case of a tiny particle is as small in proportion to the gravitation of

bodies to the Earth as the diameter of the particle is small in proportion to the diameter

of the whole Earth ; & is thus insensible altogether. Hence Newton admitted another

force in the case of cohesion, decreasing in a greater ratio than the inverse square of the

distances ; also many of the followers of Newton have admitted a force corresponding to

the formula, a'x3 + b'x2 ; in this the first term is immensely less than the second, when x

is immensely greater than some distance assumed as unit distance ; & immensely greater,

when x is immensely less. By this means, at sufficiently great distances the first part

practically vanishes & the force remains very approximately in the inverse ratio of the squares

of the distances x ; whilst, at very small distances, it is very nearly in the inverse ratio

of the cubes of the distances. Thus indeed, not even amongst the followers of Newton has

the inverse ratio of the squares of the distances been altogether rigidly adhered to.

122. Now Newton proved, in the case of planetary elliptic orbits, that that which The law follows
Astronomers call the apsidal line, i.e., the axis of the ellipse, would have a very great motion, not7

,

if the ratio of the forces varied to any great extent from the inverse ratio of the squares from the apheiia of
of the distances ; & since as far as could be observed the lines of apses were stationary

(f) This is required because in Mechanics it is shown that the area of a curve, whose abscissa r'present distances
y ordinates forces, represents the increase or decrease of the square of the velocity. Hence in order that the forces
should be capable of destroying any velocity however great, this area must be greater than any finite area.

io6 PHILOSOPHIC NATURALIS THEORIA

in earum orbitis apsidum linese, intulit, earn rationem observari omnino in gravitate. At
id nequaquam evincit, accurate servari illam legem, sed solum proxime, neque inde ullum
efficax argumentum contra meam Theoriam deduci potest. Nam inprimis nee omnino
quiescunt illae apsidum lineae, sive, quod idem est, aphelia planetarum, sed motu exiguo
quidem, at non insensibili prorsus, moventur etiam respectu fixarum, adeoque motu non
tantummodo apparente, sed vero. Tribuitur is motus perturbationi virium ortae ex mutua
planetarum actione in se invicem ; at illud utique hue usque nondum demonstratum est,
ilium motum accurate respondere actionibus reliquorum planetarum agentium in ratione
reciproca duplicata distantiarum ; neque enim adhuc sine contemptibus pluribus, &
approximationibus a perfectione, & exactitudine admodum remotis solutum est problema,
quod appellant, trium corporum, quo quasratur motus trium corporum in se mutuo
agentium in ratione reciproca duplicata distantiarum, & utcunque projectorum, ac illae
ipsae adhuc admodum imperfectae solutiones, quae prolatae hue usque sunt, inserviunt
tantummodo particularibus quibusdam casibus, ut ubi unum corpus sit maximum, &
remotissimum, quemadmodum Sol, reliqua duo admodum minora & inter se proxima, ut
est Luna, ac Terra, vel remota admodum a majore, & inter se, ut est Jupiter, & Saturnus.
Hinc nemo hucusque accuratum instituit, aut etiam instituere potuit calculum pro actione
perturbativa omnium planetarum, quibus si accedat actio perturbativa cometarum, qui,
nee scitur, quam multi sint, nee quam longe abeant ; multo adhuc magis evidenter patebit,
nullum inde confici posse argumentum [56] pro ipsa penitus accurata ratione reciproca
duplicata distantiarum.

I23- Clairautius quidem in schediasmate ante aliquot annos impresso, crediderat, ex
autem hanc legem ipsis motibus Kneje apsidum Lunae colligi sensibilem recessum a ratione reciproca duplicata
auantum iftmerit[m distantiae, & Eulerus in dissertatione De Aberrationibus Jovis, W Saturni, quas premium
retulit ab Academia Parisiensi an. 1748, censuit, in ipso Jove, & Saturno haberi recessum
admodum sensibilem ab ilia ratione ; sed id quidem ex calculi defectu non satis product!
sibi accidisse Clairautius ipse agnovit, ac edidit ; & Eulero aliquid simile fortasse accidit :
nee ullum habetur positivum argumentum pro ingenti recessu gravitatis generalis a ratione
duplicata distantiarum in distantia Lunae, & multo magis in distantia planetarum. Vero
nee ullum habetur argumentum positivum pro ratione ita penitus accurata, ut discrimen
sensum omnem prorsus effugiat. At & si id haberetur ; nihil tamen pati posset inde
Theoria mea ; cum arcus ille meae curvae postremus VT possit accedere, quantum libuerit,
ad arcum illius hyperbolae, quae exhibet legem gravitatis reciprocam quadratorum dis-
tantiae, ipsam tangendo, vel osculando in punctis quotcunque, & quibuscunque ; adeoque
ita possit accedere, ut discrimen in iis majoribus distantiis sensum omnem effugiat, &
effectus nullum habeat sensibile discrimen ab effectu, qui responderet ipsi legi gravitatis ;
si ea accurate servaret proportionem cum quadratis distantiarum reciproce sumptis.

Difficuitas a Mau- 124. Nee vero quidquam ipsi meae virium Theorias obsunt meditationes Maupertuisii,

tionemaxfma^Nlw- ingeniosae illae quidem, sed meo judicio nequaquam satis conformes Natune legibus circa

tonianae legis. legem virium decrescentium in ratione reciproca duplicata distantiarum, cujus ille perfec-

tiones quasdam persequitur, ut illam, quod in hac una integri globi habeant eandem virium

legem, quam singulae particulae. Demonstravit enim Newtonus, globos, quorum singuli

paribus a centre distantiis homogenei sint, & quorum particulae minimae se attrahant in

ratione reciproca duplicata distantiarum, se itidem attrahere in eadem ratione distantiarum

reciproca duplicata. Ob hasce perfectiones hujus Theoriae virium ipse censuit hanc legem

reciprocam duplicatam distantiarum ab Auctore Naturae selectam fuisse, quam in Natura

esse vellet.

Prima responsio : 125. At mihi quidem inprimis nee unquam placuit, nee placebit sane unquam in

n!^Jf 8Ts^rwt8 investieatione Naturae causarum fmalium usus, quas tantummodo ad meditationem quandam,

onmcs, *x jjcricui~ o f i • i • t XT 1*1* * "VT

iones, ac seiigi et- contemplationemque, usui esse posse abitror, ubi leges JNaturse aliunde innotuennt. JNam
^J$£L*£5* nee perfectiones omnes innotescere nobis possunt, qui intimas rerum naturas nequaquam

III grH.ilclIU pcrlcC~ *• f . * a C

tionum. inspicimus, sed externas tantummodo propnetates quasdam agnoscimus, & lines omnes,

quos Naturae Auctor sibi potuit [57] proponere, ac proposuit, dum Mundum conderet,

A THEORY OF NATURAL PHILOSOPHY 107

in the orbits of each, he deduced that the ratio of the inverse square of the distances was
exactly followed in the case of gravitation. But he only really proved that that law was
very approximately followed, & not that it was accurately so ; nor from this can any
valid argument against my Theory be brought forward. For, in the first place these lines
of apses, or what comes to the same thing, the aphelia of the planets are not quite stationary ;
but they have some motion, slight indeed but not quite insensible, with respect to the fixed
stars, & therefore move not only apparently but really. This motion is attributed to
the perturbation of forces which arises from the mutual action of the planets upon one
another. But the fact remains that it has never up till now been proved that this motion
exactly corresponds with the actions of the rest of the planets, where this is in accordance
with the inverse ratio of the squares of the distances. For as yet the problem of three bodies,
as they call it, has not been solved except by much omission of small quantities & by
adopting approximations that are very far from truth and accuracy ; in this problem is
investigated the motion of three bodies acting mutually upon one another in the inverse
ratio of the squares of the distances, & projected in any manner. Moreover, even these
still only imperfect solutions, such as up till now have been published, hold good only
in certain particular cases ; such as the case in which one of the bodies is very large & at
a very great distance, the Sun for instance, whilst the other two are quite small in comparison
& very near one another, as are the Earth and the Moon, or at a large distance from the
greater & from one another as well, as Jupiter & Saturn. Hence nobody has hitherto
made, nor indeed could anybody make, an accurate calculation of the disturbing influence
of all the other planets combined. If to this is added the disturbing influence of the comets,
of which we neither know the number, nor how far off they are ; it will be still more evident
that from this no argument can be built up in favour of a perfectly exact observance of
the inverse ratio of the squares of the distances.

123. Clairaut indeed, in a pamphlet printed several years ago, asserted his belief that The same thing is
he had obtained from the motions of the line of apses for the Moon a sensible discrepancy J? ^ ^duced from

, , . r i i • AIT--I • i • i • • r>^r • • T the rest of astro-

from the inverse square of the distance. Also Euler, in his dissertation De Aberratiombus nomy ; moreover
Jovis, y Saturni, which carried off the prize given by the Paris Academy, considered that thls Iaw of .mi1e

•/, ,. T . „ <-, , r ° . ' .. , ,. *', can approximate

in the case of Jupiter & Saturn there was quite a sensible discrepancy from that ratio, to the other as

But Clairaut found out, & proclaimed the fact, that his result was indeed due to a defect nearly as is desired.

in his calculation which had not been carried far enough ; & perhaps something similar

happened in Euler's case. Moreover, there is no positive argument in favour of a large

discrepancy from the inverse ratio of the squares of the distances for universal gravitation

in the case of the distance of the Moon, & still more in the case of the distances of the planets.

Neither is there any rigorous argument in favour of the ratio being so accurately observed

that the difference altogether eludes all observation. But even if this were the case, my

Theory would not suffer in the least because of it. For the last arc VT of my curve can

be made to approximate as nearly as is desired to the arc of the hyperbola that represents

the law of gravitation according to the inverse squares of the distances, touching the latter,

or osculating it in any number of points in any positions whatever ; & thus the approximation

can be made so close that at these relatively great distances the difference will be altogether

unnoticeable, & the effect will not be sensibly different from the effect that would

correspond to the law of gravitation, even if that exactly conformed to the inverse ratio

of the squares of the distances.

124. Further, there is nothing really to be objected to my Theory on account of the Objection arising
meditations of Maupertuis ; these are certainly most ingenious, but in my opinion in no p°r™ction fccord*
way sufficiently in agreement with the laws of Nature. Those meditations of his, I mean, ing to Maupertuis,
with regard to the law of forces decreasing in the inverse ratio of the squares of the distances ; j^fw»the Newtoman
for which law he strives to adduce certain perfections as this, that in this one law alone

complete spheres have the same law of forces as the separate particles of which they are
formed. For Newton proved that spheres, each of which have equal densities at equal
distances from the centre, & of which the smallest particles attract one another in the
inverse ratio of the squares of the distances, themselves also attract one another in the same
ratio of the inverse squares of the distances. On account of such perfections as these in
this Theory of forces, Maupertuis thought that this law of the inverse squares of the distances
had been selected by the Author of Nature as the one He willed should exist in Nature.

125. Now, in the first place I was never satisfied, nor really shall I ever be satisfied, First reply to this ;
with the use of final causes in the investigation of Nature ; these I think can only be employed perfections™^ 'not
for a kind of study & contemplation, in such cases as those in which the laws of Nature known ; and even
have already been ascertained from other methods. For we cannot possibly be acquainted ^sdlcted^fo^'fhe
with all perfections ; for in no wise do we observe the inmost nature of things, but all we sake of greater per-
know are certain external properties. Nor is it at all possible for us to see & know all fl

the intentions which the Author of Nature could and did set before Himself when He founded

io8 PHILOSOPHIC NATURALIS THEORIA

videre, & nosse omnino non possumus. Quin immo cum juxta ipsos Leibnitianos inprimis,
aliosque omnes defensores acerrimos principii rationis sufficients, & Mundi perfectissimi,
qui inde consequitur, multa quidem in ipso Mundo sint mala, sed Mundus ipse idcirco
sit optimus, quod ratio boni ad malum in hoc, qui electus est, omnium est maxima ; fieri
utique poterit, ut in ea ipsius Mundi parte, quam hie, & nunc contemplamur, id, quod
electum fuit, debuerit esse non illud bonum, in cujus gratiam tolerantur alia mala, sed
illud malum, quod in aliorum bonorum gratiam toleratur. Quamobrem si ratio reciproca
duplicata distantiarum esset omnium perfectissima pro viribus mutuis particularum, non
inde utique sequeretur, earn pro Natura fuisse electam, & constitutam.

Eandem legem nee I26. At nee revera perfectissima est, quin immo meo quidem judicio est omnino

pcrfcctam esse, nee • r 0 • v 1 • i • • ...

in corporibus, non imperfecta, & tarn ipsa, quam aliae plunmse leges, quas requirunt attractionem immmutis
utique accurate distantiis crcscentcm in ratione reciproca duplicata distantiarum, ad absurda deducunt
' plurima, vel saltern ad inextricabiles difficultates, quod ego quidem turn alibi etiam, turn
inprimis demonstravi in dissertatione De Lege Firium in Natura existentium a num. 59. (g)
Accedit autem illud, quod ilia, qua; videtur ipsi esse perfectio maxima, quod nimirum
eandem sequantur legem globi integri, quam particulae minimae, nulli fere usui est in
Natura ; si res accurate ad exactitudinem absolutam exigatur ; cum nulli in Natura sint
accurate perfecti globi paribus a centre distantiis homogenei, nam praeter non exiguam
inaequalitatem interioris textus, & irregularitatem, quam ego quidem in Tellure nostra
demonstravi in Opere, quod de Litteraria Expeditione per Pontificiam ditionem inscripsi,
in reliquis autem planetis, & cometis suspicari possumus ex ipsa saltern analogia, prater
scabritiem superficiei, quaj utique est aliqua, satis patet, ipsa rotatione circa proprium
axem induci in omnibus compressionem aliquam, quae ut ut exigua, exactam globositatem
impedit, adeoque illam assumptam perfectionem maximam corrumpit. Accedit autem
& illud, quod Newtoniana determinatio rationis reciprocal duplicatae distantiarum locum
habet tantummodo in globis materia continua constantibus sine ullis vacuolis, qui globi
in Natura non existunt, & multo minus a me admitti possunt, qui non vacuum tantummodo
disseminatum in materia, ut Philosophi jam sane passim, sed materiam in immenso vacuo
innatantem, & punctula a se invicem remota, ex quibus, qui apparentes globi fiant, illam
habere proprietatem non possunt rationis reciprocal duplicatae distantiarum, adeoque nee
illius perfectionis creditas maxime perfectam, absolutamque applicationem.

o ex prae- \<:$\ \2j. Demum & illud nonnullis difficultatem parit summam in hac Theoria

juuiv-.w pro impul- £~ * ' . . . . . . f i. • i i ..

sione, & ex testi- Virium, quod censeant, phaenomena omnia per impulsionem explicari debere, & immedi-
monio sensuum : atum contactum, quern ipsum credant evidenti sensuum testimonio evinci ; hinc huiusmodi

responsio ad hanc . • r n « „ XT i •

posteriorem. nostras vires immechamcas appellant, & eas, ut & Newtomanorum generalem gravitatem,

vel idcirco rejiciunt, quod mechanicae non sint, & mechanismum, quem Newtoniana
labefactare coeperat, penitus evertant. Addunt autem etiam per jocum ex serio argumento
petito a sensibus, baculo utendum esse ad persuadendum neganti contactum. Quod ad
sensuum testimonium pertinet, exponam uberius infra, ubi de extensione agam, quae eo
in genere habeamus praejudicia, & unde : cum nimirum ipsis sensibus tribuamus id,
quod nostrae ratiocinationis, atque illationis vitio est tribuendum. Satis erit hie monere
illud, ubi corpus ad nostra organa satis accedat, vim repulsivam, saltern illam ultimam,
debere in organorum ipsorum fibris excitare motus illos ipsos, qui excitantur in
communi sententia ab impenetrabilitate, & contactu, adeoque eundem tremorem ad
cerebrum propagari, & eandem excitari debere in anima perceptionem, quae in
communi sententia excitaretur ; quam ob rem ab iis sensationibus, quae in hac ipsa
Theoria Virium haberentur, nullum utique argumentum desumi potest contra ipsam,
quod ullam vim habeant utcunque tenuem.

Felicius explicari 128. Quod pertinet ad explicationem phaenomenorum per impulsionem immediatam,

sione*- "eam^nus- rnonui sane superius, quanto felicius, ea prorsus omissa, Newtonus explicarit Astronomiam,

quam positive pro- & Opticam ; & patebit inferius, quanto felicius phaenomena quaeque praecipua sine ulla

immediata impulsione explicentur. Cum iis exemplis, turn aliis, commendatur abunde

ea ratio explicandi phsenomena, quae adhibet vires agentes in aliqua distantia. Ostendant

(g) Qute hue pertinent, (J continentur novem numeris ejus Dissertations incipiendo a 59, habentur in fine Supplem.
§4-

A THEORY OF NATURAL PHILOSOPHY 109

the Universe. Nay indeed, since in the doctrine of the followers of Leibniz more especially,
and of all the rest of the keenest defenders of the principle of sufficient reason, and a most
perfect Universe which is a direct consequence of that idea, there may be many evils in the
Universe, and yet the Universe may be the best possible, just because the ratio of
good to evil, in this that has been chosen, is the greatest possible. It might certainly happen
that in this part of the Universe, which here & now we are considering, that which was
chosen would necessarily be not that goodness in virtue of which other things that are
evil are tolerated, but that evil which is tolerated because of the other things that are good.
Hence, even if the inverse ratio of the squares of the distances were the most perfect of all
for the mutual forces between particles, it certainly would not follow from that fact that
it was chosen and established for Nature.

126. But this law as a matter of fact is not the most perfect of all; nay rather, in This law is neither
my opinion, it is altogether imperfect. Both it, & several other laws, that require £0^ec^Tori0Dodt-
attraction at very small distances increasing in the inverse ratio of the squares of the distances ies that are not
lead to very many absurdities ; or at least, to insuperable difficulties, as I showed in the exactly spherical,
dissertation De Lege Virium in Natura existentium in particular, as well as in other places. (g)

In addition there is the point that the thing, which to him seems to be the greatest
perfection, namely, the fact that complete spheres obey the same law as the smallest
particles composing them, is of no use at all in Nature ; for there are in Nature no exactly
perfect spheres having equal densities at equal distances from the centre. Besides the
not insignificant inequality & irregularity of internal composition, of which I proved the
existence in the Earth, in a work which I wrote under the title of De Litteraria Ex-peditione
per Pontificiam ditionem, we can assume also in the remaining planets & the comets (at
least by analogy), in addition to roughness of surface (of which it is sufficiently evident that
at any rate there is some), that there is some compression induced in all of them by the
rotation about their axes. This compression, although it is indeed but slight, prevents
true sphericity, & therefore nullifies that idea of the greatest perfection. There is too
the further point that the Newtonian determination of the inverse ratio of the squares
of the distances holds good only in spheres made up of continuous matter that is free from
small empty spaces ; & such spheres do not exist in Nature. Much less can I admit
such spheres ; for I do not so much as admit a vacuum disseminated throughout matter,
as philosophers of all lands do at the present time, but I consider that matter as it were
swims in an immense vacuum, & consists of little points separated from one another.
These apparent spheres, being composed of these points, cannot have the property of the
inverse ratio of the squares of the distances ; & thus also they cannot bear the true &
absolute application of that perfection that is credited so highly.

127. Finally, some persons raise the greatest objections to this Theory of mine, because Objection founded
they consider that all the phenomena must be explained by impulse and immediate contact ; "mpui^and'on the
this they believe to be proved by the clear testimony of the senses. So they call forces testimony of the
like those I propose non-mechanical, and reject them, just as they also reject the universal th?s latter. rep'y t0
gravitation of Newton, for the alleged reason that they are not mechanical, and overthrow

altogether the idea of mechanism which the Newtonian theory had already begun to
undermine. Moreover, they also add, by way of a joke in the midst of a serious argument
derived from the senses, that a stick would be useful for persuading anyone who denies
contact. Now as far as the evidence of the senses is concerned, I will set forth below,
when I discuss extension, the prejudices that we may form in. such cases, and the origin
of these prejudices. Thus, for instance, we may attribute to the senses what really ought
to be attributed to the imperfection of our reasoning and inference. It will be enough
just for the present to mention that, when a body approaches close enough to our organs,
my repulsive force (at any rate it is that finally), is bound to excite in the nerves of those
organs the motions which, according to the usual idea, are excited by impenetrability and
contact ; & that thus the same vibrations are sent to the brain, and these are bound to
excite the same perception in the mind as would be excited in accordance with the usual
idea. Hence, from these sensations, which are also obtained in my Theory of Forces, no
argument can be adduced against the theory, which will have even the slightest validity.

128. As regards the explanation of phenomena by means of immediate contact I, hsaver^thineg is^°^
indeed, mentioned above how much more happily Newton had explained Astronomy and without the idea of
Optics by omitting it altogether ; and it will be evident, in what follows, how much more |^^lse^ nowhere
happily every one of the important phenomena is explained without any idea of immediate rigorously proved
contact. - Both by these instances, and by many others, this method of explaining phenomena, to exist-

by employing forces acting at a distance, is strongly recommended. Let objectors bring

(g) That which refers to this point, & which is contained in nine articles of the dissertation commencing with Art. 59,
is to bf found at the end of this work as Supplement IV,

no PHILOSOPHISE NATURALIS THEORIA

isti vel unicum exemplum, in quo positive probare possint, per immediatam impulsionem
communicari motum in Natura. Id sane ii praestabunt nunquam ; cum oculorum testi-
monium ad excludendas distantias illas minimas, ad quas primum crus repulsivum pertinet,
& contorsiones curvae circa axem, quae oculos necessario fugiunt, adhibere non possint ; cum
e contrario ego positive argumento superius excluserim immediatum contactum omnem,
& positive probaverim, ipsum, quern ii ubique volunt, haberi nusquam.

Vires hujus Theo- I2g j)e nominibus quidem non esset, cur solicitudinem haberem ullam ; sed ut &

rise pertineread ve- ...,•*,.., . ~t . . , ' . . ,. ...

rum, nee occuitum in nsdem aliquid prasjudicio cmdam, quod ex communi loquendi usu provenit, mud
mechanismum. notandum duco, Mechanicam non utique ad solam impulsionem immediatam fuisse
restrictam unquam ab iis, qui de ipsa tractarunt, sed ad liberos inprimis adhibitam contem-
plandos motus, qui independenter ab omni impulsione habeantur. Quae Archimedes de
aequilibrio tradidit, quse Galilaeus de li-[59]-bero gravium descensu, ac de projectis, quae
de centralibus in circulo viribus, & oscillationis centre Hugenius, quae Newtonus generaliter
de motibus in trajectoriis quibuscunque, utique ad Mechanicam pertinent, & Wolfiana
& Euleriana, & aliorum Scriptorum Mechanica passim utique ejusmodi vires, & motus inde
ortos contemplatur, qui fiant impulsione vel exclusa penitus, vel saltern mente seclusa.
Ubicunque vires agant, quae motum materiae gignant, vel immutent, & leges expandantur,
secundum quas velocitas oriatur, mutetur motus, ac motus ipse determinetur ; id omne
inprimis ad Mechanicam pertinet in admodum propria significatione acceptam. Quam-
obrem ii maxime ea ipsa propria vocum significatione abutuntur, qui impulsionem unicam
ad Mechanismum pertinere arbitrantur, ad quern haec virium genera pertinent multo magis,
qu33 idcirco appellari jure possunt vires Mechanic*?, & quidquid per illas fit, jure affirmari
potest fieri per Mechanismum, nee vero incognitum, & occuitum, sed uti supra demonstra-
vimus, admodum patentem, a manifestum.

Discrimen inter j -m Eodem etiam pacto in omnino propria significatione usurpare licebit vocem con-

contactum mathe- J . . .. * T i i •

maticum, & physi- tactus ; licet intervallum semper remaneat aliquod ; quanquam ego ad aequivocationes evi-
cum : hunc did tandas soleo distinguere inter contactum Mathematicum, in quo distantia sit prorsus nulla,

proprie contactum. ni • • j- • a. • o • 1 •

& contactum Physicum, in quo distantia sensus effugit omnes, & vis repulsiva satis magna
ulteriorem accessum per nostras vires inducendum impedit. Voces ab hominibus institutae
sunt ad significandas res corporeas, & corporum proprietates, prout nostris sensibus subsunt,
iis, quae continentur infra ipsos, nihil omnino curatis. Sic planum, sic laeve proprie dicitur
id, in quo nihil, quod sensu percipi possit, sinuetur, nihil promineat ; quanquam in communi
etiam sententia nihil sit in Natura mathematice planum, vel laeve. Eodem pacto & nomen
contactus ab hominibus institutum est, ad exprimendum physicum ilium contactum tantum-
modo, sine ulla cura contactus mathematics, de quo nostri sensus sententiam ferre non
possunt. Atque hoc quidem pacto si adhibeantur voces in propria significatione ilia, quae
ipsarum institutioni respondeat ; ne a vocibus quidem ipsis huic Theoriae virium invidiam
creare poterunt ii, quibus ipsa non placet.

extensionis sit orta.

Transitus ab ob- j^j. Atque haec de iis, quae contra ipsam virium legem a me propositam vel objecta

Theoriam virium sunt hactenus, vel objici possent, sint satis, ne res in infinitum excrescat. Nunc ad ilia
ad objections con- transibimus, quae contra constitutionem elementorum materiae inde deductam se menti

tra puncta. .•*.... i . j.

oiferunt, in quibus itidem, quae maxime notatu digna sunt, persequar.

Objectio ab idea 132. Inprimis quod pertinet ad hanc constitutionem elementorum materise, sunt

puncti inextensi, multi, qui nullo pacto in animum sibi possint inducere, ut admittant puncta prorsus

qua caremus : re- ,r i n T 11 • j A -J

sponsio : unde idea mdi-[6o]-visibiha, & mextensa, quod nullam se dicant habere posse eorum ideam. At id
a- hominum genus praejudiciis quibusdam tribuit multo plus aequo. Ideas omnes, saltern
eas, quae ad materiam pertinent, per sensus hausimus. Porro sensus nostri nunquam
potuerunt percipere singula elementa, quae nimirum vires exerunt nimis tenues ad movendas
fibras, & propagandum motum ad cerebrum : massis indiguerunt, sive elementorum
aggregatis, quae ipsas impellerent collata vi. Haec omnia aggregata constabant partibus,
quarum partium extremae sumptae hinc, & inde, debebant a se invicem distare per aliquod
intervallum, nee ita exiguum. Hinc factum est, ut nullam unquam per sensus acquirere
potuerimus ideam pertinentem ad materiam, quae simul & extensionem, & partes, ac
divisibilitatem non involverit. Atque idcirco quotiescunque punctum nobis animo sistimus,
nisi reflexione utamur, habemus ideam globuli cujusdam perquam exigui, sed tamen globuli
rotundi, habentis binas superficies oppositas distinctis.

A THEORY OF NATURAL PHILOSOPHY in

forward but a single instance in which they can positively prove that motion in Nature
is communicated by immediate impulse. Of a truth they will never produce one ; for
they cannot use the testimony of the eyes to exclude those very small distances to which
the first repulsive branch of my curve refers & the windings about the axis ; for these
necessarily evade ocular observation. Whilst I, on the other hand, by the rigorous argument
given above, have excluded all idea of immediate contact ; & I have positively proved
that the thing, which they wish to exist everywhere, as a matter of fact exists nowhere.

129. There is no reason why I should trouble myself about nomenclature ; but, as The forces in this
in that too there is something that, from the customary manner of speaking, gives rise to ^j^/^ot to an
a kind of prejudice, I think it should be observed that Mechanics was certainly never occult mechanism,
restricted to immediate impulse alone by those who have dealt with it ; but that in the

first place it was employed for the consideration of free motions, such as exist quite
independently of any impulse. The work of Archimedes on equilibrium, that of Galileo
on the free descent of heavy bodies & on projectiles, that of Huygens on central forces
in a circular orbit & on the centre of oscillation, what Newton proved in general for
motion on all sorts of trajectories ; all these certainly belong to the science of Mechanics.
The Mechanics of Wolf, Euler & other writers in different lands certainly treats of such
forces as these & the motions that arise from them, & these matters have been accomplished
with the idea of impulse excluded altogether, or at least put out of mind. Whenever
forces act, & there is an investigation of the laws in accordance with which velocity is
produced, motion is changed, or the motion itself is determined ; the whole of this belongs
especially to Mechanics in a truly proper signification of the term. Hence, they greatly
abuse the proper signification of terms, who think that impulse alone belongs to the science
of Mechanics ; to which these kinds of forces belong to a far greater extent. Therefore
these forces may justly be called Mechanical ; & whatever comes about through their
action can be justly asserted to have come about through a mechanism ; & one too that
is not unknown or mysterious, but, as we proved above, perfectly plain & evident.

130. Also in the same way we may employ the term contact in an altogether special Distinction be-
sense ; the interval may always remain something definite. Although, in order to avoid ticainandmph^eskai
ambiguity, I usually distinguish between mathematical contact, in which the distance is contact ; the latter
absolutely nothing, & -physical contact, in which the distance is too small to affect our Caned™orftact>.per y
senses, and the repulsive force is great enough to prevent closer approach being induced

by the forces we are considering. Words are formed by men to signify corporeal things
& the properties of such, as far as they come within the scope of the senses ; & those
that fall beneath this scope are absolutely not heeded at all. Thus, we properly call a
thing plane or smooth, which has no bend or projection in it that can be perceived by the
senses ; although, in the general opinion, there is nothing in Nature that is mathematically
plane or smooth. In the same way also, the term contact was invented by men to express
•physical contact only, without any thought of mathematical contact, of which our senses
can form no idea. In this way, indeed, if words are used in their correct sense, namely,
that which corresponds to their original formation, those who do not care for my Theory
of forces cannot from those words derive any objection against it.

131. I have now said sufficient about those objections that either up till now have Passing on from
been raised, or might be raised, against the law of forces that I have proposed ; otherwise ^y^Theorf ""of
the matter would grow beyond all bounds. Now we will pass on to objections against forces to objections
the constitution of the elements of matter derived from it, which present themselves to the agamst P°mts-
mind ; & in these also I will investigate those that more especially seem worthy of remark.

132. First of all, as regards the constitution of the elements of matter, there are indeed Potion to^the
many persons who cannot in any way bring themselves into that frame of mind to admit tended points,
the existence of points that are perfectly indivisible and non-extended ; for they say that which we postu-

*• < ' • T* i r 1_ 13. tc , reply , tiic

they cannot form any idea of such points. But that type of men pays more heed than origin of the idea

is right to certain prejudices. We derive all our ideas, at any rate those that relate to of extension.

matter, from the evidences of our senses. Further, our senses never could perceive single

elements, which indeed give forth forces that are too slight to affect the nerves & thus

propagate motion to the brain. The senses would need masses, or aggregates of the elements,

which would affect them as a result of their combined force. Now all these aggregates are

made up of parts ; & of these parts the two extremes on the one side and on the^ other

must be separated from one another by a certain interval, & that not an insignificant

one. Hence it comes about that we could never obtain through the senses any idea relating

to matter, which did not involve at the same time extension, parts & divisibility. So,

as often as we thought of a point, unless we used our reflective powers, we should get the

idea of a sort of ball, exceedingly small indeed, but still a round ball, having two distinct

and opposite faces.

"2 PHILOSOPHISE NATURALIS THEORIA

idea m puncti 133. Quamobrem ad concipiendum punctum indivisibile, & inextensum ; non debemus

refl^xionemT'quo- consulere ideas> quas immediate per sensus hausimus ; sed earn nobis debemus efformare

modo ejus idea per reflexionem. Reflexione adhibita non ita difficulter efformabimus nobis ideam ejusmodi.

negativa acqmra- Nam inprimi s ubi & extensionem, & partium compositionem conceperimus ; si utranque

negemus ; jam inextensi, & indivisibilis ideam quandam nobis comparabimus per negati-

onem illam ipsam eorum, quorum habemus ideam ; uti foraminis ideam habemus utique

negando existentiam illius materias, quas deest in loco foraminis.

Quomodo ejus idea 134. Verum & positivam quandam indivisibilis, & inextensi puncti ideam poterimus

posfit^per itmlte" comParare n°bis ope Geometrias, & ope illius ipsius ideas extensi continui, quam per sensus
& limitum inter- hausimus, & quam inferius ostendemus, fallacem esse, ac fontem ipsum fallacies ejusmodi
aperiemus, quas tamen ipsa ad indivisibilium, & inextensorum ideam nos ducet admodum
claram. Concipiamus planum quoddam prorsus continuum, ut mensam, longum ex. gr.
pedes duos ; atque id ipsum planum concipiamus secari transversum secundum longitudinem
ita, ut tamen iterum post sectionem conjungantur partes, & se contingant. Sectio ilia
erit utique limes inter partem dexteram & sinistram, longus quidem pedes duos, quanta
erat plani longitude, at latitudinis omnino expers : nam ab altera parte immediate motu
continue transitur ad alteram, quse, si ilia sectio crassitudinem haberet aliquam, non esset
priori contigua. Ilia sectio est limes secundum crassitudinem inextensus, & indivisibilis,
cui si occurrat altera sectio transversa eodem pacto indivisibilis, & inextensa ; oportebit
utique, intersectio utriusque in superficie plani concepti nullam omnino habeat extensionem
in partem quamcumque. Id erit punctum peni-[6i]-tus indivisibile, & inextensum, quod
quidem punctum, translate piano, movebitur, & motu suo lineam describet, longam quidem,
sed latitudinis expertem.

Natura inextensi, j^c. Quo autem melius ipsius indivisibilis natura concipi possit ; quasrat a nobis

quod non potest . /" r , . £ , . ^. . . ' ".

esse inextenso con- quispiam, ut aliam faciamus ejus planae massas sectionem, quas priori ita sit proxima, ut
tiguum in Uneis. nihil prorsus inter utramque intersit. Respondebimus sane, id fieri non posse : vel enim
inter novam sectionem, & veteram intercedet aliquid ejus materias, ex qua planum con-
tinuum constare concipimus, vel nova sectio congruet penitus cum praecedente. En
quomodo ideam acquiremus etiam ejus naturas indivisibilis illius, & inextensi, ut aliud
indivisibile, & inextensum ipsi proximum sine medio intervallo non admittat, sed vel cum
eo congruat, vel aliquod intervallum relinquat inter se, & ipsum. Atque hinc patebit
etiam illud, non posse promoveri planum ipsum ita, ut ilia sectio promoveatur tantummodo
per spatium latitudinis sibi asqualis. Utcunque exiguus fuerit motus, jam ille novus
sectionis locus distabit a praecedente per aliquod intervallum, cum sectio sectioni contigua
esse non possit.

Eademin punctis : 136. Hasc si ad concursum sectionum transferamus, habebimus utique non solum ideam

idea puncti eeo- ..,...,.,.„. . , . ,. ... v j -i •

metricf transiata puncti indivisibilis, & inextensi, sed ejusmodi naturae puncti ipsius, ut aliud punctum sibi
ad physicum, & contiguum habere non possit, sed vel congruant, vel aliquo a se invicem intervallo distent.
Et hoc pacto sibi & Geometrae ideam sui puncti indivisibilis, & inextensi, facile efformare
possunt, quam quidem etiam efformant sibi ita, ut prima Euclidis definitio jam inde incipiat :
•punctum est, cujus nulla •pars est. Post hujusmodi ideam acquisitam illud unum intererit
inter geometricum punctum, & punctum physicum materiae, quod hoc secundum habebit
proprietates reales vis inertias, & virium illarum activarum, quas cogent duo puncta ad se
invicem accedere, vel a se invicem recedere, unde net, ut ubi satis accesserint ad organa
nostrorum sensuum, possint in iis excitare motus, qui propagati ad cerebrum, perceptiones
ibi eliciant in anima, quo pacto sensibilia erunt, adeoque materialia, & realia, non pure
imaginaria.

Punctorum exist- j-- gn jgjtur per reffexionem acquisitam ideam punctorum realium, materialium,

entiam aliunde . .. ,J.( °. . . , . . r r.

demonstrari : per indivisibilium, inextensorum, quam inter ideas ab infantia acquisitas per sensus mcassum
ideam acquisitam quaerimus. Idea ejusmodi non evincit eorum existentiam. Ipsam quam nobis exhibent

ea tantum concipi. ^ . . J . , , . . ..*-."«...

positiva argumenta superms facta, quod mmirum, ne admittatur in colhsione corporum
saltus, quern & inductio, & impossibilitas binarum velocitatum diversarum habendarum
omnino ipso momento, quo saltus fieret, excludunt, oportet admittere in materia vires,
quas repulsivae sint in minimis distantiis, & iis in infinitum imminutis augeantur in infinitum ;

A THEORY OF NATURAL PHILOSOPHY 113

133. Hence for the purpose of forming an idea of a point that is indivisible & non- The idea of a point
extended, we cannot consider the ideas that we derive directly from the senses ; but we ^"refleTti obtaihrxed
must form our own idea of it by reflection. If we reflect upon it, we shall form an idea a negative>nidea°of
of this sort for ourselves without much difficulty. For, in the first place, when we have con- rt may ^ ac(iuired-
ceived the idea of extension and composition by parts, if we deny the existence of both, then

we shall get a sort of idea of non-extension & indivisibility by that very negation of the
existence of those things of which we already have formed an idea. For instance, we have
the idea of a hole by denying the existence of matter, namely, that which is absent from
the position in which the hole lies.

134. But we can also get an idea of a point that is indivisible & non-extended, by HOW a positive idea
the aid of geometry, and by the help of that idea of an extended continuum that we derive ^^ ^of^bourf
from the senses ; this we will show below to be a fallacy, & also we will open up the very daries, and inter-
source of this kind of fallacy, which nevertheless will lead us to a perfectly clear idea of ^g°ns of boun"
indivisible & non-extended points. Imagine some thing that is perfectly plane and
continuous, like a table-top, two feet in length ; & suppose that this plane is cut across

along its length ; & let the parts after section be once more joined together, so that they
touch one another. The section will be the boundary between the left part and the right
part ; it will be two feet in length (that being the length of the plane before section), &
altogether devoid of breadth. For we can pass straightaway by a continuous motion
from one part to the other part, which would not be contiguous to the first part if the section
had any thickness. The section is a boundary which, as regards breadth, is non-extended
& indivisible ; if another transverse section which in the same way is also indivisible &
non-extended fell across the first, then it must come about that the intersection of the
two in the surface of the assumed plane has no extension at all in any direction. It will
be a point that is altogether indivisible and non-extended ; & this point, if the plane
be moved, will also move and by its motion will describe a line, which has length indeed
but is devoid of breadth.

135. The nature of an indivisible itself can be better conceived in the following way. The nature of a
Suppose someone should ask us to make another section of the plane mass, which shall lie °h°in~gextwIhich
so near to the former section that there is absolutely no distance between them. We cannot he next to
should indeed reply that it could not be done. For either between the new section & ^ S

the old there would intervene some part of the matter of which the continuous plane was concerned,
composed ; or the new section would completely coincide with the first. Now see how
we acquire an idea also of the nature of that indivisible and non-extended thing, which
is such that it does not allow another indivisible and non-extended thing to lie next to it
without some intervening interval ; but either coincides with it or leaves some definite
interval between itself & the other. Hence also it will be clear that it is not possible
so to move the plane, that the section will be moved only through a space equal to its own
breadth. However slight the motion is supposed to be, the new position of the section
would be at a distance from the former position by some definite interval ; for a section
cannot be contiguous to another section.

136. If now we transfer these arguments to the intersection of sections, we shall truly Th.e same thing for
have not only the idea of an indivisible & non-extended point, but also an idea of the ^geometrical point
nature of a point of this sort ; which is such that it cannot have another point contiguous transferred to a
to it, but the two either coincide or else they are separated from one another by some interval. riafpoLt^

In this way also geometricians can easily form an idea of their own kind of indivisible &
non-extended points ; & indeed they do so form their idea of them, for the first defi-
nition of Euclid begins : — A -point is that which has no parts. After an idea of this sort has
been acquired, there is but one difference between a geometrical point & a physical point
of matter ; this lies in the fact that the latter possesses the real properties of a force of
inertia and of the active forces that urge the two points to approach towards, or recede
from, one another ; whereby it comes about that when they have approached sufficiently
near to the organs of our senses, they can excite motions in them which, when propagated
to the brain, induce sensations in the mind, and in this way become sensible, & thus
material and real, & not imaginary.

137. See then how by reflection the idea of real, material, indivisible, non-extended The existence of
points can be acquired ; whilst we seek for it in vain amongst those ideas that we have o^herwise^demon-
acquired since infancy by means of the senses. But an idea of this sort about things does strated ; they can
not prove that these things exist. That is just what the rigorous arguments given above through ^cquir-
point out to us ; that is to say, because, in order that in the collision of solids a sudden ing an idea of them,
change should not be admitted (which change both induction & the impossibility of

there being two different velocities at the same instant in which the change should take
place), it had to be admitted that in matter there were forces which are repulsive ^at very
small distances, & that these increased indefinitely as the distances were diminished.

I

ii4 PHILOSOPHIC NATURALIS THEORIA

unde fit, ut duse particulae materiae sibi [62] invicem contiguae esse non possint : nam illico
vi ilia repulsiva resilient a se invicem, ac particula iis constans statim disrumpetur, adeoque
prima materiae elementa non constant contiguis partibus, sed indivisibilia sunt prorsus,
atque simplicia, & vero etiam ob inductionem separabilitatis, ac distinctionis eorum, quae
occupant spatii divisibilis partes diversas, etiam penitus inextensa. Ilia idea acquisita per
reflexionem illud praestat tantummodo, ut distincte concipiamus id, quod ejusmodi rationes
ostendunt existere in Natura, & quod sine reflexione, & ope illius supellectilis tantummodo,
quam per sensus nobis comparavimus ab ipsa infantia, concipere omnino non liceret.

Ceterum simplicium, & inextensorum notionem non ego primus in Physicam
aiiis quoque ad- induco. Eorum ideam habuerunt veteres post Zenonem, & Leibnitiani monades suas &
"rastare "hanc simP^ces utique volunt, & inextensas ; ego cum ipsorum punctorum contiguitatem auferam,
eorum theoriam. & distantias velim inter duo quaelibet materiae puncta, maximum evito scopulum, in quern
utrique incurrunt, dum ex ejusmodi indivisibilibus, & inextensis continuum extensum
componunt. Atque ibi quidem in eo videntur mini peccare utrique, quod cum simplicitate,
& inextensione, quam iis elementis tribuunt, commiscent ideam illam imperfectam, quam
sibi compararunt per sensus, globuli cujusdam rotundi, qui binas habeat superficies a se
distinctas, utcumque interrogati, an id ipsum faciant, omnino sint negaturi. Neque enim
aliter possent ejusmodi simplicibus inextensis implere spatium, nisi concipiendo unum
elementum in medio duorum ab altero contactum ad dexteram, ab altero ad laevam, quin
ea extrema se contingant; in quo, praeter contiguitatem indivisibilium, & inextensorum
impossibilem, uti supra demonstravimus, quam tamen coguntur admittere, si rem altius
perpenderint ; videbunt sane, se ibi illam ipsam globuli inter duos globules inter jacentis
ideam admiscere.

impugnatur con- 139. Nee ad indivisibilitatem, & inextensionem elementorum conjungendas cum

formats ^b^inex- continua extensione massarum ab iis compositarum prosunt ea, quae nonnulli ex Leibniti-

tensis petita ab anorum familia proferunt, de quibus egi in una adnotatiuncula adjecta num. 13. dissertationis
impenetrabiiitate. j)g Mater its Divisibilitate, £? Principiis Corporum, ex qua, quae eo pertinent, hue libet
transferre. Sic autem habet : Qui dicunt, monades non compenetrari, quia natura sua
impenetrabiles sunt, ii difficultatem nequaquam amovenf ; nam si e? natura sua impenetrable s
sunt, y continuum debent componere, adeoque contigua esse ; compenetrabuntur simul, W non
compenetrabuntur, quod ad absurdum deducit, W ejusmodi entium impossibilitatem evincit.
Ex omnimodfs inextensionis, & contiguitatis notione evincitur, compenetrari debere argumento
contra Zenonistas institute per tot stecula, £if cui nunquam satis responsum est. Ex natura,
qua in [63] iis supponitur, ipsa compenetratio excluditur, adeoque habetur contradictio, &
absurdum.

inductionem a 140'. Sunt alii, quibus videri poterit, contra haec ipsa puncta indivisibilia, & inextensa

sensibihbus com- ,1 ., T. . , ^ . . . . r. . r. K .

positis, & extensis adniberi posse mductionis prmcipmm, a quo contmuitatis legem, & alias propnetates
haud vaiere contra derivavimus supra, quae nos ad haec indivisibilia, & inextensa puncta deduxerunt. Videmus

puncta simplicia, & t* *. . . ... ... ,. . .. ...

inextensa. enim in matena omni, quae se uspiam nostns objiciat sensibus, extensionem, divisibihtatem,

partes ; quamobrem hanc ipsam proprietatem debemus transferre ad elementa etiam per
inductionis principium. Ita ii : at hanc difficultatem jam superius praeoccupavimus, ubi
egimus de inductionis principio. Pendet ea proprietas a ratione sensibilis, & aggregati, cum
nimirum sub sensus nostros ne composita quidem, quorum moles nimis exigua sit, cadere
possint. Hinc divisibilitatis, & extensionis proprietas ejusmodi est ; ut ejus defectus, si
habeatur alicubi is casus, ex ipsa earum natura, & sensuum nostrorum constitutione non
possit cadere sub sensus ipsos, atque idcirco ad ejusmodi proprietates argumentum desumptum
ab inductione nequaquam pertingit, ut nee ad sensibilitatem extenditur.

Per ipsam etiam 141. Sed etiam si extenderetur, esset adhuc nostrae Theoriae causa multo melior in eo,

tensT^Hn^uctioms q110^ circa, extensionem, & compositionem partium negativa sit. Nam eo ipso, quod

habitam ipsum ex- continuitate admissa, continuitas elementorum legitima ratiocinatione excludatur, excludi

omnino debet absolute ; ubi quidem illud accidit, quod a Metaphysicis, & Geometris

nonnullis animadversum est jam diu, licere aliquando demonstrare propositionem ex

A THEORY OF NATURAL PHILOSOPHY 115

From this it comes about that two particles of matter cannot be contiguous ; for thereupon
they would recoil from one another owing to that repulsive force, & a particle composed
of them would at once be broken up. Thus, the primary elements of matter cannot be
composed of contiguous parts, but must be perfectly indivisible & simple ; and also on
account of the induction from separability & the distinction between those that occupy
different divisible parts of space, they must be perfectly non-extended as well. The idea
acquired by reflection only yields the one result, namely, that through it we may form
a clear conception of that which reasoning of this kind proves to be existent in Nature ;
of which, without reflection, using only the equipment that we have got together for
ourselves by means of the senses from our infancy, we could not have formed any
conception.

138. Besides, I was not the first to introduce the notion of simple non-extended points Simple and
into physics. The ancients from the time of Zeno had an idea of them, & the followers are^admitt
of Leibniz indeed suppose that their monads are simple & non-extended. I, since I do others as well ; but
not admit the contiguity of the points themselves, but suppose that any two points of ^m is "the7 best."
matter are separated from one another, avoid a mighty rock, upon which both these others
come to grief, whilst they build up an extended continuum from indivisible & non-extended
things of this sort. Both seem to me to have erred in doing so, because they have mixed
up with the simplicity & non-extension that they attribute to the elements that imperfect
idea of a sort of round globule having two surfaces distinct from one another, an idea they
have acquired through the senses ; although, if they were asked if they had made this
supposition, they would deny that they had done so. For in no other way can they fill up
space with indivisible and non-extended things of this sort, unless by imagining that one
element between two others is touched by one of them on the right & by the other on
the left. If such is their idea, in addition to contiguity of indivisible & non-extended
things (which is impossible, as I proved above, but which they are forced to admit if they
consider the matter more carefully) ; in addition to this, I say, they will surely see that they
have introduced into their reasoning that very idea of the two little spheres lying between
two others.

I3Q. Those arguments that some of the Leibnitian circle put forward are of no use The deduction from

, i ~ r • • T • -i •!• o • r i i -i. • impenetrability of

for the purpose of connecting indivisibility & non-extension of the elements with continuous a conciliation of

extension of the masses formed from them. I discussed the arguments in question in extension ^j1 ^

a short note appended to Art. 13 of the dissertation De Materies Divisibilitate and extendeTthings.

Principiis Corporum ; & I may here quote from that dissertation those things that concern

us now. These are the words : — Those, who say that monads cannot be corn-penetrated, because

they are by nature impenetrable, by no means remove the difficulty. For, if they are both by

nature impenetrable, & also at the same time have to make up a continuum, i.e., have to be

contiguous, then at one & the same time they are compenetrated & they are not compenetrated ;

y this leads to an absurdity \3 proves the impossibility of entities of this sort. For, from the

idea of non-extension of any sort, & of contiguity, it is proved by an argument instituted

against the Zenonists many centuries ago that there is bound to be compenetration ; & -this

argument has never been satisfactorily answered. From the nature that is ascribed to them,

this compenetration is excluded. Thus there is a contradiction 13 an absurdity.

140. There are others, who will think that it is possible to employ, for the purpose induction derived
of opposing the idea of these indivisible & non-extended points, the principle of induction, ^T'senslSf3 <£m*-
by which we derived the Law of Continuity & other properties, which have led us to pound, and ex-
these indivisible & non-extended points. For we perceive (so they say) in all matter, avauedforrthefpur°
that falls under our notice in any way, extension, divisibility & parts. Hence we must pose of opposing
transfer this property to the elements also by the principle of induction. Such is their

argument. But we have already discussed this difficulty, when we dealt with the principle
of induction. The property in question depends on a reasoning concerned with a sensible
body, & one that is an aggregate ; for, in fact, not even a. composite body can come within
the scope of our senses, if its mass is over-small. Hence the property of divisibility &
extension is such that the absence of this property (if this case ever comes about), from
the very nature of divisibility & extension, & from the constitution of our senses, cannot
fall within the scope of those senses. Therefore an argument derived from induction will
not apply to properties of this kind in any way, inasmuch as the extension does not reach
the point necessary for sensibility.

141. But even if this point is reached, there would only be all the more reason for our Extension
Theory from the fact that it denies extension and composition by parts. For, from the very exclusion of
fact that, if continuity be admitted, continuity of the elements is excluded by legitimate exte^seio
argument, it follows that continuity ought to be absolutely excluded in all cases. For in duCtion.
that case we get an instance of the argument that has been observed by metaphysicists

and some geometers for a very long time, namely, that a proposition may sometimes be

n6 PHILOSOPHIC NATURALIS THEORIA

assumpta veritate contradictoriae propositionis ; cum enim ambae simul verae esse non
possint, si ab altera inferatur altera, hanc posteriorem veram esse necesse est. Sic nimirum,
quoniam a continuitate generaliter assumpta defectus continuitatis consequitur in materiae
elementis, & in extensione, defectum hunc haberi vel inde eruitur : nee oberit
quidquam principium inductionis physicae, quod utique non est demonstrativum, nee vim
habet, nisi ubi aliunde non demonstretur, casum ilium, quern inde colligere possumus,
improbabilem esse tantummodo, adhuc tamen haberi, uti aliquando sunt & falsa veris
probabiliora.

Cujusmodi con- 142. Atque hie quidem, ubi de continuitate seipsam excludente mentio injecta est,

TheoiSadrnittatur n°tandum & illud, continuitatis legem a me admitti, & probari pro quantitatibus, quae
quid sit spatium, magnitudinem mutent, quas nimirum ab una magnitudine ad aliam censeo abire non posse,
& tempus. njg- transeant per intermedias, quod elementorum materiae, quse magnitudinem nee mutant,

nee ullam habent variabilem, continuitatem non inducit, sed argumento superius facto
penitus summovet. Quin etiam ego quidem continuum nullum agnosco coexistens, uti &
supra monui ; nam nee spatium reale mihi est ullum continuum, sed [64] imaginarium
tantummodo, de quo, uti & de tempore, quae in hac mea Theoria sentiam, satis luculenter
exposui in Supplementis ad librum i. Stayanae Philosophise (*). Censeo nimirum quodvis
materiae punctum, habere binos reales existendi modos, alterum localem, alterum tem-
porarium, qui num appellari debeant res, an tantummodo modi rei, ejusmodi litem, quam
arbitror esse tantum de nomine, nihil omnino euro. Illos modos debere admitti, ibi ego
quidem positive demonstro : eos natura sua immobiles esse, censeo ita, ut idcirco ejusmodi
existendi modi per se inducant relationes prioris, & posterioris in tempore, ulterioris, vel
citerioris in loco, ac distantiae cujusdam deter minatae, & in spatio determinatae positionis
etiam, qui modi, vel eorum alter, necessario mutari debeant, si distantia, vel etiam in spatio
sola mutetur positio. Pro quovis autem modo pertinente ad quodvis punctum, penes
omnes infinites modos possibiles pertinentes ad quodvis aliud, mihi est unus, qui cum eo
inducat in tempore relationem coexistentiae ita, ut existentiam habere uterque non possit,
quin simul habeant, & coexistant ; in spatio vero, si existunt simul, inducant relationem
compenetrationis, reliquis omnibus inducentibus relationem distantiae temporarise, vel
localis, ut & positionis cujusdam localis determinatae. Quoniam autem puncta materiae
existentia habent semper aliquam a se invicem distantiam, & numero finita sunt ; finitus est
semper etiam localium modorum coexistentium numerus, nee ullum reale continuum
efformat. Spatium vero imaginarium est mihi possibilitas omnium modorum localium
confuse cognita, quos simul per cognitionem praecisivam concipimus, licet simul omnes
existere non possint, ubi cum nulli sint modi ita sibi proximi, vel remoti, ut alii viciniores,
vel remotiores haberi non possint, nulla distantia inter possibiles habetur, sive minima
omnium, sive maxima. Dum animum abstrahimus ab actuali existentia, & in possibilium
serie finitis in infinitum constante terminis mente secludimus tarn minimae, quam maximae
distantiae limitem, ideam nobis efformamus continuitatis, & infinitatis in spatio, in quo
idem spatii punctum appello possibilitatem omnium modorum localium, sive, quod idem
est, realium localium punctorum pertinentium ad omnia materiae puncta, quae si existerent,
compenetrationis relationem inducerent, ut eodem pacto idem nomino momentum tem-
poris temporarios modos omnes, qui relationem inducunt coexistentiae. Sed de utroque
plura in illis dissertatiunculis, in quibus & analogiam persequor spatii, ac temporis
multiplicem.

Ubi habeat con- [65] 143. Continuitatem igitur agnosco in motu tantummodo, quod est successivum
uibilitaffee1ctetNatUra ^u^' non coexistens, & in eo itidem solo, vel ex eo solo in corporeis saltern entibus legem
continuitatis admitto. Atque hinc patebit clarius illud etiam, quod superius innui,
Naturam ubique continuitatis legem vel accurate observare, vel affectare saltern. ^ Servat in
motibus, & distantiis, affectat in aliis casibus multis, quibus continuity, uti etiam supra
definivimus, nequaquam convenit, & in aliis quibusdam, in quibus haberi omnino non pptest
continuitas, quae primo aspectu sese nobis objicit res non aliquanto intimius inspectantibus,
ac perpendentibus : ex. gr. quando Sol oritur supra horizontem, si concipiamus Solis discum

(h) Binte dissertatiunculis, qua hue pertinent, inde excerptte habentur hie Supplementorum § I, 13 2, quarum mentio
facta est etiam superius num. 66, W 86.

con-

A THEORY OF NATURAL PHILOSOPHY 117

proved by assuming the truth of the contradictory proposition. For since both propositions
cannot be true at the same time, if from one of them the other can be inferred, then the latter
of necessity must be the true one. Thus, for instance, because it follows, from the
assumption of continuity in general, that there is an absence of continuity in the elements
of matter, & also in the case of extension, we come to the conclusion that there is this
absence. Nor will any principle of physical induction be prejudicial to the argument,
where the induction is not one that can be proved in every case ; neither will it have any
validity, except in the case where it cannot be proved in other ways that the conclusion
that we can come to from the argument is highly improbable but yet is to be held as
true ; for indeed sometimes things that are false are more plausible than the true facts.

142. Now, in this connection, whilst incidental mention has been made of the exclusion xhe sort of

of continuity, it should be observed that the Law of Continuity is admitted by me, & tinuum that is
proved for those quantities that change their magnitude, but which indeed I consider Th^r^fthe^ature
cannot pass from one magnitude to another without going through intermediate stages ; of sPace and time,
but that this does not lead to continuity in the case of the elements of matter, which neither
change their magnitude nor have anything variable about them ; on the contrary it proves
quite the opposite, as the argument given above shows. Moreover, I recognize no co-
existing continuum, as I have already mentioned ; for, in my opinion, space is not any
real continuum, but only an imaginary one ; & what I think about this, and about time
as well, as far as this Theory is concerned, has been expounded clearly enough in the
supplements to the first book of Stay's Philosophy. (A) For instance, I consider that any
point of matter has two modes of existence, the one local and the other temporal ; I do
not take the trouble to argue the point as to whether these ought to be called things, or
merely modes pertaining to a thing, as I consider that this is merely a question of terminology.
That it is necessary that these modes be admitted, I prove rigorously in the supplements
mentioned above. I consider also that they are by their very nature incapable of being
displaced ; so that, of themselves, such modes of existence lead to the relations of before
& after as regards time, far & near as regards space, & also of a given distance &
a given position in space. These modes, or one of them, must of necessity be changed,
if the distance, or even if only the position in space is altered. Moreover, for any one
mode belonging to any point, taken in conjunction with all the infinite number of possible
modes pertaining to any other point, there is in my opinion one which, taken in conjunction
with the first mode, leads as far as time is concerned to a relation of coexistence ; so that
both cannot have existence unless they have it simultaneously, i.e., they coexist. But,
as far as space is concerned, if they exist simultaneously, the conjunction leads to a relation
of compenetration. All the others lead to a relation of temporal or of local distance, as
also of a given local position. Now since existent points of matter always have some distance
between them, & are finite in number, the number of local modes of existence is also
always finite ; & from this finite number we cannot form any sort of real continuum.
But I have an ill-defined idea of an imaginary space as a possibility of all local modes, which
are precisely conceived as existing simultaneously, although they cannot all exist simul-
taneously. In this space, since there are not modes so near to one another that there
cannot be others nearer, or so far separated that there cannot be others more so, there
cannot therefore be a distance that is either the greatest or the least of all, amongst those
that are possible. So long as we keep the mind free from the idea of actual existence &, in
a series of possibles consisting of an indefinite number of finite terms, we mentally exclude
the limit both of least & greatest distance, we form for ourselves a conception of continuity
& infinity in space. In this, I define the same point of space to be the possibility of all
local modes, or what comes to the same thing, of real local points pertaining to all points
of matter, which, if they existed, would lead to a relation of compenetration ; just as I
define the same instant of time as all temporal modes, which lead to a relation of coexistence.
But there is a fuller treatment of both these subjects in the notes referred to ; & in them
I investigate further the manifold analogy between space & time.

143. Hence I acknowledge continuity in motion only, which is something successive where there is con-

i TJ . . .° . , ' , f . V °. . tmuity in Nature ;

and not co-existent ; & also in it alone, or because or it alone, in corporeal entities at any Where Nature does
rate, lies my reason for admitting the Law of Continuity. From this it will be all the no more than at-
more clear that, as I remarked above, Nature accurately observes the Law of Continuity, jteml
or at least tries to do so. Nature observes it in motions & in distance, & tries to in many
other cases, with which continuity, as we have defined it above, is in no wise in agree-
ment ; also in certain other cases, in which continuity cannot be completely obtained. This
continuity does not present itself to us at first sight, unless we consider the subjects somewhat
more deeply & study them closely. For instance, when the sun rises above the horizon,

(h) The two notes, which refer to this matter, have been quoted in this work as supplements IS- II : these have
been already referred to in Arts. 66 & 86 above.

n8 PHILOSOPHISE NATURALIS THEORIA

ut continuum, & horizontem ut planum quoddam ; ascensus Solis fit per omnes magnitudines
ita, ut a primo ad postremum punctum & segmenta Solaris disci, & chordae segmentorum
crescant transeundo per omnes intermedias magnitudines. At Sol quidem in mea Theoria
non est aliquid continuum, sed est aggregatum punctorum a se invicem distantium, quorum
alia supra illud imaginarium planum ascendunt post alia, intervallo aliquo temporis inter-
posito semper. Hinc accurata ilia continuitas huic casui non convenit, & habetur tantummodo
in distantiis punctorum singulorum componentium earn massam ab illo imaginario piano.
Natura tamen etiam hie continuitatem quandam affectat, cum nimirum ilia punctula ita
sibi sint invicem proxima, & ita ubique dispersa, ac disposita, ut apparens quaedam ibi etiam
continuitas habeatur, ac in ipsa distributione, a qua densitas pendet, ingentes repentini
saltus non riant.

Exempla continu- 144. Innumera ejus rei exempla liceret proferre, in quibus eodem pacto res pergit.

it at is apparent gjc jn fluviorum alveis, in frondium flexibus, in ipsis salium, & crystallorum, ac aliorum

tantum : unde ea ..... . ,., . • •••*.«,

ortum ducat. corporum angulis, in ipsis cuspidibus unguium, quae acutissimae in quibusdam ammalibus
apparent nudo oculo ; si microscopio adhibito inspiciantur ; nusquam cuspis abrupta
prorsus, nusquam omnino cuspidatus apparet angulus, sed ubique flexus quidam, qui
curvaturam habeat aliquam, & ad continuitatem videatur accedere. In omnibus tamen iis
casibus vera continuitas in mea Theoria habetur nusquam ; cum omnia ejusmodi corpora
constent indivisibilibus, & a se distantibus punctis, quse continuam superficiem non efformant,
& in quibus, si quaevis tria puncta per rectas lineas conjuncta intelligantur ; triangulum
habebitur utique cum angulis cuspidatis. Sed a motuum, & virium continuitate accurata
etiam ejusmodi proximam continuitatem massarum oriri censeo, & a casuum possibilium
multitudine inter se collata, quod ipsum innuisse sit satis.

Motuum omnium 145- Atque hinc fiet manifestum, quid respondendum ad casus quosdam, qui eo

continuitas in -pertinent, & in quibus violari quis crederet F661 continuitatis legem. Quando piano aliquo

line is continuis r .*. fr . n • • n • •

nusquam inter- speculo lux excipitur, pars relrmgitur, pars renectitur : in renexione, & retractione, uti earn
ruptis, aut mutatis. olim creditum est fieri, & etiamnum a nonnullis creditur, per impulsionem nimirum, &
incursum immediatum, fieret violatio quaedam continui motus mutata linea recta in aliam ;
sed jam hoc Newtonus advertit, & ejusmodi saltum abstulit, explicando ea phenomena per
vires in aliqua distantia agentes, quibus fit, ut quaevis particula luminis motum incurvet
paullatim in accessu ad superficiem re flectentem, vel refringentem ; unde accessuum, &
recessuum lex, velocitas, directionum flexus, omnia juxta continuitatis legem mutantur.
Quin in mea Theoria non in aliqua vicinia tantum incipit flexus ille, sed quodvis materiae
punctum a Mundi initio unicam quandam continuam descripsit orbitam, pendentem a
continua ilia virium lege, quam exprimit figura I , quae ad distantias quascunque protenditur ;
quam quidem lineae continuitatem nee liberae turbant animarum vires, quas itidem non nisi
juxta continuitatis legem exerceri a nobis arbitror ; unde fit, ut quemadmodum omnem
accuratam quietem, ita omnem accurate rectilineum motum, omnem accurate circularem,
ellipticum, parabolicum excludam ; quod tamen aliis quoque sententiis omnibus commune
esse debet ; cum admodum facile sit demonstrare, ubique esse perturbationem quandam,
& mutationum causas, quae non permittant ejusmodi linearum nobis ita simplicium accuratas
orbitas in motibus.

Apparens saltus in 146. Et quidem ut in iis omnibus, & aliis ejusmodi Natura semper in mea Theoria

diffusione reflexi, accuratissimam continuitatem observat, ita & hie in reflexionibus, ac refractionibus luminis.

ac refracti luminis. . ,. , . ..'..,. , , , '. . ,

At est ahud ea in re, in quo continuitatis violatio quaedam haben videatur, quam, qui rem
altius perpendat, credet primo quidem, servari itidem accurate a Natura, turn ulterius
progressus, inveniet affectari tantummodo, non servari. Id autem est ipsa luminis diffusio,
atque densitas. Videtur prima fronte discindi radius in duos, qui hiatu quodam intermedio
a se invicem divellantur velut per saltum, alia parte reflexa, ali refracta, sine ullo intermedio
flexu cujuspiam. Alius itidem videtur admitti ibidem saltus quidam : si enim radius
integer excipiatur prismate ita, ut una pars reflectatur, alia transmittatur, & prodeat etiam
e secunda superficie, turn ipsum prisma sensim convertatur ; ubi ad certum devenitur in
conversione angulum, lux, quae datam habet refrangibilitatem, jam non egreditur, sed
reflectitur in totum ; ubi itidem videtur fieri transitus a prioribus angulis cum superficie
semper minoribus, sed jacentibus ultra ipsam, ad angulum reflexionis aequalem angulo

A THEORY OF NATURAL PHILOSOPHY 119

if we think of the Sun's disk as being continuous, & the horizon as a certain plane ; then
the rising of the Sun is made through all magnitudes in such a way that, from the first to
the last point, both the segments of the solar disk & the chords of the segments increase by
passing through all intermediate magnitudes. But, in my Theory, the Sun is not something
continuous, but is an aggregate of points separate from one another, which rise, one after
the other, above that imaginary plane, with some interval of time between them in all
cases. Hence accurate continuity does not fit this case, & it is only observed in the case
of the distances from the imaginary plane of the single points that compose the mass of the
Sun. Yet Nature, even here, tries to maintain a sort of continuity ; for instance, the
little points are so very near to one another, & so evenly spread & placed that, even in
this case, we have a certain apparent continuity, and even in this distribution, on which
the density depends, there do not occur any very great sudden changes.

144. Innumerable examples of this apparent continuity could be brought forward, in Examples of con-
which the matter comes about in the same manner. Thus, in the channels of rivers, the ^"reiy apparent'3
bends in foliage, the angles in salts, crystals and other bodies, in the tips of the claws that its origin,
appear to the naked eye to be very sharp in the case of certain animals ; if a microscope

were used to examine them, in no case would the point appear to be quite abrupt, or the
angle altogether sharp, but in every case somewhat rounded, & so possessing a definite
curvature & apparently approximating to continuity. Nevertheless in all these cases
there is nowhere true continuity according to my Theory ; for all bodies of this kind are
composed of points that are indivisible & separated from one another ; & these cannot
form a continuous surface ; & with them, if any three points are supposed to be joined
by straight lines, then a triangle will result that in every case has three sharp angles. But
I consider that from the accurate continuity of motions & forces a very close approximation
of this kind arises also in the case of masses ; &, if the great number of possible cases are
compared with one another, it is sufficient for me to have just pointed it out.

145. Hence it becomes evident how we are to refute certain cases, relating to this The. continuity of
matter, in which it might be considered that the Law of Continuity was violated. When ™uous lines ""is
light falls upon a plane mirror, part is refracted & part is reflected. In reflection & nowhere inter-
refraction, according to the idea held in olden times, & even now credited by some people, rup e
namely, that it took place by means of impulse & immediate collision, there would be

a breach of continuous motion through one straight line being suddenly changed for
another. But already Newton has discussed this point, & has removed any sudden change
of this sort, by explaining the phenomena by means of forces acting at a distance ; with
these it comes about that any particle of light will have its path bent little by little as it
approaches a reflecting or refracting surface. Hence, the law of approach and recession,
the velocity, the alteration of direction, all change in accordance with the Law of Continuity.
Nay indeed, in my Theory, this alteration of direction does not only begin in the immediate
neighbourhood, but any point of matter from the time that the world began has described
a single continuous orbit, depending on the continuous law of forces, represented in Fig. i,
a law that extends to all distances whatever. I also consider that this continuity of path
is undisturbed by any voluntary mental forces, which also cannot be exerted by us except
in accordance with the Law of Continuity. Hence it comes about that, just as I exclude
all idea of absolute rest, so I exclude all accurately rectilinear, circular, elliptic, or parabolic
motions. This too ought to be the general opinion of all others ; for it is quite easy to show
that there is everywhere some perturbation, & reasons for alteration, which do not allow
us to have accurate paths along such simple lines for our motions.

146. Just as in all the cases I have mentioned, & in others like them, Nature always Apparent discon-

'.-,,, J .. i • i • i i • i tinuity in diffusion

in my Theory observes the most accurate continuity, so also is this done here in the case Of renected and re-

of the reflection and refraction of light. But there is another thing in this connection, fracted light.

in which there seems to be a breach of continuity ; & anyone who considers the matter

fairly deeply, will think at first that Nature has observed accurate continuity, but on further

consideration will find that Nature has only endeavoured to do so, & has not actually

observed it ; that is to say, in the diffusion of light, & its density. At first sight the ray

seems to be divided into two parts, which leave a gap between them & diverge from one

another as it were suddenly, the one part being reflected & the other part refracted

without any intermediate bending of the path. It also seems that another sudden change

must be admitted ; for suppose that a beam of light falls upon a prism, & part of it is

reflected & the rest is transmitted & issues from the second surface, and that then the

prism is gradually rotated ; when a certain angle of rotation is reached, light, having

a given refrangibility, is no longer transmitted, but is totally reflected. Here also it

seems that there is a sudden transition from the first case in which the angles made^with

the surface by the issuing rays are always less than the angle of incidence, & lie on

the far side of the surface, to the latter case in which the angles of reflection are equal to

120 PHILOSOPHIC NATURALIS THEORIA

incidentiae, & jacentem citra, sine ulla reflexione in angulis intermediis minoribus ab ipsa
superficie ad ejusmodi finitum angulum.

Apparens concili- 14.7. Huic cuidam velut laesioni continuitatis videtur responderi posse per illam lucem

Unuitafe pel radios qua3 reflectitur, vel refrin-[67]-gitur irregulariter in quibusvis angulis. Jam olim enim
irregulariter disper- observatum est illud, ubi lucis radius reflectitur, non reflecti totum ita, ut angulus
reflexionis aequetur angulo incidentiae, sed partem dispergi quaquaversus ; quam ob causam
si Solis radius in partem quandam speculi incurrat, quicunque est in conclavi, videt, qui sit
ille locus, in quern incurrit radius, quod utique non fieret, nisi e solaribus illis directis radiis
etiam ad oculum ipsius radii devenirent, egressi in omnibus iis directionibus, quae ad omnes
oculi positiones tendunt ; licet ibi quidem satis intensum lumen non appareat, nisi in
directione faciente angulum reflexionis aequalem incidentiae, in qua resilit maxima luminis
pars. Et quidem hisce radiis redeuntibus in angulis hisce inaequalibus egregie utitur
Newtonus in fine Opticae ad explicandos colores laminarum crassarum : & eadem irregularis
dispersio in omnes plagas ad sensum habetur in tenui parte, sed tamen in aliqua, radii
refracti. Hinc inter vividum ilium reflexum radium, & refractum, habetur intermedia
omnis ejusmodi radiorum series in omnibus iis intermediis angulis prodeuntium, & sic etiam
ubi transitur a refractione ad reflexionem in totum, videtur per hosce intermedios angulos
res posse fieri citissimo transitu per ipsos, atque idcirco illaesa perseverare continuitas.

Cur ea apparens 148. Verum si adhuc altius perpendatur res ; patebit in ilia intermedia serie non haberi

dSitio1 pe^contiii- accuratam continuitatem, sed apparentem quandam, quam Natura affectat, non accurate

ujtatem yiae cujus- servat illaesam. Nam lumen in mea Theoria non est corpus quoddam continuum, quod

vis puncti diffundatur continue per illud omne spatium, sed est aggregatum punctorum a se invicem

disjunctorum, atque distantium, quorum quodlibet suam percurrit viam disjunctam a

proximi via per aliquod intervallum. Continuitas servatur accuratissime in singulorum

punctorum viis, non in diffusione substantiae non continuae, & quo pacto ea in omnibus iis

motibus servetur, & mutetur, mutata inclinatione incidentiae, via a singulis punctis descripta

sine saltu, satis luculenter exposui in secunda parte meae dissertationis De Lumine a num. 98.

Sed haec ad applicationem jam pertinent Theoriae ad Physicam.

QUO pacto servetur 149. Haud multum absimiles sunt alii quidam casus, in quibus singula continuitatem

bu^dam^casibusTui observant, non aggregatum utique non continuum, sed partibus disjunctis constans.
quibus videtur tedi. Hujusmodi est ex. gr. altitude cujusdam domus, quae aedificatur de novo, cui cum series
nova adjungitur lapidum determinatae cujusdam altitudinis, per illam additionem repente
videtur crescere altitude domus, sine transitu per altitudines intermedias : & si dicatur id
non esse Naturae opus, sed artis ; potest difficultas transferri facile ad Naturae opera, ut ubi
diversa inducuntur glaciei strata, vel in aliis incrustationibus, ac in iis omnibus casibus, in
quibus incrementum fit per externam applicationem partium, ubi accessiones finitae videntur
acquiri simul totae sine [68] transitu per intermedias magnitudines. In iis casibus
continuitas servatur in motu singularum partium, quae accedunt. Illae per lineam quandam
continuam, & continua velocitatis mutatione accedunt ad locum sibi deditum : quin immo
etiam posteaquam eo advenerunt, pergunt adhuc moveri, & nunquam habent quietem nee
absolutam, nee respectivam respectu aliarum partium, licet jam in respectiva positione
sensibilem mutationem non subeant : parent nimirum adhuc viribus omnibus, quae
respondent omnibus materiae punctis utcunque distantibus, & actio proximarum partium,
quae novam adhaesionem parit, est continuatio actionis, quam multo minorem exercebant,
cum essent procul. Hoc autem, quod pertineant ad illam domum, vel massam, est aliquid
non in se determinatum, quod momento quodam determinato fiat, in quo saltus habeatur,
sed ab aestimatione quadam pendet nostrorum sensuum satis crassa ; ut licet perpetuo
accedant illae partes, & pergant perpetuo mutare positionem respectu ipsius massae ; turn
incipiant censeri ut pertinentes ad illam domum, vel massam : cum desinit respectiva
mutatio esse sensibilis, quae sensibilitatis cessatio fit ipsa etiam quodammodo per gradus
omnes, & continue aliquo tempore, non vero per saltum.

Generate responsio ISO- Hinc distinctius ibi licebit difHcultatem omnem amovere dicendo, non servari

de emta.3 similes m" mutationem continuam in magnitudinibus earum rerum, quae continuae non sunt, &

magnitudinem non habent continuam, sed sunt aggregata rerum disjunctarum ; vel in iis

rebus, quae a nobis ita censentur aliquod totum constituere, ut magnitudinem aggregati non

A THEORY OF NATURAL PHILOSOPHY 121

the angles of incidence & lie on the near side of the surface, without any reflection for
rays at intermediate angles with the surface less than a certain definite angle.

147. It seems that an explanation of this apparent breach of continuity can be given Apparent recontiii-
by means of light that is reflected or refracted irregularly at all sorts of angles. For long ago of Continuity6 effect
it was observed that, when a ray of light is reflected, it is not reflected entirely in such a *ed fay means of
manner that the angle of reflection is equal to the angle of incidence, but that a part of it

is dispersed in all directions. For this reason, if a ray of light from the Sun falls upon some
part of a mirror, anybody who is in the room sees where the ray strikes the mirror ; &
this certainly would not be the case, unless some of the solar rays reached his eye directly
issuing from the mirror in all those directions that reach to all positions that the eye might
be in. Nevertheless, in this case the light does not appear to be of much intensity, unless
the eye is in the position facing the angle of reflection equal to the angle of incidence, along
which the greatest part of the light rebounds. Newton indeed employed in a brilliant
way these rays that issue at irregular angles at the end of his Optics to explain the colours
of solid laminae. The same irregular dispersion in all directions takes place as far as can
be observed in a small part, but yet in some part, of the refracted ray. Hence, in between
the intense reflected & refracted rays, we have a whole series of intermediate rays of this sort
issuing at all intermediate angles. Thus, when the transition is effected from refraction
to total reflection, it seems that it can be done through these intermediate angles by an
extremely rapid transition through them, & therefore continuity remains unimpaired.

148. But if we inquire into the matter yet more carefully, it will be evident that in Why this is only an
that intermediate series there is no accurate continuity, but only an apparent continuity ; atimT* the^true
& this Nature tries to maintain, but does not accurately observe it unimpaired. For, reconciliation is
in my Theory, light is not some continuous body, which is continuously diffused through t!nurtyhof ^ath^or
all the space it occupies ; but it is an aggregate of points unconnected with & separated any point of light,
from one another ; & of these points, any one pursues its own path, & this path is separated

from the path of the next point to it by a definite interval. Continuity is observed perfectly
accurately for the paths of the several points, not in the diffusion of a substance that is
not continuous ; & the manner in which continuity is preserved in all these motions,
& the path described by the several points is altered without sudden change, when the angle
of incidence is altered, I have set forth fairly clearly in the second part of my dissertation
De Lumine, Art. 98. But in this work these matters belong to the application of the
Theory to physics.

140. There are certain cases, not greatly unlike those already given, in which each The manner in

/ i _j« i_ • '• V j r which continuity

part preserves continuity, but not so the whole, which is not continuous but composed ot is maintained m

separate parts. For an instance of this kind, take the height of a new house which is being certain cases in

built ; as a fresh layer of stones of a given height is added to it, the height of the house ^

on account of that addition seems to increase suddenly without passing through intermediate

heights. If it is said that that is not a work of Nature, but of art ; then the same difficulty

can easily be transferred to works of Nature, as when different strata of ice are formed, or

in other incrustations, and in all cases in which an increment is caused by the external

application of parts, where finite additions seem to be acquired all at once without any

passage through intermediate magnitudes. In these cases the continuity is preserved in

the motions of the separate parts that are added. These reach the place allotted to them

along some continuous line & with a continuous change of velocity. Further, after they

have reached it, they still continue to move, & never have absolute rest ; no, nor even

relative rest with respect to the other parts, although they do not now suffer a sensible

change in their relative positions. Thus, they still submit to the action of all the forces

that correspond to all points of matter at any distances whatever ; and the action of the

parts nearest to them, which produces a new adhesion, is the continuation of the action

that they exert to a far smaller extent when they are some distance away. Moreover, in

the fact that they belong to that house or mass, there is something that is not determinate

in itself, because it happens at a determinate instant in which the sudden change takes

place ; but it depends on a somewhat rough assessment by our senses. So that, although

these parts are continually being added, & continually go on changing their position

with respect to the mass, they both begin to be thought of as belonging to that house or

mass, & the relative change ceases to be sensible ; also this cessation of sensibility itself

also takes place to some extent through all stages, and in some continuous interval of time,

& not by a sudden jump.

KO. From this consideration we may here in a clearer manner remove all difficulty

-' * . , . - •jrt,'U* simuar cd.b

by saying that a continuous change is not maintained in the magnitudes ot those tmngs, derived from this,
which are not themselves continuous, & do not possess continuous magnitude, but are
aggregates of separate things. That is to say, in those things that are thus considered as
forming a certain whole, in such a way that the magnitude of the aggregate is not determined

122

PHILOSOPHIC NATURALIS THEORIA

determinent distantias inter eadem extrema, sed a nobis extrema ipsa assumantur jam alia,
jam alia, quae censeantur incipere ad aggregatum pertinere, ubi ad quasdam distantias
devenerint, quas ut ut in se juxta legem continuitatis mutatas, nos a reliquis divellimus per
saltum, ut dicamus pertinere eas partes ad id aggregatum. Id accidit, ubi in objectis
casibus accessiones partium novae fiunt, atque ibi nos in usu vocabuli saltum facimus ; ars,
& Natura saltum utique habet nullum.

Alii casus in quibus 151. Non idem contingit etiam, ubi plantas, vel animantia crescunt, succo se insinuante

'uibusUr' hab'etur Per tubulos fibrarum, & procurrente, ubi & magnitude computata per distantias punctorum
soium proxima, non maxime distantium transit per omnes intermedias ; cum nimirum ipse procursus fiat
accurata contmm- pef omnes intermedias distantias. At quoniam & ibi mutantur termini illi, qui distantias
determinant, & nomen suscipiunt altitudinis ipsius plantas ; vera & accurata continuitas ne
ibi quidem observatur, nisi tantummodo in motibus, & velocitatibus, ac distantiis singularum
partium : quanquam ibi minus recedatur a continuitate accurata, quam in superioribus. In
his autem, & in illis habetur ubique ilia alia continuitas quasdam apparens, & affectata
tantummodo a Natura, quam intuemur etiam in progressu substantiarum, ut incipiendo ab
inanima-[69]-tis corporibus progressu facto per vegetabilia, turn per quasdam fere
semianimalia torpentia, ac demum animalia perfectiora magis, & perfectiora usque ad simios
homini tarn similes. Quoniam & harum specierum, ac existentium individuorum in quavis
specie numerus est finitus, vera continuitas haberi non potest, sed ordinatis omnibus in
seriem quandam, inter binas quasque intermedias species hiatus debet esse aliquis necessario,
qui continuitatem abrumpat. In omnibus iis casibus habentur discretas quasdam quantitates,
non continues ; ut & in Arithmetica series ex. gr. naturalium numerorum non est continua,
sed discreta ; & ut ibi series ad continuam reducitur tantummodo, si generaliter omnes
intermedias fractiones concipiantur ; sic & in superiore exemplo quasdam velut continua
series habebitur tantummodo ; si concipiantur omnes intermedias species possibiles.

uitatem.

Conciusio pertinens 152. Hoc pacto excurrendo per plurimos justmodi casus, in quibus accipiuntur

ad ea, quse veram, aggregata rerum a se invicem certis intervallis distantium, & unum aliquid continuum non

(X CcL, CJ1.13E cLttCCtcl" OO O •••iill**

tam habent contin- constituentium, nusquam accurata occurret continuitatis lex, sed per quandam dispersionem
quodammodo affectata, & vera continuitas habebitur tantummodo in motibus, & in iis, quas
a motibus pendent, uti sunt distantiae, & vires determinatas a distantiis, & velocitates a
viribus ortae ; quam ipsam ob causam ubi supra num. 39 inductionem pro lege continuitatis
assumpsimus, exempla accepimus a motu potissimum, & ab iis, quae cum ipsis motibus
connectuntur, ac ab iis pendent.

153. Sed jam ad aliam difficultatem gradum faciam, quae non nullis negotium ingens
3ito facessit, & obvia est etiam, contra hanc indivisibilium, & inextensorum punctorum Theoriam ;
' & quod nimirum ea nullum habitura sint discrimen a spiritibus. Ajunt enim, si spiritus
ejusmodi vires habeant, praestituros eadem phaenomena, tolli nimirum corpus, & omnem
corporeae substantiae notionem sublata extensione continua, quae sit prascipua materias
proprietas ita pertinens ad naturam ipsius ; ut vel nihil aliud materia sit, nisi substantia
praedita extensione continua ; vel saltern idea corporis, & materiae haberi non ppssit ; nisi
in ea includatur idea extensionis continuae. Multa hie quidem congeruntur simul, quae
nexum aliquem inter se habent, quae hie seorsum evolvam singula.

Difficultates petitae
a discrimine debito
inter materiam
spiritum.

DifferrehKcpuncta 154. Inprimis falsum omnino est, nullum esse horum punctorum discrimen a spiritibus.

fm^netrabUttlte^ Discrimen potissimum materiae a spiritu situm est in hisce duobus, quod _ materia_ est

™nSitatem,a e£- sensibilis, & incapax cogitationis, ac voluntatis, spiritus nostros sensus non afficit, & cogitare

capadtatem cogit- pOtest) ac velle. Sensibilitas autem non ab extensione continua oritur, sed ab impene-

trabilitate, qua fit, ut nostrorum organorum fibrae tendantur a corporibus, quae ipsis

sistuntur, & motus ad cerebrum pro-[7o]-pagetur. Nam si extensa quidem essent corpora,

sed impenetrabilitate carerent ; manu contrectata fibras non sisterent, nee motum ullum

in iis progignerent, ac eadem radios non reflecterent, sed liberum intra se aditum luci

prasberent. Porro hoc discrimen utrumque manere potest integrum, & manet inter mea

indivisibilia hasc puncta, & spiritus. Ipsa impenetrabilitatem habent, & sensus nostros

afficiunt, ob illud primum crus asymptoticum exhibens vim illam repulsivam primam ;

spiritus autem, quos impenetrabilitate carere credimus, ejusmodi viribus itidem carent, &

sensus nostros idcirco nequaquam afficiunt, nee oculis inspectantur, nee ^manibus palpari

possunt. Deinde in meis hisce punctis ego nihil admitto aliud, nisi illam virium legem cum

inertias vi conjunctam, adeoque ilia volo prorsus incapacia cogitationis, & voluntatis.

A THEORY OF NATURAL PHILOSOPHY

I23

by the distances between the same extremes all the time, but the extremes we take are
different, one after another ; & these are considered to begin to belong to the aggregate
when they attain to certain distances from it ; &, although in themselves changed in
accordance with the Law of Continuity, we separate them from the rest in a discontinuous
manner, by saying that these parts belong to the aggregate. This comes about, whenever
in the cases under consideration fresh additions of parts take place ; & then we make a
discontinuity in the use of a term ; art, as well as Nature, has no discontinuity.

151. It is not the same thing however in the case of the growth of plants or animals,
which is due to a life-principle insinuating itself into, & passing along the fine tubes of the
fibres ; here the magnitude, calculated by means of the distance between the points furthest
from one another, passes through all intermediate distances ; for the flow of the life-principle
takes place indeed through all intermediate distances. But, since here also the extremes
are changed, which determine the distances, & denominate the altitude of the plant ;
not even in this case is really accurate continuity observed, except only in the motions &
velocities and distances of the separate parts ; however there is here less departure from
accurate continuity, than there was in the examples given above. In both there is indeed
that kind of apparent continuity, which Nature does no more than try to maintain ; such
as we also see in the series of substantial things, which starting from inanimate bodies,
continues through vegetables, then through certain sluggish semianimals, & lastly, through
animals more & more perfect, up to apes that are so like to man. Also, since the number
of these species, & the number of existent individuals of any species, is finite, it is impossible
to have true continuity ; but if they are all ordered in a series, between any two intermediate
species there must necessarily be a gap ; & this will break the continuity. In all these
cases we have certain discrete, & not continuous, quantities ; just as, for instance, the
arithmetical series of the natural numbers is not continuous, but discrete. Also, just as the
series is reduced to continuity only by mentally introducing in general all the intermediate
fractions ; so also, in the example given above a sort of continuous series is obtained, if
& only if all intermediate possible species are so included.

152. In the same way, if we examine a large number of cases of the same kind, in which
aggregates of things are taken, separated from one another by certain definite intervals,
& not composing a single continuous whole, an accurate continuity law will never be
met with, but only a sort of counterfeit depending on dispersion. True continuity will
only be obtained in motions, & in those things that depend on motions, such as distances
& forces determined by distances, & velocities derived from such forces. It was for
this very reason that, when we adopted induction for the proof of the Law of Continuity
in Art. 39 above, we took our examples mostly from motion, & from those things which
are connected with motions & depend upon them.

153. Now I will pass on to another objection, which some people have made a great
to-do about, and which has also been raised in opposition to this Theory of indivisible &
non-extended points ; namely, that there will be no difference between my points &
spirits. For, they say that, if spirits were endowed with such forces, they would show the
same phenomena as bodies, & that bodies & all idea of corporeal substance would be
done away with by denying continuous extension ; for this is one of the chief properties of
matter, so pertaining to Nature itself ; so that either matter is nothing else but substance
endowed with continuous extension, or the idea of a body and of matter cannot be obtained
without the inclusion of the idea of continuous extension. Here indeed there are many
matters all jumbled together, which have no connection with one another ; these I will
now separate & discuss individually.

154. First of all it is altogether false that there is no difference between my points &
spirits. The most important difference between matter & spirit lies in the two facts,
that matter is sensible & incapable of thought, whilst spirit does not affect the senses,
but can think or will. Moreover, sensibility does not arise from continuous extension,
but from impenetrability, through which it comes about that the fibres of our organs are
subjected to stress by bodies that are set against them & motions are thereby propagated
to the brain. For if indeed bodies were extended, but lacked impenetrability, they would
not resist the fibres of the hand when touched, nor produce in them any motion ; nor
would they reflect light, but allow it an uninterrupted passage through themselves.
Further, it is possible that each of these distinctions should hold good independently ;
& they do so between these indivisible points of mine & spirits. My points have
impenetrability & affect our senses, because of that first asymptotic branch representing that
first repulsive force ; but spirits, which we suppose to lack impenetrability, lack also forces
of this kind, and therefore can in no wise affect our senses, nor be examined by the eyes,
nor be felt by the hands. Then, in these points of mine, I admit nothing else but the
law of forces conjoined with the force of inertia ; & hence I intend them to be incapable

Cases in which
there is a breach of
continuity ; others
in which the con-
tinuity is only very
nearly, but not
accurately, ob-
served.

Conclusion as re-
gards those things
that possess true
continuity, and
those that have a
counterfeit continu-
ity.

Objections derived
from the distinc-
tion that has to be
made between
matter & spirit.

These points differ
from spirits on
account o f their
impenet rability,
their being sen-
sible, & their inca-
pacity for thought.

I24

PHILOSOPHIC NATURALIS THEORIA

Si possibilis sub-

earn nee
materiam
spiritum.

Quamobrem discrimen essentiae illud utrumque, quod inter corpus, & spiritum
agnoscunt omnes, id & ego agnosco, nee vero id ab extensione, & compositione continua
desumitur, sed ab iis, quae cum simplicitate, & inextensione aeque conjungi possunt, &
cohaerere cum ipsis.

155. At si substantiae capaces cogitationis & voluntatis haberent ejusmodi virium legem,
an non eosdem praestarent effectus respectu nostrorum sensuum, quos ejusmodi puncta ?
capax cogitationis ; Respondebo sane, me hie non quaerere, utrum impenetrabilitas, & sensibilitas, quae ab iis
fnec viribus pendent, conjungi possint cum facultate cogitandi, & volendi, quae quidem quaestio
eodem redit, ac in communi sententia de impenetrabilitate extensorum, ac compositorum
relata ad vim cogitandi, & volendi. Illud ajo, notionem, quam habemus partim ex
observationibus tarn sensuum respectu corporurh, quam intimae conscientiae respectu
spiritus, una cum reflexione, partim, & vero etiam circa spiritus potissimum, ex principiis
immediate revelatis, vel connexis cum principiis revelatis, continere pro materia
impenetrabilitatem, & sensibilitatem, una cum incapacitate cogitationis, & pro spiritu
incapacitatem afHcicndi per impenetrabilitatem nostros sensus, & potentiam cogitandi, ac
volendi, quorum priores illas ego etiam in meis punctis admitto, posteriores hasce in
spiritibus ; unde fit, ut mea ipsa puncta materialia sint, & eorum massae constituant
corpora a spiritibus longissime discrepantia. Si possibile sit illud substantiae genus, quod
& hujusmodi vires activas habeat cum inertia conjunctas, & simul cogitare possit, ac velle ;
id quidem nee corpus erit, nee spiritus, sed tertium quid, a corpore discrepans per capacitatem
cogitationis, & voluntatis, discrepans autem a spiritu per inertiam, & vires hasce nostras,
quae impenetrabilitatem inducunt. Sed, ut ajebam, ea quaestio hue non pertinet, & aliunde
resolvi debet ; ut aliunde utique debet resolvi quaestio, qua quaeratur, an substantia extensa,
& impenetrabilis [71] hasce proprietates conjungere possit cum facultate cogitandi,
volendique.

Nihil amitti, 156. Nee vero illud reponi potest, argumentum potissimum ad evincendum, materiam

amisso argumento cogitare non posse, deduci ab extensione, & partium compositione, quibus sublatis, omne id

eorum, qui a com- r to , . r, . • • VT • i

positione partium lundamentum prorsus corruere, & ad materialismum sterm viam. JNam ego sane non video,
deducunt incapaci- quid argument! peti possit ab extensione, & partium compositione pro incapacitate cogitandi,
& volendi. Sensibilitas, praecipua corporum, & materiae proprietas, quae ipsam adeo a
spiritibus discriminat, non ab extensione continua, & compositione partium pendet, uti
vidimus, sed ab impenetrabilitate, quae ipsa proprietas ab extensione continua, & compositione
non pendet. Sunt qui adhibent hoc argumentum ad excludendam capacitatem cogitandi
a materia, desumptum a compositione partium : si materia cogitaret ; singulae ejus partes
deberent singulas cogitationis partes habere, adeoque nulla pars objectum perciperet ; cum
nulla haberet earn perceptionis partem, quam habet altera. Id argumentum in mea Theoria
amittitur ; at id ipsum, meo quidem judicio, vim nullam habet. Nam posset aliquis
respondere, cogitationem totam indivisibilem existere in tota massa materiae, quae certa
partium dispositione sit praedita, uti anima rationalis per tarn multos Philosophos, ut ut
indivisibilis, in omni corpore, vel saltern in parte corporis aliqua divisibili existit, & ad
ejusmodi praesentiam praestandam certa indiget dispositione partium ipsius corporis, qua
semel laesa per vulnus, ipsa non potest ultra ibi esse ; atque ut viventis corporei, sive animalis
rationalis natura, & determinatio habetur per materiam divisibilem, & certo modo
constructam, una cum anima indivisibili ; ita ibi per indivisibilem cogitationem inhaerentem
divisibili materise natura, & determinatio cogitantis haberetur. Unde aperte constat eo
argumento amisso, nihil omnino amitti, quod jure dolendum sit.

Etiam si quidpiam 157. Sed quidquid de eo argumento censeri debeat, nihil refert, nee ad infirmandam

iam^poStive ^prob- Theoriam positivis, & validis argumentis comprobatam, ac e solidissimis principiis directa

ari, & in ea manere ratiocinatione deductani, quidquam potest unum, vel alterum argumentum amissum, quod

inte™m™ter1ainme& a^ probandam aliquam veritatem aliunde notam, & a revelatis principiis aut directe, aut

spiritum. indirecte confirmatam, ab aliquibus adhibeatur, quando etiam vim habeat aliquam, quam,

uti ostendi, superius allatum argumentum omnino non habet. Satis est, si ilia Theoria cum

ejusmodi veritate conjungi possit, uti haec nostra cum immaterialitate spirituum con-

jungitur optime, cum retineat pro materia inertiam, impenetrabilitatem, sensibilitatem,

incapacitatem cogitandi, & pro spiritibus retineat incapacitatem afHciendi sensus nostros

per impenetrabilitatem, & facultatem cogitandi, ac volendi. [72] Ego quidem in ipsius

A THEORY OF NATURAL PHILOSOPHY 125

of thought or will. Wherefore I also acknowledge each of those essential differences between
matter and spirit, which are acknowledged by everyone ; but by me it is not deduced from
extension and continuous composition, but, just as correctly, from things that can be
conjoined with simplicity & non-extension, & can combine with them.

155. Now if there were substances capable of thought & will that also had a law of if it were possible
forces of this kind, is it possible that they would produce the same effects with respect to substanc<Pthatwas
our senses, as points of this sort ? Truly, I will answer that I do not seek to know in this both endowed with
connection, whether impenetrability & sensibility, which depend on these forces, can capsTbieofthoughT
be conjoined with the faculty of thinking & willing ; indeed this question comes to the it would be neither
same thing as the general idea of the relations of impenetrability of extended & composite matter nor sP'nt-
things to the power of thinking & willing. I will say but this, that we form our ideas,

partly from observations, of the senses in the case of bodies, & of the inner consciousness
in the case of spirits, together with reflections upon them, partly, & indeed more especially
in the case of spirits, from directly revealed principles, or matters closely connected with
revealed principles ; & these ideas involve for matter impenetrability, sensibility, combined
with incapacity for thought, & for spirit an incapacity for affecting our senses by means
of impenetrability, together with the capacity for thinking and willing. I admit the former
of these in the case of my points, & the latter for spirits ; so that these points of mine
are material points, & masses of them compose bodies that are far different from spirits.
Now if it were possible that there should be some kind of substance, which has both active
forces of this kind together with a force of inertia & also at the same time is able to
think and will ; then indeed it will neither be body nor spirit, but some third thing, differing
from a body in its capacity for thought & will, & also from spirit by possessing inertia
and these forces of mine, which lead to compenetration. But as I was saying, that question
does not concern me now, & the answer must be found by other means. So by other
means also must the answer be found to the question, in which we seek to know whether
a substance that is extended & impenetrable can conjoin these two properties with the
faculty of thinking and willing.

156. Now it cannot be ignored that an argument of great importance in proving that Nothing is lost
matter is incapable of thought is deduced from extension & composition by parts ; & ^n ^g^^1"1^
if these are denied, the whole foundation breaks down, & the way is laid open to materialism, those who deduce
But really I do not see what in the way of argument can be derived from extension & i? °t&%~*yj;°r

... , *, i • i • i *ii* n *i "v v i_ • f lutJU5i*^ irom com-

composition by parts, to support incapacity for thinking and willing, bensibmty, the cruel position by parts.

property of bodies & of matter, which is so much different from spirits, does not depend

on continuous extension & composition by parts, as we have seen, but on impenetrability ;

& this latter property does not depend on continuous extension & composition. There

are some, who use the following argument, derived from composition by parts, to exclude

from matter the capacity for thought : — If matter were to think, then each of its parts

would have a separate part of the thought, & thus no part would have perception of the

object of thought ; for no part can have that part of the perception that another part has.

This argument is neglected in my Theory ; but the argument itself, at least so I think, is

unsound. For one can reply that the complete thought exists as an indivisible thing in

the whole mass of matter, which is endowed with a certain arrangement of parts, in^the

same way as the rational soul in the opinion of so many philosophers exists, although it is

indivisible, in the whole of the body, or at any rate in a certain divisible part of the body ;

& to maintain a presence of this kind there is need for a definite arrangement of the parts of

the body, which if at any time impaired by a wound would no longer exist there. Thus,

just as from the nature of a living body, or of a rational animal, determination arises from

matter that is divisible & constructed on a definite plan, in conjunction with an indivisible

mind ; so also in this case by means of indivisible thought inherent in the nature of divisible

matter, there is a propensity for thought. From this it is very plain that, if this

argument is dismissed, there will be nothing neglected that we have any reason to

regret.

157. But whatever opinion we are to form about this argument, it makes no difference, Even a something
nor can it weaken a Theory that has been corroborated by direct & valid arguments, & iheVheorrcln Cbe
deduced from the soundest principles by a straightforward chain of reasoning, if we leave P^J^. in&a dt^t
out one or other of the arguments, which have been used by some for the purpose of ^f^fi remain in
testing some truth that is otherwise known & confirmed by revealed principles either j^^fj*^
directly or indirectly ; even when the argument has some validity, which, as I have shown, matter & spirit.
that adduced above has not in any way. It is sufficient if that theory can be conjoined

with such a truth ; just as this Theory of mine can be conjoined in an excellent manner
with the immateriality of spirits. For it retains for matter inertia, impenetrability,
sensibility, & incapacity for thinking, & for spirits it retains the incapacity for affecting
our senses by impenetrability, & the faculty of thinking or willing. Indeed I assume the

126 PHILOSOPHIC NATURALIS THEORIA

materiae, & corporeae substantias definitione ipsa assumo incapacitatem cogitandi, & volendi,
& dico corpus massam compositam e punctis habentibus vim inertiae conjunctam cum
viribus activis expressis in fig. i,& cum incapacitate cogitandi, ac volendi, qua definitione
admissa, evidens est, materiam cogitare non posse ; quae erit metaphysica quaedam conclusio,
ea definitione admissa, certissima : turn ubi solae rationes physicae adhibeantur, dicam, haec
corpora, quae meos afficiunt sensus, esse materiam, quod & sensus afficiant per illas utique
vires, & non cogitent. Id autem deducam inde, quod nullum cogitationis indicium
praestent ; quae erit conclusio tantum physica, circa existentiam illius materiae ita definitae,
aeque physice certa, ac est conclusio, quae dicat lapides non habere levitatem, quod nunquam
earn prodiderint ascendendo sponte, sed semper e contrario sibi relict! descenderint.

Sensus omnino fain 158. Quod autem pertmet ad ipsam corporum, & materiae ideam, quae videtur exten-

^nultat^in^xten- si°nem continuam, & contactum partium involvere, in eo videntur mihi quidem Cartesian!
sionis, quam nobis inprimis, qui tantopere contra prasjudicia pugnare sunt visi, praejudiciis ipsis ante omnes
alios indulsisse. Ideam corporum habemus per sensus ; sensus autem de continuitate
accurata judicare omnino non possunt, cum minima intervalla sub sensus non cadant. Et
quidem omnino certo deprehendimus illam continuitatem, quam in plerisque corporibus
nobis objiciunt sensus nostri, nequaquam haberi. In metallis, in marmoribus, in vitris,
& crystallis continuitas nostris sensibus apparet ejusmodi, ut nulla percipiamus in iis vacua
spatiola, nullos poros, in quo tamen hallucinari sensus nostros manifesto patet, turn ex
diversa gravitate specifica, quae a diversa multitudine vacuitatum oritur utique, turn ex
eo, quod per ilia insinuentur substantiae plures, ut per priora oleum diffundatur, per
posteriora liberrime lux transeat, quod quidem indicat, in posterioribus hisce potissi-
mum ingentem pororum numerum, qui nostris sensibus delitescunt.

Fons prajudici- 159- Quamobrem jam ejusmodi nostrorum sensuum testimonium, vel potius noster

orum : haberi pro eorum ratiociniorum usus, in hoc ipso genere suspecta esse debent, in quo constat nos

nulhs in se, quas , .... . • 11- • v • • -i_

sunt nuiia in nostris decipi. Suspican igitur licet, exactam continuitatem sine urns spatiolis, ut in majonbus
sensibus : eorum corporibus ubique deest, licet sensus nostri illam videantur denotare, ita & in minimis
quibusvis particulis nusquam haberi, sed esse illusionem quandam sensuum tantummodo,
& quoddam figmentum mentis, reflexione vel non utentis, vel abutentis. Est enim
solemne illud hominibus, atque usitatum, quod quidem est maximorum praejudiciorum
fons, & origo praecipua, ut quidquid in nostris sensibus est nihil, habeamus pro nihilo
absolute. Sic utique per tot saecula a multis est creditum, & nunc etiam a vulgo creditur,
[73] quietem Telluris, & diurnum Solis, ac fixarum motum sensuum testimonio evinci,
cum apud Philosophos jam constet, ejusmodi qusestionem longe aliunde resolvendam esse,
quam per sensus, in quibus debent eaedem prorsus impressiones fieri, sive stemus & nos, &
Terra, ac moveantur astra, sive moveamur communi motu & nos, & Terra, ac astra
consistant. Motum cognoscimus per mutationem positionis, quam objecti imago habet
in oculo, & quietem per ejusdem positionis permanentiam. Tarn mutatio, quam
permanentia fieri possunt duplici modo : mutatio, primo si nobis immotis objectum movea-
tur ; & permanentia, si id ipsum stet : secundo, ilia, si objecto stante moveamur nos ; haec, si
moveamur simul motu communi. Motum nostrum non sentimus, nisi ubi nos ipsi motum
inducimus, ut ubi caput circumagimus, vel ubi curru delati succutimur. Idcirco habemus
turn quidem motum ipsum pro nullo, nisi aliunde admoneamur de eodem motu per causas,
quae nobis sint cognitae, ut ubi provehimur portu, quo casu vector, qui jam diu assuevit idese
littoris stantis, & navis promotae per remos, vel vela, corrigit apparentiam illius, terrceque
urbesque recedunt, & sibi, non illis, motum adjudicat.

Eorum correctio 160. Hinc Philosophus, ne fallatur, non debet primis hisce ideis acquirere, quas e

ubi deprehendatur, sensationibus haurimus, & ex illis deducere consectaria sine diligent! perquisitione, ac in ea

modoalc°umtlsaenn quae ab infantia deduxit, debet diligenter inquirere. Si inveniat, easdem illas sensuum

suum apparentia perceptiones duplici modo aeque fieri posse ; peccabit utique contra Logicae etiam naturalis

leges, si alterum modum prze altero pergat eligere, unice, quia alterum antea non viderat,

& pro nullo habuerat, & idcirco alteri tantum assueverat. Id vero accidit in casu nostro :

A THEORY OF NATURAL PHILOSOPHY 127

incapacity for thinking & willing in the very definition of matter itself & corporeal
substance ; & I say that a body is a mass composed of points endowed with a force of
inertia together with such active forces as are represented in Fig. i, & an incapacity for
thinking & willing. If this definition is taken, it is clear that matter cannot think ; &
this will be a sort of metaphysical conclusion, which will follow with absolute certainty
from the acceptation of the definition. Again, where physical arguments are alone employed,
I say that such bodies as affect our senses are matter, because they affect the senses
by means of the forces under consideration, & do not think. I also deduce the same
conclusion from the fact that they afford no evidence of thought. This will be a conclusion
that is solely physical with regard to the existence of matter so defined ; & it will be just
as physically true as the conclusion that says that stones do not possess levity, deduced from
the fact that they never display such a thing by an act of spontaneous ascent, but on the
contrary always descend if left to themselves.

158. With regard to the idea of bodies & matter, which seems to involve continuous The senses are
extension, it seems to me indeed that in this matter the Cartesians in particular, who have altogether at fault

. • r i i_ m the greatness of

appeared to impugn pre judgments with so much vigour, have given themselves up to these the continuity of

prejudgments more than anyone else. We obtain the idea of bodies through the senses ; f^^'^beiieve5''

and the senses cannot in any way judge on a matter of accurate continuity ; for very small

intervals do not fall within the scope of the senses. Indeed we quite take it for granted

that the continuity, which our senses meet with in a large number of bodies, does not really

exist. In metals, marble, glass & crystals there appears to our senses to be continuity,

of such sort that we do not perceive in them any little empty spaces, or pores ; but in this

respect the senses have manifestly been deceived. This is clear, both from their different

specific gravities, which certainly arises from the differences in the numbers of the empty

spaces ; & also from the fact that several substances will insinuate themselves through

their substance. For instance, oil will diffuse itself through the former, & light will pass

quite freely through the latter ; & this indeed indicates, especially in the case of the

latter, an immense number of pores ; & these are concealed from our senses.

159. Hence such evidence of our senses, or rather our employment of such arguments, The origin of pre-
must now lie open to suspicion in that class, in which it is known that we have been deceived, j^fjdered : as^o-
We may then suspect that accurate continuity without the presence of any little empty thing, which are
spaces — such as is certainly absent from bodies of considerable size, although our senses SSe1srases0ar^con-
seem to remark its presence — is also nowhere existent in any of their smallest particles ; cemed ; examples
but that it is merely an illusion of the senses, & a sort of figment of the brain through its °

not using, or through misusing, reflection. For it is a customary thing for men (& a
thing that is frequently done) to consider as absolutely nothing something that is nothing
as far as the senses are concerned ; & this indeed is the source & principal origin of
the greatest prejudices. Thus for many centuries it was credited by many, & still is
believed by the unenlightened, that the Earth is at rest, & that the daily motions of the
Sun & the fixed stars is proved by the evidence of the senses ; whilst among philosophers
it is now universally accepted that such a question has to be answered in a far different
manner from that by means of the senses. Exactly the same impressions are bound to be
obtained, whether we & the Earth stand still & the stars are moved, or we & the
Earth are moved with a common motion & the stars are at rest. We recognize motion
by the change of position, which the image of an object has in the eye ; and rest by the
permanence of that position. Now both the change & the permanence can come about
in two ways. Firstly, if we remain at rest, there is a change of position if the object is
moved, & permanence if it too is at rest ; secondly, if we move, there is a change if the
object is at rest, & permanence if we & it move with a motion common to both. We
do not feel ourselves moving, unless we ourselves induce the motion, as when we turn the
head, or when we are jolted as we are borne in a vehicle. Hence we consider that the
motion is nothing, unless we are made to notice in other ways that there is motion by causes
that are known to us. Thus, when " we leave the harbour" a passenger who has for some time
been accustomed to the idea of a shore remaining still, & of a ship being propelled by
oars or sails, corrects the apparent motion of the shore ; &, as " the land & buildings recede"
he attributes the motion to himself and not to them.

160. Hence, the philosopher, to avoid being led astray, must not seek to obtain from ^ctionjrf ^
these primary ideas that we derive from the senses, or deduce from them, consequential known that the
theorems, without careful investigation; & he must carefully study those things that matter^ ^annot^ be
he has deduced from infancy. If he find that these very perceptions by the senses can agreement with
come about in two ways, one of which is as probable as the other ; then he will certainly £hattheis £JV£ e£
commit an offence against the laws of natural logic, if he should proceed to choose one some other way.
method in preference to the other, solely for the reason that previously he had not seen

the one & took no account of it, & thus had become accustomed to the other. Now

128

PHILOSOPHIC NATURALIS THEORIA

sensationes habebuntur eaedem, sive materia constet punctis prorsus inextensis, & distantibus
inter se per intervalla minima, quae sensum fugiant, ac vires ad ilia intervalla pertinentes
organorum nostrorum fibras sine ulla sensibili interruptione afficiant, sive continua sit, &
per immediatum contactum agat. Patebit autem in tertia hujusce operis parte, quo pacto
proprietates omnes sensibiles corporum generales, immo etiam ipsorum prsecipua discrimina,
cum punctis hisce indivisibilibus conveniant, & quidem multo sane melius, quam in communi
sententia de continua extensione materiae. Quamobrem errabit contra rectae ratiocinationis
usum, qui ex praejudicio ab hujusce conciliationis, & alterius hujusce sensationum nostrarum
causae ignoratione inducto, continuam extensionem ut proprietatem necessariam corporum
omnino credat, & multo magis, qui censeat, materialis substantive ideam in ea ipsa continua
extensione debere consistere.

Ordo idearum, quas

esse per tactum.

161. Verum quo magis evidenter constet horum prsejudiciorum origo, afferam hie
dissertationis De Materia Divisibilita-\j4\-te, & Principiis Corporum, numeros tres inci-
piendo a 14, ubi sic : " utcunque demus, quod ego omnino non censeo, aliquas esse innatas
ideas, & non per sensus acquisitas ; illud procul dubio arbitror omnino certum, ideam
corporis, materiae, rei corporeae, rei materialis, nos hausisse ex sensibus. Porro ideas prims
omnium, quas circa corpora acquisivimus per sensus, fuerunt omnino eae, quas in nobis
tactus excitavit, & easdem omnium frequentissimas hausimus. Multa profecto in ipso
materno utero se tactui perpetuo offerebant, antequam ullam fortasse saporum, aut odorum,
aut sonorum, aut colorum ideam habere possemus per alios sensus, quarum ipsarum, ubi eas
primum habere ccepimus, multo minor sub initium frequentia fuit. Idese autem, quas per
tactum habuimus, ortae sunt ex phsenomenis hujusmodi. Experiebamur palpando, vel
temere impingendo resistentiam vel a nostris, vel a maternis membris ortam, quae cum
nullam interruptionem per aliquod sensibile intervallum sensui objiceret, obtulit nobis ideam
impenetrabilitatis, & extensionis continuae : cumque deinde cessaret in eadem directione,
alicubi resistentia, & secundum aliam directionem exerceretur ; terminos ejusdem quanti-
tatis concepimus, & figurse ideam hausimus."

Quae fuerint turn
consideranda : in-
fantia ad eas re-
flexiones, inepta : in
quo ea sita sit.

162. " Porro oriebantur haec phsenomena a corporibus e materia jam efformatis, non a
singulis materiae particulis, e quibus ipsa corpora componebantur. Considerandum
diligenter erat, num extensio ejusmodi esset ipsius corporis, non spatii cujusdam, per quod
particulae corpus efformantes diffunderentur : num ea particulse ipsae iisdem proprietatibus
essent praeditae : num resistentia exerceretur in ipso contactu, an in minimis distantiis sub
sensus non cadentibus vis aliqua impedimento esset, quae id ageret, & resistentia ante ipsum
etiam contactum sentiretur : num ejusmodi proprietates essent intrinsecae ipsi materiae, ex
qua corpora componuntur, & necessariae : an casu tantum aliquo haberentur, & ab extrinseco
aliquo determinante. Haec, & alia sane multa considerate diligentius oportuisset : sed erat
id quidem tempus maxime caliginosum, & obscurum, ac reflexionibus minus obviis minime
aptum. Praster organorum debilitatem, occupabat animum rerum novitas, phaenomenorum
paucitas, & nullus, aut certe satis tenuis usus in phaenomenis ipsis inter se comparandis, &
ad certas classes revocandis, ex quibus in eorum leges, & causas liceret inquirere & systema
quoddam efformare, quo de rebus extra nos positis possemus ferre judicium. Nam in hac
ipsa phaenomenorum inopia, in hac efformandi systematis difficultate, in hoc exiguo
reflexionum usu, magis etiam, quam in organorum imbecillitate, arbitror, sitam esse
infantiam."

inde [75] 163. "In hac tanta rerum caligine ea prima sese obtulerunt animo, quae^ minus
orta extensionis jta jndagine, minus intentis reflexionibus indigebant, eaque ipsa ideistoties repetitis altius

continuae ut essen- . . .-1

tiaiis, odorum, &c., impressa sunt, & tenacius adhaeserunt, & quendam veluti campum nacta prorsus vacuum,
ut accidentaiium. & acjhuc immunem, suo quodammodo jure quandam veluti possessionem inierunt. Inter-
valla, quae sub sensum nequaquam cadebant, pro nullis habita : ea, quorum ideae^ semper
simul conjunctae excitabantur, habita sunt pro iisdem, vel arctissimo, & necessario^ nexu
inter se conjunctis. Hinc illud effectum est, ut ideam extensionis continuae, ideam

A THEORY OF NATURAL PHILOSOPHY 129

that is just what happens in the case under consideration. The same sensations will be
experienced, whether matter consists of points that are perfectly non-extended & distant
from one another by very small intervals that escape the senses, & forces pertaining to
those intervals affect the nerves of our organs without any sensible interruption ; or
whether it is continuous and acts by immediate contact. Moreover it will be clearly shown,
in the third part of this work, how all the general sensible properties of bodies, nay even
the principal distinctions between them as well, will fit in with these indivisible points ;
& that too, in a much better way than is the case with the common idea of continuous
extension of matter. Wherefore he will commit an offence against the use of true reasoning,
who, from a prejudgment derived from this agreement & from ignorance of this alter-
native cause for our sensations, persists in believing that continuous extension is an
absolutely necessary property of bodies ; and much more so, one who thinks that
the very idea of material substance must depend upon this very same continuous
extension.

161. Now in order that the source of these prejudices may be the more clearly known, Order of the ideas
I will here quote, from the dissertation De Materice Divisibilitate & Princi-pii Corporum, ^£ b^ies^tte
three articles, commencing with Art. 14, where we have : — " Even if we allow (a thing quite first ideas come
opposed to my way of thinking) that some ideas are innate & are not acquired through o^ifch the SenSe
the senses, there is no doubt in my mind that it is quite certain that we derive the idea

of a body, of matter, of a corporeal thing, or a material thing, through the senses. Further,
the very first ideas, of all those which we have acquired about bodies through the senses,
would be in every circumstance those which have excited our sense of touch, & these
also are the ideas that we have derived on more occasions than any other ideas. Many
things continually present themselves to the sense of touch actually in the very womb of
our mothers, before ever perchance we could have any idea of taste, smell, sound, or colour,
through the other senses ; & of these latter, when first we commenced to have them,
there were to start with far fewer occasions for experiencing them. Moreover the ideas
which we have obtained through the sense of touch have arisen from phenomena of the
following kind. We experienced a resistance on feeling, or on accidental contact with, an
object ; & this resistance arose from our own limbs, or from those of our mothers. Now,
since this resistance offered no opposition through any interval that was perceptible to the
senses, it gave us the idea of impenetrability & continuous extension ; & then when
it ceased in the original direction at any place & was exerted in some other direction,
we conceived the boundaries of this quantity, & derived the idea of figure."

162. " Furthermore, these phenomena will have arisen from bodies already formed from Such things de-
matter, not from the single particles of matter of which the bodies themselves were composed. S^time ^tae tf
It would have to be considered carefully whether such extension was a property of the tude of inf'ancy^for
body itself, & not of some space through which the particles forming the body were su.fh. reflection ; on

j-rc JIT -11 i r i • i i • wnat they maY be

diffused ; whether the particles themselves were endowed with the same properties ; founded,
whether the resistance was exerted only on actual contact, or whether, at very small
distances such as did not fall within the scope of the senses, some force would act as a
hindrance & produce the same effect, and resistance would be felt even before actual
contact ; whether properties of this kind would be intrinsic in the matter of which the
bodies are composed, & necessary to its existence ; or only possessed in certain cases,
being due to some external influence. These, & very many other things, should have
been investigated most carefully ; but the period was indeed veiled in mist & obscurity
to a great degree, & very little fitted for aught but the most easy thought. In addition
to the weakness of the organs, the mind was occupied with the novelty of things & the
rareness of the phenomena ; & there was no, or certainly very little, use made of comparisons
of these phenomena with one another, to reduce them to definite classes, from which it
would be permissible to investigate their laws & causes & thus form some sort of system,
through which we could bring the judgment to bear on matters situated outside our own
selves. Now, in this very paucity of phenomena, in this difficulty in the matter of forming
a system, in this slight use of the powers of reflection, to a greater extent even than in the
lack of development of the organs, I consider that infancy consists."

163. " In this dense haze of things, the first that impressed themselves on the mind Thence Pr«J'u<^-

i 1*1 • -i i 11 i • • •• n i • mcnis> di c uci i vcu.

were those which required a less deep study & less intent investigation ; & these, since that continuity of
the ideas were the more often renewed, made the greater impression & became fixed J^^J1^ «S
the more firmly in the mind, & as it were took possession of, so to speak, a land that they continuity of odours
found quite empty & hitherto immune, by a sort of right of discovery. Intervals, which &c- * accidental.
in no wise came within the scope of the senses, were considered to be nothing ; those things,
the ideas of which were always excited simultaneously & conjointly, were considered
as identical, or bound up with one another by an extremely close & necessary bond.
Hence the result is that we have formed the idea of continuous extension, the idea of

130 PHILOSOPHISE NATURALIS THEORIA

impenetrabilitatis prohibentis ulteriorem motum in ipso tantum contactu corporibus
affinxerimus, & ad omnia, quae ad corpus pertinent, ac ad materiam, ex qua ipsum constat,
temere transtulerimus : quse ipsa cum primum insedissent animo, cum frequcntissimis, immo
perpetuis phaenomenis, & experimentis confirmarentur ; ita tenaciter sibi invicem
adhseserunt, ita firmiter ideae corporum immixta sunt, & cum ea copulata ; ut ea ipsa pro
primis corporibus, & omnium corporearum rerum, nimirum etiam materiae corpora compo-
nentis, ejusque partium proprietatibus maxime intrinsecis, & ad naturam, atque essentiam
earundem pertinentibus, & turn habuerimus, & nunc etiam habeamus, nisi nos praejudiciis
ejusmodi liberemus. Extensionem nimirum continuam, impenetrabilitatem ex contactu,
compositionem ex partibus, & figuram, non solum naturae corporum, sed etiam corporeae
materiae, & singulis ejusdem partibus, tribuimus tanquam proprietates essentiales : csetera,
quae serius, & post aliquem reflectendi usum deprehendimus, colorem, saporem, odorem
sonum, tanquam accidentales quasdam, & adventitias proprietates consideravimus."

propositiones 164. Ita ego ibi, ubi Theoriam virium deinde refero, quam supra hie exposui, ac ad

Theoriam°continen? Pr3ecipuas corporum proprietates applico, quas ex ilia deduco, quod hie praestabo in parte
tis. tertia. Ibi autem ea adduxeram ad probandam primam e sequentibus propositionibus,

quibus probatis & evincitur Theoria mea, & vindicatur : sunt autem hujusmodi : i. Nullo
prorsus argumento evincitur materiam habere extensionem continuam, W non potius constare e
punctis prorsus indivisibilibus a se per aliquod intervallum distantibus ; nee ulla ratio seclusis
pr&judiciis suadet extensionem ipsam continuam potius, quam compositionem e punctis prorsus
indivisibilibus, inextensis, y nullum continuum extensum constituentibus. 2. Sunt argumenta,
y satis valida ilia quidem, qua hanc compositionem e punctis indivisibilibus evincant extensioni
ipsi continues pr&ferri oportere.

Quo pacto con- 165. At quodnam extensionis genus erit istud, quod e punctis inextensis, & spatio

coaiescan^lnmassas imaginario, sive puro nihilo [76] constat ? Quo pacto Geometria locum habere poterit,
tenaces: transitus ubi nihil habetur reale continue extensum? An non punctorum ejusmodi in vacuo
damPartem secun" innatantium congeries erit, ut quaedam nebula unico oris flatu dissolubilis prorsus sine ulla
consistent! figura, solidate, resistentia ? Haec quidem pertinent ad illud extensionis ,&
cohaesionis genus, de quo agam in tertia parte, in qua Theoriam applicabo ad Physicam, ubi
istis ipsis difficultatibus faciam satis. Interea hie illud tantummodo innuo in antecessum, me
cohaesionem desumere a limitibus illis, in quibus curva virium ita secat axem, ut a repulsione
in minoribus distantiis transitus fiat ad attractionem in majoribus. Si enim duo puncta
sint in distantia alicujus limitis ejus generis, & vires, quae immutatis distantiis oriuntur, sint
satis magnae, curva secante axem ad angulum fere rectum, & longissime abeunte ab ipso ;
ejusmodi distantiam ea puncta tuebuntur vi maxima ita, ut etiam insensibiliter compressa
resistant ulteriori compressioni, ac distracta resistant ulteriori distractioni ; quo pacto si
multa etiam puncta cohaereant inter se, tuebuntur utique positionem suam, & massam
constituent formae tenacissimam, ac eadem prorsus phsenomena exhibentem, quae exhiberent
solidae massulae in communi sententia. Sed de hac re uberius, uti monui, in parte tertia :
nunc autem ad secundam faciendus est gradus.

A THEORY OF NATURAL PHILOSOPHY 131

impenetrability preventing further motion only on the absolute contact of bodies ; &
then we have heedlessly transferred these ideas to all things that pertain to a solid body,
and to the matter from which it is formed. Further, these ideas, from the time when they
first entered the mind, would be confirmed by very frequent, not to say continual, phenomena
& experiences. So firmly are they mutually bound up with one another, so closely are
they intermingled with the idea of solid bodies & coupled with it, that we at the time
considered these two things as being just the same as primary bodies, & as peculiarly
intrinsic properties of all corporeal things, nay further, of the very matter from which
bodies are composed, & of its parts ; indeed we shall still thus consider them, unless we
free ourselves from prejudgments of this nature. To sum up, we have attributed continuous
extension, impenetrability due to actual contact, composition by parts, & shape, as if
they were essential properties, not only to the nature of bodies, but also to corporeal matter
& every separate part of it ; whilst others, which we comprehend more deeply & as
a consequence of some considerable use of thought, such as colour, taste, smell & sound,
we have considered as accidental or adventitious properties."

164. Such are the words I used ; & then I stated the Theory of forces which I have A pair of proposi-
expounded in the previous articles of this work, and I applied the theory to the principal tation0f containing
properties of bodies, deducing them from it ; & this I will set forth in the third part the whole of ™y
of the present work. In the dissertation I had brought forward the arguments quoted *"

in order to demonstrate the truth of the first of the following theorems. If these theorems
are established, then my Theory is proved & verified; they are as follows : — i. There is
absolutely no argument that can be brought forward to prove that matter has continuous extension,
y that it is not rather made up of perfectly indivisible points separated from one another by
a definite interval ; nor is there any reason apart from prejudice in favour of continuous extension
in preference to composition from points that are perfectly indivisible, non-extended, & forming
no extended continuum of any sort. 2. There are arguments, W fairly strong ones too, which
will prove that this composition from indivisible points is preferable to continuous extension.

165. Now what kind of extension can that be which is formed out of non-extended The manner in

o • • t i • 5 TT /-i 1111 which groups of

points & imaginary space, i.e., out of pure nothing ? How can Geometry be upheld points coalesce into
if no thing is considered to be actually continuously extended ? Will not groups of points, tenacious masses :

n • • t i • 11-1 i i T i • • i i i n & then we pass on

floating in an empty space of this sort be like a cloud, dissolving at a single breath, & to the second part.

absolutely without a consistent figure, or solidity, or resistance ? These matters pertain

to that kind of extension & cohesion, which I will discuss in the third part, where I apply

my Theory to physics & deal fully with these very difficulties. Meanwhile I will here

merely remark in anticipation that I derive cohesion from those limit-points, in which the

curve of forces cuts the axis, in such a way that a transition is made from repulsion at smaller

distances to attraction at greater distances. For if two points are at the distance that

corresponds to that of any of the limit-points of this kind, & the forces that arise when

the distances are changed are great enough (the curve cutting the axis almost at right angles

& passing to a considerable distance from it), then the points will maintain this distance

apart with a very great force ; so that when they are insensibly compressed they will resist

further compression, & when pulled apart they resist further separation. In this way

also, if a large number of points cohere together, they will in every case maintain their

several positions, & thus form a mass that is most tenacious as regards its form ; & this

mass will exhibit exactly the same phenomena as little solid masses, as commonly understood,

exhibit. But I will discuss this more fully, as I have remarked, in the third part ; for now

we must pass on to the second part.

[77] PARS II
Theories *Applicato ad Mechanicam

Ante appHcatipnem 166. Considerabo in hac secunda parte potissimum generates quasdam leges aequilibrii
consideratio'curvs! & motus tam punctorum, quam massarum, quae ad Mechanicam utique pertinent, & ad
plurima ex iis, quae in elementis Mechanics passim traduntur, ex unico principio, & adhibito
constant! ubique agendi modo, demonstranda viam sternunt pronissimam. Sed prius
praemittam nonnulla quae pertinent ad ipsam virium curvam, a qua utique motuum,
phaenomena pendent omnia.

Quid in ea con- 167. In ea curva consideranda sunt potissimum tria, arcus curvae, area comprehensa
siderandum. inter axemj & arcum, quam general ordinata continue fluxu, ac puncta ilia, in quibus

curva secat axem.

Diversa arcnum 1 68. Quod ad arcus pcrtinet, alii dici possunt repulsivi, & alii attractivi, prout nimirum

asymptotic! "tiam Jacent ac* partes cruris asymptotici ED, vel ad contrarias, ac terminant ordinatas exhibentes

numero infiniti. vires repulsivas, vel attractivas. Primus arcus ED debet omnino esse asymptoticus ex

parte repulsiva, & in infinitum productus : ultimus TV, si gravitas cum lege virium

reciproca duplicata distantiarum protenditur in infinitum, debet itidem esse asymptoticus

ex parte attractiva, & itidem natura sua in infinitum productus. Reliquos figura I exprimit

omnes finitos. Verum curva Geometrica etiam ejus naturae, quam exposuimus, posset habere

alia itidem asymptotica crura, quot libuerit, ut si ordinata mn in H abeat in infinitum.

Sunt nimirum curvae continuae, & uniformis naturae, quae asymptotes habent plurimas,

& habere possunt etiam numero infinitas. (')

Arcus intermedii. [78] 169. Arcus intermedii, qui se contorquent circa axem, possunt etiam alicubi,
ubi ad ipsum devenerint, retro redire, tangendo ipsum, atque id ex utralibet parte, &
possent itidem ante ipsum contactum inflecti, & redire retro, mutando accessum in recessum,
ut in fig. i. videre est in arcu P^R.

Arcus prostremus 170. Si gravitas gencralis legem vis proportionalis inverse quadrate distantiae, quam
36 non accurate servat, sed quamproxime, uti diximus in priore parte, retinet ad sensum non
mutatam solum per totum planetarium, & cometarium systema, fieri utique poterit, ut
curva virium non habeat illud postremum crus asymptoticum TV, habens pro asymptoto
ipsam rectam AC, sed iterum secet axem, & se contorqueat circa ipsum.(*) Turn vero inter

(i) S»* ex. gr. in fig. 12. cyclois continua CDEFGH (3e., quam generet punctum peripheries circuli continue revoluti
supra rectam AB, qute natura sua protenditur utrinque in infinitum, adeoque in infinitis punctis C, E, G, I, &c. occurrit
basi AB. Si ubicunque ducatur qutevis ordinata PQ, productaturque in R ita, ut sit PR tertia post PQ, y datam quampiam
rectam ; punctum R frit ad curvam continuum constantem totidem ramis MNO, VXY, yr., quot erunt arcus Cycloidales
CDE, EFG, i3c,, quorum ramorum singuli habebunt bina crura asymptotica, cum ordinata PQ in accessu ad omnia puncta,
C, E, G, &c. decrescat ultra quoscunque Unites, adeoque ordinata PR crescat ultra limites quoscunque. Erunt hie quidem
omnes asymptoti CK, EL, GS &c. parallels inter se, & perpendiculares basi AB, quod in aliis curvis non est necessarium,
cum etiam divergentes utcunque possint esse. Erunt autem y totidem numero, quot puncta. ilia C, E, G &c., nimirum
infinite. Eodem autem pacto curvarum quarumlibet singuli occursus cum axe in curvis per eas hac eadem lege genitis
bina crura asymptotica generant, cruribus ipsis jacentibus, vel, ut hie, ad eandem axis partem, ubi curva genetrix ab eo
regreditur retro post appulsum, vel etiam ad partes oppositas, ubi curva genetrix ipsum secet, ac transiliat : cumque possit
eadem curva altiorum generum secari in punctis plurimis a recta, vel contingi ; poterunt utique haberi y rami asymptotici
in curva eadem continua, quo libuerit data numero.

(k)Nam ex ipsa Geometrica continuitate, quam persecutus sum in dissertatione De Lege Continuitatis, y in dissertatione
De Transformatione Locorum Geometricorum adjecta Sectionibus Conicis, exhibui necessitatem generalem secundi
illius cruris asymptotici redeuntis ex infinite. Quotiescunque enim curva aliqua saltern algebraica habet asymptoticum
crus aliquod, debet necessario habere y alterum ipsi respondens, y habens pro asymptoto eandem rectam : sed id habere

132

A THEORY OF NATURAL PHILOSOPHY

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PHILOSOPHIC NATURALIS THEORIA

PART II
^Application of the Theory to Mechanics

1 66. I will consider in this second part more especially certain general laws of Consideration of
equilibrium, & motions both of points & masses ; these certainly belong to the science of proceeding w^t'h
Mechanics, & they smooth the path that is most favourable for proving very many of those tne application to
theorems, that are everywhere expounded in the elements of Mechanics, from a single

principle, & in every case by the constant employment of a single method of dealing with
them. But, before I do that, I will call attention to a few points that pertain to the curve
of forces itself, upon which indeed all the phenomena of motions depend.

167. With regard to the curve, there are three points that are especially to be considered ; The points we have
namely, the arcs of the curve, the area included between the axis & the curve swept out regard'tolt.1

by the ordinate by its continuous motion, & those points in which the curve cuts the axis.

1 68. As regards the arcs, some may be called repulsive, & others attractive, according The different kinds
indeed as they lie on the same side of the axis as the asymptotic branch ED or on the opposite totkfarc's may even
side, & terminate ordinates that represent repulsive or attractive forces. The first arc be infinite in num-
ED must certainly be asymptotic on the repulsive side of the axis, & continued indefinitely. r'

The last arc TV, if gravity extends to indefinite distances according to a law of forces in
the inverse ratio of the squares of the distances, must also be asymptotic on the attractive
side of the axis, & by its nature also continued indefinitely. All the remaining arcs are
represented in Fig. I as finite. But a geometrical curve, of the kind that we have expounded,
may also have other asymptotic branches, as many in number as one can wish ; for instance,
suppose the ordinate mn at H to go away to infinity. There are indeed curves, that are
continuous & uniform, which have very many asymptotes, & such curves may even
have an infinite number of asymptotes. («')

169. The intermediate arcs, which wind about the axis, can also, at any point where intermediate arcs,
they reach it, return backwards & touch it ; and they can do this on either side of it ; they

may also be reflected and recede before actual contact, the approach being altered into a
recession, as is to be seen in Fig. i with regard to the arc P^/yR.

170. If universal gravity obeys the law of a force inversely proportional to the square of The ultimate arc

the distance (which, as I remarked in the first part, it only obeys as nearly as possible, but [ tPyeSposs

not exactly), sensibly unchanged only throughout the planetary & cometary system, it will asymptotic,
certainly be the case that the curve of forces will not have the last arm PV asymptotic with
the straight line AC as the asymptote, but will again cut the axis & wind about it. (*) Then

(i) Let, for example, in Fig. 12, CDEFGH &c. be a. continuous cycloid, generated by a point on the circumference
of a circle rolling continuously along the straight line AB ; this by its nature extends on either side to infinity, W thus
meets the base AB in an infinite number of points such as C, E, G, I, &c. // at every point there is drawn an ordinate
such as PQ, and this is produced to R, so that PR is a third proportional to PQ W some given straight line ; then the point
R will trace out a continuous curve consisting of as many branches, MNO, VXY, &c., as there are cycloidal arcs, CDE,
EFG, &c. ; each of these branches will have a pair of asymptotic arms, since the ordinate PQ on approaching any
one of the points C,E,G, &c., will decrease beyond all limits, (3 thus the ordinate PR will increase beyond all limits.
In this curve then there will be CK, EL, GS, &c., all asymptotes parallel to one another & perpendicular to the base
AB ; this is not necessarily the case in other curves, since they may be also inclined to one another in any manner.
Further they will be as many in number as there are points such as C, E, G, &c., that is to say, infinite. Again, in
a similar way, the several intersections of any curves you please with the axis give rise to a pair of asymptotic arms
in curves derived from them according to the same law ; and these arms lie, either on the same side of the axis, as
in this case, where the original curve leaves the axis once more after approaching it, or indeed on opposite sides of the
axis, where the original curve cuts W crosses it. Also, since it is possible for the same curve of higher orders to be
cut in a large number of points, or to be touched, there will possibly be also asymptotic arms in this same continuous
curve equal to any given number you please.

(k) For, from the principle of geometrical continuity itself, which I discussed in my dissertation De Lege Continuitatis
and in the dissertation De Transformatione Locorum Geometricorum appended to my Sectionum Conicarum
Elementa, / showed the necessity for the second asymptotic arm returning from infinity. For as often as an algebraical
curve has at least one asymptotic arm, it must also have another that corresponds to it y has the same straight line

135

136 PHILOSOPHIC NATURALIS THEORIA

alios casus innumeros, qui haberi possent, unum censeo speciminis gratia hie non omitten-
dum ; incredibile enim est, quam ferax casuum, quorum singuli sunt notatu dignissimi,
unica etiam hujusmodi curva esse possit.

shnufum curTserte I7I> Si in %• H *n axe C'C sint segmenta AA', A'A" numero quocunque, quorum

Mundoru'm mag- posteriora sint in immensum majora respectu praecedentium, & per singula transeant,
donaikfm propor" asympto-[79]-ti AB, A'B', A"B" perpendiculares axi ; possent inter binas quasque asymptotes
esse curvae ejus formae, quam in fig. I habuimus, & quae exhibetur hie in DEFI &c., D'E'F'F,
&c., in quibus primum crus ED esset asymptoticum repulsivum, postremum SV attractivum,
in singulis vero intervallum EN, quo arcus curvae contorquetur, sit perquam exiguum
respectu intervalli circa S, ubi arcus diutissime perstet proximus hyperbolae habenti
ordinatas in ratione reciproca duplicata distantiarum, turn vero vel immediate abiret
in arcum asymptoticum attractivum, vel iterum contorqueretur utcunque usque ad
ejusmodi asymptoticum attractivum arcum, habente utroque asymptotico arcu aream
infinitam ; in eo casu collocate quocunque punctorum numero inter binas quascunque
asymptotes, vel inter binaria quotlibet, & rite ordinato, posset exurgere quivis, ut ita
dicam, Mundorum numerus, quorum singuli essent inter se simillimi, vel dissimillimi,
prout arcus EF&cN, E'F'&cN' essent inter se similes, vel dissimiles, atque id ita, ut quivis
ex iis nullum haberet commercium cum quovis alio ; cum nimirum nullum punctum
posset egredi ex spatio incluso iis binis arcubus, hinc repulsive, & inde attractive ; & ut
omnes Mundi minorum dimensionum simul sumpti vices agerent unius puncti respectu
proxime majoris, qui constaret ex ejusmodi massulis respectu sui tanquam punctualibus,
dimensione nimirum omni singulorum, respectu ipsius, & respectu distantiarum, ad quas
in illo devenire possint, fere nulla ; unde & illud consequi posset, ut quivis ex ejusmodi
tanquam Mundis nihil ad sensum perturbaretur a motibus, & viribus Mundi illius majoris,
sed dato quovis utcunque magno tempore totus Mundus inferior vires sentiret a quovis
puncto materiae extra ipsum posito accedentes, quantum libuerit, ad aequales, & parallelas
quae idcirco nihil turbarent respectivum ipsius statum internum.

Omissis subiimiori- 172. Sed ea jam pertinent ad applicationem ad Physicam, quae quidem hie innui

areas pr0greSSUS ad tantumm°do, ut pateret, quam multa notatu dignissima considerari ibi possent, & quanta
sit hujusce campi fcecunditas, in quo combinationes possibiles, & possibiles formae sunt
sane infinities infinitae, quarum, quae ab humana mente perspici utcunque possunt, ita
sunt paucae respectu totius, ut haberi possint pro mero nihilo, quas tamen omnes unico
intuitu prsesentes vidit, qui Mundum condidit, DEUS. Nos in iis, quae consequentur,
simpliciora tantummodo qusedam plerumque consectabimur, quae nos ducant ad phaeno-
mena iis conformia, quae in Natura nobis pervia intuemur, & interea progrediemur ad
areas arcubus respondentes.

Cuicunque axis 173. Aream curvae propositae cuicunque, utcunque exiguo, axis segmento respondentem

aream e "respondere Posse esse utcunque magnam, & aream respondentem cuicunque, utcunque magno, [80]

utcunque magnam posse esse utcunque parvam, facile patet. Sit in fig. 15, MQ segmentum axis utcunque

secundjT"1 de^non- parvum, vel magnum ; ac detur area utcunque magna, vel parva. Ea applicata ad MQ

stratio. exhibebit quandam altitudinem MN ita, ut, ducta NR parallela MQ, sit MNRQ aequalis

areae datae, adeoque assumpta QS dupla QR, area trianguli MSQ erit itidem aequalis areae

datae. Jam vero pro secundo casu satis patet, posse curvam transire infra rectam NR,

uti transit XZ, cujus area idcirco esset minor, quam area MNRQ ; nam esset ejus pars.

potest vel ex eadem parte, vel ex opposita ; W crus ipsum jacere potest vel ad easdem plagas partis utriuslibet cum priore
crure, vel ad oppositas, adeoque cruris redeuntis ex infinite poshiones quatuor esse possunt. Si in fig. 13 crus ED abeat
in infinitum, existente asymptoto ACA', potest regredi ex parte A vel ut HI, quod crus facet ad eandem plagam, velut
KL, quod, facet ad oppositam ; y ex parte A', vel ut MN, ex eadem plaga, vel ut OP, ex opposita. In posteriore ex
iis duabus dissertationibus profero exempla omnium ejusmodi regressuum ; ac secundi, ($ quarti casus exempla exhibet
etiam superior genesis, si curva generans contingat axem, vel secet, ulterius progressa respectu ipsius. Inde autem fit, ut
crura asymptotica rectilineam babentia asymptotum esse non possint, nisi numero part, ut & radices imaginarite in
eequationibus algebraicis.

Verum hie in curva virium, in qua arcus semper debet progredi, ut singulis distantiis, sive abscissis, singula vires,
sive ordinatts respondeant, casus primus, & tertius haberi non possunt. Nam ordinata RQ cruris DE occurreret alicubi
in S, S' cruribus etiam HI, MN ,• adeoque relinquentur soli quartus, & secundus, quorum usus erit infra.

A THEORY OF NATURAL PHILOSOPHY

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PHILOSOPHIC NATURALIS THEORIA

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A THEORY OF NATURAL PHILOSOPHY 139

there is one, out of an innumerable number of other cases that may possibly happen, which
I think for the sake of an example should not be omitted here ; for it is incredible how
prolific in cases, each of which is well worth mentioning, a single curve of this kind can be.

171. If, in Fig. 14, there are any number of segments AA', A'A", of which each that A series of similar
follows is immensely great with regard to the one that precedes it ; & if through each c"rve.s- wlth a s^ies
point there passes an asymptote, such as AB, A'B', A"B", perpendicular to the axis ; then tionai in magnitude,
between any two of these asymptotes there may be curves of the form given in Fig. i.
These are represented in Fig. 14 by DEFI &c., D'E'F'I' &c. ; & in these the first arm E
would be asymptotic & repulsive, & the last SV attractive. In each the interval EN,
where the arc of the curve is winding, is exceedingly small compared with the interval
near S, where the arc for a very long time continues closely approximating to the form
of the hyperbola having its ordinates in the inverse ratio of the squares of the distances ;
& then, either goes off straightway into an asymptotic & attractive arm, or once more
winds about the axis until it becomes an asymptotic attractive arc of this kind, the area
corresponding to either asymptotic arc being infinite. In such a case, if a number of points
are assembled between any pair of asymptotes, or between any number of pairs you please,
£ correctly arranged, there can, so to speak, arise from them any number of universes,
each of them being similar to the other, or dissimilar, according as the arcs EF . . . . N,
E'F' .... N' are similar to one another, or dissimilar ; & this too in such a way that
no one of them has any communication with any other, since indeed no point can possibly
move out of the space included between these two arcs, one repulsive & the other
attractive ; & such that all the universes of smaller dimensions taken together would
act merely as a single point compared with the next greater universe, which would
consist of little point-masses, so to speak, of the same kind compared with itself, that is
to say, every dimension of each of them, compared with that universe & with respect to
the distances to which each can attain within it, would be practically nothing. From
this it would also follow that any one of these universes would not be appreciably influenced
in any way by the motions & forces of that greater universe ; but in any given time,
however great, the whole inferior universe would experience forces, from any point of matter
placed without itself, that approach as near as possible to equal & parallel forces ; these
therefore would have no influence on its relative internal state.

172. Now these matters really belong to the application of the Theory to physics ; & Leaving out more
indeed I only mentioned them here to show how many things there may be well worth abstruse matters,

• j. . -i • • a i .,..,. ' - 9 . n i i r • •• • we pass on to areas.

considering in that section, & how great is the fertility of this field of investigation, m
which possible combinations & possible forms are truly infinitely infinite ; of these, those
that can be in any way comprehended by the human intelligence are so few compared
with the whole, that they can be considered as a mere nothing. Yet all of them were seen
in clear view at one gaze by GOD, the Founder of the World. We, in what follows, will
for the most part investigate only certain of the more simple matters which will lead us
to phenomena in conformity with those things that we contemplate in Nature as far as
our intelligence will carry us ; meanwhile we will proceed to the areas corresponding to
the arcs.

173. It is easily shown that the area corresponding to any segment of the axis, however To any segment of
small, can be anything, no matter how great ; & the area corresponding to any segment, corrt'spo^a'rfy
however great, can be anything, no matter how small. In Fig. 1 5 , let MQ be a segment of the area, however
axis, no matter how small, or great; & let an area be given, no matter how great, or SSi ; proof^the
small. If this area is applied to MQ a certain altitude MN will be given, such that, if NR second part of this
is drawn parallel to MQ, then MNRQ will be equal to the given area ; & thus, if QS is a

taken equal to twice QR, the area of the triangle MSQ will also be equal to the given area.
Now, for the second case it is sufficiently evident that a curve can be drawn below the
straight line NR, in the way XZ is shown, the area under which is less than the area MNRQ ;

as its asymptote ; & this can take place with either the same part of the line or with the other part ; also the arm
itself can lie either on the same side of either of the two parts, or on the opposite side. Thus there may be four positions
of the arm that returns from infinity. If, in Fig. 13, the arm ED goes off to infinity, the asymptote being ACA,
it may return from the direction of A, either like HI, wheie the arm lies on the same side of the asymptote or as KL
which lies 'on the opposite side of it ; or from the direction of A', either as MN, on the same side, or as, DP, on the
opposite side. In the second of these two dissertations, I have given examples of all regressions of this sort ; y the
method of generation given above will yield examples of the second W fourth cases, if the generating curve touches
the axis, or cuts it & passes over beyond it. Further, it thus comes about that asymptotic arms having a rectilinear
asymptote cannot exist except in pairs, just like imaginary roots in algebraical equations.

But here in the curve of forces, in which the arc must always proceed in such a manner that to each distance or
abscissa there corresponds a single force or ordinate, the first £tf third cases cannot occur. For the ordinate RQ of the
arm DE would meet somewhere, in S, S', the branches HI, MN as well. Hence only the fourth & second cases are
left ; W these we will make use of later.

140

PHILOSOPHIC NATURALIS THEORIA

Demonst ratio
primse.

Aream asympto-
ticam posse esse
infinitam, vel fini-
tam magnitudinis
cujuscunque.

Areas exprimere
incrementa, vel
decrementa quad-
ati velocitatis.

Quin immo licet ordinata QV sit utcunque magna ; facile patet, posse arcum MaV ita
accedere ad rectas MQ, QV ; ut area inclusa iis rectis, & ipsa curva, minuatur infra
quoscunque determinatos limites. Potest enim jacere totus arcus intra duo triangula
QaM, QaV, quorum altitudines cum minui possint,
quantum libuerit, stantibus basibus MQ, QV, potest
utique area ultra quoscunque limites imminui. Pos-
set autem ea area esse minor quacunque data ;
etiamsi QV esset asymptotus, qua de re paullo
inferius.

174. Pro primo autem casu vel curva secet axem
extra MQ, ut in T, vel in altero extremo, ut in M ;
fieri poterit, ut ejus arcus TV, vel MV transeat per
aliquod punctum V jacens ultra S, vel etiam per
ipsum S ita, ut curvatura ilium ferat, quemad-
modum figura exhibet, extra triangulum MSQ, quo
casu patet, aream curvae respondentem intervallo MQ
fore majorem, quam sit area trianguli MSQ, adeoque
quam sit area data ; erit enim ejus trianguli area
pars areae pertinentis ad curvam. Quod si curva
etiam secaret alicubi axem, ut in H inter M, & Q,
turn vero fieri posset, ut area respondens alteri e
segmentis MH, QH esset major, quam area data .

simul, & area alia assumpta, qua area assumpta esset minor area respondens segmento,
alteri adeoque excessus prioris supra posteriorem remaneret major, quam area data.

175. Area asymptotica clausa inter asymptotum, & ordinatam quamvis, ut in fig. I
BA#g, potest esse vel infinita, vel finita magnitudinis cujusvis ingentis, vel exiguae. Id
quidem etiam geometrice demonstrari potest, sed multo facilius demonstratur calculo
integrali admodum elementari ; & in Geometriae sublimioris elementis habentur theoremata,
ex quibus id admodum facile deducitur 0. Generaliter nimi-[8l]-rum area ejusmodi
est infinita ; si ordinata crescit in ratione reciproca abscissarum simplici, aut majore : &
est finita ; si crescit in ratione multiplicata minus, quam per unitatem.

176. Hoc, quod de areis dictum est, necessarium fuit ad applicationem ad Mechanicam,
ut nimirum habeatur scala quaedam velocitatum, quae in accessu puncti cujusvis ad aliud
punctum, vel recessu generantur, vel eliduntur ; prout ejus motus conspiret directione vis,
vel sit ipsi contrarius. Nam, quod innuimus & supra in adnot. (/) ad num. 118., ubi vires
exprimuntur per ordinatas, & spatia per abscissas, area, quam texit ordinata, exprimit
incrementum, vel decrementum quadrati velocitatis, quod itidem ope Geometrise demon-
stratur facile, & demonstravi tam in dissertatione De Firibus Vivis, quam in Stayanis
Supplements ; sed multo facilius res conficitur ope calculi integralis. («)

M H

FIG. 15.

(1) Sit Aa in Fig. I =x, ag=y ; ac sit #"y = I ; erit y = *-">/", y dx elementum areee=x~m/*dx, cujus integrate

— *fn» + A, addita constanti A, sive ob x~*>">=y, habebitur —?—xy + A. Quoniam incipit area in A, in
n~m " n-m

origine abscissarum ; si n—m fuerit numerus positivus, adeoque n major, quam m ; area erit finita, ac valor A =o;
area vero erit ad rectangulum AaXag, ut in ad n — m, quod rectangulum, cum ag possit esse magna, & parva, ut libuerit,
potest esse magnitudinis cujusvis. Is valor fit infinitus, si facto m =n, divisor evaaat—Q; adeoque multo magis fit
infinitus valor area, si m sit major, quam n. Unde constat, aream fore infiniiam, quotiescunque ordinatte crescent in
ratione reciproca simplici, y majore ; secus fore finitam.

(m) Sit u vis, c celeritas, t tempus, s spatium : erit u at = dc, cum celeritatis incrementum sit proportionale vi, W
tempusculo ; ac erit c dt = ds, cum spatiolum confectwm respondeat velocitati, & tempusculo. Hinc eruitur dt =— ,

W pariter dt =—, adeoque—- =— W c dc = u ds. Porro 2c dc est incrementum quadrati vekcitatis cc, i3 u ds

c u c

in bypotbesi, quod ordinata sit w, & spatium s sit abscissa, est areola respondens spatiolo ds confecto. Igitur incrementum
quadrati velocitatis conspirante vi, adeoque decrementum vi contraria, respondet arete respondent spatiolo percurso quovis
infinitesimo tempusculo ; & proinde tempore etiam quovis finito incrementum, vel decrementum quadrati velocitatis
respondet arece pertinenti ad partem axis referentem spatium percursum.

Hinc autem illud sponte consequitur : si per aliquod spatium vires in singulis punctis eeedem permaneant, mobile autem
adveniat cum velocitate quavis ad ejus initium ; diferentiam quadrati velocitatis finalis a quadrate velocitatis initialis
fore semper eandem, quts idcirco erit tola velocitas finalis in casu, in quo mobile initio illius spatii haberet velocitatem
nullam. Quare, quod nobis erit inferius usui, quadratum velocitatis finalis, conspirante vi cum directione motus, tzquabitur
binis quadratis binarum velocitatum, ejus, quam babuit initio, W ejus,.quam acquisivisset in fine, si initio ingressum fuisset
sine ulla velocitate.

A THEORY OF NATURAL PHILOSOPHY

141

for it is part of it. Again, although the ordinate QV may be of any size, however great,
it is easily shown that an arc MoV can approach so closely to the straight lines MQ,
QV that the area included between these lines & the curve shall be diminished beyond
any limits whatever. For it is possible for the curve to lie within the two triangles QaM,
QaV ; & since the altitudes of these can be diminished as much as you please, whilst the
bases MQ, QV remain the same, therefore the area can indeed be diminished beyond all
limits whatever. Moreover it is possible for this area to be less than any given area, even
although QV should be an asymptote ; we will consider this a little further on.

174. Again, for the first case, either the curve will cut the axis beyond MQ, as at T,
or at either end, as at M. Then it is possible for it to happen that an arc of it, TV or MV,
will pass through some point V lying beyond S, or even through S itself, in such a way
that its curvature will carry it, as shown in the diagram, outside the triangle MSQ ; in
this case it is clear that the area of the curve corresponding to the interval MQ will be
greater than the area of the triangle MSQ, & therefore greater than the given area,
for the area of this triangle is part of the area belonging to the curve. But if the curve
should even cut the axis anywhere, as at H, between M & Q, then it would be possible
for it to come about that the area corresponding to one of the two segments MH, QH would
be greater than the given area together with some other assumed area ; & that the area
corresponding to the other segment should be less than this assumed area ; and thus the
excess of the former over the latter would remain greater than the given area.

175. An asymptotic area, bounded by an asymptote & any ordinate, like BAag in
Fig. i, can be either infinite, or finite of any magnitude either very great or very small.
This can indeed be also proved geometrically, but it can be demonstrated much more
easily by an application of the integral calculus that is quite elementary ; & in the elements
of higher geometry theorems are obtained from which it is derived quite easily. 0 In
general, it is true, an area of this kind is infinite ; namely when the ordinate increases in
the simple inverse ratio of the abscissse, or in a greater ratio ; and it is finite, if it increases
in this ratio multiplied by something less than unity.

176. What has been said with regard to areas was a necessary preliminary to the
application of the Theory to Mechanics ; that is to say, in order that we might obtain a
diagrammatic representation of the velocities, which, on the approach of any point to
another point, or on recession from it, are produced or destroyed, according as its motion
is in the same direction as the direction of the force, or in the opposite direction. For,
as we also remarked above, in note (/) to Art. 118, when the forces are represented by
ordinates & the distances by abscissae, the area that the ordinate sweeps out represents
the increment or decrement of the square of the velocity. This can also be easily proved
by the help of geometry ; & I gave the proof both in the dissertation De Firibus Fivis
& in the Supplements to Stay's Philosophy ; but the matter is much more easily made
out by the aid of the integral calculus. («)

Proof
part.

of the first

An a s y m p totic
area may be either
infinite or equal to
any finite area
whatever.

The areas represent
the increments or
decrements of the
square of the velo-
city.

(1) In Fig. iletAa = x,ag = y; y let xmy" = I. Then will y — x~

the element of area y dx = x~m/*

dx : the integral of this is - x <»-"»/"+ A, where a constant A is added ; or, since x~m/*=y, we shall have-^— Xv + A

n-m n-m '

Now, since the area is initially A, at the origin of the abscissa, if n-m happened to be a positive number, y
thus n greater than m, then the area will be finite, y the value of A will be = o. Also the area will be to
the rectangle Aa.ag as n is to n-m ; y this rectangle, since ag can be either great or small, as you please, may be
of any magnitude whatever. The value is infinite, if by making m equal to n the divisor becomes equal to zero ; &
thus the value of the area becomes all the more infinite, if m is greater than n. Hence it follows that the area will
be infinite, whenever the ordinates increase in a simple inverse ratio, or in a greater ratio ; otherwise it will be finite.

(m) Let u be the force, c the velocity, t the time, y s the distance. Then will u dt — dc, since the increment
of the velocity is proportional to the force, y to the small interval of time. Also c dt = ds, since the distance traversed
corresponds with the velocity W the small interval of time. Hence it follows that dt = dc/u, y similarly dt — ds/c,
y therefore dc/u = ds/c, & c dc — u ds. Further, ^c dc is the increment of the square of the velocity c', y u ds,
on the hypothesis that the ordinate represents u, y the abscissa the distance s, is the small area corresponding to the
small distance traversed. Hence the increment of the square of the velocity, when in the direction of the force, y
the decrement when opposite in direction to the force, is represented by the area corresponding to ds, the small distance
traversed in any infinitely short time. Hence also, in any finite interval of time, the increment or decrement of the
square of the velocity will be represented by the area corresponding to that part of the axis which represents the distance
traversed.

Hence also it follows immediately that, if through any distance the force on each of the points remains as before,
but the moving body arrives at the beginning of it with any velocity, then the difference between the square of the final
velocity y the square of the initial velocity will always be the same ; y this therefore will be the total final velocity,
in the case where the moving body had no velocity at the beginning of the distance. Hence, the square of the final
velocity, when the motion is in the same direction as the force, will be equal to the sum of the squares of the velocity which
it had at the beginning y of the velocity it would have acquired at the end, if it had at the beginning started without
any velocity ; a theorem that we shall make use of later.

I42

PHILOSOPHIC NATURALIS THEORIA

Atque id ips u m,
licet segmenta axis
sint dimidia spatio-
rum percursorum a
singulis punctis.

Si arese sint partim
attractivae, partim
repulsivae, assumen-
dam esse differen-
tiam earundem.

177. Duo tamen hie tantummodo notanda sunt ; primo quidem illud : si duo
puncta ad se invicem accedant, vel a se invicem recedant in ea recta, quae ipsa conjungit,
segmenta illius [82] axis, qui exprimit distantias, non expriment spatium confectum ; nam
moveri debebit punctum utrumque : adhuc tamen ilia segmenta erunt proportionalia ipsi
spatio confecto, eorum nimirum dimidio ; quod quidem satis est ad hoc, ut illae areae adhuc
sint proportionales incrementis, vel decrementis quadrati velocitatum, adeoque ipsa
exprimant.

178. Secundo loco notandum illud, ubi areae respondentes dato cuipiam spatio sint
partim attractive, partim repulsivae, earum differentiam, quae oritur subtrahendo summam
omnium repulsivarum a summa attractivarum, vel vice versa, exhibituram incrementum
illud, vel decrementum quadrati velocitatis ; prout directio motus respectivi conspiret
cum vi, vel oppositam habeat directionem. Quamobrem si interea, dum per aliquod majus
intervallum a se invicem recesserunt puncta, habuerint vires directionis utriusque ; ut
innotescat, an celeritas creverit, an decreverit & quantum ; erit investigandum, an areas
omnes attractivae simul, omnes repulsivas simul superent, an deficiant, & quantum ; inde
enim, & a velocitate, quae habebatur initio, erui poterit quod quaeritur.

^e arcubus, & areis ; nunc aliquanto diligentius considerabimus
tangentis: sectio- ilia axis puncta, ad quae curva appellit. Ea puncta vel sunt ejusmodi, ut in iis curva axem
ducT enera UmltUm secet> cujusmodi in fig. I sunt E,G,I, &c., vel ejusmodi, ut in iis ipsa curva axem contingat
tantummodo. Primi generis puncta sunt ea, in quibus fit transitus a repulsionibus ad
attractiones, vel vice versa, & hsec ego appello limites, quod nimirum sint inter eas opposi-
tarum directionum vires. Sunt autem hi limites duplicis generis : in aliis, aucta distantia,
transitur a repulsione ad attractionem : in aliis contra ab attractione ad repulsionem.
Prioris generis sunt E,I,N,R ; posterioris G,L,P : & quoniam, posteaquam ex parte
repulsiva in una sectione curva transiit ad partem attractivam ; in proxime sequent! sectione
debet necessario ex parte attractiva transire ad repulsivam, ac vice versa ; patet, limites
fore alternatim prioris illius, & hujus posterioris generis.

t P°rro linrites prioris generis, a limitibus posterioris ingens habent inter se dis-
differant': limites crimen. Habent illi quidem hoc commune, ut duo puncta collocata in distantia unius
cohaesionls' & n°n h'111^8 cujuscunque nullam habeant mutuam vim, adeoque si respective quiescebant, pergant
itidem respective quiescere. At si ab ilia respectiva quiete dimoveantur ; turn vero in
limite primi generis ulteriori dimotioni resistent, & conabuntur priorem distantiam recu-
perare, ac sibi relicta ad illam ibunt ; in limite vero secundi generis, utcunque parum
dimota, sponte magis fugient, ac a priore distantia statim recedent adhuc magis. Nam
si distantia minuatur ; habebunt in limite prioris generis vim repulsivam, quae obstabit
uteriori accessui, & urgebit puncta ad mutuum recessum, quern sibi relicta acquirent, [83]
adeoque tendent ad illam priorem distantiam : at in limite secundi generis habebunt
attractionem, qua adhuc magis ad se accedent, adeoque ab ilia priore distantia, quae erat
major, adhuc magis sponte fugient. Pariter si distantia augeatur, in primo limitum genere
a vi attractiva, quse habetur statim in distantia majore ; habebitur resistentia ad ulteriorem
recessum, & conatus ad minuendam distantiam, ad quam recuperandam sibi relicta tendent
per accessum ; at in limitibus secundi generis orietur repulsio, qua sponte se magis adhuc
fugient, adeoque a minore ilia priore distantia sponte magis recedent. Hinc illos prioris
generis limites, qui mutuse positionis tenaces sunt, ego quidem appellavi limites coh&sionis,
& secundi generis limites appellavi limites non cobasionis.

Duo genera
tactuum.

181. Ilia puncta, in quibus curva axem tangit, sunt quidem terminus quidam virium,
quae ex utraque parte, dum ad ea acceditur, decrescunt ultra quoscunque limites, ac demum
ibidem evanescunt ; sed in iis non transitur ab una virium directione ad aliam. Si con-
tactus fiat ab arcu repulsive ; repulsiones evanescunt, sed post contactum remanent itidem
repulsiones ; ac si ab arcu attractive, attractionibus evanescentibus attractiones iterum
immediate succedunt. Duo puncta collocata in ejusmodi distantia respective quiescunt ;

A THEORY OF NATURAL PHILOSOPHY

'43

177. However, there are here two things that want noting only. The first of them The same result
is this, that if two points approach one another or recede from one another in the straight holds good even

,...., , r i •,. , ,. i ° when the segments

line joining them, the segments of the axis, which expresses distances, do not represent of the axis are the
the distances traversed ; for both points will have to move. Nevertheless the segments 'ialves of the dis-

•11 -11 i • i i T i i i if f • i . . i •, tances traversed by

will still be proportional to the distance traversed, namely, the half of it ; & this indeed is single points,
sufficient for the areas to be still proportional to the increments or decrements of the
squares of the velocities, & thus to represent them.

178. In the second place it is to be noted that, where the areas corresponding to any if the areas are
given interval are partly attractive & partly repulsive, their difference, obtained by p^ti*tt2SS2 &
subtracting the sum of all those that are repulsive from the sum of those that are attractive, their difference
or vice versa, will represent the increment, or the decrement, of the square of the velocity, must be taken-
according as the direction of relative motion is in the same direction as the force, or in

the opposite direction. Hence, if, during the time that the points have receded from
one another by some considerable interval, they had forces in each direction ; then
in order to ascertain whether the velocity had been increased or decreased, & by how
much, it will have to be considered whether all the attractive areas taken together are
greater or less than all the repulsive areas taken together, & by how much. For from this,
& from the velocity which initially existed, it will be possible to deduce what is required.

179. So much for the arcs & the areas; now we must consider in a rather more careful Approach of the
manner those points of the axis to which the curve approaches. These points are either ^en it cSa^or
such that the curve cuts the axis in them, for instance, the points E, G, I, &c. in Fig. I : touches it; two
or such that the curve only touches the axis at the points. Points of the first kind are u^ns^/'ihnit-
those in which there is a transition from repulsions to attractions, or vice-versa ; & these points.

I call limit-points or boundaries, since indeed they are boundaries between the forces acting
in opposite directions. Moreover these limit-points are twofold in kind ; in some, when
the distance is increased, there is a transition from repulsion to attraction ; in others, on
the contrary, there is a transition from attraction to repulsion. The points E, I, N, R
are of the first kind, and G, L, P are of the second kind. Now, since at one intersection,
the curve passes from the repulsive part to the attractive part, at the next following
intersection it is bound to pass from the attractive to the repulsive part, & vice versa.
It is clear then that the limit-points will be alternately of the first & second kinds.

1 80. Further, there is a distinction between limit-points of the first & those of the in what they agree
second kind. The former kind have this property in common ; namely that, if two points *iffj£ . w^ u*?ty
are situated at a distance from one another equal to the distance of any one of these limit- points of cohesion
points from the origin, they will have no mutual force ; & thus, if they are relatively & of non-cohesic«i.
at rest with regard to one another, they will continue to be relatively at rest. Also, if

they are moved apart from this position of relative rest, then, for a limit-point of the first
kind, they will resist further separation & will strive to recover the original distance, &
will attain to it if left to themselves ; but, in a limit-point of the second kind, however
small the separation, they will of themselves seek to get away from one another & will
immediately depart from the original distance still more. For, if the distance is diminished,
they will have, in a limit-point of the first kind, a repulsive force, which will impede further
approach & impel the points to mutual recession, & this they will acquire if left to
themselves ; thus they will endeavour to maintain the original distance apart. But in a
limit-point of the second kind they will have an attraction, on account of which they will
approach one another still more ; & thus they will seek to depart still further from the
original distance, which was a greater one. Similarly, if the distance is increased, in
limit-points of the first kind, due to the attractive force which is immediately obtained
at this greater distance, there will be a resistance to further recession, & an endeavour
to diminish the distance ; & they will seek to recover the original distance if left to
themselves by approaching one another. But, in limit-points of the second class, a repulsion
is produced, owing to which they try to get away from one another still further ; & thus
of themselves they will depart still more from the original distance, which was less. On
this account indeed I have called those limit-points of the first kind, which are tenacious
of mutual position, limit-points of cohesion, & I have termed limit-points of the second
kind limit-points of non-cohesion.

181. Those points in which the curve touches the axis are indeed end-terms of series Two kinds of con-
of forces, which decrease on both sides, as approach to these points takes place, beyond tactt

all limits, & at length vanish there ; but with such points there is no transition from
one direction of the forces to the other. If contact takes place with a repulsive arc, the
repulsion vanishes, but after contact remains still a repulsion. If it takes place with an
attractive arc, attraction follows on immediately after a vanishing attraction. Two points
situated such a distance remain in a state of relative rest ; but in the first case they will

144

PHILOSOPHIC NATURALIS THEORIA

pro forma curvae
prope sectionem.

sed in prime casu resistunt soli compressioni, non etiam distractioni, £ in secundo resistunt
huic soli, non illi.

l^2' Limites cohsesionis possunt esse validissimi, & languidissimi. Si curva ibi quasi
ad pcrpendiculum secat axem, & ab eo longissime recedit ; sunt validissimi : si autem
ipSum secet in angulo perquam exiguo, & parum ab ipso recedat ; erunt languidissimi.
Primum genus limitum cohsesionis exhibet in fig. I arcus tNy, secundum cNx. In illo
assumptis in axe Nz, NM utcunque exiguis, possunt vires zt, uy, & areae Nzt, Nwy esse
utcumque magnas, adeoque, mutatis utcunque parum distantiis, possunt haberi vires ab
ordinatis expressae utcunque magnae, quae vi comprimenti, vel distrahenti, quantum libuerit,
valide resistant, vel areae utcunque magnae, quae velocitates quantumlibet magnas
respectivas elidant, adeoque sensibilis mutatio positionis mutuae impediri potest contra
utcunque magnam vel vim prementem, vel celeritatem ab aliorum punctorum actionibus
impressam. In hoc secundo genere limitum cohaesionis, assumptis etiam majoribus
segmentis Nz, Nw, possunt & vires zc,ux, & areae Nzf , N«tf, esse quantum libuerit exiguae,
& idcirco exigua itidem, quantum libuerit, resistentia, quae mutationem vetet.

P°ssunt autem hi Hmites esse quocunque, utcunque magno numero ; cum
ro, utcunque proxi- demonstratum sit, posse curvam in quotcunque, & quibuscunque punctis axem secare.
mos, vel remotes possunt idcirco etiam esse utcunque inter se proximi, vel remoti, ut [84] alicubi intervallum
originis' abscissa- inter duos proximos limites sit etiam in quacunque ratione majus, quam sit distantia
ordme praecedentis ab origine abscissarum A ; alibi in intervallo vel exiguo, vel ingenti sint quam-
plurimi inter se ita proximi, ut a se invicem distent minus, quam pro quovis assumpto,
aut dato intervallo. Id evidenter fluit ex eo ipso, quod possint sectiones curvae cum axe
haberi quotcunque, & ubicunque. Sed ex eo, quod arcus curvae ubicunque possint habere
positiones quascunque, cum ad datas curvas accedere possint, quantum libuerit, sequitur,
quod limites ipsi cohaesionis possint alii aliis esse utcunque validiores, vel languidiores,
atque id quocunque ordine, vel sine ordine ullo ; ut nimirum etiam sint in minoribus
distantiis alicubi limites validissimi, turn in majoribus languidiores, deinde itidem in
majoribus multo validiores, & ita porro ; cum nimirum nullus sit nexus necessarius inter
distantiam limitis ab origine abscissarum, & ejus validitatem pendentem ab inclinatione,
& recessu arcus secantis respectu axis, quod probe notandum est, futurum nimirum usui
ad ostendendum, tenacitatem, sive cohaesionem, a densitate non pendere.

similes.

Quse positio rectae jg^.. In utroque limitum genere fieri potest, ut curva in ipso occursu cum axe pro

infinite3 rarissima! tangente habeat axem ipsum, ut habeat ordinatam, ut aliam rectam aliquam inclinatam.
quae frequentissima. Jn primo casu maxime ad axem accedit, & initio saltern languidissimus est limes ; in secundo
maxime recedit, & initio saltern est validissimus ; sed hi casus debent esse rarissimi, si
uspiam sunt : nam cum ibi debeat & axem secare curva, & progredi, adeoque secari in
puncto eodem ab ordinata producta, debebit habere flexum contrarium, sive mutare
directionem flexus, quod utique fit, ubi curva & rectam tangit simul, & secat. Rarissimos
tamen debere esse ibi hos flexus, vel potius nullos, constat ex eo, quod flexus contrarii puncta
in quovis finito arcu datae curvae cujusvis numero finite esse debent, ut in Theoria curvarum
demonstrari potest, & alia puncta sunt infinita numero, adeoque ilia cadere in intersectiones
est infinities improbabilius. Possunt tamen saepe cadere prope limites : nam in singulis
contorsionibus curvae saltern singuli flexus contrarii esse debent. Porro quamcunque
directionem habuerit tangens, si accipiatur exiguus arcus hinc, & inde a limite, vel
maxime accedet ad rectam, vel habebit curvaturam ad sensum aequalem, & ad sensum
aequali lege progredientem utrinque, adeoque vires in aequali distantia exigua a limite
erunt ad sensum hinc, & inde aequales ; sed distantiis auctis poterunt & diu aequalitatem
retinere, & cito etiam ab ea recedere.

Transitus per infi- 185. Hi quidem sunt limites per intersectionem curvae cum axe, viribus evanescentibus

astlm"toticisribUS m *PSO limite- At possunt [85] esse alii limites, ac transitus ab una directione virium ad

aliam non per evanescentiam, sed per vires auctas in infinitum, nimirum per asymptoticos

A THEORY OF NATURAL PHILOSOPHY 145

resist compression only, & not separation ; and in the second case the latter only, but not
the former.

182. Limit-points may be either very strong or very weak. If the curve cuts the axis The limit-points of
at the point almost at right angles, & goes off to a considerable distance from it, they o°h^eak ?ccord£f
are very strong. But if it cuts the axis at a very small angle & recedes from it but little, to the form of the
then they will be very weak. The arc *Ny in Fig. i represents the first kind of limit- ^Hint * iVater-
points of cohesion, and the arc cNx the second kind. At the point N, if Nz, N« are section.

taken along the axis, no matter how small, the forces zt, uy, & the areas Nzt, N«y may
be of any size whatever ; & thus, if the distances are changed ever so little, it is possible
that there will be forces represented by ordinates ever so great ; & these will strongly
resist the compressing or separating force, be it as great as you please ; also that we shall
have areas, ever so large, that will destroy the relative velocities, no matter how great they
may be. Thus, a sensible change of relative position will be hindered in opposition to
any impressed force, however great, or against a velocity generated by the actions upon
them of other points. In the second kind of limit-points of cohesion, if also segments Nz,
Nw are taken of considerable size even, then it is possible for both the forces zc, ux, &
the areas Nzc, Nux to be as small as you please ; & therefore also the resistance that
opposes the change will be as small as you please.

183. Moreover, there can be any number of these limit-points, no matter how great ; The limit-points
for it has been proved that the curve can cut the axis in any number of points, & anywhere. are m<Jefimte as

rrM F • • • i i r i i • i i f i i 6 g cl i Cl S HUTU DCr,

I herefore it is possible for them to be either close to or remote from one another, without their proximity to

any restriction whatever, so that the interval between any two consecutive limit-points one^another^&^th^

at any place shall even bear to the distance of the first of the two from A, the origin of order of their occur-

abscissae, a ratio that is greater than unity. In other words, in any interval, either very ^toe <SgVof Pab-

small or very large, there may be an exceedingly large number of them so close to one scissae.

another, that they are less distant from one another than they are from any chosen or given

interval. This evidently follows from the fact that the intersections of the curve with

the axis can happen any number of times & anywhere. Again, from the fact that arcs

of the curve can anywhere, owing to their being capable of approximating as closely as

you please to given curves, have any positions whatever, it follows that these limit-points

of cohesion can be some of them stronger than others, or weaker, in any manner ; &

that too, in any order, or without order. So that, for instance, we may have at small

distances anywhere very strong limit-points, then at greater distances weaker ones, &

then again at still greater distances much stronger ones, & so on. That is to say, since

there is no necessary connection between the distance of a limit-point from the origin of

abscissae and its strength, which depends on the inclination of the intersecting arc & the

distance it recedes from the axis. It is well that this should be made a note of ; for indeed

it will be used later to prove that tenacity or cohesion does not depend on density.

184. In each of these kinds of limit-points it may happen that the curve, where it What position of
meets the axis, may have the axis itself as its tangent, or the ordinate, or any other straight touchkig^he* curve
line inclined to the axis. In the first case it approximates very closely to the axis, & at a limit-point is
close to the point at any rate it is a very weak limit-point ; in the second case, it departs ^at magt^fr*
from the axis very sharply, & close to the point at any rate it is a very strong limit-point, quent ; small arcs
But these two cases must be of very rare occurrence, if indeed they ever occur. For, since fTm^tVo'in t* are
at the point the curve is bound to cut the axis & go on, & thus be cut in the same point equal & similar,
by the ordinate produced, it is bound to have contrary-flexure ; that is to say, a change

in the direction of its curvature, such as always takes place at a point where the curve both
touches a straight line & cuts it at the same time. Yet, that these flexures must occur
very rarely at such points, or rather never occur at all, is evident from the fact that in any
finite arc of any given curve the number of points of contrary-flexure must be finite, as can
be proved in the theory of curves ; & other points are infinite in number ; hence that the
former should happen at the points of intersection with the axis is infinitely improbable.
On the other hand they may often fall close to the limit-points ; for in each winding of
the curve about the axis there must be at least one point of contrary-flexure. Further,
whatever the direction of the tangent, if a very small arc of the curve is taken on each side
of the limit-point, this arc will either approximate very closely to the straight line, or will
have its curvature the same very nearly, & will proceed very nearly according to the same
law on each side ; & thus the forces, at equal small distances on each side of the limit-
point will be very nearly equal to one another ; but when the distances are increased,
they can either maintain this equality, for some considerable time, or indeed very soon
depart from it.

185. The limit-points so far discussed are those obtained through the intersection Passage through
of the curve with the axis, where the forces vanish at the limit-point. But there

may be other limit-points ; the transition from one direction of the forces to another

L

146

PHILOSOPHIC NATURALIS THEORIA

curvse arcus. Diximus supra num. 168. adnot. (i), quando crus asymptoticum abit in
infinitum, debere ex infinite regredi crus aliud habens pro asymptote eandem rectam, &
posse regredi cum quatuor diversis positionibus pendentibus a binis partibus ipsius rectae,
& binis plagis pro singulis rectae partibus ; sed cum nostra curva debeat semper progredi,
diximus, relinqui pro ea binas ex ejusmodi quatuor positionibus pro quovis crure abeunte
in infinitum, in quibus nimirum regressus fiat ex plaga opposita. Quoniam vero, progre-
diente curva, abire potest in infinitum tarn crus repulsivum, quam crus attractivum , jam
iterum fiunt casus quatuor possibiles, quos exprimunt figurae 16, 17, 18, & 19, in quibus
omnibus est axis ACS, asymptotus DCD', crus recedens in infinitum EKF, regrediens
ex infinite GMH.

D

A

I C

B

FIG. 1 6.

I C

D

B

D

FIG. 17.

Quatuor eorum 186. In fig. 16. cruri repulsivo EKF succedit itidem repulsivum GMH ; in fig. 17

f^nXntesTontac- repulsivo attractivum ; in 1 8 attractive attractivum; in 19 attractive repulsivum. Primus

tibus, bini Hmiti- & tertius casus respondent contactibus. Ut enim in illis evanescebat vis ; sed directionem

ionU, alaHerCOhnon non mutabat ; ita & hie abit quidem in infinitum, sed directionem non mutat. Repulsioni

cohaesionis. IK in fig. 1 6 succedit repulsio LM ; & attractioni in fig. 18 attractio. Quare ii casus non

habent limites quosdam. Secundus, & quartus habent utique limites ; nam in fig. 17

repulsion! IK succedit attractio LM ; & in fig. 19 attractioni repulsio ; atque idcirco

secundus continet limitem cobasionis, quartus limitem non cohcesionis.

Nuiium in Natura

vero eum
utcunque.

187. Ex istis casibus a nostra curva censeo removendos esse omnes praeter solum
quartum ; & in hoc ipso removenda omnia crura, in quibus ordinata crescit in ratione
ipsum minus, quam simplici reciproca distantiarum a limite. Ratio excludendi est, ne haberi
aliquando vis infinita possit, quam & per se se absurdam censeo, & idcirco praeterea, quod
infinita vis natura sua velocitatem infinitam requirit a se generandam finito tempore. Nam
in primo, & secundo casu punctum collocatum in ea distantia ab alio puncto, quam habet
I, ab origine abscissarum, abiret ad C per omnes gradus virium auctarum in infinitum,
& in C deberet habere vim infinitam ; in tertio vero idem accideret puncto collocate in
distantia, quam habet L. At in quarto casu accessum ad C prohibet ex parte I attractio
IK, & ex parte L repulsio LM. Sed quoniam, si eae crescant in ratione reciproca minus,
quam simplici distantiarum CI,CL ; area FKICD, vel GMLCD erit finita, adeoque
punctum impulsum versus C velocitate majore, quam quae respondeat illi areae, debet
transire per omnes virium magnitudines usque ad vim absolute infinitam in C, quae ibi
[86] praeterea & attractiva esse deberet, & repulsiva, limes videlicet omnium & attracti-
varum, & repulsivarum ; idcirco ne hie quidem casus admitti debet, nisi cum hac
conditione, ut ordinata crescat in ratione reciproca simplici distantiarum a C, vel etiam
majore, ut nimirum area infinita evadat, & accessum a puncto C prohibeat.

A THEORY OF NATURAL PHILOSOPHY

may occur, not with evanescence of the forces, but through the forces increasing indefinitely,
that is to say through asymptotic arcs of the curve. We said above, in Note (») to Art.
1 68, when an asymptotic arm goes off to infinity, there must be another asymptotic arm
returning from infinity having the same straight line for an asymptote ; & it may return
in four different positions, which depend on the two parts of the straight line & the two
sides of each part of the straight line. But, since our curve must always go forward, we
said that for it there remained only two out of these four positions, for any arm going off
to infinity ; that is to say, those in which the return is made on the opposite side of the
straight line. However, since, whilst the curve goes forward, either a repulsive or an
attractive arm can go off to infinity, here again we must have four possible cases, represented
in Figs. 16, 17, 18, 19, in all of which ACB is the axis, DCD' the asymptote, EKF the
arm going off to infinity, & GMH the arm returning from infinity.

I C

B

n cr

FIG 19.

1 86. In Fig. 1 6, to a repulsive arm EKF there succeeds an arm that is also repulsive;
in Fig. 17, to a repulsive succeeds an attractive ; in Fig. 18, to an attractive succeeds an
attractive ; and in Fig. 19, to an attractive succeeds a repulsive. The first & third cases
correspond to contacts. For, just as in contact, the force vanished, but did not change
its direction, so here also the force indeed becomes infinite but does not change its direction.
In Fig. 1 6, to the repulsion IK there succeeds the repulsion LM, & in Fig. 18 to an
attraction an attraction ; & thus these two cases cannot have any limit-points. But
the second & fourth cases certainly have limit-points ; for, in Fig. 17, to the repulsion
IK there succeeds the attraction LM, & in Fig. 19 to an attraction a repulsion ; &
thus the second case contains a limit-point of cohesion, & the fourth a limit-point of
non-cohesion.

187. Out of these cases I think that all except the last must be barred from our curve ;
& even with that all arms must be rejected for which the ordinates increase in a ratio
less than the simple reciprocal of the distances from the limit-point. My reasons for
excluding these are to avoid the possibility of there being at any time an infinite force
(which of itself I consider to be impossible), & because, in addition to that, an infinite
force, by its very nature necessitates the creation by it of an infinite velocity in a finite time.
For, in the first & second cases, a point, situated at the distance from another point equal
to that which I has from the origin of abscissae, would go off to C through all stages of
forces increased indefinitely, & at C would be bound to have an infinite force. In the
third case, too, the same thing would happen to a point situated at a distance equal to that
of L. Now, in the fourth case, the approach to C is restrained, from the side of I by the
attraction IK, & from the side of L by the repulsion LM. However, since, if these
forces increase in a ratio that is less than the simple reciprocal ratio of the distances CI,
CL, then the area FKICD or the area GMLCD will be finite ; £ thus the point, being
impelled towards C with a velocity that is greater than that corresponding to the area,
must pass through all magnitudes of the forces up to a force that is absolutely infinite at
C ; and this force must besides be both attractive & repulsive, the limit so to speak of all
attractive & repulsive forces. Hence not even this case is admissible, unless with the
condition that the ordinate increases in the simple reciprocal ratio of the distances from C,
or in a greater ; that is to say, the area must turn out to be infinite and so restrain the
approach towards the point C.

Four kinds of
them ; two corre-
sponding to contact,
& two to limit-
points, of which the
one is a limit-point
of cohesion & the
other of non-cohe-
sion.

None of these ex-
cept the last admis-
sible in Nature ; &
not even that in
general.

148 PHILOSOPHIC NATURAL! S THEORIA

Transitus per eum 188. Quando habeatur hie quartus casus in nostra curva cum ea conditione ; turn

bills: teqaibai quidem nullum punctum collocatum ex alters parte puncti C poterit ad alteram transilire,
distantiis constet, quacunque velocitate ad accessum impellatur versus alterum punctum, vel ad recessum

eum non haben. u • • j • • i • • £ • i • £ • • TJ

ab ipso, impediente transitum area repulsiva mnnita, vel innnita attractiva. Inde vero
facile colligitur, eum casum non haberi saltern in ea distantia, quae a diametris minimarum
particularum conspicuarum per microscopia ad maxima protenditur fixarum intervalla
nobis conspicuarum per telescopia : lux enim liberrime permeat intervallum id omne.
Quamobrem si ejusmodi limites asymptotici sunt uspiam, debent esse extra nostrae sensibi-
litatis sphaeram, vel ultra omnes telescopicas fixas, vel citra microscopicas moleculas.

Transitus ad puncta 189. Expositas hisce, quae ad curva virium pertinebant, aggrediar simpliciora quaadam,

iae, & massas. maxime notatu digna sunt, ac pertinent ad combinationem punctorum primo quidem

duorum, turn trium, ac deinde plurium in massa etiam coalescentium, ubi & vires mutuas,
& motus quosdam, & vires, quas in alia exercent puncta, considerabimus.

in limitibus ; icp. Duo puncta posita in distantia aequali distantiae limitis cujuscunque ab origine

?1 ^ ' abscissarum, ut in fig. i. AE, AG, AI, &c, (immo etiam si curva alicubi axem tangat, aequali
distantiae contactus ab eodem), ac ibi posita sine ulla velocitate, quiescent, ut patet, quia
nullam habebunt ibi vim mutuam : posita vero extra ejusmodi limites, incipient statim
ad se invicem accedere, vel a se invicem recedere per intervalla aequalia, prout fuerint sub
arcu attractivo, vel repulsive. Quoniam autem vis manebit semper usque ad proximum
limitem directionis ejusdem ; pergent progredi in ea recta, quae ipsa urgebat prius, usque
ad distantiam limitis proximi, motu semper accelerate, juxta legem expositam num. 176,
ut nimirum quadrata velocitatum integrarum, quae acquisitae jam sunt usque ad quodvis
momentum (nam velocitas initio ponitur nulla) respondeant areis clausis inter ordinatam
respondentem puncto axis terminanti abscissam, quae exprimebat distantiam initio motus,
& ordinatam respondentem puncto axis terminanti abscissam, quae exprimit distantiam
pro eo sequent! momento. Atque id quidem, licet interea occurrat contactus aliquis ;
quamvis enim in eo vis sit nulla, tamen superata distantia per velocitatem jam acquisitam,
statim habentur iterum [87] vires ejusdem directionis, quae habebatur prius, adeoque
perget acceleratio prioris motus.

Motus post proxi- lqI prOximus limes erit ems generis, cuius generis diximus limites cohaesioms, in quo

mum limitem super- . . 7 . ,. . ,J. i j ^ • • • i

atum, & osciiiatio. nimirum si distantia per repulsionem augebatur, succedet attractio ; si vero minuebatur
per attractionem, succedet e contrario repulsio, adeoque in utroque casu limes erit ejusmodi,
ut in distantiis minoribus repulsionem, in majoribus attractionem secum ferat. In eo
limite in utroque casu recessus mutui, vel accessus ex praecedentibus viribus, incipiet,
velocitas motus minui vi contraria priori, sed motus in eadem directione perget ; donee
sub sequent! arcu obtineatur area curvae aequalis illi, quam habebat prior arcus ab initio
motus usque ad limitem ipsum. Si ejusmodi aequalitas obtineatur alicubi sub arcu
sequente ; ibi, extincta omni praecedenti velocitate, utrumque punctum retro reflectet
cursum ; & si prius accedebant, incipient a se invicem recedere ; si recedebant, incipient
accedere, atque id recuperando per eosdem gradus velocitates, quas amiserant, usque ad
limitem, quern fuerant prsetergressa ; turn amittendo, quas acquisiverant usque ad dis-
tantiam, quam habuerant initio ; viribus nimirum iisdem occurrentibus in ingressu, &
areolis curvae iisdem per singula tempuscula exhibentibus quadratorum velocitatis incre-
menta, vel decrementa eadem, quae fuerant antea decrementa, vel incrementa. Ibi autem
iterum retro cursum reflectent, & oscillabunt circa ilium cohaesionis limitem, quern fuerant
praetergressa, quod facient hinc, & inde perpetuo, nisi aliorum externorum punctorum
viribus perturbentur, habentia velocitatem maximam in plagam utramlibet in distantia
ipsius illius limitis cohaesionis.

Casus

osdiiationis jo.2. Quod si ubi primum transgressa sunt proximum limitem cohaesionis, offendant

' S arcum ita minus validum praecedente, qui arcus nimirum ita minorem concludat aream,

quam praecedens, ut tota ejus area sit aequalis, vel etiam minor, quam ilia praecedentis

arcus area, quae habetur ab ordinata respondente distantiae habitse initio motus, usque ad

A THEORY OF NATURAL PHILOSOPHY 149

1 88. When, if ever, this fourth case occurs in our curve, then indeed no point situated Passage through a
on either side of the point C will be able to pass through it to the other side, no matter L^dTslmpo'ssibiel
what the velocity with which it is impelled to approach towards, or recede from, the other distances at which
point ; for the infinite repulsive area, or the infinite attractive area, will prevent such $,ere are^iio such
passage. Now, it can easily be derived from this, that this case cannot happen at any rate limit-points.

in the distance lying between the diameters of the smallest particles visible under the
microscope & the greatest distances of the stars visible to us through the telescope ; for
light passes with the greatest freedom through the whole of this interval. Therefore, if
there are ever any such asymptotic limit-points, they must be beyond the scope of our
senses, either superior to all telescopic stars, or inferior to microscopic molecules.

1 89. Having thus set forth these matters relating to the curve of forces, I will now w? n°w pass on to
discuss some of the simpler things that are more especially worth mentioning with regard ^^s.of matter> &
to combination of points ; & first of all I will consider a combination of two points, then

of three, & then of many, coalescing into masses ; & with them we will discuss their
mutual forces, & certain motions, and forces, which they exercise on other points.

190. Two points situated at a distance apart equal to the distance of any limit-point Rest at Hmit-
from the origin of abscissae, like AE, AG, AI, &c. in Fig. I (or indeed also where the r

curve touches the axis anywhere, equal to the distance of the point of contact from the without them.

origin), & placed in that position without any velocity, will be relatively at rest ; this is

evident from the fact that they have then no mutual force ; but if they are placed at any

other distance, they will immediately commence to move towards one another or away

from one another through equal intervals, according as they lie below an attractive or a

repulsive arc. Moreover, as the force always remains the same in direction as far as the

next following limit-point, they continue to move in the same straight line which contained

them initially as far as the distance apart equal to the distance of the next limit-point

from the origin, with a motion that is continually accelerated according to the law given

in Art. 176 ; that is to say, in such a manner that the squares of the whole velocities which

have been already acquired up to any instant (for the velocity at the commencement is

supposed to be nothing) will correspond to the areas included between the ordinate

corresponding to the point of the axis terminating the abscissa which the distance traversed

since motion began and the ordinate corresponding to the point on the axis terminating

the abscissa which expresses the distance for the next instant after it. This is still the case,

even if a contact should occur in the meantime. For, although at a point where contact

occurs the force is nothing, yet, this distance being passed by the velocity already acquired,

immediately afterwards there will be forces having the same direction as before ; and thus

the acceleration of the former motion will proceed.

191. The next limit-point will be one of the kind we have called limit-points of cohesion, Motion after the
namely, one in which, if the distance is increased by repulsion, then attraction follows ; passed^osc^Sion5
but if the distance is diminished by attraction, then on the contrary repulsion will follow ;

& thus, in either case, the limit-point will be of such a kind, that it gives a repulsion at
smaller distances & an attraction at larger. In this limit-point, in either case, the separation
or approach, due to the forces that have preceded, will be changed, & the velocity of motion
will begin to be diminished by a force opposite to the original force, but the motion will
continue in the same direction ; until an area of the curve under the arc that follows the
limit-point becomes equal to the area under the former arc from the commencement of
the motion as far as the limit-point. If equality of this kind is obtained somewhere under
the subsequent arc, then, the whole of the preceding velocity being destroyed, both the
points will return along their paths ; & if at the start they approached one another, they
will now begin to recede from one another, or if they originally receded from one another,
they will now commence to approach ; and as they do this, they will regain by the same
stages the velocities which they lost, as far as the limit-point which they passed through ;
then they will lose those which they had acquired, until they reach the distance
apart which they had at the commencement. That is to say, the same forces occur on
the return path, & the same little areas of the curve for the several short intervals of time
represent increments or decrements of the squares of the velocities which are the same
as were formerly decrements or increments. Then again they will once more retrace their
paths, & they will oscillate about the limit-point of cohesion which they had passed through ;
& this they will do, first on this side & then on that, over & over again, unless they are disturbed
by forces due to other points outside them ; & their greatest velocity in either direction
will occur at a distance apart equal to that of the limit-point of cohesion from the origin.

192. But if, when they first passed through the nearest limit-point of cohesion, they The case of a larger
happened to come to an arc representing forces so much weaker than those of the preceding

•ti-ii f • , ° 11 i ft T

arc that the whole area of it was equal to, or even less than, the area of the preceding arc,
reckoning from the ordinate corresponding to the distance apart at the commencement

150 PHILOSOPHIC NATURALIS THEORIA

limitem ipsum ; turn vero devenient ad distantiam alterius limitis proximi priori, qui
idcirco erit limes non cohaesionis. Atque ibi quidem in casu sequalitatis illarum arearum
consistent, velocitatibus prioribus elisis, & nulla vi gignente novas. At in casu, quo tota
ilia area sequentis arcus fuerit minor, quam ilia pars areae praecedentis, appellent ad dis-
tantiam ejus limitis motu quidem retardate, sed cum aliqua velocitate residua, quam
distantiam idcirco praetergressa, & nacta vires directionis mutatse jam conspirantes cum
directione sui motus, non, ut ante, oppositas, accelerabunt motum usque ad distantiam
limitis proxime sequentis, quam praetergressa precedent, sed motu retardato, ut in priore ;
& si area sequentis arcus non sit par extinguendae ante suum finem toti [88] velocitati,
quae fuerat residua in appulsu ad distantiam limitis praecedentis non cohaesionis, & quae
acquisita est in arcu sequent! usque ad limitem cohsesionis proximum ; turn puncta
appellent ad distantiam limitis non cohaesionis sequentis, ac vel ibi sistent, vel progredientur
itidem, eritque semper reciprocatio quaedam motus perpetuo accelerati, turn retardati ;
donee deveniatur ad arcum ita validum, nimirum qui concludat ejusmodi aream, ut tota
velocitas acquisita extinguatur : quod si accidat alicubi, & non accidat in distantia alicujus
limitis ; cursum reflectent retro ipsa puncta, & oscillabunt perpetuo.

Velocitatis muta- 193. Porro in hujusmodi motu patet illud, dum itur a distantia limitis cohaesionis

^"^abeat^maxU a^ distantiam limitis non cohaesionis, velocitatem semper debere augeri ; turn post

mum, & minimum transitum per ipsam debere minui, usque ad appulsum ad distantiam limitis non cohaesionis,

extmgui possit adeOque habebitur semper in ipsa velocitate aliquod maximum in appulsu ad distantiam

limitis cohaesionis, & minimum in appulsu ad distantiam limitis non cohaesionis. Quamo-

brem poterit quidem sisti motus in distantia limitis hujus secundi generis ; si sola existant

ilia duo puncta, nee ullum externum punctum turbet illorum motum : sed non poterit

sisti in distantia limitis illius primi generis ; cum ad ejusmodi distantias deveniatur semper

motu accelerate. Praeterea patet & illud, si ex quocunque loco impellantur velocitatibus

aequalibus vel alterum versus alterum, vel ad partes oppositas, debere haberi reciprocationes

easdem auctis semper aeque velocitatibus utriusque, dum itur versus distantiam limitis

primi generis, & imminutis, dum itur versus distantiam limitis secundi generis.

oscMatlo°S ma/**? X94' Patet & illud, si a distantia limitis primi generis dimoveantur vi aliqua, vel non

esse debeat, & unde ita uigenti velocitate impressa, oscillationem fore perquam exiguam, saltern si quidam
"

CJUS mag vah"dus fuerit limes ; nam velocitas incipiet statim minui, & ei vi statim vis contraria
invenietur, ac puncta parum dimota a loco suo, turn sibi relicta statim retro cursum reflect-
ent. At si dimoveantur a distantia limitis secundi generis vi utcunque exigua ; oscillatio
erit multo major, quia necessario debebunt progredi ultra distantiam sequentis limitis
primi generis, post quern motus primo retardari incipiet. Quin immo si arcus proximus
hinc, & inde ab ejusmodi limite secundi generis concluserit aream ingentem, ac majorem
pluribus sequentibus contrariae directionis, vel majorem excessu eorundem supra areas
interjacentes directionis suae ; turn vero oscillatio poterit esse ingens : nam fieri poterit,
ut transcurrantur hinc, & inde limites plurimi, antequam deveniatur ad arcum ita validum,
ut velocitatem omnem elidat, & motum retro reflectat. Ingens itidem oscillatio esse
poterit, si cum ingenti vi dimoveantur puncta a distantia limitum generis utriuslibet ; ac
res tota pendet a velocitate initiali, & ab areis, quae post oc-[8Q]-currunt, & quadratum
velocitatis vel augent, vel minuunt quantitate sibi proportionali.

Accessum debere 195. Utcunque magna sit velocitas, qua dimoveantur a distantia limitum ilia duo

swt^ saltern a^pmno puncta> utcunque validos inveniant arcus conspirantes cum velocitatis directione, si ad
recess um posse se invicem accedunt, debebunt utique alicubi motum retro reflectere, vel saltern sistere,
cas^'^o^bms 1™ saltern advenient ad distantias illas minimas, quae respondent arcui asymptotico,
exiguae differentiae cujus area est capax cxtinguendse cujuscunque velocitatis utcunque magnae. At si
velocitatis ingentis. rece(jant a se mvicem, fieri potest, ut deveniant ad arcum aliquem repulsivum validissimum,
cujus area sit major, quam omnis excessus sequentium arearum attractivarum supra repul-

A THEORY OF NATURAL PHILOSOPHY 151

of the motion up to the limit-point ; then indeed they will arrive at a distance apart equal
to that of the limit-point next following the first one, which will therefore be a limit-point of
non-cohesion. Here they will stop, in the case of equality between the areas in question ;
for the preceding velocities have been destroyed & no fresh ones will be generated. But
in the case when the whole of the area under the second arc is less than the said part of
the first area, they will reach a distance apart equal to that of the limit-point with a motion
that is certainly diminished ; but some velocity will be left, & this distance will therefore
be passed, & the points, coming under the influence of forces changed in direction so that they
now act in the same sense as their own motion, will accelerate their motion as far as the
next following limit-point ; & having passed through this they will go on, but with
retarded motion as in the first case. Then, if the area of the subsequent arc is not capable
before it ends of destroying the whole of the velocity which remained on attaining the
distance of the preceding limit-point of non-cohesion, & that which was acquired in the
arc that followed it up to the next limit-point of cohesion, then the points will move to a
distance apart equal to that of the next following limit-point from the origin, & will either
stop there or proceed ; & there will always be a repetition of the motion, continually
accelerated & retarded. Until at length it comes to an arc so strong, that is to say, one
under which the area is such, that the whole velocity acquired is destroyed ; & when this
happens anywhere, & does not happen at a distance equal to that of any limit-point, then
the points will retrace their paths & oscillate continuously.

193. Further in this kind of motion it is clear that along the path from the distance Alternate changes
of a limit-point of cohesion to a limit-point of non-cohesion the velocity is bound to be of velocity ; where
always increasing ; then after passing through the latter it must decrease up to its arrival at the value! & ""a^mln?
distance of a limit-point of non-cohesion. Thus, there will always be in the velocity a Pum value ; where
maximum on arrival at a distance equal to that of a limit-point of cohesion, & a minimum ' maybe estr°yed.
on arrival at a distance of a limit-point of non-cohesion. Hence indeed the motion may

possibly cease at a limit-point of this second kind, if the two points exist by themselves,
& no other point influences their motion from without. But it cannot cease at a distance
of a limit-point of the first kind ; for it will always arrive at distances of this kind with
an accelerated motion. Moreover it is also clear that, if they are urged from any given
position with equal velocities, either towards one another or in opposite directions, the
same alternations must be had as before, the velocities being increased equally for each
point whilst they are moving up to a distance of a limit-point of the first kind, & diminished
whilst they are moving up to a distance of a limit-point of the second kind.

194. It is evident also that, if the points are moved from a distance apart equal to that of The limit-points
a limit-point of the first kind by some force (especially when the velocity thus impressed oscfflation mus^be
is not extremely great), then the oscillation will be exceedingly small, at least so long as the krger; & the thing
limit-point is a fairly strong one. For the velocity will commence to be diminished tude"
immediately, & to the force another force will be obtained at once, acting in opposition

to it ; & the points, being moved but little from their original position, will immediately
afterwards retrace their paths if left to themselves. But if they are moved from a distance
apart equal to that of a limit-point of the second kind by any force, no matter how small,
then the oscillation will be much greater ; for, of necessity, they are bound to go on beyond
the distance equal to that of the next following limit-point of the first kind ; & not until
this has been done, will the motion begin to be retarded. Nay, if the next arc on each
side of such a limit-point of the second kind should include a very large area, and one that
is greater than several of those subsequent to them, which are opposite in direction, or
greater than the excess of these over the intervening areas that are in the same direction,
then indeed the oscillation may be exceedingly large. For it may be that very many
limit-points on either side are traversed before an arc is arrived at, which is sufficiently
strong to destroy the whole of the velocity & reverse the direction of motion. A very
large oscillation will also be possible, if the points are moved from a distance apart equal to
that of a limit-point of either kind by an exceedingly large force. The whole thing depends
on the initial velocity & the areas which occur subsequently, & either increase or decrease
the square of the velocity by a quantity that is proportional to the areas themselves.

195. However great the velocity may be, with which the two points are moved from Approach is bound
a distance equal to that of any limit-point, no matter how strong are the arcs they come *^gS \* ttahney %££
upon, which are in the same direction as that of the velocity ; yet, if they approach one repulsive arc, but
another, they are bound somewhere to have their motion reversed, or at least to come onpaSde°nnitety ; Sa
to rest ; for, at all events, they must finally attain to those very small distances that correspond noteworthy case
to an asymptotic arm, the area of which is capable of destroying any velocity whatever, arSerencTfar aSvery
no matter how great. But, if they recede from one another, it may happen that they come great velocity.

to some very strong repulsive arc, the area of which is greater than the whole of the excess
of the subsequent attractive arcs above those that are repulsive, as far as the very weak

152

PHILOSOPHIC NATURALIS THEORIA

sivas, usque ad languidissimum ilium arcum postremi cruris gravitatem exhibentis. Turn
vero motus acquisitus ab illo arcu nunquam poterit a sequentibus sisti, & puncta ilia recedent
a se invicem in immensum : quin immo si ille arcus repulsivus cum sequentibus repulsivis
ingentem habeat areae excessum supra arcus sequentes attractivos ; cum ingenti velocitate
pergent puncta in immensum recedere a se invicem ; & licet ad initium ejus tarn validi
arcus repulsivi deveniant puncta cum velocitatibus non parum diversis ; tamen velocitates
recessuum post novum ingens illud augmentum erunt parum admodum discrepantes a
se invicem : nam si ingentis radicis quadrate addatur quadratum radicis multo minoris,
quamvis non exiguae ; radix extracta ex summa parum admodum differet a radice priore.

Demonstratio ad-
modum simplex.

A

BD

FIG. 20.

Quid accidat binis
punctis, cum sunt
sola, quid possit
accidere actionibus
aliorum externis.

Si limites sint a se
invicem r e m o t i,
m u t a t a multum
distantia r e d i ri
retro : secus, si
sint proximi.

196. Id quidem ex Euclidea etiam Geometria manifestum fit. Sit in fig. 20 AB
linea longior, cui addatur ad perpendiculum BC, multo minor, quam fit ipsa ; turn centre A,
intervallo AC, fiat semicirculus occurrens AB hinc, & inde in E, D. Quadrate AB
addendo quadratum BC habetur quadratum AC, sive AD ; & tamen haec excedit prsece-
dentem radicem AB per solam BD, quae semper est minor, quam BC, & est ad ipsam, ut
est ipsa ad totam BE. Exprimat AB velocitatem, quam in punctis quiescentibus gigneret
arcus ille repulsivus per suam aream, una cum differentia omnium sequentium arcuum
repulsivorum supra omnes sequentes attractivos : exprimat autem BC velocitatem, cum
qua advenitur ad distantiam respondentem

initio ejus arcus : exprimet AC velocitatem,
qu33 habebitur, ubi jam distantia evasit major,
& vis insensibilis, ac ejus excessus supra
priorem AB erit BD, exiguus sane etiam re-
spectu BC, si BC fuerit exigua respectu AB,
adeoque multo magis respectu AB ; & ob
eandem rationem perquam exigua area sequentis
cruris attractivi ingentem illam jam acqui-
sitam velocitatem nihil ad sensum mutabit, quae
permanebit ad sensum eadem post recessum in
immensum.

197. Haec accident binis punctis sibi relictis, vel impulsis [90] in recta, qua junguntur,
cum oppositis velocitatibus aequalibus, quo casu etiam facile demonstratur, punctum,
quod illorum distantiam bifariam secat, debere quiescere ; nunquam in hisce casibus
poterit motus extingui in adventu ad distantiam limitis cohaesionis, & multo minus poterunt
ea bina puncta consistere extra distantiam limitis cujuspiam, ubi adhuc habeatur vis aliqua
vel attractiva, vel repulsiva. Verum si alia externa puncta agant in ilia, poterit res multo
aliter se habere. Ubi ex. gr. a se recedunt, & velocitates recessus augeri deberent in accessu
ad distantiam limitis cohaesionis ; potest externa compressio illam velocitatem minuere,
& extinguere in ipso appulsu ad ejusmodi distantiam. Potest externa compressio cogere
ilia puncta manere immota etiam in ea distantia, in qua se validissime repellunt, uti duae
cuspides elastri manu compressae detinentur in ea distantia, a qua sibi relictas statim
recederent : & simile quid accidere potest vi attractivae per vires externas distrahentes.

198. Turn vero diligenter notandum discrimen inter casus varies, quos inducit varia
arcuum curvae natura. Si puncta sint in distantia alicujus limitis cohassionis, circa quern
sint arcus amplissimi, ita, ut proximi limites plurimum inde distent, & multo magis etiam,
quam sit tota distantia proximi citerioris limitis ab origine abscissarum ; turn poterunt
externa vi comprimente, vel distrahente redigi ad distantiam multis vicibus minorem,
vel majorem priore ita, ut semper adhuc conentur se restituere ad priorem positionem
recedendo, vel accedendo, quod nimirum semper adhuc sub arcu repulsive permaneat, vel
attractive. At si ibi frequentissimi limites, curva saepissime secante axem ; turn quidem
post compressionem, vel distractionem ab externa vi factam, poterunt sisti in multo minore,
vel majore distantia, & adhuc esse in distantia alterius limitis cohaesionis sine ullo conatu
ad recuperandum priorem locum.

Superiorum usus in 199. Haec omnia aliquanto fusius considerare libuit, quia in applicatione ad Physicam

magno usui erunt infra haec ipsa, & multo magis hisce similia, quae massis respondent
habentibus utique multo uberiores casus, quam bina tantummodo habeant puncta. Ilia
ingens agitatio cum oscillationibus variis, & motibus jam acceleratis, jam retardatis, jam
retro reflexis, fermentationes, & conflagrationes exhibebit : ille egressus ex ingenti arcu

A THEORY OF NATURAL PHILOSOPHY 153

arc of the last branch which represents gravity. Then indeed the velocity acquired through
that arc can never be stopped by the subsequent arcs, & the points will recede from one
another to an immense distance. Nay further, if that repulsive arc taken together with
the subsequent repulsive arcs has a very great excess of area over the subsequent attractive
arcs, then the points will continue to recede to an immense distance from one another
with a very great velocity ; &, although points arrive at this repulsive arc, which is so strong,
with considerably different velocities, yet the velocities after this fresh & exceedingly great
increase will be very little different from one another. For, if to the square of a very
great number there is added the square of a number that is much less, although
not in itself very small, the square root of the sum differs very little from the first
number.

196. This indeed is very evident from Euclidean geometry even. In Fig. 20, let The demonstration
AB be a fairly long line, to which is added, perpendicular to it, BC, which is much less ls Perfectly sunPIe-
than AB. Then, with centre A, & radius AC, describe a semicircle meeting AB on either

side in E & D. On adding the square on BC to the square on AB, we get the square on
AC or AD ; & yet this exceeds the former root AB by BD only, which is always less than
BC, bearing the same ratio to it as BC bears to the whole length BE. Suppose that
AB represents the velocity which the repulsive arc, owing to the area under it, would
generate in points initially at rest, together with the difference for all the subsequent
repulsive arcs over all the subsequent attractive arcs ; also let BC represent the velocity
with which the distance corresponding to the beginning of this arc is reached ; then AC
will represent the velocity which is obtained when the distance has already become of
considerable amount, & the force insensible. Now the excess of this above the former
velocity AB will be represented by BD ; & this is really very small compared with BC,
if BC were very small compared with AB ; & therefore much more so with regard to AB.
For the same reason, the very small area under the subsequent attractive branch will not
sensibly change the very great velocity acquired so far ; this will remain sensibly the same
after recession to a huge distance.

197. These things will take place in the case of two points left to themselves, or impelled What may happen
along the straight line joining them with velocities that are equal & opposite ; in such the^areT^then?
a case it can be easily proved that the middle point of the distance between them is bound selves ; what may
to remain at rest. The motion in the cases we have discussed can never be destroyed ^teif11 under ththe
altogether on arrival at a distance equal to that of a limit-point of cohesion, & much less actions of other
will the two points be able to stop at a distance apart that is not equal to that of some them3 extemal to
limit-point, as far as which there is some force acting, either attractive or repulsive. But

if other external points act upon them, we may have altogether different results. For
instance, in a case where they recede from one another, & the velocities would therefore
be bound to be increased as they approached a distance equal to that of a limit-point of
cohesion, an external compression may diminish that velocity, & completely destroy it
as it approaches the distance of that limit-point. An external compression may even
force the points to remain motionless at a distance for which they repel one another very
strongly ; just as the two ends of a spring compressed by the hands are kept at a distance
from which if left to themselves they will immediately depart. A similar thing may come
about in the case of an attractive force when there are external tensile forces.

198. Now, a careful note must be made of the distinctions between the various cases, if the limit-points
which arise from the various natures of the arcs of the curve. If our points are at a distance fje far apart, there

.,_. i • i 111 is a tenaency to

of any limit-point of cohesion, on each side of which the arcs are very wide, so that the return if the dis-
nearest limit-points are very far distant from it, & also much more so than the nearest j^sfderabSH-han^e;
limit-point to the left is distant from the origin of abscissae ; they may, under the action but this is not the'
of an external force causing either compression or tension, be reduced after many alternations l^tsare very'dose
to a distance, either less, or greater, than the original distance, in such a way that they together,
will always strive however to revert to their old position by receding from or approaching
towards one another ; for indeed they will still always remain under a repulsive, or an
attractive arc. But if, near the limit-point in question, the limit-points on either side
occur at very frequent intervals ; then indeed, after compression, or separation, caused
by an external force, they may stop at a much less, or a much greater, distance apart, &
still be at a distance equal to that of another limit-point of cohesion, without there being
any endeavour to revert to their original position.

199. All these considerations I have thought it a good thing to investigate somewhat The use of the
at length ; for they will be of great service later in the application of the Theory to physics, ^°ve facts "* phy"
both these considerations, & others like them to an even greater degree ; namely those

that correspond to masses, for which indeed there are far more cases than for a system
of only two points. The great agitation, with its various oscillations & motions that are
sometimes accelerated, sometimes retarded, & sometimes reversed, will represent fermentations

154 PHILOSOPHIC NATURALIS THEORIA

repulsive cum velocitatibus ingentibus, quas ubi jam ad ingentes deventum est distantias,
parum admodum a se invicem differant, nee ad sensum mutentur quidquam per immensa
intervalla, luminis emissionem, & propagationem uniformem., ac ferme eandem celeritatem
in quovis ejusdem speciei radio fixarum, Solis, flammse, cum exiguo discrimine inter
diversos coloratos radios ; ilia vis permanens post compressionem ingentem, vel diffractionem
elasticitati explicandae in-[9i]-serviet ; quies ob frequentiam limitum, sine conatu ad
priorem recuperandam figuram, mollium corporum ideam suggeret ; quae quidem hie
innuo in antecessum, ut magis haereant animo, prospicienti jam nine insignes eorum usus.

Motus binorum 2OO. Quod si ilia duo puncta proiiciantur oblique motibus contrarns, & aequalibus

punctorum oblique j- • • -11 j i rr •

projectorum. Per directiones, quae cum recta jungente ipsa ilia duo puncta angulos aequales efficiant ;

turn vero punctum, in quo recta ilia conjungens secatur bifariam, manebit immotum ;
ipsa autem duo puncta circa id punctum gyrabunt in curvis lineis aaqualibus, & contrariis,
quae data lege virium per distantias ab ipso puncto illo immoto (uti daretur, data nostra
curva virium figurae i, cujus nimirum abscissae exprimunt distantias punctorum a se invicem,
adeoque eorum dimidiae distantias a puncto illo medio immoto) invenitur solutione pro-
blematis a Newtono jam olim soluti, quod vocant inversum problema virium centralium,
cujus problematis generalem solutionem & ego exhibui syntheticam eodem cum Newtoniana
recidentem, sed non nihil expolitam, in Stayanis Supplementis ad lib. § 19.

Casus, in quo duo 201. Hie illud notabo tantummodo, inter infinita curvarum genera, quae describi

sc1Hberedebe&pira<ies Possunt5 cum nulla sit curva, quas assumpto quovis puncto pro centre virium describi
circa medium im- non possit cum quadam virium lege, quae definitur per Problema directum virium
centralium, esse innumeras, quas in se redeant, vel in spiras contorqueantur. Hinc fieri
potest, ut duo puncta delata sibi obviam e remotissimis regionibus, sed non accurate in
ipsa recta, quae ilia jungit (qui quidem casus accurati occursus in ea recta est infinities
improbabilior casu deflexionis cujuspiam, cum sit unicus possibilis contra infinites), non
recedant retro, sed circa punctum spatii medium immotum gyrent perpetuo sibideinceps
semper proxima, intervallo etiam sub sensus non cadente ; qui quidem casus itidem
diligenter notandi sunt, cum sint futuri usui, ubi de cohaesione, & mollibus corporibus
agendum erit.

Theorema de statu 202. Si utcunque alio modo projiciantur bina puncta velocitatibus quibuscunque ;

enSfterm1nU'maSs P°test facile ostendi illud : punctum, quod est medium in recta jungente ipsa, debere

centri gravitatis quiescere, vel progredi uniformiter in directum, & circa ipsum vel quietum, vel uniformiter

progrediens, debere haberi vel illas oscillationes, vel illarum curvarum descriptiones.

Verum id generalius pertinet ad massas quotcunque, & quascunque, quarum commune

gravitatis centrum vel quiescit, vel progreditur uniformiter in directum a viribus mutuis

nihil turbatum. Id theorema Newtonus proposuit, sed non satis demonstravit. Demon-

strationem accuratissimam, ac generalem simul,& non per casuum inductionem tantummodo,

inveni, ac in dissertation e De Centra Gravitatis proposui, quam ipsam demonstrationem

hie etiam inferius exhibebo.

Accessum aiterius [92] 203. Interea hie illud postremo loco adnotabo, quod pertinet ad duorum

quodvis ad aiterius punctorum motum ibi usui futurum : si duo puncta moveantur viribus mutuis tantummodo,

aequari recessui ex & ultra ipsa assumatur planum quodcunque ; accessus aiterius ad illud planum secundum

directionem quamcunque, aequabitur recessui aiterius. Id sponte consequitur ex eo,

quod eorum absoluti motus sint aequales, & contrarii ; cum inde fiat, ut ad directionem

aliam quamcunque redacti aequales itidem maneant, & contrarii, ut erant ante. Sed de

aequilibrio, & motibus duorum punctorum jam satis.

Transitus ad syste- 204. Deveniendo ad systema trium punctorum, uti etiam pro punctis quotcunque,

trium "binagene™ res> si generaliter pertractari deberet, reduceretur ad haac duo problemata, quorum alterum
alia probiemata. pertinet ad vires, & alterum ad motus : I. Data positions, £5? distantia mutua eorum punc-
torum, invenire magnitudinem, & directionem vis, qua urgetur quodvis ex ipsis, composites a
viribus, quibus urgetur a reliquis, quarum singularum virium lex communis datur per curvam
figure primce. 2. Data ilia lege virium figures -primes invenire motus eorum punctorum,
quorum singula cum datis velocitatibus projiciantur ex datis locis cum datis directionibus.
Primum facile solvi potest, & potest etiam ope curva; figurae i determinari lex virium

A THEORY OF NATURAL PHILOSOPHY 155

& conflagrations. The starting forth from a very large repulsive arc with very great
velocities, which, as soon as very great distances have been reached, are very little different
from one another ; nor are they sensibly changed in the slightest degree for very great
intervals ; this will represent the emission & uniform propagation of light, & the approximately
equal velocities in any ray of the same kind from the stars, the sun, and a flame, with a
very slight difference between rays of different colours. The force persisting after
compression, or separation, will serve to explain elasticity. The lack of motion due to
the frequent occurrence of limit-points, without any endeavour towards recovering the
original configuration, will suggest the idea of soft bodies. I mention these matters here
in anticipation, in order that they may the more readily be assimilated by a mind that
already sees from what has been said that there is an important use for them.

200. But if the two points are projected obliquely with velocities that are equal and The motion of two
opposite to one another, in directions making equal angles with the straight line joining obikmei pro^ected
the two points ; then, the point in which the straight line joining them is bisected

will remain motionless ; the two points will gyrate about this middle point in equal curved
paths in opposite directions. Moreover, if the law of forces is given in terms of the distances
from that motionless point (as it will be given when our curve of forces in Fig. i is given,
where the abscissae represent the distances of the points from one another, & therefore
the halves of these abscissae represent -the distances from the motionless middle point),
then we arrive at a solution of the problem already solved by Newton some time ago, which
is called the inverse -problem of central forces. Of this problem I also gave a general synthetic
solution that was practically the same thing as that of Newton, not altogether devoid of
neatness, in the Supplements to Stay's Philosophy, Book 3, Art. 19.

201. At present I will only remark that, amongst the infinite number of different The case in which
curves that can be described, there are an innumerable number which will either re-enter boumT^o^teswibe
their paths, or wind in spirals ; for there is no curve that, having taken any point whatever spirals about the
for the centre of forces, cannot be described with some law of forces, which is determined j^°^°nless mlddle
by the direct problem of central forces. Hence it may happen that two points approaching

one another from a long way off, but not exactly in the straight line joining them — and
the case of accurate approach along the straight line joining them is infinitely more improbable
than the case in which there is some deviation, since the former is only one possible case
against an infinite number of others — then the points will not reverse their motion and
recede, but will gyrate about a motionless middle point of space for evermore, always
remaining very near to one another, the distance between them not being appreciable by
the senses. These cases must be specially noted ; for they will be of use when we come
to consider cohesion & soft bodies.

202. If two points are projected in any manner whatever with any velocities whatever, Theorem on the
it can readily be proved that the middle point of the line joining them must remain at steady state of the

1 . , r , . , , . r , , , ,J . P , , . . central point &,

rest or move uniformly in a straight line ; and that about this point, whether it is at rest more generally,
or is moving uniformly, the oscillations or descriptions of the curved paths, referred to of the .cen,tre of

i ' T> i • 11 • i • r c gravity in the case

above, must take place. But this, more generally, is a property relating to masses, of any of masses.

number or kind, for which the common centre of gravity is either at rest or moves uniformly

in a straight line, in no wise disturbed by the mutual forces. This theorem was enunciated

by Newton, but he did not give a satisfactory proof of it. I have discovered a most rigorous

demonstration, & one that is at the same time general, & I gave it in the dissertation

De Centra Gravitatis ; this demonstration I will also give here in the articles that

follow.

203. Lastly, I will here mention in passing something that refers to the motion of The approach of
two points, which will be of use later, in connection with that subject. If two points poLts towards^y
move subject to their mutual forces only, & any plane is taken beyond them both, then plane is equal to
the approach of one of them to that plane, measured in any direction, will be equal to the other^rom1 it 'on
recession of the other. This follows immediately from the fact that their absolute motions account of 'the
are equal & opposite ; for, on that account, it comes about that the resolved parts in any mutual force-
other direction also remain equal & opposite, as they were to start with. However, I

have said enough for the present about the equilibrium & motions of two points.

204. When we come to consider systems of three points, as also systems of any number Extension to a
of points, the whole matter in general will reduce to these two problems, of which the system of three

tt 11 . T, . . . , points ; two general

one refers to forces and the other to motions, i. Being given the position and the mutual problems.

distance of the points, it is required to find the magnitude and direction of the force, to which

any one of them is subject ; this force being the resultant of the forces due to the remaining

points, and each of these latter being found by a general law which is given by the curve of Fig. i.

2. Being given the law of forces represented by Fig. I, it is required to find the motions of

the points, when each of them is projected with known velocities from given initial positions

in given directions. The first of these problems is easily solved ; and also, by the aid of

156 PHILOSOPHIC NATURALIS THEORIA

generaliter pro omnibus distantiis assumptis in quavis recta positionis datae,' a que id tarn
geometrice determinando per puncta curvas, quae ejusmodi legem exhibeant, ac determinent
sive magnitudinem vis absolutae, sive magnitudines binarum virium, in quas ea concipiatur
resoluta, & quarum altera sit perpendicularis data? illi rectse, altera secundum illam agat ;
quam exhibendo tres formulas analyticas, quas id praestent. Secundum omnino generaliter
acceptum, & ita, ut ipsas curvas describendas liceat definire in quovis casu vel constructione,
vel caculo, superat (licet puncta sint tantummodo tria) vires methodorum adhuc cognit-
arum : & si pro tribus punctis substituantur tres massae punctorum, est illud ipsum
celeberrimum problema quod appellant trium corporum, usque adeo qusesitum per haec
nostra tempora, & non nisi pro peculiaribus quibusdam casibus, & cum ingentibus limita-
tionibus, nee adhuc satis promoto ad accurationem calculo, solutum a paucissimis nostri
asvi Geometris primi ordinis, uti diximus num. 122.
Theorema de motu 205. Pro hoc secundo casu illud est notissimum, si tria puncta sint in fig. 21 A, C, B,

puncti habentis ac- „ •,. J . . T. i •>• • IT- • T-> i

tionem cum aiiis & distantia AB duorum divisa semper bifanam in D, ac ducta CD, & assumpto ejus
binis- triente DE, utcunque moveantur eadem puncta

motibus compositis a projectionibus quibus-

cunque, & mutatis viribus ; punctum E debere

vel quiescere semper, vel progredi in directum

motu uniformi. Pendet id a general! theore-

mate de centre gravitatis, cujus & superius

injecta est mentio, & de quo age-[93]-mus

infra pro massis quibuscunque. Hinc si sibi re-

linquantur, accedet C ad E, & rectae AB

punctum medium D ibit ipsi obviam versus

ipsum cum velocitate dimidia ejus, quam ipsum

habebit, vel contra recedent, vel hinc, aut inde

movebuntur in latus, per lineas tamen similes,

atque ita, ut C, & D semper respectu puncti E

immoti ex adverse sint, in quo motu tam directio « —. _

rectae AB, quam directio rectae CD, & ejus incli- " _ *^

natio ad AB, plerumque mutabitur.
Determmatio vis 2o6. Quod pertinet ad inveniendam vim pro quacunque positione puncti C respectu

ejusdem composite A „ T> r -i • • • T r • •

e binis viribus. punctorum A, & B, ea facile sic mvemetur. In fig. i assumendae essent abscissae in axe
asquales rectis AC, BC figurae 21, & erigendae ordinatas ipsis respondentes, quae vel ambae
essent ex parte attractiva, vel ambae ex parte repulsiva ; vel prima attractiva, & secunda
repulsiva ; vel prima repulsiva & secunda attractiva. In primo casu sumendae essent CL,
CK ipsis aequales (figura 21 exhibet minores, nenimis excrescat) versus A, &B ; in secundo
CN, CM ad partes oppositas A,B : in tertio CL versus A, & CM ad partes oppositas B ; in
quarto CN ad partes oppositas A, & CK versus B. Tam complete parallelogrammo LCKF,
vel MCNH, vel LCMI, vel KCNG, diameter CF, vel CH, vel CI, vel CG exprimeret
directionem, & magnitudinem vis compositae, qua urgetur C a reliquis binis punctis.

Methodus constru- 207. Hinc si assumantur ad arbitrium duo loca qusecunque punctorum A, & B, ad

expi? <lu3e referendum sit tertium C ; ducta quavis recta DEC indefinita, ex quovis ejus puncto

mat vim ejusmodi. posset erigi recta ipsi perpendicularis, & asqualis illi diametro, ut CF in primo casu, ac
haberetur curva exprimens vim absolutam puncti in eo siti, & solicitati a viribus, quas
habet cum ipsis A, & B. Sed satis esset binas curvas construere, alteram, quae exprimeret
vim redactam ad directionem DC per perpendiculum FO, ut CO ; alteram, quae exprimeret
vim perpendicularem OF : nam eo pacto haberentur etiam directiones vis absolutae ab
iis compositae per ejusmodi binas ordinatas. Oporteret autem ipsam ordinatam curvas
utriuslibet assumere ex altera plaga ipsius CD, vel ex altera opposita ; prout CO jaceret
versus D, vel ad plagam oppositam pro prima curva; & prout OF jaceret ad alteram partem
rectae DC, vel ad oppositam, pro secunda.

Expressio magis 208. Hoc pacto datis locis A, B pro singulis rectis egressis e puncto medio D duas

*115 Per super" haberentur diversae curvae, quae diversas admodum exhiberent virium leges ; ac si quasre-
retur locus geometricus continuus, qui exprimeret simul omnes ejusmodi leges pertinentes
ad omnes ejusmodi curvas, sive indefinite exhiberet omnes vires pertinentes ad omnia

A THEORY OF NATURAL PHILOSOPHY 157

the curve given in Fig. i, the law of forces can be determined in general for any assumed
distances along any straight line given in position. Moreover, this can be effected either
by constructing geometrically curves through sets of points, which represent a law of this
sort & give either the magnitude of the absolute force, or the magnitudes of the pair of
forces into which it may be considered to be resolved, the one acting perpendicularly to
the given straight line & the other in its direction ; or else by writing down three analytical
formulae, which will represent its value. The second, if treated perfectly generally, &
in such a manner that the curves to be described can be assigned in any case whatever,
either by construction or by calculation, is (even when there are only three points in question)
beyond the power of all methods known hitherto. Further, if instead of three points
we have three masses of points, then we have the well-known problem that is called " the
problem of three bodies." The solution of this problem is still sought after in our own
times ; & has only been solved in certain special cases, with great limitations by a very
few of the geometricians of our age belonging to the highest rank, & even then with insufficient
accuracy of calculation ; as was pointed out in Art. 122.

205. As for this second case, it is very well known that, if in Fig. 21, A,C,B, are three Theorem with re-
points, & the distance between two of them, A & B, is always bisected at D, & CD is joined, g'Vyjj m°^r
& DE is taken equal to one third of DC, then, however these points move under the influence the action of two
of the forces compounded from the forces of any projection whatever & the mutual forces, other P°mts-

the point E must always remain at rest or proceed in a straight line with uniform motion.
This depends on a general theorem with regard to the centre of gravity, about which
passing mention has already been made, & with which we shall deal in what follows for the
case of any masses whatever. From this it follows that, if they are left to themselves,
the point C will approach the point E, & D, the middle point of the straight line AB, will
move in the opposite direction towards E with half the velocity of C ; or, on the contrary,
both C & D will recede from E ; or they will move, one in one direction & the other in
the opposite direction : nevertheless they will follow similar paths, in such a manner that
C & D will always be on opposite sides of the stable point E ; & in this motion, the direc-
tion of the straight line AB, that of the straight line DE, & the inclination of the latter
to AB will usually be altered.

206. As regards the determination of the force for any position of the point C with Determination o f
regard to the points A & B, that is easily effected in the following manner. Take, in Fig. I, compound^
abscissa measured along the axis equal to the straight lines AC & BC of Fig. 21 ; draw two forces.

the ordinates corresponding to them, which may be either both on the attractive side of
the axis, or both on the repulsive side ; or the first on the attractive & the second on the
repulsive ; or the first on the repulsive & the second on the attractive side. In the first
case, take CL, CK, equal to these ordinates (in Fig. 21 they are reduced so as to prevent
the figure from being too large) ; let them be taken in the direction of A & B ; similarly,
in the second case, take CN & CM in the opposite directions to those of A & B ; and, in
the third case, take CL in the direction of A, & CM in the direction opposite to that of B ;
whilst, in the fourth case, take CN in the direction opposite to that of A, & CK in the
direction of B. Then, completing the parallelogram LCKF, or MCNH, or LCMI, or
KCNG, the diagonal CF, or CH, or CI, or CG, will represent the direction & the magnitude
of the resultant force, which is exerted upon the point C by the remaining two points.

207. Hence, if any two positions are taken at random as those of the points A & B, The method of
& to these the third point C is referred ; & if any straight line DEC is drawn of indefinite curve' wWch^m in
length ; then from any point of it a straight line can be erected perpendicular to it, & general express a
equal to the diagonal of the parallelogram, for instance CF in the first case. From these force of this sort'
perpendiculars a curve will be obtained, which will represent the absolute force on a point

situated in the straight line DEC, & under the action of the forces exerted upon it by the
points A & B. However, it would be more satisfactory if two curves were constructed ;
one of which would represent the force resolved along the direction DC by means of a
perpendicular FO, such as CO ; & the other to represent the perpendicular force OF.
For, in this way, we should also obtain the directions of the absolute forces compounded
from these resolved parts, by means of the two ordinates of this kind. Moreover,
we ought to take these ordinates of either of the curves on the one side or the other of
the straight line CD, according as CO would be towards D, or away from it, in the first
curve, & according as OF would be away from the straight line CD, on the one side or on the
other, in the second curve.

208. In this way, given the positions of A & B, for each straight line drawn through A m°re general
the point D, we should obtain distinct curves ; & these would represent altogether different of ^ surface.71
laws of forces. If then a continuous geometrical locus is required, which would
simultaneously represent all the laws of this kind relating to every curve of this sort,

or express in general all the forces pertaining to all points such as C, wherever they might

I58

PHILOSOPHIC NATURALIS THEORIA

puncta C, ubicunque collocata ; oporteret erigere in omnibus punctis C rectas normales
piano ACB, alteram aequalem CO, [94] alteram OF, & vertices ejusmodi normalium
determinarent binas superficies quasdam continuas, quarum altera exhiberet vires in
directione CD attractivas ad D, vel repulsivas respectu ipsius, prout, cadente O citra, vel
ultra C, normalis ilia fuisset erecta supra, vel infra planum ; & altera pariter vires perpen-
diculares. Ejusmodi locus geometricus, si algebraice tractari deberet, esset ex iis, quos
Geometrse tractant tribus indeterminatis per unicam aequationem inter se connexis ; ac
data aequatione ad illam primam curvam figurse I, posset utique inveniri tam sequatio ad
utramlibet curvam respondentem singulis rectis DC, constans binis tantum indeterminatis,
quam sequatio determinans utramlibet superficiem simul indefinite per tres indetermin-
atas. (»)

Methpdus determi- [gel 20Q. Si pro duobus punctis tantummodo agentibus in tertium daretur numerus

nandi vim composi- L7TJ . r . , . . . . .,

tam ex viribus re- quicunque punctorum positorum in datis locis, ac agentium in idem punctum, posset utique
spicientibus puncta constructione simili inveniri vis, qua sineula agunt in ipsum collocatum in quovis assumpto

quotcunque. , . . ' ~*. . .. ° . r , ., . .?

loci puncto, ac vis ex ejusmodi viribus composita denniretur tam directione, quam
magnitudine, per notam virium compositionem. Posset etiam analysis adhiberi ad expri-
mendas curvas per asquationes duarum indeterminatarum pro rectis quibuscunque, & (")
si omnia puncta jaceant in eodem piano, superficies per asquationem trium. [96] Mirum
autem, quanta inde diversarum legum combinatio oriretur. Sed & ubi duo tantummodo
puncta agant in tertium, incredibile dictu est, quanta diversitas legum, & curvarum inde
erumpat. Manente etiam distantia AB, leges pertinentes ad diversas inclinationes rectae
DC ad AB, admodum diversse obveniunt inter se : mutata vero punctorum A, B distantia

(n) Stantibus in fig. 22 punctis ADBCKFLO, ut in fig. 21, ducantur perpendicula BP, AQ in CD, qute dabuntur
data inclinations DC, y punctis B, A, ac pariter dabuntur y DP, DQ. Dicatur prtsterea DC = x, y dabuntur analytice
CQ, CP. Quare ob angulos rectos P, Q, dabuntur etiam analytice CB, CA. Denominentur CK=«, CL =z, CF =y.
Quoniam datur AB, y dantur analytice AC, CB ; dabitur analytice ex applicatione Algebrte ad Trigonometriam
sinus anguli ACB per x, y datas quantitates, qui est idem, ac sinus anguli CKF complements ad duos rectos. Datur
autem idem ex datis analytice valoribus CK = «, KF = CL =z, CF =y ; quare habetur ibi una tequatio per x, y,
z, u, y constantes. Si pr<eterea valor CB ponatur pro valore abscissae in tequatione curvte figurte I ; acquiritur altera
tequatio per valores CK, CB, sive per x,u, y constantes. Eodem facto invenietur ope tequationis curvte figure I tertia
tequatio per AC, & CL, adeoque per x, z, y constantes. Quare jam babebuntur tequationes tres per x,u,z,y, y con-
stantes, qute, eliminates u, y z, reducentur ad unicam per x,y, & constantes, ac ea primam illam curvam definiet.

Quod si queeratur' tequatio ad secundam curvam, cujus ordinata est CO, vel tertiam, cujus ordinata OF,

T>p

inveniri itidem poterit. Nam datur analytice sinus anguli DCB = ™, W *» trianguk FCK datur analytice

\sD
•pTT

sinus FCK ===- X sin CKF. Quare datur analytice etiam sinus
Cr

differentite OCF, adeoque & ejus cosinus, & inde, ac ex CF datur
analytice OF, vel CO. Sz igitur altera ex illis dicatur p, acquiri-
tur nova tequatio, cujus ope una cum superioribus eliminari
poterit pristerea una alia indeterminata ; adeoque eliminata
CF =y, habebitur unica tequatio per x,p, y constantes, qua
exhibebit utramlibet e reliquis curvis determinantibus legem
virium CO, vel OF.

Pro tequatione cum binis indeterminatis, quts exhibebit locum-
ad superficiem, ducatur CR perpendicularis ad AB, y dicatur
DR —x, RC = q, denominatis, ut prius, CK =«, CL = z,
CF = v ; y quoniam dantur AD, DB ; dabuntur analytice per x,
y constantes AR, RB, adeoque per x, q, & constantes AC, CB, W
factis omnibus reliquis, ut prius, kabebuntur quatuor tequationes
per x,q,u,z,y,p, y constantes, qute eliminatis valoribus u,z,y,
reducentur ad unicam datam per constantes, y tres indeterminatas
x,p,q, sive DR, RC, y CO, vel OF, qute exhibebit qutesitum
locum ad superficiem.

Calculus quidem esset immensus, sed patet methodus, qua deveniri possit ad tequationem qutesitam. Mirum autem,
quanta curvarum, y superficierum, adeoque y legum virium varietas obvenerit, mutata tantummodo distantia AB binorum
punctorum agentium in tertium, qua mutata, mutatur tola lex, y tequatio.

(o) Htec conditio punctorum jacentium in eodem piano necessaria fuit pro loco ad superficiem, y pro tequatione, qute
legem virium exhibeat per tequationem indeterminatarum tantummodo trium : at si puncta sint plura, y in eodem piano
non jaceant, quod punctis tantummodo tribus accidere omnino non potest ; turn vero locus ad superficiem, y tequatio trium
indeterminatarum non sufficit, sed ad earn generaliter exprimendam legem Geometria omnis est incapax, y analysis indiget
tequatione indeterminatarum quatuor. Primum patet ex eo, quod si manentibus punctis A, B, exeat punctum C ex data
quodam piano, pro quo constructus sit locus ad superficiem ; liceret converters circa rectam AB planum illud cum superficie
curva legem virium determinate, donee ad punctum C deveniret planum ipsum : turn enim erecto perpendiculo usque ad
superficiem illam curvam, definiretur per ipsum vis agens secundum rectam CD, vel ipsi perpendicularis, prout locus ille
ad curvam superficiem constructus fuerit pro altera ex iis.

A THEORY OF NATURAL PHILOSOPHY 159

be situated ; we should have to erect at every point C normals to the plane ACB, one of
them equal to CO & the other to OF. The ends of these normals would determine two
continuous surfaces ; & of these, the one would represent the forces in the direction CD,
attractive or repulsive with respect to the point D, according as the normal was erected
above or below this plane, whether C fell on the near side or on the far side of D ; &
similarly the other would represent the perpendicular forces. A geometrical locus of this
kind, if it has to be treated algebraically, is such as geometricians deal with by means of
three unknowns connected together by a single equation ; &, if the equation to the primary
curve of Fig. i is given, it would in all cases be possible to find, not only the equations to
the two curves corresponding to each & every straight line DC, involving only two unknowns,
but also the equations for both the surfaces corresponding to the general determination,
by means of three unknowns. («)

209. If instead of only two points acting upon a third we are given any number of The method of
points situated in given positions, & acting on the same point, it would be possible, by a force ""compounded
similar construction in each case, to find the force, with which each acts on the point from the forces due
situated in any chosen position ; & the force compounded from forces of this kind would points7 The1 great
be determined, both in position & magnitude, by the well-known method for composition £"]^sr & variety
of forces. Also analysis could be employed to represent the curves by equations involving
two unknowns for any straight lines ; & (») provided that all the points were in the same
plane, the surface could be represented by an equation involving three unknowns. But
it is marvellous what a huge number of different laws arise. But, indeed, it is incredible,
even when there are only two points acting on a third, how great a number of different
laws & curves are produced in this way. Even if the distance AB remains the same, the
laws with respect to different inclinations of the straight line CD to the straight line AB,
come out quite different to one another. But when the distance of the points A & B from

(n) In Fig. 22, let the -points A,D,B,C,K,F,L,O be in the same positions as in Fig. 21, y let BP, AQ be drawn
•perpendicular to CD ; then these will be known, if the inclination of CD y the positions of A y B are known: y
so also will DP W DQ be known. Further, suppose DC = x, then CQ y CP will be given analytically. Hence on
account of the right angles at P y Q, CB y CA will also be given analytically. Suppose CK = w, CL = z, CF = y.
Since AB is known, y AC, CB are given analytically, by an application of algebra to trigonometry, the sine of the
angle ACB is also known analytically in terms of x y known quantities ; y this is the same thing as the sine of
the supplementary angle CKF. Moreover the same thing will be given in terms of the known analytical values of
3K = u, KF = CL = z, CF = y. Hence there is obtained in this case an equation involving x,y,z,u, y constants.
If, in addition, the value CB is substituted for the value of the abscissa in the equation of the curve in Fig. I, another
equation will be obtained in terms of the values of CK, CB, i.e. in terms of x, u, y constants. In a similar way by the help
of the equation of the curve of Fig. I, there can be found a third equation in terms of AC y CL, i.e., in terms of
#,z, y constants. Now, snce there will be' thus obtained three equations in terms of x,y,z,u, y constants, these, on
eliminating u,z, will reduce to a single equation involving x,y, y constants ; y this will be the equation defining
the first curve.

Again, «/ the equation to the second curve is required, of which the ordinate is CO, or of a third curve for which
the ordinate is CF, it will be possible to find either of these as well. For the sine of the angle DCB is analytically
given, being equal to BP/CB ; y from the triangle FCK, the sine of the angle FCK is given, being equal to
«'»CKF.(FK/CF). There fore the sine of the difference OOP is also given analytically, y therefore also its cosine; y
from this y the value of CF, the value of OF or CO will be given analytically. If then one or the other of them is
denoted by p, a new equation will be obtained: by the help of this y one of the equations given above, another of the
unknowns can be eliminated. If then, we eliminate CF = y, a single equation will be obtained in terms of x,p, y
constants, which will be that of one or other of the remaining curves determining the law of forces for CO or OF.

For an equation in three unknowns, which will represent the surface, draw CR perpendicular to AB, y let DR=#
RC = q ; y, as before, let CK = it, CL = z, CF = y. Then, since AD, DB are given, AR y RB are also given
analytically in terms of x y constants : y therefore AC y CB are given in terms of x,q, y constants : y if all
the rest of the work is done as before, four equations will be obtained in terms of x,q,u,z,y,p, y constants. These, on
eliminating the values u,z,y, will reduce to a single equation in terms of constants y the three unknowns x,p,q, or DR,
RC, y CO or OF ; this equation will represent the surface required.

The calculation would indeed be enormous ; but the method, by which the required equation might be obtained is
perfectly clear. But it is wonderful what a great number of curves y surfaces, y therefore of laws of force, would be
met with, if merely the distance between A y B, the two points which act upon the third, is changed ; for if this
alone is changed, the whole law is altered y so too is the equation.

(o) This condition, that the points should all lie in the same plane, is necessary for the determination of the surface,
y for the equation, which will express the law of the forces by an equation involving only three unknowns. If the points
are numerous, y they do not all lie in the same plane (which is quite impossible in the case of only three points), then
indeed a surface locus, y an equation in three unknowns, will not be sufficient; indeed, to express the law generally,
the whole of geometry is powerless, y analysis requires an equation in four unknowns. The first point is clear from
the fact that if, whilst the points A y B remain where they were, the point C moves out of the given plane, with
regard to which the construction for the surface locus was made, it would be right to rotate about the straight line AB
that plane together with its curved surface, which determines the law of forces, until the plane passes through the point
C. For then, if a perpendicular is drawn to meet the curved surface, this would define the force acting along the
straight line CD, or perpendicular to it, according as the locus to the curved surface had been constructed for the one
or for the other of them.

160

PHILOSOPHIC NATURALIS THEORIA

a se invicem, leges etiam pertinentes ad eandem inclinationem DC differunt inter se
plurimum ; & infinitum esset singula persequi ; quanquam earum variationum cognitio,
si obtineri utcunque posset, mirum in modum vires imaginationis extenderet, & objiceret
discrimina quamplurima scitu dignissima, & maximo futura usui, atque incredibilem
Theoriae foecunditatem ostenderet.

distantiis1 2IO> ^8° ^c simpliciora quaedam, ac faciliora, & usum habitura in sequentibus, ac in

ac ejus usus pro applicatione ad Physicam inprimis attingam tantummodo ; sed interea quod ad generalem
nu'iiaVinUs s^mma Pertinet determinationem expositam, duo adnotanda proponam. Primo quidem in ipsa
virium simpiicium. trium punctorum combinatione occurrit jam hie nobis praeter vim determinantem ad
accessum, & recessum, vis urgens in latus, ut in fig. 21, praeter vim CF, vel CH, vis CI,
vel CG. Id erit infra magno usui ad explicanda solidorum phaenomena, in quibus,
inclinato fundo virgse solidae, tola virga, & ejus vertex moventur in latus, ut certam ad
basim positionem acquirant. Deinde vero illud : haec omnia curvarum, & legum discrimina
tam quae [97] pertinent ad diversas directiones rectarum DC, data distantia punctorum
A, B, quam quae pertinent ad diversas distantias ipsorum punctorum A, B, data etiam
directione DC, ac hasce vires in latus haberi debere in exiguis illis distantiis, in quibus
curva figurae I circa axem contorquetur, ubi nimirum mutata parum admodum distantia,
vires singulorem punctorum mutantur plurimum, & e repulsivis etiam abeunt in attractivas,
ac vice versa, & ubi respectu alterius puncti haberi possit attractio, respectu alterius repulsio,
quod utique requiritur, ut vis dirigatur extra angulum ACB, & extra ipsi ad verticem
oppositum. At in majoribus distantiis, in quibus jam habetur illud postremum crus
figurae I exprimens arcum attractivum ad sensum in ratione reciproca duplicata distantiarum,
vis in punctum C a punctis A, B inter se proximis, utcunque ejusmodi distantia mutetur,
& quaecunque fuerit inclinatio CD ad AB, erit semper ad sensum eadem, directa ad sensum
ad punctum D, ad sensum proportionalis reciproce quadrato distantiae DC ab ipso puncto
D, & ad sensum dupla ejus, quam in curva figurae i requireret distantia DC.

At secundum sit manifestum ex eo, quod si puncta agenda sint etiam omnia in eodem piano, y punctum, cufus vis
composita quteritur, in quavis recta posita extra ipsum planum, relationes omnes distantiarum a reliquis punctis, ac
directionum, a quibus pendent vires singulorum, y compositio ipsarum virium, longe alia essent, ac in quavis recta in eodem
piano posita, uti facile videre est. Hinc pro quovis puncto loci ubicunque assumpto sua responderet vis composita, y quarta
aliqua plaga, seu dimensio, prater longum, latum, & profundum, requireretur ad ducendas ex omnibus punctis spatii rectas
• Us viribus proportionales, quarum rectarum vertices locum continuum aliqucm exhiberent determinantem virium legem.

Sed quod Geometria non assequitur, assequeretur quarta alia dimensio mente concepta, ut si conciperetur spatium totum
plenum materia continua, quod in mea sententia cogitatione tantummodo effingi potest, W ea esset in omnibus spatii punctis
densitatis diverse, vel diversi pretii ; turn ilia diversa densitas, vel illud pretium, vel quidpiam ejusmodi, exhibere posset
legem virium ipsi respondentium, ques nimirum ipsi essent proportionales. Sed ibi iterum ad determmandam directtonem
vis composite non esset satis resolutio in duas vires, alteram secundum rectam transcuntem per datum punctum ; altcram
ipsi perpendicularem ; ed requirerentur tres, nimirum vel omnes secundum tres datas directiones, vel tendentes per rectas,
qua per data tria puncta transeant, vel quavis alia certa lege definitas : adeoque tria loca ejusmodi ad spatium, quarta
aliqua dimensione, vel qualitate affectum requirerentur, qu<e tribus ejusmodi plusquam Geometricis legibus vis composite
legem definirent, turn quod pertinet ad ejus magnitudinem, turn quod ad directionem.

quod non assequitur Geometria, assequeretur Analysis ope aquationis quatuor indeterminatarum : si enim
planum, quod libuerit, ut ACB, y in eo quavis recta AB, ac in ipsa recta quodvis punitum D ; turn quovis

Ferum

conciperetur . M

hujus segmento DR appellate x, quavis recta RC ipsi perpendiculari y, quavis tertia perpendicular! ad totum planum z,
per hasce tres indeterminatas involveretur positio puncti spatii cujuscumque, in quo collocatum esset punctum materiel,
cufus vis quteritur.

Punctorum agentium utcunque collocatorum ubicunque vel intra id planum, vel extra, possent definiri positiones per
ejusmodi tres rectas, datas utique pro singulis, si eorum positiones dentur. Per eas, y per illas x,y,z, posset utique haberi
distantia cujuscumque ex Us punctis agentibus, y positione datis, a puncto indefinite accepto ; adeoque ope aquationis
figurtz I posset haberi analytice per aquationes quasdam, ut supra, vis ad singula agentia puncta pertinens, y per easdem
rectas ejus etiam directio resoluta in tres parallelas illis x,y,z. Hinc haberetur analytice omnium summa pro singulis
ejusmodi directionibus per aliam aquationem derivatam ab ejus summa denominatione, ea nimirum facia = u, ac expunctis
omnibus subsidiariis valoribus, methodo non absimili ei, quam adhibuimus superius pro loco ad superficiem, deveniretur ad
unam aquationem constitutam illis quatuor indeterminatis x,y,z,u, y constantibus ; ac tres ejusmodi aquationes pro tribus
directionibus vim omnem compositam definirent. Sed hac innuisse sit satis, qua nimirum y altiora sunt, y ob ingentem
complicationcm casuum, ac nostra humantf mentis imbecillitatem nulli nobis inferius futura sunt usui.

A THEORY OF NATURAL PHILOSOPHY 161

one another is also changed, the laws corresponding to the same inclination of DC are
altogether different to one another ; & it would be an interminable task to consider them
all, case by case. However, a comprehensive insight into their variations, if it could be
obtained, would enlarge the powers of imagination to a marvellous extent ; it would bring
to the notice a very large number of characteristics that would be well worth knowing &
most useful for further work ; & it would give a demonstration of the marvellous fertility
of my Theory.

210. First of all, therefore, I will here only deal slightly with certain of the more simple The lateral force at
cases, such as will be of use in what follows, & later when considering the application to tances, Tits use \n
Physics. But meanwhile, I will enunciate two theorems, applying to the general deter- t]}e consideration
mination set forth above, which should be noted. Firstly, in the case of the combination absence^ this

of three points, we have here already met with, in addition to a force inducing approach f?rce at great
. . ' . T-,. . IT. ,. X-,T-< ,"ITT ^-.T 2Z^~.rr distances, the sum

& recession, i.e., in rig. 21, in addition to a force CF or CH, a force CI or CG, urging of the simple forces

the point C to one side. This will be of great service to us in explaining certain phenomena in the latter case-

of solids ; for instance, the fact that, if the bottom of a solid rod is inclined, the whole

rod, including its top, is moved to one side & takes up a definite position with respect to

the base. Secondly, there is the fact that we are bound to have all these differences of

curves & laws, not only those corresponding to different directions of the straight lines DC

when the distance between the points A & B is given, but also those corresponding to

different distances of the points A & B when the direction of DC is given ; & that we are

bound to have these lateral forces for very small distances, for which the curve in Fig. I

twists about the axis ; for then indeed, if the change in distance is very slight, the change

in the forces corresponding to the several points is very great, & even passes from repulsion

to attraction & vice versa ; & also there may be attraction for one point & repulsion for

another ; & this must be the case if the direction of the force has to be without the angle

ACB, or the angle vertically opposite to it. But, at distances that are fairly large, for

which we have already seen that there is a final branch of the curve of Fig. i that represents

attraction approximately in the ratio of the inverse square of the distance, the force on the

point C, due to two points A & B very near to one another, will be approximately the

same, no matter how this distance may be altered, or what the inclination of CD to AB

may be ; its direction is approximately towards D ; & its magnitude will be approximately

in inverse proportion to the square of DC, its distance from the point D ; that is to say, it

will be approximately double of that to which in Fig. I the distance DC would correspond.

The second point is evident from the fact that, if all the points acting are all in the same plane, £5? the point for
which the resultant farce is required, lies in any straight line situated without that plane, even then all the relations
between the distances from the remaining points as well as between their directions, will be altogether different from
those for any straight line situated in the same plane, as can be easily seen. Hence, for any point of space chosen at
random there would be a corresponding force ; W a fourth region, or dimension, in addition to length, breadth, & depth,
would be required, in order to draw through each point of space straight lines proportional to these forces, the ends of
which straight lines would give a continuous locus determining the law for the forces.

But 'what can not be attained by the use of geometry, could be attained, by imagining another, a fourth, dimension
(just as if the whole of space were imagined to be full of eontinuous matter, which in my opinion can only be a mental
fiction) ; W this would be of different density, or different value, at all points of space. Then the different density, or
value, or something of that kind, might represent the law of forces corresponding to it, these indeed being proportional
to it. But here again, in order to find the direction of the resultant force, resolution into two forces, the one along the
straight line passing through the given -point, y the other perpendicular to it, would, not be sufficient. Three resolved
parts would be required, either all in three given directions, or along straight lines passing through three given points,
or defined by some other fixed law. Thus, three regions of this kind in space possessed of some fourth dimension or quality
would be required ; y these would define, by three ultra-geometrical laws of this sort, the law of the resultant force
both as regards magnitude & direction.

But what cannot be obtained with the help of geometry could be obtained by the aid of analysis by employing an
equation with four unknowns. For, if we take any arbitrary plane, as ACB, y in it any straight line AB, y in this
straight line any point D / then, calling any segment of it x, any straight line perpendicular to it y, y any third
straight line perpendicular to the whole plane z, there would be contained in these three unknowns the position of any
point in space, at which is situated a point of matter, for which the force is required.

The positions of the acting points, however W wherever they may be situated, either within the plane or without
it, could be defined by three straight lines of this sort ; y these would in all cases be known for each point, if the positions
of the points are given. By means of these, y the former straight lines denoted by x,y,z, there could he obtained in
all cases the distance of each of the acting points, that are given in position, from any point assumed indefinitely. Thus
by the help of the equation to the curve of Fig. I, there could be obtained analytically, by means of certain equations
similar to those above, the force corresponding to each of the acting points ; also from the same straight lines, its
direction as well, by resolving along three parallels to x, y, y z. Hence there could be obtained analytically the sum
of all of them for each of these directions, by means of another equation derived from the symbol used for the sum (for
instance, let this be called u] ; y, eliminating all the subsidiary values, by a method not unlike that which was used
above for the surface locus, we should arrive at a single equation in terms of the four unknowns, x, y, z, u, y constants.
Three equations of this sort, one for each of the three directions, would determine the resultant force completely. But
let it suffice merely to have mentioned these things ; for indeed they are too abstruse, y, on account of the enormous
Complexity of cares, y the disability of the human intelligence, will not be of any use to us later.

1 62

PHILOSOPHIC NATURALIS THEORIA

Demonstratio post-

remi theorematis.

2u. Id quidem facile demonstratur. Si enim AB respectu DC sit perquam exigua,

, . ,-,« ^ . . /-.T^ i i • r • i • A n

anguius AL.D erit perquam exiguus, & a recta CL) ad sensum bitanam sectus : distantias AC,
CB erunt ad se invicem ad sensum in ratione sequalitatis, adeoque & vires CL, CK ambae
attractive debebunt ad sensum aequales esse inter se, & proinde LCKF ad sensum rhombus,
diametro CF ad sensum secante angulum LCK bifariam, quae rhombi proprietas est, &
ipsa CF congruente cum CO, ac (ob angulum FCK insensibilem, & CKF ad sensum
aequalem duobus rectis) aequali ad sensum binis CK, KF, sive CK, CL, simul sumptis ;
quae singulae cum sint quam proxime in ratione reciproca duplicata distantiarum CB,
BA ; erunt & eadem, & earum summa ad sensum in ratione reciproca duplicata distantiae
CD.

suia

summa
tarum

massae,

ingens 212. Porro id quidem commune est etiam massulis constantibus quocunque punctorum

qu;is mas- _ _ -^ _ •... • • »••**«* •

exercet in numero. Mutata alarum combmatione, vis composita a vinbus singulorum agens in
oM- punctum distans a massula ipsa per intervallum perquam exiguum, nimirum ejusmodi,
in remo- in quo curva figurae I circa axem contorquetur, debet mutare plurimum tarn intensitatem
^us^ quse suanij quam directionem, & fieri utique potest, quod infra etiam in aliquo simpliciore
& reciproce, casu trium punctorum videbimus, ut in alia combinatione punctorum massulae pro eadem
dlS distantia a medio repulsiones praevaleant, in alia attractiones, in alia oriatur vis in latus ad
perpendiculum, ac in eadem constitutione massulae pro diversis directionibus admodum
diversae sint vires pro eadem etiam distantia a medio. At in magnis illis distantiis, in
quibus singulorum punctorum vires jam attractive sunt omnes, & directiones, ob molem
massulae tarn exiguam respectu ingentis distantias, ad sensum conspirant, vis com-[98]
-posita ex omnibus dirigetur necessario ad punctum aliquod intra massulam situm, adeoque
ad sensum ejus directio erit eadem, ac directio rectae tendentis ad mediam massulam, &
aequabitur vis ipsa ad sensum summae virium omnium punctorum constituentium ipsam
massulam, adeoque erit attractiva semper, & ad sensum proportionalis in diversis etiam
massulis numero punctorum directe, & quadrate distantias a medio massulae ipsius reciproce ;
sive generaliter erit in ratione composita ex directa simplici massarum, & reciproca duplicata
distantiarum. Multo autem majus erit discrimen in exiguis illis distantiis, si non unicum
punctum a massula ilia solicitetur, sed massula alia, cujus vis componatur e singulis viribus
singulorum suorum punctorum, quod tamen in massula etiam respectu massulae admodum
remotae evanescet, singulis ejus punctis vires habentibus ad sensum aequales, & agentes
in eadem ad sensum directione ; unde net, ut vis motrix ejus massulae solicitatae, orta ab
actionibus illius alterius remotae massulae, sit ad sensum proportionalis numero punctorum,
quas habet ipsa, numero eorum, quae habet altera, & quadrate distantiae, quaecunque sit
diversa dispositio punctorum in utralibet, quicunque numerus.

Unde necessaria 213. Mirum sane, quantum in applicatione ad Physicam haec animadversio habitura

unTf^mitTsTn s^ usum '•> nam inde constabit, cur omnia corporum genera gravitatem acceleratricem
gravitate, differ- habeant proportionalem massae, in quam tendunt, & quadrato distantiae, adeoque in
a- superficie Terrae aurum, & pluma cum aequali celeritate descendant seclusa resistentia, vim
autem totam, quam etiam pondus appellamus, proportionalem praeterea massae suae, adeoque
in ordine ad gravitatem nullum sit discrimen, quascunque differentia habeatur inter corpora,
quae gravitant, & in quae gravitant, sed ad solam demum massam, & distantiam res omnis
deveniat ; at in iis proprietatibus, quae pendent a minimis distantiis, in quibus nimirum
fiunt reflexionis lucis, & refractiones cum separatione colorum pro visu, vellicationes fibrarum
palati pro gustu, incursus odoriferarum particularum pro odoratu, tremor communicatus
particulis aeris proximis, & propagatus usque ad tympanum auriculare pro auditu, asperitas,
ac aliae sensibiles ejusmodi qualitates pro tactu, tot cohaesionum tarn diversa genera,
secretiones, nutritionesque, fermentationes, conflagrationes, displosiones, dissolutiones.
prascipitationes, ac alii effectus Chemici omnes, & mille alia ejusmodi, quae diversa corpora
a se invicem discernunt, in iis, inquam, tantum sit discrimen, & vires tarn variae, ac tarn

A THEORY OF NATURAL PHILOSOPHY 163

211. The latter theorem can be easily demonstrated. For, if AB is very small compared S[oof of thelattcr
with DC, the angle ACB will be very small, & will be very nearly bisected by the straight

line CD. The distances AC, CB will be approximately equal to one another ; & thus
the forces CL, CK, which are both attractive, must be approximately equal to one another.
Hence, LCKF is approximately a rhombus, & the diagonal CF very nearly bisects the
angle LCK, that being a property of a rhombus ; CF will fall along CO, &, because the
angle FCK is exceedingly small & CKF very nearly two right angles, CF will be very
nearly equal to CK & KF, or CK & CL, taken together. Now each of these are as
nearly as possible in the inverse ratio of the square of the distances CB, CA ; & these will
be the same, & their sum therefore approximately inversely proportional to the square
of the distance DC.

212. Further this theorem is also true in general for little masses consisting of points, J.*iere is .a hufe

,. T-,, , jjf it- difference in the

whatever their number may be. Ine force compounded from the several forces acting forces which a small
on a point, whose distance from the mass is very small, i.e., such a distance as that for which, mass, exerts on, a

™r .-11 7 11 i T i i • • small mass that

in Fig. i, the curve is twisted about the axis, must be altered very greatly if the combination is very near to it;

of the points is altered ; & this is so, both as regards its intensity, & as regards its direction. possibie^nrformH*

It may even happen, as will be seen later in the more simple case of three points, that in in the forces due

one combination of the points forming the little mass, & for one & the same distance from these'var6 dlrectf'

the mean point, repulsions will preponderate, in another case attractions, & in another as the masses, &

case there will arise a perpendicular lateral force. Also for the same constitution of the squaref'of the dls6

mass, for the same distance from the mean point, there may be altogether different forces tances.

for different directions. But, for considerable distances, where the forces due to the several

points are now attractive, & their directions practically coincide owing to the dimensions

of the little mass being so small compared with the greatness of the distance, the force

compounded from all of them will necessarily be directed towards some point within the

mass itself ; & thus its direction will be approximately the same as the straight line drawn

through the mean centre of the mass ; & the force itself will be equal approximately to

the sum of all the forces due to the points composing the little mass. Hence, it will always

be an attractive force ; & in different masses, it will be approximately proportional to the

number of points directly, & to the square of the distance from the mean centre of the mass

inversely. That is, in general, it will be in the ratio compounded of the simple direct

ratio of the masses & the inverse duplicate ratio of the distances. Further, the differences

will be far greater, in the case of very small distances, if not a single point alone, but

another mass, is under the action of the little mass under consideration ; for in this case,

the force is compounded from the several forces on each of the points that constitute it ;

& yet these differences will also disappear in the case of a mass acted on by a mass considerably

remote from it, since each of the points composing it is under the influence of forces that

are approximately equal & act in practically the same direction. Hence it comes about that

the motive force of the mass acted upon, which is produced by the action of the other

mass remote from it, is approximately proportional to the number of points in itself, to

the number of points in the other mass, & to the square of the distance between them,

whatever the difference in the disposition of the points, or their number, may be for either

mass.

213. It is indeed wonderful what great use can be made of this consideration in the Heice we have
application of my Theory to Physics ; for, from it it will be clear why all classes of bodies bodies? uniformity
have an accelerating gravity, proportional to the mass on which they act, & to the square m ^e c*se ?f
of the distance [inversely] ; & hence that, on the surface of the Earth, a piece of gold & a uniformity in the
feather will descend with equal velocity, when the resistance of the air is eliminated. It cafe of numerous
will be clear also that the whole force, which we call the weight, is in addition proportional

to the mass itself ; & thus, without exception, there is no difference as regards gravity,
no matter what difference there may be between the bodies which gravitate, or towards
which they gravitate ; the whole matter reducing finally to a consideration of mass &
distance alone. However, for those properties that depend on very small distances, for
instance, where we have reflection of light, & refraction with separation of colours, with
regard to sight, the titillation of the nerves of the palate, with regard to taste, the inrush
of odoriferous particles where smell is concerned, the quivering motion communicated to
the nearest particles of the air & propagated onwards till it reaches the drum of the ear
for sound, roughness & other such qualities as may be felt in the case of touch, the large
number of kinds of cohesion that are so different from one another, secretion, nutrition,
fermentation, conflagration, explosion, solution, precipitation, & all the rest of the
effects met with in Chemistry, & a thousand other things of the same sort, which
distinguish different bodies from one another ; for these, I say, the differences become
as great, the forces and the motions become as different, as the differences in the phenomena,

1 64

PHILOSOPHISE NATURALIS THEORIA

vis in duo puncta

puacti positi in

recta jungente
ipsa. vei in recta

secante hanc bi-

fariam, & ad angu-
los rectos directa

secundum eandem

rectam.

varii motus, qui tarn varia phaenomena, & omnes specificas tot corporum differentias
inducunt, consensu Theoriae hujus cum omni Natura sane admirabili. Sed hsec, quas
hue usque dicta sunt ad massas pertinent, & ad amplicationem ad Physicam : interea
peculiaria quaedam persequar ex innumeris iis, quas per-[99]-tinent ad diversas leges binorum
punctorum agentium in tertium.

214. Si libeat considerare illas leges, quas oriuntur in recta perpendiculari ad AB

, T _ . . , _. . . . ° • T- . . ...... r r. , ... , ,.

ducta per D, vel m ipsa AB hmc, & inde producta, mprimis facile est videre mud, direc-
tionem vis compositas utrobique fore eandem cum ipsa recta sine ulla vi in latus, & sine ulla

. . . " , . . . _ , *, . . _

declinatione a recta, quas tendit ad ipsum D, vel ab ipso. Pro recta AB res constat per
sese . nam vjres Qjgg qU33 ad bina ea puncta pertinent, vel habebunt directionem eandem,

, • t » *« . . r ,. • -1 j

vel oppositas, jacente ipso tertio puncto in directum cum utroque e pnonbus : unde
fit, ut vis composita asquetur summae, vel differentias virium singularum componentium,
quae in eadem recta remaneat. Pro recta perpendiculari facile admodum demonstratur.
Si enim in fig. 23 recta DC fuerit perpendicularis ad AB sectam bifariam in D, erunt AC,
BC aequales inter se. Quare vires, quibus C agitatur ab A, & B, sequales erunt, & proinde
vel ambae attractivae, ut CL, CK, vel ambae repulsivae, ut CN, CM. Quare vis composita
CF, vel CH, erit diameter rhombi, adeoque secabit bifariam angulum LCK, vel NCM ;
quos angulos cum bifariam secet etiam recta DC, ob asqualitatem triangulorum DCA, DCB,
patet, ipsas CF, CH debere cum eadem congruere. Quamobrem in hisce casibus evane-
scit vis ilia perpendicularis FO, quae in prsecedentibus binis figuris habebatur, ac in iis
per unicam aequationem res omnis absolvitur (f), quarum ea, quae ad posteriorem casum
pertinet, admodum facile invenitur.

exhfbentis°
casus posterioris.

2I5' ^egem pro recta perpendiculari rectae jungenti duo puncta, & asque distanti ab
utroque exhibet fig. 24, quse vitandae confusionis causa exhibetur, ubi sub numero 24
habetur littera B, sed quod ad ejus constructionem pertinet, habetur separatim, ubi sub
num. 24 habetur littera A ; ex quibus binis figuris fit unica ; si puncta XYEAE' censeantur
utrobique eadem. In ea X, Y sunt duo materiae puncta, & ipsam XY recta CC* secat
bifariam in A. Curva, quae vires compositas ibi exhibet per ordinatas, constructa est ex
fig. I, quod fieri potest, inveniendo vires singulas singulorum punctorum, turn vim com-
positam ex iis more consueto juxta [100] generalem constructionem numeri 205 ; sed
etiam sic facilius idem praestatur ; centro Y intervallo cujusvis abscissae Ad figurae I in-
veniatur in figura 24 sub littera A in recta CC' punctum d, sumaturque de versus Y
aequalis ordinatae dh figuras i , ductoque ea perpendiculo in CA, erigatur eidem CA itidem
perpendicularis dh dupla da versus plagam electam ad arbitrium pro attractionibus, vel
versus oppositam, prout ilia ordinata in fig. I attractionem, vel repulsionem expresserit,
& erit punctum h ad curvam exprimentem legem virium, qua punctum ubicunque
collocatum in recta C'C solicitatur a binis X, Y.

de

proprietates. °

Demonstratio facilis est : si enim ducatur dX, & in ea sumatur dc aequalis de,
ac compleatur rhombus debc ; patet fore ejus verticem b in recta dA secante angulum
XdY bifariam, cujus diameter db exprimet vim compositam a binis de, dc, quae bifariam
secaretur a diametro altera ec, & ad angulos rectos, adeoque in ipso illo puncto a ; & dh,
dupla da, aequabitur db exprimenti vim, quae respectu A erit attractiva, vel repulsiva, prout
ilia dh figurae I fuerit itidem attractiva, vel repulsiva.

2I7- Porro ex ipsa constructione patet, si centro Y, intervallis AE, AG, AI figuras i
inveniantur in recta CAC' hujus figurae positae sub littera B puncta E, G, I, &c, ea fore
limites respectu novae curvas ; & eodem pacto reperiri posse limites E', G', Y, &c. ex parte
opposita A ; in iis enim punctis evanescente de figuras ejusdem positae sub A, evadit nulla
da, & db. Notandum tamen, ibi in figura posita sub B mutari plagam attractivam in

(p) Ducta enim LK in Fig. 23. ipsam FC secabit alicubi in I bifariam, W ad angulos rectos ex rhombi natura.

Dicatur CD = x, CF = y, DB = a, W erit CB = Vaa + xx, fcf CD = *.CB = Vaa + xx : : CI = Jy.CK =—

\/aa + xx, quo valore posito in tequatione curvie figura I pro valore ordinata, y vaa + xx ffo valore abscissa, habebitur
immediate cfquatio nova per x, y, W constants, qua ejusmodi curvam deUrminabit,

A THEORY OF NATURAL PHILOSOPHY

165

0

O

0

i66

PHILOSOPHIC NATURALIS THEORIA

o

^>

A THEORY OF NATURAL PHILOSOPHY 167

& all the specific differences between the large number of bodies which they yield ; the
agreement between the Theory & the whole of Nature is truly remarkable. But what
has so far been said refers to masses, & to the application of the Theory to Physics. Before
we come to this, however, I will discuss certain particular cases, out of an innumerable
number of those which refer to the different laws concerning the action of two points on
a third.

214. If we wish to consider the laws that arise in the case of a straight line drawn The force exerted
through D perpendicular to AB, or in the case of AB itself produced on either side, first by. two points on a
of all it is easily seen that the direction of the resultant force in either case will coincide the?* straight*1 line
with the line itself without any lateral force or any declination from the straight line which Joining them, or in
is drawn towards or away from D. In the case of AB itself the matter is self-evident ; whicifblsects it'at
for the forces which pertain to the two points either have the same direction as one another, "8ht angles.

or are opposite in direction, since the third point lies in the same straight line as each of
the two former points. Whence it comes about that the resultant force is equal to the
sum, or the difference, of the two component forces ; & it will be in the same straight
line as they. In the case of the line at right angles, the matter can be quite easily
demonstrated. For, if in Fig. 23 the straight line DC were perpendicular to AB, passing
through its middle point, then will AC, BC be equal to one another. Hence, the forces,
by which C is influenced by A & B, will also be equal ; secondly, they will either be both
attractive, as CL, CK, or they will be both repulsive, as CN, CM. Hence the resultant
force, CF, or CH, will be the diagonal of a rhombus, & thus it will bisect the angle LCK,
or NCM. Now since these angles are also bisected by the straight line DC, on account
of the equality of the triangles DCA, DCB, it is evident that CF, CH must coincide with
DC. Therefore, in these cases the perpendicular force FO, which was obtained in the
two previous figures, will vanish. Also in these cases, the whole matter can be represented
by a single equation (?) ; & the one, which refers to the latter case, can be found quite
easily.

215. The law in the case of the straight line perpendicular to the straight line joining Construction for
the two points, & equally distant from each, is graphically given in Fig. 24 ; to avoid *he curve . eiv^s

. . . ^ 1r.' . • -r- i -i i r • r • • • .the law in the

confusion the curve itself is given in rig. 243, whilst the construction for it is given separately second case.

in Fig. 24A. These two figures are but one & the same, if the points X,Y,E,A,E' are

supposed to be the same in both. Then, in the figure, X,Y are two points of matter, &

the straight line CC' bisects XY at A. The curve, which here gives the resultant forces

by means of the ordinates drawn to it, is constructed from that of Fig. i : & this can be

done, by finding the forces for the points, each for each, then the force compounded from

them in the usual manner according to the general construction given in Art. 205. But

the same thing can be more easily obtained thus :, — With centre Y, & radius equal to any

abscissa Ad in Fig. i, construct a point d in the straight line CC', of Fig. 24A, & mark off

de towards Y equal to the ordinate db in Fig. i ; draw ea perpendicular to CA, & erect

a perpendicular, dh, to the same line CA also, so that dh = 2ae ; this perpendicular should

be drawn towards the side of CA which is chosen at will to represent attractions, or towards

the opposite side, according as the ordinate in Fig. i represents an attraction or a repulsion ;

then the point h will be a point on the curve expressing the law of forces, with which a

point situated anywhere on the line CC' will be influenced by the two points X & Y.

. 216. The demonstration is easy. For, if dX is drawn, & in it dc is taken equal to de, Proof of the fore-
& the rhombus debc is completed, then it is clear that the point b will fall on the straight g°»ng construction,
line dA. bisecting the angle X/Y ; & the diagonal of this rhombus represents the resultant
of the two forces de, dc. Now, this diagonal is bisected at right angles by the other diagonal
ec, & thus, at the point a in it. Also dh, being double of da, will be equal to db, which
expresses the resultant force ; this will be attractive with respect to A, or repulsive, according
as the ordinate dh in Fig. I is also attractive or repulsive.

217. Further, from the construction, it is evident that, if with centre Y & radii Further properties
respectively equal to AE, AG, AI in Fig. i, there are found in the straight line CAC' of sort. °'
Fig. 248 the points E, G, I, &c, then these will be limit-points for the new curve ; &
that in the same way limit-points E', G', I', &c. may be found on the opposite side of A.
For, since at these points, in Fig. 24A, de vanishes, it follows that da & db become nothing
also. Yet it must be noted that, in this case, in Fig. 248, there is a change from the attractive

(p) For, if in Fig. 23, LK is drawn, it will cut FC somewhere, in I say ; & it will be at right angles to it
on account of the nature of a rhombus. Sup-pose CD = x, CF = y, DB = a ; then CB = •\/(az+ x2), W toe have

CD (or x) : CB (or ^(a* + x*) = CI (or Jy) : CK, /. CK = y.yV + x*)/2x ;

y if this value is substituted in the equation of the curve in Fig. I instead of the ordinate, W ^/ (az + xz) for the
abscissa, we shall get straightaway a new equation in x, y, \£ constants ; & Ms will determine a curve of the kind
under consideration.

1 68 PHILOSOPHIC NATURALIS THEORIA

repulsivam, & vice versa ; nam in toto tractu CA vis attractiva ad A habet directionem
CC', & in tractu AC' vis itidem attractiva ad A habet directionem oppositam C'C. Deinde
facile patebit, vim in A fore nullam, ubi nimirum oppositae vires se destruent, adeoque
ibi debere curvam axem secare ; ac licet distantiae AX, AY fuerint perquam exiguse, ut
idcirco repulsiones singulorum punctorum evadant maximse ; tamen prope A vires erunt
perquam exiguae ob inclinationes duarum virium ad XY ingentes, & contrarias ; & si ipsae
AY, AX fuerint non majores, quam sit AE figurae I ; postremus arcus EDA erit repulsivus ;
secus si fuerint majores, quam AE, & non majores, quam AG, atque ita porro ; cum vires
in exigua distantia ab A debeant esse ejus directionis, quam in fig. I requirunt abscissas
paullo majores, quam sit haec YA. Postrema crura T/>V,T"/>'V, patet, fore attractiva ;
& si in figura I fuerint asymptotica, fore asymptotica etiam hie ; sed in A nullum erit
asymptoticum crus.

2l8> At curva Cluae exhibet in fig. 25 legem virium pro recta CC' transeunte per duo
casus prioris. puncta X, Y, est admodum diversa a priore. Ea facile construitur : satis est pro quovis

ejus puncto d assumere in fig. I duas abscissae aequales, alteram Yd hujus figurae, alteram
Xd ejusdem, & sumere hie db aequalem [101] summae, vel differentiae binarum ordinatarum
pertinentium ad eas abscissas, prout fuerint ejusdem directionis, vel contrariae, & earn
ducere ex parte attractiva, vel repulsiva, prout ambae ordinatae figurae I, vel earum major,
attractiva fuerit, vel repulsiva. Habebitur autem asymptotus bYc, & ultra ipsam crus
asymptoticum DE, citra ipsam autem crus itidem asymptoticum dg attractivum respectu
A, cui attractivum, sed directionis mutatas respectu CC', ut in fig. superiore diximus, ad
partes oppositas A debet esse aliud g'd', habens asymptotum c'V transeuntem per X ;
ac utrumque crus debet continuari usque ad A, ubi curva secabit axem. Hoc postremum
patet ex eo, quod vires oppositae in A debeant elidi ; illud autem prius ex eo, quod si a
sit prope Y, & ad ipsum in infinitum accedat, repulsio ab Y crescat in infinitum, vi, quae
provenit ab X, manente finita ; adeoque tam summa, quam differentia debet esse vis
repulsiva respectu Y, & proinde attractiva respectu A, quae imminutis in infinitum distantiis
ab Y augebitur in infinitum. Quare ordinata ag in accessu ad bYc crescet in infinitum ;
unde consequitur, arcum gd fore asymptoticum respectu Yc ; & eadem erit ratio pro a'g',
& arcu g'd' respectu b'Xc'.

Ejus curvae pro- 219. Poterit autem etiam arcus curvae interceptus asymptotis bYc, b'Xc' sive cruribus

at mutata
puncto-

*

mutata ^S> ^'g' secare alicubi axem, ut exhibet figura 26 ; quin immo & in locis pluribus, si nimirum
distantia puncto- AY sit satis major, quam AE figurae i, ut ab Y habeatur alicubi citra A attractio, & ab X

curva casus*aiterius! repulsio, vel ab X repulsio major, quam repulsio ab Y. Ceterum sola inspectione
postremarum duarum figurarum patebit, quantum discrimen inducat in legem virium, vel
curvam, sola distantia punctorum X, Y. Utraque enim figura derivata est a figura I, & in
fig. 25 assumpta est XY sequalis AE figurae I, in fig. 26 aequalis AI, ejusdem quae variatio
usque adeo mutavit figurse genitae ductum ; & assumptis aliis, atque aliis distantiis punc-
torum X, Y, aliae, atque aliae curvae novae provenirent, quae inter se collatae, & cum illis,
quae habentur in recta CAC' perpendiculari ad XAY, uti est in fig. 24 ; ac multo magis
cum iis, quae pertinentes ad alias rectas mente concipi possunt, satis confirmant id, quod
supra innui de tanta multitudine, & varietate legum provenientium a sola etiam duo-
rum punctorum agentium in tertium dispositione diversa ; ut & illud itidem patet ex
sola etiam harum trium curvarum delineatione, quanta sit ubique conformitas in arcu illo
attractive TpV, ubique conjuncta cum tanto discrimine in arcu se circa axem contorquente.

genera hujus 220. Verum ex tanto discriminum numero unum seligam maxime notatu dignum,
Usima!0 'g & maximo nobis usui futurum inferius. Sit in fig. 2jC — 'AC axis idem, ac in fig. i, & quin-

que arcus consequenter accept! alicubi GHI, IKL, LMN, NOP, PQR sint aequales
prorsus inter se, ac similes. Ponantur autem bina puncta B', B hinc, & inde ab A in fig. 28
[102] ad intervallum aequale dimidiae amplitudini unius e quinque iis arcubus, uti uni
GI, vel IL ; in fig. 29 ad intervallum aequale integrae ipsi amplitudini ; in fig. 30 ad
intervallum aequale duplae ; sint autem puncta L, N in omnibus hisce figuris eadem, &
quaeratur, quae futura sit vis in quovis puncto g in intervallo LN in hisce tribus posi-
tionibus punctorum B', B.

A THEORY OF NATURAL PHILOSOPHY

169

1 7o

PHILOSOPHISE NATURALIS THEORIA

A THEORY OF NATURAL PHILOSOPHY 171

side to the repulsive side, & vice versa. For along the whole portion CA, the force of
attraction towards A has the direction CC', whilst for the portion AC', the force of attraction
also towards A has the direction C'C. Secondly, it will be clear,/ seen that the force at
A will be nothing ; for there indeed the forces, being equal & opposite, cancel one another,
& so the curve cuts the axis there ; & although the distances AX, AY would be very small,
& thus the repulsions due to each of the two points would be Immensely great, nevertheless,
close to A, the resultants would be very small, on account of the inclinations of the two
forces to XY being extremely great & oppositely inclined. Also if AY, AX were not greater
than AE in Fig. i, the last arc would be repulsive ; & attractive, if they were greater than
AE, but not greater than AG, & so on ; for the forces at very small distances from A must
have their directions the same as that required in Fig. I for abscissae that are slightly greater
than YA. The final branches TpV, T'p'V will plainly be attractive ; &, if in Fig. i they
were asymptotic, they would also be asymptotic in this case ; but there will not be an
asymptotic branch at A.

218. But the curve, in Fig. 25, which expresses the law of forces for the straight line Construction fo-
CC', when it passes through the points X,Y, is quite different from the one just considered. j£| the^aw^'tte
It is easily constructed ; it is sufficient, for any point d upon it, to take, in Fig. i, two first case,
abscissae, one equal to Yd, & the other equal to Xd ; & then, for Fig. 25, to take dh equal

to the sum or the difference of the two ordinates corresponding to these abscissas, according
as they are in the same direction or in opposite directions ; &, according as each ordinate,
or the greater of the two, in Fig. i, is attractive or repulsive, to draw dh on the attractive
or repulsive side of CC'. Moreover there will be obtained an atymptote bYc ; on the
far side of this there will be an asymptotic branch DE, & on the near side of it there will
also be an asymptotic branch dg, which will be attractive with respect to A ; & with respect
to this part, there must be another branch g'd', which is attractive but, since the direction
with regard to CC' is altered, as we mentioned in the case of the preceding figure, falling
on the opposite side of CC' ; this has an asymptote c'b' passing through X. Also each
branch must be continuous up to the point A, where it cuts thVaxis. This last fact is
evident from the consideration that the equal & opposite forces at A must cancel one another ;
& the former is clear from the fact that, if a is very near to Y, & approaches indefinitely
near to it, the repulsion due to Y increases indefinitely, whilst the force due to X remains
finite. Thus, both the sum & the difference must be repulsive with respect to Y, & therefore
attractive with respect to A ; & this, as the distance from Y is diminished indefinitely, will
increase indefinitely. Hence the ordinate ag, when approaching bYc, increases indefinitely :
& it thus follows that the arc gd will be asymptotic with respect to Yc ; & the reasoning
will be the same for a'g', & the arc g'd', with respect to b'Xc',

219. Again, it is even possible that the arc intercepted between the asymptotes bYc, The properties of
b'Xc', i.e., between the branches dg, d'g', to cut the axis somewhere, as is shown in Fig. 26 ; encescor^espo'n'dSg
nay rather, it may cut it in more places than one, for instance, if AY is sufficiently greater to changed dis-
than AE in Fig. i ; so that, at some place on the near side of A, there is obtained an attraction p^s ^"clrnpari6
from the point Y & a repulsion from the point X, or a repulsion from X greater than the son with the curve
repulsion fiom Y. Besides, by a mere inspection of the last two figures, it will be evident other case. in the
how great a difference in the law of forces, & the curve, may be derived from the mere

distance apart of the points X & Y. For both figures are derived from Fig. I, &, in Fig. 25,
XY is taken equal to AE in Fig. i , whilst, in Fig. 26, it is taken equal to AI of Fig. i ; &
this variation alone has changed the derived figure to such a degree as is shown. If other
distances, one after another, are taken for the points X & Y, fresh curves, one after the
other, will be produced. If these are compared with one another, & with those that are
obtained for a straight line CAC' perpendicular to XAY, like the one in Fig 24, nay, far
more, if they are compared with those, referring to other straight lines, that can be imagined,
will sufficiently confirm what has been said above with regard to the immense number &
variety of the laws arising from a mere difference of disposition of the two points that act
on the third. Also, from the drawing of merely these three curves, it is plainly seen
what great uniformity there is in all cases for the attractive arc TpV, combined always
with a great dissimilarity for the arc that is twisted about the axis.

_ 220. But I will select, from this great number of different cases, one which is worth T^Tee classes of
notice in a high degree, which also will be of the greatest service to us later. In Fig. 27, weu
let CAC' be the same axis as in Fig. i, & let the five arcs, GHI, IKL, LMN, NOP, PQR
taken consecutively anywhere along it, be exactly equal & like one another. Moreover,
in Fig. 28, let the two points B & B', one on each side of A, be taken at a distance equal
to half the width of one of these five arcs, i.e., half of the one GL, or LI ; in Fig. 29, at
3. distance equal to the whole of this width ; &, in Fig. 30, at a distance equal to double
the width ; also let the points L,N be the same in all these figures. It is required to find
the force at any point g in the interval LN, for these three positions of the points B & B'.

I72

PHILOSOPHISE NATURALIS THEORIA

Determinatip vis
compositaa in iis-
dem.

221. Si in Fig. 27 capiantur hinc, & inde ab ipso g intervalla sequalia intervallis AB',
AB reliquarum trium figurarum ita, ut ge, gi respondeant figurae 28 ; gc, gm figures 29 ;
ga, go figurae 30 ; patet, intervallum ei fore aequale amplitudini LN, adeoque Le, Ni
aequales fore dempto communi Lz, sed puncta e, i debere cadere sub arcus proximos
directionum contrariarum ; ob arcuum vero aequalitatem fore aequalem vim ef vi contrariae
il, adeoque in fig. 28 vim ab utraque compositam, respondentem puncto g, fore nullam.
At quoniam gc, gm integrae amplitudini aequantur ; cadent puncta c, m sub arcus IKL,
NOP, conformes etiam directione inter se, sed directionis contraries respectu arcus LMN,
eruntque asquales wzN, cl ipsi gL, adeoque attractiones mn, cd, & repulsioni gh aequales,
& inter se ; ac idcirco in figura 29 habebitur vis attractiva gh composita ex iis binis dupla
repulsivae figurae 27. Demum cum ga, go sint sequales duplae amplitudini, cadent puncta
a, o sub arcus GHI, PQR conformis directionis inter se, & cum arcu LMN, eruntque
pariter binae repulsiones ab, op aequales repulsioni gh, & inter se. Quare vis ex iis com-
positae pro fig. 30 erit repulsio gh dupla repulsionis gh figurae 27, & aequalis attraction!
figurae 29.

vhn'in tractu*'0116 222< ^^ igitur jam patet, loci geometric! exprimentis vim compositam, qua bina

tinuo nuiiam, in puncta B', B agunt in tertium, partem, quae respondet intervallo eidem LN, fore in prima
aha attractionem, e tribus eorum positionibus propositis ipsum axem LN, in secunda arcum attractivum

in aha repulsionem, T , ,,., . . , . i i • , r

manente distantia ; LMN, in tertia repulsivum, utroque reccdente ab axe ubique duplo plus, quam in fig.
Physica 27 ; ac pro quovis situ puncti g in toto intervallo LN in primo e tribus casibus fore prorsus
nullam, in secundo fore attractionem, in tertio repulsionem aequalem ei, quam bina puncta
B', B exercerent in tertium punctum situm in g, si collocarentur simul in A, licet in omnibus
hisce casibus distantia puncti ejusdem g a medio systematis eorundem duorum punctorum,
sive a centre particulae constantis iis duobus punctis sit omnino eadem. Possunt autem
in omnibus hisce casibus puncta B', B esse simul in arctissimis limitibus cohaesionis inter
se,- adeoque particulam quandam constantis positionis constituere. Aequalitas ejusmodi
accurata inter arcus, & amplitudines, ac limitum distantias in figura I non dabitur uspiam ;
cum nullus arcus curvae derivatae utique continuae, deductae nimirum certa lege a curva
continua, possit congruere accurate cum recta ; at poterunt ea omnia ad sequalitatem
accedere, quantum [103] libuerit ; poterunt haec ipsa discrimina haberi ad sensum per
tractus continues aliis modis multo adhuc pluribus, immo etiam pluribus in immensum,
ubi non duo tantummodo puncta, sed immensus eorum numerus constituat massulas,
quae in se agant, & ut in hoc simplicissimo exemplo deprompto e solo trium punctorum
systemate, multo magis in systematis magis compositas, & plures idcirco variationes admit-
tentibus, in eadem centrorum distantia, pro sola varia positione punctorum componentium
massulas ipsas vel a se mutuo repelli, vel se mutuo attrahere, vel nihil ad sensum agere in
se invicem. Quod si ita res habet, nihil jam mirum accidet, quod quaedam substantial
inter se commixtse ingentem acquirant intestinarum partium motum per effervescentiam,
& fermentationem, quas deinde cesset, particulis post novam commixtionem respective
quiescentibus ; quod ex eodem cibo alia per secretionem repellantur, alia in succum
nutrititium convertantur, ex quo ad eandem prseterfluente distantiam alia aliis partibus
solidis adhaereant, & per alias valvulas transmittantur, aliis libere progredientibus. Sed
adhuc multa supersunt notatu dignissima, quae pertinent ad ipsum etiam adeo simplex
trium punctorum systema.

Alius casus vis nul-
lius trium puncto-
rum positorum in
directum e x dis-
tantiis limitum :
tres alii in quorum
binis vis nulla ex
elisione contrari-
arum.

223. Jaceant in figura 31 tria puncta A,D,B, in directum : ea poterunt respective
quiescere, si omnibus mutuis viribus careant, quod fieret, si tres distantiae AD, DB, AB
omnes essent distantiae limitum ; sed potest haberi etiam quies respectiva per elisionem
contrariarum virium. Porro virium mutuarum casus diversi tres esse poterunt : vel enim
punctum medium D ab utroque extremorum A, B attrahitur, vel ab utroque repellitur,
vel ab altero attrahitur, ab altero repellitur. In hoc postremo casu, patet, non haberi
quietem respectivam ; cum debeat punctum medium moveri versus extremum attrahens
recedendo simul ab altero extremo repellente. In reliquis binis casibus poterit utique

A THEORY OF NATURAL PHILOSOPHY

173

c'

C

B'A B

FIG. 28.

C'

B' A B

FIG. 29.

cV

1

I

g

C

B'

T

B

L N

FIG. 30.

'74

PHILOSOPHIC NATURALIS THEORIA

H

R C

C'

FIG. 27.

Ln. ," 5

B'A B

FIG. 28.

B' A B

B7

FIG .29.

B L N

FIG. 30.

A THEORY OF NATURAL PHILOSOPHY 175

221. If, in Fig. 27, we take, on either side of this point g, intervals that are equal to Determination of
the intervals AB', AB of the other three figures ; so that ge, gi correspond to Fig. 28 ;

gc, gm to Fig. 29 ; & ga, go to Fig. 30 ; then it is plain that the interval ei will be equal
to the width LN, & thus, taking away the common part Lz, we have L£ & Ni equal to one
another, but the points e & i must fall under successive arcs of opposite directions. Now,
on account of the equality of the arcs, the force ef will be equal to the opposite force il ;
thus, in Fig. 28, the force compounded from the two, corresponding to the point g, will
be nothing. Again, in Fig. 29, since gc, gm are each equal to the whole width of an arc,
the points c & m fall under arcs IKL, NOP, which lie in the same direction as one another,
but in the opposite direction to the arc LMN. Hence, mN, c\ will be equal to gL ; &
thus the attractions mn, cd will be equal to the repulsion gb, & to one another. Therefore,
in Fig. 29, we shall have an attractive force, compounded of these two, which is double
of the repulsive force in Fig. 27. • Lastly, in Fig. 30, since ga, go are equal to double the
width of an arc, the points a & o will fall beneath arcs GHI, PQR, lying in the same direction
as one another, & as that of the arc LMN as well. As before, the two repulsions, ab, op
will be equal to the repulsion gb, & to one another. Hence, in Fig. 30, the force compounded
from the two of them will be a repulsion gh which is double of the repulsion gh in Fig. 27,
& equal to the attraction in Fig. 29.

222. Therefore, from the preceding article, it is now evident that the part of the in one arrange-

geometrical locus representing the resultant force, with which two points B', B act " '*

region
upon a third, corresponding to the same interval LN, will be the axis LN itself in the first no force at ail, in

of the three stated positions of the points ; in the second position it will be an attractive attraction"5* ^a

arc LMN, & in the third a repulsive arc ; each of these will recede from the axis at all third a repulsion

points along it to twice the corresponding distance shown in Fig. 27. So, for any position maining^constant •

of the point g in the whole interval LN, the force will be nothing at all in the first of the this result is of the

three cases, an attraction in the second, & a repulsion in the third. This latter will be -physics. l

equal to that which the two points B', B would exert on the third point, if they were both

situated at the same time at the point A. And yet, in all these three cases, the distance

of the point g under consideration remains absolutely the same, measured from the centre

of the system of the same two points, or from the mean centre of a particle formed from

them. Moreover, in all three cases, the points B',B may be in the positions defining the

strongest limits of cohesion with regard to one another, & so constitute a particle fixed

in position. Now we never can have such accurate equality as this between the arcs, the

widths, & the distances of the limit-points ; for no arc of the derived curve, which is every-

where continuous because it is obtained by a given law from a continuous curve, can possibly

coincide accurately with a straight line ; but there could be an approximation to equality

for all of them, to any degree desired. The same distinctions could be obtained,

approximately for continuous regions in very many more different ways, nay the number

of ways is immeasurable ; in which the number of points constituting the little masses

is not two only, but a very large number, acting upon one another ; &, as in this very simple

case derived from a consideration of a single system of three points, so, much more in systems

that are more complex & on that account admitting of more variations, corresponding to

a single variation of the points composing the masses, whilst the distance between the

masses themselves remains the same, there may be either mutual repulsion, mutual attraction,

or no mutual action to any appreciable extent. But, that being the case, there is

nothing wonderful in the fact that certain substances, when mixed together, acquire a

huge motion of their inmost parts, as in effervescence & fermentation ; this motion ceasing

& the particles attaining relative rest after rearrangement. There is nothing wonderful

in the fact that from the same food some things are repelled by secretion, whilst others

are converted into nutritious juices ; & that from these juices, though flowing past at

exactly the same distances, some things adhere to some solid parts & some to others ; that

some are transmitted through certain little passages, some through others, whilst some

pass along uninterruptedly. However, there yet remain many things with regard to this ever

so simple system of three points ; & these are well worth our attention.

223. In Fig. 31, let A,D,B be three points in a straight line. These will be at rest Another instance

• i 11 i n i • ., , of no force in the

with regard to one another if they lack all mutual forces ; & this would be the case, it the case of three points
three distances AD, DB, AB were all distances corresponding to limit-points. In addition, sitV?;,tf.d i"a

i • 111 i • i • ,...*-» in • f ft. straight line at the

relative rest could be obtained owing to elimination of equal & opposite iorces. .further, distances corre-
there will be three different cases with regard to the mutual forces. For, either the middle spending to Hmit-

. -i r i i • > T> •• 11 ii i r V. points. Ihree

point D is attracted by each of the outside points A & B, or it is repelled by each of them, others, in two of
or it is attracted by one of them & repelled by the other. In the last case, it is evident ^^res^tenl'farce
that relative rest could not obtain ; for the middle point must then be moved towards the arises from an eii-
outside point that is attracting it, & recede from the other outside point which is repelling
it at the same time. But in the other two cases, it is at least possible that there may be

PHILOSOPHIC NATURALIS THEORIA

In eorum altero
nisus ad recuper-
andam positionem,
in altero ad magis
ab ea recedendum,
si incipiant inde
removeri.

res haberi : nam vires attractive, vel repulsive, quas habet medium punctum, possunt
esse aequales ; turn autem extrema puncta debebunt itidem attrahi a medio in primo casu,
repelli in secundo ; quae si se invicem e contrario aeque repellant in casu primo, attrahant
in secundo ; poterunt mutuse vires elidi omnes.

224. Adhuc autem ingens est discrimen inter hosce binos casus. Si nimirum puncta
ilia a directione rectae lineae quidquam removeantur, ut nimirum medium punctum D
distet jam non nihil a recta AB, delatam in C, in secundo casu adhuc magis sponte recedet
inde, & in primo accedet iterum ; vel si vi aliqua externa urgeatur, conabitur recuperare
positionem priorem, & ipsi urgenti vi resi-
stet. Nam binae repulsiones CM, CN adhuc
habebuntur in secundo casu in ipso primo
recessu a D (licet ese mutatis jam satis distan-
tiis BD, AD inBC, AC, evadere possint at-
tractiones) & vim com-[i opponent direc-
tam per CH contrariam directioni tendenti
ad rectam AB. At in primo casu habebuntur
attractiones CL, CK, quae component vim
CF directam versus AB, quo casu attractio
AP cum repulsione AR, et attractio BV,
cum repulsione BS component vires AQ, BT,
quibus puncta A, B ibunt obviam puncto C
redeunti ad rectam transituram per illud

T"V /"*

FIG. 31.

Theoria generalior
indicata : t r i u m
punctorum jacen-
tium in directum :
vis maxima ad
conservandam dis-
tantiam.

M

R A

B S

punctum E, quod est in triente rectae DC,
& de quo supra mentionem fecimus num. 205.

225. Haec Theoria generaliter etiam non rectilineae tantum, sed & cuivis position!
trium massarum applicari potest, ac applicabitur infra, ubi etiam generale simplicissimum,
ac fcecundissimum theorema eruetur pro comparatione virium inter se : sed hie interea
evolvemus nonnulla, quae pertinent ad simpliciorem hunc casum trium punctorum. In-
primis fieri utique potest, ut ejusmodi tria puncta positionem ad sensum rectilineam
retineant cum prioribus distantiis, utcunque magna fuerit vis, quae ilia dimovere tentet,
vel utcunque magna velocitas impressa fuerit ad ea e suo respectivo statu deturbanda.
Nam vires ejusmodi esse possunt, ut tarn in eadem directione ipsius rectas, quam in
directione ad earn perpendicular!, adeoque in quavis obliqua etiam, quae in eas duas resolvi
cogitatione potest, validissimus exurgat conatus ad redeundum ad priorem locum, ubi inde
discesserint puncta. Contra vim impressam in directione ejusdem rectae satis est, si pro
puncto medio attractio plurimum crescat, aucta distantia ab utrolibet extreme, & plurimum
decrescat eadem imminuta ; ac pro utrovis puncto extreme satis est, si repulsio decrescat
plurimum aucta distantia ab extreme, & attractio plurimum crescat, aucta distantia a
medio, quod secundum utique fiet, cum, ut dictum est, debeat attractio medii in ipsum
crescere, aucta distantia. Si haec ita se habuerint, ac vice versa ; differentia virium vi
extrinsecae resistet, sive ea tenet contrahere, sive distrahere puncta, & si aliquod ex iis
velocitatem in ea directione acquisiverit utcunque magnam, poterit differentia virium
esse tanta, ut extinguat ejusmodi respectivam velocitatem tempusculo, quantum libuerit,
parvo, & post percursum spatiolum, quantum libuerit, exiguum.

Quid ubi vis exter-

& virgae flexiiis.

226. Quod si vis urgeat perpendiculariter, ut ex.gr. punctum medium D moveatur
per rectam DC perpendicularem ad AB ; turn vires CK, CL possunt utique esse ita validae,
ut vis composita CF sit post recessum, quantum libuerit, exiguum satis magna ad ejusmodi
vim elidendam, vel ad extinguendam velocitatem impressam. In casu vis, quas constanter
urgeat, & punctum D versus C, & puncta A, B ad partes oppositas, habebitur inflexio ; ac
in casu vis, quae agat in eadem directione rectae jungentis puncta, habebitur contractio,
seu distractio ; sed vires resistentes ipsis poterunt esse ita validae, ut & inflexio, & contractio,
vel distractio, sint prorsus insensibiles ; [105] ac si actione externa velocitas imprimatur
punctis ejusmodi, quae flexionem, vel contractionem, aut distractionem inducat, turn
ipsa puncta permittantur sibi libera ; habebitur oscillatio quasdam, angulo jam in alteram
plagam obverso, jam in alteram oppositam, ac longitudine ejus veluti virgae constantis iis
tribus punctis jam aucta, jam imminuta, fieri poterit ; ut oscillatio ipsa sensum omnem
effugiat, quod quidem exhibebit nobis ideam virgae, quam vocamus rigidam, & solidam,
contractionis nimirum, & dilatationis incapacem, quas proprietates nulla virga in Natura

[The reader should draw a more general figure for Art. 224 & 227, taking AD, DB
unequal and CD not at right angles to AB.]

A THEORY OF NATURAL PHILOSOPHY 177

relative rest ; for the attractive, or repulsive, forces which are acting on the middle point
may be equal. But then, in these cases, the outside points must be respectively attracted,
or repelled by the middle point ; & if they are equally & oppositely repelled by one another
in the first case, & attracted by one another in the second case, then it will be possible for
all the mutual forces to cancel one another.

224. Further, there is also a very great difference between these two cases. For in one of these
instance, if the points are moved a small distance out of the direct straight line, so that endeavour6 towards
the middle point D, say, is now slightly off the straight line AB, being transferred to C, a recovery of posi-
then, if left to itself, it will recede still further from it in the first case, & will approach t^warts^iurther
it once more in the second case. Or, if it is acted on by some external force, it will endeavour recession from it,
to recover its position & will resist the force acting on it. For two repulsions, CM, CN, moved o

will at first be obtained in the second case, at the first instant of motion from the position position.
D ; although indeed these may become attractions when the distances BD, AD are
sufficiently altered into the distances BC, AC. These will give a resultant force
acting along CH in a direction away from the straight line AB. But in the first
case we shall have two attractions CL, CK ; & these will give a force directed
towards AB. In this case, the attraction AP combined with the repulsion AR, &
the attraction BV combined with the repulsion BY, will give resultant forces, AQ, BT,
under the action of which the points A,B will move in the opposite direction to that of
the point C, as it returns to the straight line passing through that point E, which is a third
of the way along the straight line DC, of which mention was made above in Art. 205.

225. This Theory can also be applied more generally, to include not only a position Enunciation of a
of the three points in a straight line but also any position whatever. This application more general theory
will be made in what follows, where also a general theorem, of a most simple & fertile nature ly^g ^ a straight
will be deduced for comparison of forces with one another. But for the present we will line ; possibility of
consider certain points that have to do with this more simple case of three points. First tendSg^conser^
of all, it may come about that three points of this kind may maintain a position practically vation of distance,
in a straight line, no matter how great the force tending to drive them from it may be,

or no matter how great a velocity may be impressed upon them for the purpose of disturbing
them from their relative positions. For there may be forces of such a kind that both in
the direction of the straight line, & perpendicular to it, & hence in any oblique direction
which may be mentally resolved into the former, there may be produced an extremely
strong endeavour towards a return to the initial position as soon as the points had departed
from it. To counterbalance the force impressed in the direction of the same straight line
itself, it is sufficient if the attraction for the middle point should increase by a large amount
when the distance from either of the outside points is increased, & should be decreased
by a large amount if this distance is decreased. For either of the outside points it is sufficient
if the repulsion should greatly decrease, as the distance is increased, from the outside point,
and the attraction should greatly increase, as the distance is increased, from the middle
point ; & this second requirement will be met in every case, since, as has been said, and
attraction on it of the middle point will necessarily increase when the distance is increased.
If matters should turn out to be as stated, or vice versa, then the difference of the forces
will resist the external force, whether it tries to bring the points together or to drive them
apart ; & if any one of them should have acquired a velocity in the direction of the straight
line, no matter how great, there will be a possibility that the difference of the forces may
be so great that it will destroy any relative velocity of this kind, in any interval of time,
no matter how short the time assigned may be ; & this, after passing over any very small
assigned space, no matter how small.

226. But if the force acts perpendicularly, so that, for instance, the point D moves what happens if
along the line DC perpendicular to AB, then the forces CK, CL, can in any case be so ^es n^tTct ah^g
strong that the resultant force CF may become, after a recession of any desired degree the straight line ;
of smallness, large enough to eliminate any force of this kind, or to destroy any impressed

velocity. In the case of a force continually urging the point D towards C, & the points
A & B in the opposite direction, there will be some bending ; & in the case of a force acting
in the same direction as the straight line joining the two points, there will be some contraction
or distraction. But the forces resisting them may be so strong that the bending, the
contraction, or the distraction will be altogether inappreciable. If by external action a
velocity is impressed on points of this kind, & if this induces bending, contraction or distraction,
& if the points are then left to themselves, there will be produced an oscillation, in which
the angle will jut out first on one side & then on the other side ; & the length of, so to
speak, the rod consisting of the three points will be at one time increased & at another
decreased ; & it may possibly be the case that the oscillation will be totally unappreciable ;
& this indeed will give us the idea of a rod, such as we call rigid & solid, incapable of
being contracted or dilated ; these properties are possessed by no rod in Nature perfectly

N

I78

PHILOSOPHLE NATURALIS THEORIA

habet accurata tales, sed tantummodo ad sensum. Quod si vires sint aliquanto debiliores,
turn vero & inflexio ex vi externa mediocri, & oscillatio, ac tremor erunt majores, & jam
hinc ex simplicissimo trium punctorum systemate habebitur species quaedam satis idonea
ad sistendum animo discrimen, quod in Natura observatur quotidie oculis. inter virgas
rigidas, ac eas, quae sunt flexiles, & ex elasticitate trementes.

Systemate inflexo 227. Ibidem si binse vires, ut AQ, BT fuerint perpendiculares ad AB, vel etiam

vlsr ^ncti^medii utcunque parallels inter se, tertia quoque erit parallela illis, & aequalis earum summae,
contraria extremis, sed directionis contrariae. Ducta enim CD parallela iis, turn ad illam KI parallela BA,
" ' erit ob CK> yB sequales, triangulum CIK aequale simili BTV, sive TBS, adeoque CI squalls

BT, IK aequalis BS, sive AR, vel QP. Quare si sumpta IF aequali AQ ducatur KF ; erit
triangulum FIK aequale AQP, ac proinde FK aequalis, & parallela AP, sive LC, & CLFK
parallelogrammum, ac CF, diameter ipsius, exprimet vim puncti C utique parallelam
viribus AQ, BT, & asqualem earum summae, sed directionis contrariae. Quoniam vero
est SB ad BT, ut BD ad DC ; ac AQ ad AR, ut DC ad DA ; erit ex aequalitate perturbata
AQ ad BT, ut BD ad DA, nimirum vires in A, & B in ratione reciproca distantiarum AD,
DB a recta CD ducta per C secundum directionem virium.

&

summae.

Postremum theo-
rema generate, ubi
etiam tria puncta
non jaceant in di-
rectum.

Equilibrium trium
punctorum non in
directum jacentium
impossible sine vi
externa, nisi sint
in distantiis limi-
tum : cum iis qui
nisus ad retinen-
dam formam syste-
matis.

228. Ea, quas hoc postremo numero demonstravimus, aeque pertinent ad actiones
mutuas trium punctorum habentium positionem mutuam quamcunque, etiam si a rectilinea
recedat quantumlibet ; nam demonstratio generalis est : sed ad massas utcunque inaequales,
& in se agentes viribus etiam divergentibus, multo generalius traduci possunt, ac traducentur
inferius, & ad aequilibrii leges, & vectem, & centra oscillationis ac percussionis nos deducent.
Sed interea pergemus alia nonnulla persequi pertinentia itidem ad puncta tria, quae in
directum non jaceant.

229. Si tria puncta non jaceant in directum, turn vero sine externis viribus non poterunt
esse in aequilibrio ; nisi omnes tres distantiae, quae latera trianguli constituunt, sint dis-
tantiae limitum figurae i. Cum enim vires illae mutuae non habeant [106] directiones
oppositas ; sive unica vis ab altero e reliquis binis punctis agat in tertium punctum, sive
ambae ; haberi debebit in illo tertio puncto motus, vel in recta, quae jungit ipsum cum
puncto agente, vel in diagonali parallelogrammi, cujus latera binas illas exprimant vires.
Quamobrem si assumantur in figura I tres distantiae limitum ejusmodi, ut nulla ex iis sit
major reliquis binis simul sumptis, & ex ipsis constituatur triangulum, ac in singulis angu-
lorum cuspidibus singula materiae puncta collocentur ; habebitur systema trium punctorum
quiescens, cujus punctis singulis si imprimantur velocitates aequales, & parallelae ; habebitur
systema progrediens quidem, sed respective quiescens ; adeoque istud etiam systema
habebit ibi suum quemdam limitem, sed horum quoque limitum duo genera erunt : ii,
qui orientur ab omnibus tribus limitibus cohaesionis, erunt ejusmodi, ut mutata positione,
conentur ipsam recuperare, cum debeant conari recuperare distantias : ii vero, in quibus
etiam una e tribus distantiis fuerit distantia limitis non cohaesionis, erunt ejusmodi, ut
mutata positione : ab ipsa etiam sponte magis discedat systema punctorum eorundem. Sed
consideremus jam casus quosdam peculiares, & elegantes, & utiles, qui hue pertinent.

Eiegans theoria 230. Sint in fig. 32 tria puncta A,E,B ita collocata, ut tres distantise AB, AE, BE sint
ratto eiupsTs binis distantiae limitum cohaesionis, & postremae duae
aiiis occupantibus sint aequales. Focis A, B concipiatur ellipsis

transiens per E, cujus axis transversus sit FO,

conjugatus EH, centrum D : sit in fig. I AN

aequalis semiaxi transverse hujus DO, sive BE,

vel AE, ac sit DB hie minor, quam in fig. I

amplitude proximorum arcuum LN, NP, & sint

in eadem fig. i arcus ipsi NM, NO similes, &

aequales ita, ut ordinatae uy, zt, aeque distantes

ab N, sint inter se aequales. Inprimis si punctum

materiae sit hie in E ; nullum ibi habebit vim,

cum AE, BE sint aequales distantiae AN limitis

N figurae I ; ac eadem est ratio pro puncto

collocate in H. Quod si fuerit in O, itidem

erit in aequilibrio. Si enim assumantur in fig. I

Az, AM aequales hisce BO, AO ; erunt Nz,

foco : vis nulla in
verticibus axium.

illius aequales DB, DA hujus, adeoque & inter se. Quare & vires illius zt, uy erunt aequales
inter se, quae cum pariter oppositae directionis sint, se mutuo elident ; ac eadem ratio est
pro collocatione in F. Attrahetur hie utique A, & repelletur B ab O ; sed si limes, qui
respondet distantiae AB, sit satis validus ; ipsa puncta nihil ad sensum discedent a focis

A THEORY OF NATURAL PHILOSOPHY 179

accurately, but only approximately. But if the forces are somewhat more feeble, then
indeed, under the action of a moderate external force, the bending, the oscillation & the
vibration will all be greater ; & from this extremely simple system of three points we now
obtain several kinds of cases that are adapted to giving us a mental conception of the differences,
that meet our eyes every day in Nature, between rigid rods & those that are flexible &
elastically tremulous.

227. At the same time, if the two forces, represented by AQ, BT, were perpendicular In * system dis-

A T> n i i_ • ^i. ^L- j f u f i n i torted by parallel

to AB, or parallel to one another in any manner, then the third force would also be parallel forces the force on
to them, equal to their sum, but in the opposition direction. For, if CD is drawn parallel the middle point is
to the forces, & KI parallel to BA to meet CD in I, then, since CK & VB are equal to direction t° thaTof
one another, the triangle CIK will be equal to the similar triangle BTV, or to the triangle the outside forces,
TBS ; & therefore CI will be equal to BT, IK to BS or AR or QP. Hence if IF is taken sumT*
equal to AQ & KF is drawn, then the triangle FIK will be equal to AQP, & thus FK
will be equal £ parallel to AP or LC, CLFK will be a- parallelogram, & its diagonal CF
will represent the force for the point C, in every case parallel to the forces AQ, BT, &
equal to their sum, but opposite in direction. But, because SB : BT : : BD : DC, &
AQ : AR : : DC : DA ; hence, ex cequali we have AQ : BT : : BD : DA, that is to say, the
forces on A & B are in the inverse ratio of the distances AD & DB, drawn from the straight
line CD in the direction of the forces.

228. What has been proved in the last article applies equally to the mutual actions The last theorem in
of three points having any relative positions whatever, even if it departs from a rectilinear fhe tiTree^point^do
position to any extent you may please. For the demonstration is general ; &, further, the not He in a straight
results can be deduced much more generally for masses that are in every manner unequal, line-

& that act upon one another even with diverging forces ; & they will be thus deduced
later ; & these will lead us to the laws of equilibrium, the lever, & the centres of oscillation
& percussion. But meanwhile we will go straight on with our consideration of some
matters relating in the same manner to three points, which do not lie in a straight line.

229. If the three points do not lie in a straight line, then indeed without the presence Equilibrium of
of an external force they cannot be in equilibrium ; unless all three distances, which form ^o^no*0^ in^a
the sides of the triangle, are those corresponding to the limit-points in Fig. I. For, since straight line isim-
the mutual forces do not have opposite directions, either a single force from one of the F^^^^^1^0^

. i i • i i TT i iri presence 01 an

remaining two points acts on the third, or two such forces. Hence there must be for that external force,
third point some motion, either in the direction of the straight line joining it to the acting "?Bless3l. thHisl^«

' o j o o are at distances

point, or along the diagonal of the parallelogram whose sides represent those two forces, corresponding to

Therefore, if in Fig. i we take three limit-distances of such a kind, that no one of them is 8 '

in his
greater than the other two taken together, & if from them a triangle is formed & at each case, to 'conserve

vertical angle a material point is situated, then we shall have a system of three points at rest, sygte^"11 of the

If to each point of the system there is given a velocity, and these are all equal & parallel to one

another, we shall have a system which moves indeed, but which is relatively at rest. Thus

also that system will have a certain limit of its own ; moreover, of such limits there are also

two kinds. Namely, those that arise from all three limit-points being those of cohesion

which will be such that, if the relative position is altered, they will strive to recover it ;

for they are bound to try to restore the distances. Secondly, those in which one of the

three distances corresponds to a limit-point of non-cohesion, which will be such that, if

the relative position is altered, the system will of its own accord depart still more from it.

However, let us now consider certain special cases, that are both elegant & useful, for which

this is the appropriate place.

230. In Fig. 32, let the three points A,E,B be so placed that the three distances AB, An elegant theory
AE, BE correspond to limit-points of cohesion, & let the two last be equal to one another. i^the'periineter of
Suppose that an ellipse, whose foci are A & B, passes through E ; let the transverse axis of an ellipse, each of
this be FO, & the conjugate axis EH, & the centre D. In Fig. i, let AN be equal to behVpiaced irTa
the transverse semiaxis DO of Fig. 32, that is equal to BE or AE ; also in the latter figure focus ; no force at
let DB be less than the width of the successive arcs LN, NP of Fig. i ; also, in Fig. i, let en
the arcs NM, NO be similar & equal, so that the ordinates uy, zt, which are equidistant
from N, are equal to one another. Then, first of all, if in Fig. 32, the point of matter
is situated at E, there will be no force upon it ; for AE, BE are equal to the distance AN
of the limit-point N in Fig. i ; & the argument is the same for a point situated at H.
Further, if it is at O, it will in like manner be in equilibrium. For, if in Fig. i we take
Az, Au equal to AO, BO of Fig. 32, then Nz, NM of the former figure will be equal to DB,
DA of the latter ; & thus equal also to one another. Hence also the forces in that figure,
zt & uy, will be equal to one another ; & since they are likewise opposite in direction, they
will cancel one another ; & the argument is the same for a point situated at F. Here
in every case A is attracted & B is repelled from O ; but if the limit-point, which corresponds
to the distance AB is strong enough, the points will not depart to any appreciable extent

i8o

PHILOSOPHISE NATURALIS THEORIA

In reliquis puncti
perimetri vis direc-
ta per ipsam peri-
metrum versus ver-
tices axis conju-
gati.

Analogia verticum
binorum axium
cum limitibus cur-
vae virium.

Quando limites
contrario m o d o
positi : casus ele-
gantissimi alterna-
tionis p 1 u r i u m
limitum in peri-
metro ellipseos.

N

ellipseos, in quibus fuerant collocata, vel si debeant discedere ob limitem minus validum,
considerari poterunt per externam vim ibidem immota, ut contemplari liceat solam
relationem tertii puncti ad ilia duo.

231. Manet igitur immotum, ac sine vi,
punctum collocatum tarn in verticibus axis con-
jugati ejus ellipseos, quam in verticibus axis
transversi ; & si ponatur in quovis puncto C
[107] perimetri ejus ellipseos, turn ob AC, CB
simul aequales in ellipsi axi transverse, sive duplo
semiaxi DO ; erit AC tanto longior, quam ipsa
DO, quanto BC brevior ; adeoque si jam in fig.
I sint AM, Az aequales hisce AC, BC ; habe-
buntur ibi utique uy, zt itidem aequales inter se.
Quare hie attractio CL sequabitur repulsioni
CM, & LIMC erit rhombus, in quo inclinatio 1C
secabit bifariam angulum LCM ; ac proinde si
ea utrinque producatur in P, & Q ; angulus ACP,
qui est idem, ac LCI, erit aequalis angulo BCQ,
qui est ad verticem oppositus angulo ICM. Quse
cum in ellipsi sit notissima proprietas tangentis

relatae ad focos ; erit ipsa PQ tangens. Quamobrem dirigetur vis puncti C in latus secundum
tangentem, sive secundum directionem arcus elliptici, atque id, ubicunque fuerit punctum
in perimetro ipsa, versus verticem propiorem axis conjugati, & sibi relictum ibit per ipsam
perimetrum versus eum verticem, nisi quatenus ob vim centrifugam motum non nihil
adhuc magis incurvabit.

232. Quamobrem hie jam licebit contemplari in hac curva perimetro vicissitudinem
limitum prorsus analogorum limitibus cohaesionis, & non cohaesionis, qui habentur in axe
rectilineo curvae primigeniae figures I. Erunt limites quidam in E, in F, in H, in O, in
quibus nimirum vis erit nulla, cum in omnibus punctis C intermediis sit aliqua. Sed in

E, & H erunt ejusmodi, ut si utravis ex parte punctum dimoveatur, per ipsam perimetrum,
debeat redire versus ipsos ejusmodi limites, sicut ibi accidit in limitibus cohaesionis ; at in

F, & O erit ejusmodi, ut in utramvis partem, quantum libuerit, parvum inde punctum
dimotum fuerit, sponte debeat inde magis usque recedere, prorsus ut ibi accidit in limitibus
non cohaesionis.

233. Contrarium accideret, si DO aequaretur distantiae limitis non cohaesionis : turn
enim distantia BC minor haberet attractionem CK, distantia major AC repulsionem CN,
& vis composita per diagonalem CG rhombi CNGK haberet itidem directionem tangentis
ellipseos ; & in verticibus quidem axis utriusque haberetur limes quidam, sed punctum
in perimetro collocatum tenderet versus vertices axis transversi, non versus vertices axis
conjugati, & hi referrent limites cohaesionis, illi e contrario limites non cohaesionis. Sed
adhuc major analogia in perimetro harum ellipsium habebitur cum axe curvae primigeniae
figurae I ; si fuerit DO asqualis distantiae limitis cohaesionis AN illius, & DB in hac major,
quam in fig. i amplitude NL, NP ; multo vero magis, si ipsa hujus DB superet plures
ejusmodi amplitudines, ac arcuum aequalitas maneat hinc, & inde per totum ejusmodi
spatium. Ubi enim AC hujus figurae fiet aequalis abscissae AP illius, etiam BC hujus fiet
pariter aequalis AL illius. Quare in ejusmodi loco habebitur limes, & ante ejusmodi locum
versus A distantia [108] longior AC habebit repulsionem, & BC brevior attractionem,
ac rhombus erit KGNC, & vis dirigetur versus O. Quod si alicubi ante in loco adhuc
propriore O distantiae AC, BC aequarentur abscissis AR, AI figurae i ; ibi iterum esset
limes ; sed ante eum locum rediret iterum repulsio pro minore distantia, attractio pro
majore, & iterum rhombi diameter jaceret versus verticem axis conjugati E. Generaliter
autem ubi semiaxis transversus aequatur distantiae cujuspiam limitis cohaesionis, & distantia
punctorum a centre ellipseos, sive ejus eccentricitas est major, quam intervallum dicti
limitis a pluribus sibi proximis hinc, & inde, ac maneat aequalitas arcuum, habebuntur in
singulis quadrantibus perimetri ellipeos tot limites, quot limites transibit eccentricitas
hinc translata in axem figurae I, a limite illo nominato, qui terminet in fig. i semiaxem
transversum hujus ellipseos ; ac praetererea habebuntur limites in verticibus amborum
ellipseos axium ; eritque incipiendo ab utrovis vertice axis conjugati in gyrum per ipsam
perimetrum is limes primus cohaesionis, turn illi proximus esset non cohaesionis, deinde

A THEORY OF NATURAL PHILOSOPHY 181

from the foci of the ellipse, in which they were originally situated ; or, if they are forced
to depart therefrom owing to the insufficient strength of the limit-point, they may be
considered to be kept immovable in the same place by means of an external force, so that
we may consider the relation of the third point to those two alone.

2i>i. A point, then, which is situated at one of the vertices of the conjugate axis of At remaining points

, ,.J. , ' , • r i_ • • • i o j -L of tne perimeter

the ellipse or at one of the vertices of the transverse axis remains motionless & under the the force directed
action of no force. If it is placed at any point C in the perimeter of the ellipse, then, since alons the perimeter

A ^i /~.T> i • i IT i i • Till .is towards the ver-

AC, CB taken together are m the ellipse equal to the transverse axis, or double the semi- tices of the conju-

axis DO, AC will be as much longer than DO as BC is shorter. Hence, if in Fig. i AM, 8ate axis-

Az are equal to these lines AC, BC, we shall have in every case, in Fig. I, uy, zt also equal

to one another. Therefore, in Fig. 32, the attraction CL will be equal to the repulsion

CM, & LIMC will be a rhombus, in which the inclination 1C will bisect the angle LCM.

Hence if it is produced on either side to P & Q, the angle ACP, which is the same as the

angle LCI will be equal to the angle BCQ, which is vertically opposite to the angle ICM.

Now this is a well-known property with respect to the tangent referred to the foci in the

case of an ellipse ; & therefore PQ is the tangent. Hence the force on the point C is directed

laterally along the tangent, i.e., in the direction of the arc of the ellipse ; & this is true,

no matter where the point is situated on the perimeter, & the force is towards the nearest

vertex of the conjugate axis ; if left to itself, the point will travel along the perimeter

towards that vertex, except in so far as its motion is disturbed somewhat in addition, owing

to centrifugal force.

232. Hence we can consider in this curved perimeter the alternation of limit-points Analogy between
as being perfectly analogous to those of cohesion & non-cohesion, which were obtained in two ^xes63* the
the rectilinear axis of the primary curve of Fig. I. There will be certain limit-points at limit-points of the
E, F, H, O, in which there is no force, whilst in all intermediate points such as C there c

will be some force. But at E & H they will be such that, if the point is moved towards
either side along the perimeter, it must return towards such limit-points, just as it has to
do in the case of limit-points of cohesion in Fig. I. But at F & O, the limit-point would
be such that, if the point is moved therefrom to either side by any amount, no matter
how small, it must of its own accord depart still further from it ; exactly as it fell out in
Fig. i for the limit-points of non-cohesion.

233. Just the contrary would happen, if DO were equal to the distance corresponding when the limit
to a limit-point of non-cohesion. For then the smaller distance BC would have an Pomif are disposed

ATT- i T A ^i i • /^XT i i r -i , in the opposite

attraction CK, & the greater distance AC a repulsion CJN ; the resultant force along the way ; most elegant
diagonal CG of the rhombus CNGK would in the same way have its direction along the instances of aiter-

,,,. . . r • -i • i 111 .,..9 nation of several

tangent to the ellipse, & at the vertices of either axis there would be certain limit-points ; limit-points in the

but a point situated in the perimeter would tend towards the vertices of the transverse g^™6*" of the

axis, & not towards the vertices of the conjugate axis ; & the latter are of the nature of

limit-points of cohesion & the former of non-cohesion. However, a still greater analogy

in the case of the perimeter of these ellipses with the axis of the primary curve of Fig. i

would be obtained, if DO were taken equal to the distance corresponding to the limit-point

of cohesion AN in that figure, & in the present figure DB were taken greater than the

width of NL, NP in Fig. i ; much more so, if DB were greater than several of these widths,

& the equality between the areas on one side & the other held good throughout the whole

of the space taken. For where AC in the present figure becomes equal to the abscissa AP

of the former, BC in the present figure will likewise become equal to AL in the former.

Hence at a position of this kind there will be a limit-point ; & before a position of this

kind, towards O, the longer distance AC will have a repulsion & the shorter distance BC

an attraction, KGNC will be a rhombus, & the force will be directed towards O. But if

at some position, on the side of O, & still nearer to O, the distances AC, BC were equal

to the abscissae AR, AI of Fig. I, then again there would be a limit-point ; but before

that position there would return once more a repulsion for the smaller distance & an

attraction for the greater, & once more the diagonal of the rhombus would lie in the direction

of E, the vertex of the conjugate axis. Moreover, in general, whenever the transverse

semiaxis is equal to the distance corresponding to any limit-point of cohesion, & the distance

of the points from the centre of the ellipse, i.e., its eccentricity, is greater than the interval

between the said limit-point & several successive limit-points on either side of it, & the

equality of the arcs holds good, then for each quadrant of the perimeter of the ellipse there

will be as many limit-points as the number of limit-points in the axis of Fig. I that the

eccentricity will cover when transferred to it from the present figure, measured from that

limit-point mentioned as terminating in Fig. I the transverse semiaxis of the ellipse of the

present figure ; in addition there will be limit-points at the vertices of both axes of the

ellipse. Beginning at either vertex -of the conjugate axis, & going round the perimeter,

the first limit-point will be one of cohesion, then the next to it one of non-cohesion, then

PHILOSOPHISE NATURALIS THEORIA

alter cohaesionis, & ita porro, donee redeatur ad primum, ex quo incceptus fuerit gyrus,
vi in transitu per quemvis ex ejusmodi limitibus mutante directionem in oppositam. Quod
si semiaxis hujus ellipseos aequetur distantiae limitis non cohsesionis figurae i ; res ecdem
ordine pergit cum hoc solo discrimine, quod primus limes, qui habetur in vertice semiaxis
conjugati sit limes non cohaesionis, turn eundo in gyrum ipsi proximus sit cohsesionis limes,
deinde iterum non cohaesionis, & ita porro.

Perimetn piunum 2, . Verum est adhuc alia quaedam analogia cum iis limitibus ; si considerentur

elhpsium aequiva- JT .. . .... , . ° .. '.....

lentes limitibus. plures ellipses nsdem illis iocis, quarum semiaxes ordine suo aequentur distantns, in altera
cujuspiam e limitibus cohaesionis figuras I, in altera limitis non cohaesionis ipsi proximi,
& ita porro alternatim, communis autem ilia eccentricitas sit adhuc etiam minor quavis
amplitudine arcuum interceptorum limitibus illis figurse I, ut nimirum singulae ellipsium
perimetri habeant quaternos tantummodo limites in quatuor verticibus axium. Ipsae
ejusmodi perimetri totae erunt quidam veluti limites relate ad accessum, & recessum a
centro. Punctum collocatum in quavis perimetro habebit determinationem ad motum
secundum directionem perimetri ejusdem ; at collocatum inter binas perimetros diriget
semper viam suam ita, ut tendat versus perimetrum definitam per limitem cohaesionis
figurae I, & recedat a perimetro definita per limitem non cohaesionis ; ac proinde punctum
a perimetro primi generis dimotum conabitur ad illam redire ; & dimotum a perimetro
secundi generis, sponte illam adhuc magis fugiet, ac recedet.

Demonstrate. 235. Sint enim in fig. 33. ellipsium FEOH, F'E'O'H', F"E"O"H" semiaxes DO,

D'O', D"O" aequales primus di-[iO9]-stantiae AL limitis non cohaesionis figurae i ; secundus
distantiae AN limitis cohaesionis ; tertius distantiae AP limitis iterum non cohaesionis, &
primo quidem collocetur C aliquanto ultra perimetrum mediam F'E'O'H' : erunt AC,
BC majores, quam si essent in perimetro, adeoque in fig. I factis AM, Az majoribus, quam
essent prius, decrescet repulsio zt, crescet attractio uy ; ac proinde hie in parallelogram mo
LCMI erit attractio CL major, quam repulsio CM, & idcirco accedet directio diagonalis
CI magis ad CL, quam ad CM, & inflectetur introrsum versus perimetrum mediam.
Contra vero si C' sit intra perimetrum mediam, factis BC', AC' minoribus, quam si essent
in perimetro media ; crescet repulsio C'M', & decrescet attractio C'L', adeoque directio
C'l' accedet magis ad priorem C'M', quam ad posteriorem C'L', & vis dirigetur extrorsum
versus eandem mediam perimetrum. Contrarium autem accideret ob rationem omnino
similem in vicinia primae vel tertias perimetri : atque inde patet, quod fuerat propositum.

blematum s e g e s,
sed minus utilis :
immensa combina-
tionum varietas.

Alias curvas eiiip- 236. Quoniam arcus hinc, & inde a quovis limite non sunt prorsus aequales ; quanquam,

das13; ampfa^pro- ut suPra observavimus num. 184, exigui arcus ordinatas ad sensum aequales hinc, & inde
habere debeant ; curva, per cujus tangentem perpetuo dirigatur vis, licet in exigua eccen-
tricitate debeat esse ad sensum ellipsis, tamen nee in iis erit ellipsis accurate, nee in
eccentricitatibus majoribus ad ellipses multum accedet. Erunt tamen semper aliquae
curvae, quae determinent continuam directionem virium, & curvse etiam, quae trajectoriam
describendam definiant, habita quoque ratione vis centifugae : atque hie quidem uberrima
seges succrescit problematum Geometrise, & Analysi exercendae aptissimorum ; sed omnem
ego quidem ejusmodi perquisitionem omittam, cujus nimirum ad Theoriae applicationem
usus mihi idoneus occurrit nullus ; & quae hue usque vidimus, abunde sunt ad ostendendam
elegantem sane analogiam alternationis in directione virium agentium in latus, cum virium
primigeniis simplicibus, ac harum limitum cum illarum limitibus, & ad ingerendam animo
semper magis casuum, & combinationum diversarum ubertatem tantam in solo etiam
trium punctorum systemate simplicissimo ; unde conjectare liceat, quid futurum sit, ubi
immensus quidam punctorum numerus coalescat in massulas constituentes omnem hanc
usque adeo inter se diversorum corporum multitudinem sane immensam.

Conversio t o t i u s
systematis illaesi :
impulsu per peri-
metrum ellip sees
oscil latio: idea
liquationis, & con-
glaciationis.

237. At praeterea est & alius insignis, ac magis determinatus fructus, quern ex ejusmodi
contemplationibus capere possumus, usui futurus etiam in applicatione Theoriae ad
Physicam. Si nimirum duo puncta A, & B sint in distantia limitis cohaesionis satis validi,
& punctum tertium collocatum in vertice axis conjugati in E distantiam a reliquis habeat,
quam habet limes itidem cohaesionis satis validus ; poterit sane [no] vis, qua ipsum
retinetur in eo vertice, esse admodum ingens pro utcunque exigua dimotione ab eo loco,

A THEORY OF NATURAL PHILOSOPHY

183

FIG. 33.

i84

PHILOSOPHISE NATURALIS THEORIA

FIG. 33.

A THEORY OF NATURAL PHILOSOPHY 185

another of cohesion, & so on, until we arrive at the first of them, from which the circuit
was commenced ; & the force changes direction as we pass through each of the limit-points
of this kind to the exactly opposite direction. But if the semiaxis of this ellipse is equal
to the distance corresponding to a limit-point of non-cohesion in Fig. i, the whole matter
goes on as before, with this difference only, namely, that the first limit-point at the vertex
of the conjugate semiaxis becomes one of non-cohesion ; then, as we go round, the next
to it is one of cohesion, then again one of non-cohesion, & so on.

234. Now there is yet another analogy with these limit-points. Let us consider a The perimeters of
number of ellipses having the same foci, of which the semiaxes are in order equal to the sev<rral ellipses

,. ,....,-,. , - . . £ equivalent to limit-

distances corresponding to limit-points in .tig. I, namely to one of cohesion for one, to points.

that of non-cohesion next to it for the second, & so on alternately ; also suppose that the
eccentricity is still smaller than any width of the arcs between the limit-points of Fig. I,
so that each of the elliptic perimeters has only four limit-points, one at each of the four
vertices of the axes. The whole set of such perimeters will be somewhat of the nature
of limit-points as regards approach to, or recession from the centre. A point situated in
any one of the perimeters will have a propensity for motion along that perimeter. If it
is situated between two perimeters, it will always direct its force in such a way that it will
tend towards a perimeter corresponding to a limit-point of cohesion in Fig. i, & will
recede from a perimeter corresponding to a limit-point of non-cohesion. Hence, if a point
is disturbed out of a position on a perimeter of the first kind, it will endeavour to return
to it ; but if disturbed from a position on a perimeter of the second kind, it will of its
own accord try to get away from it still further, & will recede from it.

235. In Fig. 33, of the semiaxes DO, DO', DO" of the ellipses FEOH, F'E'O'H', Demonstration.
F"E"O"H". let the first be equal to the distance corresponding to AL, a limit-point of
non-cohesion in Fig. I, the second to AN, one of cohesion, the third to AP, one of non-
cohesion. In the first place, let the point C be situated somewhere outside the middle
perimeter F'E'O'H' ; then AC, BC will be greater than if they were drawn to the perimeter.

Hence, in Fig. I, since AM, Az would be made greater than they were formerly, the repulsion
zt would decrease, & the attraction uy would increase. Therefore, in Fig. 33, in the
parallelogram LCMI, the attraction CL will be greater than the repulsion CM, & so the
direction of the diagonal CI will approach more nearly to CL than to CM, & will be turned
inwards towards the middle perimeter. On the other hand, if C' is within the middle
perimeter, BC', AC' are made smaller than if they were drawn to the middle perimeter ;
the repulsion C'M' will increase, & the attraction C'L' will decrease, & thus the direction
of CT will approach more nearly to the former, C'M', than to the latter, C'L' ; & the
force will be directed outwards towards the middle perimeter. Exactly the opposite would
happen in the neighbourhood of the first or third perimeter, & the reasoning would be
similar. Hence, the theorem enunciated is evidently true.

236. Now, since the arcs on either side of any chosen limit-point are not exactly equal, other curves to
& yet, as has been mentioned above in Art. 184, very small arcs on either side are bound b,f. substltuted for

i • IT i 11 i • ellipses ; an ample

to nave approximately equal ordmates ; the curve, along the tangent to which the force crop of theorems,

is continually directed, although for small eccentricity it must be practically an ellipse, b"ga"ot °v^|eth USoi

yet will neither be an ellipse accurately in this case, nor approach very much to the form combinations.

of an ellipse for larger eccentricity. Nevertheless, there will always be certain curves

determining the continuous direction of the force, & also curves determining the path

described when account is taken of the centrifugal force. Here indeed there will spring

up a most bountiful crop of problems well-adapted for the employment of geometry &

analysis. But I am going to omit all discussion of that kind ; for I can find no fit use for

them in the application of my Theory. Also those which we have already seen are quite

suitable enough to exhibit the truly elegant analogy between the alternation in direction

of forces acting in a lateral direction & the simple primary forces, between the limit-points

of the former & those of the latter ; also for impressing on the mind more & more the

great wealth of cases & different combinations to be met with even in the single very simple

system of three points. From this it may be conjectured what will happen when an

immeasurable number of points coalesce into small masses, from which are formed all that

truly immense multitude of bodies so far differing from one another.

237. In addition to the above, there is another noteworthy & more determinate result Rotation of the
to be derived from considerations of this kind, & one that will be of service in the application ™** °' e sy.,s,t ® m

t 1 »-r>i T.I • i-i • •!• -i A n T> T T mtact ) oscillation

oi the 1 neory to rnysics. r or instance, it the two points A & is are at a distance corresponding along the perimeter
to a limit-point of cohesion that is sufficiently strong, & the third point situated at the of the ellipse due to

T-, £ , . . . , . ' 1-1 an impulse ; the

vertex r, oi the conjugate axis is at a distance from the other two which corresponds to idea of liquefaction
a limit-point of cohesion that is also sufficiently strong, then the force retaining the point & congelation,
at that vertex might be great enough, for any slight disturbance from that position, to
prevent it from being moved any further, unless through the action of a huge external

1 86 PHILOSOPHIC NATURALIS THEORIA

ut sine ingenti externa vi inde magis dimoveri non possit. Turn quidem si quis impediat
motum puncti B, & circa ipsum circumducat punctum A, ut in fig. 34 abeat in A' ; abibit
utique & E versus E', ut servetur forma trianguli AEB, quam necessario requirit conver-
satio distantiarum, sive laterum inducta a limitum validitate, &
in qua sola poterit respective quiescere systema, ac habebitur
idea quaedam soliditatis cujus & supra injecta est mentio. At
si stantibus in fig. 32 punctis A, B per quaspiam vires externas,
quae eorum motum impediant, vis aliqua exerceatur in E ad
ipsum a sua positione deturbandum ; donee ea fuerit medio-
cris, dimovebit illud non nihil ; turn, ilia cessante, ipsum se resti-
tuet, & oscillabit hinc, & inde ab illo vertice per perimetrum
curvae cujusdam proximse arcui elliptico. Quo major fuerit vis
externa dimovens, eo major oscillatio net ; sed si non fuerit
tanta, ut punctum a vertice axis conjugati recedens deveniat ad
verticem axis transversi ; semper retro cursus reflectetur, & de-
scribetur minus, quam semiellipsis. Verum si vis externa coegerit
percurrere totum quadrantem, & transilire ultra verticem axis
transversi ; turn verogyrabit punctum circumquaque per totam FIG. 34.

perimetrum motu continue, quern a vertice axis conjugati ad

verticem transversi retardabit, turn ab hoc ad verticem conjugati accelerabit, & ita porro,
nee sistetur periodicus conversionis motus, nisi exteriorum punctorum impedimentis
occurrentibus, quae sensim celeritatem imminuant, & post ipsos ejusmodi motus periodicos
per totum ambitum reducant meras oscillationes, quas contrahant, & pristinam debitam
positionem restituant, in qua una haberi potest quies respectiva. An non ejusmodi aliquid
accidit, ubi solida corpora, quorum partes certam positionem servant ad se invicem, ingenti
agitatione accepta ab igneis particulis liquescunt, turn iterum refrigescentes, agitatione
sensim cessante per vires, quibus igneae particulae emittuntur, & evolant, positionem prio-
rem recuperant, ac tenacissime iterum servant, & tuentur ? Sed haec de trium punctorum
systemate hucusque dicta sint satis.
Systema punctorum 238. Quatuor, & multo magis plurium, punctorum systemata multo plures nobis

quatuor, in eodem . .J , .. . . ,',..., .r -p,

piano cum distan- vanationes objicerent ; si rite ad examen vocarentur ; sed de us id unum innuam. H,a
tiis hmitum, suao quidem in piano eodem possunt positionem mutuam tueri tenacissime ; si singulorum

forma; tenax. f. . r ,. . ....... . ,. , ,.

distantiae a reliquis sequentur distantns hmitum satis validorum tigurae I : neque emm
in eodem piano positionem respectivam mutare possunt, aut aliquod ex iis exire e piano
ducto per reliqua tria, nisi mutet distantiam ab aliquo e reliquis, cum datis trium punctorum
distantiis mutuis detur triangulum, quod constituere debent, turn datis distantiis quarti
a duobus detur itidem ejus positio respectu eorum in eodem piano, & detur distantia ab
eorum tertio, quae, si id punctum exeat e [in] priore piano, sed retineat ab iis duobus
distantiam priorem, mutari utique debet, ut facili negotio demonstrari potest.

Alia ratio system- 239. Quin immo in ipsa ellipsi considerari possunt puncta quatuor, duo in focis, &

quatuor ^^eodem a^a ^uo nmc> & 'm<^e a vertice axis conjugati in ea distantia a se invicem, ut vi mutua

piano cum idea repulsiva sibi invicem elidant vim, qua juxta praecedentem Theoriam urgentur in ipsum

flexiiis •"Systema verticem ; quo quidem pacto rectangulum quoddam terminabunt, ut exhibet fig. 35, in

eorundem forms punctis A, B, C, D. Atque inde si supra angulos quadratae basis assurgant series ejusmodi

nes^rifmrticufa- punctorum exhibentium series continuas rectangulorum, habebitur quaedam adhuc magis

rum pyramidaiium. praecisa idea virgae solidae, in qua si basis ima inclinetur ; statim omnia superiora puncta

movebuntur in latus, ut rectangulorum illorum positionem retineant,& celeritas conversionis

erit major, vel minor, prout major fuerit, vel minor vis ilia in latus, quae ubi fuerit aliquanto

languidior, multo serius progredietur vertex, quam fundum, .

& inflectetur virga, quae inflexio in omni virgarum genere

apparet adhuc multo magis manifesta, si celeritas conversionis C O

fuerit ingens. Sed extra idem planum possunt quatuor puncta
collocata ita, ut positionem suam validissime tueantur, etiam
ope unicae distantiae limitis unici satis validi. Potest enim fieri
pyramis regularis, cujus latera singula triangularia habeant

ejusmodi distantiam. Turn ea pyramis constituet particulam • •

quandam suae figurae tenacissimam, quae in puncta, vel pyra- /\ 3

mides ejusmodi aliquanto remotiores ita poterit agere, ut ejus FlG 35

puncta respectivum situm nihil ad sensum mutent. Ex quatuor

ejusmodi particulis in aliam majorem pyramidem dispositis fieri poterit particula secundi
ordinis aliquanto minus tenax ob majorem distantiam particularum primi earn componen-

A THEORY OF NATURAL PHILOSOPHY 187

force. In that case, if the motion of the point B were prevented, & the point A were set
in motion round B, so that in Fig. 34 it moved to A', then the point E would move off
to E' as well, so as to conserve the form of the triangle AEB, as is required by the conservation
of the sides or distances which is induced by the strength of the limits ; & the system can
be relatively at rest in this form only ; thus we get an idea of a certain solidity, of which
casual mention has" already been made above. But if, in Fig. 2, whilst the points A,B,
are kept stationary by means of an external force preventing their motion, some force is
exerted on the point at E to disturb it from its position, then, as long as the force is only
moderate, it will move the point a little ; & afterwards, when the force ceases, the point
will recover its position, & will then oscillate on each side of the vertex along a perimeter
of the curve that closely approximates to an elliptic arc. The greater the external force
producing the motion, the greater the oscillation will be ; but if it is not so great as to make
the point recede from the vertex of the conjugate axis until it reaches the vertex of the
transverse axis, its path will always be retraced, & the arc described will be less than a
semi-ellipse. But if the external force should compel the point to traverse a whole quadrant
& pass through the vertex of the transverse axis, then indeed the point will make a complete
circuit of the whole perimeter with a continuous motion ; this will be retarded from the
vertex of the conjugate axis to that of the transverse axis, then accelerated from there
onwards to the vertex of the conjugate axis, & so on ; there will not be any periodic reversal
of motion, unless there are impediments met with from external points that appreciably
diminish the speed ; in which case, following on such periodic motions round the whole circuit,
there will be a return to mere oscillations ; & these will be shortened, & the original position
restored, the only one in which there can possibly be relative rest. Probably something
of this sort takes place, when solid bodies whose parts maintain a definite position with
regard to one another, if subjected to the enormous agitation produced by fiery particles,
liquefy ; & once more freezing, as the agitation practically ceases on account of forces due
to the action of which the fiery particles are driven out & fly off, recover their initial position
& again keep & preserve it most tenaciously. But let us be content with what has been said
above with regard to a system of three points for the present.

238. Systems of four, & much more so for more, points would yield us many more varia- A system of four
tions, if they were examined carefully one after the other ; but I will only mention one thing jj^^ces^orre*
about such systems. It is possible that such systems, in one plane, may conserve their rela- spending to Hmit-
tive positions very tenaciously, if the distances of each from the rest are equal to the dis- ^ves^tslform.0011
tances in Fig. i corresponding to limit-points of sufficient strength. For neither can they

change their relative position in the plane, nor can any one of them leave the plane drawn
through the other three ; since, if the distances of three points from one another is given,
then we are given the triangle which they must form ; & then being given the distances
of the fourth point from two of these, we are also given the position of this fourth point
from them, & therefore also the distance from the third of them. If the point should
depart from the plane mentioned, & yet preserve its former distances from the two points
the distance from the third point must be changed in any case, as can be easily proved.

239. Again, we may consider the case of four points in the ellipse, two being at the A further consider-
foci, & the other two on either side of a vertex of the conjugate axis at such a distance from a{io"ou°f ^i^f6,™
one another, that the mutual repulsive force between them will cancel the force with which connection with
they are urged towards that vertex, according to the preceding theorem. Thus, they are the idea of rigid &

f i . T->- i • A T, V-i T-V flexible rods; a.

at the vertices of a rectangle, as is shown in rig. 35, where they occupy the points A,B,U,D. system of four
Hence, if we have a series of points of this kind to stand above the four angles of the quadratic P°mts m the fo.rm

r i 11 i • r i • ^ • • of a pyramid;

base, so as to represent continuous series of rectangles, we shall obtain from this supposition different arrange-
a more precise idea than hitherto has been possible of a solid rod, in which, if the lowest ments °f particular

r r • • • v i n • T i 1-1 pyramids.

set or points is inclined, all the points above are immediately moved sideways, but
so that they retain the positions in their rectangles ; & the speed of rotation will be greater
or less according as the force sideways was greater or less ; even where this force is somewhat
feeble, the top will move considerably later than the base & the rod will be bent ; & the
amount of bending in every kind of rod will be still more apparent if the speed of rotation is
very great. Again, four points not in the same plane can be so situated that they preserve
their relative position very tenaciously ; & that too, when we make use of but a single
distance corresponding to a limit-point of sufficient strength. For they can form a regular
pyramid, of which each of the sides of the triangles is of a length equal to this distance.
Then this pyramid will constitute a particle that is most tenacious as regards its form ;
& this will be able to act upon points, or pyramids of the same kind, that are more remote,
in such a manner that its points do not alter their relative position in the slightest degree
for all practical purposes. From four particles of this kind, arranged to form a larger
pyramid, we can obtain a particle of the second order, somewhat less tenacious of form on
account of the greater distance between the particles of the first order that compose it ;

1 88 PHILOSOPHIC NATURALIS THEORIA

tium, qua fit, ut vires in easdem ab externis punctis impressae multo magis inaequales inter se
sint,^quam fuerint in punctis constituentibus particulas ordinis primi ; ac eodem pacto ex
his secundi ordinis particulis fieri possunt particulse ordinis tertii adhuc minus tenaces figurae
suae, atque ita porro, donee ad eas deventum sit multo majores, sed adhuc multo magis
mobiles, atque variabiles, ex quibus pendent chemica; operationes, & ex quibus haec ipsa
crassiora corpora componuntur, ubi id ipsum accideret, quod Newtonus in postrema Optics
questione proposuit de particulis suis primigeneis, & elementaribus, alias diversorum ordi-
num particulas efformantibus. Sed de particularibus hisce systematis determinati punc-
torum numeri jam satis, ac ad massas potius generaliter considerandas faciemus gradum.

Transitus ad 240. In massis primum nobis se offerunt considerandas elegantissimse sane, ac £ foccund-

massas : quid cen- • m **t* ' • • • • j T->I

trum gravitatis : issimae, & utilissimae propnetates centn gravitatis, quse quidem e nostra 1 heona sponte
theoremata hk de propemodum fluunt, aut saltern eius ope evidentissime demonstrantur. Porro centrum

eo demonstrando. L *••. • . • •.., . . . . . , .

gravitatis a gravium aequihbno nomen accepit suum, a quo etiam ejus consideratio ortum

duxit ; sed id quidem a gravi-[ii2]-tate non pendet, sed ad massam potius pertinet.
Quamobrem ejus definitionem proferam ab ipsa gravitate nihil omnino pendentem, quan-
quam & nomen retinebo, & innuam, unde originem duxerit ; turn demonstrabo accuratissime,
in quavis massa haberi aliquod gravitatis centrum, idque unicum, quod quidem passim
omittere solent, & perperam ; deinde ad ejus proprietatem praecipuam exponendam
gradum faciam, demonstrando celeberrimum theorema a Newtono propositum, centrum
gravitatis commune massarum, sive mihi punctorum quotcunque, & utcunque disposi-
torum, quorum singula moveantur sola inertiae vi motibus quibuscunque, qui in singulis
punctis uniformes sint, in diversis utcunque diversi, vel quiescere, vel moveri uniformiter
in directum : turn vero mutuas actiones quascunque inter puncta quaelibet, vel omnia
simul, nihil omnino turbare centri communis gravitatis statum quiescendi vel movendi
uniformiter in directum, unde nobis & actionis, ac reactionis aequalitas in massis quibusque,
& principia collisiones corporum definientia, & alia plurima sponte provenient. Sed
aggrediamur ad rem ipsam.

Definitio centri 241. Centrum igitur commune gravitatis punctorum quotcunque. & utcunque

gravitatis non j- • n f -j j • j i

pendens ab idea dispositorum, appellabo id punctum, per quod si ducatur planum quodcunque ; summa

gravitatis : ejus distantiarum perpendicularium ab eo piano punctorum omnium jacentium ex altera
idea8communi.C * ejusdem parte, sequatur summa distantiarum ex altera. Id quidem extenditur ad quas-
cunque, & quotcunque massas ; nam eorum singulae punctis utique constant, & omnes
simul sunt quaedam punctorum diversorum congeries. Nomen traxit ab aequilibrio
gravium, & natura vectis, de quibus agemus infra : ex iis habetur illud, singula
pondera ita connexa per virgas inflexiles, ut moveri non possint, nisi motu circa aliquem
horizontalem axem, exerere ad conversionem vim proportionalem sibi, & distantiae perpen-
diculari a piano verticali ducto per axem ipsum ; unde fit, ut ubi ejusmodi vires, vel, ut
ea vocant, momenta virium hinc, & inde asqualia fuerint, habeatur aequilibrium. Porro
ipsa pondera in nostris gravibus, in quibus gravitatem concipimus, ac etiam ad sensum
experimur, proportionalem in singulis quantitati materiae, & agentem directionibus inter
se parallelis, proportionalia sunt massis, adeoque punctorum eas constituentium numero ;
quam ob rem idem est, ea pondera in distantias dncere, ac assumere summam omnium
distantiarum omnium punctorum ab eodem piano. Quod si igitur respectu aggregati
cujuscunque punctorum materiae quotcunque, & quomodocunque dispositorum sit aliquod
punctum spatii ejusmodi, ut, ducto per ipsum quovis piano, summa distantiarum ab illo
punctorum jacentium ex parte altera aequetur summse distantiarum jacentium ex altera ;
concipiantur autem singula ea puncta animata viribus aequalibus, & parallelis, cujusmodi sunt
vires, quas in nostris gravibus concipimus ; illud utique consequitur, [113] suspense utcunque
ex ejusmodi puncto, quale definivimus gravitatis centrum, omni eo systemate, cujus
systematis puncta viribus quibuscunque, vel conceptis virgis inflexibilus, & gravitate
carentibus, positionem mutuam, & respectivum statum, ac distantias omnino servent, id
systema fore in aequilibrio ; atque illud ipsum requiri, ut in aequilibrio sit. Si enim vel
unicum planum ductum per id punctum sit ejusmodi, ut summae illae distantiarum non
sint aequales hinc, & inde ; converse systemate omni ita, ut illud planum evadat verticale,
jam non essent aequales inter se summae momentorum hinc, & inde, & altera pars alteri
prseponderaret. Verum haec quidem, uti supra monui, fuit occasio quaedam nominis
imponendi ; at ipsum punctum ea lege determinatum longe ulterius extenditur, quam

A THEORY OF NATURAL PHILOSOPHY 189

for from this fact it comes about that the forces impressed upon these from external points
are much more unequal to one another. than they would be for the points constituting
particles of the first order. In the same manner, from these particles of the second order
we might obtain particles of the third order, still less tenacious of form, & so on ; until
at last we reach those which are much greater, still more mobile, & variable particles, which
are concerned in chemical operations ; & to those from which are formed the denser bodies,
with regard to which we get the very thing set forth by Newton, in his last question in
Optics, with respect to his primary elemental particles, that form other particles of different
orders. We have now, however, said enough concerning these systems of a definite number
of points, & we will proceed to consider masses rather more generally.

240. In dealing with masses, the first matters that present themselves for our considera- Passing on to
tion are certain really very elegant, as well as most fertile & useful properties of the centre of "ntr? of "gravity^
gravity. These indeed come forth almost spontaneously from my Theory, or at least are Theorems to be
demonstrated most clearly by means of it. Further, the centre of gravity derived its name

from the equilibrium of heavy (gravis) bodies, & the first results in connection with the former
were developed by means of the latter ; but in reality it does not depend on gravity, but rather is
related to masses. On this account, I give a definition of it, which in no way depends on
gravity, although I will retain the name, & will mention whence it derived its origin. Then
I will prove with the utmost rigour that in every body there is a centre of gravity, & one
only (a thing which is usually omitted by everybody, quite unjustifiably). Then I will
proceed to expound its chief property, by proving the well-known theorem enunciated by
Newton ; that the centre of gravity of masses, or, in my view, of any number of points in
any positions, each of which is moved in any manner by the force of inertia alone, this
force being uniform for the separate points but maybe non-uniform to any extent for
different points, will be either at rest or will move uniformly in a straight line. Finally,
I will show that any mutual action whatever between the points, or all of them taken
together, will in no way disturb the state of rest or of uniform motion in a straight line of
the centre of gravity. From which the equality of action & reaction in all bodies, & the
principles governing the collision of solids, & very many other things will arise of them-
selves. However let us set to work on the matter itself.

241. Accordingly, I will call the common centre of gravity of any number of points, Definition of the
situated in any positions whatever, that point which is such that, if through it any plane in^tndent^fTn7
is drawn, the sum of the perpendicular distances from the plane of all the points lying on idea of gravitation ;
one side of it is equal to the sum of the distances of all the points on the other side of it. ^ dtfaiSorT
The definition applies also to masses, of any sort or number whatever ; for each of the the usual idea,
latter is made up of points, & all of them taken together are certain groups of different

points. The name is taken from the equilibrium of weights (gravis), & from the principle
of the lever, with which we shall deal later. Hence we obtain the principle that each of
the weights, connected together by rigid rods in such a manner that the only motion possible
to them is one round a horizontal axis, will exert a turning force proportional to itself &
to its perpendicular distance from a vertical plane drawn through this axis. From which
it comes about that, when the forces of this sort (or, as they are called, the moments of the
forces) are equal to one another on this side & on that, then there is equilibrium. Further,
the weights in our heavy bodies, in which we conceive the existence of gravity (& indeed
find by experience that there is such a thing) proportional in each to the quantity of matter,
& acting in directions parallel to one another, are proportional to the masses, & thus to
the number of points that go to form them. Therefore, the product of the weights into
the distances comes to the same thing as the sum of all the distances of all the points from
the plane. If then, for an aggregate of points of matter, of any sort & number whatever,
situated in any way, there is a point of space of such a nature that, for any plane drawn
through it, the sum of the distances from it of all points lying on one side of it is equal to the
sum of the distances of all the points lying on the other side of it ; if moreover each of the
points is supposed to be endowed with a force, & these forces are all equal & parallel to one
another, & of such a kind as we conceive the forces in our weights to be ; then it follows
directly that, if the whole of this system is suspended in any way from a point of the sort
we have defined the centre of gravity to be, the points of the system, on account of certain
assumed forces or rigid weightless rods, preserving their mutual position, their relative
state & their distances absolutely unchanged, then the system will be in equilibrium. Such a
point is to be found, in order that the system may be in equilibrium. For, if any one plane
can be drawn through the point, such that the sum of the distances on the one side are
not equal to those on the other side, & thewhole system is turned so that this plane becomes
vertical, then the sums of the moments will not be equal to one another on each
side, but one part will outweigh the other part. This indeed, as I said above, was the idea
that gave rise to the term centre of gravity ; but the point determined by this rule has

190

PHILOSOPHIC NATURALIS THEORIA

Corollarium g e it-
erate pertinens ad
summas distanti-
arum omnium
punctorum massse
a piano transeunte
per centrum gravi-
tatis xquales utrin-
que.

Bi.n a theoremata
per tinentia ad
planum parallel urn
piano distantiarum
aequalium cum
eorum demonstra-
tiouibus.

Com pie me n turn
demonstrationis ut
e x t e n d[a t u r ad
omnes casus.

ad solas massas animatas viribus asqualibus, & parallelis, cujusmodi concipiuntur a nobis
in nostris gravibus, licet ne in ipsis quidem accurate sint tales. Quamobrem assumpta
superiore definitione, quae a gravitatis, & sequilibrii natura non pendet, progrediar ad
deducenda inde corollaria quaaedam, quae nos ad ejus proprietates demonstrandas deducant.

242. Primo quidem si aliquod fuerit ejusmodi planum, ut binae summae distantiarum
perpendicularium punctorum omnium hinc & inde acceptorum aequenter inter se :
aequabuntur & summae distantiarum acceptarum secundum quancunque aliam directionem
datam, & communem pro omnibus. Erit enim quaevis distantia perpendicularis ad quanvis
in dato angulo inclinatam semper in eadem ratione, ut patet. Quare & sunimae illarum
ad harum summas erunt in eadem ratione, ac asqualitas summarum alterius binarii utriuslibet
secum trahet aequalitatem alterius. Quare in sequentibus, ubi distantias nominavero,
nisi exprimam perpendiculares, intelligam generaliter distantias acceptas in quavis
directione data.

243. Quod si assumatur planum aliud quodcunque parallelum piano habenti aequales
hinc, & inde distantiarum summas ; summa distantiarum omnium punctorum jacentium
ex parte altera superabit summam jacentium ex altera, excessu aequali distantiae planorum
acceptae secundum directionem eandem ductae in nwmerum punctorum : & vice versa si
duo plana parallela sint, ac is excessus alterius summas supra summam alterius in altero
ex iis aequetur eorum distantiae ductae in numerum punctorum ; planum alterum habebit
oppositarum distantiarum summas aequales. Id quidem facile concipitur ; si concipiatur,
planum distantiarum aequalium moveri versus illud alterum planum motu parallelo
secundum earn directionem, secundum quam sumuntur distantiae. In eo motu distantiae
singulse ex altera parte crescunt, ex altera decrescunt continue tantum, quantum promo-
vetur planum, & si aliqua distantia evanescit interea ; jam eadem deinde incipit tantundem
ex parte contraria crescere. Quare patet excessum omnium citeriorum [114] distantiarum
supra omnes ulteriores aequari progressui plani toties sumpto, quot puncta habentur, &
in regressu destruitur e contrario, quidquid in ejusmodi progressu est factum, atque idcirco
ad aequalitatem reditur. Verum ut demonstratio

quam accuratissima evadat, exprimat in fig. 36 recta
AB planum distantiarum aequalium, & CD planum
ipsi parallelum, ac omnia puncti distribui poterunt
in classes tres, in quorum prima sint omnia jacentia
citra utrumque planum, ut punctum E ; in secunda
omnia puncta jacentia inter utrumque, ut F, in tertia
omnia puncta adhuc jacentia ultra utrumque, ut G.
Rectae autem per ipsa ductae in directione data
quacunque, occurrant rectae AB in M, 'H, K, &
rectae CD in N, I, L ; ac sit quaedam reacta direc-
tionis ejusdem ipsis AB, CD occurrens in O, P.
Patet, ipsam OP fore aequalem ipsis MN, HI, KL.
Dicatur jam summa omnium punctorum E primae
classis E, & distantiarum omnium EM summa e ;
punctorum F secundae classis F, & distantiarum / ;
punctorum G tertiae classis summa G, & distantiarum
earundem g ; distantia vero OP dicatur O. Patet, sum-
mam omnium MN fore E X O ; summam omnium
HI fore F X O ; summam omnium KL fore G X O ; erit autem quaevis EN — EM +MN ;
quaevis FI = HI — FH ; quaevis GL = KG — KL. Quare summa omnium EN erit
e + E xO ; summa omnium FI = F x O — /, & summa omnium GL = g — G X O ;
adeoque summa omnium distantiarum punctorum jacentium citra planum CD, primae nimi-
rum, ac secundae classis, erit e + E xO+F X O — /, & summa omnium jacentium ultra,
nimirum classis tertiae, erit g — G X O. Quare excessus prioris summae supra secundam
erit e + E X O -f F xO — / — g+ G xO; adeoque si prius fuerit e = f + g ;
delete e — f— g, totus excessus erit E x O + F X O + G X O, sive (E + F + G) X O,
summa omnium punctorum ducta in distantiam planorum ; & vice versa si is excessus
respectu secundi plani CD fuerit aequalis huic summas ductae in distantiam O, oportebit
e — f — /aequetur nihilo, adeoque sit e = f -\- g, nimirum respectu primi plani AB summas
distantiarum hinc, & inde aequales esse.

244. Si aliqua puncta sint in altero ex iis planis, ea superioribus formulis contineri
possunt, concepta zero singulorum distantia a piano, in quo jacent ; sed & ii casus involvi
facile possent, concipiendo alias binas punctorum classes ; quorum priora sint in priore
piano AB, posteriora in posteriore CD, quae quidem nihil rem turbant : nam prioris classis

FIG. 36.

A THEORY OF NATURAL PHILOSOPHY 191

a far wider application than to the single cases of mass endowed with equal & parallel forces
such as we have assumed to exist in our heavy bodies ; & indeed such do not exist accurately
even in the latter. Hence, taking the definition given above, which is independent of
gravity & the nature of equilibrium of weights, I will proceed to deduce from it certain
corollaries, which will enable us to demonstrate the properties of the centre of gravity.

242. First of all, then, if there should be any plane such that the two sums of the General corollary
perpendicular distances of all the points on either side of it taken together are equal to Of ^h^d^ances'of
one another, then the sums of the distances taken together in any other given direction, ail the points of a
that is the same for all of them, will also be equal to one another. For, any perpendicular ^slng™ •hrVu'g'h
distance will evidently be in the same ratio to the corresponding distance inclined at a the centre of grav-
given angle. Hence the sums of the former distances will bear the same ratio to the sums e^he/sicfeoTit! °
of the latter distances ; & therefore the equality of the sums in either of the two cases

will involve the equality of the sums for the other also. Consequently, in what follows,
whenever I speak of distances, I intend in general distances in any given direction, unless
I expressly say that they are perpendicular distances.

243. If now we take any other plane parallel to the plane for which the sums of the Two. theorems
distances on either side are equal, then the sum of the distances of all the points lying pearaiief °o P the
on the one side of it will exceed the sum for those lying on the other side by an amount P|ane o f equal
equal to the distance between the two planes measured in the like direction multiplied demonstrations,
by the number of all the points. Conversely, if there are two parallel planes, & if the

excess of the sum of the distances from one of them over the sum of the distances from
the other is equal to the distance between the planes multiplied by the number of the
points, then the second plane will have the sums of the opposite distances equal to one
another. This is easily seen to be true ; for, if the plane of equal distances is assumed to
be moved towards the other plane by a parallel motion in the direction in which the distances
the measured, then as the plane is moved each of the distances on the one side increase,
& those on the other side decrease by just the amount through which the plane is moved ;
& should any distance vanish in the meantime, there will be an increase on the other side
of just the same amount. Thus, it is evident that the excess of all the distances on the
near side above the sum of all the distances on the far side will be equal to the distance
through which the plane has been moved, taken as many times as there are points. On
the other hand, when the plane is moved back again, this excess is destroyed, namely exactly
the amount that was produced as the plane moved forward, & consequently equality will
be restored. But to give a more rigorous demonstration, let the straight line AB, in Fig. 36,
represent the plane of equal distances, & let CD represent a plane parallel to it. Then
all the points can be grouped into three classes ; let the first of these be that in which we
have every point that lies on the near side of both the planes, as E ; let the second be that
in which every point lies between the two planes, as F ; & the third, every point lying
on the far side of both planes, as G. Let straight lines, drawn in any given direction whatever,
through the points meet AB in M, H, K, & the straight line CD in N, I, L ; also let any
straight line, drawn in the same direction, meet AB, CD in O & P. Then it is clear that
OP will be equal to MN, HI, or KL. Now, let us denote the sum of all the points of the
first class, like E, by the letter E, & the sum of all the distances like EM by the letter e ;
& those of the second class by the letters F & / ; those of the third class by G & g ; &
the distance OP by O. Then it is evident that the sum of all the MN's will be E X O ;
the sum of all the Hi's will be F X O ; the sum of all the KL's will be G X O ; also
in every case, EN = EM + MN, FI = HI — FH, & GL = KG — KL. Hence the sum
of the EN's will be e + E X O, the sum of the FI's will be F X O— /, & the sum of the
GL's will be g — G X O. Hence, the sum of all the distances of the points lying on the
near side of the plane CD, that is to say, those belonging to the first & second classes, will
be equal to<?+ExO + Fx O— / ; & the sum of all those lying on the far side, that
is, of the third class, will be equal to g — G X O. Hence, the excess of the former over
the latter will be equal to ^+ExO+FxO — / — g+GxO. Therefore, if at
first we had e = / + g, then, on omitting e — f — g, we have the total excess equal to
ExO+FxO + GxO,or(E+F-fG)xO, i.e., the sum of all the points multiplied
by the distance between the planes. Conversely, if the excess with respect to the second
plane CD were equal to this sum multiplied by the distance O, it must be that e — f — g
is equal to nothing, & thus e = f -\- g; in other words the sum of the distances with respect
to the first plane AB must be equal on one side & the other.

244. If any of the points should be in one or other of the two planes, these may also C°^lestio"s°tf0 ^e
be included in the foregoing formulae, if we suppose that the distance for each of them ^°^e' au apos°ib"e
is zero distance from the plane in which they lie. Then these cases may also be included cases.

by considering that there are two fresh classes of points ; namely, first those lying in the
first plane AB, & secondly those lying in the second plane CD ; & these classes will in

192

PHILOSOPHIC NATURALIS THEORIA

Theoremata pro
piano posito ultra
omnia pun eta:
eorum extensio ad
qua; vis plana.

Cuivis piano in-
veniri posse paral-
lelum planum dis-
tantiarum aequa-
lium.

Thoorema prseci-
puum si tria plana
distantiarum aequa-
lium habeant uni-
cum punctum
commune ; rcliqua
ornnia por id tran-
seuntia erunt ejus-
modi.

Demonstratio ejus-
dem.

distantiae a priore piano erunt omnes simul zero, & a posteriore sequabuntur distantiae O
ductae in eorum numerum, quae summa accedit priori summae punctorum jacentium citra ;
posterioris autem classis distantiaa a priore erant prius simul aequales summaa ipsorum
ductae itidem in O, & deinde fiunt nihil ; adeoque [115] summse distantiarum punctorum
jacentium ultra, demitur horum posteriorum punctorum summa itidem ducta in O, &
proinde excessui summse citeriorum supra summam ulteriorum accedit summa omnium
punctorum harum duarum classium ducta in eandem O.

245. Quod si planum parallelum piano distantiarum aequalium jaceat ultra omnia
puncta ; jam habebitur hoc theorema : Summa omnium distantiarum punctorum omnium ab
eo piano cequabitur distantly planorum ducta in omnium punctorum summam, & si fuerint duo
plana parallela ejusmodi, ut alterum jaceat ultra omnia puncta, fcsf summa omnium distanti-
arum ab ipso cequetur distantice planorum ductce in omnium punctorum numerum ; alterum illud
planum erit planum distantiarum cequalium. Id sane patet ex eo, quod jam secunda sum-
ma pertinens ad puncta ulteriora, quae nulla sunt, evanescat, & excessus totus sit sola prior
summa. Quin immo idem theorema habebit locum pro quovis piano habente etiam ulteriora
puncta, si citeriorum distantiae habeantur pro positivis, & ulteriorum pro negativis ; cum
nimirum summa constans positivis, & negativis sit ipse excessus positivorum supra negativa ;
quo quidem pacto licebit considerare planum distantiarum aequalium, ut planum, in quo
summa omnium distantiarum sit nulla, negativis nimirum distantiis elidentibus positivas.

246. Hinc autem facile jam patet, data cuivis piano haberi aliquod planum parallelum,
quod sit planum distantiarum cequalium ; quin immo data positione punctorum, & piano illo
ipso, facile id alterum definitur. Satis est ducere a singulis punctis datis rectas in data
directione ad planum datum, quae dabuntur ; turn a summa omnium, quae jacent ex parte
altera, demere summam omnium, si quae sunt, jacentium ex opposita, ac residuum dividere
per numerum punctorum. Ad earn distantiam ducto piano priori parallelo, id erit planum
quaesitum distantiarum aequalium. Patet autem admodum facile & illud ex eadem
demonstratione, & ex solutione superioris problematis, dato cuivis piano non nisi unicum
esse posse planum distantiarum aequalium, quod quidem per se satis patet.

247. Hisce accuratissime demonstratis, atque explicatis, progrediar ad demonstrandum
haberi aliquod gravitatis centrum in quavis punctorum congerie, utcunque dispersorum,
& in quotcunque massas ubicunque sitas

coalescentium. Id net ope sequentis
theorematis ; si per quoddam punctum tran-
seant tria plana distantiarum cequalium se
non in eadem communi aliqua recta secan-
tia ; omnia alia plana transeuntia per illud
idem punctum erunt itidem distantiarum
esqualium plana. Sit enimin fig. 37, ejus-
modi punctum C, per quod transeant tria
plana GABH, XABY, ECDF, qua; om-
nia sint plana distantiarum aequalium,
ac sit quodvis aliud planum KICL tran-
[i i6]-siens itidem per C, ac secans pri-
mum ex iis recta CI quacunque ; opor-
tet ostendere, hoc quoque fore planum
distantiarum aequalium, si ilia priora
ejusmodi sint. Concipiaturquodcunque
punctum P ; & per ipsum P concipiatur
tria plana parallela planis DCEF, ABYX,
GABH, quorum sibi priora duo mutuo
occurrant in recta PM, postrema duo
in recta PV, primum cum tertio in
recta PO : ac primum occurrat piano
GABH in MN, secundum vero eidem
in MS, piano DCEF in QR, ac piano CIKL in SV, ducaturque ST parallela rectis QR, MP,
quas, utpote parallelorum planorum intersectiones, patet fore itidem parallelas inter se, uti
& MN, PO, DC inter se, ac MS, PTV, BA inter se.

248. Jam vero summa omnium dis antiarum a piano KICL secundum datam direc-
tionem BA erit summa omnium PV, quae resolvitur in tres summas, omnium PR, omnium
RT, omnium TV, sive eae, ut figura exhibet in unam colligendss sint, sive, quod in aliis
plani novi inclinationibus posset accidere, una ex iis demenda a reliquis binis, ut habeatur
omnium PV summa. Porro quaevis PR est distantia a piano DCEF secundum eandem
earn directionem ; quaevis RT est aequalis QS sibi respondenti, quae ob datas directiones
laterum trianguli SCQ est ad CQ, aequalem MN, sive PO, distantiae a piano XABY secundum

A THEORY OF NATURAL PHILOSOPHY 193

no way cause any difficulty. For the distances of the points of the first class from the first
plane, all together, will be zero, & their distances from the second plane will, all together,
be equal to the distance O multiplied by the number of them ; & this sum is to be added
to the former sum for the points lying on the near side. Again, the distances of the points
of the second class from the first plane were, all together, at first equal to the distance
O multiplied by their number, & then are nothing for the second plane. Hence from the
sum of the distances of the points lying on the far side, we have to take away the sum of
these last points also multiplied by the distance O ; & thus, to the excess of the sum of the
points on the near side over the sum of the points on the far side we have to add the sum
of all the points in these two classes multiplied by the same distance O.

245. Now, if the plane parallel to the plane of equal distances should lie on the far Theorems for a
side of all the points then the following theorem is obtained. The sum of all the distances P1,3;"6 lvins beyond

/ M i • i 7 • i -77 7 77- 7 7 i , . , . , , a 1 1 » t h e points ;

of all the -points from this plane will be equal to the distance between the planes multiplied by extension of these

the sum of all the points ; y if there were two parallel planes, such that one of them lies beyond theorems to ^any

all the points, £ff if the sum of all the distances from this plane is equal to the distance between

the planes multiplied by the number of points, then the other plane will be the plane of equal

distances. This is perfectly clear from the fact that in this case the second sum relating

to the points that lie beyond the planes vanishes, for there are no such points, & the whole

excess corresponds to the first sum alone. Further, the same theorem holds good for any

plane even if there are points beyond it, if the distances of points on the near side of it

are reckoned as positive & those on the far side as negative ; for the sum formed from the

positives & the negatives is nothing else but the excess of the positives over the negatives.

In precisely the same manner, we may consider the plane of equal distances to be a plane

for which the sum of all the distances is nothing, that is to say, the positive distances cancel

the negative distances.

246. From the foregoing theorem it is now clear that for any given plane there exists Given any plane,
another plane parallel to it, which is a plane of equal distances ; further, if we are given the a ^fane" of equal
position of the points, y also the plane is given, then the parallel plane is easily determined, distances, parallel
It is sufficient to draw from each of the points straight lines in a given direction to the

given plane, & then these are all given ; then from the sum of all of them that lie on the
one side to take away the sum of all those that lie on the other side, if any such there are ;
& lastly to divide the remainder by the number of the points. If a plane is drawn parallel to
the first plane, & at a distance from it equal to the result thus found, then this plane will
be a plane of equal distances, as was required. Moreover it can be seen quite clearly,
& that too from the very demonstration just given, that to any given plane there can cor-
respond but one single plane of equal distances ; indeed this is sufficiently self-evident
without proof.

247. Now that the foregoing theorems have received rigorous demonstrations & The important
explanation, I will proceed to prove that there is a centre of gravity for any set of points, three^i'anes' o'f
no matter how they are dispersed or what the number of masses may be into which they equal distances
coalesce, or where these masses may be situated. The proof follows from the theorem : — p^ft( ^heiT^ny
// through any point there pass three planes of equal distances that do not all cut one another in other plane through
some common line then all other planes passing through this same point will also be planes of equal \^ ^^e nature! °
distances. In Fig. 37, let C be a point of this sort, & through it suppose that three planes,

GABH, XABY, ECDF, pass ; also suppose that all the planes are planes of equal distances.
Let KICL be any other plane passing through C also, & cutting the first of the three planes
in any straight line CI ; we have to prove that this latter plane is a plane of equal dis-
tances, if the first three are such planes. Take any point P ; & through P suppose three
planes to be drawn parallel to the planes DCEF, ABYX, GABH ; let the first two of
these meet one another in the straight line PM, the last two in the straight line PV, & the
first & third in the straight line PO. Also let the first meet the plane GABH in the
straight line MN, the second meet this same plane in MS, & the plane DCEF in QR,
the plane CIKL in SV, & let ST be drawn parallel to the straight lines QR & MP, which,
since they are intersections with parallel planes, are parallel to one another ; similarly MN,
PO, DC are parallel to one another, as also are MS, PTV & BA parallel to one another.

248. Now, the sum of all the distances from the plane KICL, in the given direction Proof of the theo-
BA, will be equal to the sum of all the PV's ; & this can be resolved into the three sums, r

that of all the PR's, that of all the RT's, & that of all the TV's ; whether these, as are
shown in the figure, have to be all collected into one whole, or, as may happen for other
inclinations of a fresh plane, whether one of the sums has to be taken away from the other
two, to give the sum of all the PV's. Now each PR is the distance of a point P from the
plane DCEF, measured in the given direction ; & eachRT is equal to the QS that corresponds
to it, which, on account of the given directions of the sides of the triangle SCQ bears a
given ratio to CQ , the latter being equal to MN or PO, the distance of P from the plane

194 PHILOSOPHISE NATURALIS THEORIA

datam directionem DC, in ratione data ; & quaevis VT est itidem in ratione data ad TS
aequalem PM, distantiae a piano GABH secundum datam directionem EC ; ac idcirco
etiam nulla ex ipsis PR, RT, TV poterit evanescere, vel directione mutata abire e positiva
in negativam, aut vice versa, mutato situ puncti P, nisi sua sibi respondens ipsius puncti
P distantia ex iis PR, PO, PM evanescat simul, aut directionem mutet. Quamobrem &
summa omnium positivarum vel PR, vel RT, vel TV ad summam omnium positivarum
vel PR, vel PO, vel PM, & summa omnium negativarum prioris directionis ad summam
omnium negativarum posterioris sibi respondentis, erit itidem in ratione data ; ac proinde
si omnes positivae directionum PR, PO, PM a suis negativis destruuntur in illis tribus
aequalium distantiarum planis, etiam omnes positivae PR, RT, TV a suis negativis destru-
entur, adeoque & omnes PV positivae a suis negativis. Quamobrem planum LCIK erit
planum distantiarum aequalium. Q.E.D.
[Haberi semper 2A.Q. Demonstrato hoc theoremate iam sponte illud consequitur, in quavis punclorum

aliquod pravitatis • J 7- 77- ?•:;•_.•

centrum, atque id fongene, adeoque massarum utcunque dispersarum summa, haben semper aiiquod gravitatis

esse unicum.] centrum, atque id esse unicum, quod quidem data omnium -punctorum positione facile determin-

abitur. Nam assumpto puncto quovis ad arbitrium ubicunque, ut puncto P, poterunt duci
per ipsum tria plana quaecunque, ut OPM, RPM, RPO. Turn singulis poterunt per
num. 246 inveniri plana parallela, [117] quae sint plana distantiarum sequalium, quorum
priora duo si sint DCEF, XABY, se secabunt in aliqua recta CE parallela illorum inter-
section! MP ; tertium autem GABH ipsam CE debebit alicubi secare in C ; cum planum
RPO secet PM in P : nam ex hac sectione constat, hanc rectam non esse parallelam huic
piano, adeoque nee ilia illi erit, sed in ipsum alicubi incurret. Transibunt igitur per
punctum C tria plana distantiarum aequalium, adeoque per num. 247 & aliud quodvis
planum transiens per punctum idem C erit planum aequalium distantiarum pro quavis
directione, & idcirco etiam pro distantiis perpendicularibus ; ac ipsum punctum C juxta
definitionem num. 241, erit commune gravitatis centrum omnium massarum, sive omnis
congeriei punctorum, quod quidem esse unicum, facile deducitur ex definitione, & hac
ipsa demonstratione ; nam si duo essent, possent utique per ipsa duci duo plana parallela
directionis cujusvis, & eorum utrumque esset planum distantiarum aequalium, quod est
contra id, quod num. 246 demonstravimus.

^nm^nlaberiseml 25O- Demonstr<indum necessario fuit, haberi aliquod gravitatis centrum, atque id

per centrum gravi- esse unicum ; & perperam id quidem a Mechanicis passim omittitur ; si enim id non
ubique adesset, & non esset unicum, in paralogismum incurrerent quamplurimae Mechanic-
orum ipsorum demonstrationes, qui ubi in piano duas invenerunt rectas, & in solidis tria
plana determinantia aequilibrium, in ipsa intersectione constituunt gravitatis centrum, &
supponunt omnes alias rectas, vel omnia alia plana, quae per id punctum ducantur, eandem
aequilibrii proprietatem habere, quod utique fuerat non supponendum, sed demonstrandum.
Et quidem facile est similis paralogismi exemplum praebere in alio quodam, quod magni-
tudinis centrum appellare liceret, per quod nimirum figura sectione quavis secaretur in
duas partes asquales inter se, sicut per centrum gravitatis secta, secatur in binas partes
aequilibratas in hypothesi gravitatis constantis, & certam directionem habentis piano
secanti parallelam.

mapiltudinis^noii 2SI- Erraret sane, qui ita defmiret centrum magnitudinis, turn determinaret id ipsum

semper haberi. in datis figuris eadem ilia methodo, quae pro centri
gravitatis adhibetur. Is ex. gr. pro triangulo ABG
in fig. 38 sic ratiocinationem institueret. Secetur
AG bifariam in D, ducaturque BD, quse utique
ipsum triangulum secabit in duas partes aequales.
Deinde, secta AB itidem bifariam in E, ducatur GE,

quam itidem constat, debere secare triangulum in / C"^^ \C* \

partes aequales duas. In earum igitur concursu C •" ' '"•^AVx

habebitur centrum magnitudinis. Hoc invento si
progrederetur ulterius, & haberet pro aequalibus
partes, quae alia sectione quacunque facta per C
obtinentur ; erraret pessime. Nam ducta ED, jam
constat, fore ipsam ED parallelam BG, & ejus dimi-
diam ; adeoque similia fore triangula [118] ECD,
BCG, & CD dimidiam CB. Quare si per C ducatur FH parallela AG ; triangulum FBH,
erit ad ABG, ut quadratum BC ad quadratum BD, seu ut 4 ad 9, adeoque segmentum
FBH ad residuum FAGH est ut 4 ad 5, & non in ratione sequalitatis.

Ubi haec primo 252. Nimirum quaecunque punctorum, & massarum congeries, adeoque & figura

demonstrata • • • • • p • e •

quaevis, in qua concipiatur punctorum numerus auctus in innnitum, donee ngura ipsa
evadat continua, habet suum gravitatis centrum ; centrum magnitudinis infinites earum
non habent ; & illud primum, quod hie accuratissime demonstravi, demonstraveram jam

A THEORY OF NATURAL PHILOSOPHY 195

XABY, measured in the given direction DC ; lastly, VT is also in a given ratio to TS, the
latter being equal to PM, the distance of the point P from the plane GABH, measured in
the given direction EC. Hence, none of the distances PR, RT, TV can vanish or, having
changed their directions, pass from positive to negative, or vice versa, by a change in the
position of the point P, unless that one of the distances PR, PO, PM, of the point P, which
corresponds to it vanishes or changes its direction at the same time. Therefore also the
sum of all the positives, whether PR, or RT, or TV to the sum of all the positives, PR,
or PO, or PM, & the sum of all the negatives for the first direction to the sum of all the
negatives for the second direction which corresponds to it, will also be in a given ratio.
Thus, finally, if all the positives out of the direction PR, PO, PM are cancelled by the
corresponding negatives in the case of the three planes of equal distances ; then also all
the positive PR's, RT's, TV's are cancelled by their corresponding negatives, & therefore
also all the positive PV's are cancelled by their corresponding negatives. Consequently,
the plane LCIK will be a plane of equal distances. Q.E.D.

249. Now that we have demonstrated the above theorem, it follows immediately There is always
from it that, for any group of -points, tj therefore also for a set of masses scattered in any manner, ty^^^ty1^
there exists a centre of gravity, W there is but one ; W this can be easily determined when the

position of each of the points is given. For if a point is taken at random anywhere, like the
point P there could be drawn through it any three planes, OPM, RPM, RPO. Then
corresponding to each of these there could be found, by Art. 245, a parallel plane, such
that these planes were planes of equal distances. If the first two of these are DCEF &
XABY, they will cut one another in some straight line CE parallel to their intersection
MP ; also the third plane GABH must cut this straight line CE somewhere in C ; for
the plane RPO will cut PM in P, & from this fact it follows that the latter line is not parallel
to the latter plane, & therefore the former line is not parallel to the former plane, but will
cut it somewhere. Hence three planes of equal distances will pass through the point C,
& therefore, by Art. 247, any other plane passing through this point C will also be a plane
of equal distances for any direction, & thus also for perpendicular distances. Hence, according
to the definition of Art. 241, the point C will be the common centre of gravity of all the
masses, or of the whole group of points. That there is only one can be easily derived from
the definition & the demonstration given ; for, if there were two, there could in every
case be drawn through them two parallel planes in any direction, & each of these would
be a plane of equal distances ; which is contrary to what we have proved in Art. 246.

250. It was absolutely necessary to prove that there always exists a centre of gravity, The need for
& that there is only one in every case ; & this proof is everywhere omitted by Mechanicians, ^centre^f gray6
quite unjustifiably. For, if there were not one in every case, or if it were not unique, ity in every case,
very many of the proofs given by these Mechanicians would result in fallacious argument.

Where, for instance, they find two straight lines, in the case of a plane, & in the case of solids
three planes, determining equilibrium, & suppose that all other lines, & all other planes,
which are drawn through the point to have the same property of equilibrium ; this in
every case ought not to be a matter of supposition, but of proof. Indeed it is easy to give
a similar example of fallacious argument in the case of something else, which we may call
the centre of magnitude ; for instance, where a figure is cut, by any section, into two parts
equal to one another ; just as when the section passes through the centre of gravity it is
cut into two parts that balance one another, on the hypothesis of uniform gravitation
acting in a fixed direction parallel to the cutting plane.

251. He would indeed be much at fault, who would so define the centre of magnitude For there is not
& then proceed to determine it in given figures by the same method as that used for the m^aitude061
centre of gravity. For example, the reasoning he would use for the triangle ABG, in Fig. 38,

would be as follows. Let AB be bisected in D, & through D draw BD ; this will certainly
divide the triangle into two equal parts. Then, having bisected AB also in E, draw GE ;
it is true that this also divides the triangle into two equal parts. Hence their point of
intersection C will be the centre of magnitude. If then, having found this, he proceeded
further, & said that those parts were equal, which were obtained by any other section made
through C ; he would be very much in error. For, if ED is drawn, it is well known that
we now have ED parallel to BG & equal to half of it ; & therefore the triangles BCD, BCG
would be similar, & CD half of CB. Hence, if FH is drawn through C parallel to AG,
the triangle FBH will be to the triangle ABG, as the square on BC is to the square on BD,
or as 4 is to 9 ; & thus the segment FBH is to the remainder FAGH as 4 is to 5, & not
in a ratio of equality.

252. Thus, any group of points or masses, & therefore any figure in which the number Where the first
of points is supposed to be indefinitely increased until the figure becomes continuous, *™° c
possesses a centre of gravity ; but there are an infinite number of them which have not

got a centre of magnitude. The first of these, of which I have here given a rigorous

196

PHILOSOPHISE NATURALIS THEORIA

olim methodo aliquanto contractiore in dissertatione De Centra Gravitatis ; hujus vero
secundi exemplum hie patet, ac in dissertatione De Centra Magnitudinis, priori illi addita
in secunda ejusdem impressione, determinavi generaliter, in quibus figuris centrum
magnitudinis habeatur, in quis desk ; sed ea ad rem praesentem non pertinent.

Inde ubi sit cen-
t r u m commune
massarum duarum.

Inde & communis
methodus pro quot-
cunque massis.

Inde & theorema,
ope cujus investi-
gatur id in figuris
continuis.

253. Ex hac general! determinatione centri gravitatis facile colligitur illud, centrum
commune binarum massarum jacere in directum cum centris gravitatis singularum, &
horum distantias ab eodem esse reciproce, ut ipsas massas. Sint enim binae massae, quarum
centra gravitatis sint in fig. 39 in A, & B. Si per rectam AB ducatur planum quodvis, id
debet esse planum distantiarum sequalium re-

spectu utriuslibet. Quare etiam respectu
summae omnium punctorum ad utrumque
simul pertinentium distantiae omnes hinc, &
inde acceptae sequantur inter se ; ac proinde id
etiam respectu summae debet esse planum dis-
tantiarum aequalium, & centrum commune FlG 3g
debet esse in quovis ex ejusmodi planis, ade-

oque in intersectione duorum quorumcunque ex iis, nimirum in ipsa recta AB. Sit
id in C, & si jam concipiatur per C planum quodvis secans ipsam AB ; erit summa omnium
distantiarum ab eo piano secundum directionem AB punctorum pertinentium ad massam
A, si a positivis demantur negativae, aequalis per num. 243 numero punctorum massae A
ducto in AC, & summa pertinentium ad B numero punctorum in B ducto in BC ; quae
producta aequari debent inter se, cum omnium distantiarum summae positivae a negativis
elidi debeant respectu centri gravitatis C. Erit igitur AC ad CB, ut numerus punctorum
in B ad numerum in A, nimirum in ratione massarum reciproca.

254. Hinc autem facile deducitur communis methodus inveniendi centrum gravitatis
commune plurium massarum. Conjunguntur prius centra duarum, &? eorum distantia dividitur
in ratione reciproca ipsarum. Turn harum commune centrum sic inventum conjungitur cum
centra tertics, tsf dividitur distantia in ratione reciproca summa massarum priorum ad massam
tertiam, & ita porro. Quin immo possunt seorsum inveniri centra gravitatis binarum
quarumvis, ternarum, denarum quocunque [119] ordine, turn binaria conjungi cum ternariis,
denariis, aliisque, ordine itidem quocunque, & semper eadem methodo devenitur ad centrum
commune gravitatis masses totius. Id patet, quia quotcunque massae considerari possunt
pro massa unica, cum agatur de numero punctorum massae tantummodo, & de summa
distantiarum punctorum omnium ; summae massarum constituunt massam, & summae
distantiarum summam per solam conjunctionem ipsarum. Quoniam autem ex generali
demonstratione superius facta devenitur semper ad centrum gravitatis, atque id centrum
est unicum ; quocunque ordine res peragatur, ad illud utique unicum devenitur.

255. Inde vero illud consequitur, quod est itidem commune, si plurium massarum
centra gravitatis sint in eadem aliqua recta, fore etiam in eadem centrum gravitatis summce
omnium ; quod viam sternit ad investiganda gravitatis centra etiam in pluribus figuris
continuis. Sic in fig. 38 centrum commune gravitatis totius trianguli est in illo puncto,
quod a recta ducta a vertice anguli cujusvis ad mediam basim oppositam relinquit trientem
versus basim ipsam. Nam omnium rectarum basi parallelarum, quae omnes a recta BD
secantur bifariam, ut FH, centra gravitatis sunt in eadem recta, adeoque & areae ab iis
contextae centrum gravitatis est tarn in recta BD, quam in recta GE ob eandem rationem,
nempe in illo puncto C. Eadem methodus applicatur aliis figuris solidis, ut pyramidibus ;
at id, ut & reliqua omnia pertinentia ad inventionem centri gravitatis in diversis curvis
lineis, superficiebus, solidis, hinc profluentia, sed meae Theoriae communia jam cum
vulgaribus elementis, hie omittam, & solum illud iterum innuam, ea rite procedere, ubi
jam semel demonstratum fuerit, haberi in massis omnibus aliquod gravitatis centrum, &
esse unicum, ex quo nimirum hie & illud fluit, areas FAGH, FBH licet inaequales, habere
tamen aequales summas distantiarum omnium suorum punctorum ab eadem recta FH.

Difficuitas demon- 2c6. In communi methodo alio modo se res habet. Posteaquam inventum est in

strationis in com- „ . • A n T. • • T\/~> o

muni methodo. fig. 40 centrum gravitatis commune massis A, & B, juncta pro tertia massa DC, & secta
in F in ratione massarum D, & A + B reciproca, habetur F pro centro communi omnium
trium. Si prius inventum esset centrum commune E massarum D, B, & juncta AE, ea
secta fuisset in F in ratione reciproca massarum A, & B + D ; haberetur itidem illud

A THEORY OF NATURAL PHILOSOPHY 197

demonstration, I proved some time ago in a somewhat shorter manner in my dissertation
De Centra Gravitatis ; & a case of the second is here clearly shown ; & in the dissertation
De Centra Magnitudinis, which was added as a supplement to the former in the second
edition, I determined in general the figures in which there existed a centre of magnitude
& those in which there was none ; but such things have no Hearing on the matter now
in question.

253. From this general determination of the centre of gravity it is readily deduced that Hence to determine
the common centre of two masses lies in the straight line joining the centres of each of the

masses, & that the distances of the masses from this point will be reciprocally proportional two masses.

to the masses themselves. For suppose we have two masses, & that their centres of gravity

are, in Fig. 39, at A & B. If through the straight line AB any plane is drawn, it must be

a plane of equal distances for either of the masses. Therefore also, with regard to the

sum of the points of both masses taken together, all the distances taken on one side & on

the other side will be equal to one another. Hence also with regard to this sum it must

be a plane of equal distances ; the common centre must lie in any one of these planes, &

therefore in the line of intersection of any two of them, that is to say, in the straight line

AB. Suppose it is at C ; & suppose that any plane is drawn through C to cut AB. Then

the sum of all the distances from this plane in the direction AB of all the points belonging

to the mass A, the negatives being taken from the positives, will by Art. 243 be equal to

the number of points in the mass A multiplied by AC ; & the sum of those belonging to

the mass B to the number of points in the mass B multiplied by BC. These products

must be equal to each other, since the positives in the sum of all the distances must be

cancelled by the negatives with regard to the centre of gravity C. Hence AC is to CB

as the number in B is to the number of points in A, i.e., in the reciprocal ratio of the masses.

254. Further, from the foregoing theorem can be readily deduced the usual method Hence, the usual
of finding the common centre of gravity of several masses. First of all the centres of two of ™mtec Offmassesy
them are joined, & the distance between them is divided in the reciprocal ratio of the masses.

Then the common centre of these two masses, thus found, is joined to the centre of a third, & the
distance is divided in the reciprocal ratio of the sum of the first two masses to the third mass ;
W so on. Indeed, we may find the centres of gravities of any groups of two, three, or ten, in
any order, & then the groups of two may be joined to the threes, the tens, or what not, also in
any order whatever ; & in every case, in precisely the same manner, we shall arrive at the
common centre of gravity of the whole mass. This is evidently the case, for the reason that
any number of masses can be reckoned as a single mass, since it is only a question of the
number of points in the mass & the sum of the distances of all the points ; the sum of the
masses constitute a mass, & the sums of the distances a sum of distances, merely by taking
them as a whole. Moreover, since, by the general demonstration given above, a centre of
gravity is always obtained, & since this centre is unique, it follows that, no matter in what
order the operations are performed, the same centre is arrived at in every case.

255. From the above we have a theorem, which is also well known, namely : — // the Hence, a theorem,
centres of gravity of several masses all lie in one & the same straight line, then the centre of Which "the "centre
gravity of the whole set will also lie in the same straight line. This indicates a method for of gravity for con-
investigating the centres of gravity also in the case of many continuous figures. Thus,

in Fig. 38, the centre of gravity of the whole triangle is at that point, which cuts off, from
the straight line drawn through the vertex of any angle to the middle point of the base
opposite to it, one-third of its length on the side nearest to the base. For, the centre of
gravity of every line drawn parallel to the base, such as FH, since each of them is bisected
by BD, lies in this latter straight line. Hence the centre of gravity of the area formed
from them lies in this straight line BD ; as it also does in GE for a similar reason ; that
is to say, it is at the point C. The same method can be applied to some solid figures, such
as pyramids. But I omit all this here, just as I do all the other matters relating to the
finding of the centre of gravity for diverse curved lines, surfaces & solids, to be derived from
what has been proved, but in which my theory is in agreement with the usual fundamental
principles ; I will only remark once again that these all will follow in due course when once
it has been shown that for all masses there exists a centre of gravity, & that there is only
one ; and from this indeed there follows also the theorem that, although the areas
FAGH, FBH are unequal, yet the sums of the distances from the straight line FH of all
the points forming them are equal to one another.

256. In the ordinary method it is quite another thing. Afterthat, in Fig. 40, the The difficulty of
common centre of gravity of the masses A & B has been found, for the third mass, whose ^ary' method.
centre is D, join DC and divide it at F in the reciprocal ratio of D to A + B, then

F is obtained as the common centre for all three masses. If, first of all, the common
centre E of the masses D & B had been found, & AE were joined, & the latter
divided at F in the reciprocal ratio of the masses A & B + D ; then the point of section,

198

PHILOSOPHISE NATURALIS THEORIA

Similis difficultas
in summa, & mul-
tiplicatione plurium
numerorum, & in
vi composita ex
pluribus : methodus
componendi simul
omnes.

sectionis punctum pro centre gravitatis. Nisi
generaliter demonstratum fuisset, haberi sem-
per aliquod, & esse unicum gravitatis cen-
trum ; oporteret hie iterum demonstrare id
novum sectionis punctum fore idem, ac illud
prius : sed per singulos casus ire, res infi-
nita esset, cum diversae rationes conjungendi
massas eodem redeant, quo diversi ordines
litterarum conjungendarum in voces, de qua-
rum multitudine immensa in exiguo etiam ter-
minorum numero mentionem fecimus num. 1 14.
[120] 257. Atque hie illud quidem accidit,
quod in numerorum summa, & multiplica-
tione experimur, ut nimirum quocunque ordine

Consensus e j u s
methodi cum com-
muni per parallelo-
gramma.

FIG. 40.

Demonstr a t i o
generalis methodi.

accipiantur numeri, vel singuli, ut

addantur numero jam invento, vel ipsum multiplicent, vel plurium aggregata seorsum
addita, vel multiplicata ; semper ad eundem demum deveniatur numerum post omnes,
qui dati fuerant, adhibitos semel singulos ; ac in summa patet facile deveniri eodem, & in
multiplication potest res itidem demonstrari etiam generaliter, sed ea hue non pertinent.
Pertinet autem hue magis aliud ejusmodi exemplum petitum a compositione virium, in
qua itidem si multse vires componantur communi methodo componendo inter se duas per
diagonalem parallelogrammi, cujus latera eas exprimant, turn hanc diagonalem cum tertia,
& ita porro ; quocunque ordine res procedat, semper ad eandem demum post omnes adhibitas
devenitur. Hujusmodi compositione plurimarum virium generali jam indigebimus, & ad
absolutam demonstrationem requiritur generalis expressio compositionis virium quotcunque,
qua uti soleo. Compono nimirum generaliter motus, qui sunt virium effectus, & ex effectu
composite metior vim, ut e spatiolo, quod dato tempusculo vi aliqua percurreretur, solet
ipsa vis simplex quselibet sestimari. Assumo illud, quod & rationi est consentaneum, &
experimentis constat, & facile etiam demonstratur consentire cum communi methodo com-
ponendi vires, ac motus per parallelogramma, nimirum punctum solicitatum simul initio
cujusvis tempusculi actione conjuncta virium quarumcunque, quarum directio, & magnitude
toto tempusculo perseveret eadem, fore in fine ejus tempusculi in eo loci puncto, in quo esset,
si singulae eadem intensitate, & directione egissent aliae post alias totidem tempusculis, quot
sunt vires, cessante omni nova solicitatione, & omni velocitate jam producta a vi qualibet post
suum tempusculum : turn rectam, quae conjungit primum illud punctum cum hoc postremo,
assume pro mensura vis ex omnibus compositse, quae cum eadem perseveret per totum
tempusculum ; punctum mobile utique per unicam illam eandem rectam abiret. Quod
si & velocitatem aliquam habuerit initio illius tempusculi jam acquisitam ante ; assume
itidem, fore in eo puncto loci, in quo esset, si altero tempusculo percurreret spatiolum, ad
quod determinatur ab ilia velocitate, altero spatiolum, ad quod determinatur a vi, sive aliis
totidem tempusculis percurreret spatiola, ad quorum singula determinatur a viribus singulis.

258. Hue recidere methodum compon-
endi per parallelogramma facile constat ; si
enim in fig. 41 componendi sint plures motus,
vel vires expressae a rectis PA, PB, PC, &c, &
incipiendo a binis quibusque PA, PB, eae com-
ponantur per parallelogrammum PAMB, turn
vis composita PM cum tertia PC per parallelo-
grammum PMNC, & ita porro ; [121] patet,
ad idem loci punctum N per haec parallelo-
gramma definitum debere devenire punctum
mobile, quod prius percurrat PA, turn AM par-
allelam, & aequalem PB ; turn MN parallelam,
& aequalem PC, atque ita porro additis quot-
cunque aliis motibus, vel viribus, quae per

FIG. 41.

N

nova parallela, & aequalia parallelogrammorum latera debeant componi.

259. Deveniretur quidem ad idem punctum N, si alio etiam ordine componerentur
ii motus, vel vires, ut compositis viribus PA, PC per parallelogrammum PAOC, turn vi
PO cum vi PB per novum parallelogrammum, quod itidem haberet cuspidem in N ; sed
eo deveniretur alia via PAON. Hoc autem ipsum, quod tarn multis viis, quam multas diversae
plurium'compositiones motuum, ac virium exhibere possunt, eodem semper deveniri debeat, sic
generaliter demonstro. Si assumantur ultra omnia puncta, ad quse per ejusmodi compositiones
deveniri potest, planum quodcunque ; ubi punctum mobile percurrit lineolam pertinentem
ad quencunque determinatum motum, habet eundem perpendicularem accessum ad id
planum, vel recessum ab eo, quocunque tempusculo id fiat, sive aliquo e prioribus, sive

A THEORY OF NATURAL PHILOSOPHY 199

F, would again be obtained as the centre of gravity. Now, unless it had been already proved
in general that there always was one centre of gravity, & only one, it would be necessary
here to demonstrate afresh that the new point of section was the same as the first one.
But to do this for every single instance would be an endless task ; for diverse ways of joining
the masses come to the same thing as diverse orders of joining up letters to form words ;
& I have already, in Art. 114, remarked upon the immense number of these even with a
small number of letters.

257. Indeed the same thing happens in the case of addition & multiplication; for A similar difficulty
we find, for instance, no matter what the order is in which the numbers are taken, whether a^ulnOT^roduct
they are taken singly, & added to the number already obtained, or multiplied, or whether of several num-
the addition or multiplication is made with a group of several of them ; the same number ^y ^)i^Iso in, *
is arrived at finally after all those that have been given have been used each once. Now from several forces ;
in addition it is easily seen that the result obtained is the same ; & for multiplication also the m(*hod of

. ' .. , , . . t compounding them

the matter can be easily demonstrated ; but we are not concerned with these proofs here, all at one time.

Moreover, there is another example of this sort that is far more suitable for the present

occasion, derived from the composition of forces. In this, if several forces are compounded

in the ordinary manner, by compounding two of them together by means of the diagonal

of the parallelogram whose sides represent the forces, & then this diagonal with a third

force, & so on. In whatever order the operations are performed we always arrive at the

same force finally, after all the given forces have been used. We shall now need a general

composition of very many forces, & for rigorous proof we must have a general representation

for the composition of any number of forces, such as the one I usually employ. Thus, I

in general compound the motions, which are the effects of the forces, & measure the force

from the resultant of the effects ; so that any simple force is usually estimated by the small

interval of space through which the force moves its point of application in a given short

interval of time. I make an assumption, which is not only a reasonable one, but is also

verified by experiment, & further one which can be easily shown to agree with the usual

method for the composition of forces & motions by means of the parallelogram. Thus,

I assume that a point, which is influenced simultaneously, at the beginning of any short

interval of time by the joint action of any forces whatever, whose directions & magnitudes

continue unchanged during the whole of the interval, will be at the end of the interval

in the same position in space, as if each of the forces had acted independently, one after

another, with the same intensity & in the same direction, during as many intervals of time

as there are forces ; where each fresh influence & the velocity already produced by any

one of the forces ceases at the end of the interval that corresponds to it. Then I take the

straight line which joins the initial point to the final point as the measure of the force that

is the resultant of them all, & that this force will be represented by this same straight line

during the whole of the interval of time, & that the moving point will traverse in every

case that straight line & that one only. But if, moreover, at the beginning of the interval

of time, the point should have a velocity previously acquired, then I also assume that it

would occupy that position in space that it would have occupied if during another interval

of time it had passed over an .interval of space, determined by this other velocity, which

is itself determined by the force ; or if it had passed over as many intervals of spaces in

as many intervals of time as there are forces determining the initial velocity.

258. It is easily seen that the method of composition by means of the parallelogram Agreement of this
comes to the same thing. For, if, in Fig. 41, the several motions or forces to be compounded J^^j^^f*11 ^
are represented by PA, PB, PC, &c. ; &, beginning with any two of them, PA & PB, these of the6 pL™Uek>S
are compounded by means of the parallelogram PAMB, then the resultant force PM is gram-
compounded with a third PC by means of the parallelogram PMNC, & so on ; it is clear

that the moving point must reach the same point of space, N, determined by these
parallelograms, as it would have done if it had traversed PA, then AM parallel & equal to
PB, & then MN parallel & equal to PC ; & so on, for any number of additional motions
or forces, which have to be compounded by fresh straight lines equal & parallel to the sides
of the parallelograms.

259. Now the same point N would be reached also, if these motions or forces were General proof of
compounded in another order, say, by first compounding PA & PC by means of the
parallelogram PAOC, then the force PO with the force PB by another parallelogram, which

has its fourth vertex at N, although the point is reached by another path PAON. The
fact that the same point is bound to be reached, by each of the many paths that correspond
to the many different orders of compounding several motions or forces, I prove in general
as follows. Imagine a plane drawn beyond any point that could be reached owing to
compositions of this kind ; then, when a moving point traverses a short path corresponding
to any given motion, there is the same perpendicular approach towards the plane, or recession
from it, in whichever of the short intervals of time it takes place, whether one of those at

200 PHILOSOPHIC NATURALIS THEORIA

aliquo e postremis, vel mediis. Nam ea lineola ex quocunque puncto discedat, ad quod
deventum jam sit, habet semper eandem & longitudinem, & directionem, cum eidem e
componentibus parallela esse debeat, & sequalis. Quare summa ejusmodi accessuum, ac
summa recessuum erit eadem in fine omnium tempusculorum, quocunque ordine dispon-
antur lineolae hae parallels, & sequales lineolis componentibus, adeoque etiam id, quod
prodit demendo recessuum summam a summa accessuum, vel vice versa, erit idem, & distantia
puncti postremi, ad quod deventum est ab illo eodem piano, erit eadem. Inde autem
sponte jam fruit id, quod demonstrandum erat, nimirum punctum illud esse idem semper.
Si enim ad duo puncta duabus diversis viis deveniretur, assumpto piano perpendiculari ad
rectam, quae ilia duo puncta jungeret, distantia perpendicularis ab ipso non esset utique
eadem pro utroque, cum altera distantia deberet alterius esse pars.

- Porro similis admodum est etiam methodus, qua utor ad demonstrandum
manente etiam ubi praeclarissimum Newtoni theorema, in quod coalescunt simul duo, quae superius innui, &
v^esntmutusenqac ^uc reducuntur. Si quotcunque materice puncta utcunque disposita, 6? in quotcunque utcunque
ejus demonstra- disjunctas massas coalescentia habeant velocitates quascunque cum directionibus quibuscunque,
y -prceterea urgeantur viribus mutuis quibuscunque, quce in binis quibusque punctis cequaliter
agant in plagas oppositas ; centrum commune gravitatis omnium vel quiescet, vel movebitur
uniformiter in directum eodem motu, quern haberet, si nulla adesset mutua punctorum actio in
se invicem. Hoc autem theorema sic generaliter, & admodum facile, ac luculenter demon-
stratur. [122] Concipiamus vires singulas per quodvis determinatum tempusculum
servare directiones suas, & magnitudines : in fine ejus tempusculi punctum materiae
quodvis erit in eo loci puncto, in quo esset, si singularum virium effectus, vel effectus
velocitatis ipsius illi tempusculo debitus, haberentur cum eadem sua directione, & magnitudine
alii post alios totidem tempusculis, quot vires agunt. Assumantur jam totidem tempuscula,
quot sunt punctorum binaria diversa in ea omni congerie, & praeterea unum, ac primo
tempusculo habeant omnia puncta motus debitos velocitatibus illis suis, quas habent initio
ipsius, singula singulos ; turn assignato quovis e sequentibus tempusculis cuivis binario,
habeat binarium quodvis tempusculo sibi respondente motum debitum vi mutuae, quae
agit inter bina ejus puncta, ceteris omnibus quiescentibus. In fine postremi tempusculi
omnia puncta materiae erunt in hac hypothesi in iis punctis loci, in quibus revera esse debent
in fine unici primi tempusculi ex actione conjuncta virium omnium cum singulis singulorum
velocitatibus.

- Concipiatur jam ultra omnia ejusmodi puncta planum quodcunque. Primo
ex illis tot assumptis tempusculis alia puncta accedent, alia recedent ab eo piano, & summa
omnium accessuum punctorum omnium demptis omnibus recessibus, si qua superest, vel
vice versa summa recessuum demptis accessibus, divisa per numerum omnium punctorum,
aequabitur accessui perpendiculari ad idem planum, vel recessui centri gravitatis communis ;
cum summa distantiarum perpendicularium tarn initio tempusculi, quam in fine, divisa
per eundem numerum exhibeat ipsius communis centri gravitatis distantiam juxta num.
246. Sequentibus autem tempusculis manebit utique eadem distantia centri gravitatis
communis ab eodem piano nunquam mutata ; quia ob sequales & contraries punctorum
motus, alterius accessus ab alterius recessu aequali eliditur. Quamobrem in fine omnium
tempusculorum ejus distantia erit eadem, & accessus ad planum erit idem, qui esset, si
solae adfuissent ejusmodi velocitates, quae habebantur initio ; adeoque etiam cum omnes
vires simul agunt, in fine illius unici tempusculi habebitur distantia, quas haberetur, si
vires illae mutuae non egissent, & accessus aequabitur summae accessuum, qui haberentur ex
solis velocitatibus, demptis recessibus. Si jam consideretur secundum tempusculum in
quo simul agant vires mutuse, & velocitates ; debebunt considerari tria genera motuum :
primum eorum, qui proveniunt a velocitatibus, quae habebantur initio primi tempusculi ;
secundum eorum, qui proveniunt a velocitatibus acquisitis actione virium durante per
primum tempusculum ; tertium eorum, qui proveniunt a novis actionibus virium mutuarum,
quae ob mutatas jam positiones concipiantur aliis directionibus agere per totum secundum
tempusculum. Porro quoniam hi posteriorum duorum generum motus [123] sunt in
singulis punctorum binariis contrarii, & aequales ; illi itidem distantiam centri gravitatis
ab eodem piano, & accessum, vel recessum debitum secundo tempusculo non mutant ;

A THEORY OF NATURAL PHILOSOPHY 201

the commencement, or one of those at the end, or one in the middle. For the short line,
whatever point it has for its beginning & whatever point it finally reaches, must always
have the same length & direction ; for it is bound to be parallel & equal to the same one
of the components. Hence the sum of these approaches, & the sum of these recessions,
will be the same at the end of the whole set of intervals of time, no matter in what order .
these little lines, which are parallel & equal to the component lines, are disposed. Hence
also, the result obtained by taking away the sum of the recessions from the sum of the
approaches, or conversely, will be the same ; & the distance of the ultimate point reached
from the plane will be the same. Thus there follows immediately what was required to
be proved, namely, that the point is the same point in every case. For, if two points could
be reached by any two different paths, & a plane is taken perpendicular to the line joining
those two points, then it is impossible for the perpendicular distance from this plane to be
exactly the same for both points, since the one distance must be a part of the other.

260. Further, the method, which I make use of to prove a most elegant theorem Theorem relating
of Newton, is exactly similar ; in it the two noted above are combined, & come to the *?_J;h,?, P?rman?*t

,. - . It- I 1- J • C J T

same thing. // any number of points of matter, disposed in any manner, t5 coalescing of gravity even

to form any number of separate masses in any manner , have any velocities in any direction; JJ^utu offeree's

^f *'/, in addition, the points are under the influence of any mutual forces whatever, these forces acting ; the first

acting on each pair of points equally in opposite directions ; then the common centre of gravity steps of thc Proof-

of the whole is either at rest, or moves uniformly in a straight line with the same motion as it

would have if there were no mutual action of the points upon one another. Now this theorem

is quite easily & clearly proved in all generality as follows. Suppose that each force maintains

its direction & magnitude during any given short intervals of time ; at the end of the interval

any point of matter will occupy that point of space, which it would occupy if the effects

for each of the forces (i.e., the effect of each velocity corresponding to that interval of time)

were obtained, one after another, in as many intervals of time as there are forces acting,

whilst each maintains its own direction & magnitude the same as before. Now take

as many small intervals of time as there are different pairs of points in the whole

group, & one interval in addition ; & in the first interval of time let all the points have

the motions due to the velocities that they have at the beginning of the interval of time

respectively. Then, any one of the subsequent intervals of time being assigned to any

chosen pair of points, let any pair have, in the interval of time proper to it, that motion

which is due to the mutual force that acts between the two points of that pair, whilst all

the others remain at rest. Then at the end of the last of these intervals of time, each

point of matter will be, according to this hypothesis, at that point of space which it is

bound to occupy at the end of a single first interval of time, under the conjoint action of

all the mutual forces, each having its corresponding velocity.

261. Now imagine a plane situated beyond all points of this kind. Then, in the first Continuation of
place, for these little intervals of time of which we have assumed the number stated, some the demonstratlon-
of the points will approach towards, & some recede from the plane ; & the sum of all these
approaches less the sum of all the recessions, if the former is the greater, & conversely, the

sum of the recessions less the sum of the approaches, divided by the number of all the points,
will be equal to the perpendicular approach of the common centre of gravity to the plane,
or the recession from it. For, by Art. 246, the sum of the perpendicular distances,
both at the beginning & at the end of the interval of time will represent the distance of
the common centre of gravity itself. Further, in subsequent intervals, this distance of
the common centre of gravity from the plane will remain in every case quite unchanged ;
because, on account of the equal & opposite motions of pairs of points, the approach of the
one will be cancelled by the equal recession of the other. Hence, at the end of all the intervals
the distance of the centre of gravity will be the same, & its approach towards the plane
will be the same, as it would have been if there had existed no velocities except those
which it had at the beginning of the interval ; thus, too, when all the forces act together,
at the end of the single interval of time there will be obtained that distance, which would
have been obtained if the mutual forces had not been acting ; & the approach will be equal
to the sum of the approaches, less the recessions, acquired from the velocities alone. If
now we would consider a second interval of time, in which we have acting the mutual forces,
& the velocities ; we shall have to consider three kinds of motions. Firstly, those that come
from the velocities which exist at the beginning of the interval ; secondly, those which
arise from the velocities acquired through the action of forces lasting throughout the first
interval ; & thirdly, those which arise from the new actions of the mutual forces, which may
be assumed to be acting in fresh directions, due to the change in the positions of the points
during the whole of this second interval. Further, since the latter of the last two kinds,
of motion are equal & opposite for each pair of points, these two kinds also will not change
the distance of the centre of gravity from the plane & the approach towards it or recession

202 PHILOSOPHIC NATURALIS THEORIA

sed ea habentur, sicuti haberentur, si semper durarent solae illse velocitates, quae habebantur
initio primi tempusculi ; & idem redit argumentum pro tempusculo quocunque : singulis
advenientibus tempusculis accedet novum motuum genus durantibus cum sua directione,
& magnitudine velocitatibus omnibus inductis per singula praecedentia tempuscula, ex
quibus omnibus, & ex nova actione vis mutuae, componitur quovis tempusculo motus
puncti cujusvis : sed omnia ista inducunt motus contraries, & sequales, adeoque summa
accessuum, vel recessuum ortam ab illis solis initialibus velocitatibus non mutant.

Progressus ulterior. 262. Quod si jam tempusculorum magnitudo minuatur in infinitum, aucto itidem

in infinitum intra quodvis finitum tempus eorundem numero, donee evadat continuum
tempus, & continua positionum, ac virium mutatio ; adhuc centrum gravitatis in fine
continui temporis cujuscunque, adeoque & in fine partium quarumcunque ejusdem
temporis, habebit ab eodem piano distantiam perpendicularem, quam haberet ex solis
velocitatibus habitis initio ejus temporis, si nullae deinde egissent mutuae vires ; & accessus
ad illud planum, vel recessus ab eo, aequabitur summae omnium accessuum pertinentium
ad omnia puncta demptis omnibus recessibus, vel vice versa. Is vero accessus, vel recessus
assumptis binis ejus temporis partibus quibuscunque, erit proportionalis ipsis temporibus.
Nam singulorum punctorum accessus, vel recessus orti ab illis velocitatibus initialibus
perseverantibus, adeoque ab motu aequabili, sunt in ratione eadem earundem temporis
partium ; ac proinde & eorum summae in eadem ratione sunt.

Demonstration is 2&3' In.de vero prona jam est theorematis demonstratio. Ponamus enim, centrum

finis- gravitatis quiescere quodam tempore, turn moveri per aliquod aliud tempus. Debebit

utique aliquo momento temporis esse in alio loci puncto, diverse ab eo, in quo erat initio
motus. Sumatur pro prima e duabus partibus temporis continui pars ejus temporis, quo
punctum quiescebat, & pro secunda tempus ab initio motus usque ad quodvis momentum,
quo centrum illud gravitatis devenit ad aliud aliquod punctum loci. Ducta recta ab
initio ad finem hujusce motus, turn accepto piano aliquo perpendiculari ipsi productae
ultra omnia puncta, centrum gravitatis ad id planum accederet secunda continui ejus
temporis parte per intervallum aequale illi rectae, & nihil accessisset primo tempore, adeoque
accessus non fuissent proportionales illis partibus continui temporis. Quamobrem ipsum
commune gravitatis centrum vel semper quiescit, vel movetur semper. Si autem movetur,
debet moveri in directum. Si enim omnia puncta loci, per quje transit, non jacent in
directum, sumantur tria in dire-[i24]-ctum non jacentia, & ducatur recta per prima duo,
quas per tertium non transibit, adeoque per ipsam duci poterit planum, quod non transeat
per tertium, turn ultra omnem punctorum congeriem planum ipsi parallelum. Ad id
secundum nihil accessisset illo tempore, quo a primo loci puncto devenisset ad secundum,
& eo tempore, quo ivisset a secundo ad tertium, accessisset per intervallum sequale distantiae
a priore piano, adeoque accessus iterum proportionales temporibus non fuissent. Demum
motus erit aequabilis. Si enim ultra omnia puncta concipiatur planum perpendiculare
rectae, per quam movetur ipsum centrum commune gravitatis, jacens ad earn partem, in
quam id progreditur, accessus ad ipsum planum erit totus integer motus ejusdem centri ;
adeoque cum ii accessus debeant esse proportionales temporibus ; erunt ipsis temporibus
proportionales motus integri ; & idcirco non tantum rectilineus, sed & uniformis erit motus ;
unde jam evidentissime patet theorema totum.

Coraiiarium de 264. Ex eodem fonte, ex quo profluxit hoc generale theorema, sponte fluit hoc aliud

quantitate motus . .' r. „, , ,r

in eandem piagam ut consectanum : quantitas motus in Mundo conservatur semper eadem, si ea computetur
conservata in secunaum directionem quacunque ita, ut motus secundum directionem oppositam consider etur

Mundo. , . •* . 7 . c • •

ut negativus, e-jusmodi motuum contranorum summa subtracta a summa directorum. 01 enim
consideretur eidem direction! perpendiculare planum ultra omnia materiae puncta,
quantitas motus in ea directione est summa omnium accessuum, demptis omnibus recessibus,
quae summa tempusculis aequalibus manet eadem, cum mutuae vires inducant accessus,
& recessus se mutuo destruentes ; nee ejusmodi conservation! obsunt liberi motus ab anima
nostra producti, cum nee ipsa vires ullas possit exerere, nisi quae agant in partes oppositas
aequaliter juxta num. 74.

^Equaiitas actionis 265. Porro ex illo Newtoniano theoremate statim jam profluit lex actionis, & reactionis

& reactionis in ggqualium pro massis omnibus. Nimirum si duae massae quaecunque in se invicem agant

massis inde orta. .\. X . „ . . , i_« •* vi_ i • -11

viribus quibuscunque mutuis, & inter smgula punctorum binana aequalibus ; bmas illae

A THEORY OF NATURAL PHILOSOPHY 203

from it corresponding to the second interval. Hence, these will be the same as they would
have been, if those velocities that existed at the beginning of the first interval had persisted
throughout ; & the same argument applies to any interval whatever. Each interval as it
occurs will yield a fresh kind of motions, all the velocities induced during each of the preceding
intervals remaining the same in direction & magnitude ; & from all of these, & the fresh
action of the mutual force, there is compounded for any interval the motion of any point.
But all the latter induce equal & opposite motions in pairs of points ; & thus the sum of
the approaches or recessions arising from the velocities alone are unchanged by the mutual
forces.

262. Now if the length of the interval of time is indefinitely diminished, the number Further steps in
of intervals in any given finite time being thus indefinitely increased, until we acquire the
continuous time, & continuous change of position & forces ; still the centre of gravity at

the end of any continuous time, & thus also at the end of any parts of that time, will have
that perpendicular distance from the plane, which it would have had, due to the velocities
that existed at the beginning of the time, if no mutual forces had been acting. The approach
towards the plane, or the recession from it, will be equal to the sum of all the approaches
corresponding to all the points less the sum of all the recessions, or vice versa. Indeed,
any two parts of the time being taken, this approach or recession will be proportional to these
parts of the time. For the approach or recession, for each of the points, arising from the
velocities that persist throughout & thus also from uniform motion, is proportional for all
parts of the time ; & hence also, their sums are proportional.

263. The complete proof now follows immediately from what has been said above. Conclusion of the
For, let us suppose that the centre of gravity is at rest for a certain time, & then moves demonstration,
for some other time. Then at some instant of time it is bound to be at some other

point of space different from that in which it was at the beginning of the motion. Of
two parts of continuous time, let us take as the first part of the time, that in which the
point is at rest ; & for the second part, the time between the beginning of the motion &
the instant when the centre of gravity reaches some other point of space. Draw a straight
line from the beginning to the end of this motion, & take any plane perpendicular to this
line produced beyond all the points ; then the centre of gravity would approach towards
the plane, in the second part of the continuous time, through an interval equal to the
straight line, but in the first part of the time there would have been no approach at all ;
hence the approaches would not have been proportional to those parts of the continuous
time. Hence the centre of gravity is always at rest, or is always in motion. Further,
if it is in motion, it must move in a straight line. For, if all points of space, through
which it passes, do not lie in a straight line, take three of them which are not collinear ;
& draw a straight line through the first two, which does not pass through the third ; then
it will be possible to draw through this straight line a plane which will not pass through
the third point ; & consequently, a plane parallel to it beyond the whole group of points.
To this second plane there will be no approach at all for the time, during which the
centre of gravity would travel from the first point of space to the second ; & for that
time, during which it would go from the second point to the third, there would be an
approach through an interval equal to its distance from the first plane ; & thus, once
again, the approaches would not be proportional to the times. Lastly, the motion will
be uniform. For, if we imagine a plane drawn beyond all the points, perpendicular to
the straight line along which the centre of gravity moves, & on that side to which there
is approach, then the approach to that plane will be the whole of the entire motion of
the centre ; hence, since these approaches must be proportional to the times, the whole
motions must be proportional to the times ; & therefore the motion must not only be
rectilinear, but also uniform. Thus, the whole theorem is now perfectly clear.

264. From the same source as that from which we have drawn the above general theorem, Corollary with
there is obtained immediately the following also, as a corollary. The quantity of motion conservation* of *«ie
in the Universe is maintained always the same, so long as it is computed in some given direction quantity of motion
in such a way that motion in the opposite direction is considered negative, £5" the sum of the ^given^rection "*
contrary motions is subtracted from the sum of the direct motions. For, if we consider a plane
perpendicular to this direction lying beyond all points of matter, the quantity of motion

in this direction is the sum of all the approaches with the sum of the recessions subtracted ;
this sum remains the same for equal times, since the mutual forces induce approaches &
recessions that cancel one another. Nor is such conservation affected by free motions
that are the result of our will ; since it cannot exert any forces either, except such as act
equally in opposite directions, as was proved in Art. 74.

265. Further, from the Newtonian theorem, we have immediately the law of equal Equality of action
action & reaction for all masses. Thus, if any two masses act upon one another with any massesa the° result
mutual forces, which are also equal for each pair of points, the two masses will acquire, of this theorem.

204

PHILOSOPHIC NATURALIS THEORIA

massae acquirent ab actionibus mutuis summas motuum aequales in partes contrarias, &
celeritates acquisitae ab earum centris gravitatis in partes oppositas, componendae cum
antecedentibus ipsarum celeritatibus, erunt in ratione reciproca massarum. Nam centrum
commune gravitatis omnium a mutuis actionibus nihil turbabitur per hoc theorema, &
sive ejusmodi vires agant, sive non agant, sed solius inertiae effectus habeantur ; semper
ab eodem communi gravitatis centro distabunt ea bina gravitatis centra hinc, & inde in
directum ad distantias reciproce proportionales massis ipsis per num. 253. Quare si praeter
priores motus ex vi inertiae uniformes, ob actionem mutuam adhuc magis ad hoc commune
centrum accedet alterum ex iis, vel ab eo recedet ; accedet & alterum, [125] vel recedet,
accessibus, vel recessibus reciproce proportionalibus ipsis massis. Nam accessus ipsi, vel
recessus, sunt differentiae distantiarum habitarum cum actione mutuarum virium a distantiis
habendis sine iis, adeoque erunt & ipsi in ratione reciproca massarum, in qua sunt totae
distantiae. Quod si per centrum commune gravitatis concipiatur planum quodcumque,
cui quaepiam data directio non sit parallela ; summa accessuum, vel recessuum punctorum
omnium massae utriuslibet ad ipsum secundum earn directionem demptis oppositis, quae
est summa motuum secundum directionem eandem, aequabitur accessui, vel recessui centri
gravitatis ejus massae ducto in punctorum numerum ; accessus vero, vel recessus alterius
centri ad accessum, vel recessum alterius in directione eadem, erit ut secundus numerus
ad primum ; nam accessus, & recessus in quavis directione data sunt inter se, ut accessus,
vel recessus in quavis alia itidem data ; & accessus, ac recessus in directione, quae jungit
centra massarum, sunt in ratione reciproca ipsarum massarum. Quare productum accessus,
vel recessus centri primae massae per numerum punctorum, quae habentur in ipsa, aequatur
producto accessus, vel recessus secundae per numerum punctorum, quae in ipsa continentur ;
nimirum ipsae motuum summae in ilia directione computatorum aequales sunt inter se,
in quo ipsa actionis, & reactionis aequalitas est sita.

molhbus.

inde leges coliisi- 266. Ex hac actionum, & reactionum aequalitate sponte profluunt leges collisionis

^L : i/^^P corporum, quas ex hoc ipso principio Wrennus olim, Hugenius, & Wallisius invenerunt

m corpon- • * * • i XT i XT * T* • • •

bus eiasticis, & simul, ut in hac ipsa lege Naturae exponenda Newtonus etiam memorat Pnncipiorum
jjj^ T _ Ostendam autem, quo pacto generales formulae inde deducantur tarn pro directis
collisionibus corporum mollium, quam pro perfecte, vel pro imperfecte elasticorum.
Corpora mollia dicuntur ea, quae resistunt mutationi figurae, seu compressioni, sed compressa
nullam exercent vim ad figuram recuperandam, ut est cera, vel sebum : corpora elastica,
quae figuram amissam recuperare nituntur ; & si vis ad recuperandam sit aequalis vi ad
non amittendam ; dicuntur perfecte elastica, quae quidem, ut & perfecte mollia, nulla,
ut arbitror, sunt in Natura ; si autem imperfecte elastica sunt, vis, quae in amittenda,
ad vim, quae in recuperanda figura exercetur, datam aliquam rationem habet. Addi solet
& tertium corporum genus, quae dura dicunt, quae nimirum figuram prorsus non mutent ;
sed ea itidem in Natura nusquam sunt juxta communem sententiam, & multo magis nulla
usquam in hac mea Theoria. Adhuc qui ipsa velit agnoscere, is mollia consideret, quae
minus, ac minus comprimantur, donee compressio evadat nulla ; & ita quae de mollibus
dicentur, aptari poterunt duris multo meliore jure, quam alii elasticorum leges ad ipsa
transferant, considerando elasticitatem infinitam ita, ut figura nee mutetur, nee se restituat ;
[126] nam si figura non mutetur, adhuc concipi poterit, impenetrabilitatis vi amissus
motus, ut amitteretur in compressione ; sed ad supplendam vim, quae exeritur ab eiasticis
in recuperanda figura, non est, quod concipi possit, ubi figura recuperari non debet. Porro
unde corpora mollia sint, vel elastica hie non quaero ; id pertinet ad tertiam partem,
quanquam id ipsum innui superius num. 199 ; sed leges quae in eorum collisionibus observari
debent, & ex superiore theoremate fluunt, expono. Ut autem simplicior evadat res,
considerabo globes, atque hos ipsos circumquaque circa centrum, in eadem saltern ab
ipso centro distantia, homogeneos, qui primo quidem concurrant directe ; nam deinde
ad obliquas etiam collisiones faciemus gradum.

Praeparatio pro col- 267. Porro ubi globus in globum agit, & ambo paribus a centro distantiis homogenei
pianorum,8l0^Trc™-' sunt, facile constat, vim mutuam, quse est summa omnium virium, qua singula alterius
puncta agunt in singula puncta alterius, habituram semper directionem, quae jungit centra ;

lorum-

A THEORY OF NATURAL PHILOSOPHY 205

as a result of the mutual actions, sums of motions that are equal in opposite directions ;
& the velocities acquired by their centres of gravity in opposite directions, being compounded
of the foregoing velocities, will be in the inverse ratio of the masses. For, by the theorem,
the common centre of gravity of the whole will not be disturbed in the slightest degree
by the mutual actions, whether such forces act or whether they do not, but only the effects
of inertia will be obtained ; hence the two centres of gravity will always be distant from
this common centre of gravity, one on each side of it, in a straight line with it, at distances
that are reciprocally proportional to the masses, as was proved in Art. 253. Hence, if in
addition to the former uniform motions due to the force of inertia, one of the two masses,
on account of the mutual action, should approach still nearer to the common centre, or
recede still further from it ; then the other will either approach towards it or recede from
it, the approaches or recessions being reciprocally proportional to the masses. For these
approaches or recessions are the differences between the distances that are obtained when
there is action of mutual forces & the distances when there is not ; & thus, they too will
be in the inverse ratio of the masses, such as the whole distances are. But if we imagine
a plane drawn through the common centre of gravity, & that some given direction is not
parallel to it, then the sum of the approaches or recessions of all the points of either of the
masses with respect to this plane, the opposites being subtracted (which is the same thing
as the sum of the motions in this direction) will be equal to the approach or the recession
of the centre of gravity of that mass multiplied by the number of points in it. But the
approach or recessions of the centre of the one is to the approach or recession of the centre
of the other, in the same direction, as the second number is to the first ; for the approaches
or recessions in any given direction are to one another as the approaches or recessions in any
other given direction ; & the approaches or recessions along the line joining the two masses
are inversely proportional to the masses. Therefore the product of the approach or recession
of the centre of the first mass, multiplied by the number of points in it, is equal to the
approach or recession of the centre of the second mass, multiplied by the number of points
that are contained in it. Thus the sums of the motions in the direction under consideration
are equal to one another ; & in this is involved the equality of action & reaction.

266. From this equality of action & reaction there immediately follow the laws for Hence the laws for
collision of bodies, which some time ago Wren, Huygens & Wallis derived from this very Jetton ' be tween
principle at about the same time, as Newton also mentioned in the first book of the Principia, ^e forc?s for elas-
when expounding this law of Nature. Now I will show how general formulas may be bodies"*16
derived from it, both for the direct collision of soft bodies, & also for perfectly or imperfectly

elastic bodies. By soft bodies are to be understood those, which resist deformation of
their shapes, or compression ; but which, when compressed, exert no force tending to
restore shape ; such as wax or tallow. Elastic bodies are those that endeavour to recover
the shape they have lost ; & if the force tending to restore shape is equal to that tending
to prevent loss of shape, the bodies are termed perfectly elastic ; &, just as there are no
perfectly soft bodies, there are none that are perfectly elastic, according to my thinking,
in Nature. Lastly, they are imperfectly elastic, if the force exerted against losing shape
bears to the force exerted to restore it some given ratio. It is usual to add a third class of
bodies, namely, such as are called hard ; & these never alter their shape at all ; but these
also, even according to general opinion, never occur in Nature ; still less can they
exist in my Theory. Yet, if anyone wishes to take account of such bodies, they could
consider them as soft bodies which are compressed less & less, until the compression finally
becomes evanescent ; in this way, whatever is said about soft bodies could be adapted to
hard bodies with far more justification than there is for applying some of the laws of
elastic bodies to them, by considering that there is infinite elasticity of such a nature that
the figure neither suffers change nor seeks to restore itself. For, if the figure remains
unchanged, it is yet possible to consider the motion lost due to the force of impenetrability,
& that thus it would be lost in compression ; but to supply the force which in elastic bodies
is exerted for the recovery of shape, there is nothing that can be imagined, when there
is necessarily no recovery of shape. Further, what are the causes of soft or elastic bodies,
I do not investigate at present ; this relates to the third part, although I have indeed mentioned
it above, in Art. 199. But I set forth the laws which have to be observed in collisions
between them, these laws coming out immediately from the theorem given above. Moreover
to make the matter easier, I consider spheres, & these too homogeneous round about the
centre, at any rate for the same distance from that centre ; & these indeed will in the first
place collide directly ; for from direct collision we can proceed to oblique impact also.

267. Now, where one sphere acts upon another, & both of them are homogeneous ^^era'tlo^ "'f
at equal distances from their centres, it is readily shown that the mutual force, which is collisions of spheres,
the sum of all the forces with which each of the points of the one acts on- each of the points Planes & cirdes-
of the other, must always be in the direction of the line joining the two centres. For,

206 PHILOSOPHISE NATURALIS THEORIA

nam in ea recta jacent centra ipsorum globorum, quse in eo homogeneitatis casu facile
constat, esse centra itidem gravitatis globorum ipsorum ; & in eadem jacet centrum com-
mune gravitatis utriusque, ad quod viribus illis mutuis, quas alter globis exercet in alterum,
debent ad se invicem accedere, vel a se invicem recedere ; unde fit, ut motus, quos
acquirunt globorum centra ex actione mutua alterius in alterum, debeant esse in directione,
quae jungit centra. Id autem generaliter extendi potest etiam ad casum, in quo concipiatur,
massam immensam terminatam superficie plana, sive quoddam immensum planum agere
in globum finitum, vel in punctum unicum, ac vice versa : nam alterius globi radio in
infinitum aucto superficies in planum desinit ; & radio alterius in infinitum imminuto,
globus abit in punctum. Quin etiam si massa quaevis teres, sive circa axem quendam
rotunda, & in quovis piano perpendicular! axi homogenea, vel etiam circulus simplex,
agat, vel concipiatur agens in globum, vel punctum in ipso axe constitutum ; res eodem
redit.

Formulae pro cor- 268. Praecurrat jam globus mollis cum velocitate minore, quem alius itidem mollis

pore moih incur- consequatur cum maiore ita, ut centra ferantur in eadem recta, quae ilia coniungit, & hie

rente in molle . * . .1, ,. . ,,..,. T . ., . .J, ° r

lentius progrediens demum mcurrat in ilium, quae dicitur colhsio directa. Is incursus mini quidem non net
m eandem piagam. per immediatum contactum, sed antequam ad contactum deveniant, vi mutua repulsiva
comprimentur partes posteriores praecedentis, & anteriores sequentis, qua; compressio net
semper major, donee ad aequales celeritates devenerint ; turn enim accessus ulterior desinet,
adeoque & ulterior compressio ; & quoniam corpora sunt mollia, nullam aliam exercent
vim mutuam post ejusmodi compressionem, sed cum aequali ilia velocitate pergunt moveri
porro. Haec aequalitas velocitatis, ad quam reducuntur ii duo glo-[i27]-bi, una cum
asqualiate actionis, & reactionis aequalium, rem totam perficient. Sit enim massa, sive
quantitas materiae, globi prascurrentis = q, insequentis = Q ; celeritas illius = c, hujus
= C : quantitas motus illius ante collisionem erit cq, hujus CQ ; nam celeritas ducta per
numerum punctorum exhibet summam motuum punctorum omnium, sive quantitatem
motus ; unde etiam fit, ut quantitas motus per massam divisa exhibeat celeritatem. Ob
actionem, & reactionem aequales, haec quantitas erit eadem etiam post collisionem, post
quam motus totus utriusque massae, erit CQ + cq. Quoniam autem progrediuntur cum
aequali celeritate ; celeritas ilia habebitur ; si quantitas motus dividatur per totam

quantitatem materias ; quae idcirco erit -Q— — . Nimirum ad habendam velocitatem

communem post collisionem, oportebit ducere singulas massas in suas celeritates, &
productorum summam dividere per summam massarum.

Ejus extensio ad 269. Si alter globus 'q quiescat ; satis erit illius celeritatem c considerare = o : & si

rTufs "amissa 'vlei moveatur rnotu contrario motui prioris globi ; satis erit illi valorem negativum tribuere ;

acquisita. ut adeo & hie, & in sequentibus formula inventa pro illo primo casu globorum in eandem

progredientium piagam, omnes casus contineat. In eo autem si libeat invenire celeritatem

amissam a globo Q, & celeritatem acquisitam a globo q, satis erit reducere singulas

formulas

c CQ+cg kCQ+c9_
" "

ad eundem denominatorem, ac habebitur

Cg —eg, CQ— cQ

Q + q ' Q + q '

ex quibus deducitur hujusmodi theorema : ut summa massarum ad massam alteram, ita
differentia celeritatum ad celeritatem ab altera acquisitam, quae in eo casu accelerabit motum
praecurrentis & retardabit motum consequentis.

Transitus ad eias- 270. Ex hisce, quae pertinent ad corpora mollia, facile est progredi ad perfecte elastica.

ies' In iis post compressionem maximam, & mutationem figurse inductam ab ipsa, quae habetur,
ubi ad aequales velocitates est ventum, agent adhuc in se invicem bini globi, donee deveniant
ad figuram priorem, & haec actio duplicabit effectum priorem. Ubi ad sphaericam figuram
deventum fuerit, quod fit recessu mutuo oppositarum superficierum, quae in compressione
ad se invicem accesserant, pergent utique a se invicem recedere aliquanto magis eaedem
superficies, & figura producetur, sed opposita jam vi mutua inter partes ejusdem globi
incipient retrahi, & productio perget fieri, sed usque lentius, donee ad maximam quandam
productionem de-[i28]-ventum fuerit, quae deinde incipiet minui, & globus ad sphaericam
accedet iterum, ac iterum comprimetur quodam oscillatorio, ac partium trepidatione
hinc, & inde a figura sphasrica, ut supra vidimus etiam duo puncta circa distantiam limitis

A THEORY OF NATURAL PHILOSOPHY 207

in that straight line lie the centres of the two spheres ; & these in the case of homogeneity
are easily shown to be also the centres of gravity of the spheres. Also in this straight line
lies the common centre of gravity of both spheres ; & to, or from, it the spheres must approach
or recede mutually, owing to the action of the mutual forces with which one sphere acts
upon the other. Hence it follows that the motion, which the centres of the spheres acquire
through the mutual action of one upon the other, is bound to be along the line which joins the
centres. The argument can also be extended generally, even to include the case in which
it is supposed that an immense mass bounded by a plane surface, or an immense plane
acts upon a finite sphere, or on a single point, or vice versa ; for, if the radius of either of
the spheres is increased indefinitely, the surface ultimately becomes a plane, & if the radius of
either becomes indefinitely diminished, the sphere degenerates into a point. Moreover, if any
round mass, or one contained by a surface of rotation round an axis and homogeneous in any
plane perpendicular to that axis, or even a simple circle, act, or is supposed to act upon a
sphere or point situated in the axis ; it comes to the same thing.

268. Now suppose that a soft body proceeds with a less velocity than another soft F°™uiae for a soft

i • i • f 1 1 • • • i • • i i i • body impinging

body which is following it with a greater velocity, in such a manner that their centres are upon another soft

travelling in the same straight line, namely that which joins them ; & finally let the latter more^iow^in^thf

impinge upon the former ; this is termed direct impact. This impact, in my opinion same direction.

indeed, does not come about by immediate contact, but, before they attain actual contact,

the hinder parts of the first body & the foremost parts of the second body are compressed

by a mutually repulsive force ; & this compression becomes greater & greater until finally

the velocities become equal. Then further approach ceases, & therefore also further

compression ; &, since the bodies are soft, they exercise no further mutual force after

such compression, but continue to move forward with that equal velocity. This equality

in the velocity, to which the two spheres are reduced, together with the equality of action

& reaction, finishes off the whole matter. For, supposing that the mass or quantity of

matter of the foremost sphere is equal to q, that of the latter to Q ; the velocity of the

former equal to c, & that of the latter to C. Then the quantity of motion of the former

before impact is cq, & that of the latter is CQ ; for the velocity multiplied by the number

of points represents the sum of the motions of all the points, i.e., the quantity of motion,

& in the same way the quantity of motion divided by the mass gives the velocity. Now,

since the action & reaction are equal to one another, this quantity will be the same even

after impact ; hence after impact the whole motion of both the masses together will be

equal to CQ + cq. Further, since they are travelling with a common velocity, this velocity

will be the result obtained on dividing the quantity of motion by the whole quantity of

matter ; & it will therefore be equal to (CQ + ^?)/(Q + ?)• That is to say, to obtain

the common velocity after impact, we must multiply each mass by its velocity, & divide

the sum of these products by the sum of the masses.

269. If one of the two spheres is at rest, all that need be done is to put its velocity c Extension to ail
equal to zero ; also, if it is moving in a direction opposite to that of the first sphere, we ordained!0"*7 k
need only take the value of c as negative. Thus, both here & subsequently, the formula

found for the first case, in which the spheres are moving forward in the same direction,
includes all cases. Again, if in this case, we wish to find the velocity lost by the sphere
Q, & the velocity gained by the sphere q, we need only reduce the two formulae
C — (CQ + f?)/(Q + ?) & (CQ + cq)/(Q + q) — f to a common denominator, when we
shall obtain the formulae (Cq— cq)/(Q + q) & (CQ — cQ)/(Q + q). From these there
can be derived the theorem : — The sum of the masses is to either of the masses as the difference
between the velocities is to the velocity acquired by the other mass ; in the present case there
will be an increase of velocity for the foremost body & a decrease for the hindmost.

270. From these theorems relating to soft bodies we can easily proceed to those that Transition to im-
are perfectly elastic. For such bodies, after the maximum compression has taken place, tic-0 bodies^6

& the alteration in shape consequent on this compression, which is attained when equality
of the velocities is reached, the two spheres still continue to act upon one another, until
the original shape is attained ; & this action will duplicate the effect of the first action. When
the spherical shape is once more attained, as this takes place through a mutual recession
of the opposite surfaces of the spheres, which during compression had approached one
another, these same surfaces in each sphere will continue to recede from one another still
somewhat further, & the shape will be elongated ; but the mutual force between the parts of
each sphere is now changed in direction & the surfaces begin to be drawn together again.
Hence elongation will continue, but more slowly, until a certain maximum elongation is
attained ; this then begins to be diminished & the sphere once more returns to a spherical
shape, once more is compressed with a sort of oscillatory motion & forward & backward
vibration of its parts about the spherical shape ; exactly as was seen above in the case of
two points oscillating to & fro about a distance equal to that corresponding to a limit-point

208 PHILOSOPHIC NATURALIS THEORIA

cohassionis oscillare hinc, & inde ; sed id ad collisionem, & motus centrorum gravitatis
nihil pertinebit, quorum status a viribus mutuis nihil turbatur ; actio autem unius globi
in alterum statim cessabit post regressum ad figuram sphaericam, post quern superficies
alterius postica & alterius antica in centra jam retracts ulteriore centrorum discessu a
se invicem incipient ita distare, ut vires in se invicem non exerant, quarum effectus sentiri
possit ; & hypothesis perfecte elasticorum est, ut tantus sit mutuae actionis effectus in
recuperanda, quantus fuit in amittenda figura.

- Duplicate igitur effectu, globus ammittet celeritatem *Cq~ 2~J , & globus q

acquiret celeritatem * Q~2fQ- Quare illius celeritas post collisionem erit C - - 2C?
slve CQi«f. hujus?vero

eandem plagam, vel globus alter quiescet, vel fient in plagas oppositas ; prout determinatis
valoribus Q,q, C,c, formulae valor evaserit positivus, nullus, vel negativus.
Formula? pro im- 272. Quod si elasticitas fuerit imperfecta, & vis in amittenda ad vim in recuperanda

perfecte elasticis. c t • • i- • j • rr • • i rr • • • •>

ngura fuerit in aliqua ratione data, ent & effectus pnons ad effectum postenons itidem in
ratione data, nimirum in ratione subduplicata prioris. Nam ubi per idem spatium agunt
vires, & velocitas oritur, vel extinguitur tota, ut hie respectiva velocitas extinguitur in
compressione, oritur in restitutione figurae, quadrata velocitatum sunt ut areae, quas
describunt ordinatae viribus proportionales juxta num. 176, & hinc areae erunt in ratione
virium, si, viribus constantibus, sint constantes & ordinatae, cum inde fiat, ut scalae celeri-
tatum ab iis descriptae sint rectangula. Sit igitur rationis constantis illarum virium ratio
subduplicata m ad «, & erit effectus in amittenda figura ad summam effectuum in tota

collisione, ut m ad m -\- n, quae ratio si ponantur esse i ad r, ut sit r = - - satis erit,

effectus illos inventos pro globis mollibus, sive celeritatem ab altero amissam, ab altero
acquisitam, non duplicare, ut in perfecte elasticis, sed multiplicare per r, ut habeantur
velocitates acquisitae in partes contrarias, & componendse cum velocitatibus [129] prioribus.

Erit nimirum ilia quae pertinet ad globum Q = — ^— ^, & quae pertinet ad globum
q, erit = — j^ —i adeoque velocitas illius post congressum erit C — '—ft- —•> &

hujus c -\- — ; quae formulae itidem reducuntur ad eosdem denominatores ;

ac turn ex hisce formulis, turn e superioribus quam plurima elegantissima theoremata
deducuntur, quae quidem passim inveniuntur in elementaribus libris, & ego ipse aliquanto
uberius persecutus sum in Supplements Stayanis ad lib. 2, § 2 ; sed hie satis est, fundamenta
ipsa, & primarias formulas derivasse ex eadem Theoria, & ex proprietatibus centri gravitatis,
ac motuum oppositorum sequalium, deductis ex Theoria eadem ; nee nisi binos, vel ternos
evolvam casus usui futures infra, antequam ad obliquam collisionem, ac reflexionem motuum
gradum faciam.

Casus, in quo 27'?. Si elobus perfecte elasticus incurrat in globum itidem quiescentem, erit, c = o,

globus perfecte 2r _ '

elasticus mcurrit in adeoque velocitas contrana priori pertmens ad incurrentem, quae erat —£• - ", erit

alium. Q + q

zCq , . . . . 2CQ — 2<rQ . 2CQ

^r— - ; velocitas acquisita a quiescente, quae erat — , erit ~— - ; unde

U + y Q + q U + y

habebitur hoc theorema : ut summa massarum ad duflam massam quiescentis, vel incurrentis,
ita celeritas incurrentis, ad celeritatem amissam a secundo, vel acquisitam a primo ; & si
massae aequales fuerint, fit ea ratio aequalitatis ; ac proinde globus incurrens totam suam
velocitatem amittit, acquirendo nimirum aequalem contrariam, a qua ea elidatur, & globus
quiescens acquirit velocitatem, quam ante habuerat globus incurrens.

Casus triplex globi 274. Si globus imperfecte elasticus incurrat in globum quiescentem immensum, &

numI^nimob\ie.pla" °lui habeatur pro absolute infinite, cujus idcirco superficies habetur pro plana, in formula

velocitatis acquisitae a globo quiescente ~?r- — , cum evanescat Q respectu q absolute

infiniti, & proinde ^ - evadat = o, tota formula evanescit, adeoque ipse haberi potest
pro piano immobili. In formula vero velocitatis, quam in partem oppositam acquiret
globus incurrens, -—-^ —, evadit f= o, [130] & Q evanescit itidem respectu q.

Hinc habetur ^', sive rC, nimirum ob r = ^? fit C^±-"\X C, cujus prima pars — x C,

A THEORY OF NATURAL PHILOSOPHY 209

of cohesion. However, this has nothing to do with the impact or the motion of the centres
of gravity, nor are their states affected in the slightest by the mutual forces. Again, the
action of one sphere on the other will cease directly after return to the spherical shape ;
for after that the hindmost surface of the one & the foremost surface of the other, being
already withdrawn in the direction of their centres, will through a further recession of
the centres from one another begin to be so far distant from one another that they will
not exert upon one another any forces of which the effects are appreciable. We are left
with the hypothesis, for perfectly elastic solids, that the effect of their action on one another
is exactly the same in amount during alteration of shape & recovery of it.

271. Hence, the effect being duplicated, the sphere Q will lose a velocity equal to Formula for per-
(iCq - 2cq)/(Q + ?), & the sphere q will gain a velocity equal to (zCQ — 2Cy)/(Q + q). fect'yelastic **"«••
Hence, the velocity of the former after impact will be C — (zCq — 2tq)/(Q + <?) or

(CQ — Cq + 2cq)/(Q + ?), & the velocity of the latter will be c + (zCQ — 2«rQ)/(Q + q)
or (cq — cQ + 2CQ)/(Q -j- q). The motions will be in the original direction, or one of the
spheres may come to rest, or the motions maybe in opposite directions, according as formula,
given by the values of Q, q, C, & c, turns out to be positive, zero, or negative.

272. But if the elasticity were imperfect, & the force during loss of shape were in Formula; for impei -
some given ratio to the force during recovery of shape, then the effect corresponding to fectly elastic bodies.
the former would also be in a given ratio to the effect due to the latter, namely, in the
subduplicate ratio of the first ratio. For, when forces act through the same interval of

space, & velocity is generated, or is entirely destroyed, as here the relative velocity is
destroyed during compression & generated during recovery of shape, the squares of the
velocities are proportional to the areas described by the ordinates representing the forces,
as was proved in Art. 176. Hence these areas are proportional to the forces, if, the forces
being constant, the ordinates also are constant ; for from that it is easily seen that the
measures of the velocities described by them are rectangles. Suppose then that the
subduplicate ratio of the constant ratio of the forces be m : n ; then the ratio of the effect
during loss of shape to the sum of the effects during the whole of the impact will be m : m + n.
If we call this ratio I : r, so that r = (m -f- ri)/m, we need only, instead of doubling the effects
found for soft bodies, or the velocity lost by one sphere or gained by the other, multiply
these effects by r, in order to obtain the velocities acquired in opposite directions, which
are to be compounded with the original velocities. Thus, that for the sphere Q will be
equal to (rCq — rcg)/(Q -f- q), & that for the sphere q will be (rCQ — r<rQ)/(Q + q).
Hence, the velocity of the former after impact will be C — (rCq — rcq)/(Q -f- q) & the
velocity of the latter will be c + (VCQ — rcQ)/(Q + q) ; & these formulje also can be
reduced to common denominators. From these formula, as well as from those proved
above, a large number of very elegant theorems can be derived, such as are to be found
indeed everywhere in elementary books. I myself have followed the matter up somewhat
more profusely in the Supplements to Stay's Philosophy, in Book II, § 2. But here it is
sufficient that I should have derived the fundamentals themselves, together with the primary
formulae, from one & the same Theory, & from the properties of the centre of gravity &
of equal & opposite motions, which are also derived from the same theory. Except that
I will consider below two or three cases that will come in useful in later work, before I pass
on to oblique impact & reflected motions.

273. If a perfectly elastic sphere strikes another, & the second sphere is at rest, then Case of a perfectly
c = o, & the velocity, in the direction opposite to the original velocity, for the striking f^mrfhe™ St"k"
body, which was (zCq — 2cq)/(Q + q), will in this case be 2Cq/(Q -f- q) ; whilst the velocity

gained by the body that was at rest, which was shown to be (2CQ — 2cQ)/(Q + q), will
be 2CQ/(Q -|- q). Hence we have the following theorem. As the sum of the masses is
to twice the mass of the body at rest, or to the body that impinges upon it, so is the velocity of
the impinging body to the velocity lost by the second body, or to that gained by the first. If
the masses were equal to one another, this ratio would be one of equality ; hence in this
case the impinging body loses the whole of its velocity, that is to say it acquires an equal
opposite velocity which cancels the original velocity ; & the sphere at rest acquires a velocity
equal to that which the impinging sphere had at first.

274. If an imperfectly elastic sphere impinges on an immense sphere at rest, which Threefold case of
may be considered as absolutely infinite, & therefore its surface may be taken to be a plane ; onPanre immovable
then, in the formula for the velocity gained by a sphere at rest, (rCQ — rcQ)/(Q + q), plane.

since Q vanishes in comparison with q which is absolutely infinite, & thus Q/(Q + q) = o,
the whole formula vanishes, & therefore the immense sphere can be taken to be an immovable
plane. Now, in the formula for the velocity which the impinging sphere acquires in the
opposite direction to its original motion, namely, (rCq — rcq)/(Q -j- q), we have c = o,
& Q also vanishes in comparison with q. Hence we obtain rCq/q, or rC ; that is to say,
since r = (m -f ri)/m, we have C X (m + n)/m, of which the first part, C X m/m, or C,

210

PHILOSOPHIC NATURALIS THEORIA

Summa quadra-
torum velocitatis
ductorum in massas
manens in perfecte
elasticis.

sive C, est ilia, quse amittitur, sive acquiritur in partem oppositam in comprimenda figura,
& — X C est ilia, quse acquiritur in recuperanda, ubi si fit n = o, quod accidit nimirum
in perfecte mollibus ; habetur sola pars prima ; si m = n, quod accidit in perfecte elasticis,
est - - x C = C, secunda pars aequalis primae ; & in reliquis casibus est, ut m ad n, ita

ilia pars prima C, sive praecedens velocitas, quae per primam partem acquisitam eliditur,
ad partem secundam, quae remanet in plagam oppositam. Quamobrem habetur ejusmodi
theorema. Si incurrat ad perpendiculum in planum immobile globus perfecte mollis, acquirit
velocitatem contrariam cequalem suce priori, & quiescat ; si perfecte elasticus, acquirit duplam
suce, nimirum cequalem in compressione, qua motus omnis sistitur, & cequalem in recuperanda
figura, cum qua resilit ; si fuerit imperfecte elasticus in ratione m ad n, in ilia eadem ratione
erit velocitas priori suce contraria acquisita, dum figura mutatur, quce priorem ipsam velocitatem
extinguit, ad velocitatem, quam acquirit, dum figura restituitur, W cum qua resilit.

275. Est & aliud theorema aliquanto operosius, sed generale, & elegans, ab Hugenio
inventum pro perfecte elasticis, quod nimirum summa quadratorum velocitatis ductorum
in massas post congressum remaneat eadem, quae fuerat ante ipsum. Nam velocitates

C - ^- X (C - c), & c + ,,**- X (C-f) ; quadrata ducta

post congressum sunt

in massas continent singula ternos terminos : primi erunt QCC+ qcc ; secundi erunt

(-CC+ CO X

cc) X

4Q?

Q+ q

postremi
Q + q

2Cc + cc), sive simul

&(cC-cc) X
4Q??

erunt

X

quorum summa evadit (— CC

(CC - 2CC + CC),

(Q + ?)

[131] X (CC - 2Cc + cc), vel _

X

2Cc—

(CC-

(Q+f)

(Q + g)« *---- " *~ ' "" ' - QT~, ^C

quod destruit summam secundi terminorum binarii, remanente sola ilia

QCC + qcc,
Sed haec aequalitas nee

Collisionis obliquae
communis metho-
du s per virium
resolutionem.

Compositio virium
resolu t i o n i s u b-
stituta.

summa quadratorum velocitatum prsecedentium ducta in massas.
habetur in mollibus, nee in imperfecte elasticis.

276. Veniendo jam ad congressus obliques, deveniant dato tempore bini globi A, C
in fig. 42 per rectas quascunque AB, CD, quae illorum velocitates metiantur, m B, & D ad
physicum contactum, in quo jam sensibi-

lem effectum edunt vires mutuae. Com-

muni methodo collisionis effectus sic de-

finitur. Junctis eorum centris per rectam

BD, ducantur, ad earn productam, qua

opus est, perpendicula AF, CH, & com-

pletis rectangulis AFBE, CHDG resolvan-

tur singuli motus AB, CD in binos ; ille

quidem in AF, AE, sive EB, FB, hie vero

in CH, CG, sive GD, HD. Primus utro-

bique manet illaesus ; secundus FB, & HD

collisionem facit directam. Inveniantur

per legem collisionis directas velocitates BI,

DK, quse juxta ejusmodi leges superius

expositas haberentur post collisionem di-

versae pro diversis corporum speciebus,

& componantur cum velocitatibus expositis

per rectas BL, DQ jacentes in directum cum EB, GD, & illis aequales. His peractis

expriment BM, DP celeritates, ac directiones motuum post collisionem.

277. Hoc pacto consideratur resolutio motuum, ut vera quaedam resolutio in duos,
quorum alter illaesus perseveret, alter mutationem patiatur, ac in casu, quern figura exprimit,
extinguatur penitus, turn iterum alius producatur. At sine ulla vera resolutione res
vere accidit hoc pacto. Mutua vis, quse agit in globos B, D, dat illis toto collisionis tempore
velocitates contrarias BN, DS aequales in casu, quern figura exprimit, binis illis, quarum
altera vulgo concipitur ut elisa, altera ut renascens. Ese compositae cum BO, DR jacentibus
in directum cum AB, CD, & aequalibus iis ipsis, adeoque exprimentibus effectus integros
praecedentium velocitatum, exhibent illas ipsas velocitates BM, DP. Facile enim patet,
fore LO aequalem AE, sive FB, adeoque MO aequalem NB, & BNMO fore parallelogram-
mum ; ac eadem demonstratione est itidem parallelogrammum DRPS. Quamobrem
nulla ibi est vera resolutio, sed sola compositio motuum, perseverante nimirum velocitate
priore per vim inertiae, & ea composita cum nova velocitate, quam generant vires, quse
agunt in collisione.

FIG. 42.

A THEORY OF NATURAL PHILOSOPHY 211

is the part that is lost, or acquired in the opposite direction to the original velocity, during
the compression, & C X n/m is the part that is acquired during the recovery of shape.
In this, if n = o, which is the case for perfectly soft bodies, there is only the first part ;
if m — n, which is the case for perfectly elastic bodies, then C X n/m will be equal to C, and
the second part is equal to the first part ; & in all other cases as m is to n, so is the first part
C, or the original velocity, which is cancelled by the first part of the acquired velocity,
to the second part, which is the final velocity in the opposite direction. Hence we have
the following theorem. // a perfectly soft sphere impinges perpendicularly upon an immovable
plane, it will acquire a velocity equal & opposite to its original velocity, & will be brought
to rest. If the body is perfectly elastic, it will acquire a velocity double of its original velocity
but in the opposite direction, that is to say, an equal velocity during compression, by which the
whole of the motion ceases, & an equal velocity during recovery of shape, with which it rebounds.
If it were imperfectly elastic, the ratio being equal to that of m to n, the velocity acquired in
the opposite direction to its original velocity whilst the shape is being changed, by which the
original velocity is cancelled, will bear this same ratio to the velocity acquired whilst the shape
is being restored, that is, the velocity with which it rebounds.

275. There is also another theorem, which is rather more laborious, but it is a general The sum of the
& elegant theorem, discovered by Huygens for perfectly elastic solids. Namely, that the Velocities. ° each
sum of the squares of the velocities, each multiplied by the corresponding mass, remains multiplied by the
the same after the impact as it was before it. Now, the velocities after impact are remains'11 unaltered

C — 2^ X (C — c), & c + £~ X (C — f) ; the squares of these, multiplied by the perfectly elastic

^i + q {J. ~T- q bodies.

masses contain three terms each ; the first are QCC & qcc : the second are

( — CC + Cf) X & (fC - cc) X -> & the sum of these reduces to

(- CC + aCr - «) X : the last are (CC - 2Q + «), & - X

(CC-2Cc + w)» or added together 4(Q + $ * Q? X (CC-2Cc+cc), or
X (CC — 2Cc 4- cc), which will cancel the sum of the second terms ; hence all

that remains is QCC + qcc, the sum of the squares of the original velocities, each multiplied
by the corresponding mass. This equality does not hold good for soft bodies, nor yet for
imperfectly elastic bodies.

276. Coming now to oblique impacts, suppose that, in Fig. 42, the two spheres A & The usual method
C at some given time, moving along any straight lines AB, CD, which measure their velocities,

come into physical contact in the positions B & D, where the mutual forces now produce lution of forces.

a sensible effect. In the usual method the effect of the impact is usually determined as

follows. Join their centres by the line BD, & to this line, produced if necessary, draw

the perpendiculars AF, CH, & complete the rectangles AFBE, CHDG ; resolve each of

the motions AB, CD in two, the former into AF, AE, or EB, FB, & the latter into CH,

CG, or GD, HD. In either pair, the first remains unaltered ; the second, FB, & HD,

give the effect of direct impact. The direct velocities BI, DK are found by the law of

impact ; & these, according to laws of the kind set forth above, will after impact be different

for different kinds of bodies. They are compounded with velocities represented by the

straight lines BL, DQ, which are in the same straight lines as EB, GD respectively, & equal

to them. This being done, BM, DP will represent the velocities & the directions of motion

after collision.

277. In this method, there is considered to be a resolution of motions, as if there were Composition of
a certain real resolution into two parts, of which the one part persisted unchanged, & the *orces substituted

•T j i „ r. r 1 • i • A ? , for resolution.

other part suffered alteration ; & m the case, for which the figure has been drawn, the
latter is altogether destroyed & a fresh motion is again produced. But the matter really
proceeds without any real resolution in the following manner. The mutual force acting
upon the spheres B, D, gives to them during the complete time of impact opposite velocities
BN, DS, which are also equal, in the case for which the figure is drawn, to those two, of
which the one is considered to be destroyed & the other to be produced. These motions,
compounded with BO, DR, drawn in the directions of AB, CD & equal to them, & thus
representing the whole effects of the original velocities, will represent the velocities BM,
DP. For it is easily seen that LO is equal to AE, or FB ; & thus MO is equal to NB, &
BNMO will be a parallelogram ; in the same manner it can be shown that DRPS is a
parallelogram. Therefore, there is in reality no true resolution, but only a composition
of motions, the original velocity persisting throughout on account of the force of inertia ; &
this is compounded with the new velocity generated by the forces which act during the impact.

212

PHILOSOPHIC NATURALIS THEORIA

M

tiTiU°Sltsubstft°uta 278- Jdem etiam mihi^accidit, ubi oblique globus incurrit in planum, sive consideretur

etiam ubi globus motus, qui haberi debet deinde, sive percussionis obliquae energia respectu perpendicularis
immobile.11 1 ' Deveniat in fig. 43 globus A cum directione obliqua AB ad planum [132] CD consideratum

ut immobile, quod contingat physice in N,& concipiatur planum GI parallelum priori ductum

per centrum B, ad quod appellet ipsum centrum, & a quo resiliet, si resilit. Ducta AF

perpendiculari ad GI, & completo par-

allelogrammo AFBE, in communi

methodo resolvitur velocitas AB in duas

AF, AE ; sive FB, EB, primam dicunt

manere illaesam, secundam destrui a

resistentia plani : turn perseverare illam

solam per BI aequalem ipsi FB ; si corpus

incurrens sit perfecte molle, vel componi

cum alia in perfecte elasticis BE aequali

priori EB, in imperfecte elasticis Be,

quae ad priorem EB habeat rationem

datam, & percurrere in primo casu BI,

in secundo BM, in tertio EOT. At in

mea Theoria globus a viribus in ilia

minima distantia agentibus, quae ibi sunt

repulsivae, acquirit secundum direc-

tionem NE perpendicularem piano re-

pellenti CD in primo casu velocitatem

BE, aequalem illi, quam acquireret, si

cum velocitate EB perpendiculariter advenisset per EB, in secundo BL ejus duplam, in

tertio BP, quae ad ipsam habeat illam rationem datam r ad I, sive m -f- n ad m, & habet

deinde velocitatem compositam ex velocitate priore manente, ac expressa per BO aequalem

AB, & positam ipsi in directum, ac ex altera BE, BL, BP, ex quibus constat, componi illas

ipsas BI, BM, Bwz, quas prius ; cum ob IO aequalem AF, sive EB, & IM, Im aequales BE,

BI?, sive EL, EP, totae etiam BE, BP, BL totis OI, OM, Om sint aequales, & parallelae.
ubique in hac 279. Res mihi per compositionem virium ubique eodem redit, quo in communi
tionerrf resolution! methodo per earum resolutionem. Resolutionem solent vulgo admittere in motibus,
substitui, easque quos vocant impeditos, ubi vel planum subiectum, vel ripa ad latus procursum impediens,

sibi mvicem aequi- ,. • n • i • 1 ci • j i -n • -i_

vaiere. ut m nuviorum alveis, vel filum, aut virga sustentans, ut in pendulorum oscillatiombus,

impedit motum secundum earn directionem, qua agunt velocitates jam conceptas, vel
vires ; ut & virium resolutionem agnoscunt, ubi binae, vel plures etiam vires unius cujusdam
vis alia directione agentis effectum impediunt, ut ubi grave a binis obliquis planis sustinetur,
quorum utrumque premit directione ipsi piano perpendiculari, vel ubi a pluribus filis
elasticis oblique sitis sustinetur. In omnibus istis casibus illi velocitatem, vel vim agnoscunt
vere resolutam in duas, quarum utrique simul ilia unica velocitas, vel vis aequivaleat, ex
illis veluti partibus constituta, quarum si altera impediatur, debeat altera perseverare, vel
si impediatur utraque, suum utraque effectum edat seorsum. At quoniam id impedi-
mentum in mea Theoria nunquam habebitur ab immediato contactu plani rigidi subjecti,
nee a virga vere rigida, & inflexili sustentante, sed semper a viribus mutuis repulsivis in
primo casu, attractivis in secundo ; semper habebitur nova velocitas, vel vis aequalis, &
contraria illi, quam communis methodus elisam dicit, quae cum [133] tota velocitate, vel
vi obliqua composita eundem motum, vel idem aequilibrium restituet, ac idem omnino
erit, in effectuum computatione considerare partes illas binas, & alteram, vel utramque
impeditam, ac considerare priorem totam, aut velocitatem, aut vim, compositam cum iis
novis contrariis, & aequalibus illi parti, vel illis partibus, quae dicebantur elidi. In id autem,
quod vel inferne, vel superne motum massae cujuspiam impedit, vel vim, non aget pars
ilia prioris velocitatis, vel illius vis, quae concipitur resoluta, sed velocitas orta a vi mutua,
& contraria velocitati illi novae genitae in eadem massa, a vi mutua, vel ipsa vis mutua, quae
semper debet agere in partes contrarias, & cui occasionem praebet ilia determinata distantia
major, vel minor, quam sit, quae limites, & aequilibrium constitueret.

"mom 2%°- ^ quidem abunde apparet in ipso superiore exemplo. Ibi in fig. 43 globus

incurrente in pia- (quem concipamus mollem) advenit oblique per AB, & oblique impeditur a piano ejus
progressus. Non est velocitas perpendicularis AF, vel EB, quae extinguitur, durante AE,
vel FB, uti diximus ; nee ilia ursit planum CD. Velocitas AB occasionem dedit globo
accedendi ad planum CD usque ad earn exiguam distantiam, in qua vires variae agerent ;

A THEORY OF NATURAL PHILOSOPHY

213

278. The same thing comes about in my theory, when a sphere impinges obliquely
on a plane, whether the motion which it must have after impact is under consideration,
or whether we are considering the energy of oblique percussion with regard to the
perpendicular to the plane. Thus, in Fig. 43, suppose a sphere A to move along the oblique
direction AB & to arrive at the plane CD, which is considered to be immovable, & with
which the sphere makes physical contact at the point N. Now imagine a plane GI, parallel
to the former, to be drawn through the centre B ; to this plane the centre of the sphere
will attain, & rebound from it, if there is any rebound. After drawing AF perpendicular
to GI & completing the parallelogram AFBE, the usual method continues by resolving
the velocity AB into the two velocities AF, AE, or FB, EB ; of these, the first is stated
to remain constant, whilst the second is destroyed by the resistance of the plane ; & all
that remains after impact is represented by BI, which is equal to FB, if the body is soft ;
or that this is compounded with another represented by BE, equal to the original velocity
EB, in the case of perfectly elastic bodies ; and in the case of imperfectly elastic bodies,
it is compounded with Bi?, which bears a given ratio to the original EB. Then the sphere
will move off, in the first case along BI, in the second case along BM, & in the third case
along Em. But, according to my Theory the sphere, on account of the action of forces
at those very small distances, which are in that case repulsive, acquires in the direction
NE perpendicular to the repelling plane CD, in the first case a velocity BE equal to that
which it would have acquired if it had travelled along EB with a velocity EB at right
angles to the plane ; in the second case, it acquires a velocity double of this, namely BL,
& in the third a velocity BP, which bears to BE the given ratio r to I, i.e., m -f- n : m.
After impact it has a velocity compounded of the original velocity which persists, expressed
by BO equal to AB, & drawn in the same direction as AB, with another velocity, either
BE, BL, or BP ; from which it is easily shown that there results either BI, BM, or EOT,
just as in the usual method. For, since IO, AF, or EB, & IM, Im are respectively equal
to BE, Be, or EL, EP ; hence the wholes BE, BP, BL are also respectively equal to the
wholes OI, OM, Om, & are parallel to them.

279. The matter, in my hands, comes to the same thing in every case with composition
of forces, as in the usual method is obtained by resolution. In the usual method it is customary
to admit resolution for motions which are termed impeded, for instance, when a bordering
plane, or a bank, impedes progress to one side, as in the channels of rivers ; a string, or a
sustaining rod, as in the oscillations of pendulums hinders motion in the direction in which
the velocities or forces are in that case supposed to be acting. In a similar manner, they
recognize resolution of forces, when two, or even more forces impede the effect of some
one force acting in another direction ; for instance, when a heavy body is sustained by two
inclined planes, each of which exerts a pressure on the body in a direction perpendicular
to itself, or when such a body is suspended by several elastic strings in inclined positions.
In all these cases, the velocity of force is taken to be really resolved into two ; to both of
these taken together the single velocity or force will be equivalent, being as it were compounded
of these parts, of which if one is impeded, the other will still persist, or if both are impeded,
they will each produce their own effect separately. Now, since in my Theory there never
is such impediment, caused by an immediate contact with the bordering plane, nor by
a truly rigid or inflexible sustaining rod, but always considered to be due to mutual forces,
that are repulsive in the first case & attractive in the second case, a new velocity or force,
equal & opposite to that which is in the usual theory supposed to be destroyed, is obtained.
This velocity, or force, combined with the whole oblique velocity or force, will give the
same motion or the same equilibrium ; & it will come to exactly the same thing, when
computing the effects, if we consider the two velocities, or forces, either one or the other,
or both, to be impeded, as it would to consider the original velocity, or force, to be com-
pounded with the new velocities, or forces, which are opposite in direction & equal
to that part or parts which are said to be destroyed. Moreover, upon the object which
hinders the motion, or force, of any mass upwards or downwards, it is not the part of the
original velocity, or force, which is said to be resolved, that will act ; but it is the velocity
arising from the mutual force, opposite in direction to that velocity which is newly generated
in the mass by the mutual force, or the mutual force itself. This must always act in opposite
directions ; & is governed by the given distance, greater or less than that which gives the
limit-points & equilibrium.

280. This fact indeed is seen clearly enough in the example given above. There, in Fig.
43, the sphere, which we will suppose to be soft, travels obliquely along AB, & its progress is
impeded, also obliquely, by the plane. It is not true that the perpendicular velocity AF, or
EB is destroyed, whilst AE, or FB persists, as we have already proved ; nor was there any
direct pressure from it on the plane CD. The velocity AB made the sphere approach the
plane CD to within a very small distance from it, at which various forces come into action ;

Composition sub-
stituted also for
resolution i n the
case of a sphere
impinging on an
immovable plane.

In every case, in
my Theory, com-
position is used
instead of resolu-
tion ; & these are
equivalent to one
another.

A case in point
where a soft sphere
impinges on an
immovable plane.

2I4

PHILOSOPHISE NATURALIS THEORIA

Aliud globi descen-
dentis per planum
inclinatum.

Aliud in pendulo.

Alia ratio com-
ponondi vires in
eodem casu.

Aliud exemplum in
globo sustentato a
binis planis. Diffi-
cultas communis
methodi in eodem.

FIG. 44.

donee ex omnibus actionibus conjunctis impediretur accessus ad ipsum planum, sive
perpendicularis distantiae ulterior diminutio. Illae vires agent simul in directione perpen-
dicular! ad ipsum planum juxta num. 266 : debebunt autem, ut impediant ejusmodi
ulteriorem accessum, producere in ipso globo velocitatem, quae composita cum tota BO
perseverante in eadem directione AB, exhibeat velocitatem per BI parallelam CD. Quoniam
vero triangula rectangula AFB, BIO aequalia erunt necessario ob AB, BO aequales ; erit
BEIO parallelogrammum, adeoque velocitas perpendicularis, quae cum priore velocitate
BO debeat componere velocitatem per rectam parallelam piano, debebit necessario esse
contraria, & aequalis illi ipsi EB perpendiculari eidem piano, in quam resolvunt vulgo
velocitatem AB. Interea vero vis, quae semper agit in partes contrarias aequaliter, urserit
planum tantundem, & omnes in eo produxerit effectus illos, qui vulgo tribuuntur globo
advenienti cum velocitate ejusmodi, ut perpendicularis ejus pars sit EB.

281. Idem accidet etiam in reliquis omnibus casibus superius memoratis. Descendat
globus gravis per planum inclinatum CD (fig. 44) oblique, quod in communi sententia
continget hunc in modum. Resolvunt gravita-

tem BO in duas, alteram BR perpendicularem
piano CD, qua urgetur ipsum planum, quod eum
sustinet ; alteram BI, parallelam eidem piano,
quse obliquum descensum accelerat. In mea
Theoria gravitas cogit globum semper magis ac-
cedere ad planum CD ; donee distantia ab eodem
evadat ejusmodi ; ut vires mutuae [134] repul-
sivae agant, & ilia quidem, quae agit in B, sit
ejusmodi ut composita cum BO exhibeat BI
parallelam piano ipsi, adeoque non inducentem
ulteriorem accessum, sit autem perpendicularis
piano ipsi. Porro ejusmodi est BE, jacens in
directum cum RB, & ipsi aequalis, cum nimirum
debeat esse parallela, & aequalis OI. Vis autem
aequalis ipsi, & contraria, adeoque expressa per RB, urgebit planum.

282. Quod si grave suspensum in fig. 45 filo, vel virga BC debeat oblique descendere
per arcum circuli BD ; turn vero in communi methodo gravitatem BO itidem resolvunt
in duas BR, BI, quarum prima filum, vel virgam tendat, & elidatur, secunda acceleret
descensum obliquum, qui fieret ex velocitate concepta per rectam BA perpendicularem
BC, ac praeterea etiam tensionem fili agnoscunt ortum

a vi centrifuga, quae exprimitur per DA perpendicu-
larem tangenti. At in mea Theoria res hoc pacto
procedit. Globus ex B abit ad D per vires tres com-
positas simul cum velocitate praecedente ; prima e
viribus est vis gravitatis BO ; secunda attractio versus
C orta a tensione fili, vel virgae, expressa per BE paral-
lelam, & aequalem OI, adeoque RB, quae solae compo-
nerent vim BI ; tertia est attractio in C expressa per
BH aequalem AD orta itidem a tensione fili respond-
ente vi centrifugae, & incurvante motum. Adest prae-
terea velocitas praecedens, quam exprimit BK aequalis
IA, ut sit BI aequalis KA. His viribus cum ea veloci-
tate simul agentibus erit globus in D in fine ejus tem-
pusculi,cui ejusmodi effectus illarum virium respond-
ent. Nam ibi debet esse, ubi esset, si aliae ex illis causis agerent post alias : gravitate agente
veniret per BO, vi BE abiret per OI, velocitate BK abiret per IA ipsi aequalem, vi BH
abiret per AD. Quamobrem res tota itidem peragitur sola compositione virium, & motuum.

283. Porro si sumatur EG aequalis BH ; turn tota attractio orta a tensione fili erit
BG, quae prius considerata est tanquam e binis partibus in directum agentibus composita,
ac res eodem redit ; nam si prius componantur BH, & BE in BG (quo casu tota BG ut
unica vis haberetur), turn BO, ac demum BK, ad idem punctum D rediretur juxta generalem
demonstrationem, quam dedi num. 259. Jam vero vi expressa per totam BG attraheretur
ad centrum suspensionis C ab integra tensione fili, ubi pars EG, vel BH ad partem BE
habet proportionem pendentem a celeritate BK, ab angulo RBO, ac a radio CB ; sed ista
meae Theoriae cum omnium usitatis Mechanicae elementis communia sunt, posteaquam
compositionis hujus cum ilia resolutione aequivalentia est demonstrata.

284. Quae de motu diximus facto vi oblique, sed non penitus impedita, eadem in
aequilibrio habent locum, ubi omnis impeditur motus. Innitatur globus gravis B in fig.
46 binis planis AC, CD, quae accurate, vel in mea Theoria [135] physice solum, contingat

B

FIG. 45.

A THEORY OF NATURAL PHILOSOPHY 215

then, under the combined actions of all the forces further approach toward the plane,
or further diminution of the perpendicular distance from it, is impeded. The forces act
together in the direction perpendicular to the plane, as was shown in Art. 266 ; & they
must, in order to impede further approach of this kind, produce in the sphere itself a
velocity which, compounded with the whole velocity that persists throughout, namely BP,
in the direction of AB, will give a velocity represented by BI parallel to CD. But, since
the right-angled triangles AFB, BIO are necessarily congruent on account of the equality
of AB & BO, it follows that BEIO is a parallelogram. Hence, the perpendicular velocity,
which has, when combined with the original velocity BO, to give a resultant represented
by a straight line parallel to the plane, must of necessity be equal & opposite to that represented
by EB, also perpendicular to the plane, into which commonly the velocity AB is resolved.
Meanwhile, the force, which always acts equally in opposite directions, would act on the
plane to precisely the same extent, & all those effects would be produced on it, which are
commonly attributed to the sphere striking it with a velocity of such sort that its perpendicular
part is EB.

281. The same thing happens also in the rest of the cases mentioned above. Let a Another case in
heavy sphere descend along the inclined plane CD, in Fig. 44 ; the descent takes place, ^ere descending
according to the customary idea, in the following manner. Gravity, represented by BO, along an inclined
is resolved into two parts, the one, BR, perpendicular to the plane CD, acts upon the plane plane'

& is resisted by it ; the other, BI, parallel to the plane, accelerates the oblique descent.
According to my Theory, gravity forces the sphere to approach the plane CD ever nearer
& nearer, until the distance from it becomes such as that for which the repulsive forces
come into action ; that which acts on B is such that, when combined with BO, will give
a velocity represented by BI parallel to the plane, & thus does not induce further approach ;
moreover it is perpendicular to the plane. Further, it is such as BE, lying in the same
straight line as RB, & equal to it, because indeed it must be parallel & equal to OI. Lastly,
a force that is equal & opposite, & so represented by BR, will act upon the plane.

282. But if, in Fig. 45, a heavy body is suspended by a string or rod BC, it is bound to The pendulum is
descend obliquely along the circular arc BD. Now, in the usual method, gravity, represented ^°^CT C*SG m
by BO, is again resolved into two parts, BR & BI ; the first of these exerts a pull on the

string or rod & is destroyed ; the second accelerates the oblique descent, which would
come about through a velocity supposed to act along BA perpendicular to BC ; in addition
to these, account is taken of the tension of the string arising from a centrifugal force, which
is represented by DA perpendicular to the tangent. But, according to my Theory, the
matter goes in this way. The sphere passes from B to D, under the action of three forces
compounded with the original velocity. The first of these forces is gravity, BO ; the second
is the attraction towards C arising from the tension of the string or rod, & represented
by BE, parallel & equal to OI, & thus also to RB, these two alone, taken together, give a
force BI. The third is an attraction towards C, represented by BH, equal to AD, arising
also from the tension of the string corresponding to the centrifugal force & incurving the
motion. In addition to these, we have the original velocity, represented by BK, equal
to IA, so that BI is equal to KA. If such forces as these act together with this velocity,
the sphere will arrive at D at the end of the interval of time to which such effects of the
forces correspond. For it must reach that point at which it would be, if all these causes
acted one after the other ; &, with gravity acting, it would travel along BO ; with
the force BE acting it would pass along OI ; with the velocity BK, it would traverse IA,
which is equal to BK ; & with the force BH acting, it would go from A to D. Hence,
in this case also, the whole matter is accomplished with composition alone, for forces
& motions.

283. Further, if EG is taken equal to BH ; then the whole attraction arising from Another manner of
the tension of the string will be BG, which previously was considered only as being com- f °™epsouin'dS1|
pounded of two parts acting in the same straight line ; & it comes to the samejthing just considered.

as before. For, if BH & BE are first of all compounded into BG (in which case BG is reckoned
as a single force), then BO is taken into account, & finally BK ; we shall be led to the same
point D, according to the general demonstration I gave in Art. 259. Now we have an
attraction to the centre of suspension C due to a force expressed by the whole BG, where
the part of it, EG, or BH, bears to the part BH a ratio that depends upon the velocity
BK, the angle RBO, & the radius CB. The results of my Theory are in agreement with
the elementary principles of Mechanics accepted by everyone else, as soon as the equivalence
of my composition with their resolution has been demonstrated.

284. The same things hold good in the case of equilibrium, where all motion is impeded, ^^ su^
as those we have already spoken of with respect to motion derived from a force acting by two planes
obliquely, but not altogether impeded. In Fig. 46, a heavy sphere is supported by two i^ai^
planes AC, CD, which actually, or in my Theory physically only, it touches at H & F ; this case.

216 PHILOSOPHISE NATURALIS THEORIA

in H, & F, & gravitatem referat recta verticalis BO, ac ex puncto O ad rectas BH, BF
ducantur rectse OR, OI parallels ipsis BF, BH, & producta sursum BK tantundem, ducantur
ex K ipsis BF, BH parallels KE, KL usque ad easdem BH, BF ; ac patet, fore rectas BE,
BL aequales, & contrarias BR, BI. In communi methodo resolutionis virium concipitur
gravitas BO resoluta in binas BR, BI, quarum prima urgeat planum AC, secunda DC ;
& quoniam si angulus HCF fuerit satis acutus ; erit itidem satis acutus angulus R, qui
ipsi aequalis esse debet, cum uterque sit complementum HBF ad duos rectos, alter ob
parallelogrammum, alter ob angulos BHC, BFC rectos ; fieri potest, ut singula latera
BR, RO, sive BI, sint, quantum libuerit, longiora quam BO ; vires singulas, quas urgent
ilia plana, possunt esse, quantum libuerit, majores, quam sola gravitas : mirantur multi,
fieri posse, ut gravitas per solam ejusmodi applicationem tantum quodammodo supra se
assurgat, & effectum tanto majorem edat.

Soiutio in ipsa 285. DifEcultas ejusmodi in communi etiam sententia evitari facile potest exemplo

methodo communi * ,. • i • * • i *.... « . i* • 11

in hac Theoria vectls> de quo agemus infra, in quo sola applicatio vis in multo majore distantia collocatae
nuiium ipsi difficul- multo majorem effectum edit. Verum in mea Theoria ne ullus quidem difficultati est
locus. Non resolvitur revera gravitas in duas vires BR, BI, quarum singulae plana urgeant,
sed gravitas inducit ejusmodi accessum ad ea plana, in quo vires repulsivse perpendiculares
ipsis planis agentes in globum componant vim BK aequalem, & contrariam gravitati BO,
quam sustineat, & ulteriorem accessum impediat. Ad id praestandum requiruntur illas
vires BE, BL aequales, & contrarias hisce BR, BI, quae rem conficiunt. Sed quoniam vires
sunt mutuae, habebuntur repulsiones agentes in ipsa plana contrarias, & aequales illis ipsis
BE, BL, adeoque agent vires expressae per illas ipsas BR, BI, in quas communis methodus
gravitatem resolvit.

Aliud in giobo sus- 286. Quod si globus gravis P in fig. 47 e filo BP pendeat, ac sustineatur ab obliquis

>iqms. £jjg ^g^ j-j-g^ exprimat autem BH gravitatem, & sit BK ipsi contraria, & aequalis, ac sint
HI, KL parallelae DB, & HR, KE parallelae filo AB ; communis methodus resolvit gravi-
tatem BH in duas BR, BI, quas a filis sustineantur, & ilia tendant ; sed ego compono vim
BK gravitati contrariam, & aequalem e viribus BE, BL, quas exerunt attractivas puncta
fili, quae ob pondus P delatum deorsum sua gravitate ita distrahuntur a se invicem, donee
habeantur vires attractivae componentes ejusmodi vim contrariam, & aequalem gravitati.

Conciusio gene- 287. Quamobrem per omnia casuum diversorum genera pervagati jam vidimus, nullam
rails pro hac esse USpiam ;n mea Theoria veram aut virium, aut motuum resolutionem, sed omnia

theona, quae omnia r , , , . . , r ,-.

per solam prorsus phasnomena pendere a sola compositione virium, & motuum, adeo-[i3oj-que

compositionem. naturam eodem ubique modo simplicissimo agere, componendo tantummodo vires, &
motus plures, sive edendo simul eum effectum, quern ederent illae omnes causae ; si aliae
post alias effectus ederent suos aequales, & eandem habentes directionem cum iis, quos
singulae, si solae essent, producerent. Et quidem id generale esse Theorise meae, patet
vel ex eo, quod nulli possunt esse motus ex parte impediti, ubi nullus est immediatus
contactus, sed in libero vacuo spatio punctum quodvis liberrime movetur parendo simul
velocitati, quam habet jam acquisitam, & viribus omnibus, quae ab aliis omnibus pendent
materiae punctis.

Resolutio tantum 288. Quanquam autem habeatur revera sola compositio virium ; licebit adhuc vires

mente concept a imaginatione nostra resolvere in plures, quod saepe demonstrationes theorematum, &

ssepe utilis ad con- -°. .. ix. . i T • jjoi

trahendas solu- solutionem problematum contrahet mirum in modum, ac expeditiores reddet, & elegant-
tiones. iores ; nam licebit pro unica vi assumere vires illas, ex quibus ea componeretur. Quoniam

enim idem omnino effectus oriri debet, sive adsit unica vis componens, sive reapse habeantur
simul plures illae vires componentes ; manifestum est, substitutione harum pro ilia nihil
turbari conclusiones, quae inde deducuntur : & si post resolutionem ejusmodi inveniatur
vis contraria, & aequalis alicui e viribus, in quas vis ilia data resolvitur ; ilia haberi potest
pro nulla consideratis solis reliquis, si in plures resoluta fuit, vel sola altera reliqua, si
resoluta fuit in duas. Nam componendo vim, quae resolvitur, cum ilia contraria uni ex iis,
in quas resolvitur, eadem vis provenire debet omnino, quae oritur componendo simul
reliquas, quae fuerant in resolutione sociae illius elisae, vel retinendo unicam illam alteram
reliquam, si resolutio facta est in duas tantummodo ; atque id ipsum constat pro resolu-
tione in duas ipsis superioribus exemplis, & pro quacunque resolutione in vires quotcunque
facile demonstratur.

A THEORY OF NATURAL PHILOSOPHY

217

FIG. 47.

218

PHILOSOPHIC NATURALIS THEORIA

FIG. 47.

A THEORY OF NATURAL PHILOSOPHY

219

let the vertical line BO represent gravity, & draw from the point O, to meet the straight
lines BH, BF, the straight lines OR, OI parallel to BF, BH ; also producing BK upwards
to the same extent, draw through K the straight line KE, KL, parallel to BF, BH to meet
BH & BF. Then it is clear that BE, BL will be equal & opposite to BR, BI. Now,
according to the usual method by means of resolution of forces, the gravity BO is supposed
to be resolved into the two parts BR, BI, of which the first acts upon the plane AC & the
second upon DC. Also if the angle HCF is sufficiently acute, then the angle at R is also
sufficiently acute ; for these angles must be equal to one another. For, each is the supple-
ment of the angle HBF, the one in the parallelogram, the other on account of BHC
& BFC being right angles. This being so, it may happen that each of the sides BR, RO,
or BI, will be greater than BO, to any desired extent. Thus each of the forces, which
act upon the planes, may be greater than gravity alone, to any desired extent. Many
will wonder that it is possible that gravity, by a mere application of this kind, surpasses
itself to so great an extent, & gives an effect that is so much greater.

285. A difficulty of this kind even according to the ordinary opinion is easily avoided Answer to the
by comparing the case of the lever, with which we will deal later ; in it the mere application ortoar'y' mltho'd!
of a force situated at a much greater distance gives a far greater effect. But with my in my Theory
Theory there is no occasion for any difficulty of the sort. For there is no actual resolution for ^ny difficulty?11
of gravity into the two parts BR, BI, each acting on one of the planes ; but gravity induces

an approach to the planes, to within the distance at which repulsive forces acting perpen-
dicular to the planes upon the sphere compound into a force BK, equal & opposite to the grav-
ity BO ; this force sustains the sphere & impedes further approach to the planes. To represent
this, the forces BE, BL are required ; these are equal & opposite to BR, BI ; & that is all there
need be said about the matter. Now, since the forces are mutual, there are repulsions acting
upon the planes, & these repulsions are equal & opposite to BE, BL ; & thus the forces acting
are represented by BR, BI, which are those into which the ordinary method resolves gravity.

286. But if, in Fig. 47, a heavy sphere P is suspended by a string BP, & this is held Explanation in^he
up by inclined strings AB, DB, & gravity is represented by BH ; let BK be equal & opposite £uspendedabyP i™
to it, & let HI, KL be parallel to the string DB, & HR, KE parallel to the string AB. The clined strings,
ordinary method resolves the gravity BH into the two parts BR, BI, which are sustained

by the strings & tend to elongate them. On the other hand, I compound the force BK,
equal & opposite to gravity from the two forces BE, BL ; these attractive forces are put
forth by the points of the string, which, owing to the heavy body P suspended beneath
are drawn apart by its gravity to such a distance that attractive component forces are
obtained such as will give a force that is equal & opposite to the gravity of P.

287. Having thus considered all sorts of different cases, we now see that there is nowhere General summing
in my Theory any real resolution either of forces or of motions ; but that all phenomena "his Theory?* which
depend on composition of forces & motions alone. Thus, nature in all cases acts in the gives everything
same most simple manner, by compounding many forces & motions only; that is to say, akrae°mp0

by producing at one time that effect, which all the causes would produce, if they acted
one after the other, & each produced that effect which was equal & in the same direction
as that which it would produce if it alone acted. That this is a general principle of my
Theory is otherwise evident from the fact that no motions can be in part impeded, where
there is no immediate contact ; on the contrary, any point can move in a free empty space
in the freest manner, subject to the combined action of the velocity it has already acquired,
& to all the forces which come from all other points of matter.

288. Now, although as a matter of fact we can only have compositions of forces, yet Resolution, ai-
we may mentally resolve our forces into several ; & this will often shorten the proofs of m°ntii fiction, is
theorems & the solution of problems in a wonderful manner, & render them more elegant yet often useful in
& less cumbrous ; for we may assume instead of a single force the forces from which it is 8
compounded. For, since the same effect must always be produced, whether a single
component force is present, or whether in fact we have the several component forces taken

all together, it is plain that the conclusions that are derived will in no way be disturbed
by the substitution of the latter for the former. If after such resolution a force is found,
equal & opposite to any one of the forces into which the given force is resolved, then these
two can be taken together as giving no effect ; & only the rest need be considered if the
given force was resolved into several parts, or only the other force if the given force was
resolved into two parts. For, by compounding the force which was resolved with that
force which is equal & opposite to the one of the forces into which it was resolved, the
same force must be obtained as would arise from compounding all the other forces which
were partners of the cancelled force in the resolution, or from retaining the single remaining
force when the resolution was into two parts only. This has been shown to be the case
for resolution in the two examples given above, & can be easily proved for any sort of
resolution into forces of any number whatever.

220 PHILOSOPHIC NATURALIS THEORIA

Methodus generalis 289. Porro quod pertinct ad resolutionem in plures vires, vel motus, facile est ex

SiM^qwtCTiaqne. "s> <luse dicta sunt num- 257 definire legem, quae ipsam resolutionem rite dirigat, ut
habeantur vires, quae datam aliquam componant. Sit in fig. 48, vis quaecunque, vel motus
AP, & incipiendo ab A ducantur quotcunque, & cujuscunque longitudinis rectae AB, BC,
CD, DE, EF, FG, GP, continue inter se connexse ita, ut incipiant ex A, ac desinant in P ;
& si ipsis BC, CD, &c. ducantur parallels, & aequales Ac, Ad, &c. ; vires omnes AB, Ac,
Ad, Ae, A/, Ag, Ap component vim AP ; unde patet illud : ad componendam vim
quamcunque posse assumi vires quotcunque, & quascunque, quibus assumptis determinari
poterit una alia praeterea, quae compositionem perficiat ; nam poterunt duci rectae AB,
BC, CD, &c. parallelae, & aequales datis quibuscunque, & ubi postremo deventum fuerit
ad aliquod punctum G, satis erit addere vim expressam per GP.

Eyoiutio resolu- [137] 200. Eo autem generali casu continetur particularis casus resolutionis in vires

tionis in duas L J/J , , ~ , r. ....

tantum. tantummodo duas, quae potest hen per duo quaevis latera tnanguli cujuscunque, ut in

fig. 49, si datur vis AP, & fiat quodcunque triangulum ABP ; vis resolvi potest in duas
AB, BP, & data illarum altera, datur & altera, quod quidem constat etiam ex ipsa com-
positione, seu resolutione per parallelogrammum ABPC, quod semper compleri potest,
& in quo AC est parallela, & aequalis BP, ac bins vires AB, AC componunt vim AP : atque
idem dicendum de motibus.

Cur vis composita 291. Ejusmodi resolutio illud etiam palam faciet ; cur vis composita a viribus non

sit minor com- m directum iacentibus, sit minor ipsis componentibus, quae nimirum sunt ex parte sibi

ponentibus simul . •«_!•• •• * 1-1 • • •

sumptis. invicem contranae, & elisis mutuo contrarns, & aequalibus, remanet in vi composita summa

virium conspirantium, vel differentia oppositarum pertinentium ad componentes. Si
enim in fig. 50, 51, 52 vis AP componatur ex viribus AB, AC, quae sint latera parallelogrammi
ABPC, & ducantur in AP perpendicula BE, CF, cadentibus E, & F inter A, & P in fig. 50,
in A, & P in fig. 51, extra in fig. 52 ; satis patet, fore in prima, & postrema acqualia triangula
AEB, PFC, adeoque vires EB, FC contrarias, & aequales elidi ; vim vero AP in primo
casu esse summam binarum virium conspirantium AE, AF, aequari unicae AF in secundo,
& fore differentiam in tertio oppositarum AE, AF.

292' ^n res°luti°ne quidem vis crescit quodammodo ; quia mente adjungimus alias
lutione : nihii inde oppositas, & aequales, quae adjunctae cum se invicem elidant, rem non turbant. Sic in
vdivlUCi pr° fi§- 52 resolvendo Ap in bmas AB, AC, adjicimus ipsi AP binas AE, PF contrarias, &
praeterea in directione perpendiculari binas EB, FC itidem contrarias, & aequales. Cum
resolutio non sit realis, sed imaginaria tantummodo ad faciliorem problematum solutionem ;
nihil inde difHcultatis afferri potest contra communem methodum concipiendi vires, quas
hue usque consideravimus, & quae momento temporis exercent solum nisum, sive pressionem ;
unde etiam fit, ut dicantur vires mortuae, & idcirco solum continue durantes tempore
sine contraria aliqua vi, quae illas elidat, velocitatem inducunt, ut causae velocitatis ipsius
inductae : nee inde argumentum ullum desumi poterit pro admittendis illis, quas Leibnitius
invexit primus, & vires vivas appellavit, quas hinc potissimum necessario saltern concipiendas
esse arbitrantur nonnulli, ne nimirum in resolutione virium habeatur effectus non aequalis
suae causae. Effectus quidem non aequalis, sed proportionalis esse debet, non causae, sed
actioni causae, ubi ejusmodi actio contraria aliqua actione non impeditur vel tota, vel ex
parte, quod accidit, uti vidimus, in obliqua compositione : ac utcunque & aliae responsiones
sint in communi etiam sententia pro casu resolutionis ; [138] in mea Theoria, cum ipsa
resolutio realis nulla sit, nulla itidem est, uti monui, difficultas.

us.

Satis patere ex hac 2g, £t quidem tarn ex iis, quae hue usque demonstrata sunt, quam ex

Theoria, Vires . i • fl T • • • • j- •

Vivas in Natwa quae consequentur, satis apparebit, nullum usquam esse ejusmodi virium vivarum indicium,
nuttas esse. nullam necessitatem ; cum omnia Naturae phaenomena pendeant a motibus, & sequilibrio,

adeoque a viribus mortuis, & velocitatibus inductis per earum actiones, quam ipsam ob
causam in ilia dissertatione De Viribus Fivis, quae hujus ipsius Theoriae occasionem mihi
praebuit ante annos 13, affirmavi, Fires Vivas in Natura nullas esse, & multa, quae ad eas
probandas proferri solebant, satis luculenter exposui per solas velocitates a viribus non
vivis inductas.

A THEORY OF NATURAL PHILOSOPHY

221

FIG. 48.

FIG. 49.

FIG. 50.

FIG. 51.

EL.

FIG. 52.

222

PHILOSOPHIC NATURALIS THEORIA

P

B

FIG. 48.

FIG. 49.

FIG. 50.

FIG. 51.

FIG. 52.

A THEORY OF NATURAL PHILOSOPHY 223

289. Further, as regards resolution into several forces or motions, it is easy, from A general method
what has been said in Art. 257, to determine a law which will rightly govern such resolution, ^to^yTu^ber^f
so that the forces which compose any given force may be obtained. In Fig. 48, let AP other forces.

be any force, or motion ; starting from A, draw any number of straight lines of any length,
AB, BC, CD, DE, EF, FG, GP, continuously joining one another so that they start from
A & end up at P. Then, if to these lines AB, BC, &c., straight lines Ac, Ad, &c., are drawn,
equal & parallel, all the forces represented by AB, Ac, Ad, Ae, A/, Ag, Ap, will compound
into a force AP. From this it is clear that, to make up any force, it is possible to assume
any forces, & any number of them, & these being taken, it is possible to find one other force
which will complete the composition. For, the straight lines AB, BC, CD, &c., can be
drawn parallel & equal to any given lines whatever, & when finally they end up at some
point G, it will be sufficient to add the force represented by GP.

290. Moreover the particular case of resolution into two forces only is contained in Derivation of the
the general case. This can be accomplished by means of any two sides of any triangle, t^in'6 two ^co-
in Fig. 49, if AP is the given force, & any triangle ABP is constructed, then the force AP tions only.

can be resolved into the two parts AB, BP ; & if one of these is given, the other also is
given. This indeed is manifest even from the composition itself, or from resolution by
means of the parallelogram ABPC, which can be completed in every case ; in this AC
is parallel & equal to BP, & the two forces AB, AC will compound into the force AP. The
same may be said with regard to motions.

291. Such a resolution also brines out clearly the reason why a force compounded Why the resultant

.. f ,.., •if-i-L-Ltr. force is less than

from forces that do not lie in the same straight line is less than the sum of these components. the two component
These are indeed partly opposite to one another; &, when the equals & opposites have forces taken to-
cancelled one another, there remains in the force compounded of them the sum of the g
forces that agree in direction or the difference of the opposites which relate to the components.
For, in Figs. 50, 61, 62, if the force AP is compounded from the forces AB, AC, which are
sides of the parallelogram ABPC, & BE, CF are drawn perpendicular to AP, E & F falling
between A & P in Fig. 50, at A & P in Fig. 51, & beyond them in Fig. 52 ; then it is
plain that, in the first & last cases the triangles AEB, PFC are equal, & thus the forces EB
& FC, which are equal & opposite, cancel one another. But the force AP in the first case
is the sum of the two forces AE, AF acting in the same direction ; it is equal to the single
force AF in the second case ; & in the third case it is equal to the difference of the opposite
forces AE, AF.

292. In resolution there is indeed some sort of increase of force. The reason for The reason why the
this is that mentally we add on other equal & opposite forces, which taken together cancel ij^rJase^n6 resoiu°
one another, & thus do not have any disturbing effect. Thus, in Fig. 52, by resolving tion: no argument
the force AP into the two forces AB, AC, we really add to AP the two equal & opposite ^ "derfved^f rom
forces AE, PF, &, in addition, in a direction at right angles to AP, the two forces EB, FC, this."

which are also equal & opposite. Now, since resolution is not real, but only imaginary,
& merely used for the purpose of making the solution of problems easier ; no exception
can be taken on this account to the usual method of considering forces such as we have
hitherto discussed, such as exert for an instant of time merely a stress or a pressure ; for
which reason they are termed dead forces, & because, whilst they last for a continuous
time without any contrary force to cancel them, they yet only produce velocity, they are
looked upon as the causes of the velocity produced. Nor from this can any argument be
derived in favour of admitting the existence of those forces, which were first introduced
by Leibniz, & called by him living forces. These forces some people consider must at
least be supposed to exist, in order that in the resolution of forces, for instance, there should
not be obtained an effect unequal to its cause. Now the effect must be proportional, &
not equal ; also it must be proportional, not to the cause, but to the action of the cause,
where an action of this kind is not impeded, either wholly or in part, by some equal &
opposite action, which happens, as we have seen, in oblique composition. But, whatever
may be the various arguments, according to the usual opinion, to meet the difficulties in
the case of resolution, since, in my Theory, there is no real resolution, there is no difficulty,
as I have already said.

293. Indeed it will be sufficiently evident, both from what has already been proved, it is sufficient to
as well as from what will follow, that there is nowhere any sign of such living forces, nor xheory that" there
is there any necessity. For all the phenomena of Nature depend upon motions & equilibrium, do not exist in
& thus from dead forces & the velocities induced by the action of such forces. For this forces.e
reason, in the dissertation De Firibus Vivis, which was what led me to this Theory thirteen

years ago, I asserted that there are no living forces in Nature, & that many things were
usually brought forward to prove their existence, I explained clearly enough by velocities
derived solely from forces that were not living forces.

224 PHILOSOPHIC NATURALIS THEORIA

Tiobf Ce"a stl cT to 294" Unum hie proferam, quod pertinet ad collisionem globorum elasticorum obliquam,

quatuor giobos, quae compositionem resolutioni substitutam illustrat. Sint in fig. 53 triangula ADB.
BH9> GML rectan§ula in D> H> M «a, ut latera BD, GH, LM, sint aequalia singula
dimidiae basi AB, ac sint BG, GL, LQ parallels AD, BH, GM. Globus A cum velocitate
AB = 2 incurrat in B in globum C sibi aequalem jacentem in DB producta : ex collisione
obliqua dabit illi velocitatem CE = I, aequalem suae BD, quam amittet, & progredietur
per BG cum velocitate = AD = \/ 3. Ibi eodem pacto si inveniat globum I, dabit ipsi
velocitatem IK = I, amissa sua GH, & progredietur per GL cum V 2 ; turn ibi dabit,
globo O velocitatem OP = I, amissa sua LM, & abibit cum LQ = I, quam globo R,
directe in eum incurrens, communicabit. Quare, ajunt, ilia vi, quam habebat cum veloci-
tate = 2, communicabit quatuor globis sibi aequalibus vires, quse junguntur cum
velocitatibus singulis = I ; ubi si vires fuerint itidem singulas = i, erit summa virium =
4, quae cum fuerit simul cum velocitate = 2, vires sunt, non ut simplices velocitates in
massis sequalibus, sed ut quadrata velocitatum.

Ejus explicate in 295. At in mea Theoria id argumentum nullam sane vim habet. Globus A non

!dribulheviVis t« transfer! in globum C partem DB suae velocitatis AB resolutse in duas DB, TB, & cum

soiam compositi- ea partem suae vis. Agit in globos vis nova mutua in partes oppositas, quae alteri imprimit

velocitatem CE, alteri BD. Velocitas prior globi A expressa per BF positam in directum

cum AB, & ipsi aequalem, componitur cum hac nova accepta BD, & oritur velocitas BG

minor ipsa BF ob obliquitatem compositionis. Eodem pacto nova vis mutua agit in globos

in G, & I, in L, & O, in Q, & R, & velocitates novas primi globi GL, LQ, zero, componuht

velocitates GH, & GN ; LM, & LS ; LQ, & QL, sine ulla aut vera resolutione, aut

translatione vis vivae, Natura in omni omnino casu, & in omni corporum genere agente

prorsus eodem pacto.

Quid notandum 296. Sed quod attinet ad collisiones corporum, & motus [139] reflexes, unde digressi

sunTSobTcontimli1 eramus '•> inprimis illud monendum duco ; cum nulli mihi sint continui globi, nulla plana
autpianacontinua', continua ; pleraque ex illis, quae dicta sunt, habebunt locum tantummodo ad sensum,
contactushematl°US & Proxime tantummodo, non accurate ; nam intervalla, quae habentur inter puncta,
inducent inaequalitates sane multas. Sic etiam in fig. 43. ubi globus delatus ad B incurrit
in CD, mutatio viae directionis non fiet in unico puncto B, sed per continuam curvaturam ;
ac ubi globus reflectetur, ipsa reflexio non fiet in unico puncto B, sed per curvam quandam.
Recta AB, per quam globus adveniet, non erit accurate recta, sed proxime ; nam vires ad
distantias omnes constant! lege se extendunt, sed in majoribus distantiis sunt insensibiles ;
nisi massa, in quam tenditur, sit enormis, ut est totius Terrae massa in quam sensibili vi
tendunt gravia. At ubi globus advenerit satis prope planum CD ; incipiet incurvari
etiam via centri, quae quidem, jam attracto, jam repulso globo, serpet etiam, donee alicubi
repulsio satis praevaleat ad omnem ejus perpendicularem velocitatem extinguendam (utar
enim imposterum etiam ego vocabulis communibus a virium resolutione petitis, uti &
superius aliquando usu fueram, & nunc quidem potiore jure, posteaquam demonstravi
aequipollentiam verse compositionis virium cum imaginaria resolutione), & retro etiam
motum reflectat.

Lex reflexionis 297. Et quidem si vires in accessu ad planum, ac in recessu a piano fuerint prorsus
^q113!68 inter se ; dimidia curva ab initio sensibilis curvaturae usque ad minimam distantiam
a piano erit prorsus aequalis, & similis reliquae dimidise curvae, quae habebitur inde usque
ad finem curvaturae sensibilis, ac angulus incidentiae erit sequalis angulo reflexionis. Id
in casu, quem exprimit fig. 43, curva ob insensibilem ejus tractum considerata pro unico
puncto, pro perfecte elasticis patet ex eo, quod in triangulis rectangulis AFB, MIB latera
aequalia circa angulos rectos secum trahant aequalitatem angulorum ABF, MBI, quorum
alter dicitur angulus incidentiae, & alter reflexionis, ubi in imperfecte elasticis non habetur
ejusmodi aequalitas, sed tantummodo constans ratio inter tangentem anguli incidentiae,
& tangentem anguli reflexionis, quae nimirum ad radios sequales BF, BI sunt FA, & Im,
& sunt juxta denominationem, quam supra adhibuimus num. 272, & retinuimus hue usque,
ut 772 ad n.

A THEORY OF NATURAL PHILOSOPHY

225

FIG. 53-

Q

226

PHILOSOPHIC NATURALIS THEORIA

FIG. 53.

A THEORY OF NATURAL PHILOSOPHY

227

294. I will bring forward here one example, which deals with the oblique impact of
elastic spheres ; this will illustrate the substitution of composition for resolution. In
Fig. 53, let ADB, BHG, GML, be right-angled triangles such that the sides BD, GH, LM
are each equal to half the base AB, & let BG, GL, LQ be parallel to AD, BH, GM. Suppose
the sphere A, moving with a velocity = 2, to impinge at B upon a sphere C, equal to itself,
lying in DB produced. From the oblique impact, it will impart to C a velocity CE = i,
which is equal to its own velocity BD, which it loses ; & it itself will go on along BG with
a velocity equal to AD = -\/3- It will then come to the sphere I, will give to it a velocity
IK = I, losing its own velocity GH, & will go on along GL with a velocity equal to \/2.
Then it will give the sphere O a velocity OP = i, losing its own velocity LM, & will go on
with a velocity LQ = I. This it will give up to the sphere R, on which it impinges directly.
Wherefore, they contest, by means of the force which it had in connection with a velocity

= 2, it will communicate to four spheres equal to itself forces, each of which is conjoined
with a velocity = i ; hence, since, if each of the forces were also equal to I, their sum
would be equal to 4, & this sum was at the same time connected with a velocity = 2, it
must be that the forces are not in the simple ratio of the velocities in equal masses but as
their squares.

295. But in my Theory this argument has no weight at all. The sphere A does not
transfer to the sphere C that part DB of its velocity AB resolved into the two parts DB,
TB ; & with it part of its force. There acts on the spheres a new mutual force in opposite
directions, which gives the velocity CE to the one sphere, & the velocity BD to the other.
The previous velocity of the sphere A, represented by BF lying in the same direction as,
and equal to, AB, is compounded with the newly received velocity BD, and the velocity
BG, less than BF on account of the obliquity of the composition, is the result. In the
same way, a new mutual force acts upon the spheres at G & I, at L & O, at Q & R, & the
new velocities of the first sphere, GL, LQ & zero, are the resultants of the velocities
GH & GN, LM & LS, & LQ & QL respectively ; & there is not either any real resolution,
or transference of living force. 'Nature in every case without exception, & for all classes
of bodies acts in exactly the same manner.

296. But we have digressed from the consideration of impact of bodies & reflected
motions. Returning to them, I will first of all bring forward a point to be noted carefully.
Since, to my idea, there are no such things as continuous spheres or continuous planes,
many of the things that have been said are only true as far as we can observe, & only very
approximately & not accurately ; for the intervals, which exist between the points, induce
a large number of inequalities. So also, in Fig. 43, where the sphere carried forward to
B impinges upon the plane CD, the change in the direction of the path will not take place
at the single point B, but by means of a continuous curvature. Also in the case where
the sphere is reflected, the reflection will not occur at the single point B, but along a certain
curve. The straight line AB, along which the sphere is approaching, will not accurately
be a straight line, but approximately so ; for the forces extend to all distances according
to a fixed law, but at fairly great distances are insensible, unless the mass it is approaching
is enormous, as in the case of the whole Earth, to which heavy bodies tend to approach
with a sensible force. But as soon as the sphere comes sufficiently near to the plane CD,
the path to the centre will begin to be curved, & indeed, as the sphere is first attracted
& then repelled, the path will be winding, until it reaches a distance at which the repulsion
will be strong enough to destroy all its perpendicular velocity (for in future I also will use
the usual terms derived from resolution of forces, as I did once or twice in what has been
given above ; & this indeed I shall now do with greater justification seeing that I have
proved the equivalence between true composition & imaginary resolution), & also will
reflect the motion.

297. Indeed, if the forces during the approach towards the plane & those during the
recession from it were exactly equal to one another, then the half of the curve starting
from the beginning of sensible curvature up to the least distance from the plane would
be exactly equal & similar to the other half of the curve from this point to the end of sensible
curvature, & the angle of incidence would be equal to the angle of reflection. This, in
the case for which Fig. 43 is drawn, where on account of the insensible length of its arc
the curve is considered as a single point, is evidently true for perfectly elastic bodies, from
the fact that in the right-angled triangles AFB, MIB, the equal sides about the right angles
involve the equality of the angles ABF, MBI, of which the first is called the angle of incidence
& the second that of reflection ; whereas, in imperfectly elastic bodies, there is no such
equality, but only a constant ratio between the tangents of the angle of incidence & the
tangent of the angle of reflection. For instance, these are, measured by the equal radii
BF, BI, equal to FA, Im ; & these latter are, according to the notation used above in Art.
272, & retained thus far, in the proportion of m to n.

Oblique impact of
a sphere on four
sph e r e s, an ex-
ample usually
brought forward in
support of living
forces.

Its explanation in
my Theory without
living forces by
means of compo-
sition alone.

It is therefore to
be noted that there
are no continuous
spheres or con-
tinuous planes, nor
such a thing as
mathematical con-
tact.

Law of reflection
for perfectly &
imperfectly elastic
bodies.

228 PHILOSOPHIC NATURALIS THEORIA

Eadem facta vi 2g8. Curvaturam in reflexione exhibet figura 154, ubi via puncti mobilis repulsi a piano

agente in ahqua /~1/~. f \-n/-\-r^-\/r • T> i • • ... JT .. ., r . . . ,

distantia, consider- <~<~> Ml ArSvjJJM, quae circa D, ubi vires mcipiunt esse sensibiles, incipit ad sensum mcurvan,
atacurvatura & desinit in eadem distantia circa D. Ea

quidem, si habeatur semper repulsio,

incurvatur perpetuo in eandem plagam,

ut figura exhibet ; si vero & attractio

repulsionibus interferatur, serpit, uti

monui ; sed si paribus a piano distantiis

vires aequales sunt ; satis patet, & accu-

ratissime demonstrari [140] etiam pos-
set, ubi semel deventum sit alicubi, ut

in Q, ad directionem parallelam piano,

debere deinceps describi arcum QD

prorsus aequalem, & similem arcui QB,

& ita similiter positum respectu plani

CO, ut ejus inclinationes ad ipsum

planum in distantiis aequalibus ab eo,
& a Q hinc, & inde sint prorsus aequales ;

R/I
'*'

quam ob causam tangentes BN, DP,
quae sunt quasi continuationes rectarum AB, MD, angulos faciunt ANC, MPO aequales,
qui deinde habentur pro angulis incidentiae, & reflexionis.

Quid, si planum sit 299. Si planum sit asperum, ut Figura exhibet, & ut semper contingit in Natura ;

catk^rene^Fonem aequalitas ilia virium utique non habetur. At si scabrities sit satis exigua respectu ejus
lucU. distantiae, ad quam vires sensibiles protenduntur ; inaequalitas ejusmodi erit perquam

exigua, & anguli incidentiae, & reflexionis aequales erunt ad sensum. Si enim eo intervallo
concipiatur sphaera VRTS habens centrum in puncto mobili, cujus segmentum RTS jaceat
ultra planum ; agent omnia puncta constituta intra illud segmentum, adeoque monticuli
prominentes satis exigui respectu totius ejus massae, satis exiguam inaequalitatem poterunt
inducere ; & proinde sensibilem aequalitatem angulorum incidentias, & reflexionis non
turbabunt, sicut & nostri terrestres montes in globo oblique projecto, & ita ponderante,
ut a resistentia aeris non multum patiatur, sensibiliter non turbant parabolicum motum
ipsius, in quo bina crura ad idem horizontale planum eandem ad sensum inclinationem
habent. Secus accideret, si illi monticuli ingentes essent respectu ejusdem sphaerae. Atque
haec quidem, qui diligentius perpenderit, videbit sane, & lucem a vitro satis laevigato resilire
debere cum angulo reflexionis aequali ad sensum angulo incidentiae ; licet & ibi pulvisculus
quo poliuntur vitra, relinquat sulcos, & monticules, sed perquam exiguos etiam respectu
distantiae, ad quam extenditur sensibilis actio vitri in lucem ; sed respectu superficierum,
quae ad sensum scabrae sunt, debere ipsam lucem irregulariter dispergi quaqua versus.
Quid in impactu 300. Pariter ubi globus non elasticus deveniat per AB in eadem ilia fig. 43, & deinde

Hs^Tn ^pianum1" ^ebeat sme reflexione excurrere per BQ, non describet utique rectam lineam accurate,
veiocitas amissa, sed serpet, & saltitabit non nihil : erit tamen recta ad sensum : velocitas vero mutabitur
m"^u™atvira "con* *ta ' ut s^ vel°citas P"01" AB ad posteriorem BI, ut radius ad cosinum inclinationis OBI
tinua. rectae BO ad planum CD, ac ipsa velocitas prior ad velocitatum differentiam, sive ad partem

velocitatis amissam, quam exprimit IQ determinata ab arcu OQ habente centrum in B,
erit ut radius ad sinum versum ipsius inclinationis. Quoniam autem imminuto in infinitum
angulo, sinus versus decrescit in infinitum etiam respectu ipsius arcus, adeoque summa
omnium sinuum versorum pertinentium ad omnes inflexiones infinitesimas tempore finito
factas adhuc in infinitum decrescit ; ubi inflexio evadat [141] continua, uti fit in curvis
continuis, ea summa evanescit, & nulla fit velocitas amissio ex inflexione continua orta, sed
vis perpetua, quae tantummodo ad habendam curvaturam requiritur perpendicularis ipsi
curvae, nihil turbat velocitatem, quam parit vis tangentialis, si qua est, quae motum perpetuo
acceleret, vel retardet ; ac in curvilineis motibus quibuscunque, qui habeantur per quas-
cunque directiones virium, semper resolvi potest vis ilia, quae agit, in duas, alteram
perpendicularem curvas, alteram secundum directionem tangentis, & motus in curva per
hanc tangentialem vim augebitur, vel retardabitur eodem modo, quo si eaedem vires agerent,
& motus haberetur in eadem recta linea constanter. Sed hasc jam meae Theoriae communia
sunt cum Theoria vulgari.

Theoremata pro 301. Communis est itidem in fig. 44, & 45 ratio gravitatis absolutae BO ad vim BI, quse

scen^um^vef retar- obliquum descensum accelerat, vel ascensum retardat, quae est, ut radius ad sinum anguli
dante ascensum in BOI, vel OBR, sive cosinum OBI. Angulum OBI est in fig. 44, quem continet directio
& BI> quse est eadem, ac directio plani CD, cum linea verticali BO, adeoque angulus OBR
est aequalis inclinationi plani ad horizontem, & angulus idem OBR in fig. 45 est is, quem
continet cum verticali BO recta CB jungens punctum oscillans cum puncto suspensionis.
Quare habentur haec theoremata : Fis accelerans descensum, vel retardans ascensum in flanis

A THEORY OF NATURAL PHILOSOPHY 229

298. Fig. 54 illustrates the curvature in reflection ; here we have the path of a moving The case of a force
point repelled by a plane CO represented by ABQDM ; this, near B, where the forces sSbie^tance •
begin to be sensible, begins to be appreciably curved, & leaves off at the same distance consideration of
from the plane, near the point D. The path, indeed, if there is always repulsion, will be the path™
continuously incurved towards the same parts, as is shown in the figure ; but if attraction

alternates with repulsion, the path will be winding, as I mentioned. However, if the
forces at equal distances from the plane are equal to one another, it is sufficiently clear,
& indeed it could be rigorously proved, that as soon as some point such as Q was reached
where the direction of the path was parallel to the plane, it must thereafter describe an
arc QD exactly equal & similar to the arc QB ; & therefore similarly placed with .respect
to the plane CO ; so that the inclinations of the parts at equal distances from the plane,
& fromQ on either side, are -exactly equal. Hence, the tangents BN, DP, which are as it
were continuations of the straight lines AB, MD, will make the angles ANC, MPO equal to
one another ; & these may then be looked upon as the angles of incidence & reflection.

299. If the plane is rough, as is shown in the figure, & such as always occurs in Nature, What if the plane
there will in no case be this equality of forces. But if the roughness is sufficiently slight tion^to

in comparison with that distance, over which sensible forces are extended, such inequality tion of light.

will be very slight, & the angle of incidence will be practically equal to the angle of reflection.

For if with a radius equal to that distance we suppose a sphere VRTS to be drawn, having

its centre at the position of the moving point, & a segment RTS lying on the other side of

the plane ; then all the points contained within that segment exert forces ; &, if therefore

the little prominences are sufficiently small compared with the whole mass, they can only

induce quite a slight inequality. Hence, they will not disturb the sensible equality of the

angles of incidence & reflection ; just as the mountains on our Earth, acting on a sphere

projected in a direction inclined to the vertical, & of such a weight that it does not suffer

much from the resistance of the air, do not sensibly disturb its parabolic motion, in which

the two parts of the parabola have practically the same inclination to the same horizontal

plane. It would be quite another matter, if the little prominences were of large size

compared with the sphere. Anyone who will study these matters with considerable care

will perceive clearly that light also must rebound from a sufficiently well polished piece of

glass with the angle of reflection to all intents equal to the angle of incidence. Although

it is true that the powder with which glasses are polished leaves little furrows & prominences ; ,

but these are always very slight compared with the distance over which the sensible action

of glass on light extends. However, for surfaces that are sensibly rough, it will be perceived

that light must be scattered irregularly in all directions.

300. Similarly, when a non-elastic sphere travels along AB, in Fig. 43, & then without What happens in
reflection has to continue along BQ, it will not describe a perfectly accurate straight line, SlLS?9 ~« »bh3"ft

...... , , , .. .-M i 11 • • i ""pact in a soit

but will wind irregularly to some extent ; yet the line will be to all intents a straight sphere ; the veio-

line. Moreover, the velocity will be changed in such a way that the previous velocity ma^ns'^unTm'pairwi

AB will be to the new velocity BI, as the radius is to the cosine of OBI the inclination in continuous cur-

of the straight line BO to the plane CD ; & the previous velocity is to the difference vature-

between the velocities, i.e., to the velocity that is lost, which is represented by IQ

determined by the arc OQ having its centre at B, as the radius is to- the versine of the

same angle. Now, since, when the angle is indefinitely diminished, the versine decreases

indefinitely with respect to the arc itself, & thus the sum of all the versines belonging to

all the infinitesimal inflections made in a finite time still decreases indefinitely ; it follows

that, when the inflexion becomes continuous, as is the case with continuous curves, this

sum vanishes, & therefore there is no loss of velocity arising from continuous inflection.

There is a perpetual force, which is required for the purpose of keeping up the curvature,

perpendicular to the curve itself, & therefore not disturbing the velocity at all ; the

velocity arises from a tangential force, if there is any, & this continuously accelerates

or retards the motion. In curvilinear motions of all kinds, due to forces in all kinds of

directions, it is always possible to resolve the force acting into two parts, one of them

perpendicular to the curve, & the other along the tangent ; the motion along the curve

will be increased or retarded by the tangential force, in precisely the same manner as if these

same forces acted & the motion was constantly in the same straight line. But all these

matters are common to my theory and the usual theory.

301. In Fig. 44, 45, there is a common ratio between the absolute gravity BO & the force The o'rems with

T>T i • i 11 „ i • • • i i r i regard to the force

Bl, which accelerates the descent or retards the ascent; & this ratio is equal to that of the accelerating de-
radius to the sine of the angle BOI, or OBR, or the cosine of OBI. The angle OBI is, in scent or retarding

TI. t ,.,. -ill T • T>T i-i-i IT • r ascent in the cases

rig. 44, that which is contained by the direction BI, which is the same as the direction of Of the inclined
the plane CD, with the vertical line BO ; & thus the angle OBR is equal to the inclination Planf & ° f the
of the plane to the horizon ; & the same angle OBR, in Fig. 45, is that which is contained r
by the vertical BO with the straight line CB, which joins the point of oscillation with the
point of suspension. Hence, we have the following theorems. The force accelerating descent,

230 PHILOSOPHIC NATURALIS THEORIA

inclinatis, vel ubi oscillatio fit in arcu circulari, est ad gravitatem absolutam, ibi quidem ut
sinus inclinationis ipsius plani, hie vero ut sinus anguli, quern cum verticali linea continet recta
jungens punctum oscillans cum puncto suspensionis, ad radium. E quorum theorematum
priore fluunt omnia, quae Galilaeus tradidit de descensu per plana inclinata ; ac e posteriore
omnia, quae pertinent ad oscillationes in circulo ; quia immo etiam ad oscillationes factas
in curvis quibuscunque pondere per filum suspense, & curvis evolutis applicato ; ac eodem
utemur infra in definiendo centre oscillationis.

Appiicatio Theoriae 302. Hisce perspectis, applicanda est etiam Theoria ad motuum refractionem, ubi

tres rasus^veioci- continentur elementa mechanica pro refractione luminis, & occurrit elegantissimum
tatis normaiis ex- theorema a Newtono inventum hue pertinens. Sint in fig. 55 binae superficies AB, CD
' l t£e> parallelae inter se, & punctum mobile quodpiam extra ilia plana nullam sentiat vim, inter
ipsa vero urgeatur viribus quibuscunque, quae tamen & semper habeant directionem
perpendicularem ad ipsa plana, & in asqualibus distantiis ab altero ex iis asquales sint ubique ;
ac mobile deferatur ad alterum ex iis, ut AB, directione quacunque GE. Ante appulsum
feretur motu rectilineo, & sequabili, cum nulla urgeatur vi : ejus velocitatem exprimat EH,
quas erecta ER, perpendiculari ad AB, resolvi poterit in duas, alteram perpendicularem
ES, alteram parallelam HS. Post ingressum inter alia duo [142] plana incurvabitur motus
illis viribus, sed ita, ut velocitas parallela ab iis nihil turbetur, velocitas autem perpendicularis
vel minuatur, vel augeatur ; prout vires tendent versus planum citerius AB, vel versus
ulterius CD. Jam vero tres casus haberi hinc possunt ; vel enim iis viribus tota velocitas
perpendicularis ES extinguitur, antequam deveniatur ad planum ulterius CD ; vel perstat
usque ad appulsum ad ipsum CD, sed imminuta, vi contraria praevalente viribus eadem
directione agentibus ; vel perstat potius aucta.

Primo reflexionem 303. In primo casu, ubi primum velocitas perpendicularis extincta fuerit alicubi in

X, punctum mobile reflectet cursum retro per XI, & iisdem viribus agentibus in regressu,
quae egerant in progressu, acquiret velocitatem perpendicularem IL asqualem amissae ES,
quas composita cum parallela LM, sequali priori HS, exhibebit obliquam IM in recta nova
IK, quam describet post egressum, & erunt aequales anguli HIL, MES, adeoque & anguli
KIB, GEA ; quod congruit cum iis, quae in fig. 54. sunt exhibita, & pertinent ad
reflexionem.

- 3°4" ^n secundo casu prodibit ultra superficiem ulteriorem CD, sed ob velocitatem

cessu ad superficiem perpendicularem OP minorem priore ES, parallelam vero PN sequalem priori HS, erit
iUd^enterefractio° angulus ONP minor, quam EHS, adeoque inclinatio VOD ad superficiem in egressu minor
nem, sed cum inclinatione GEA in ingressu. Contra vero in tertio casu ob op majorem ES, angulus
udD erit major. In utroque autem hoc casu differentia quadratorum velocitatis ES, &
OP vel op, erit constans, per num. 177 in adn. m, quscunque fuerit inclinatio GE in
ingressu, a qua inclinatione pendet velocitas perpendicularis SE.

sufuV°angu?i1Suici- 3°5- Inde autem facile demonstratur, fore sinum anguli incidentiae HES, ad sinum
dentiae, ad sinum anguli refracti PON (& quidquid dicitur de iis, quae designantur litteris PON, erunt com-
munia iis, quae exprimuntur litteris pon) in ratione constanti, quaecunque fuerit inclinatio
rectae incidentis GE. Sumatur enim HE constans, quae exprimat velocitatem ante
incidentiam : exprimet HS velocitatem parallelam, quae erit aequalis rectae PN exprimenti
velocitatem parallelam post refractionem ; ac ES, OP expriment velocitates perpendiculares
ante, & post, quarum quadrata habebunt differentiam constantem. Sed ob HS, PN semper
aequales, differentia quadratorum HE, ON aequatur differentiae quadratorum ES, OP.
Igitur etiam differentia quadratorum HE, ON erit constans ; cumque ob HE constantem
debeat esse constans ejus quadratum ; erit constans etiam quadratum ON, adeoque constans
etiam ipsa ON, & proinde constans erit & ratio HE ad ON ; quas quidem ratio est eadem,
ac sinus anguli NOP ad sinum HES : cum enim sit in quovis triangulo rectangulo radius
ad latus utrumvis, ut basis ad sinum anguli oppositi ; in diversis triangulis rectangulis
sunt sinus, ut latera opposita divisa per [143] bases, sive directe ut latera, & reciproce ut
bases, & ubi latera sunt sequalia, ut hie HS, PN, erunt reciproce ut bases.

A THEORY OF NATURAL PHILOSOPHY

231

232

PHILOSOPHISE NATURALIS THEORIA

R

H S

L M

FIG. 55.

A THEORY OF NATURAL PHILOSOPHY 233

or retarding ascent, on inclined planes, or where there is oscillation in a circular arc, is to the
absolute gravity, in the first case as the sine of the inclination of the plane to the radius, & in
the second case as the sine of the angle between the vertical i3 the line joining the oscillating
'point to the -point of suspension, is to the radius. From the first of these theorems there
follow immediately all that Galileo published on the descent along inclined planes ; &
from the second, all matters relating to oscillations in a circle. Moreover, we have also
all matters that relate to oscillations made in curves of all sorts by a weight suspended by a
string wrapped round in volute curves ; & we shall make use of the same idea later to define
the centre of oscillation.

302. These matters being investigated, we now have to apply the Theory to the refraction Application of the
of motions, in which are contained the mechanical principles of the refraction of light ; Son^the "three
here also we find a most elegant theorem discovered by Newton, referring to the subject, cases in which the
In Fig. 55, let AB, CD be two surfaces parallel to one another ; & let a moving point feel

the action of no force when outside those planes, but when between the two planes diminished, or

suppose it is subject to any forces, so long as these always have a direction perpendicular increased-

to the planes, & they are always equal at equal distances from either of them. Suppose

the point to approach one of the planes, AB say, in any direction GE. Until it reaches

AB it will travel with rectilinear & uniform motion, since it is acted upon by no force ;

let EH represent its velocity. Then, if ER is erected perpendicular to the plane AB, the

velocity can be resolved into two parts, the one, ES, perpendicular to, & the other, HS,

parallel to, the plane AB. After entry into the space between the two planes the motion

will be incurved owing to the action of the forces ; but in such a manner that the velocity

parallel to the plane will not be affected by the forces ; whereas the perpendicular velocity

will be diminished or increased, according as the forces act towards the plane AB, or towards

the plane CD. Now there are three cases possible ; for, the whole of the perpendicular

velocity may be destroyed before the point reaches the further plane CD, or it may persist

right up to contact with the plane CD, but diminished in magnitude, owing to a force

existing contrary to the forces in that direction, or it may continue still further increased.

303. In the first case, where the perpendicular velocity was first destroyed at a point in the first case
X, the moving point will follow a return path along XI ; & as the same forces act in the d^edCti°n ** '""
backward motion as in the forward motion, the point will acquire a perpendicular velocity

IL, equal to ES, that which it lost ; this, compounded with the parallel velocity LM,
equal to the previous parallel velocity HS, will give a velocity IM, in an oblique direction
along the new straight line IK, along which the point will move after egress. Now the
angles HIL, MES will be equal, & therefore also the angles KIB, GEA ; this agrees with
what is represented in Fig. 54, & pertains to reflection.

304. In the second case, the point will proceed beyond the further surface CD ; but, In *he second case

• 1-11- /^vV> • 11 i • T-«I i •» i it i we have refraction

since the perpendicular velocity Or is now less than the previous one ES, whilst the parallel & nearer approach
velocity is the same as the previous one HS, the angle ONP will be less than the angle EHS, *°rfthe. re.fracti?g
& therefore the inclination to the surface, VOD, on egress, will be less than the inclination, third, refraction &
GEA, on ingress. On the other hand, in the third case, since op is greater than ES, the recession from the
angle uoD will be greater than the angle GEA. But in either case, we here have the difference
between the squares of the velocity ES, & that of OP, or op, constant, as was shown in
Art. 177, note m, whatever may be the inclination on ingress, made by GE with the plane,
upon which inclination depends the perpendicular velocity SE.

305. Further, from this it is easily shown that the sine of the angle of incidence HES The constant ratio
is to the sine of the angle of refraction HON (& whatever is said with regard to these angles, alguT o^taddence
denoted by the letters PON, will hold good for the angles denoted by the letters pan}, in to the sine of the
a constant ratio, whatever the inclination of the line of incidence, GE, may be. For, "**"
suppose HE, which represents the velocity before incidence, to be constant ; then HS,
representing the parallel velocity, will be equal to PN, which represents the parallel velocity

after refraction. Now, if ES, OP represent the perpendicular velocities before & after
refraction, they will have the difference between their squares constant. But, since HS,
PN are equal, the difference between the squares of HE, ON will be equal to the difference
between the squares of ES, OP. Hence the difference of the squares of HE, ON will be
constant. But, since HE is constant, its square must also be constant ; therefore the
square of ON, & thus also ON itself, must be constant. Therefore also the ratio of HE
to ON is constant ; & this ratio is the same as that of the sine of the angle NOP to the
sine of the angle HES. For, since in any right-angled triangle the ratio of the radius to
either side is that of the base to the angle opposite, in different right-angled triangles,
the sines vary as the sides opposite them divided by the bases, or directly as the sides &
inversely as the bases ; & where the sides are equal, as HS, PN are in this case, the sines
vary as the bases.

234

PHILOSOPHIC NATURALIS THEORIA

&nrati?ve°io-
citatum reciproca
ratbnis sinuum.

3°6' °-uamobrem in refractionibus, quae hoc modo fiant motu libero per intervallum
inter duo plana parallela, in quo vires paribus distantiis ab altero eorum pares sint, ratio
sinus anguli incidentiae, sive anguli, quern facit via ante incursum cum recta perpendiculari
piano, ad sinum anguli refracti, quern facit via post egressum itidem cum vertical!, est
constans, quaecunque fuerit inclinatio in ingressu. Praeterea vero habetur & illud, fore
celeritates absolutas ante, & post in ratione reciproca eorum sinuum. Sunt
ejusmodi velocitates ut HE, ON, quae sunt reciproce ut illi sinus.

enim

Haec q11^6111 ad luminis refractiones explicandas viam sternunt, ac in Tertia
open' occasionem Parte videbimus, quo pacto hypothesis hujusce theorematis applicetur particulis luminis.
Sed interea considerabo vires mutuas, quibus in se invicem agant tres massas, ubi habebuntur
generalius ea, quae pertinent etiam ad actiones trium punctorum, & quae a num. 225, &
228 hue reservavimus. Porro si integrae vires alterius in alteram diriguntur ad ipsa centra
gravitatis, referam hie ad se invicem vires ex integris compositas ; sed etiam ubi vires aliam
directionem habeant quancunque ; si singulae resolvantur in duas, alteram, quae se dirigat
a centre ad centrum ; alteram, quae sit ipsi perpendicularis, vel in quocunque dato angulo
obliqua ; omnia in prioribus habebunt itidem locum.

Consideratio direc-

se mutuo agunt.

308. Agant in se invicem in fig. 56 tres massae, quarum centra gravitatis sint A, B, C,
yi"bus mutuis ad ipsa centra directis, & considerentur inprimis directiones virium. Vis
puncti C ex utraque CV, Cd attractiva erit Ce ; ex utraque repulsiva CY, Ca, erit CZ,
& utriusque directio saltern ad partes oppositas producta ingreditur triangulum, & secat
ilia angulum internum ACB, haec ipsi ad verticem oppositum aCY. Vi CV attractiva in
B, ac CY repulsiva ab A, habetur CX ; & vi Cd attractiva in A, ac Ca repulsiva a B, habetur
Cb, quarum utraque abit extra triangulum, & secat angulos ipsius externos. Primae Ce,
cum debeant respondere attractiones BP, AG, respondent cum attractionibus mutuis
BN, AE, vires BO, AF, vel cum repulsionibus BR, AI, vires BQ, AH, ac tarn priores binae,
quam posteriores, jacent ad eandem partem lateris AB, & vel ambae ingrediuntur triangulum
tendentes versus ipsum, vel ambae extra ipsum etiam productae abeunt, & tendunt ad
partes oppositas directionis Ce respectu AB. Secundae CZ debent respondere repulsiones
BT, AL, quae cum repulsionibus BR, AI, constituunt BS, AK, cum attractionibus BN,
AE constituunt BM, AD, ac tarn priores binae, quam posteriores jacent ad eandem plagam
respectu AB, & ambarum [144] directiones vel productae ex parte posteriore ingrediuntur
triangulum, sed tendunt ad partes ipsi contrarias, ut CZ, vel extra triangulum utrinque
abeunt ad partes oppositas direction! CZ respectu AB. Quod si habeatur CX, quam
exponunt CV, CY, turn illi respondent BP, & AL, ac si prima conjungitur cum BN, jam
habetur BO ingrediens triangulum ; si BR, turn habetur quidem BQ, cadens etiam ipsa
extra triangulum, ut cadit ipsa CX ; sed secunda AL jungetur cum AI, & habebitur AK,
quae producta ad partes A ingredietur triangulum. Eodem autem argumento cum vi Cb
vel conjungitur AF ingrediens triangulum, vel BS, quae producta ad B triangulum itidem
ingreditur. Quamobrem semper aliqua ingreditur, & turn de reliquis binis redeunt, quae
dicta sunt in casu virium Ce, CZ.

um.

Theorema pertinens 309. Habetur igitur hoc thcorema. Quando tres masses in se invicem agunt viribus
ir directis ad, centra gravitatis, vis composita saltern unius babet directionem, quez saltern producta

ad •partes oppositas secat angulum internum trianguli, i3 ipsum ingreditur : reliquce autem
duce vel simul ingrediuntur, vel simul evitant, W semper diriguntur ad eandem plagam respectu
lateris jungentis earum duarum massarum centra ; ac in primo casu vel omnes tres tendunt ad
interiora trianguli jacendo in angulis internis, vel omnes tres ad exteriora in partes triangulo
oppositas jacendo in angulis ad verticem oppositis ; in secundo vero casu respectu lateris
jungentis eas binas massas tendunt in plagas oppositas ei, in quam tendit vis ilia prioris masses.

Theorema elegan- 310. Sed est adhuc elegantius theorema, quod ad directionem pertinet, nimirum :

nens cum^eju^de- Omnium trium compositarum virium directiones utrinque products transeunt per idem punctum :

monstratione. y si id jaceat intra triangulum ; vel omnes simul tendunt ad ipsum, vel omnes simul ad partes

ipsi contrarias : si vero jaceat extra triangulum ; bince, quarum directiones non ingrediuntur

A THEORY OF NATURAL PHILOSOPHY

235

H

FIG. 56.

236

PHILOSOPHIC NATURALIS THEORIA

H

FIG. 56.

A THEORY OF NATURAL PHILOSOPHY 237

306. Hence, in refractions, which arise in this way from a free motion between two T.he ratio of ihe
parallel planes, where the forces at equal distances from one or the other of them are equal, the * ratioCC"oSfta^he
the ratio of the sine of the angle of incidence, or the angle made by the path before refraction, velocities the
with a straight line perpendicular to the plane, to the sine of the angle of refraction, or the^ines.

the angle made after refraction with the vertical also, is constant, whatever may be the
inclination at ingress. We also obtain the theorem that the absolute velocities before and
after refraction are in the inverse ratio of the sines. For such velocities are represented
by HE, ON ; & these are inversely as the sines in question.

307. These facts suggest a method for explaining refraction of light ; & in the Third Passing on to the

T> * u n ,0, • L* L »L '1. *u -rut, -.I. u TJ theorem which

Part we shall see the manner in which the hypothesis of the above theorem may be applied gave rise to this

to particles of light. In the meanwhile, I will consider the mutual forces, with which work-

three masses act upon one another ; here we shall obtain more generally all those things

that relate to the actions of three points also, such as I reserved from discussion

in Art. 225, 228 until now. Further, if the total forces of the one or the other are directed

towards their centres of gravity, I will here take account of the mutual forces compounded

of these wholes. But, where the forces have any directions whatever, if each of them is

resolved into two parts, of which one is directed from centre to centre & the other is

perpendicular to this line, or makes some given inclination with it, then also all things

that are true for the former hold good also in this case.

308. In Fig. 56, let three masses, whose centres of gravity are at A, B, C, act upon investigation of
one another with mutual forces directed to their centres ; & first of all let the directions ^ fo^e^with
of the forces be considered. The force on the point C, from the two attractive forces which three masses
CV, Cd will be Ce ; that from CY, Ca, both repulsive, will be CZ ; & the direction of ^ther P°n °De an"
both of these, produced backwards in one case, will fall within the triangle, the former

dividing the angle ACB, & the latter the vertically opposite angle aCY, into two parts.
But, from CV,. attractive towards B, & CY, repulsive from A, we obtain CX ; & from Cd,
attractive towards A, & Ca, repulsive from B, we have Cb ; & the direction of each of these
will fall without the triangle, & divide its exterior angles into two parts. To Ce, the
first of these, since we must have the corresponding attractions BP, AG, there correspond
the forces BO, AF, from combination with the mutual attractions BN, AE ; or the forces
BQ,'AH, from combination with the mutual repulsions BR, AI. Both the former of these
pairs, & the latter, lie on the same side of AB ; either both will fall within the triangle
& tend in its direction, or both will, even if produced, fall without it ; in each case, they
will tend in the opposite direction to that of Ce with respect to AB. To CZ, the second
of the forces on C, there must correspond the repulsions BT, AL ; these, combined with
the repulsions BR, AI, give the forces BS, AK ; & with the attractions BN, AE, the forces
BM, AD. Both the former of these, & both the latter, lie on the same side of AB ; &
the directions of the two, either when produced backwards will fall within the triangle
but tend in opposite directions to that of CZ with respect to it, or they will fall without
the triangle & tend off on either side in directions opposite to that of CZ with respect to
AB. Now if CX is obtained, given by CV, CY, then there will correspond to it BP & AL ;
&, if the first of these is compounded with BN, we shall then have BO falling within the
triangle ; or if compounded with BR, we shall have BQ, falling also without the triangle,
just as CX does ; but, in that case, the second action AL will be compounded with AI,
& AK will be obtained, & this when produced in the direction of A will fall within the
triangle. By the same argument, with the force Cb there will be associated the force AF
falling within the triangle, or the force BS, which when produced in the direction of B
will also fall within the triangle. Hence, in all cases, some one of the forces falls within
the triangle ; & then what has been said in the case of Ce, CZ will apply to the other two
forces.

309. We therefore have the following theorem. When three masses act upon one another Theorern relating

• .it ' ]• . j . 1^1- 1 • i i i • ri to the directions of

with forces directed towards their centres of gravity, the resultant force, in at least one case, the forces.
will have a direction which, produced backwards if necessary, will divide an internal angle
of the triangle into two parts, W fall within the triangle. Also the remaining two forces will
either both fall within, or both without, the triangle W will in all cases be directed towards
the same side of the line joining the centres of the two masses. In the first case, all three forces
either tend towards the interior of the triangle, falling within the interior angles, or outwards
away from the triangle, falling within the angles that are vertically opposite to the interior
angles. In the second case, on the other hand, they tend to opposite sides, of the line joining
the two masses, to that towards which the force on the third mass tends.

310. But there is a still more elegant theorem with regard to the directions of the A still more^eie-
forces, namely : — The directions of all three resultant forces, when produced each way, pass ^^ regard^ the
through the same point. If this point lies within- the triangle, all three forces tend towards directions of the
it, or all three away from it ; but, if it lies without the triangle, those two forces, which do not ^onstration.

238 PHILOSOPHIC NATURALIS THEORIA

triattgulum, tendunt ad ipsum, ac tertia, cujus directio triangulum ingreditur, tend.it ad Cartes
ipsi contrariias ; vel illce bints ad partes ipsi contr arias, W tertia ad ipsiim.

Prima pars, quod omnes transeant per idem punctum, sic demonstratur. In figura
quavis a 57 ad 62, quae omnes casus exhibent, vis pertinens ad C sit ea, quas triangulum
ingreditur, ac reliquse binas HA, QB concurrant in D : oportet demonstrare, vim etiam,
quae pertinet ad C, dirigi ad D. Sint CV, Cd vires componentes, ac ducta CD, ducatur
VT parallela CA, occurrens CD in T ; & si ostensum fuerit, ipsam fore aequalem Cd •
res erit perfecta : ducta enim dT remanebit CVTW parallelogramrnum, per cujus diagonalem
CT dirigetur vis composita ex CV, Cd. Ejusmodi autem aequalitas demonstrabitur
considerando rationem CV ad Cd compositam ex quinque intermediis, CV ad BP, BP ad
PQ, PQ, sive BR ad AI, AI, sive HG ad AG, AG ad [145] Cd. Prima vocando A, B, C
massas, quarum ea puncta sunt centra gravitatum, est ex actione, & reactione aequalibus
ratio massae B ad C ; secunda sin PQB, sive ABD, ad sin PBQ, sive CBD ; tertia A ad
B : quarta sin HAG, sive CAD, ad sin GHA, sive BAD : quinta C ad A. Tres rationes,
in quibus habentur massas, componunt rationem BxAxCadCxBxA, quas est i ad
i, & remanet ratio sin ABD x sin CAD ad sin CBD X sin BAD. Pro sin ABD, & sin
BAD, ponantur AD, & BD ipsis proportionales ; ac pro sinu CAD, & sin CBD ponantur

sin ACD X CD „ sin BCD X CD . . , ™ . . . . , , , .

._ — , & — , ipsis aequales ex ingonometna, & habebitur ratio

AD r>D

sin ACD X CD ad sin BCD X CD sive sin ACD, vel CTV, qui ipsi aequatur ob VT,
CA parallelas, ad sin BCD, sive VCT, nimirum ratio ejusdem illius CV ad VT. Quare
VT aequatur Cd, CVTd est parallelogrammum, & vis pertinens ad C, habet directionem
itidem transeuntem per D.

Secunda pars patet ex iis, quae demonstrata sunt de directione duarum virium, ubi
tertia triangulum ingreditur, & sex casus, qui haberi possunt, exhibentur totidem figuris.
In fig. 57, & 58 cadit D extra triangulum ultra basim AB, in 59, & 60 intra triangulum,
in 61, & 62 extra triangulum citra verticem ad partes basi oppositas, ac in singulorum
binariorum priore vis CT tendit versus basim, in posteriore ad partes ipsi oppositas. In
iis omnibus demonstratio est communis juxta leges transformationis locorum geometri-
corum, quas diligenter exposui, & fusius persecutus sum in dissertatione adjecta meis
Sectionum Conicarum Elementis, Elementorum tomo 3.

311- Quoniani evadentibus binis HA, O_B parallelis, punctum D abit in infinitum
paraiieiarum. & tertia CT evadit parallela reliquis binis etiam ipsa juxta easdem leges ; patet illud : Si

bince ex ejusmodi directionibus fuerint parallels inter se ; erit iisdem parallela y tertia :
ac ilia, qua jacet inter directiones virium transeuntes per reliquas binas, quce idcirco in eo
casu appellari potest media, habebit directionem oppositam directionibus reliquarum conformibus
inter se.

AHud generate ter- 312. Patet autem, datis binis directionibus virium, dari semper & tertiam. Si enim

dTtis'wnU011' 6 illae sint parallelae ; erit illis parallela & tertia : si autem concurrant in aliquo puncto ;

tertiam determinant recta ad idem punctum ducta : sed oportet, habeant illam conditionem,

ut tarn binae, quas triangulum non ingrediantur, quam quae ingrediantur, vel simul tendant

ad illud punctum, vel simul ad partes ipsi contrarias.

Theorema pracip. 313. Haec quidem pertinent ad directiones : nunc ipsas earum virium magnitudines

dine, quod^oU inter se comparabimus, ubi statim occurret elegantissimum illud theorema, de quo

Open occasionem mentionem fed num. 225 : Vires acceleratrices binarum quarumvis e tribus massis in se

s trVtio]expeditis." mutuo agentibus sunt in rations composita ex tribus, [146] nimirum ex directa sinuum angulorum

sima- quo s continet rec ta jungens ipsarum centra gravitatis cum rectisductis ab iisdem centrisad centrum

tertice mass<e ; reciproca sinuum angulorum, quos directiones ipsarum virium continent cum

iisdem rectis illas jungentibus cum tertia ; & reciproca massarum. Nam est BQ ad AH

assumptis terminis mediis BR, AI in ratione composita ex rationibus BQ, ad BR, & BR

ad AI, & AI ad AH. Prima ratio est sinus QRB, sive CBA ad sinum BQR, sive PBQ, vel

CBD : secunda massae A ad massam B : tertia sinus IHA, sive HAG, vel CAD, ad sinum

HIA, sive CAB : eae rationes, permutato solo ordine antecedentium, & consequentium,

sunt rationes sinus CBA ad sinum CAB, quae est ilia prima e rationibus propositis directa ;

sinus CAD ad sinum CBD, quae est secunda reciproca : & massae A ad massam B, quas

est tertia itidem reciproca. Eadem autem est prorsus demonstratio : si comparetur BQ,

vel AH cum CT, ac in hac demonstratione, ut & alibi ubique, ubi de sinubus angulorum

A THEORY OF NATURAL PHILOSOPHY

239

Q

B R

FIG. 57.

A, I

E

FIG. 59.

R B

E

Fio. 6c

P Q

I A

E B R

FIG, 61.

H

Fio. 62.

240

PHILOSOPHISE NATURALIS THEORIA

B R

FIG. 57.

Q P

A I

R B

FIG. 59

FIG. 60.

P Q

I A

E

FIG. 61.

B R

H

FIG. 62.

A THEORY OF NATURAL PHILOSOPHY 241

fall within the triangle, tend towards it, W the third, whose direction does not fall within the
triangle, tends away from it, or the former two tend away from the point y the third towards it.
The proof of the first part of the theorem, that the forces all pass through the same point,
is as follows. In any one of the diagrams from Fig. 57 to Fig. 62, which between them
give all possible cases, let the force which acts on C be that which falls within the triangle ;
& let the other two, HA & QB, meet in the point D ; then it has to be shown that the'
force which acts on C, also passes through D. Let CV, Cd be the component forces ; join
CD & draw VT parallel to CA to meet CD in T ; then, if it can be shown that VT is equal
to Cd, the proposition is proved ; for, if dT is joined, CVTd will be a parallelogram, &
the force compounded of CV & Cd will be directed along its diagonal. Such equality will
be proved by considering the ratio of CV to Cd, compounded of the five intermediate
ratios CV to BP ; BP to PQ ; PQ, or BR, to AI ; AI, or HG, to AG ; & AG to Cd. The
first of these, if we call the masses A, B, C, which have these points as their centres of gravity,
will, on account of the equality of action & reaction, be the ratio of the mass B to the mass
C ; the second, the ratio of the sine of PQB, or ABD, to the sine of PBQ, or CBD ; the
third, that of the mass A to the mass B ; the fourth, that of the sine of HAG, or CAD,
to the sine of GHA, or BAD ; the fifth, that of the mass C to the mass A. The three
ratios, in which the masses appear, together give the ratio BxAxCtoCxBxA,
which is that of i to i ; & there remains the ratio of sinAED X sinCAD to wzCBD X
sinEAD. For sinABD & sinEAD substitute AD & BD, which are proportional to them ;
& for sinCAD & sinCED substitute sinACD X CD/AD & sinECD X CD/BD, which are
equal to them by trigonometry. There will be obtained the ratio of sinACD X CD to
sinECD X CD, or sinACD to sinECD ; &, since VT & CA are parallel, this ratio is equal to
that of sinCTV to .rzWCT, that is, to the ratio of CV to VT. Therefore VT is equal to
Cd, CVT^ is a parallelogram, & the force on C has also its direction passing through D.
The second part is evident from what has already been proved with regard to the directions
of two forces when the third falls within the triangle ; & the six possible cases are shown
in the six figures. In Fig. 57, 58, the point D falls without the triangle on the far side
of the base AB ; in Fig. 59, 60, it falls within the triangle ; in Fig. 61, 62, outside the
triangle on the side of the vertex remote from the base ; & in the first of each pair of figures,
the force CT tends towards the base, & in the latter away from it. In all of these the
proof is the same, having regard to the laws of transformation of geometrical positions ;
these I have set forth carefully, & I investigated them more minutely in a dissertation
added as a supplement to my Sectionum Conicarum Elementa, the third volume of my
Elementa Matheseos.

311. Now, since the point D will go off to infinity, when two of the forces, HA & Corollary for the
QB, happen to be parallel, & the third also, according to the same laws, becomes parallel Directions. pa:

to the other two, we have this theorem. // two of these forces are parallel to one another,
the third also is parallel to them ; & that force, which lies between the directions of the other
two, y consequently in that case can be called the middle force, has its direction opposite to
the directions of the other two, which are in agreement with one another.

312. Further, it is clear that, when the directions of two of the forces are given, the Another general

^

direction also of the third force is given in all cases. For if the former are parallel, the tion^ot'the third
third will be parallel to them ; & if the former meet at a point, the straight line joining force is given when
the third mass to this point will determine the third direction. But this condition holds ; thee 0^£ ^ a°e
namely, that the two which do not fall within the triangle, or the pair which do fall within given.
the triangle, either both tend towards the point D, or both tend away from it.

313. So much with regard to directions ; now we will go on to compare with one Fundamental theo.
another the magnitudes of these forces. We immediately come to that most elegant Magnitude1 ° which
theorem, which has already been mentioned in Art. 225. The accelerating effects of any gave rise to the

,, T ' ,, j • ,• jjjjZ. whole of this work.

two masses out of three that mutually act upon one another are in a ratio compounded of three
ratios ; namely, the direct ratio of the sines of the angles made by the straight line joining the
centres of gravity of these two with the straight lines joining each of these to the centre of gravity
of the third mass : the inverse ratio of sines of the angles which the directions of the forces make
with the straight lines joining the two masses to the third ; W the inverse ratio of the masses. For,
if BR, AI are taken as intermediary terms, the ratio of BQ to AH is equal to the ratios com-
pounded from the ratio of BQ to BR, that of BR to AI, & that of AI to AH. The first ratio is
equal to that of the sine of QRB, or CBA, to the sine of BQR, or PBQ, or CBD ; the second
is that of the mass A to the mass B ; & the third is equal to that of the sine of IHA, or
HAG, or CAD to the sine of HIA, or CAB. These ratios are, by a simple permutation
of the antecedents & consequents, as sinCEA is to sinCAE, which is the first direct ratio
of those enunciated ; as sinCAD to sinCED, which is the second inverse ratio ; & as^ the
mass A to the mass B, which also is the third inverse ratio. Moreover the proof is precisely
similar, if the ratio of BQ, or AH, to CT is considered ; & in this proof, as also in all others,

242 PHILOSOPHIC NATURALIS THEORIA

agitur, angulis quibusvis substitui possunt, uti saepe est factum, & fiet imposterum, eorum
complementa ad duos rectos, quae eosdem habent sinus.

CoroiiaHum sim. 314. Inde consequitur, esse ejusmodi vires reciproce, ut massas ductas in, suas distantias

FpsU.pr° V" S a tertia massa, y reciproce, ut sinus, quos earum directiones continent cum iisdem rectis ;
adeoque ubi e& ad, ejusmodi rectas inclinentur in angulis (squalibus, esse tantummodo reciproce,
ut producta massarum per distantias a massa tertia. Nam ratio directa sinuum CBA, CAB
est eadem, ac distantiarum AC, BC, sive reciproca distantiarum BC, AC, qua substituta pro
ilia, habentur tres rationes reciprocje, quas exprimit ipsum theorema hie propositum.
Porro ubi anguli aequales sunt, sinus itidem sunt aequales, adeoque eorum sinuum ratio
fit I ad i.

Ratio virium mo. 315. Vires autem matrices sunt in ratione composita ex binis tantummodo, nimirum

directa sinuum angulorum, quos continent distantiee a tertia massa cum distantia a se invicem ;
y reciproca sinuum angulorum, quos continent cum iisdem distantiis directiones virium ; vel
in ratione composita ex reciproca illarum distantiarum, y reciproca horum posteriorum sinuum :
ac si inclinationes ad distantias sint eequales, in sola ratione reciproca distantiarum. Nam
vires motrices sunt summae omnium virium determinantium celeritatem in punctis
omnibus secundum earn directionem, secundum quam movetur centrum gravitatis commune,
quae idcirco sunt praeterea directe, ut massae, sive ut numeri punctorum ; adeoque ratio
directa, & reciproca massarum mutuo eliduntur.

Ratio yirium acce- 316. Praeterea vires acceleratrices, si alicubi earum directiones concurrunt, sunt ad se

dirituiatur1'adbl^if ^nv^cem ™ ratione composita ex reciproca massarum, & reciproca sinuum angulorum, quibus

quod commune inclinantur ad directionem ter tice ; y vires motrices in hac poste-[i4j]-riore tantum. Nam

punctum. o{j iatera proportionalia sinubus angulorum oppositorum, erit AC X sin CAD = CD X

sin CDA ; & pariter CB x sin CBA = CD X sin CDB. Quare ob CD communem,

sola ratio sinuum ADC, BDC, quibus directiones AD, BD inclinantur ad CD, aequatur

compositae ex rationibus sinuum CAD, CBD, & distantiarum CA, CB, quae ingrediebantur

rationem virium B, & A ; ac eodem pacto AC X sin ACD= AD x sin ADC, & AB x

sin ABD = AD X sin ADB, adeoque AC X sin ACD ad AB x sin ABD, ut sinus ADC

ad sinum ADB, quibus directiones CD, BD inclinantur ad AD ; & eadem est demonstratio

pro sinubus ADB, EDB assumpto communi latere BD.

Alia expressio tam 317. Si ducatur MO parallela DA, occurrens BD, CD in M, O, y compleatur parallelo-
qut'm11 SiSStri- grammum DMON ; erunt vires motrices in C, B, A ad se invicem, ut recta DO, DM, DN,
cium in eodem casu. y vires acceleratrices prteterea in ratione massarum reciproca. Est enim ex praecedenti vis
motrix in C ad vim in B, ut sin BDA ad sin CDA, vel ob AD, OM parallelas, ut sin DMO
ad sin DOM, nimirum ut DO ad DM, & simili argumento vis in C ad vim in A, ut DO
ad DN. ' Vires autem motrices divisae per massas evadunt acceleratrices. Quamobrem si,
tres vires agerent in idem punctum cum directionibus, quas habent eee vires motrices, y essent
Us proportionates ; binee componerent vim oppositam, y tequalem tertice, ac essent in ^equilibria.
Id autem etiam directe patet : nam vires BQ, AH componuntur ex quatuor viribus BR,
BP, AI, AG, quae si ducantur in massas suas, ut riant motrices ; evadit prima aequalis, &
contraria tertiae, quam idcirco elidit, ubi deinde AH, BQ componantur simul, & in ejusmodi
compositione remanent BP, AG, ex quarum oppositis, & aequalibus CV, Cd componitur
tertia CT.

Hie debere haberi ^jg. Hinc in hisce viribus motricibus habebuntur omnia, quae habentur in compositione

fn' compositione," virium ; dummodo capiatur [resolutio] compositse contraria. Si nimirum resolvantur

resolution virium. singulae componentes in duas, alteram secundum directionem tertiae, alteram ipsi perpen-

dicularem, hae posteriores elidentur, illae priores confident summam sequalem tertiae, ubi

ambae eandem directionem habent, uti sunt binae, quas simul ingrediantur, vel simul evitent

triangulum ; nam in iis, quarum altera ingreditur, altera evitat, tertia aequaretur differen-

tiae ; & facile tam hie, quam in ratione composita, res traducitur ad resolutionem in

aliam quamcunque directionem datam, praeter directionem tertiae, binis semper elisis, &

reliquarum accepta summa ; si rite habeatur ratio positivorum, & negativorum.

AT} y TTT~) AT?

Alia expressio -jg ]?st & juu(j utije . tres vjres motrices in C, B, A sunt inter se, ut -. ' _TA, -. _,

rationum earundem AD X BD AD

virium.

, , y acceler atrices prczterea [148] in ratione reciproca massarum. Nam ex Trigonometria
BD

* BAD' & * EAD

AB AE
communis : erit sin ADB ad sin ADE, ut =- ad ==, vel, ducendo utrunque terminum

A THEORY OF NATURAL PHILOSOPHY

Q

B R

FIG. 57.

A I

E R B

FIG. 59.

HG

E

FIG. 60.

P Q

I A

B R

M

FIG. 61.

FIG. 62.

244

PHILOSOPHIC NATURALIS THEORIA

B R

FIG. 57.

Q P

A I

R B

FIG. 59.

P Q

I A

E B R

FIG. 6r.

H

FIG. 62.

A THEORY OF NATURAL PHILOSOPHY 245

where sines of angles are considered, we can substitute for any of the angles, as often has
been done, & as will be done hereafter, their supplements ; for these have the same sines.

314. Hence we have the following corollary. Such accelerating effects are inversely as simple corollary
the -products of each of the two masses into its distance from the third mass, & inversely as the

sines of the angles between their directions & these distances ; y thus, if they are inclined
at equal angles to these distances, the effects are inversely -proportional to the -products of the
masses into the distances from the third mass only. For the direct ratio of the sines of the
angles CB A, CAB is the same as that of the distances AC, BC, or inversely as the distances
BC, AC ; & if the latter is substituted for the former, we have three inverse ratios, which
are given in the enunciation of this corollary. Further, when the angles are equal, their
sines are also equal, & their ratio is that of I to i.

315. The motive forces are in a ratio compounded of two ratios only, namely, the direct The ratio of the
ratio of the sines of the angles the line joining each to the third mass & the line joining the two motlve forces-

to one another ; fcff the inverse ratio of the sines of the angles which their directions make with
these distances ; or the ratio compounded of the inverse ratio of these distances W the inverse
ratio of the latter sines. Also, if the inclinations to the distances are equal to one another, the
ratio is the simple inverse ratio of the distances. For the motive forces are the sums of all
the forces determining velocity for all points in the direction along which the common
centre of gravity will move ; & hence they are, other things apart, directly as the masses,
or as the number of points ; & thus the direct & the inverse ratio of the masses eliminate
one another.

316. Further, the accelerations, if their directions meet at a point, are to one another in The ratio of the
the ratio compounded from the inverse ratio of the masses, & the inverse ratio of the sines of ^ey "are^directed
the angles between their directions & that of the third. The motive forces are in the latter towards some com-
ratio only. For, since the sides of a triangle are proportional to the sines of the opposite mon pomt'
angles, we have AC. wzCAD = CD. sinCDA, & similarly, CB. sinCBA. = CD. wiCDB.

Hence, since CD is common, the single ratio of the sines of ADC, BDC, the inclinations
of AD, BD, to CD, is equal to that compounded from the ratios of the sines of CAD, CBD,
& the distances CA, CB, which formed the ratio of the forces on B & A. In the same
way, AC. .rzwACD = AD. jjwADC, & AB. stnABD = AD. sinADE, & therefore AC. j««ACD
is to AB. sinABD as the sine of ADC is to the sine of ADB, the inclinations of CD, BD
to AD. The proof is the same for the sines of the angles ADB, EDB, by using the common
side DB.

317. // MO is drawn parallel to DA, meeting BD, CD in M, O respectively, & if the Another expression
parallelogram DMON is completed, then the motive forces for C, B, A will be to one another t°*e forces & the
as the straight lines DO, DM, DN ; & for the accelerations, we have in addition the inverse accelerations in the
ratio of the masses. For, from the preceding article, the motive force for C is to the motive s

force for B as ji'wBDA is to sinCDA ; that is to say, since AD, OM are parallel, as j/wDMO
is to JtwDOM, or as DO is to DM. Similarly the force for C is to the force for B as DO
is to DN. Now, the motive forces divided by the corresponding masses give the accelerations.
Hence, if three forces act at a point, having the same directions as the motive forces & propor-
tional to them, the resultant compounded from any two of these will give a force equal & opposite
to the third, W they will be in equilibrium. This is immediately evident ; for, the forces BQ,
AH are compounded from the four forces BR, BP, AI, AG ; & if these are multiplied by
the corresponding masses, so as to give the motive forces, the first of them will come out
equal & opposite to the third & will thus cancel it, when later AH, BQ are compounded
together ; & in such composition we are left with BP, AG ; & from CV & Cd, which are
equal & opposite to these, the third force CT is compounded.

318. Hence for these motive forces, we have all those things which hold good in the For these forces
composition of forces, so long as resolution is considered to be the inverse of composition. ^ostfthings^hich
Thus, if each of the components is resolved into two parts, one in the direction of the third hold good for com-
force, & the other perpendicular to it, the latter will cancel one another, & the former tton^'Tforces™55
will give a sum equal to the third, when both have the same direction, as is the case when

both of them either fall within the triangle or both of them are directed away from it ;
for those, in which one falls within the triangle & the other away from it, the third will
be equal to the difference. The matter, both in this, & in the ratio compounded of these,
is easily referred to a resolution in any chosen direction other than the direction of the
third, the two at right angles always cancelling one another & the sum being taken of those
that remain ; provided due regard is had to positives & negatives.

319. Here is another useful theorem. The three motive forces on C, B, & A are in Another expression
the ratio of AB.ED/ADVBD, AE/AD, BE/BD, W the accelerations have, in addition, **
the inverse ratio of the masses. For, by trigonometry, we have AB/BD = JtflADB/tt'ffBAD,

& AE/ED = j*«ADE/tt«EAD. Hence, since the divisors sinEAD, & ji'wEAD are equal,
it follows that sinADB is to sinADE as AB/BD is to AE/ED ; or, multiplying each term

246 PHILOSOPHIC NATURALIS THEORIA

in •£•= » ut AT-. ; ^ ad -pp- . Simili autem argumento est itidem sin BDA. sin BDE
/VL' /\U X rjU AL)

AB x ED BE
: : ADx?p • BD ; ex quo patent omma.

Expressio simpli- 320. Si punctum D abeat in infinitum, directionibus virium evadentibus parallels ;

leiumi! C " rati° rectarum ED, AD, BD, ad se invicem evadit ratio asqualitatis. Quare in eo casu

illae ties vires sunt ut AB, AE, EB, in quibus prima asquatur summae reliquarum. Conci-
piantur rectas parallelae directioni virium ductas per omnium trium massarum centra
gravitatis, quarum massarum earn, quae jacuerit inter reliquarum binarum parallelas diximus
mediam : ac si ducantur in quavis alia directione data rectae ab iis massis ad illas parallelas ;
erunt ejusmodi distantiae ab iis parallelis, ut ipsas AB, EB, ad quas erunt singular in ratione
data, ob datas directiones. Quare pro vinous parallelis habetur hujusmodi theorema :
Vires parallels matrices binarum quarunvis ex tribus massis sunt inter se reciproce ut distantice
a directione communi transeunte per tertiam : vires autem acceleratrices prtzterea in ratione
reciproca massarum, & media est directionis contraria respectu reliquarum, ac vis media
matrix tequatur reliquarum summce, utralibet vero extrema differentia.

Appiicatio ratio- 52i. Hoc theorema primo quidem exhibet centrum asquilibrii, viribus utcunque

num supenorum ad ,. J .. , r ., * n. . T» /-i /„

centrum zquUibrii. divergentibus, vel convergentibus. bi nimirum sint tres massas A, B, C (& nomine massarum
etiam intelligi possunt singula puncta), quarum binas, ut A, & B, solicitentur viribus
motricibus externis ; poterunt mutuis viribus illas elidere, ac esse in asquilibrio, & eas
elident omnino, mutatis, quantum libuerit, parum mutuis distantiis ; si fuerint ante
applicationem earum virium externarum in satis validis limiribus cohassionis, ac vis massas
C elidatur fulcro opposite in directione DC, vel suspensione contraria : dummodo binae
illas vires ductae in massas habeant conditiones requisitas in superioribus, ut nimirum ambae
tendant ad idem punctum, vel ab eodem, aut si fuerint parallelae, ambae eandem directione m
habeant, ubi simul ambae ingrediantur, vel simul ambae evitent triangulum ABC : ubi
vero altera ingrediatur triangulum, altera evitet, tendat aliera ad punctum concursus,
altera ad partes illi oppositas : vel si fuerint parallelae, habeant directiones [149] oppositas :
& si parallelas fuerint ; sint inter se, ut distantiae a directione virium transeunte per C ;
si fuerint convergentes, sint reciproce, ut sinus angulorum, quos earum directiones continent
cum recta ex C tendente ad earum concursum, vel sint in ratione reciproca sinuum
angulorum, quos continent cum rectis AC, BC, & ipsarum rectarum conjunctim.

Dcterminatio vis, 322. Determinabitur autem admodum facile per ipsa theoremata etiam vis, quam

™ ' sustinebit fulcrum C, quae in casu parallelism! aequabitur summae, vel differentiae reliquarum,

prout ibi fuerit media, vel extrema : & in casibus reliquis omnibus aequabitur summae
pariter, vel differentiae reliquarum ad suam directionem reductarum, reliquis binis in
resolutione priorum sociis se per contrariam directionem, & aequalitatem elidentibus.

Consideratio mas- *2*. Habebitur ieitur, quidquid pertinet ad aequilibrium virium agentium in eodem

sarum etiam inter- 1J-i • • n -i •• i_ •

mediarum. qua: piano, & connexarum non per virgas mnexiles carentes omni vi praster cohaesionem, uti
connectant massas eas vulgO concipiunt, sed hisce viribus mutuis. Et Theoria quidem habebit locum turn

viribus externis , . 3. ., ,. A ~n /-< • •• • J* J*

prsditas, & positas hie, turn in sequentibus ; licet massae A, B, C non agant in se invicem immediate, sed sint
in aequiiibno. ^{^ massa; intermedise, quae ipsas jungant. Nam si inter massam B, & C sint alias massae

nullis externis viribus agitatas, & positae in aequilibrio cum hisce massis, & inter se, ac prima,
quae venit post B, agat in ipsam vi motrice asquali BP, aget & B in ipsam vi asquali : quare
debebit ilia ad servandum aequilibrium urgeri a secunda, quae est post ipsam, vi aequali
in partes contrarias. Hinc asquali contraria aget tertia in secundam, ut secunda in
asquilibrio sit, & ita porro, donee deveniatur ad C, ubi habebitur vis motrix asqualis motrici,
quae erat in B, & erunt vires BP, CV acceleratrices in ratione reciproca massarum B, & C,
cum vires illas motrices asquales sint producta ex acceleratricibus ductis in massas. At si
circumquaque sint massas quotcunque cum vacuis quibuscunque, ac ubicunque interjectis,
quae connectantur cum punctis A, B, C, affectis illis tribus viribus externis, quarum una
concipitur provenire a fulcro, una solet appellari potentia, & una resistentia, ac vires illas
externas QB, HA concipiantur resolutae singulas in binas agentes secundum eas rectas,

A THEORY OF NATURAL PHILOSOPHY 247

of the ratio by ED/AD, as AB.ED/AD.BD is to AE/AD. By a similar argument we
obtain also that sinRDA is to sinEDE as AB.ED/AD.BD is to BE/BD ; from which the
whole proposition is clear.

320. If the point D goes off to infinity, & the directions of the forces thus become A more simple
parallel to one another, the ratios of the straight lines ED, AD, BD finally become ratios caTe^'para'uefem6
of equality. Hence, in that case, the three forces are to one another as AB to AE to EB ;

& the first of these is equal to the sum of the other two. Imagine straight lines drawn
parallel to the directions of the forces, through the centres of gravity of all three masses,
& let that one of the masses which lies between the parallels drawn through the other
two be called the middle mass ; then, if we draw in any given direction straight lines from
the masses to meet the parallels, the distances from the parallels measured along these lines
will be as AB, EB ; for the distances bear the same given ratio to AB, EB, on account of
the given directions. Hence for parallel forces we obtain the following theorem. Parallel
motive forces for any two out of three masses are to one another inversely as the distances from a
common direction -passing through the third ; & the accelerations have in addition the inverse
ratio of the masses. The middle acceleration is in an opposite direction to that of the others ;
y the middle motive force is equal to the sum of the other two, whilst either outside one is
equal to the difference of the other two.

321. The theorem of the preceding article will yield the centre of equilibrium for Application of the
any forces, whether diverging or converging. For instance, if A, B, C are three masses ^£t™ "^ *°*^
(& in the term masses, single points can also be understood to be included), of which two, brium.

A & B say, are acted upon by external motive forces ; then the mass will be able to eliminate
these by means of mutual forces, & remain in equilibrium, & then to eliminate the mutual
forces entirely by changing slightly their mutual distances, as required ; provided that,
before the application of those external forces, they were in positions corresponding to a
sufficiently strong limit point of cohesion, & the force on the mass C was cancelled by a
fulcrum opposite to the direction DC, or by a contrary suspension ; & so long as the two
forces multiplied each by its corresponding mass preserve the conditions stated as requisite
in the above, namely, that both tend to the same point or both away from it, or if they
are parallel both have the same direction, when they both together fall within the triangle
ABC, or both tend away from it ; or if, on the other hand, when one of them falls within the
triangle & the other away from it, the one tends to the point of intersection & the other
away from it, or if they are parallel have opposite directions. Further, if they are parallel,
they are to one another as the distances from the direction of forces which passes through
C ; if they are convergent, they are inversely as the sines of the angles between their directions
& the straight line through C to their point of intersection ; or are in the inverse ratio
of the sines of the angles between their directions & the straight lines AC, BC & the ratio
of these straight lines jointly.

322. It is moreover quite easy by means of the theorems to determine also the force Determination of
on the fulcrum placed at C ; this, in the case of parallelism, will be equal to the sum or |u]ecruf^rce on the
the difference of the other two forces according as C is the middle or one of the outside

masses. In all other cases, it will be equal to the sum or difference of the other forces,
in a similar way, if these are reduced to the direction of the force on C, the remaining pairs
of forces that are associated with the former in the resolution cancelling one another on
account of their being equal & opposite.

323. Hence may be obtained all things that relate to the equilibrium of forces acting investigation of the
in one plane, & connected, not by inflexible rods lacking all force except cohesion, but by ^e "iJJ*1^*,!^0
these mutual forces. The Theory holds good indeed, not only here, but also in what diate masses con-
follows ; that is to say, although the bodies A, B, C may not act upon one another directly, "l^"^^^ the^f
yet there are other intermediate masses which connect them. For, if between A & B ternai forces act, &
there were other masses not influenced by any external forces, & placed in equilibrium fr?um '" equUl"
with these masses & with one another, then the first mass which comes after B will act

upon B with a motive force equal to BP, & B will act upon it with an equal force ; hence,
to preserve the equilibrium, this mass must be acted upon by the second, the one which
comes next after it, with a force equal & opposite to this. Hence it follows that the third
must act on this second with a force equal & opposite to that, in order that the second
may be in equilibrium ; & so on, until we come to C, where we have a motive force equal
to that acting on B ; & the accelerations BP, CV will be in the inverse ratio of the masses
B & C, since the equal motive forces are proportional to the products of the accelerations
into the masses. Moreover, if in any positions there are any number of masses, having
any empty spaces interspersed anywhere, & these are in connection with three masses A,
B, C, which are under the influence of those three forces, of which one is assumed to be
produced by a fulcrum, one is usually termed the power, & the third the resistance ; &
if the external forces BQ, HA are considered to be resolved each into two parts acting along

248 PHILOSOPHISE NATURALIS THEORIA

ilia tria puncta conjungunt ; poterit elisis mutuo reliquis omnibus aequilibrium
constituentibus deveniri ad vires in punctis binis, ut A, & C, acceleratrices contrarias viribus
BP, BR, & reciproce proportionales massis ipsarum respectu massae B ; licet ipsae proveniant
a massis quibusvis etiam non in eadem directione sitis, & agentibus in latus : nam per
ejusmodi resolutionem, & ejusmodi virium considerationem adhuc habetur aequilibrium
totius systematis affecti in illis tribus punctis per illas tres vires, cum assumantur in iis
tantummodo vires motrices contrarias, & sequales : unde fit, ut etiam illae, qua; praeterea
ad has in illis considerandas assumuntur, & per quas connectuntur cum reliquis massis,
se mutuo elidant.

Qui motus, ubi non [1501324. Quod si vires eiusmodi non fuerint in ea ratione inter se ; non poterunt

habeatur aequili- T> o \ • -1-1 • i i- • •

brium. puncta D, oc A esse in aequuibno, sed consequetur motus secundum directionem ejus, quae

prevalet : ac si omnis motus puncti C fuerit impeditus ; habebitur conversio circa ipsum C.
Extensio ad aequi- 325. Quod si non in tribus tantummodo massis habeantur vires externae, sed in pluribus ;

imrssanim0t&UIinde li^bit considerare quanvis aliam massam carentem omni externa vi, & earn concipere
principium generate connexam cum singulis reliquarum massis, & massa C per vires mutuas, ac habebitur itidem
ratio momentorunf Theoria pro sequilibrio omnium, cum positione omnium constanter servata etiam sine
ulla figurse mutatione, quae sensu percipi possit. Quin immo si singulae vires illae externae
resolvantur in duas, quarum altera urgeat in directione rectse transeuntis per C, ac elidatur
vi proveniente a solo puncto C, & altera agat perpendiculariter ad ipsam, ut habeatur
aequilibrium in singulis ternariis ; oportebit esse singulas vires novae massae assumptae ad
vim ejus, cum qua conjungitur, in ratione reciproca distantiarum ipsarum massarum a C ;
cum jam sinus anguli recti ubique sit idem. Debebunt autem omnes vires, quae in massam
assumptam agunt directionibus contrariis, se mutuo elidere ad habendum aequilibrium.
Quare debebit summa omnium productorum earum virium, quae urgent conversione in
unam plagam, per ipsarum distantias a centre conversionis, aequari summas productorum
earum, quae urgent in plagam oppositam, per distantias ipsarum, ut habeatur aequilibrium ;
curnque arcus circulares in ea conversione descripti dato tempusculo sint illis distantiis
proportionales, & proportionales sint ipsis arcubus velocitates ; debebunt singularum
virium agentium in unam plagam producta per velocitates, quas haberent puncta, quibus
applicantur secundum suam directionem, si vincerentur, vel contra, si vincerent, simul
sumpta aequari summae ejusmodi productorum agentium in plagam oppositam. Atque
inde habetur principium pro machinis & simplicibus, & compositis, ac notio illius, quod
appellant momentum virium, deducta ex eadem Theoria.

Appiicatio ad om- 326. Casus trium tantummodo massarum exhibet vectem, cujus brachia sint utcunque

ma vectmm genera. jnflexa> Quod si tres massae jaceant in directum, efformabunt rectilineum vectem, qui
quidem applicatis viribus inflectetur semper nonnihil, ut & in superioribus casibus semper
non nihil a priore positione discedet systema novis viribus externis affectum ; sed is discessus
poterit esse utcunque exiguus, ut supra monui : si limites sint satis validi ; adeoque poterit
adhuc vectis esse ad sensum rectilineus. Turn vero vires externae debebunt esse unius
directionis, & contrariae direction! vis mediae, & binae quaevis ex iis erunt ad se invicem
reciproce, ut distantiae a tertia. Inde autem oriuntur tria genera vectium : si fulcrum,
vel hypomochlium, sit in medio in E, vis in altero extremo A, [151] resistentia in altero
B ; vis ad resistentiam est, ut BE, distantia resistentiae a fulcro, ad AE distantiam vis ab
eodem : fulcrum autem sentiet summam virium. Et quod de hoc vectis genere dicitur,
id omne ad libram pariter pertinet, quae ad hoc ipsum vectis genus reducitur. Si fulcrum
sit in altero extremo, ut in B, vis in altero, ut in A, & resistentia in medio, ut in E ; vis
ad resistentiam erit in ratione distantiae EB ad distantiam majorem AB, cujus idcirco
momentum, seu energia, augetur in ratione suae distantiae AB ad EB, ut nimirum possit
tanto majori resistentiae aequivalere. Si demum fuerit quidem fulcrum in altero extremo
B, & resistentia in A, vis prior in E ; turn e contrario erit resistentia ad vim in majore
ratione AB ad EB, decrescente tantundem hujus energia, seu momento. In utroque
autem casu fulcrum sentiet differentiam virium.

trime^'vectibus 327- Qu°d si perticae utcunque inclinatae applicetur pondus in aliquo puncto E, & bini

& principium pro humeros supponant in A, & B, sentient ponderis partes inaequales in ratione reciproca

statera ; cur totum djstantiarum aD ipso ; & si e contrario bina pondera suspendantur in A, & B utcunque

ponQus consiQ.crG~ __, , •*-, . . * \ o T> * *

tur, ut coiiectum inaequalia, assumpto autem puncto E, cujus distantiae a punctis A, & J3 sint in ratione

in centre ravitatis.

in centre gravitatis.

A THEORY OF NATURAL PHILOSOPHY 249

the lines which join the three points ; then it will be possible, all the other forces constituting
the equilibrium cancelling one another, to arrive at accelerations for the two points A & C
say, in opposite directions to the forces BP, BR, & inversely proportional to their masses
with regard to the mass B. This will be the case, even although they may proceed from
any masses not lying in the same direction, & acting to one side ; for, by means of resolution
of this kind, & a consideration of such forces, we yet have equilibrium of the whole system
affected at the three points by the three forces, since here are assumed only motive forces
such as are equal & opposite. Hence it follows that the former, which are assumed in
addition for the consideration of the latter in such cases, & by which they are connected
with the other masses, must also cancel one another.

324. But if such forces are not in this ratio to one another, the points B & A cannot The nature of the
be in equilibrium; but motion would follow in the direction of that which preponderated ; "r°um^ d o^s^no t
also if all motion of the point C were prevented, then there would be rotation about C. obtain.

325. Now if we have external forces acting, not on three masses only, but on several, Extension to the
we can consider any one mass to be without an external force, & suppose that this mass equllib"urr,?fc^

• i r i n /~* i i r rt i f-ni *ii nuniDer 01 masses ,

is connected to each of the others, & to the mass (_, by mutual forces ; & the Theory will & thence a general
hold good for the equilibrium of them all, with the position of them all constantly maintained Prin.clPle, £°r

<• r i- i T« i T 11 i i r machines & the

without any change of figure so tar as can be observed, .further, it all the external forces ratio of moments.

are resolved each into two parts, of which one acts along the straight line passing through

C, & is cancelled by a force proceeding from C alone, & the other acts perpendicularly

to this line, so that equilibrium is obtained for each set of three ; then it will be necessary

that each of the forces on the new mass chosen will be to the force of that to which it is

joined in the inverse ratio of these masses from C, since now the sines of the right angles

are everywhere the same. Also all the forces which act on the chosen mass in opposite

directions, must cancel one another to maintain equilibrium. Hence the sum of all the forces

which tend to produce rotation in one direction, each multiplied by its distance from the

centre of rotation, must be equal to the sum of the products of the forces which tend to

produce rotation in the opposite direction, multiplied by their distances, in order that

equilibrium may be maintained. Since the circular arcs in this rotation which are described

in any interval of time are proportional to the distances, & these are proportional to the

velocities in the arcs, it follows that the products of each of the forces acting in one direction

by the velocities which correspond to the points to which they are applied, in the direction

of the forces if they are overcome, & in the opposite direction if they overcome, all together

must be equal to the sum of the like products acting in the other direction. Hence is

derived a principle for machines, both simple & complex ; & also an idea of what is called

the moment of forces ; & these have been deduced from this same Theory.

326. The case of three masses only yields the case of the lever, whose arms are curved Application to ail
in any manner. But if the three masses lie in one straight line, they will form a rectilinear kmds o£ levers-
lever ; now this, on the application of forces, will always be bent to some degree ; just as,

in the cases above, the system when affected by fresh external forces always departed from
its original position to some extent. But this departure is exceedingly slight in every case,
as I mentioned above, if only the limit-points are sufficiently strong ; & thus the lever
can still be considered as sensibly rectilinear. In this case, then, the external forces must
be in the same direction, & in an opposite direction to that of the middle force, & any two
of them must be to one another in the inverse ratio of their distances from the third. Now
from this there arise three kinds of levers. If the fulcrum, or lever-support, is in the
middle at E, the force acting on one end A & the resistance at the other end B ; then the
ratio of the force to the resistance is as BE, the distance of the resistance from the fulcrum,
to AE the distance of the force from it ; & the force on the fulcrum will be the sum of
the two. What is said about this kind of lever applies equally well to the balance, which
reduces to this kind of lever. If the fulcrum should be at one end, at B say, the force at
the other, A, & the resistance in the middle, at E ; then the force is to the resistance in
the ratio of the distance EB to the greater distance AB ; & therefore the moment, or
energy, will increase in the ratio of the distance AB to EB, so that indeed it may be able
to balance a much greater resistance in proportion. Finally, if the fulcrum were at one
end, B, the resistance at A, & the former force at E ; then, on the contrary, the resistance
is to the force in the greater ratio of AB to EB, thus decreasing its energy or momentum
in the same proportion. In both these latter cases the force on the fulcrum will be equal
to the difference of the forces. Consequences of

327. Now, if to a long pole, inclined at any angle to the horizontal, a weight is applied *his ^ocffine °f
at any point E ; & if two men place their shoulders under the pole at A & B ; then they dpte ol the6 steel.
will support unequal parts of the weight, in the inverse ratio of their distances from it. ytrd'uTh? ,reason

/-i i • r , i • i r -,-,, . T. . T-I • why the whole may

Conversely, if two unequal weights of any sort are suspended from A & B, & a point E is be considered as if
taken whose distances from the points A & B are in the inverse ratio of the weights, & so collected at the

centre of gravity.

250

PHILOSOPHISE NATURALIS THEORIA

Theoriam exhibere
egregie itidem cen-
trum oscillationis.
Quid ipsum sit.

Preparatio ad solu-
tionem problematis
quaerentis ipsum
centrum.

Solutio problematis,
ac demonstratio.

reciproca ipsorum ponderum, adeoque massarum, quibus pondera proportionalia sunt,
quod idcirco erit centrum gravitatis ; suspensa per id punctum pertica, vel supposito
fulcro, habebitur aequilibrium, & in E habebitur vis aequalis summae ponderum. Quin
immo si pertica sit utcunque inflexa, & pendeant in A, & B pondera ; suspendatur autem
ipsa pertica per C ita, ut directio verticalis transeat per centrum gravitatis ; habebitur
sequilibrium, & ibi sentietur vis aequalis summae ponderum, cum ob naturam centri
gravitatis debeant esse singula pondera, seu massae ductae in suas perpendiculares distantias
a linea verticali, quam etiam vocant lineam directionis, hinc, & inde aequalia. Nam vires
ponderum sunt parallelae, & in iis juxta num. 320 satis est ad aequilibrium, si vires motrices
sint reciproce proportionales distantiis a directione virium transeunte per tertium punctum :
sentietur autem in suspensione vis aequalis summae ponderum. Atque inde fluit, quidquid
vulgo traditur de aequilibrio solidorum, ubi linea directionis transit per basim, sive fulcrum,
vel per punctum suspensionis, & simul illud apparet, cur in iis casibus haberi possit tota
massa tanquam collecta in suo centro gravitatis, & habeatur aequilibrium impedito ejus
descensu tantummodo. Gravitas omnium punctorum non applicatur ad centrum gravi-
tatis, nee ibi ipsa agit per sese ; sed ejusmodi esse debent distantiae punctorum totius
systematis, ut inter fulcrum, & punctum ipsi imminens habeatur vis quaedam aequalis
summae virium omnium parallelarum, & directa ad partes oppositas directionibus illarum.
[152] 328. At non minus feliciter ex eadem Theoria, & ex eodem illo theoremate,
fluit determinatio centri oscillationis. Pendula breviora citius oscillant, remotiora lentius.
Quare ubi connexa sunt inter se plura pondera, aliud propius axi oscillationis, aliud remotius
ab ipso, oscillatio neque fiet tarn cito, quam requirunt propiora, neque tarn lente, quam
remotiora, sed actio mutua debebit accelerare haec, retardare ilia. Erit autem aliquod
punctum, quod nee accelerabitur, nee retardabitur, sed oscillabit, tanquam si esset solum.
Illud dicitur centrum oscillationis. Determinatio illius ab Hugenio primum est facta, sed
precario, & non demonstrate principle : turn alii alias itidem obliquas inierunt vias, ac
praecipuas quasque methodos hue usque notas persecutus sum in Supplements Stayanis
§ 4 lib. 3. En autem ejus determinationem simplicissimam ope ejusdem theorematis
numeri 313.

329. Sint plures massae, quarum una A in fig. 63, mutuis viribus singulae connexae cum
P, cujus motus sit impeditus suspensione, vel fulcro, & cum massa Q jacente in quavis
recta PQ, cujus massae Q motus a mutuo nexu nihil turbetur, quae nimirum sit in centro
oscillationis. Porro hie cum massas pone in punctis spatii A, P, Q, intelligo vel puncta
singula, vel quaevis aggregata punctorum, quae concipiantur, ut compenetrata in iis punctis.
Velocitati jam acquisitae in descensu nihil obstabit is nexus,

cum ea sit proportionalis distantiae a puncto suspensionis P,
nisi quatenus per eum nexum retrahentur omnes massae a
recta tangente ad arcum circuli, sustinente puncto ipso sus-
pensionis justa num. 282 vim mutuam respondentem iis om-
nibus viribus centrifugis. Resoluta gravitate in duas partes,
quarum altera agat secundum rectam, quas jungit massam cum A
P, altera sit ipsi perpendicularis, idem punctum P sustinebit
etiam priorem illam, posterior autem determinabit massas
ad motus AN, QM, perpendiculares ipsis AP, QP, ac pro-
portionales per num. 301 sinubus angulorum APR, QPR,
existente PR verticali. Sed nexus coget describere arcus
similes, adeoque proportionales distantiis a P. Quare si sit AO
spatium, quod vi gravitatis obliquae, sed ex parte impeditas a
nexu, revera percurrat massa A ; quoniam Q non turbatur,
adeoque percurrit totum suum spatium QM ; erit QM ad
AO, ut QP ad AP. Demum actio ex A in Q ad actionem
ex Q in A proportionalem ON, erit ex theoremate numeri
3i4utestQ X QPadA X AP, & omnes ejusmodi actiones
ab omnibus massis in Q debebunt evanescere, positivis & negativis valoribus se mutuo
elidentibus. Ex illis tribus proportionibus, & hac aequalitate res omnis sic facillime
expeditur.

330. Dicatur QM = V, sinus APR = a, sinus QPR = q. Erit ex prima proportione

XV.

Q

M

FIG. 63.

q : a : : QM = V : AN =— X V. [153] Ex secunda QP. AP : : QM - V. AO=

Sed ex tertia

Quar.ON=(i--Qp)xV.

Q X QP. A X AP : :ON =( - -~ ) X V.

\ q \j," '

__
'CTxQP'

A THEORY OF NATURAL PHILOSOPHY 251

of the masses to which the weights are proportional, so that the point is their centre of
gravity ; then, if the pole is suspended by this point, or a fulcrum is placed beneath it,
there will be equilibrium, & the force at E will be equal to the sum of the two weights.
Further, if the pole were bent in any manner, & weights were suspended at A & B, & the
pole itself were suspended at C, so that the vertical direction passes through the centre
of gravity of the weights ; then there would be equilibrium, & there would be a force at
C equal to the sum of the weights. For, on account of the nature of the centre of gravity,
each of the weights, or masses, multiplied by its perpendicular distance from the vertical
line, which is also called the line of direction, must be equal on the one side & on the other.
For the forces of the weights are parallel ; & for such, according to Art. 320, it is sufficient
for equilibrium, if the motive forces are proportional inversely to the distances from the
direction of forces passing through the third point ; moreover there will be experienced
at the point of suspension a force equal to the sum of the weights. Hence is derived every-
thing that is usually taught concerning the equilibrium of solids, where a line of direction
passes through the base, or through the fulcrum, or through the point of suspension ; at
the same time we get a clear perception of the reason why in such cases the whole mass
can be considered as if it were condensed at its centre of gravity, & equilibrium can be
obtained by merely preventing the descent of this point. The gravity of all the points is
not applied at the centre of gravity, nor does it act there of itself ; but the distances of
the points of the whole system must be such that between the fulcrum & the point hanging
just over it there must be a certain force equal to the sum of all the parallel forces, & directed
so as to be opposite to their direction.

328. In a no less happy manner there follows from this same Theory, & from the The Theory affords
very same theorem, the determination of the centre of oscillation. Shorter pendulums an excellent expia-

... i i TT i i • i nation of the

oscillate more quickly, & longer ones more slowly. Hence when several weights are centre of oscillation
connected together, one nearer to the axis of oscillation, & another more remote from it, as well-
the oscillation is neither so fast as that required by the nearer, nor so slow as that required
by the more remote ; but a mutual action must accelerate the one & retard the other.
Moreover there will be one point, which will be neither accelerated nor retarded, but
will oscillate as if it were alone ; that point is called the centre of oscillation. Its deter-
mination was first made by Huygens, but from a principle that was doubtful & unproved.
After him, others came upon it indirectly, some in one way & some in another ; & I
investigated some of the best methods then known in the Supplements to Stay's Philosophy,
§ 4, Bk. 3. Now I present you with an exceedingly simple determination of it, derived
from that same theorem of Art. 313.

329. Suppose there are several masses, of which in Fig. 63 one is at A, & that each of Preparation for the
these is connected to P by mutual forces ; & let the motion of P be prevented by suspension, P°10Uye<nJl Of°find^e
or by a fulcrum ; also let A be connected with a mass Q lying in a straight line PQ, & let this centre.

the motion of this mass Q be in no way affected by the mutual connection, as will happen
if Q is at the centre of oscillation. Now, when I place masses at the points of space A,
P, Q, I intend single points of matter, or any aggregates of such points, which may be
considered as condensed at those points of space. The connection will not oppose in any
way the velocity already acquired in descent, since it is proportional to the distance from
the point of suspension P ; except in so far as all the masses are pulled out of the tangent
line into a circular arc by the connection, the point of suspension itself being under the
influence of a mutual force corresponding to all the centrifugal forces; If gravity is
resolved into two parts, one of which acts along the straight line joining the mass to P,
& the other perpendicular to it ; then the point P will sustain the former of these as well,
but the latter will give to the masses the motions AN, QM, respectively perpendicular
to AP, QP, & proportional, by Art. 301, to the sines of the angles APR, QPR, where PR
is the vertical. But the connection forces them to describe arcs that are similar, & therefore
proportional to the distances from P. Hence, if AO is the space, which under the oblique
force of gravity, but partly hindered by the connection, the mass A would really pass over ;
then, since Q is not affected, & will thus pass over the whole of its course QM, we shall
have QM to AO as QP to AP. Lastly, the action of A on Q is to the action of Q on A,
(which is proportional to ON), as Q X QP is to A X AP, by the theorem of Art. 314;
& all such actions from all the masses upon Q must vanish, the positive & negative values
cancelling one another. From the three proportions & this equality the whole question
is worked out in the easiest possible way.

330. Suppose QM = V, the sine of APR = a, the sine of QPR = q. Then, since Solution of the
from the first proportion, q : a = QM : AN, therefore AN = a.V/q ; &, since from demonstration
the second proportion, QP : AP = QM : AO, therefore AO = AP.V/QP. Hence

ON= (a/q — AP/QP).V. But, from the third proportion, Q X QP is to A X AP as
ON is to the action of A on Q. Therefore the action on Q due to the connection with A

252 PHILOSOPHIC NATURALIS THEORIA

quas erit actio in Q ex nexu cum A. At eodem pacto si esset alibi alia massa B itidem
connexa cum P, & Q, actio in Q inde orta haberetur, positis B, b loco A, a ; & ita porro
in quibusquis massis C, D, &c. Omnes autem isti valores positi = o, dividi possent per

Q— — Qp, utique commune omnibus, & deberent e valoribus conclusis intra parentheses ii,

qui sunt positivi, aequales esse negativis. Quare habebitur

a X A x AP + b x B x BP A x AP2 + B x BP2 &c.

QP

& inde OP — ^ v «• x AT -f- j> x or etc.

K rx Ax AP + b xinTBP &c.
Evoiutjo casus pon. 331. Sint jam primo omnes massse in eadem recta linea cum puncto suspensionis

derum jacentium in T> s -11 ...• • r\ o i r\r»n i •

eadem recta cum "i & cum centro oscillationis Q ; & angulus QPR aequabitur cuivis ex angulis APR, ac ejus

puncto suspension. ... A X AP2 -I- B V BP2 &r

is. sinus g smgulis smubus a, b &c. Quare pro eo casu formula evadit -j— - — —

A x AP + B x BP &c. '

quse est ipsa formula Hugeniana pro ponderibus jacentibus recta transeunte per centrum
suspensionis.

Et casus jacentium 332. Quod si jaceant extra ejusmodi rectam in piano FOR perpendiculari ad axem

rotationis transeuntem per P ; sit G centrum commune gravitatis omnium massarum,
ducanturque perpendicula AA', GG', QQ' ad PR, & erit ut radius = i ad a, ita AP ad
AA' = a x AP ; & eodem pacto QQ' = q x QP, GG' = g x GP. Substitutis AA'
pro a X AP & eodem pacto BB' (quam Figura non exprimit) pro b X BP

j ™> A x AP2 + B x BP2 &c c , . ,.

&c. evadat QP = q X^ ~A~A^~JTB — pp/fl • bed si summa massarum dicatur M,

est per num. 245 ex natura centri gravitatis, A X AA' + B X BB' &c. = M X GG' =
M X g X GP. Habebitur igitur valor QP radii nihil turbati in ea inclinatione

q A x AP2 + B x BP2 &c.

g M x GP

initium appiica- [154] 333. Is valor erit variabilis pro varia inclinatione ob valores sinuum q, & g

ones Sina<iatuSsCi Jon" variatos, nisi QP transeat per G, quo casu sit q = g ; & quidem ubi G accedit in infinitum

eodem piano!11" * ad PR, decrcscente g in infinitum, si PQ non transeat per G, manente finito q, valor -

excrescit in infinitum ; contra vero appellente QP ad PR, evadit q = o, & g remanet aliquid,

adeoque ?. evanescit. Id vero accidit, quia in appulsu G ad verticalem tntum systema

vim acceleratricem in infinitum imminuit, & lentissime acceleratur ; adeoque ut radius
PQ adhuc obliquus sit ipsi in ea particula oscillationis infinitesima isochronus, nimirum
jeque parum acceleratus, debet in infinitum produci. Contra vero appellente PQ ad PR
ipsius acceleratio minima esse debet, dum adhuc acceleratio radii PG obliqui est in
immensum major, quam ipsa ; adeoque brevitate sua ipse radius compensare debet
accelerationis imminutionem.

Finis ejusdemcum 334. Quare ut habeatur pendulum simplex constantis longitudinis, & in quacunque

formula generaii. inciinatione isochronum composito, debet radius PQ ita assumi, ut transeat per centrum

gravitatis G, quo unico casu fit constanter q = g, & formula evadit constans

OP — X x -• quae est formula generalis pro oscillationibus in latus

M xGP

massarum quotcumque, & quomodocunque collocatarum in eodem piano perpendiculari
ad axem rotationis, qui casus generaliter continet casum massarum jacentium in eadem
recta transeunte per punctum suspensionis, quern prius eruimus.

Coroliarium pro 335. Inde autem pro hujusmodi casibus plura corollaria deducuntur. Inprimis

positione centri _atet . gravitas centrum debere jacere in recta, qua a centra suspensionis ducitur -per centrum
gravitatis ex eadem oscillationis, uti demonstratum est num. 334. Sed & debet jacere ad eandem partem cum
PUnCt° *PSO centro oscillationis. Nam utcunque mutetur situs massarum per illud planum,
manentibus puncto suspensionis P, & centro gravitatis G, signum valoris quadrati cujusvis
AP, BP manebit semper idem. Quare formula valoris sui signum mutare non poterit ;

A THEORY OF NATURAL PHILOSOPHY 253

//? v A v AP A v AP2\ V

will be (- np ) x n r>P" In tne same manner, if there were another

^ 0 vj-t vi X v2-T

mass somewhere else, also connected with P & Q, the action on Q arising from its presence
would be obtained, if B & b were substituted for A & a ; & so on for any masses C, D,
&c. Now, putting all these values together equal to zero, they can be divided through
by V/(Q x QP), which is common to every one of them ; & those of the values included
in the brackets that are positive must be equal to those that are negative. Hence we have

(*xAxAP+£xBxBP + &c.)/? - (A x AP2 -f B x BP2 + &c.)/QP ;

A x AP2 + B x BP2 + &c.

and hence QP = a. - - i= - = - =-! — == -
a x A x AP + b X B X BP + &c.

331. Suppose now, first of all, that all the masses lie in one straight line with the point Derivation of the

of suspension P, & so with the point of oscillation Q ; then the angle QPR will be equal hanging0 ^"fh'e

to any one of the angles like APR, & its sine q will be equal to any one of the sines a, b, same straight line

&c. Hence for this case the formula reduces to Tus Jensen P°int °f

A x AP2 + B BP2 + &c.

A x AP + B x BP + &c. '

& this is the selfsame formula found by Huygens for weights lying in the straight line passing
through the centre of suspension.

332. But if the masses lie outside of any such line, in the plane FOR, perpendicular The case of when

,JJ. . t T\ i f-~t • i r • the masses are not

to the axis of rotation passing through P, suppose that G is the common centre of gravity On this line.
of all the masses, & let perpendiculars AA', GG', QQ' be drawn to PR. Then, since the
radius (= i) : a = AP : AA', therefore AA' = a X AP : & in a similar manner, QQ' =
q X QP, & GG' = g X GP. Now, if AA' is substituted for a X AP, & similarly BB'
(not shown in the figure) for b X BP, & so on ; the formula will become

OP= A x AP2 + B x BP2 + &c.
?>A x AA' + B x BB' + &c.

But, if the sum of the masses is denoted by M, then, by Art. 245, from the nature of the
centre of gravity, we have A X AA' + B X BB' -f &c. = M X GG' = M X g X GP ;
& therefore we obtain the value of the radius QP, in a form that is independent of the
inclination, namely,

q A
'

g' M x GP

. The value obtained will vary with various inclinations, owing to the varying Commencement of

i r i • t^-n i 1^1- i • i_ T J J -U tne application to

values of the sines q & g, unless QP passes through G ; in which case q = g. Indeed, when oscillations to one

G approaches indefinitely n'ear to PR, & g thus decrease indefinitely, if PQ does not pass side of bodies lying

through G, thus leaving q finite, the value of q/g will increase indefinitely. On the other

hand, when QP coincides with PR, q — O, & g will remain finite ; & thus q/g will vanish.

This indeed is just what does happen ; for, when G approaches the vertical the whole

system diminishes the accelerating force indefinitely, & it is accelerated exceedingly

slowly ; thus, in order that the radius PQ whilst still oblique may be isochronous during

that infinitesimally small part of the oscillation, that is to say, may be accelerated by an

equally small amount, it must be prolonged indefinitely. On the other hand, as PQ

approaches PR, its acceleration must be very small, whilst the acceleration of the radius

PG which is still oblique is immensely greater in comparison with it ; & thus the radius

PQ must by its shortness compensate for the diminution of the acceleration.

334. Hence, in order to obtain a simple pendulum of constant length, isochronous Conclusion of the

• i. . -11 • 11 i j- -nr\ L i T. ^ • same, with a gene-

at any inclination with the composite pendulum, the radius PQ must be so taken that it rai formula.
passes through the centre of gravity G, in which case alone q — g, & the formula reduces
to a constant value for QP, which

_ A x AP2 + B X BP2 + &c.

M xGP

This is a general formula for oscillations to one side of any number of masses, disposed in
any way whatever in the same plane, the plane being perpendicular to the axis of rotation ;
& this case contains in general the case of masses lying in the same straight line through
the point of suspension, which we have already solved.

335. Now for cases of this sort many corollaries can be derived from the theorem Corollary with re-
proved above. First of all, it is clear that : — The centre of gravity must lie in the straight centres of oscillation
line joining the centres of oscillation & suspension ; this has been proved in Art. 335. But & gravity on the
also it must lie on the same side of the -point of suspension as does the centre of oscillation. For

however the positions of the masses are changed in the plane, so long as the positions of
the points of suspension P & of the centre of gravity G remain unaltered, the sign of the
value of any square, such as AP, BP, will remain the same. Hence the formula cannot

254 PHILOSOPHI/E NATURALIS THEORIA

adeoque si in uno aliquo casu jaceat Q respectu P ad eandem plagam, ad quam jacet G ;

debebit jacere semper. Jacet autem ad eandem plagam in casu, in quo concipiatur, omnes

massas abire in ipsum centrum gravitatis, quo casu pendulum evadit simplex, & centrum

oscillationis cadit in ipsum centrum gravitatis, in quo sunt massae. Jacebit igitur semper

ad eandem partem cum G.

[155] 336. Deinde debet centrum gravitatis jaccre inter punctum
bhia'reiiqiTa ex"iis suspensionis, y centrum oscillationis. Sint enim in fig. 64
punctis. puncta A, P, G, Q eadem, ac in fig. 63, ducanturque AG,

AQ, & Aa perpendicularis ad PQ ; summa autem omnium

massarum ductarum in suas distantias a recta quapiam, vel

piano, vel in earum quadrata, designetur praefixa litera J soli

termino pertinente ad massam A, ut contractiores evadant

f A x AP2
demonstrationes. Erit ex formula inventa PQ =

M xGP

Porro est AG2 = AP2 + GP2 — 2 GP X Pa, adeoque
AP2 = AG2 — GP2 + 2 GP X Pa,

& J.A x GPZ est M x GP2,

ob GP constantem ; ac J.A x Pa = M X GP, cum Pa sit
sequalis distantise massae a piano perpendiculari rectae QP
transeunte per P, & eorum productorum summa sequetur distan-
tias centri gravitatis ductae in summam massarum ; adeoque
J.A x 2 GP X Pa erit = 2 M X GP2.

0 J.A x AP2 f.A x AG2 - M x GP2 + 2 M x GP2 J.A X AG2+np

MxGP MxGP MxG

f A v
Erit igitur PQ major, quam PG, excessu GQ= }

~

IVl /

Valor constans pro. 337. £x iflo excessu facile constat, mutato utcunque puncto suspensions, rectangulum

ducti ex bmis dis. i L- • j- •• • , • ^ r ... . .r r ° ~,

lantiis centri gravi- sub bmis distantns centri gravitatis ab ipso, & a centro oscillationis fore constans. Cum

tatis ab iisdem. f A V AO2 f A V ACi*

enim sit QG = Tr r-P ' erit GQ X GP = 4:pL> quod productum est constans,

1V1 /\ VJA 1VJ.

& habetur hujusmodi elegans theorema : singula: masste ducantur in quadrata suarum
distantiarum a centro gravitatis communi, y dividatur omnium ejusmodi productorum summa
•per summam massarum, ac habebitur productum sub binis distantiis centri gravitatis a centro
suspensionis y a centro oscillationis.

Manente puncto 338. Inde autem primo eruitur illud ; manente puncto suspensionis, & centro gravitatis,

centronS1g°ravitatif debere ctiam centrum oscillationis manere nibil mutatum ; utcunque totum sy sterna, servata

manere centrum respectiva omnium massarum distantia, y positione ad se invicem convertatur intra idem

planum circa ipsum gravitatis centrum ; nam ilia GP inventa eo pacto pendet tantummodo

a distantiis, quas singular massae habent a centro gravitatis.

Centrum osciiia- 330,. Sed & illud sponte consequitur : Centrum oscillationis, y centrum suspensionis

suspensionlsUIred™ reciprocari ita, ut, si fiat suspensio per id punctum, quod fuerat centrum oscillationis ; evadat

rocari- oscillationis [156] centrum illud, quod fierat punctum suspensionis ; y alterius distantia a

centro gravitatis mutata, mutetur y alterius distantia in eadem rations reciproca. Cum enim

earum distantiarum rectangulum debcat esse constans ; si pro secunda ponatur valor, quern

habuerat prima ; debet pro prima obvenire valor, quern habuerat secunda, & altera debet

sequari quantitati constanti divisae per alteram.

Altera ex iis dis- 340. Consequitur etiam illud : Altera ex Us binis distantiis evanescente, abibit altera

cente 'abire aUeram *'K infinitum, nisi omnes masses in unico puncto sint simul compenetratce. Nam sine ejusmodi

in infinitum. compenetratione summa omnium productorum ex massis, & quadratis distantiarum a

centro gravitatis, remanet semper finita quantitas : adeoque remanet finita etiam, si

dividatur per summam massarum, & quotus, manente diviso finite, crescit in infinitum ;

si divisor in infinitum decrescat.

Suspensione facta 341. Hinc vero iterum deducitur : Suspensione facia per ipsum centrum gravitatis

tatis0 imUum haberi nu^um motum consequi. Evanescit enim in eo casu distantia centri gravitatis a puncto
motum. suspensionis, adeoque distantia centri oscillationis crescit in infinitum, & celeritas

oscillationis evadit nulla.

Quae distantia cen- 342. Quoniam utraque distantia simul evanescere non potest, potest autem centrum

omniurn^'lmn'ima oscillationis abire in infinitum ; nulla erit maxima e longitudinibus penduli simplicis

pro data positione isochroni pendulo facto per suspensionem dati systematis ; sed aliqua debet esse minima,

datarum T^maxi^ Suspensione quadam inducente omnium celerrimam dati systematis oscillationem. Ea

mam haberi nuiiam. vero minima debet esse, ubi illae binas distantiae aequantur inter se : ibi enim evadit minima

earum summa, ubi altera crescente, & altera decrescente, incrementa prius minora

decrementis, incipiunt esse majora, adeoque ubi ea aequantur inter se. Quoniam autem

illae binae distantiae mutantur in eadem ratione, utut reciproca ; incrementum alterius

A THEORY OF NATURAL PHILOSOPHY 255

change the sign of its value ; & thus, if in any one case, Q lies on the same side of P as
G does, it must always lie on the same side. Now they lie on the same side for the case
in which it is supposed that all the masses go to their common centre of gravity ; for in
this case the pendulum becomes a simple pendulum, & the centre of oscillation coincides
with the centre of gravity, at which all the masses are placed. Hence it will always fall
on the same side of the centre of suspension as G does.

336. Next, the centre of gravity must lie intermediate between the centre of suspension of the three points,
W the centre of oscillation. For, in Fig. 64, let the points A, P, G, O be the same points as *hte c^e °| e™-
in Fig. 63 ; & let AG, AQ, & Aa be drawn perpendicular to PQ. Then, the sum of all tweer^The 'other
the masses, each multiplied into its distance from some chosen straight line or plane, or two-

into their squares, may be designated by the letter J prefixed to the term involving the
mass A alone, so as to make the proofs shorter. If this is done, the formula found will
become PQ = J.A x AP2 /M x GP. Now AG3 = AP2 + GP2 - aGP X Pa, &
therefore AP2 = AG2 - GP2 + 2GP X Pa ; & J.A X GP2 = M X GP2, since GP is
constant ; also /.A X 7a — M X GP, since Pa is equal to the distance of the mass A
from the plane perpendicular to the straight line QP, passing through P, & thus the sum
of these products will be equal to the distance of the centre of gravity multiplied
by the sum of the masses ; hence J.A X 2GP X Pa = 2M X GP2. Therefore
I A x AP2/M x GP - J-(A X AG2 - M x GP2 + zM x GP2) _ f.A X AG2 Qp

M xGP M xGP

Hence PQ will be greater than PG ; & the excess GQ will be equal to J.A xAG2/M xGP.

337. From the value of this excess, it is readily seen that, however the point of The value of the
suspension may be changed, the rectangle contained by the two distances of the centre distances °of the
of gravity from it & from the centre of oscillation, will be constant. For, since centre of gravity
QG = J.A x AG2/M x GP, it follows that GQ X GP =J.A X AG2/M ; & this product SS^woS^E
is constant. Hence we have the following elegant theorem : — // each of the masses is multi-
plied by the square of its distance from the common centre of gravity, y the sum of all these

products is divided by the sum of the masses, then the result obtained will be the product of the
two distances of the centre of gravity from the centres of suspension tff oscillation.

338. Now, from this theorem, we can derive first of all the following theorem. // H the centre of
the centre of gravity W the centre of suspension remain unchanged, then also the centre of centre^of1 gravity
oscillation must remain quite unchanged ; no matter how the whole system is rotated about the remain unchanged,
centre of gravity, in the same plane, so long as the mutual distances of all the masses & their centre of osciiia-
position with regard to one another are preserved. For, the value of GP found in the manner tion.

above depends solely on the distances of the several masses from their centre of gravity.

330. But there is another theorem that 'aKo follows immediately. The centre of The centre of osci!-

.,jJ7 ,, nii i • j j j • lation & the centre

oscillation C5 the centre of suspension are mutually related to one another in such a fashion Of suspension are
that, if the suspension is made from the point which formerly was the centre of oscillation, then reversible.
the new centre of oscillation will prove to be that point which was formerly the centre of suspension ;
y if the distance of either of them from the centre of gravity is changed the distance of the
other will be also changed in the same ratio inversely. For, since the rectangle contained
by their distances remains constant, if for the second there is substituted that which the
first had, then for the first there must be obtained the value which the second formerly
had ; & either of the two is equal to the constant quantity divided by the other.

340. It also follows that, if either of the distances vanishes, the other must become infinite, If one of the djs.
unless all the masses are condensed at a single point. For, unless there is condensation of other will |^c'ome
this kind, the sum of all the products formed from the masses & the squares of their distances infinite.

from their centre of gravity will always remain a finite quantity ; & thus it will still remain
finite if it is divided by the sum of the masses, & the quotient, still left finite after division,
will increase indefinitely, if its divisor decreases indefinitely.

341. Hence, again, it can be deduced that if the suspension is made from the centre of « the suspension

. . ' . ,.' . , , . 'is made from the

gravity, no motion will ensue. For, in this case, the distance of the centre ot gravity centre of gravity,
from the centre of suspension vanishes and so the distance of the centre of oscillation there is no motion,
increases indefinitely, & therefore the speed of the oscillation becomes zero.

342. Since both distances cannot vanish together, but the centre of oscillation can To find the least

aT ./-. , • 11 i_ * • i j i distance of the

go oft to infinity, there cannot be a maximum among the lengths of a simple pendulum centre of oscillation
isochronous with the pendulum made by the suspension of the given system; but there for a given position

. . . F . ' , , . c i . i -ii • of the masses with

must be a minimum, since there must be one suspensidn of the given system which will give regarti to
the greatest speed of oscillation. Indeed, this least value must occur, when the two distances
are equal to one another ; for their sum will be least when, as the one increases & the other
decreases, the increments, which were before less than the decrements, now begin to be
greater than the latter ; & thus, at the time when they are equal to one another.^ Moreover
since the two distances change in the same ratio, although inversely, the infinitesimal

PHILOSOPHIC NATURALIS THEORIA

Superiora habere
locum t an t u m-
modo, ubi omnes
massae sint in eo-
dem piano perpen-
dicular! ad axem
rotationis : transi-
tus ad centrum
percussionis.

Praeparatio ad in-
veniendum cen-
trum percussionis
massarum jacen-
tium in e a d e m
recta.

Calculus cum ejus
determinatione.

infinitesimum erit ad alterius decrementum in ratione ipsarum, nee ea aequari poterunt
inter se, nisi ubi ipsas distantias inter se aequales fiant. Turn vero illarum productum
evadit utriuslibet quadratum, & longitude penduli simplicis isochroni aequatur eorum
summse ; ac proinde habetur hujusmodi theorema : Singulee masses ducantur in quadrata suarum
distantiarum a centro gravitatis, ac productorum summa dividatur per summam massarum :
y dupla radix quadrata quoti exhibebit minimam penduli simplicis isochroni longitudinem.
Vel Geometrice sic : Pro quavis massa capiatur recta, ques ad distantiam cujusvis masses a
centro gravitatis sit in ratione subduplicata ejusdem masses ad massarum summam : inveniatur
recta, cujus quadratum eequetur quadratis omnium ejusmodi rectarum simul : & ipsius duplum
dabit quczsitum longitudinem mediam, quce brevissimam pr&stet oscillationem.

343. Haec quidem omnia locum habent, ubi omnes massae sint in unico piano perpen-
diculari ad axem rotationis, ut ni-[i57]-mirum singulae massae possint connecti cum centro
suspensionis, & centro oscillationis. At ubi in diversis sunt planis, vel in piano non per-
pendiculari ad axem rotationis, oportet singulas massas connectere cum binis punctis axis,
& cum centro oscillationis, ubi jam occurrit systema quatuor massarum in se mutuo
agentium (?) ; & relatio virium, quae in latus agant extra planum, in quo tres e massis
jaceant, quae perquisitio est operosior, sed multo foecundior, & ad problemata plurima
rite solvenda magni usus ; sed quae hucusque protuli, speciminis loco abunde sunt ; mirum
enim, quo in hujusmodi Theoria promovenda, & ad Mechanicam applicanda progredi
liceat. Sic etiam in determinando centro percussionis, virgam tantummodo rectilineam
considerabo, speciminis loco futuram, sive massas in eadem recta linea sitas, & mutuis
actionibus inter se connexas.

344. Sint in fig. 65 massae A, B, C, D connexae inter se in recta
quadam, quae concipiatur revoluta circa punctum P in ea situm, &
quaeratur in eadem recta punctum quoddam Q, cujus motu impedito
debeat impediri omnis motus earumdem massarum per mutuas ac-
tiones ; quod punctum appellatur centrum percussionis. Quoniam sys-
tema totum gyrat circa P, singulae massae habebunt velocitates Aa, B£
&c. proportionales distantiis a puncto P, adeoque singularum motus,
qui per mutua* vires motrices extingui debent, poterunt exprimi per
A X AP, B X BP &c. Quare vires motrices in iis debebunt esse pro-
portionales iis motibus. Concipiantur singulae connexae cum punctis
P, & Q, & quoniam velocitas puncti P erat nulla ; ibi omnium ac-
tionum summa debebit esse = o : summa autem earum, quae habentur
in Q, elidetur a vi externa percussionem sustinente.

345. Quoniam actiones debent esse perpendiculares eidem rectae
jungenti massas, erit per theorema numeri 314, ut PQ ad AQ, ita

actio in A = A X AP, ad actionem in P =

A X

X AQ>

sive ob

AQ = PQ — AP, erit ea actio [158]
pacto actio in P ex nexu cum B erit

xPQ-AxAfr £odem

T) *.

PQ

FIG. 65.

__ "R

2

' & ita Porro' Iis omnlbus

Determinatio vis
percussionis in ipso
centro.

"pfT"

positis=o. divisor communis PQ abit, & omnia positiva aequantur negativis. Erit
igitur A x AP x PQ+ B x BP x PQ &c. = A x AP2 + BxBP2 &c. ; quare
PQ = A X AP2 + B X BP2 &c.^ quse formuja est eadem, ac formula centri oscilla-

J\ X .A.A ~T~ -^ X IJX oCC.

tionis, ac habetur hujusmodi theorema : Distantia centri percussionis a puncto conver-
sionis eequatur distantly centri oscillationis a puncto suspensionis ; adeoque hie locum habent
in hoc casu, quaecunque de centro oscillationis superius dicta sunt.

346. Quod si quis quserat vim percussionis in Q, hie habebit

QP . AP : : A xAP.

<lU3e eT'lt

ex nexu cum A- Eodem pacto in-

. A X AP2 + B X BP2
venientur vires ex reliquis : adeoque summa virium erit ' &c.,

(q) Systema binarwm massarum cum binis punctis connexarum, W inter se, sed adhuc in eodem piano jacentium,
persecutus fueram ante aliquot annos ; quod sibi a me communicatum exhibuit in siia Synopsi Physicae Generalis
P. Benvenutus, ut ibidem ipse innuit. Id inde excerptum habetur hie in Supplements § 5.

Habetur autem post idem supplementum y Epistola, quam delatus Florentiam scripsi ad P. Scherferum, dum hoc
ipsum opus relictum Vienna ante tres menses jam ibidem imprimeretur, qua quidem adjecta est in ipsa prima editione
in -fine operis. Ibi W theoriam trium massarum extendi ad casum massarum quatuor ita ; ut inde generaliter deduct
fossit W equilibrium, & centrum oscillationis, & centrum percussionis, pro massis quotcunque, W utcunque dispositis.

A THEORY OF NATURAL PHILOSOPHY 257

increment of the one will be to the infinitesimal decrement of the other in the ratio of
the distances themselves ; & the former cannot be equal to one another, unless the distances
themselves are equal to one another. In this case their product becomes the square of
either of them, & the length of the simple isochronous pendulum will be equal to their
sum. Hence we have the following theorem : — // each mass is multiplied by the square of
the distance from the centre of gravity, y the sum of all such products is divided by the sum
of the masses ; then, twice the square root of the quotient will give the least length of a simple
isochronous pendulum. This may be expressed geometrically as follows : — For each mass,
take a straight line, which is to the distance of that mass from the centre of gravity in the
subduplicate ratio of the mass to the sum of all the masses ; find a straight line whose square
is equal to the sum of the squares on all the straight lines so found ; then the double of this
straight line will give the required mean length, which will afford the quickest oscillation.

343. These theorems hold good when all the masses are in a single plane perpendicular The theorems given
to the axis of rotation, so that each of the masses can be connected with the point of abo^e ,only ,,h?,!d

, ...... _ . . . ,.„ . .. . good when all the

suspension & the centre ot osculation. But, when they are m ditterent planes, or all in masses are in the
a plane that is not perpendicular to the axis of rotation, it is necessary to connect each *;am<f plane perpen-

-r , . , .r , • „ • i i r -11 • dicular to the axis

of the masses with a pair or points on the axis & with the centre oi oscillation : & we thus Of rotation ; let us
have the case of a system of four masses acting upon one another (q), & the relation between pass on to the centre

if 1-1 • i fit- i'ii r -i i> * rrn • ot percussion.

the forces which act to one side, out ot the plane in which three or the masses lie. Ihis
investigation is much more laborious, but also far more fertile, & of great use for the correct
solution of a large number of problems. However, I have already given enough as
examples ; for it is wonderful how far one can go in developing a Theory of this kind,
& in applying it to Mechanics. So also in determining the centre of percussion, I shall
only consider a rectilinear rod, which will serve as an example, or masses in the same straight
line, connected together by mutual actions.

344. In Fig. 65, let A,B,C,D be masses connected together, lying in one straight line, preparation for
which is supposed to be rotated about a point P situated in it ; it is required to find in findijlg the. centre

. , rf. .~ . - . r .. ,', 1^11 • r of percussion for

this straight line a point Q such that, if its motion is prevented, then the whole motion of masses lying in the

the masses is also prevented through the mutual actions. This point is called the centre same straight line.

of percussion. Now, since the whole system rotates round P, each of the masses will have

velocities, such as Aa, B£, &c., proportional to their distances from the point P ; & thus

the motions of each, which have to be destroyed by the mutual motive forces^can be

represented by A X AP, B x BP, &c. Hence, the motive forces on them must be

proportional to these motions. Suppose each of the masses to be connected with P &

Q ; then, since the velocity of the point P is zero, at P the sum of all the actions must

be equal to zero ; moreover, the sum of those that act at Q is cancelled by the external

force sustaining the percussion.

345. Since the actions must be perpendicular to the straight line joining the masses, The calculation
we shall have, by Art. 314, PQ to AQ as the action on A, which is equal to A X AP, is Kfion^

to the action on P ; hence the latter is equal to A X AP X AQ/PQ, or, since AQ = PQ — AP, centre,
this action will be equal to (A X AP x PQ— A X AP2)/PQ. In the same way, the
action on P due to the connection with B is equal to (B x BP X PQ — B X BP2)/PQ,
& so on. If all these together are put equal to zero, the common divisor
PQ goes out, & all the positives will be equal to the negatives. Therefore
A x AP X PQ + B x BP x PQ + &c. = A x AP2 + B x BP2 + &c. Hence

PQ = -j- ] — ', which is the same formula as the formula for the

A X -A..T ~\~ K X xSJr ~\- oCC.

centre of oscillation. Thus we have the following theorem : — The distance of the centre
of percussion from the point of rotation is equal to the distance of the centre of oscillation from
the centre of suspension. Hence all that has been said above concerning the centre of
oscillation holds good also for the centre of percussion.

346. Now, if the force of percussion at Q is required, we have QP is to AP as A X AP Determination

1 1 r >~\ i i • • i A i • i • i A A T>» rn/~\ the force °* Per'

is to the force on Q due to the connection with A ; hence this latter is equal to A xAPyrQ. cussion at the
In the same way we can find the forces due to the rest ; and thus the sum of all centre of percus-
the forces will be (A X AP2 + B X BP2 + &c.)/PQ. Now, since PQ is equal to s

(q) / investigated the system of two masses connected with two faints W with one another, yet all lying in the
same -plane, several years ago : y, when I had communicated the matter to Father Benvenuto, he expounded it in his
Synopsis Physicje Generalis, mentioning that he had obtained it from me. It is also included in this work, abstracted
from the above, as Supplement 5.

Moreover, after this supplement, it is also contained in a letter, which 1 wrote 'to Father Scherffer when I
reached Florence, whilst this work, which I had left in his hands at Vienna three months before, was in the press there ;
W it was added to the first edition at the end of the work. In it I have also extended the theory of three masses to the
case of fow masses, in such a manner that from it it is possible to deduce, in a perfectly general way, the equilibrium,
the centre of oscillation, y the centre of percussion for any number of masses disposed in any manner whatever.

S

of

258

PHILOSOPHIC NATURALIS THEORIA

sive ob

PQ = A X AP2 + B x BP2 &c.

summa ilia erit A X AP + B X BP &c. ;

Omitti hie multa
quae adhanc Theo-
riam pertinerent,
ad quam pertinet
universa Mcchanica.

A

!

Pressio fluidorum
si puncta sint in
recta vertical!.

Eadem p u n c t i s
utcunque dispersis,
& cum omnibus
directionibus agens.

Ax AP -f B x BP&c.

nimirum ejusmodi vis erit sequalis summse virium, quse requiruntur ad sistendos omnes
motus massarum A, B, &c., cum illis diversis velocitatibus progredientium, videlicet ejusmodi,
quas in massa percussionem excipiente possit producere quantitatem motus sequalem
toti motui, qui sistitur in massis omnibus, quod congruit cum lege actionis, & reactionis
aequalium, & cum conservatione ejusdem quantitatis motus in eandem plagam, de quibus
egimus num. 265, & 264.

347. Haberent hie locum alia sane multa, quse pertinent ad summas virium, quibus
agunt massse, compositarum e viribus, quibus agunt puncta, vel a Newtono, vel ab aliis
demonstrata, & magni usus in Mechanica, & Physica : hujusmodi sunt ea omnia, quse
Newtonus habet sectione 12, & 13 libri I Princip. de attractionibus corporum sphaericorum,
& non sphaericorum, quae componantur ex attractionibus particularum ; ubi habentur
praeclarissima theoremata tarn pro viribus quibuscunque generaliter, quam pro certis virium
legibus, ut illud, quod pertinet ad rationem reciprocam duplicatam distantiarum, in
qua globus globum trahit, tanquam si omnis materia esset compenetrata in centris
eorundem ; punctum intra [159] orbem sphaericum, vel ellipticum vacuum nullas vires
sentit, elisis contrariis ; intra globos plenos punctum habet vim directe proportionalem
distantiae a centro ; unde fit, ut in particulis exiguis ejusmodi vires fere evanescant, & ad
hoc, ut vires adhuc etiam in iis sint admodum sensibiles, debeant decrescere in ratione
multo majore, quam reciproca duplicata distantiarum. Hujusmodi etiam sunt, quse
Mac-Laurinus tradit de sphaeroide elliptico potissimum, quae Clairautius de attractionibus
pro tubulis capillaribus, quae D'Alembertus, Eulerus, aliique pluribus in locis
persecuti sunt ; quin omnis Mechanica, quse agit vel de asquilibrio, vel de moti-

bus, seclusa omni impulsione, hue pertinet, & ad diversos arcus reduci potest curvae
nostrse, qui possunt esse quantumlibet multi, habere quascunque amplitudines, sive
distantias limitum, & areas quae sint inter se in ratione quacunque, ac ad curvas
quascunque ibi accedere, quantum libuerit ; sed res in immensum abiret, & satis
est, ea omnia innuisse.

348. Addam nonnulla tantummodo, quae generaliter pertinent ad pressionem,
& velocitatem fluidorum. Tendant directione quacunque AB puncta disposita in
eadem recta in fig. 66 vi quadam externa respectu systematis eorum punctorum,
cujus actionem mutuis viribus elidant ea puncta, & sint in sequilibrio. Inter
primum punctum A, & secundum ipsi proximum debebit esse vis repulsiva, quae
sequetur vi externse puncti A. Quare urgebitur punctum secundum hac vi repul-
siva, & praeterea vi externa sua. Hinc vis repulsiva inter secundum, & tertium
punctum debebit aequari vi huic utrique, adeoque erit aequalis summse virium ex-
ternarum puncti primi, & secundi. Adjecta igitur sua vi externa tendet deorsum
cum vi sequali summae virium externarum omnium trium ; & ita porro progred-
iendo usque ad B, quodvis punctum urgebitur deorsum vi aequali summae virium FIG- 66-
externarum omnium superiorum punctorum.

349. Quod si non in directum disposita sint, sed utcunque dispersa per parallelepipedum,
cujus basim perpendicularem directioni vis externae exprimat recta FHin fig. 67, & FEGH
faciem ipsi parallelam ; adhuc facile demonstrari potest

componendo, vel resolvendo vires ; sed & per se patet, A C G

vires repulsivas, quas debebit ipsa basis exercere in par-

ticulas sibi propinquas, & ad quas vis ejus mutua perti-

nebit, fore aequales summae omnium superiorum virium

externarum ; atque id erit commune tarn solidis, quam

fluidis. At quoniam in fluidis particulae possunt ferri

directione quacunque, quod unde proveniat, videbimus

in tertia parte ; quaevis particula, ut ibidem videbimus,

in omnem plagam urgebitur viribus aequalibus, & urgebit L

sibi proximas, quse pressionem in alias propagabunt ita,

ut, quse sint in eodem piano LI, parallelo FH, in cujus

directione [160] nulla vis externa agit, vires ubique

eaedem sint. Quamobrem quaevis particula sita ubicun-

que in ea recta in N, habebit eandem vim tarn versus

planum EF, quam versus planum EG, & versus FH, quam

habet particula collocata in eadem linea in MK etiam,

ubi addantur parietes AM, CK parallel! FE, cum planis

LM, KI, parallelis FH, nimirum vi, quse respondet

altitudini MA : ac particula sita in O prope basim FH urgebitur, ut quaquaversum, ita

& versus ipsam, iisdem viribus, quibus particula sha in BD sub AC. Ipsam urgebunt

B

H

A THEORY OF NATURAL PHILOSOPHY 259

(A x AP*+B x BP2 + &c.)/(A x AP + B x BP + &c.), this sum will be equal to
A X AP -f-B X BP -f- &c. That is, the whole force will be equal to the sum of the forces,
which are required to stop all the motions of the masses A, B, &c., which are proceeding
with their several different velocities ; in other words, a force which, acting on the mass
receiving percussion, can produce a quantity of motion equal to the whole motion existing
in all the masses ; and this agrees with the law of equal action & reaction, & with the con-
servation of the same quantity of motion for the same direction, with which I dealt in
Art. 265, & 264.

347. Many other things indeed should find a place here, such as relate to the sums Man.y things per.
of forces, with which masses act, these being compounded from the forces with which Theory must here
points act ; such as have been proved by Newton & others ; & things that are of great use be omitted ; for the
in Mechanics & Physics. Of this kind are all those which Newton has in the I2th & I3th rTics pertains toCthts
sections of The First Book of the Principia concerning the attractions of spherical bodies, Theory.

& non-spherical bodies, such as are compounded from the attractions of their particles.
Here we have some most wonderful theorems, not only for forces in general, but also for
certain laws of forces like that relating to the inverse square of the distances, where a sphere
attracts another sphere as if the whole of its matter were condensed at the centre of each
of them : the theorem that a point within a spherical or elliptic hollow shell is under the
action of no force, equal & opposite forces cancelling one another ; the theorem that within
solid spheres a point is under the action of a force proportional to the distance from the
centre directly. From this it follows that in exceedingly small particles of this kind the
forces must almost vanish ; & in order that the forces even then may be quite sensible,
they must decrease in a much greater ratio than that of the inverse square of the distances.
Also we have theorems such as Maclaurin enunciated with regard to the elliptic spheroid
especially, & those which Clairaut gave with regard to attractions in the case of capillary
tubes, & those which D'Alembert, Euler, & others have investigated in many places. Nay,
the whole ot Mechanics, which deals with equilibrium, or motions, impulse being ex-
cluded, belongs here : the whole of it can be reduced to different arcs of our curve ; & these
may be as many in number as you please, they can have any amplitudes, or distances
between the limit-points, any areas, which may be in any ratio whatever to one another,
& can approach as nearly as you please to any given curves. But the matter would
become endless, & it is quite sufficient for me to have given all those that I have given.

348. I will add a few things only that in general deal with pressure & velocity of Pressure of fluids
fluids. Suppose we have a set of points, in Fig. 66, lying in a straight line, extended in any ^e^tha Vertical
direction AB, under the action of some force external to the system of points ; & suppose line.

that the action of this external force is cancelled by the mutual forces between the points,
& that the latter are in equilibrium. Then between the first point A & the next to it
there must be a repulsive force which is equal to the external force on the point A.
Then the second point will be under the action of this repulsive force in addition to the
external force on it. Hence the repulsive force between the second & third points must
be equal to both of these ; &, further, it will be equal to the sum of the external forces
on the first & second points. Hence, adding the external force on the third point, it
will tend downwards with a force equal to the sum of the external forces on all three ;
& so on, until we reach B, any point will be under the action of a force equal to the sum
of the external forces on all the points lying above it.

349. Now if the points are not all situated in a straight line, but dispersed anyhow The same for points
throughout a parallelepiped, & if, in Fig. 67, FH denotes the base of the parallelepiped, ^annei? &" acting
which is perpendicular to the direction of the external force, & FEGH is a face parallel in all directions,
thereto ; then, it can yet easily be proved, either by composition or by resolution of forces,

indeed it is self-evident, that the repulsive forces, which the base exerts on the particles next
to it, & to which its mutual force will pertain, must be equal to the sum of the external forces
on all points above it : & this will hold good for solids as well as for fluids. But, since in
fluids the particles can move in any direction (we will leave the cause of this to be seen in
the third part), any particle (as we shall also see there) will be urged in any direction with
equal forces : & each will act on the next to it & propagate the pressure to the others in
such a manner that the forces on those points which lie in the same plane LI, parallel to the
base FH, in which direction there is no external force acting, will be everywhere the same.
Hence, every particle situated anywhere in the straight line, at N say, will have the same force
towards the plane EF as towards the plane EG, & towards FH ; the same also as there
is on a particle situated in the same straight line in MK also, where the partitions AM,
CK are added parallel to FE, together with the planes LM, KI parallel to FH, namely,

one equal to a force corresponding to the altitude MA. And a particle situated close to the
base FH, at O say, will be urged in all directions & towards FH with the same forces as
a particle situated in BD which is below AC. All the particles lying in the same horizontal

260 PHILOSOPHI/E NATURALIS THEORIA

particulae in eodem piano horizontali jacentes, & accedet ad omnes fluidi, & baseos particu-
las, donee vi contraria elidatur vis ejus tota ab ejusmodi pressione derivata. Quamobrem
basis FH a fluido tanto minore FLMACKIH sentiet pressionem, quam sentiret a toto
fluido FEGH : superficies autem LM sentiet a particulis N vim aaqualem vi massae LEAM,
accedentibus ad ipsam particulis, donee vis mutua repulsiva ei vi aequetur.

fd'i ponderffieri ?5°' Hinc autem Patet' cur in fluidis nostris gravitate praaditis basis FH sentiat

ssit ingens pres- pressionem tanto majorem massae fluidse incumbentis pondere, & cur pondere perquam
- exiguo fluidi AMKC elevetur pondus collocatum supra LM etiam immane, ubi repagulum

LM sit ejusmodi, ut pressioni fluidi parere possit, quemadmodum sunt coriacea. At
totum yas FLMACKIH bilanci impositum habebit pondus aequale ponderi suo, & fluidi
content! tantummodo ; nam superficies vasis LM, KI horizontalis vi repulsiva mutua
urgebit sursum, quantum urget deorsum puncta omnia N versus O, & ilia pressio tantundem
imminuit vim, quam in bilancem exercet vas, ac tota vis ipsius habebitur dempta pressione
sursum superficiei LM, KI a pressione fundi FH facta deorsum : & pariter se mutuo elident
vires ^exercitae in parietes oppositos. Atque haec Theoria poterit applicari facile aliis etiam
figuris quibuscunque. Respondebit semper pressio superficiei, & toti ponderi fluidi, quod
habeat basim illi superficiei asqualem, & altitudinem ejusmodi, quae usque ad supremam
superficiem pertinet inde accepta in directione illius externae vis.

sone 351- Quod s.i ™es particularum repulsivae sint ejusmodi, ut ad eas multum augendas

sensibili unde requiratur mutatio distantiae, quae ad distantiam totam habeat rationem sensibilcm ; turn
provemat m hac vero compressio massae erit sensibilis, & densitas in diversis

altitudinibusadmodum diversa: sed iniisdemhorizontalibus

planis eadem. Si vero mutatio sufficiat, quae rationem habet

prorsus insensibilem ad totam distantiam ; turn vero com-

pressio sensibilis nulla erit, & massa in fundo eandem

habebit ad sensum densitatem, quam prope superficiem

supremam. Id pendet a lege virium mutua inter particu-

las, & a curva, qua; illam expri-[i6i]-mit. Exprimat in

fig. 68 AD distantiam quandam, & assumpta BD ad AB in

quacunque ratione utcunque parva, vel utcunque sensi-

bili, capiantur rectae perpendiculares DE, BF itidem in

quacunque ratione minoris inaequalitatis utcunque magna : FlG 6g

poterit utique arcus MN curvse exprimentis mutuas par-

ticularum vires transire per ilia puncta F, F, & exhibere quodcunque pressionis incre-

mentum cum quacunque pressione utcunque magna, vel utcunque insensibili.
Compressio aeris 3^2. Compressionem ingentem experimur in acre, quae in eo est proportionalis vi

a qua vi provemat : JJ. . *-, . r. . ' j. r _

aquae compressio compnmenti. Pro eo casu demonstravit Newtonus Prmc. Lib. 3. prop. 23, vim particularum
cur ad sensum repulsivam mutuam debere esse in ratione reciproca simplici distantiarum. Quare in iis

nulla: unde muta- ,.r ... . ,.

tio in vapores tam distantiis, quas nabere possunt particulas aeris perseverantis cum ejusmodi propnetate,
& formam aliam non inducentis (nam & aerem posse e volatili fieri fixum, Newtonus innuit,
ac Halesius inprimis uberrime demonstravit), oportet, arcus MN accedat ad formam arcus
hyperbolae conicae Apollonianae. At in aqua compressio sensibilis habetur nulla, utcunque
magnis ponderibus comprimatur. Inde aliqui inferunt, ipsam elastica vi carere, sed
perperam ; quin immo vires habere debet ingentes distantiis utcunque parum imminutis ;
quanquam esedem particulse debent esse prope limites, nam & distraction! resistit aqua.
Infinita sunt curvarum genera, quae possunt rei satisfacere, & satis est, si arcus EF directionem
habeat fere perpendicularem axi AC. Si curvam cognitam adhibere libeat ; satis est, ut
arcus EF accedat plurimum ad logisticam, cujus subtangens sit perquam exigua respectu
distantise AD. Demonstratur passim, subtangentem logisticae ad intervallum ordinatarum
exhibens rationem duplam esse proxime ut 14 ad 10 ; & eadem subtangens ad intervallum,
quod exhibeat ordinatas in quacunque magna ratione inaequalitatis, habet in omnibus
logistic is rationem eandem. Si igitur minuatur subtangens logisticae, quantum libuerit ;
minuetur utique in eadem ratione intervallum BD respondens cuicunque rationi ordina-
tarum BF, DE, & accedet ad aequalitatem, quantum libuerit, ratio AB ad AD, a qua pendet
compressio ; & cujus ratio reciproca triplicata est ratio densitatum, cum spatia similia sint
in ratione triplicata laterum homologorum, & massa compressa possit cum eadem nova
densitate redigi ad formam similem. Quare poterit haberi incrementum vis comprimentis

A THEORY OF NATURAL PHILOSOPHY 261

plane will act upon it & it will approach all the particles of the fluid & the base, until the
whole of its force is cancelled by a contrary force derived from pressure of this kind.
Hence the base FH would be subject, from the much smaller amount of fluid FLMACKIH,
to the same pressure as it would be subject to from the whole fluid FEGH ; & the surface
LM would be subject to a force from the particles like N equal to the force of the mass
LEAM, these particles tending to approach LM, until the mutual repulsive force is equal
to this pressure.

350. Further, from this the reason is evident, why the base FH should be subject, Hence the reason
in our fluids possessed of gravity, to a pressure so much greater than the weight of the why ™ a very small

n -i i 11-1 to -j vi ATI/TTT-/-. i • i amount of fluid

superincumbent fluid ; & why by a very small weight of fluid, like AMKC, the weight there can exist a

collected above LM can be upheld, even though this is immensely great, when the restraint verv great pressure.

LM is of such a nature that it can submit to the pressure of the fluid, leather for

example. But if the whole vessel FLMACKIH is placed on a balance it will only have a

weight equal to its own weight plus that of the fluid contained. For, the horizontal

surface LM, KI of the vessel will urge it upwards with its mutual repulsive force, just

the same amount as all the points N will urge it downwards towards O, & this pressure

will to the same extent diminish the force which the vessel exerts upon the balance ; &

the whole force will be obtained by taking away the pressure upwards on the surface LM,

KI from the pressure produced downwards on the base FH. In the same way the forces

exerted on the partitions will mutually cancel one another. The Theory can also easily

be applied to any other figures whatever. The pressure on the surface will always

correspond to the whole weight of the fluid having for its base an area equal to the surface,

& for its height that which belongs to the highest surface from it measured in the direction

of the external force.

351. Now if the repulsive forces of the particles are of such a kind that, in order to The source of
increase them to any sensible extent, a change of distance is required, which bears a sensible Pr.efsure .f°r flulds

, , '.. ' & . . * •« 1 MI i with sensible com-

ratio to the whole distance ; then the compression of the mass will also be sensible, & the pression according

density at different heights will be quite different ; nevertheless, they will still be the to thls Theory-

same throughout the same horizontal planes. However, if a change, which bears to the

whole distance a ratio that is quite insensible, is sufficient, then the mass at the bottom

will have approximately the same density as near the top surface. This depends on the

mutual law of forces between the particles, & on the curve which represents this law. In

Fig. 68, let AD be any distance, & suppose that BD is taken in AB produced, bearing to

AB any ratio however small, or however sensible ; ta*ke the perpendicular straight lines

DE, BF, also in any ratio of less inequality however great. In all cases, it will be possible

for the arc MN of the curve representing the mutual forces of the particles to pass through

the points E & F, & to represent any increment of pressure, together with any pressure

however great, or however insensible, it may be.

352. We find that in air there is great compression, & that this is proportional to The force that
the compressing force. For this case, Newton proved, in prop. 3, of the Third Book of ' *

air

his Principia, that the mutual repulsive force between the particles must be inversely the reason for the
proportional to the first power of the distance. Hence, for these distances, which the ™.??Pm?rtlllbll'tJr?,!

'if- i • • • i »i-»»i«i • i i water , i

particles of air can have as it persists with a property of this kind, & does not induce another of the change in

form (for Newton remarked that an air could from being volatile become fixed, & Hales elastlc va

especially gave a very full proof of this), the arc MN must approach the form of an arc

of the rectangular hyperbola. But in water there is no sensible compression, however

great the compressing weights may be. Hence some infer that it lacks elastic force ; but

that is not the case ; nay rather, there are bound to be immense forces if the distances

are diminished ever so slightly ; although the particles must be nea'r limit-points, for

water also resists separation. There are infinitely many classes of curves which would

satisfy the conditions ; & it is sufficient if the arc EF has a direction that is nearly perpen-

dicular to the axis AC. If it is desired to employ some known curve, it is sufficient to

know that the arc EF approximates closely to the logistic curve whose subtangent is very

small compared with the distance AD. Now it is proved that the subtangent of the

logistic curve is to the interval corresponding to a double ratio between the ordinates

very nearly as 14 is to 10 ; & the subtangent is to the interval, corresponding to a ratio of

inequality between the ordinates of any magnitude, in the same ratio for all logistic curves.

If therefore the subtangent of the logistic curve is diminished indefinitely, in every case there

is a diminution in the same ratio of the interval BD corresponding to any' ratio of the

ordinates BF, DE, & the ratio of AB to AD, upon which depends the compression, will

approach indefinitely near to equality. Now the ratio of the densities is the inverse

triplicate of this ratio : for similar parts of space are in the triplicate ratio of homologous

lengths, & the mass when compressed can be reduced to similar form having the same

new density. Thus, we can have the increment of the compressing force, increased in

262 PHILOSOPHIC NATURALIS THEORIA

in quacunque ingenti ratione auctae cum compressione utcunque exigua, & ratione densi-
tatum utcunque accedente ad aequalitatem. Verum ubi ordinata ED jam satis exigua
fuerit, debet curva recedere plurimum ab arcu logisticae, ad quern accesserat, & qui in
infinitum protenditur ex parte eadem, ac debet accedere ad axem AC, & ipsum secare,
ut habeantur deinde vires attractivae, quae ingentes etiam esse possunt ; turn post exiguum
intervallum debet haberi alius arcus [162] repulsivus, recedens plurimum ab axe, qui
exhibeat vires illas repulsivas ingentes, quas habent particulse aquese, ubi in vapores abierunt
per fermentationem, vel calorem.

Ubi pressio propor- 353. In casu densitatis non immutatae ad sensum, & virium illarum parallelarum

tionaiis aititudmi, gequalium uti eas in gravitate nostra concipimus, pressiones erunt ut bases, & altitudines ;

& unde. . to. .. . . r .. ..' , . '. . '

nam numerus particularum panbus altitudimbus respondens ent aequans, adeoque in
diversis altitudinibus erit in earum ratione ; virium autem aequalium summae erunt ut
particularum numeri. Atque id experimur in omnibus homogeneis fluidis, ut in Mercuric,
& aqua.

Quomodo fiat ac. 354* Ubi facto foramine liber exitus relinquitur ejusmodi massae particulis, erumpent

ceieratio in effluxu. ipsae velocitatibus, quas acquirent, & quae respondebunt viribus, quibus urgentur, & spatio,
quo indigent, ut recedant a particulis se insequentibus ; donee vis mutua repulsiva jam
nulla sit. Prima particula relicta libera statim incipit moveri vi ilia repulsiva, qua
premebatur a particulis proximis : utcunque parum ilia recesserit, jam secunda illi proxima
magis distat ab ea, quam a tertia, adeoque movetur in eandem plagam, differentia virium
accelerante motum ; & eodem pacto aliae post alias ita, ut tempusculo utcunque exiguo
omnes aliquem motum habeant, sed initio eo minorem, quo posteriores sunt. Eo pacto
discedunt a se invicem, & semper minuitur vis accelerans motum, donee ea evadat nulla ;
quin immo etiam aliquanto plus asquo a se invicem deinde recedunt particulae, & jam
attractivis viribus retrahuntur, accedentes iterum, non quod retro redeant, sed quod
anteriores moveantur jam aliquanto minus velociter, quam posteriores ; turn iterum aucta
vi repulsiva incipiunt accelerari magis, & recedere, ubi & oscillationes habentur quaedam
hinc, & inde.

Unde velocitas 355. Velocitates, quae remanent post exiguum quoddam deter minatum spatium, in

duplicate16 aititudT cluo v*res mutU3e> ve^ nullas jam sunt, vel aeque augentur, & minuuntur, pendent ab area
nis. curvae, cujus axis partes exprimant non distantias. a proxima particula, sed tota spatia ab

initio motus percursa, & ordinatae in singulis punctis axis exprimant vires, quas in iis habebat
particula. Velocitates in effluxu aquae experimur in ratione subduplicata altitudinum,
adeoque subduplicata virium comprimentium.
Id haberi debet, si id spatium sit ejusdem
longitudinis, & vires in singulis punctis res-
pondentibus ejus spatii sint in ratione primae
illius vis. Turn enim areae totae erunt ut ipsae
vires initiales, & proinde velocitatum quadrata,
ut ipsae vires. Infinita sunt curvarum genera,
quae rem exhibere possunt ; verum id ipsum
ad sensum exhibere potest etiam arcus al-
terius logisticae cujuspiam amplioris ilia, quae
exhibuit distantias singularum particularum.

Sit ea in fig. 69 MFIN. Tota ejus area FIG. 69:

infinita ad partes CN asymptotica a quavis

ordinata [163] sequatur producto sub ipsa ordinata, & subtangente constanti. Quare
ubi ordinata ED jam est perquam exigua respectu ordinatarum BE, HI tota area
CDEN respectu CBFN insensibilis erit, & areae CBFN, CHIN integrae accipi poterunt
pro areis FBDE, IHDE, qua; idcirco erunt, ut vires initiales BF, HI.

Quid requiritur, 35°"- Inde quidem habebuntur quadrata celeritatum proportionalia pressionibus, sive

ut velocitas sit altitudinibus. Ut autem velocitas absoluta sit aequalis illi, quam particula acquireret cadendo

habetjfr ca'dendo * superficie suprema, quod in aqua experimur ad sensum; debet praeterea tota ejusmodi

per aititudinem. area 32quari rectangulo facto sub recta exprimente vim gravitatis^ unius particulss, sive vis

repulsive, quam in se mutuo exercent binae particulae, quae se primo repellunt, sustinente

inferiore gravitatem superioris, & sub tota altitudine. Deberet eo casu esse totum pondus

BF ad illam vim, ut est altitude tota fluidi ad subtangentem logisticae, si FE est ipsius

logistics arcus. Est autem pondus BF ad gravitatem primae particulae, ut_ numerus

particularum in ea altitudine ad unitatem, adeoque ut_ eadem ilia tota altitudo ad

distantiam primarum particularum. Quare subtangens illius logisticas deberet aequan

A THEORY OF NATURAL PHILOSOPHY 263

any very great ratio in conjunction with a compression that is small to any extent, & a
ratio of densities which approaches indefinitely near to equality. But when the ordinate
ED is sufficiently small, the curve must depart considerably from an arc of the logistic
curve, to which it formerly approximated, & which proceeded to infinity in the same
direction ; it must approach the axis AC, & cut it, in order that attractive forces may
be obtained, which may also become very great. Then, after a small interval, we must
have another repulsive arc, receding far from the axis, to represent those very great
repulsive forces, which the particles of water have, when they pass into vapour through
fermentation or heat.

353. In the case of the density not being sensibly changed, & of those equal parallel Where the pressure
forces, such as we suppose our gravity to be, the pressures will be proportional to the bases ihe^hrTude0"*1 the
& the altitudes. For, the number of particles corresponding to equal altitudes will be reason for this,
equal, & therefore, in different altitudes, the numbers will be proportional to the altitudes ;

moreover the sums of the equal forces will be proportional to the numbers of particles.
We find this to be the case in all homogeneous fluids, such as mercury & water.

354. When, on making an opening, a free exit is left for the particles of a mass, they HOW acceleration
burst forth with the velocities which they acquire & which correspond to the forces urging *" efflux arises-
them, & to the space to which it is necessary for them to recede from those particles that

follow, before the mutual repulsive force becomes zero. The first particle, when left free,
immediately begins to move under the action of the repulsive force by which it is pressed
by the particles next to it. As soon as it has moved ever so little, the second particle next
to it becomes more distant from it than from the third, & thus moves in the same direction
as the difference of the forces accelerates the motion. Similarly, one after the other they
acquire motion in such a manner that in any little interval of time, no matter how brief,
all of them will have some motion ; this motion at the commencement is so much the less,
the farther back the particles are. In this way they separate from one another, & the force
accelerating the motion ever becomes less until finally it vanishes. Nay rather, to speak
more correctly, the particles still recede from one another, & come under the action of
attractive forces, & approach one another ; not indeed that they retrace their paths, but
because the more forward particles are now moving with somewhat less velocity than
those behind ; then once more the repulsive force is increased & they begin to be accelerated
more than those behind & to recede from them ; & so oscillations to & fro are obtained.

355. The velocities that are left after any determinate interval of space, in which the Why the velocity
mutual forces are either nothing or are equally increased & diminished, depend on the the^ub^u^Hcate
area of the curve, of which parts of the axis represent not the distances from the next of the height,
particle, but the whole spaces travelled from the beginning of the motion, & the ordinates

at each point of the axis represent the forces which the particle had at those points. It
is found that the velocities of effluent water are in the subduplicate ratio of the altitudes,
& thus in the subduplicate ratio of the compressing forces. Now this is what must be
obtained, if the space is of the same length, & the forces at each corresponding point of
that space are in the ratio of that first force. For, then the total areas will be as the initial
forces, & hence the squares of the velocities will be as the forces. There are an infinite
number of classes of curves which will serve to represent the case ; but this also can be
represented by the arc of another logistic curve more ample than that which represented
the distances of the single particles. Let MFIN be such a curve, in Fig. 69. The whole
area, indefinitely produced in the direction of C & N, which are asymptotic, measured
from any ordinate, will be equal to the product of that ordinate & the constant subtangent.
Therefore when the ordinate ED is now very small with respect to the ordinates BF, HI,
the whole area CDEN will be insensible with respect to the area CBFN ; & thus the whole
areas CBFN, CHIN can be taken instead of the areas FBDE, IHDE ; & therefore these
are to one another as the initial forces BF, HI.

356. From this, then, we have that the squares of the velocities are proportional to what is required
the pressures, or the altitudes. Now, in order that the absolute velocity may be equal so that the velocity

., ,, ..,.,,. , , f • r J snail be equal to

to that which the particle would acquire in falling from the upper surface, as is found that acquired in

to be approximately the case for water, we must have, in addition, that the whole of such failing from the

area must be equal to the rectangle formed by multiplying the straight line representing

the force of gravity on one particle (or the repulsive force which a pair of particles mutually

exert upon one another, when they first repel one another, the lower sustaining the

gravity of the one above) by the whole altitude. In this case, the whole weight BF

would be bound to be to the force as the whole altitude of the fluid is to the subtangent

of the logistic curve, if FE is an arc of the logistic curve. Moreover, the^ weight BF

is to the gravity of the first particle as the number of particles in the altitude is to

unity ; & thus in the ratio of the altitude to the distance between the primary particles.

Hence the subtangent of the logistic curve would have to be equal to the distance between

264 PHILOSOPHIC NATURALIS THEORIA

illi distantise primarum particularum, quae quidem subtangens erit itidem idcirco perquam

exigua.

Tentandum an in 357. An in omnibus fluidis habeatur ejusmodi absoluta velocitas & an quadrata

aotidat S Transitus vel°citatum *n effluxu respondeant altitudinibus ; per experimenta videndum est, ut
ad partem tertiam. constet, an curvse virium in omnibus sequantur superiores leges, an diversas. Sed ego jam

ab applicatione ad Mechanicam ad applicationem ad Physicam gradum feci, quam uberius

in tertia Parte persequar. Haec interea speciminis loco sint satis ad immensam quandam

hujusce campi foecunditatem indicandam utcunque.

A THEORY OF NATURAL PHILOSOPHY 265

the primary particles ; & thus the subtangent must also be itself very small on this
account.

357. Whether such an absolute velocity exists in all fluids, & whether the squares of it must be tested
the velocities with which they issue correspond to the altitudes, must be investigated wheth?r t*lsflh.^p'

11- » i • 1111 i f r 11 pens in all fluids.

experimentally ; m order that it may be shown whether the curves of forces follow the laws we will now pass
given above, or different ones. But now I will pass on from the application to Mechanics on to the third
to the application to Physics, which I will follow out more fully in the third part. These
things, in the meanwhile, may be sufficient in some sort to indicate an immense fertility
in this field of knowledge.

[164] PARS III
Applicatio Theories ad Physicaih

Agendum hie primo 358. In secunda hujusce Operis parte, dum Theoriam meam applicarem ad

prietatSus'bcorpor" Mechanicam, multa identidem immiscui, quae application! ad Physicam sterncrcnt viam,

um, turn de discrim- & vero etiam ad eandem pertinerent ; at hie, quae pertinent ad ipsam Physicam, ordinatius

species" * Va"aS Persequar ; & primo quidem de generalibus agam proprietatibus corporum, quas omnes

omnino exhibet ilia lex virium, quam initio primae partis exposui ; turn ex eadem prsecipua

discrimina deducam, quae inter diversas observamus corporum species, & mutationes,

quae ipsis accidunt, alterationes, atque transformations evolvam.

Enumeratio earum, 359. Primum igitur agam de Impenetrabilitatc, de Extensione, de Eigurabilitate,

&Co?dobUS &8etUr' de Mole> Massa, & Densitate, dc Inertia, de Mobilitate, de Continuitate motuum, de
/Equalitate Actionis & Reactionis, de Divisibilitate, & Componibilitate, quam ego divisi-
bilitati in infinitum substiluo, de Immutabilitate primorum materiae elementorum, de
Gravitate, de Cohaesione, quas quidem generalia sunt. Turn agam de Varietate Naturae,
& particularibus proprietatibus corporum, nimirum de varietate particularum, & massarum
multiplici, de Solidis, & Fluidis, de Elasticis, & Mollibus, de Principiis Chemicarum
Operationum, ubi de Dissolutione, Praecipitatione, Adhaesione, & Coalescentia, de Fermen-
tatione, & emissione Vaporum, de Igne, & emissione Luminis ; ac ipsis praecipuis Lutninis
proprietatibus, de Odore, de Sapore, de Sono, de Electricitate, de Magnetismo itidem
aliquid innuam sub finem ; ac demum ad generaliora regressus, quid Alterationes,
Corruptiones, Transformationes mihi sint, explicabo. Verum in horum pluribus rem
a mea Theoria deducam tantummodo ad communia principia, ex quibus peculiares
singulorum tractatus pendent ; ac alicubi methodum indicabo tantummodo, quae ad
rei perquisitionem aptissima mihi videatur.

impenetrabiiitas 360. Impenetrabiiitas corporum a mea Theoria omnino sponte fluit ; si enim in

Theoria ' haC mmimis distantiis agunt vires repulsivae, quae iis in infinitum imminutis crescant in infinitum
ita, ut pares sint extinguendae cuilibet velocitati utcunque magnae, utique non potest
ulla finita vis, aut velocitas efncere, ut distantia duorum punctorum evanescat, quod
requiritur ad compenetrationem ; sed ad id praestandum infinita Divina virtus, quae
infinitam vim exerceat, vel infinitam producat velocitatem, sola sufficit.

Aliud impenetra- [165] 361. Praeter hoc impenetrabilitatis genus, quod a viribus repulsivis oritur, est
priumhuk/rheona". & aliud, quod provenit ab inextensione punctorum, & quod evolvi in dissertationibus
De Spatio, W Tempore, quas ex Stayanis Supplementis hue transtuli, & habetur hie in fine
Supplementorum § i, & 2. Ibi enim ex eo, quod in spatio continue numerus punctorum
loci sit infinities infinitus, & numerus punctorum materiae finitus, erui illud : nullum
punctum materiae occupare unquam punctum loci, non solum illud, quod tune occupat
aliud materiae punctum, sed nee illud, quod vel ipsum, vel ullum aliud materiae punctum
occupavit unquam. Probatio inde petitur, quod si ex casibus ejusdem generis una classis
infinities plures contineat, quam altera, infinities improbabilius sit, casum aliquem, de
quo ignoremus, ad utram classem pertineat, pertinere ad secundam, quam ad primam.
Ex hoc autem principio id etiam immediate consequitur ; si enim una massa projiciatur
contra alteram, & ab omnibus viribus repulsivis abstrahamus animum ; numerus projec-
tionum, quae aliquod punctum massae projectae dirigant per rectam transeuntem per
aliquod punctum massae, contra quam projicitur, est utique finitus ; cum numerus
punctorum in utraque massa finitus sit ; at numerus projectionum, quae dirigant puncta
omnia per rectas nulli secundse massae puncto occurrentes, est infinities infinitus, ob puncta
spatii in quovis piano infinities infinita. Quamobrem, habita etiam ratione infinitorum
continui temporis momentorum, est infinities improbabilior primus casus secundo ; & in
quacunque projectione massae contra massam nullus habebitur immediatus occursus puncti
materiae cum altero puncto materiae, adeoque nulla compenetratio, etiam independenter
a viribus repulsivis.

266

PART III

Application of the Theory to Physics

358. In the second part of this work, in applying my Theory to Mechanics, I brought We wUi first of all
in also at the same time many things which opened the road for an application to Physics, generaf^properttes
& really even belonged to the latter. In this part I will investigate in a more ordered of bodies, & then
manner those things that belong to Physics. First of all, I will deal with general properties between^he^venu
of bodies ; & these will be given by that same law of forces that I enunciated at the beginning species.

of the first part. After that, from the same law I will derive the most important of the
distinctions that we observe between the different species of bodies, & I will discuss the
changes, alterations & transformations that happen to them.

359. First, therefore, I will deal with Impenetrability, Extension, Figurability, Volume, Enumeration of
Mass, Density, Inertia, Mobility, Continuity of Motions, the Equality of Action & deai^with" &°the
Reaction, Divisibility, & Componibility (for which I substitute infinite divisibility), the order in which they
Immutability of the primary elements of matter, gravity, & Cohesion ; all these are general wm ^ taken-
properties. Then I will consider the Variety of Nature, & special properties of bodies ;

such, for instance, as the manifold variety of particles & masses, Solids & Fluids, Elastic,
& Soft bodies ; the principles of chemical operations, such as Solution, Precipitation,
Adhesion & Coalescence, Fermentation, & emission of Vapours, Fire & the emission of
Light ; also about the principal properties of Light, Smell, Taste, Sound, Electricity
& Magnetism, I will say a few words towards the end. Finally, coming back to more
general matters, I will explain my idea of the nature of alterations, corruptions & trans-
formations. Now in most of these, I shall derive the whole matter from my Theory
alone, & reduce it to those common principles, upon which depends the special treatment
for each ; in certain cases I shall only indicate the method, which seems to me to be the
most fit for a further investigation of the matter.

360. The Impenetrability of bodies comes naturally from my Theory. For, if repulsive The origin of im-
forces act at very small distances, & these forces increase indefinitely as the distances co'rfi^g^fo this
decrease, so that they are capable of destroying any velocity however large ; then there Theory.

never can be any finite force, or velocity, that can make the distance between two points
vanish, as is required for compenetration. To do this, an infinite Divine virtue, exercising
an infinite force, or creating an infinite velocity, would alone suffice.

361. Besides this kind of Impenetrability, which arises from repulsive forces, there Another kind of
is also another kind, which comes from the inextension of the points ; this I discussed in p^cTfiar to 'this
the dissertations De Spatio, y Tempore, which I have abstracted from the Supplement Theory.

to Stay's Philosophy, & set at the end of this work as Supplements, §§1,2. From the fact
that the number of points of position in a continuous space may be infinitely infinite, whilst
the number of points of matter may be finite, I derive the following principle ; namely,
that no point of matter can ever occupy either a point of position which is at the time
occupied by another point of matter, or one which any other point of matter has ever
occupied before. The proof is derived from the argument that, if of cases of the same
nature one class of them contains infinitely more than another, then it is infinitely more
improbable that a certain case, concerning which we are in doubt as to which class it belongs,
belongs to the second class rather than to the first. It also follows immediately from this
principle ; if one mass is projected towards another, & we disallow a directive mind in all
repulsive forces, the number of the ways of projection, which direct any point of the
projected mass along a straight line passing through any point of the mass against which
it is projected, is finite ; for the number of points in each of the masses is finite. But
the number of ways of projection, which direct all points along straight lines that pass
through no point of the second mass, is infinitely infinite because the number of points
of space in any plane is infinitely infinite. Therefore, even when the infinite number
of moments in continuous time is taken into account, the first case is infinitely more
improbable than the second. Hence, in any projection whatever of mass against mass
there is no direct encounter of one point of matter with another point of matter ; &
thus there can be no compenetration, even apart from the idea of repulsive forces.

267

268 PHILOSOPHIC NATURALIS THEORIA

sine viribus repui. 362. Si vires repulsivae non adessent ; omnis massa libere transiret per aliam quanvis

comSpenetrationeem massam, ut lux per vitra, & gemmas transit, ut oleum per marmora insinuatur ; atque id

apparentem. Quid semper fieret sine ulla vera compenetratione. Vires, quffi ad aliquod intervallum extend-

tkifii^l6 ye/" quo. untur sat^s magnae, impediunt ejusmodi liberum commeatum. Porro hie duo casus

dam, potissimum si distinguendi sunt ; alter, in quo curva virium non habeat ullum arcum asymptoticum

toti.* asymP- cum asymptoto perpendicular! ad axem, praeter ilium primum, quem exhibet figura i,

cujus asymptotus est in origine abscissarum ; alter, in quo adsint alii ejusmodi arcus

asymptotici. In hoc secundo casu si sit aliqua asymptotus ad aliquam distantiam ab origine

abscissarum, quae habeat arcum citra se attractivum, ultra repulsivum cum area infinita,

ut juxta num. 188 puncta posita in minore distantia non possint acquirere distantiam

majorem, nee, quae in majore sunt, minorem ; turn vero particula composita ex punctis

in minore distantia positis, esset prorsus impenetrabilis a particula posita in majore distantia

ab ipsa, nee ulla finita velocitate posset cum ilia commisceri, & in ejus locum irrumpere ;

& si duae habeantur [166] asymptoti ejusmodi satis proximae, quarum citerior habeat ulterius

crus repulsivum, ulterior citerius attractivum cum areis infinitis, turn duo puncta collocata

in distantia a se invicem intermedia inter distantias earum asymptotorum, nee possent

ulla finita vi, aut velocitate acquirere distantiam minorem, quam sit distantia asymptoti

citerioris, nee majorem, quam sit ulterioris ; & cum eae duae asymptoti possint esse utcunque

sibi invicem proximae ; ilia puncta possent esse necessitata ad non mutandam distantiam

intervallo utcunque parvo. Si jam in uno piano sit series continua triangulorum aequi-

laterorum habentium eas distantias pro lateribus, & in singulis angulis poneretur quicunque

numerus punctorum ad distantiam inter se satis minorem ea, qua distent illae duae asymptoti,

vel etiam puncta singula ; fieret utique velum quoddam indissoluble, quod tamen esset

plicatile in quavis e rectis continentibus triangulorum latera, & posset etiam plicari in

gyrum more veterum voluminum.

Soiidum indissolu- 363. Si autem sit solidum compositum ex ejusmodi velis, quorum alia ita essent aliis

bUe. & impermea- }mpOSita, ut punctum quodlibet superioris veli terminaret pyramidem regularem habentem
pro basi unum e triangulis veli inferioris, & in singulis angulis collocarentur puncta, vel
massae punctorum ; id esset solidissimum, & ne plicatile quidem ; etiamsi crassitude
unicam pyramidum seriem admitteret. Possent autem esse dispersa inter latera illius
veli, vel hujus muri, puncta quotcunque, nee eorum ullum posset inde egredi ad distantiam
a punctis positis in angulis veli, vel muri, majorem ilia distantia ulterioris asymptoti. Quod
si praeterea ultra asymptotum ulteriorem haberetur area repulsiva infinita ; nulla externa
puncta possent perrumpere nee murum, nee velum ipsum, vel per vacua spatiola transire,
utcunque magna cum velocitate advenirent ; cum nullum in triangulo aequilatero sit
punctum, quod ab aliquo ex angulis non distet minus, quam per latus ipsius trianguli.

Alia ratio acqui- 364. Quod si ejusmodi binae asymptoti inter se proximae sint in ingenti distantia a

bmtatem,I&Pn°xum principio abscissarum, & in distantia media inter earum binas distantias ab ipso initio
per asymptotes ponantur in cuspidibus trianguli aequilateri tria puncta materise, turn in cuspide pyramidis
regularis habentis id triangulum aequilaterum pro basi ponantur quotcunque puncta, quae
inter se minus distent, quam pro distantia illarum asymptotorum ; massula constans hisce
punctis erit indissolubilis ; cum nee ullum ex iis punctis possit acquirere distantiam a
reliquis, nee reliqua inter se distantiam minorem distantia asymptoti citerioris, & majorem
distantia ulterioris, & ipsa haec particula impenetrabilis a quovis puncto externo materiae,
cum nullum ad reliqua ilia tria puncta possit ita accedere, si distat magis, vel recedere, si
minus, ut acquirat distantiam, quam habent puncta ejus massae. Ejusmodi massis ita
cohibitis per terna puncta ad maximas distantias sita posset integer constare Mundus,
qui ha-[l67]-beret in suis illis massulis, seu primigeniis particulis impenetrabilitatem
continuam prorsus insuperabilem, sine ulla extensione continua, & indissolubilitatem
itidem insuperabilem etiam sine ullo mutuo nexu inter earum puncta, per solum nexum,
quem haberent singula cum illis tribus punctis remotis.

in us & aliis casi. 365. In omnibus hisce casibus habetur in massa non continua vis ita continua, ut

ttnuaesineeco^tinuo nu^a ne apparens quidem compenetratio, & permixtio haberi possit aeque, ac in communi

faciente vim, & sententia de continua impenetrabilis materiae extensione. Quod autem in illo velo, vel

meSabUitas.imper" muro exhibuit triangulorum, & pyramidum series, idem obtineri potest per figuras alias

A THEORY OF NATURAL PHILOSOPHY

269

362. If there were no repulsive forces, every mass would pass freely through every other
mass, as light passes through glass & crystals, & as oil insinuates itself into marble ; but
such a thing as this would always happen without any true compenetration. Forces, which
extend to an interval that is sufficiently large for the purpose, prevent free passage of that
kind. Further there are here two cases to be distinguished ; one, in which the curve of
forces has not any asymptotic arc with an asymptote perpendicular to the axis, except the
first, as is shown in Fig. I, where the asymptote occurs at the origin of abscissae ; the other,
in which there are other such asymptotic arcs. In the second case, if there is an asymptote
at some distance from the origin of abscissae, which has an attractive arc on the near side of
it, & on the far side a repulsive arc with an infinite area corresponding to it, so that, as
was shown in Art. 188, points situated at a less distance cannot acquire a greater, & those
at a greater distance cannot acquire a less ; then particles that are made up of points situated
at the less distance would be quite impenetrable by a particle situated at a greater distance
from it ; nor could any finite velocity force it to mingle with it or invade its position ; and if
there are two asymptotes of the kind sufficiently near together, of which the nearer to the
origin has its further branch repulsive, & the further has its nearer branch attractive, the
corresponding areas being infinite, then two points situated at a distance from one another
that is intermediate between the distances of these asymptotes, cannot with any finite
force or velocity acquire a distance less than that of the nearer asymptote or greater than
that of the further asymptote. Now since these two asymptotes may be indefinitely
near to one another, the two points may be forced to keep their distance unchanged within
an interval of any smallness whatever. Suppose now that we have in a plane a continuous
series of equilateral triangles having these distances as sides, & that at each of the angles
there are placed any number of points at a distance from one another sufficiently less than
that of the distance between the two asymptotes, or even single points ; then, in every
case, we should have a kind of unbreakable skin, which however could be folded along any
of the straight lines containing sides of the triangles, or could even be folded in spirals
after the manner of ancient manuscripts.

363. Moreover, if we have a solid composed of such skins, one imposed upon the other
in such a manner that any point of an upper skin should terminate a regular pyramid having
for its base one of the equilateral triangles of the skin beneath, & in each of them points
were situated, or masses of points ; then that would have very great solidity, & would not
be even capable of being folded, even if its thickness only admitted of a single series of
pyramids. Further, any number of points could be scattered between the sides of the former
skin, or the wall of the latter, & none of these could get out of this position to a distance from
the points situated at the angles of the skin, or of the wall, greater than the distance of the
further asymptote. Now if, in addition to these, there happened to be beyond the further
asymptote a corresponding infinite repulsive area, no external points could break into the
skin or wall, nor could they pass through empty spaces in it, no matter how great the velocity
with which they approached it. For, there is no point within an equilateral triangle
that is at a less distance from the angular points than a side of the triangle.

364. Again, if there are two asymptotes very near one another, at a great distance
from the origin of abscissae, & at a distance intermediate between their two distances from
the origin there are placed three points of matter at the vertices of an equilateral triangle,
& then at the vertex of a regular pyramid having for its base that equilateral triangle there
are placed any number of points, which are at a less distance from one another than that
between the two asymptotes, the little mass made up of these points will be unbreakable.
For, none of these points can acquire from the rest, nor the rest from one another, a distance
less than the distance of the nearer asymptote, nor greater than that of the further
asymptote. This particle will also be impenetrable by any external point of matter ; for no
point can possibly approach those other three points so nearly, if the distance is greater, or
recede from them so far, if the distance is less, as to acquire the same distance as
that between the several points of the mass. The whole Universe may be made up of
masses of this kind restrained by sets of three points situated at very great distances ; &
it would have in the little masses forming it, or in the primary particles, a continuous
impenetrability that was quite insuperable, without any continuous extension ; it would
also have an insuperable unbreakableness without any mutual connection between the
points forming it, simply owing to the connection existing between each of its points with
the three remote points.

365. In all these cases there is obtained for a non-continuous mass a force that is
continuous in such sort that there is not even apparent compenetration ; & commingling
can be had just as well as with the usual idea of continuous extension of impenetrable
matter. Moreover, what has been represented by the skin or wall of a series of triangles
or pyramids, can be obtained by means of very many other figures ; & it can be obtained

Without repulsive
forces there must
be apparent com-
penetration. What
these forces may
give us in particles,
& a sort of skin,
especially if there
are asymptotes.

An unbreakable &
impermeable solid.

Another way in
which impenetra-
bility may be ac-
quired, & the con-
nection with asym-
ptotes that a re
remote from the
origin of abscissae.

In these & other
cases, we have
continuous resist-
ance without im-
agining a continu-
ous force, & also
absolute impene-
trability.

270 PHILOSOPHIC NATURALIS THEORIA

quamplurimas, & id multo pluribus abhuc modis obtineretur ; si non in unica, sed in
pluribus distantiis essent ejusmodi asymptotica repagula cum impenetrabilitate continua
per non continuam punctorum dispersorum dispositionem.

Sineasymptotoom. 366. At in primo illo casu, in quo nulla habetur ejusmodi asymptotus praeter primam,

nfeaWks^fore91!^ res ^onge a^° niodo sc haberet. Patet in co casu illud, si velocitas imprimi possit massae

aiiis si iis satis cuipiam satis magna ; fore, ut ea transeat per massam quancunque sine ulla perturbatione

m^gnas ^veioatas suarum part}umj & sine ulla partium alterius ; nam vires, ut agant, & motum aliquem

Exempium giobuii finitum sensibilem gignant, indigent continue tempore, quo imminuto in immensum,

n"es transeuntis.g" ut* imminuitur, si velocitas in immensum augeatur, imminuitur itidem in immensum

earum effectus. Rei ideam exhibebit globulus ferreus, qui debeat transire per planum,

in quo dispersae sint hac, iliac plurimae massae magneticae vim habentes validam satis. Si

is globus cum velocitate non ita ingenti projiciatur per directionem etiam, quae in nullam

massam debeat incurrere ; progredi ultra illas massas non poterit ; sed ejus motus sistetur

ab illarum attractionibus. At si velocitas sit satis magna, ut actiones virium magneticarum

satis exiguo tempore durare possint, praetervolabit utique, nullo sensibili damno ejus

velocitati illato.

Diversi effectus re- 367. Quin immo ibi considerandum & illud ; si velocitas eius fuerit exigua, ipsum

late ad magnetes i i " ' r ••• • • . '.. . J *.,

pro diversa veioci. globum taciic sisti, exiguo motu a vi mutua aequall, seu reactione, impresso magnetibus,
tate ejus giobuii. quo per solam plani fractionem, & mutuas eorum vires impedito, exigua in eorum position-
ibus mutatio fiat. Si velocitas impressa aliquantulum creverit ; turn mutatio in positione
magnetum major fiet, & adhuc sistetur giobuii motus ; sed si velocitas fuerit multo major,
globulus autem transeat satis prope aliquas e massis magnetifcis ; ab actione mutua inter
ipsum, & eas massas communicabitur satis ingens motus iis ipsis massis, quo possint etiam
ipsum non nihil retardatum, sed adhuc progredientem sequi, avulsae, a caeteris, quae ob
actiones in majore distantia minores, & brevitatem temporis, remaneant ad sensum immotae,
& nihil turbatae. Sed si velo-[i68]-citas ipsa adhuc augeretur, quantum est opus, eo
deveniri posset ; ut massa utcunque proxima in giobuii transitu nullum sensibilem motum
auferret illi, & ipsa sibi acquireret.

inde faciiis expiica. 368. Porro ejusmodi exemplum intueri licet, ubi globus aliquis contra obstaculum

no phsenomeni. quo vj ••• j • • i • 11 • n vrr • •

globus sciopeto ex- aliquod projicitur, quod, si satis magnam velocitatem habet, concuti totum, & diitnngit
piosus perforat ac eo majorem effectum edit, quo maior est velocitas, ut in muris arcium accidit, qui

plana mobilia, nee . i i • A i • i • i • j i •

movet : cur lumini tormentarns globis impetuntur. At ubi velocitas ad mgentem quandam magmtudmem
data tanta veioci- devenerit ; nisi satis solida sit compages obstaculi, sive vires cohaesionis satis validae ; jam
non major effectus fit, sed potius minor, foramine tantum excavate, quod aequetur ipsi
globo. Id experimur ; si globus ferreus explodatur sciopeto contra portam ligneam,
quae licet semiaperta sit, & summam habeat super suis cardinibus mobilitatem ; tamen
nihil prorsus commovetur ; sed excavatur tantummodo foramen aequale ad sensum diametro
globi, quod in mea Theoria multo facilius utique intelligitur, quam si continue nexu partes
perfecte solidae inter se complicarentur, & conjungerentur. Nimirum, ut in superiore
magnetum casu, particulae globi secum abripiunt particulas ligni, ad quas accesserunt
magis, quam ipsae ad sibi proximas accederent, & brevitas temporis non permisit viribus
illis, a quibus distantium ligni punctorum nexus praestabatur, ut in iis motus sensibilis
haberetur, qui nexum cum aliis sibi proximis a vi mutua ortum dissolveret, aut illis, &
toti portae satis sensibilem motum communicaret. Quod si velocitas satis adhuc augeri
posset ; " ne iis quidem avulsis massa per massam transvolaret, nulla sensibili mutatione
facta, & sine vera compenetratione haberetur ilia apparens compenetratio, quam habet
lumen, dum per homogeneum spatium liberrimo rectilineo motu progreditur ; quam
ipsam fortasse ob causam Divinus Naturae Opifex tarn immanem luci velocitatem voluit
imprimi, quantam in ea nobis ostendunt eclipses Jovis satellitum, & annua fixarum aberratio,
ex quibus Rcemerus, & Bradleyus deprehenderunt, lumen semiquadrante horae percurrere
distantiam aequalem distantiae Solis a Terra, sive plura milliariorum millia singulis arteriae
pulsibus.

Cur in cinere re- ^^ Ac eodem pacto, ubi herbarum forma in cinere cum tenuissimis filamentis remanet

forma* plants avo* intacta, avolantibus oleosis partibus omnibus sine ulla laasione structurae illarum, id quidem

lante parte volatili admodum facile intelligitur, qui fiat : ibi nova vis excitata ingentem velocitatem parit

brevi tempore, quae omnem alium effectum impediat virium mutuarum inter olea, &

A THEORY OF NATURAL PHILOSOPHY 271

in a much greater number of ways as well, if not only at one, but at many distances, there
were these asymptotic restraints, resulting in continuous impenetrability through a non-
continuous disposition of scattered points.

366. Now, in the first case, where there is no such asymptote besides the first, there if there were no
would be a far different result. In this case, it is evident that, if a sufficiently great velocity substances' * "would
can be given to any mass, it would pass through any other mass without any perturbation be permeable by
of its own parts, or of the parts of the other. For, the forces have no continuous time sufficiently hgre a*t
in which to act & produce any finite sensible motion ; since if this time is diminished velocity is given
immensely (as it will be diminished, if the velocity is immensely increased), the effect of an^orT'eio^'pass-
the forces is also diminished immensely. We can illustrate the idea by the example of an ing between mag-
iron ball, which is required to pass across a plane, in which lie scattered in all positions nets-

a great number of magnetic masses possessed of considerable force. If the ball is not
projected with a certain very great velocity, even if its direction is such that it is not bound
to meet any of the masses, yet it will not go beyond those masses ; but its motion will be
checked by their attractions. But if the velocity is great enough, so that the actions of
the magnetic forces only last for a sufficiently short interval of time, then it will certainly
get through & beyond them without suffering any sensible loss of velocity.

367. Lastly, there is to be considered also this point ; if the velocity of the ball were Relatively diverse
very small, the ball might easily be brought to rest, a slight motion due to an equal mutual to th^magnetsfdue
force or reaction being communicated to the magnets ; but this latter being prevented to diverse velocities
merely by the friction of the plane, the change in their positions would be very small. c

Then if the impressed velocity were increased somewhat, the change in the positions of
the magnets would become greater, & still the ball might be brought to rest. But if the
velocity was much greater, the ball may also pass near enough to some of the magnetic
masses ; & by the mutual action between it & the masses there will be communicated to
the masses a sufficiently great motion, to enable them to follow it as it goes on with its
velocity somewhat retarded ; they will be torn from the rest, which owing to the smaller
action corresponding to a greater distance, & the shortness of the time, remain approximately
motionless, & in no wise disturbed. If the velocity is still further increased, to the necessary
extent, it could become such that a mass, no matter how near it was to the path of the
ball, would communicate no velocity to it, nor acquire any from it.

368. Further, an example of this sort of thing can be seen in the case where a ball is Hence an easy

.,. -ft i • • «• • i •• ii_io explanation of the

projected against an obstacle ; if the velocity is sufficiently great, it agitates the whole & phenomenon in
breaks it to pieces ; & the effect produced is the greater, the greater the velocity, as is the ™hich a bal1

r r 111-1 iii-n i i t • i from

a cannon

r r 111-1 iii-n i i t • i

case for the walls of forts bombarded with cannon-balls. But when the velocity reaches a perforate a mov-
certain very great magnitude, unless the fabric of the obstacle is sufficiently solid or the able plane without

* i • <•?•! i -11 i rr i i moving it ; & why

forces of cohesion sufficiently great, there will now be no greater effect, rather a less, a such a great
hole only being made, equal to the size of the ball. Let us consider this ; suppose an iron Jel°tcity is given to
ball is fired from a gun against a wooden door, & this door is partly open, & it has the utmost
mobility to swing on its hinges ; nevertheless, it will not be moved in the slightest. Merely
a hole, approximately equal to the size of the ball, will be made. Now this is far more easily
understood according to my Theory, than if we assume that there are perfectly solid parts
united & joined together by a continuous connection. Indeed, as in the case of the magnets
given above, the particles of the ball carry off with them particles of the wood, which they
have approached more closely than these particles have approached to the particles of
wood next to them ; & the shortness of the time does not allow the forces, by which the
connection between the distances of the points of the wood is maintained, to give to the
particles a sensible motion in the latter, which would dissolve the connection with others
next to them arising from the mutual force, or in the former, which would also communicate
a sufficiently sensible motion in the whole door. But if the velocity is still further increased
to a sufficient extent, not even the latter particles are torn away, & one mass will pass
through the other, without any sensible change being made. Thus, without real
compenetration, we should have that apparent compenetration that we have in the case of
light, as it passes through a homogeneous space with a perfectly free rectilinear motion.
Perchance that is the reason why the Divine Founder of Nature willed that so enormous a
velocity should be given to light ; how great this is we gather from the eclipses of Jupiter's
satellites, & from the annual aberration of the fixed stars. From which Roemer & Bradley
worked out the fact that light took an eighth of an hour to pass over the distance from the Sun
to the Earth, or many thousands of miles in a single beat of the pulse. The reason Why

369. In the same way, when the form of stalks remain intact in the ash with their in the ash there
finest fibres, after that the oleose parts have all been driven off without any breaking down th™aiforr^imf"the
of their structure, what happens can be quite easily understood. Here, a new force being plant after that the
excited produces in a brief space of time a mighty velocity, which prevents all that other ^en^riverToff by
effect arising from the mutual forces between the oily & the ashy parts ; the oily particles the action of fire.

272 PHILOSOPHIC NATURALIS THEORIA

cineres^ oleaginosis particulis inter terreas cum hac apparenti compenetratione liberrime
avolantibus sine ullo immediate impactu, £ incursu.

. 37°- Quod si ita res habet ; liceret utique nobis per occlusas ingredi portas, £ per
retur, si possums durissima transvolarc murorum sc-[i69]-pta sine ullo obstaculo, £ sine ulla vera compene-
ve!odtate1mPnsaetis trati°ne, nimirum satis magnam velocitatem nobis ipsis possemus imprimere, quod si
magnam. Natura nobis permisisset, & velocitates corporum, quae habemus prae manibus, ac nostrorum

digitorum celeritates solerent esse satis magnae ; apparentibus ejusmodi continuis
compenetrationibus assueti, nullam impenetrabilitatis haberemus ideam, quam mediocritati
nostrarum virium, & velocitatum, ac experimentis hujus generis a sinu materno, & prima
infantia usque adeo frequentibus, & perpetuo repetitis debemus omnem.

371' ^x imPenetrabilitate oritur extensio. Ea sita est in eo, quod alise partes sint
extra alias : id autem necessario haberi debet ; si plura puncta idem spatii punctum simul
occupare non possint. Et quidem si nihil aliunde sciremus de distributione punctorum
materias ; ex regulis probabilitatis constaret nobis, dispersa esse per spatium extensum
in longum, latum, & profundum, atque ita constaret, ut de eo dubitare omnino non liceret,
adeoque haberemus extensionem in longum, latum, & profundum ex eadem etiam sola
Theoria deductam. Nam in quovis piano pro quavis recta linea infinita sunt curvarum
genera, quae eadem directione egressae e dato puncto extenduntur in longum, & latum
respectu ejusdem rectae, & pro quavis ex ejusmodi curvis infinitse sunt curvae, quae ex illo
puncto egressae habeant etiam tertiam dimensionem per distantiam ab ipso. Quare sunt
infinities plures casus positionum cum tribus dimensionibus, quam cum duabus solis, vel
unica, & idcirco infinities major est probabilitas pro uno ex iis, quam pro uno ex his, &
probabilitas absolute infinita omnem eximit dubitationem de casu infinite improbabili,
utut absolute possibili. Quin immo si res rite consideretur, & numeri casuum inter se
conferantur ; inveniemus, esse infinite improbabile, uspiam jacere prorsus accurate in
directum plura, quam duo puncta, & accurate in eodem piano plura, quam tria.

Extensum ejusmodi 372. Haec quidem extensio non est mathematice, sed physice tantum continua : at

mathematics6' con" de prsejudicio, ex quo ideam omnino continuae extensionis ab infantia nobis efformavimus,
tinuum : real em satjs dictum est in prima Parte a num. 158 ; ubi etiam vidimus, contra meam Theoriam
consutat" ^° * non posse afferri argumenta, quae contra Zenonistas olim sunt facta, £ nunc contra
Leibnitianos militant, quibus probatur, extensum ab inextenso fieri non posse. Nam
illi inextensa contigua ponunt, ut mathematicum continuum efforment, quod fieri non
potest, cum inextensa contigua debeant compenetrari, dum ego inextensa admitto a se
invicem disjuncta. Nee vero illud vim ullam contra me habet, quod nonnulli adhibent,
dicentes, hujusmodi extensionem nullam esse, cum constet punctis penitus inexten-[i7o]-sis,
& vacuo spatio, quod est purum nihil. Constat per me non solis punctis, sed punctis
habentibus relationes distantiarum a se invicem : eae relationes in mea Theoria non
constituuntur a spatio vacuo intermedio, quod spatium nihil est actu existens, sed est
aliquid solum possibile a nobis indefinite conceptum, nimirum est possibilitas realium
modorum localium existendi cognita a nobis secludentibus mente omnem hiatum, uti
exposui, in prima Parte num. 142, £ fusius in ea dissertatione De Spatio £ Tempore,
quam hie ad calcem adjicio ; constituuntur a realibus existendi modis, qui realem utique
relationem inducunt realiter, £ non imaginarie tantum diversam in diversis distantiis.
Porro si quis dicat, puncta inextensa, £ hosce existendi modos inextensos non posse con-
stituere extensum aliquid ; reponam facile, non posse constituere extensum mathematice
continuum, sed posse extensum physice continuum, quale ego unicum admitto, £
positivis argumentis evinco, nullo argumento favente alteri mathematice continue extenso,
quod potius etiam independenter a meis argumentis difficultates habet quamplurimas.
Id extensum, quod admitto, est ejusmodi, ut puncta materis alia sint extra alia, ac
distantias habeant aliquas inter se, nee omnia jaceant in eadem recta, nee in eodem piano
omnia, sint vero multa ita proxima, ut eorum intervalla omnem sensum effugiant. In eo
sita est extensio, quam admitto, quae erit reale quidpiam, non imaginarium, £ erit physice
continua.

A THEORY OF NATURAL PHILOSOPHY 273

fly off between the earthy particles with this apparent compenetration, in the freest manner,
without any immediate impulse or collision.

370. But if this were the case, we could walk through shut doors, or pass through the Apparent compene-
hardest walled enclosures without any resistance, & without any real compenetration; wouw be' obtained
that is to say, if we could impress upon ourselves a sufficiently great velocity. Now if if we were able to
Nature allowed us this, & the velocities of bodies which are around us, & the speed of our v'efocTt^^r^at
fingers were usually sufficiently great, we, being accustomed to such continuous apparent enough.
compenetration, should have no idea of impenetrability. We owe the whole idea of
impenetrability to the mediocrity of our forces & velocities, & to experiences of this kind,

which have happened to us from the time we were born, during infancy & up till the present
time, frequently & continually repeated.

371. From impenetrability there arises extension. It is involved in the fact that Extension nee es-
some parts are outside other parts ; & this of necessity must be the case, if several points repuLve1Sforces.°
cannot at the same time occupy the same point of space. Indeed, even if we knew nothing

from any other source about the distribution of the points of matter, it would be manifest
from the rules of probability that they were dispersed through a space extended in length,
breadth & depth ; & it would be so clear, that there could not be the slightest doubt about
it ; & thus we should obtain extension in length, breadth & depth as a consequence of
my Theory alone. For, in any plane, for any straight line in it, there are an infinite number
of kinds of curves, which starting in the same direction from a given point extend in length
& breadth with respect to this same straight line ; & for any one of these curves there are
an infinite number of curves that, starting from that point, have also a third dimension
through distance from the point. Hence, there are infinitely more cases of positions with
three dimensions than with two alone or only one ; & thus there is infinitely greater probability
in favour of one of the former than for one of the latter ; & as the probability is absolutely
infinite, it removes any doubt about a case which is infinitely improbable, though absolutely
possible. Indeed, if the matter is carefully considered, & the number of cases compared
with one another, we shall find that it is infinitely improbable that more than two points
will anywhere lie accurately in the same straight line, or more than three in the same
plane.

372. This extension is not mathematically, but only physically, continuous ; & on the s"ch extension is
matter of the prejudgment, from which we have formed for ourselves the idea of absolutely mtthematically,
continuous extension from infancy, enough has been said in the First Part, starting with continuous; it is
Art. 158. There, too, we saw that there could not be brought forward against my Theory consists!

the arguments which of old were brought against the followers of Zeno, & which now are
urged against the disciples of Leibniz, by which it is proved that extension cannot be
produced from non-extension. For these disputants assume that their non-extended points
are placed in contact with one another, so as to form a mathematical continuum ; & this
cannot happen, since things that are contiguous as well as non-extended must compenetrate ;
but I assume non-extended points that are separated from one another. Nor indeed have
the arguments, which some others use, any validity in opposition to my Theory ; when they
say that there is no such extension, since it is founded on non-extended points & empty
space, which is absolute nothing. According to my Theory, it is founded, not on points
simply, but on points having distance relations with one another ; these relations, in my
Theory, are not founded upon an empty intermediate space ; for this space has no actual
existence. It is only something that is possible, indefinitely imagined by us ; that is to
say, it is the possibility of real local modes of existence, pictured by us after we have
mentally excluded every gap, as I explained in the First Part in Art. 142, & more fully
in the dissertation on Space & Time, which I give at the end of this work. The relations
are founded on real modes of existence ; & these in every case yield a real relation which
is in reality, & not merely in supposition, different for different distances. Further, if
anyone should argue that these non-extended points, or non-extended modes of existence,
cannot constitute anything extended, the reply is easy. I say that they cannot constitute
a mathematically extended continuum, but they can a physically extended continuum.
The latter only I admit, & I prove its existence by positive arguments ; none of these
arguments being favourable to the other continuum, namely one mathematically extended.
This latter, even apart from any arguments of mine, has very many difficulties. The
extension, which I admit, is of such a nature that it has some points of matter that lie outside
of others, & the points have some distance between them, nor do they all lie on the same
straight line, nor all of them in the same plane ; but many of them are so close to one another
that the intervals between them are quite beyond the scope of the senses. In that is involved
the extension which I admit ; & it is something real, not imaginary, & it will be physically
continuous.

274 PHILOSOPHIC NATURALIS THEORIA

Quomodo existat 373. At erit fortasse, qui dicet, sublata extensione absolute mathematica tolli omnem

con^uo'Ltu^fsi? Geometriam. Respondeo, Geometriam non tolli, quae considerat relationes inter distantias,
ente. & inter intervalla distantiis intercepta, quae mente concipimus, & per quam ex hypothesibus

quibusdam conclusiones cum iis connexas ex primis quibusdam principiis deducimus.
Tollitur Geometria actu existens, quatenus nulla linea, nulla superficies mathematice
continua, nullum solidum mathematice continuum ego admitto inter ea, quae existunt ;
an autem inter ea, quse possunt existere, habeantur, omnino ignoro. Sed aliquid ejusmodi
in communi etiam sententia accidit. Nulla existit revera in Natura recta linea, nullus
circulus, nulla ellipsis, nee in ejusmodi lineis accurate talibus fit motus ullus, cum omnium
Planetarum, & Terrae in communi sententia motus habeantur in curvis admodum compli-
catis, atque altissimis, &, ut est admodum probabile, transcendentibus. Nee vero in
magnis corporibus ullam habemus superficiem accurate planam, & continuam, aut sphsericam,
aut cujusvis e curvis, quas Geometrae contemplantur, & plerique ex iis ipsis, qui solida
volunt elementa, simplices ejusmodi figuras ne in ipsis quidem elementis admittent.

Quid in ea imagi- 374. Quamobrem Geometria tota imaginaria est, & idealis, sed propositiones hypo-

narium. quid reale : i • • j j j r i a • • j- • i -11

eiegans anaiogia theticse, quae inde deducuntur, [171] sunt verse, & si existant conditiones ab ilia assumptae,
loci cum tempore existent utique & conditionata inde eruta, ac relationes inter distantias punctorum
Hatis '"mensural " imaginarias ope Geometriae ex certis conditionibus deductae, semper erunt reales, & tales,
quales eas invenit Geometria, ubi illae ipsae conditiones in realibus punctorum distantiis
existant. Ceterum ubi de realibus distantiis agitur, nee illud in sensu physico est verum,
ubi punctum interiacet aliis binis in eadem recta positis, a quibus aeque distet, binas illas
distantias fore partes distantiae punctorum extremorum juxta ea quae diximus num. 67.
Physice distantia puncti primi a secundo constituitur per puncta ipsa, & binos reales ipsorum
existendi modos, ita & distantia secundi a tertio ; quorum summa continet omnia tria
puncta cum tribus existendi modis, dum distantia primi a tertio constituitur per sola duo
puncta extrema, & duos ipsorum existendi modos, quae ablato intermedio reali puncto
manet prorsus eadem. Illas duae sunt partes illius tertiae tantummodo in imaginario, &
geometrico statu, qui concipit indefinite omnes possibiles intermedios existendi modos
locales, & per earn cognitionem abstractam concipit continua intervalla, ac eorum partes
assignat, & ope ejusmodi conceptuum ratiocinationes instituit ab assumptis conditionibus
petitas, quae, ubi demum ad aliquod reale deducunt, non nisi ad verum possint deducere,
sed quod verum sit tantummodo, si rite intelligantur termini, & explicentur. Sic quod
aliqua distantia duorum punctorum sit aequalis distantiae aliorum duorum, situm est in
ipsa natura illorum modorum, quibus existunt, non in eo, quod illi modi, qui earn individuam
dlstantiam constituunt, transferri possint, ut congruant. Eodem pacto relatio duplae,
vel triplae distantiae habetur immediate in ipsa essentia, & natura illorum modorum. Vel
si potius velimus illam referre ad distantiam aequalem ; dici poterit, earn esse duplam
alterius, quae talis sit, ut si alteri ex alterius punctis ponatur tertium novum ad aequalem
distantiam ex parte altera ; distantia nova hujus tertii a primo sit aequalis illi, quae duplae
nomen habet, & sic de reliquis, ubi ad realem statum transitur. Neque enim in statu
reali haberi potest usquam congruentia duarum magnitudinum in extensione, ut haberi
nee in tempore potest unquam ; adeoque nee aequalitas per congruentiam in statu reali
haberi potest, nee ratio dupla per partium asqualitatem. Ubi decempeda transfertur
ex uno loco in alium, succedunt alii, atque alii punctorum extremorum existendi modi,
qui relationes inducunt distantiarum ad sensum aequalium : ea aequalitas a nobis supponitur
ex causis, nimirum ex mutuo nexu per vires mutuas, uti hora hodierna ope egregii horologii
comparatur cum hesterna, itidem aequalitate supposita ex causis, sed loco suo divelli, &
ex uno die in alterum hora eadem traduci nequaquam potest. Verum haec omnia ad
Metaphysicam potius pertinent, & ea fusius cum omnibus [172] loci, ac temporis relationibus
persecutus sum in memoratis dissertationibus, quas hie in fine subjicio.

FigurabiHtas orta 375. Ex extensione oritur figurabilitas, cum qua connectitur moles, & densitas

quid6 "siV figure" C& supposita massa. Quoniam puncta disperguntur per spatium extensum in longum, latum,
quam vaga, ' & & profundum ; spatium, per quod extenduntur, habet suos terminos, a quibus figura
etiam^n* communi pendet. Porro figuram determinatam ab ipsa natura, & existentem in re, possunt agnoscere
sententia. tantummodo in elementis ii, qui admittunt elementa ipsa solida, atque compacta, & continua,

A THEORY OF NATURAL PHILOSOPHY 275

373. But perhaps some one will say that, if absolutely continuous extension is barred, How Geometry can
then the whole of Geometry is demolished. I reply, that Geometry is not demolished, existing continuum
since it deals with relations between distances, & between intervals intercepted in these is excluded,
distances ; that these we mentally conceive, & by them we derive from certain hypotheses
conclusions connected with them, by means of certain fundamental principles. Geometry,

as actually existent, is demolished ; in so far as there is no line, no surface, & no solid that
is mathematically continuous, which I admit as being among things actually existing ;
whether they are to be numbered amongst things that might possibly exist, I do not know.
But something of the sort does take place, according to the usual idea of things. As a
matter of fact, there is in Nature no such thing as a straight line, or a circle, or an ellipse ;
nor is there motion in lines that are accurately such as these ; for in the opinion of everybody,
the motions of all the planets & the Earth take place in curves that are very complicated,
having equations of a very high degree, or, as is quite possible, transcendent. Nor in large
bodies do we have any surfaces that are quite plane, & continuous, or spherical, or shaped
according to any of the curves which geometers investigate ; & very many of these men,
who accept solid elements, will not admit simple figures even in the very elements.

374. Hence the whole of geometry is imaginary ; but the hypothetical propositions The imaginary &
that are deduced from it are true, if the conditions assumed by it exist, & also the conditional Geometry ^arfeie-
things deduced from them, in every case ; & the relations between the imaginary distances gant analogy be-
of points, derived by the help of geometry from certain conditions, will always be real, *ween ^Iace & time

11 el 11 1 1 T • • e IT

& such as they are found to be by geometry, when those conditions exist for real distances of equality,
of points. Besides, when we are dealing with real distances, it is not true in a physical
sense, when a point lies between two others in the same straight line, equally distant from
either, to say that the two distances are parts of the distance between the two outside points,
according to what we have said in Art. 67. Physically speaking, the distance of the first
point from the second is fixed by the two points & their two real modes of existence, & so
also for the distance between the second & the third. The sum of these contains all three
points & their three modes of existence ; whilst the distance of the first from the third
is fixed by the two end points only, together with their two modes of existence ; & this
remains unaltered if the intermediate real point is taken away. The two distances are
parts of the third only in imagination, & in the geometrical condition, which in an indefinite
manner conceives all the possible intermediate local modes of existence ; & from that
abstract conception forms a picture of continuous intervals, & assigns parts to them ; then,
by the aid of such imagery institutes chains of reasoning founded on assumed conditions ;
& these, when at last they lead to something real, will only do so, if it is possible for them
to lead to something that is true, & something that is only true if the terms are correctly
understood & explained. Thus, the fact, that the distance between two points is equal
to the distance between two other points, rests upon the nature of their modes of existence,
& not upon the idea that the modes, which constitute the individual distances, can be
transferred, so as to agree with one another. In the same way, the idea of twice, or three
times a distance, is obtained directly from the essential nature of those modes of existence.
Or, if we prefer to refer it to the idea of equal distances, we can say that one distance is
twice another when it is such that, if beyond the second point of the latter we place a new
third point at a distance equal to that of the first point from the second, then the distance
of this new third point from the first point will be equal to that to which the name double
distance is given ; & so on for other multiples, when the matter is reduced to a consideration
of real state. For, in the real state, there never can be a congruence of two magnitudes
in extension, just as there never can be such a congruence in time ; & therefore there never
can be an equality depending on congruence in the real state, nor a double ratio through
equality of parts. When a length of ten feet is transferred from one place to another, there
follow, one after the other, different modes of existence of the end points ; & these modes
introduce relations of practically equal distances. This equality is supposed by us to be
due to causes ; for instance, to the mutual connection in consequence of mutual forces ;
just as an hour of to-day may be compared with one of yesterday by the help of an accu-
rate clock ; but the same hour cannot be disjointed from its own position & transferred
from one day to another in any way. But really, such matters have more to do with
Metaphysics ; & I have investigated them more fully, together with all the relations of
space & time, in the dissertations I have mentioned, which I add at the end of this work.

375. From extension arises the idea of figurability ; with this is connected volume &, Figurabiiity arises

, J'J , , , ., - ' £. . i from extension;

when we have conceived the idea of mass, density. Since points are scattered through the nature of shape,
extended space in length, breadth & depth, the space through which they are extended & how vague the

i_ • 1 j • , i • i T-* i • • • i_ i idea of it is, even m

has its boundaries ; & upon these boundaries depends shape, rurther, it is in the elements the opinion usually
alone that a shape, determinate by its very nature, & existing of itself, can be acknowledged held.
by those who suppose the elements to be solid, compact & continuous ; & by those who

276 PHILOSOPHIC NATURALIS THEORIA

& qui ab inextensis extensum continuum componi posse arbitrantur, ubi nimirum tota
ilia materia superficie continua quadam terminetur. Ceterum in corporibus hisce, quae
nobis sub sensum cadunt, idea figurae, quae videtur maxime distincta, est admodum vaga,
& indefinita, quod quidem diligenter exposui agens superiore anno de figura Telluris in
dissertatione inserta postremo Bononiensium Actorum tomo, in qua continetur Synopsis
mei operis de Expeditione Litteraria -per Pontificiam ditionem, ubi sic habeo ; Inprimis hoc
ipsum nomen figure terrestris, quod certam quandam, ac determinatam significationem videtur
habere, habet illam quidem admodum incertam, y vagam. Superficies ilia, quce maria, y
lacus, y fluvios, ac mantes, y campos, vallesque terminat, est ilia quidem admodum, nobis
saltern, irregularis, y vero etiam instabilis : mutatur enim quovis utcunque minima undarum,
y glebarum motu, nee de hac Telluris figura agunt, qui in figuram Telluris inquirunt : aliam
ipsi substituunt, quce regularis quodammodo sit, sit autem illi priori proxima, quce nimirum
abrasis haberetur montibus, collibusque, vallibus vero oppletis. At hcec iterum terrestris figures
notio vaga admodum est, & incerta. Uti enim infinita sunt curvarum regMlarium genera,
quce per datum datorum punctorum numerum transire possint, ita infinita sunt genera curvarum
superficierum, quce Tellurem ita ambire possint, atque concludere, ut vel omnes, vel datos
contingant in datis punctis mantes, collesque, vel si per medios transire colles, ac monies debeat
superficies qucedam ita, ut regularis sit, y tantundem materics concludat extra, quantum
vacui aeris infra sese concludat usque ad veram hanc nobis irregularem Telluris superficiem,
quam intuemur : infinites itidem, & a se invicem diverse? admodum superficies haberi possunt,
quce problemati satisfaciant, atque ece ejusmodi etiam, ut nullam, quce sensu percipi possif,
prce se ferant gibbositatem, quce ipsa vox non ita determinatam continet ideam.

Quanto magis in 376. Haec ego ibi de Telluris figura, quae omnino pertinent ad figuram corporis

cujuscunque in communi etiam sententia de continua extensione materiae : nam omnium
fere corporum superficies hie apud nos utique multo magis scabrae sunt pro ratione suae
magnitudinis, quam Terra pro ratione magnitudinis suae, & vacuitates interrias habent
quamplurimas. Ve-[i73]-rum in mea Theoria res adhuc magis indefinita, & incerta est.
Nam infinite sunt etiam superficies curvae continues, in quibus tamen omnia jacent puncta
massae cujusvis : quin immo infinitae numero curvae sunt lineae, quae per omnia ejusmodi
puncta transeant. Quamobrem mente tantummodo confingenda est qusedam superficies,
quae omnia puncta includat, vel quae pauciora, & a reliquorum coacervatione remotiora
excludat, quod aestimatione quadam morali fiet, non accurata geometrica determinatione.
Ea superficies figuram exhibebit corporis ; atque hie jam, quae ad diversa figurarum genera
pertinent ; id omne mini commune est cum communi Theoria de continua extensione
materiae.

Moles a figura 377. A figura pendet moles, quae nihil est aliud, nisi totum spatium extensum in

e]usenideain&e7n longum, latum, & profundum externa superficie conclusum. Porro nisi concipiamus

sententia communi, superficiem illam, quam innui, quae figuram determinet ; nulla certa habebitur molis

haclxhteoriaf8 " i^ea : <lum immo si superficiem concipiamus tortuosam illam, in qua jaceant puncta omnia ;

jam moles triplici dimensione praedita erit nulla ; si lineam curvam concipimus per omnia

transeuntem : nee duarum dimensionum habebitur ulla moles. Sed in eo itidem incerta

asstimatione indiget sententia communis ob interstitia ilia vacua, quae habentur in omnibus

corporibus, & scabritiem, juxta ea, quae diximus, de indeterminatione figurae. Hie autem

itidem concepta superficie extima terminante figuram ipsam, quae deinde de mole relata

ad superficiem tradi solent, mihi communia sunt cum aliis omnibus, ut illud : posse eandem

magnitudine molem terminari superficiebus admodum diversis, & forma, & magnitudine,

ac omnium minimam esse sphaericae figurae superficiem respectu molis : in figuris autem

similibus molem esse in ratione triplicata laterum homologorum, & superficiem in

duplicata, ex quibus pendent phaenomena sane multa, atque ea inprimis, quae pertinent

ad resistentiam tarn fluidorum, quam solidorum.

Massa : quid inejus 578. Massa corporis est tota quantitas materiae pertinentis ad id corpus, quae quidem

idea mcertum ob •,•'•• • • j -11 j A !_• •

matenam exteram mmi erit i?56 numerus punctorum pertmentium ad illud corpus. At hie jam ontur

immixtam. Omnia indeterminatio quaedam, vel saltern summa difficultas determinandi massae ideam, nee id

partibus1 "diversaj tantum in mea, verum etiam in communi sententia, ob illud additum punctorum pertinentium

naturae. ad muj, corpus, quod heterogeneas substantias excludit. Ea de re sic ego quidem in Stayanis

Supplementis § 10 Lib. i : Nam admodum difficile est determinare, quce sint illce substan-

tice beterogenees, quce non pertinent ad corporis constitutionem. Si materiam spectemus ; ea

y mihi, y aliis plurimis homogenea est, y solis ejus diversis combinationibus diverse oriuntur

A THEORY OF NATURAL PHILOSOPHY 277

think that an extended continuum can be formed out of non-extended points, when indeed
the whole of the matter is bounded by a continuous surface. Besides, in those bodies that
fall within the scope of our senses, the idea of figure, which seems to be very distinct, is
however quite vague & indefinite ; & I pointed this out fairly carefully, when dealing some
time ago with the figure of the Earth, in a dissertation inserted in the last volume of the
Acta Bononiensia ; this contains the synopsis of my work, Expeditio Litteraria -per Pontificiam
ditionem, & there the following words occur. Now, in the first place, this term, " the figure
of the Earth " which seems to have a certain definite & determinate meaning, is really very
vague y indefinite. The surface which bounds the seas, the lakes, the rivers, the mountains,
the plains & the valleys, is really something quite irregular, at least to us ; y moreover it is
also unstable ; for it changes with the slightest motion of the waves y the soil. But those who
investigate " the figure of the Earth," do not deal with this figure of the Earth ; they substitute
for it another figure which, although to some extent regular, yet approximates closely to the
former true figure ; that is to say, it has the mountains y the hills levelled off, whilst the valleys
are fitted up. Now once more the idea of this figure of the Earth is vague y uncertain. For,
just as there are infinite classes of regular curves that can be made to pass through a given number
of given points ; so also there are infinite classes of curved surfaces that can be made to go round
the Earth y circumscribe it in such a manner that they touch all the mountains y hills, or at
least certain given ones ; or, if you like, some surface is bound to pass through the middle of the
hills y mountains in such a way that it cuts off as much matter outside itself, as it encloses
empty air-spaces within it y our true surface of the Earth, to our eyes, so irregular. Also,
there can be an infinite number of surfaces, y these too quite different frqm one another, which
satisfy the problem ; y all of them, too, of such a kind that they have no manifest humps, as
far as can be detected ; y this term even contains no true definiteness.

376. These are my words in that dissertation with regard to the figure of the Earth ; The vagueness is
& they apply in general to the figure of any body also, if considered according to the usual jheogreater m th'S
way with regard to the continual extension of matter. For, the surfaces of nearly all bodies

here around us are in every case much rougher in comparison with their size than is the
Earth in comparison with its magnitude ; & they have many internal empty spaces. But,
in my Theory, the matter is much more indefinite & uncertain still. For there are an
infinite number of continuous curved surfaces, in which nevertheless all the points of any
mass lie ; nay, further, there are an infinite number of curved lines passing through all the
points. Therefore we can only mentally conceive a certain surface which shall include all
the points or exclude a few of them which are more remote by gathering the rest together ;
this can be done by a kind of moral assessment, but not by an accurate geometrical construc-
tion. This surface gives the shape of the body ; & with that idea, all that relates to the
different kinds of shapes of bodies is in agreement in my Theory with the usual theory of the
continual extension of matter.

377. Volume depends upon shape ; & volume is nothing else but the whole of the space, Volume depends on
extended in length, breadth & depth, which is included by the external surface. Further, thfsPtoo ^vague h!
unless we picture that surface which I mentioned as determining the shape, there can be the usual theory, &
no definite idea of volume. Nay indeed, if we think of the tortuous surface in which all ™"^eh mor'

the points lie, we shall never have a volume possessed of a third dimension ; whilst if we think
of a curved line passing through all the points, no volume will be obtained that has even
two dimensions. But in that the usual idea is also wanting, as regards indefinite assessment,
owing to those empty interstices that are present in all bodies, & the roughness, as we have
said, which arises from the indeterminateness of figure. Here again, if an outside surface is
conceived as bounding the figure, all those things that are usually enunciated about volume
in relation to figure agree in my theory with those of all others ; for instance, that the
same volume as regards magnitude can be bounded by surfaces that are quite different,
both in shape & size, & that the least surface of all having the same volume is that of a
sphere. Also that, in similar figures, the volumes are in the triplicate ratio of homologous
sides, the surfaces in the duplicate ratio ; & upon these depend a truly great number of
phenomena, & especially those which are connected with the resistance both of fluids &
of solids.

378. The mass of a body is the total quantity of matter pertaining to that body ; & Mass ; what there

J'n-ii i • • -11 , • i <• r r. is m the idea of it

in my 1 heory this is precisely the same thing as the number of points that go to form the that is indefinite
body. Here now we have a certain indefiniteness, or at least the greatest difficulty, in owin_8 to <?utside

e ' . ,,,.., , ..' . fl . , J matter mingling

forming a definite idea of mass ; & that, not only in my theory, but in the usual theory as with it. All bodies
well, on account of the addition of the words points that go to form the body ; this excludes are composed of

, i • • . i T i i <• 11 • i • i. parts of different

heterogeneous substances. On this point indeed, I made the following remarks m the natures.
Supplements to Stay's Philosophy : — For it is very difficult to define what those heterogeneous
substances may be, if they do not pertain to the constitution of a body. If we consider matter,
it is in my opinion, y in that of very many others, homogeneous ; y the different species of

2 78 PHILOSOPHISE NATURALIS THEORIA

corporum species. Quare ab ipsa materia non potest desumi discrimen illud inter substantias
pertinentes, y non pertinentes. Si autem y diver sam [174] illam combinationem spectemus,
corpora omnia, quce observamus, mixta sunt ex substantiis adm.od.um dissimilibus, quce tamen
omnes ad ejus corporis constitutionem pertinent. Id in animalium corporibus, in plant-is, in
marmoribus plerisque, oculis etiam patet, in omnibus autem corporibus Cbemia docet, quce
mixtionem illam dissohit.

piures substantial 370,. Ex alia parte tenuissima cetherea materia, quce omnino est aliqua nostro aere varior,

substantiam" "cor- a^ constitutionem masses nequaquam pertinere censetur, ut nee pro corporibus plerisque aer,

pom. qui meatibus internis interjacet. Sic aer inclusus spongice meatibus, ad ipsius constitutionem

nequaquam censetur pertinere. Idem autem ad multorum corporum constitutionem pertinet :

saltern ad fixam naturam redactus, ut Halesius demonstravit, piures y animalis regni, &

vegetabilis substantias magna sui parte constare aere fixitatem adepto. Rursus substanties

volatiles, aere ipso tenuiores multo, quce in corporum dissolutions chemica in balitus, y fumos

abeunt, y piures fortasse, quas nos nullo sensu percipimus, ad ipsa corpora pertinebant.

Nee exciudi omnia 380. Nee illud assumi potest, quidquid solidum, y fixum est, id tantummodo pertinere

fnci<u^inposse°Iqu1^ a^ corp°r^s massam ; quis enim a corporis humani massa sanguinem omnem, y tot lymphas

translate corpore excludat, a lignis resectis succos nondum concretos ? Prceterquam quod masses idea non ad

untur!PS° transfe " solida solum corpora pertinet, sed etiam ad fluida, in quibus ipsis alia tenuiora aliorum densiorum

meatibus interjacent. Nee vero did potest, pertinere ad corporis constitutionem, quidquid

materice translato corpore, simul cum ipso transfertur ; nam aer, qui intra spongiam est, partim

mutatur in ea translatione, is nimirum, qui orificio est propior, partim manet, qui nimirum

intimior, y qui aliquandiu manet, mutatur deinde.

Hinc indistinctam 381. Hcec, & alia mihi diligentius perpendenti, illud videtur demum, ideam masses non

Quid 'dwisitaseai& esse Accurate determinatam, y distinctam, sed admodum vagam, arbitrariam, y confusam.
raritas ; utranque Erit massa materia omnis ad corporis constitutionem pertinens ; sed a crassa quadam, y
poss"'inhact!ieoria arbitraria cestimatione pendebit illud, quod est pertinere ad ipsam ejus constitutionem. Hsec
in quacunque ego ibi : turn ad molem transeo, de cujus indeterminatione jam hie superius egimus, ac
deinde ad densitatem, quae est relatio massae, ad molem, eo major, quo pari mole est major
massa, vel quo pari massa est minor moles. Hinc mensura densitatis est massa divisa per
molem ; & qusecunque vulgo proferuntur de comparationibus inter massam, molem, &
densitatem, haec omnia & mihi communia sunt. Massa est ut factum ex mole & densitate ;
moles ut massa divisa per densitatem. Raritas autem etiam mihi, ut & aliis, est densitatis
inversa, ut nimirum idem sit dicere, corpus aliquod esse decuplo minus densum alio aliquo
corpore, ac dicere, esse decuplo magis rarum. Verum quod ad densitatem & raritatem
pertinet, in eo ego quidem a communi sententia discrepo, uti exposui num. 89, quod [175]
ego nullum habeo limitem densitatis & raritatis, nee maximum, nee minimum ; dum illi
minimam debent aliquam raritatem agnoscere, & maximam densitatem possibilem, utut
finitam, quse illis idcirco per saltum quendam necessario abrumpitur ; licet nullam agnoscant
raritatem maximam, & minimam densitatem. Mihi enim materiae puncta possunt &
augere distantias a se invicem, & imminuere in quacunque ratione ; cum data linea quavis,
possit ex ipsis Euclideis elementis inveniri semper alia, quae ad ipsam habeat rationem
quancunque utcunque magnam, vel parvam ; adeoque potest, stante eadem massa, augeri
moles, & minui in quacunque ratione data ; at illis potest quidem quaevis massa dividi in
quenvis numerum particularum, quae dispersae per molem utcunque magnam augeant
raritatem, & minuant densitatem in immensum ; sed ubi massa omnis ita ad contactus
immediatos devenit, ut nihil jam supersit vacui spatii ; turn vero densitas est maxima, &
raritas minima omnium, quae haberi possint, & tamen finita est, cum mensura prioris
habeatur, massa finita per finitam molem divisa, & mensura posterioris, divisa mole per
massam.

inertia massarum 382. Inertia corporum oritur ab inertia punctorum, & a viribus mutuis ; nam illud

punctorum'^ipsl demonstravimus num. 260, si puncta quaecunque vel quiescant, vel moveantur directionibus,
respondens conser. & celeritatibus quibuscunque, sed singula aequabili motu ; centrum commune gravitas
gra^tafe^&^dla ve^ quiesccre, vel moveri uniformiter in directum, ac vires mutuas quascunque inter eadem
massae unitae in puncta nihil turbare statum centri communis gravitatis sive quiescendi, sive movendi
Ipsa uniformiter in directum. Porro vis inertise in eo ipso est sita : nam vis inertiae est

A THEORY OF NATURAL PHILOSOPHY 279

bodies arise solely from different combinations of it. Hence it is impossible to take away from
matter the distinction between substances that pertain to a body W those that do not. Again
if we consider the difference of combination, all bodies that come under our observation are mixtures
of substances that are perfectly unlike one another ; & yet all of them are necessary to the
constitution of the body. We have ocular evidence of this in the bodies of animals, in plants,
in most of the marbles ; moreover, in all bodies, chemistry teaches us how to separate that mixture.

379. In another respect, that very tenuous ethereal matter, which is something indeed A !arge number of
much less dense than our air, can in no sense be considered to be a constituent part of a body ; pertafn'To the sub-
nor indeed, in the case of most bodies can the air which is contained in its internal parts. Thus stance of a body.
the air that is included in the passages of a sponge can in no sense be considered as being necessary

to the constitution of the sponge. But the same thing pertains to the constitution of many bodies ;
at least, when reduced to a fixed nature. For Hales has proved that many substances of the
animal & vegetable kingdoms in a great part consist of air that has attained fixity. Again,
volatile substances, more tenuous than air itself, which go off in vapours W fumes from bodies
chemically decomposed, y perchance many which are not perceived by any of our senses, all
pertained to these bodies.

380. Nor can it be assumed that only something solid & fixed can pertain to the mass of a Nor can ail fluids be
body. For who would exclude from the mass of the human body the whole of the blood, y the ^those 'things0]*;
large number of watery fluids, or from chips of wood the juices that are not yet congealed? included, which
Especially as the idea of mass pertains not only to solids alone but also to fluids ; fc? in these moved^r

some of the more tenuous parts lie in the interstices of the more dense. On the other hand, it with the body.

cannot be said that any kind of matter, which when the body is moved is carried with it, pertains

of necessity to the constitution of the body. For the air which is within a sponge is partly

moved by that translation, that is to say that part which is near an orifice ; whilst it partly

remains, that is to say that part which is more internal, W remains for some length of time,

£5" then is moved.

381. After carefully considering these & other matters, I have come to the conclusion Hence also the idea
that the idea of mass is not strictly definite & distinct, but that it is quite vague, arbitrary nitemasxhe5 nature

confused. Mass will be the whole of the matter pertaining to the constitution of a body ; of density, & rarity;

but what part of it actually does pertain to its constitution, will depend upon a non-scientific thi^Theorv116™ be
y arbitrary assessment. These are my words ; & after that I pass on to volume, the increased or dimin-
indefiniteness of which I have already dealt with above, & after that to density, which is lshed to anyextent-
the relation of mass to volume ; being so much the greater as in equal volume there is so
much the greater mass, or according as for equal mass there is so much the less volume.
Hence the measure of density is mass divided by volume ; & whatever is usually said about
comparisons between mass, volume & density, everything is in agreement with what I say.
Mass is, so to speak, the product of volume & density ; & volume is mass divided by density.
Rarity, with me, as well as with others, is the inverse of density ; thus it is the same thing
to say that one body is ten times less dense than another body as to say that it is ten times
more rare. But as regards the properties of rarity & density, here I indeed differ from the
usual opinion. For, as I showed in Art. 89, I have no limiting value for either density
or rarity, no maximum, no minimum ; whereas others must admit a minimum rarity, or
a maximum density, as being possible ; &, since this must be something finite, it must
of necessity involve a sudden break in continuity ; although they may not admit any maximum
rarity or minimum density. For with me the points of matter can both increase & diminish
their distances from one another in any ratio whatever ; since, given any line, it is possible,
by the elementary principles of Euclid, to find another in every case, which shall bear to
the given line any ratio however great or small. Thus, it is possible that, whilst the mass
remains the same, the volume should be increased or diminished in any ratio whatever.
But, in the case of other theories, it is indeed possible that a mass can be divided into any
number of particles, which when dispersed throughout a volume of any size however great
will increase the rarity or diminish the density to an indefinitely great extent ; but when
the whole mass has been brought into a state of immediate contact of its particles in such
a manner that there no longer exist any empty spaces between these particles, then indeed
there is a maximum density or a minimum rarity obtainable, although this is finite ; for,
a measure of the first may be obtained by dividing a finite mass by a finite volume, or of
the second by dividing volume by mass. The iner^^ oj a

382. The inertia of bodies arises from the inertia of their points & their mutual forces, mass arises from
For, in Art. 260, it was proved that, if any points are either at rest, or moving in any ^ntsTThe corre*
directions with any velocities, so long as each of the motions is uniform, then the centre spending conserva-
of gravity of the set will either be at rest or move uniformly in a straight line ; & that, thTcentre^Tgrav-
whatever mutual forces there may be between the points, these will in no way affect the tty ; the idea of
state of the common centre of gravity, whether it is at rest or whether it is moving uniformly
in a straight line. Further the force of inertia is involved in this ; for the force of inertia of gravity.

280 PHILOSOPHIC NATURALIS THEORIA

determinatio perseverandi in eodem statu quiescendi, vel movendi uniformiter in directum :
nisi externa vis cogat statum suum mutare : & cum ex mea Theoria demonstretur, earn
proprietatem debere habere centrum gravitatis massae cujuscunque compositae punctis
quotcunque, & utcunque dispositis ; patet, earn deduci pro corporibus omnibus : & hie
illud etiam intelligitur, cur concipiantur corpora tanquam collecta, & compenetrata in
ipso gravitatis centre.

Mobjiitas : quies 383. Mobilitas recenseri solet inter generales corporum proprietates, quse quidem

haberi.tolcxclnBa sPonte consequitur vel ex ipsa curva virium : cum enim ipsa exprimat suarum ordinatarum
prorsus quiete a ope determinationcs ad accessum, vel recessum, requirit necessario mobilitatem, sive
possibilitatem motuum, sine quibus accessus, & recessus ipsi haberi utique non possunt.
Aliqui & quiescibilitatem adscribunt corporibus : at ego quidem corporum quietem saltern
in Natura, uti constituta est, haberi non posse arbitror, uti exposui num. 86. Earn excludi
oportere censeo etiam infinitae improbabilitatis argumento, quo sum usus in ea dissertatione
De Spatio, y Tempore, quam toties jam nominavi, & in Supplementis hie proferam § I,
ubi [176] evinco, casum, quo punctum aliquod materiae occupet quovis momento temporis
punctum spatii, quod alio quopiam quocunque occuparit vel ipsum, vel aliud punctum
quodcunque, esse infinities improbabilem, considerate nimirum numero punctorum material
finite, numero momentorum possibilium infinite ejus generis, cujus sunt infinita puncta
in una recta, qui numerus momentorum bis sumitur, semel cum consideratur puncti dati
materise cujuscunque momentum quodvis, & iterum cum consideratur momentum quodvis,
quo aliud quodpiam materiae punctum alicubi fuerit, ac iis collatis cum numero punctorum
spatii habentis extensionem in longum, latum, & profundum, qui idcirco debet esse infinitus
ordinis tertii respectu superiorum. Deinde ab omnium corporum motu circa centrum
commune gravitatis, vel quiescens, vel uniformiter progrediens in recta linea, quies actualis
itidem a Natura excluditur.

tate 3^4- Verum ipsam quietem excludit alia mihi proprietas, quam omnibus itidem

omnium motuum : materiae punctis, & omnium corporum centris gravitatis communem censeo, nimirum
probiema generate continuitas motuum, de qua egi num. 883, & alibi. Quodvis materiae punctum seclusis

eo pertmens. ., V1 . ' . ^ ° . . ,., . . ,r ..

motibus libens, qui onuntur ab imperio liberorum spmtuum, debet describere curvam
quandam lineam continuam, cujus determinatio reducitur ad hujusmodi probiema generale :
Dato numero punctorum materiae, ac pro singulis dato puncto loci, quod occupent dato
quopiam momento temporis, ac data directione, & velocitate motus initialis, si turn primo
projiciuntur, vel tangentialis, si jam ante fuerunt in motu, ac data lege virium expressa
per curvam aliquam continuam, cujusmodi est curva figurae I, quas meam hanc Theoriam
continet, invenire singulorum punctorum trajectorias, lineas nimirum, per quas ea moventur
singula. Id probiema mechanicum quam sublime sit, quam omnem humanae mentis
excedat vim, ille satis intelliget, qui in Mechanica versatus non nihil noverit, trium etiam
corporum motus, admodum simplici etiam vi praeditorum, nondum esse generaliter definitos,
uti monui num. 204, & consideret immensum punctorum numerum, ac altissimam curvae
virium tantis flexibus circa axem circumvolutae elevationem.

Quid curvae de- 385. Sed licet ejusmodi probiema vires omnes humanae mentis excedat ; adhuc tamen

no^iSbean^Pro! unusquisque Geometra videbit facile, probiema esse prorsus determinatum, & curvas

biema inversum ejusmodi fore omnes continuas sine ullo saltu, si in lege virium nullus sit saltus. Quin

s^ptis1tenvpusScufo immo & ^u^ arbitror, in ejusmodi curvis nee ullas usquam cuspides occurrere ; nam nodos

utcunque parvo. nullos esse consequitur ex eo, quod nullum materiae punctum redeat ad idem punctum

spatii, in quo ipsum aliquando fuerit, adeoque nullus habeatur regressus, qui tamen ad

nodum est necessarius. Hujusmodi curvae necessariae essent omnes, & mens, [177] <!U3e

tantum haberet vim, quanta requiritur ad ejusmodi problemata rite tranctanda, & intimius

perspiciendas solutiones (quas quidem mens posset etiam finita esse, si finitus sit punctorum

numerus, & per finitam expressionem sit data notio curvse exprimentis legem virium) posset

ex arcu continue descripto tempore etiam utcunque exiguo a punctis materiae omnibus

derivare ipsam virium legem, cum quidam finiti tantummodo positionum numeri fmitos

determinare possint numeros punctorum curvae virium, & arcus continuus legem ipsam

continuam : & fortasse solae etiam positiones omnium punctorum cum dato arcu continuo

percurso ab unico etiam puncto motu continuo, exiguo etiam aliquo tempusculo ad rem

prsestandam satis essent. Cognita autem lege virium & positione, ac velocitate, & directione

punctorum omnium dato tempore, posset ejusmodi mens praevidere omnes futures neces-

saries motus, ac status, & omnia Naturae phenomena necessaria, ab iis utique pendentia,

atque prsedicere : & ex unico arcu descripto a quovis puncto, tempore continuo utcunque

A THEORY OF NATURAL PHILOSOPHY 281

consists in a propensity for staying in a state of rest or of maintaining a uniform state of motion
in a straight line, unless some external force compels a change of this state. Now, since by
my Theory it is proved that the centre of gravity of any mass, composed of any number of
points disposed in any manner whatever, is bound to have this property, it is clear that
the same property can be deduced for all bodies ; £ by this it can also be understood why
bodies can be conceived to be collected & condensed at their centres of gravity.

383. Mobility is usually considered as one. of the general properties of bodies; & Mobility; quiesci-
indeed it follows immediately from the curve of forces. For, since this curve, by means of ^ty ^^P0^^
its ordinates, represents the propensity to approach or recede, it necessarily requires mobility, such thing in
or the possibility of motion, without which approach or recession can certainly not be ^ure as absolute
obtained. Now there are some, who ascribe quiescibility to bodies ; but I consider that

absolute rest, at any rate in Nature as it is at present constituted, is impossible, as I explained
in Art. 86. I think also that it must be excluded by the argument of infinite improbability,
which I used in the dissertation De Spatio, & Tempore, which I have mentioned so many
times already, & which I quote in this work as Supplement, § I ; in it I prove that the
case in which any point of matter occupies at any instant of time a point of space, which
at any other instant whatever either it or any other point whatever would occupy, is infinitely
improbable ; this, by considering the finite number of points of matter, & the infinite
number of instants of time possible, of that class for which there are an infinite number of
points in the same straight line ; this number of instants is considered twice, once when
any instant for any given point of matter is considered, & again when any instant is considered
in which any other point of matter was somewhere else ; when these are compared with
the number of points of a space which has extension in length, breadth &. depth, the latter
must be infinite of the third order with respect to those mentioned above. Finally, by
the motion of all bodies about a common centre of gravity, whether this is at rest or travelling
uniformly in a straight line, absolute rest is excluded from Nature.

384. In my opinion also, there is another property that excludes absolute rest, one Absolute rest is
which I consider is common also to all points of matter & to the centres of gravity of all the^continuity 'of
bodies ; namely, continuity of motion, with which I dealt in Art. 88 & elsewhere. Any all motions ; gene-
point of matter, setting aside free motions that arise from the action of arbitrary will, respect°to^t1 Wth
must describe some continuous curved line, the determination of which can be reduced to

the following general problem. Given a number of points of matter, & given, for each of
them, the point of space that it occupies at any given instant of time ; also given the direction
& velocity of the initial motion if they were projected, or the tangential velocity if they
are already in motion ; & given the law of forces expressed by some continuous curve,
such as that of Fig. i, which contains this Theory of mine ; it is required to find the path
of each of the points, that is to say, the line along which each of them moves. How difficult
this mechanical problem may become, how it may surpass all powers of the human mind,
can be easily enough understood by anyone who is versed in Mechanics & is not quite unaware
that the motions of even three bodies only, & these possessed of a perfectly simple law of
force, have not yet been completely determined in general, & then will consider an immense
number of points, & the extremely high degree of a curve of forces twisting round the axis
with so many sinuosities.

385. Now, although a problem of such a kind surpasses all the powers of the human What does not
intellect, yet any geometer can easily see thus far, that the problem is determinate, & that described byThe
such curves will all be continuous without any break in them, so long as there is no discon- points. The inverse
tinuity in the law of forces, Indeed, I think that, in such curves, there never occur curves"1' Described
any cusps ; for, it follows that there are no nodes, from the fact that no point of matter in any interval of
returns to the same point of space that it occupied at any time ; & thus there is none of

that regression which is necessary for a node. All the curves must be of this kind ; & a mind
which had the powers requisite to deal with such a problem in a proper manner & was
brilliant enough to perceive the solutions of it (& such a mind might even be finite, provided
the number of points were finite, & the notion of the curve representing the law of forces
were given by a finite representation), such a mind, I say, could, from a continuous arc
described in an interval of time, no matter how small, by all points of matter, derive the law
of forces itself ; for, any merely finite number of positions can determine a finite number
of points on the curve of forces, & a continuous arc the continuous law. Perhaps even
the positions of all the points, together with a given continuous arc traversed with continuous
motion by but a single one of them, & that too in an interval of time no matter how small,
would be sufficient to obtain a solution of the problem. Now, if the law of forces were
known, & the position, velocity & direction of all the points at any given instant, it would
be possible for a mind of this type to foresee all the necessary subsequent motions & states,
& to predict all the phenomena that necessarily followed from them. It would be possible
from a single arc described by any point in an interval of continuous time, no matter how

282

PHILOSOPHIC NATURALIS THEORIA

parvo, quern aliqua mens satis comprehenderet, eadem determinare posset reliquum omnem
ejusdem continuae curvae tractum utraque e parte in infinitum productum.
Cur ab humana 386. Nos eo aspirare non possumus, turn ob nostrae mentis imbecillatatem, turn quia

mente solvi non • . . r • i /

possit. Quid offi- ignoramus numerum, & positionem, ac motum punctorum smgulorum (nam nee motus
cjat ei determina- absolutos intuemur, sed respectivos tantummodo- respectu Telluris, vel ad summum

tioni hbertas : Har- • i . .r . • r • \

monia; pra:stabiiit<e respectu systematis planetani, vel systematis rixarum omnium) turn etiam, quia curvas
impugnatio. illas turbant liberi motus, quos producunt spirituales substantiae. Harmonia praestabilita

Leibnitianorum ejusmodi perturbationem tollit omnem, saltern respectu animae nostrae,
cum omne immediatum commercium demat inter corpus, & animam ; & id, quod tantopere
improbatum est in Theoria Cartesiana, quse bruta redegerat ad automata, ad homines
etiam ipsos transfer!, quorum motus a machina provenire omnes, & necessaries esse in ea
Theoria, facile constat : & quidem idcirco etiam mihi Theoria displicet plurimum, quam
praeterea si admitterem, nullam sane viderem, ne tenuissimam quidem rationem, quae
mihi suadere posset, praeter animam meam, cujus ideae per se, & sine ullo immediate nexu
cum corpore evolvantur, me habere aliquod corpus, quod motus ullos habeat, & multo
minus, ejusmodi motus esse conformes iis ideis, aut ullos alios esse homines, ullam naturam
corpoream extra me ; ad quae omnia, & multo adhuc pejora, mentem suis omnia momentis
librantem deducat omnino oportet ejusmodi sententia, quam promoveri passim, & vero
etiam recipi, ac usque adeo gliscere, quin & omnino tolerari, semper miratus sum.

saltu.

Motus Hberos om- 387. Censeo ieitur, & id intima vi, qua anima suarum [178! idearum naturam, &

nino ab anima • • • i • vi • i

progigni, sed non proprietates quasdam, atque ongmem novit, constare arbitror, motus liberos corpons ab
imprimi, nisi aequa- anima provenire : ac quemadmodum virium lex necessaria, in ipsa fortasse materiae natura

liter in partes* • i* « t • • 11 ^ • • i

oppositas, & sine slta> ejusmodi est ; ut juxta earn bma materiae puncta debeant ad se invicem accedere,
vel a se invicem recedere, determinata & quantitate motus, & directione per distantias ;
ita esse alias leges virium liberas animae, secundum quas debeant quaedam puncta materias
habentia ejusmodi dispositionem, quae ad vivum, & sanum corpus organicum requiritur,
ad ipsius animae nutum moveri ; sed hujusmodi leges itidem censeo requirere illud, ut
nulli materice puncto imprimatur motus aliquis, nisi alicui alteri imprimatur alius contrarius,
& aequalis, quod constat ex ipso nisu, quern semper exercemus in partes contrarias, juxta ea,
quae diximus num. 74 : ac itidem arbitror, & id ipsum diligent! observatione, & reflexione
facile colligitur, ejusmodi quoque motus imprimi non posse, nisi servata lege continuitatis
sine ullo saltu, quod si ab omnibus spiritibus observari debeat ; discedent quidem veri
motus a curvis illis necessariis, & a libera voluntatis determinatione pendebunt curvae
descriptae ; sed motuum continuitas nequaquam turbabitur.

Conclusion^ de- 388. Porro inde constat, cur in motibus nullum uspiam deprehendamus saltum, cur
exdu1ioPquieti™Um nullum materiae punctum ab uno loci puncto abeat ad aliud punctum loci sine transitu
per intermedia, cur nulla densitas mutetur per saltum, cur & motus reflexi, £ refracti fiant
per curvaturam continuam, ac alia ejusmodi, quas hue pertinent. Verum simul patebit
& illud, in cujus gratiam haec congessimus, nullam fore absolutam quietam, in qua nimirum
continuatus ille curvae descriptae ductus abrumpatur ea continuitate laesa nihilo minus,
quam laederetur, si curva continua desineret alicubi in rectam.

Aequalitas action-
is, & reactionis, &
ejus consectaria.

389. Jam vero ad actionis, & reactionis asqualitatem gradu facto, earn abunde deduximus
a num. 265. pro binis quibusque corporibus ex actione, & reactione aequalibus in punctis
quibuscunque. Cum nimirum mutuae vires nihil turbent statum centri gravitatis com-
munis, & centra gravitatis binarum massarum debeant cum ipso communi centre jacere
in directum ad distantias hinc, & inde reciproce proportionales ipsis massis, ut ibidem
demonstravimus ; consequitur illud, motus quoscunque, quos ex mutua actione habebunt
binarum massarum centra gravitatis, debere fieri in lineis similibus, & proportionalibus
distantiae singularum ab ipso gravitatis centro communi, adeoque reciproce proportionalibus
ipsis massis ; & quod inde consequitur, summam motuum computatorum secundum
directionem quancunque, quam ex mutuis actionibus acquiret altera massa, fore semper
aequalem summae motuum computatorum secundum oppositam, quam massa altera acquiret
simul, in quo ipso sita est actionis & reactionis aequalitas, ex qua corporum [179] collisiones
deduximus in secunda parte, & ex qua multa phsenomena pendent, in Astronomia inprimis.

A THEORY OF NATURAL PHILOSOPHY 283

small, which was sufficient for a mind to grasp, to determine the whole of the remainder
of such a continuous curve, continued to infinity on either side.

386. We cannot aspire to this, not only because our human intellect is not equal to the why the problem
task, but also because we do not know the number, or the position & motion of each of these cannot be solved
points (for we do not observe absolute motions, but merely relative motions with respect intellect ;U what
to the Earth, or at most those with respect to the planetary system or the system of all obstacle to its

.1 t~ J \ o ^u i_ i i. t. £ • j j determination is

the fixed stars) ; & there is yet another reason, namely that the free motions produced due to freedom ;

by spiritual substances affect these curves. The " pre-established harmony " of the followers argument against

of Leibniz abrogates all such disturbing effect, at least as far as regards our will, since it does harmony!"

not admit any direct intercourse between body & spirit. What was so strongly condemned

in the theory of Descartes, which reduced animals to automata, is transferred to men as well ;

& it is easily shown that all their motions arise from a mechanism, & that these are necessary

upon that theory. For this reason, indeed, I am very much against the Cartesian theory ;

for, besides other things, if I admitted its principles, I should not be able to see any real

reason, nay, not of the slightest kind, which would lead me to think that, in addition to

my mind, ideas about which are evolved of itself & without any direct connection with

the body, I had a body that had motions ; much less, that these motions conformed to those

ideas, or that there were any other men, or any corporeal nature outside myself. Such a

philosophy must of necessity lead a mind that puts everything in the scales of its own

impulses to such absurdities, & still worse ; & I have always been astonished that this

philosophy has gained ground & has even been accepted everywhere, & up to the present

has been growing ; I am amazed that it should have been tolerated at all.

387. I think, therefore, that the free motions of bodies arise from the mind; & that Free motions are
this is due to an inner force, by which the mind knows the nature, certain properties & the byrtth"lymmddUbut
origin of its ideas, I think can be easily established. Just as we must have a law of forces, are not impressed
perhaps involved in the very nature of matter, of such a kind that according to it two points opposite ^irec'aons1
of matter must approach towards, or recede from, one another with a motion determined & without breach
in magnitude & direction by the distance between the points ; so there must be other of contmulty
free laws for the mind, according to which any points that have that disposition which a

living & healthy body requires, must obey the command of the mind. But such laws, I
also think, require the condition that a motion cannot be impressed on any point of matter,
unless an equal & opposite motion is impressed on some other point of matter ; this follows
from the stress that we always exert in opposite directions, according to what has been
said in Art. 74. Lastly, I consider, & the fact can be derived by diligent observation &
reflection, that such motion can not be impressed, unless it follows a law of continuity
without any break ; & if this law is bound to be observed by all object-souls, the real motions
will truly depart from the necessary curves, & the curves actually described will depend
on a free determination of the will ; but the continuity of the motions will not thereby
be affected.

388. Further, it is hence evident why we nowhere get any discontinuity in motions, Conclusions de-
why no point of matter can ever pass from one position to another without passing through £ h^eiciusion^o3!
all intermediate positions, why density can in no case be suddenly changed, why reflected absolute rest.

& refracted motions come about through continuous curvature, & other things of the sort
relating to the matter in hand. But, in particular, there will at the same time be evident
the fact, which is the purpose of all we have just done, namely, that there is no such thing
as absolute rest ; that is to say, such a thing as the sudden breaking off of the continuous
drawing of the curve described, the continuity being destroyed just as much as it
would be if a continuous curve finally became a straight line after reaching a certain
point.

389. Passing on to the equality of action & reaction, we have already, in Art. 265, Equality of action
fully proved its truth for any two bodies from the equality of the action & reaction between & reaction ; its

' J , _ . ' . « » * i ' rr i f consequences.

any two points, bor instance, since the mutual forces do not in any way affect the state of
the common centre of gravity, & the centres of gravity of two masses must lie in a straight line
with the common centre of the two, at distances on each side of the latter that are inversely
proportional to the masses, as was also proved in the same article ; it must follow that any
motions, which owing to mutual action are possessed by the centres of gravity of the two
masses, must take place along lines that are similar & proportional to the distances of each
from the common centre of gravity, & thus inversely proportional to the masses. Also it
then follows that the sum of the motions, reckoned in any direction, acquired by either of
the masses on account of the mutual actions, must always be equal to the sum of the motions
in the directly opposite direction, acquired simultaneously by the other mass ; & in this
is involved the equality of action & reaction ; & from it we deduced the laws of the
collisions of bodies in the second part ; & upon it depend many phenomena, especially in
Astronomy.

284

PHILOSOPHISE NATURALIS THEORIA

inde an motus

an ab externis.

390. Illud unum hie adnotandum censeo, per hanc ipsam legem comprobari plurimum
*Psas v*res mutuas inter materiae particulas, & deveniri ad originem motuum plurimorum,
' quae inde pendet ; si nimirum particulae massae cujuslibet ingentem habeant motum
reciprocum hac, iliac, & interea centrum commune gravitatis iisdem iis motibus careat ;
id sane indicio est, eos motus provenire ab internis viribus mutuis inter puncta ejusdcm
massae. Id vero accidit inprimis in fermentationibus, quae habentur post quarundam
substantiarum permixtionem, quarum particulse non omnes simul jam in unam feruntur
plagam, jam in aliam, sed singillatim motibus diversissimis, & inter se etiam contrariis,
quos idcirco motus omnes illarum centra gravitatis habere non possunt ; ii motus provenire
omnino debent a mutuis viribus, & commune gravitatis centrum interea quiescet respectu
ejus vasis, in quo fermentatio sit, & Terrae, respectu cujus quiescit vas.

Divisibmtas in in- 391. Quod ad divisibilitatem pertinet, earn quidem in infinitum progredientem sine

tinu!1"1 immaterial u^° ^mite m spatio continue ille solus non agnoscet, qui Geometriae etiam elementaris
itidem si sit con- vim non sentiat, a qua pro ejusmodi divisibilitate in infinitum tarn multa, & simplicia, &
Perspicua sane argumenta desumuntur. Ubi ad materiam sit transitus ; si, ubi de ea agitur,
quae distinctas occupant loci partes, distincta etiam sunt ; ab ilia spatii continui divisibilitate
in infinitum, materiae quoque divisibilitas in infinitum consequitur evidentissime, &
utcunque prima materiae elementa atomos, sive Naturae vi insectilia censeant multi, ut
& Newtonus ; adhuc tamen absolutam eorum divisibilitatem agnoscunt passim illi ipsi.

virtuaiem exten- 392. Materiae elementa extensa per spatium divisibile, sed omnino simplicia, & carentia

swnem non haben. partjijUS) admiserunt nonnulH e Peripateticis, & est etiam nunc, qui recentiorem Philoso-
phiam professus admittat ; at earn sententiam non ex praejudicio quodam, quanquam id
etiam est ingens, & commune, sed ex inductionis principio, & analogia impugnavi in prima
parte num. 83. Quamobrem arbitror, si quid corporeum extensionem habeat per totum
quodpiam continuum spatium, id ipsum debere absolute habere partes, & esse divisibile
in infinitum aeque, ac illud ipsum est spatium.

Puncta esse indi-
visibilia ; massas
divisibiles usque ad
certum limit em
singulas.

Componibilitas i n
infinitum.

Ejus

in infinitum.

393- At in mea Theoria, in qua prima elementa materiae mihi sunt simplicia, ac inex-
tensa, nullam, eorum divisibilitatem haberi constat. Massae autem, queecunque actu
existant, sunt mihi congeries punctorum ejusmodi numero finitae. Hinc eae congeries
dividi utique possunt in partes, sed non plures, quam sit ipse punctorum numerus massam
constituentium, cum nulla pars minus continere possit, quam unum ex iis punctis. Nee
Geometrica argumenta quidquam probant in mea Theo-[i8o]ria pro divisibilitate ultra
eum limitem ; posteaquam enim deventum fuerit ad intervalla minora, quam sit distantia
duorum punctorum, sectiones ulteriores secabunt intervalla ipsa vacua, non materiam.

394. Verum licet ego non habeam divisibilitatem in infinitum, habeo tamen componi-
bilitatem, ut appellare soleo, in infinitum. In quovis dato spatio habebitur quidem semper
certus quidam punctorum numerus, qui idcirco etiam finitus erit ; neque enim ego admitto
infinitum ullum in Natura, aut in extensione, neque infinite parvum in se determinatum,
quod ego positiva demonstratione exclusi primum in mea Dissertatione de Natura Iff usu
infinitorum, & infinite parvorum ; turn & aliis in locis ; quod tamen requireretur ad hoc,
ut intra finitum spatium contineretur punctorum numerus indefinitus : at longe aliter se
res habet ; si consideremus, qui numerus punctorum in dato spatio possit existere : turn
enim nullus est numerus finitus ita magnus, ut alius adhuc finitus ipso major haberi in eo
spatio non possit. Nam inter duo puncta quaecunque potest in medio interseri aliud,
quod quidem neutrum continget ; aliter enim etiam ea duo se contingerent mutuo, &
non distarent, sed compenetrarentur. Potest autem eadem ratione inter hoc noyum, &
priora ilia interseri novum utrinque, & ita porro sine ullo limite : adeoque deveniri potest
ad numerum punctorum quovis determinato utcunque magno majorem in unica etiam
recta, & proinde multo magis in spatio extenso in longum, latum, & profundum. Hanc
ego voco componibilitatem in infinitum. Numerus, qui in quavis data massa existit,
finitus est ; sed dum eum Naturae Conditor determinare voluit, nullos habuit limites,
quos non potuerit praetergredi, nullum ultimum habente terminum serie ilia possibilium
finitorum in infinitum crescentium.

sequivaientia ^95. Haec componibilitas in infinitum aequivalet divisibilitati in ordine ad explicanda

Naturae phaenomena. Posita divisibilitate materiae in infinitum, solvitur facile illud

A THEORY OF NATURAL PHILOSOPHY 285

390. I consider that in this connection it should be remarked that by means of this Hence, the point as
law especially the existence of these mutual forces between particles of matter is established, rn'otion^f"'! mass
& that in it we attain to the source of most of the motions, which arises from it. For arises from internal
instance, considering that the particles of a mass may have an immense reciprocal motion, or external forces,
whilst the common centre of gravity is without any such motion, surely that is a token

that these motions come from mutual internal forces between the particles of the mass.
Now, this takes place, in particular, in fermentations, such as are obtained after making
a mixture of certain substances ; here the particles of the substances are not all at the same
time moving first in one direction, then in another, but each of them separately in the
most widely diverging directions, & even in opposite directions, to one another. Hence,
as the centres of gravity cannot have all these motions, the motions must arise from mutual
forces ; &, besides, the common centre of gravity is at rest with regard to the vessel in
which the fermentation takes place, & also with regard to the Earth, with respect to which
the vessel is at rest.

391. Now, as concerning divisibility, that this can be carried on indefinitely without infinite divisibility
any limit in continuous space will be denied only by one who does not feel the force of o™^?1}.^ «"„?»" f

. r / / i i • i • i s>Pace • tne same ot

the most elementary principles of geometry; for, from it may be derived so many simple matter, if it is
& perfectly clear arguments in favour of such infinite divisibility. When we come to outtlvktuai&exten-
consider matter, if in dealing with it, we take it that what occupies a distinct part of space sion.
is itself distinct, then, from the infinite divisibility of continuous space, the infinite
divisibility of matter also follows very clearly ; &, although there are many who think that
the primary elements of matter are atoms, that is to say, things that are incapable of further
division by any Natural force, as Newton also thought, yet even they must still in all cases
admit their absolute divisibility.

392. Some of the Peripatetics admitted elements of matter extended through divisible Virtual extension is
space, but quite simple & without parts ; & at the present day there is one professing a E

more modern philosophy who admits such elements. This idea, in Art. 83 of the first
part of this work, I contradicted, not by the employment of any prejudgment, although
there certainly exists one that is very forcible & generally acknowledged, but by the
employment of the principle of induction & analogy. Hence, I think that, if anything
has corporeal extension throughout the whole of any continuous space, it must also absolutely
have parts & must be infinitely divisible, in exactly the same manner as the space is infinitely
divisible.

393. Now, in my Theory, in which the primary elements of matter are simple & non- Points are indivi-
extended, it is easily seen that there can be no divisibility of the elements. Also masses, sibie, whilst every

, ' ,, . _'..-.. mass is divisible up

in so far as they actually exist, are to me merely sets of such points finite in number, to a certain limit.

Hence these sets of points can at any rate be divided into parts, but not into a greater number

of points than that given by the number of points constituting the mass, since no part can

contain less than one of these points. Nor do geometrical arguments prove anything,

as far as my Theory is concerned, in favour of divisibility beyond this limit ; for, as soon

as we reach intervals that are less than the distance between two points, further sections

will cut these empty intervals & not matter.

394. Now, although I do not hold with infinite divisibility, yet I do admit infinite infinite componi-
componibility, as it is usually called. In any given space we can always have a certain

number of points ; & hence this number is finite. For, I do not admit anything infinite
in Nature, or in extension, or a self-determined infinitely small. Such a thing I excluded
by direct proof, for the first time in my dissertation De Natura, y usu infinitorum, y infinite
parvorum ; & later, in other writings ; this, however, is required, if an indefinite number
of points is to be included within a finite space. But the facts of the matter are quite
different, if we consider how great a number of points can exist within a given space ; for,
then there is no finite number so great, but that a still greater finite number can be had
within the space. For, between any two points it is possible to insert another midway,
which will touch neither of the former ; if this is not the case, then the two former points
must touch one another, & not be at a distance from one another, but compenetrated.
Further, in the same manner, between the new point & the first two points, we can insert
a new one on either side ; & so on without any limit. Thus we could arrive at a number
of points greater than any given number, no matter how large, all of them even lying in
a single straight line ; much more then would this be the case in space extended in length,
breadth & depth. This I call infinite componibility. The number of points present in
any given mass is finite ; but when the Creator of the Universe willed what that number
was to be, he had no limits ; for the series of possible finites increasing indefinitely has no
last term.

395. This infinite componibility is equivalent to divisibility for the purpose of explaining The equivalence of

i v r XT rr i • f • T • -1 •!• f componibility to

the phenomena of Nature, If we postulate infinite divisibility for matter, we have an easy infinite divisibility

286 PHILOSOPHIC NATURAL1S THEORIA

problema : Datam massam utcunque parvam, ita distribuere -per datum spatium utcunque
magnum, ut in eo nullum sit spatiolum majus dato quocunque utcunque parvo penitus vacuum,
y sine ulla ejus materice particula. Concipitur enim numerus, quo illud magnum spatium
datum continere possit hoc spatiolum exiguum, qui utique finitus est, & in se determinatus :
concipitur in totidem particulas divisa massula, & singulse particulse destinantur singulis
spatiolis ; qua iterum dividi possunt, quantum libuerit, ut parietes spatioli sui convestiant,
qui utique ad unam ejus transversam sectionem habent finitam rationem, adeoque continua
sectione planis parallelis facta possunt ipsi parietes convestiri segmentis suas particulse,
vel possunt ejus particulas segmenta iterum per illud spatiolum utcunque dispergi. In
[181] mea Theoria substituitur hujusmodi aliud problema : Intra datum spatiolum collocare
eum punctorum numerum, qui deinde disiribui possit per spatium utcunque magnum ita, ut
in eo nullum sit spatiolum cubicum majus dato quocunque utcunque parvo penitus vacuum, &
quod in se non habeat numerum punctorum utcunque magnum.

Demonstrate ea 3^5 Quocl jn ordine ad explicanda phaenomena hoc secundum problema sequivaleat

illi primo, patet utique : nam solum deest convestitio parietum continua mathematice :
sed illi succedit continuatio physica, cum in singulis parietibus collocari possit ejus ope
quicunque numerus utcunque magnus, distantiis idcirco imminutis utcunque. Quod in
mea Theoria secundum illud problema solvi possit ope expositse componibilitatis in infinitum,
patet : quia ut inveniatur numerus, qui ponendus est in spatiolo dato, satis est, ut numerus
vicium, quo ingens spatium datum continet illud spatiolum posterius multuplicetur per
numerum punctorum, quern velimus collocari in hoc ipso quovis posteriore spatiolo post
dispersionem, & auctor Naturae potuit utique intra illud spatiolum primum hunc punctorum
numerum collocare.

Divisibiiitas in 397. Jam quod pertinet ad divisibilitatem immanem, quam nobis ostendunt Naturas

Natura immams; phaenomena in coloratis quibusdam corporibus, immanem molem aquae inficientibus eodem
colore, in auro usque adeo ductili, in odoribus, & ante omnia in lumine, omnia mihi cum
aliis communia erunt ; & quoniam nulla ex observationibus nobis potest ostendere divisi-
bilitatem absolute infinitam, sed ingentem tantummodo respectu divisionum, quibus
plerumque assuevimus ; res ex meo problemate aeque bene explicabitur per componibilitatem
ac in communi Theoria ex illo alio per divisibilitatem materiae in infinitum.

immutabiiitas pri- *gg_ prima materiae elementa volunt plerunque immutabilia, & eiusmodi, ut atteri,

morum elemen- J' . . . . . \ . * , ,J „ XT

torum materiae : or- atque conirmgi ommno non possint, ne mrmrum pnaenomenorum ordo, & tota Naturae
dines diversi parti- facies commutetur. At elementa mea sunt sane eiusmodi, ut nee immutari ipsa, nee

cularum minus, ac , . . ,. , . . . J ., ,,

minus immuta- legem suam vinum, ac agendi modum in compositiombus commutare ullo modo possmt ;
cum nimirum simplicia sint, indivisibilia, & inextensa. Ex iis autem juxta ea, quse diximus
num. 239 ad distantias perquam exiguas collocatis in limitibus virium admodum validis
oriri possunt primae particulae minus jam tenaces suae formae, quam simplicia elementa,
sed ob ingentem illam viciniam adhuc tenacissimae idcirco, quod alia particula quasvis
ejusdem ordinis in omnia simul ejus puncta fere aequaliter agat, & vires mutuae majores
sint, quam sit discrimen virium, quibus diversa ejus puncta solicitantur ab ilia particula.
Ex hisce primi ordinis particulis possunt constare particulae ordinis secundi ; adhuc minus
tenaces, & ita porro ; quo enim plures compositiones sunt, & majores distantiae, eo facilius
fieri potest, ut inaequalitas [182] virium, quas sola mutuam positionem turbat, incipiat
esse major, quam sint vires mutuae, quae tendunt ad conservandam mutuam positionem,
& formam particularum ; & tune jam alterationes, & transformationes habebuntur, quas
videmus in corporibus hisce nostris, & quae habentur etiam in pluribus particulis postremorum
ordinum, haec ipsa nova corpora componentibus. Sed prima materiae elementa erunt
omnino immutabilia, & primorum etiam ordinum particulae formas suas contra externas
vires validissime tuebuntur.

Gravitas exhibita
a postremo arcu

curvffi accedens ad 399. Gravitas etiam inter generales proprietates a Newtonianis inprimis numeratur,

p^xfmT^oTs™ quibus assentior ; dummodo ea reipsa non habeat rationem reciprocam duplicatam
nostro concipiendi distantiarum extensam ad omnes distantias, sed tantum ad distantias ejusmodi, cujusmodi
' eae, quse interjacent inter distantiam nostrorum corporum a parte multo maxima

A THEORY OF NATURAL PHILOSOPHY 287

solution of the following problem. Distribute a given mass, however small, within a given
space, however large, in such a manner that there shall be no little space in it greater than any
given one, no matter how small, that shall be quite empty, W without any particle of that matter.
For we assume a certain number to represent the number of times the large given space
can contain the exceedingly small space, this number being in every case finite & self-
determined ; we assume the mass to be divided into the same number of particles, & one
of the particles to be placed in each of the small spaces. The former can again be divided,
as much as is desired, so that the new parts of each particle cover the boundary walls of
the corresponding small space ; & these in every case bear a finite ratio to one transverse
section of it, so that, by making continuous sections with parallel planes, these boundary
walls can be covered each with segments of the particle corresponding to it ; or the segments
of a particle can be scattered in any manner throughout the small space, repeating the
above process. In my Theory another problem is substituted, such as the following : —
Place within a given small space such a number of points, that these can then be distributed
throughout any space, however great, in such a manner that there shall be no little cubical space
in it greater than any given one, however small, that shall be quite empty, & which does not
contain in itself any number of points however great.

396. It is quite clear that, for the purpose of explaining the phenomena of Nature, Demonstration.
the second problem is equivalent to the first ; for, the only thing that is wanting in it is

a continuous covering of the boundary walls, in a strictly mathematical sense ; & instead
of this we have a physical continuity, since in each of the walls there can be placed by means
of it any number of particles, however great, & therefore at distances from one another
which are indefinitely diminished. It is also clear that, in my Theory, the second problem
can be solved by the employment of the infinite componibility that I have explained ;
for, in order to find the number to be placed in a given small space, it is sufficient that the
number of times that the large given space contains the latter small space should be multiplied
by the number of points which we desire to be placed in this latter small space after
dispersion ; & certainly the Author of Nature was able to place this number of points within
that first small space.

397. Now, as regards the immense divisibility, which the phenomena of Nature present The immense divi-
to us in certain coloured bodies, when they stain an immense volume of water with the same Slbl Nature.
colour, in the extremely great ductility of gold, in odours, & more than all in light, everything

will be in agreement in my Theory with the theories of others. Moreover, since no
observations can show us any divisibility that is absolutely infinite, but only such as is
immensely great when compared with such divisions as we are for the most part accustomed
to ; it follows that the matter can be explained just as well from my problem by means
of componibility, as in the usual theory it can be from the other problem by the infinite
divisibility of matter.

398. The primary elements of matter are considered by most people to be immutable, immutability of
& of such a kind that it is quite impossible for them to be subject to attrition or fracture, ments^o^matte'ri
unless indeed the order of phenomena & the whole face of Nature were changed. Now, different kinds of
my elements are really such that neither themselves, nor the law of forces can be changed ; ^immutable3 &
& the mode of action when they are grouped together cannot be changed in any way ; for

they are simple, indivisible & non-extended. From these, by what I have said in Art.

239, when collected together at very small distances apart, in sufficiently strong limit-points

on the curve of forces, there can be produced primary particles, less tenacious of form

than the simple elements, but yet, on account of the extreme closeness of its parts, very

tenacious in consequence of the fact that any other particle of the same order will act

simultaneously on all the points forming it with almost the same strength, & because the

mutual forces are greater than the difference between the forces with which the different

points forming it are affected by the other particle. From such particles of the first order

there can be formed particles of a second order, still less tenacious of form ; & so on. For

the greater the composition, & the larger the distances, the more readily can it come about

that the inequality of forces, which alone will disturb the mutual position, begins to be

greater than the mutual forces which endeavour to maintain that mutual position, i.e. the

form of the particles. Then indeed we shall have changes & transformations, such as we

see in these bodies of ours, & which are also obtained in most of the particles of the last

orders, which compose these new bodies. But the primary elements of matter will be Gravity, as repre-

quite immutable, & particles of the first orders will preserve their forms in opposition to sente.d ,,bv the last

, ' - . , arc of the curve, ap-

even very strong forces from without. proximates to that

399. Gravity also is counted as a general property, especially by followers of Newton ; siv<rn by the New-

, T J'' , , ' .. ° . . * '' i }. . .' toman law; possi-

& 1 am ot the same opinion, so long as it is not supposed to be in the inverse ratio of biiity of its being

the squares of the distances for all distances, but merely for distances such as those that lie ^^nthet Sam1e>

n
between the distance of our bodies from the far greatest part of the mass of the Earth, hypothesis.

288 PHILOSOPHIC NATURALIS THEORIA

massae terrestris, & distantias a Sole apheliorum pertinentium ad cometas remotissimos,
& dummodo in hoc ipso tractu sequatur non accuratissime, sed, quam libuerit, proxime,
rationem ipsam reciprocam duplicatam, juxta ea, quae diximus num. 121. Ejusmodi
autem gravitas exhibetur ab arcu illo postremo meae curvae figurae i, qui, si gravitas exten-
ditur cum eadem ilia lege ad sensum, vel cum aliqua simili, in infinitum, erit asymptoticus.
Posset quidem, ut monui num. 119, concipi gravitas etiam accurate talis, quae extendatur
ad quascunque distantias cum eadem lege, & praeterea alia quaedam vis exposita per aliam
curvam, in quam vim, & in gravitatem accurate reciprocam quadratis distantiae resolvatur
lex virium figurae I ; quae quidem vis in illis distantiis, in quibus gravitas sequitur quam
proxime ejusmodi legem, esset insensibilis ; in aliis autem distantiis plurimis ingens esset :
ac ubi figura I exhibet repulsiones, deberet esse vis hujus alterius conceptae legis itidem
repulsiva tanto major, quam vis legis primitivae figurae i, quanta esset gravitas ibi concepta,
quae nimirum ab illo additamento vis repulsivae elidi deberet. Sed haec jam a nostro
concipiendi modo penderent, ac in ipsa mea lege primitiva, & reali, gravitas utique est
generalis materiae, ac legem sequitur rationis reciprocae duplicatas distantiarum, quanquam
non accurate, sed quamproxime, nee ad omnes extenditur distantias ; sed illas, quas exposui.

i irftoto 4°°' Ceterum gravitatem generalem haberi in toto planetario systemate, ego

soiari systemate, quidem arbitror omnino evinci iisdem argumentis ex Astronomia petitis, quibus utuntur
pressi(^SSfluidrbm Newtoniani, quae hie non repeto, cum ubique prostent, & quae turn alibi ego quidem
congessi pluribus in locis, turn in Adnotationibus ad poema P. Noceti De Aurora Boreali.
Illud autem arbitror evidentissimum, ilium accessum ad Solum cometarum, & planetarum
primariorum, ac secundariorum ad primaries, quem videmus in descensu a recta tangente
ad arcum curvae, & multo magis alios motus a mutua gravitate pendentes haberi omnino
[183] non posse per ullius fluidi pressionem ; nam ut alia praetermittam sane multa, id
fluidum, quod sola sua pressione tantum possit in ejusmodi globos, multo plus utique posset
occursu suo contra illorum tangentialem velocitatem, quae omnino deberet imminui per
ejusmodi resistentiam, cum ingenti perturbatione arearum, & totius Astronomiae Mechanicae
perversione ; adeoque id fluidum vel resistentiam ingentem deberet parere planetae, aut
cometae progredienti, vel ne pressione quidem ullum ipsi sensibilem imprimit motum.

Theoria respondere 4ai- Ejus autem praecipuae leges sunt, ut directe respondeat massae, & reciproce
massae directe, & quadratis distantiarum a singulis punctis massae ipsius. quod in mea Theoria est admodum

quadrate distantiae •/• • i • j MI

reciproce. mamfestum ita esse debere ; nam ubi ventum est ad arcum mum meae curvae, qui gravitatem

refert, vires omnes jam sunt attractivae, & eandem illam ad sensum sequuntur legem,
adeoque aliae alias non elidunt contrariis directionibus, sed summa earum respondet ad
sensum summae punctorum ; nisi quatenus ob inaequalem punctorum distantiam, &
positionem, ad habendam accurate ipsam summam, ubi moles sunt aliquanto majores,
opus erit ilia reductione, qua Mechanic! utuntur passim, & cujus ope inveniuntur leges,
secundum quas punctum in data distantia, & positione situm respectu massae habentis
datam figuram, ab ipsa attrahitur ; ubi, quemadmodum indicavimus num. 347, globus
in globum ita gravitat, ut gravitaret, si totae eorum massae essent compenetratae in eorum
centris : at in aliis figuris longe aliae leges obveniunt.

Th," datio 402. Verum hie illud maxime Theoriam commendat meam, quod num. 212 notandum

i neon**; ex con- f » • ...«. »» 1»

formitate omnium dixi, quod videamus tantam hanc conformitatem in vi gravitatis in omnibus massis ; licet
S?^mur. rn6?,^ eaedem in ordine ad alia phaenomena, quae a minoribus distantiis pendent, tantum discrimen

A *i**l 1 TVT

aliis. habeant, quantum habent diversa corpora in duntie, colore, sapore, odore, sono. JNam

diversa combinatio punctorum materiae inducit summas virium admodum diversas pro
iis distantiis, in quibus adhuc curva virium contorquetur circa axem ; proinde exigua
mutatio distantiae vires attractivas mutat in repulsivas, ac vice versa summis differentias
substituit ; dum in distantiis illis, in quibus gravitas servat quamproxime leges, quas diximus,
curva ordinatas omnes ejusdem directionis habet, & vero etiam distantia parum mutata,-
fere easdem ; quod necessario inducit tanta priorum casuum discrimina, & tantam in
hoc postremo conformitatem.

Omnia fere a gravi-

tate pendentia sunt . ,

communia huic 403. Distinctio gravitatis (quae est ut massa, in quam tenditur, directe, & quadratum

Theory cum ^com- j;,,^^ reciproce) a pondere (quod est praeterae ut massa, quae gravitat) est mihi eadem,
rum in ea faciiior ac Newtonianis, & omnibus Mechanicis ; £ ilia vim acceleratricem exhibet, hoc vim

i

deductio.

A THEORY OF NATURAL PHILOSOPHY 289

& the distances from the Sun of the aphelia of the most remote comets ; & so long as in
this region it is not assumed to follow the law of the inverse squares exactly, but only very
approximately to any desired degree of closeness, as I said in Art. 121. Now gravity of
this kind is represented by the last arc of my curve in Fig. I ; & this, if gravity goes on
indefinitely according to this same or any similar law, will be an asymptotic branch. Indeed,
it may be, as I remarked in Art. 119, assumed that gravity is even accurately as the inverse
square, & that it extends to all distances according to the same law, but that in addition
there is some other force represented by another curve ; then the law of forces of Fig. I
is to be resolved into this force & into gravity reckoned as being exactly as the inverse square
of the distance. This force, then, at those distances, for which gravity follows very
approximately such a law, will be an insensible force ; but at most other distances it would
be very great. Where Fig. i gives repulsions, the force that is assumed to follow this other
law would also have to be repulsive, & greater than the force, given by the law of the primitive
curve of Fig. i, by an amount equal to the supposed value of gravity at that place ; & this
must be cancelled by the addition of this repulsive force. However, this would depend
upon our manner of assumption ; & in this my own primitive & actual law, I consider that
gravity is indeed universal & follows the law of the inverse squares of the distances, although
not exactly, but very closely ; I consider that it does not extend to all distances, but only
to those I have set forth.

400. For the rest, that gravity exists universally throughout the whole planetary Gravity exists
system, I think is thoroughly demonstrated by those arguments derived from Astronomy whcTeUsoW° system6
which are used by the disciples of Newton ; these I do not repeat here, since they are set & it cannot possibly
forth everywhere ; I too have discussed them in several places, besides including them in
Adnotationes ad poema P. Noceti De Aurora Boreali. But I consider that it is most evident

that the approach to the Sun of the comets & primary planets, & that of the secondaries
to the primaries, such as we see in the descent from the rectilinear tangent to the arc of
the curve, & to a far greater degree other motions depending on mutual gravitation
cannot possibly be due to fluid pressure. For, to omit other reasons truly numerous,
the fluid that can avail so much in its action on spheres of this kind merely by its pressure,
would certainly have a much greater effect upon their tangential velocities, by its opposi-
tion ; these would in every case be bound to be diminished by such resistance, with a huge
perturbation of areas,- & the perversion of the whole of astronomical mechanics. Thus
the fluid would either be bound to set up a huge resistance to 'the progress of a planet or
a comet, or else it does not even by its pressure impress any sensible motion upon it.

401. Now, the principal laws of gravitation are that it varies directly as the mass & Gravitation, ac
inversely as the square of the distances from each of the points of that mass ; & in my xheo'ry varies"^!
Theory it is quite clear that this must be the case. For, as soon as we reach the arc of the mass directly
my curve that represents gravitation, all the forces are attractive, & to all intents obey *f "

the same law ; & so some of them do not cancel others in opposite directions, but their inversely.

sum approximately corresponds to the number of points. Except in so far as, on account

of the inequality between the distances of the points, & their relative positions, there will

be need, in order to obtain the sum of the forces accurately when the volumes are somewhat

large, to make use of the reduction usually employed by mechanicians ; by the aid of which

are found the laws according to which a point situated at a given distance & in a given position

from a mass of given shape is attracted by that mass. Here, as I indicated in Art. 347,

one sphere gravitates towards another sphere in the manner that it would if the whole

of their masses were for each condensed at their respective centres ; whilst for other figures

we meet with altogether different laws.

402. But the greatest support for my Theory lies in a statement in Art. 212, which I j"^0/^ ^ me
said ought to be noticed ; namely, in the fact that we see so much uniformity in all masses Theory from the
with regard to the force of gravity ; in spite of the fact that these same masses, for the conformity of ail

,• i i 6 i ' v r i 11 T i vrr bodies in having

purpose of other phenomena depending on the smaller distances apart, have differences gravitation, whilst
so great as those possessed by different bodies as regards hardness, colour, taste, smell & there are so many

„ ,.£ /. . . . . . . 11 vrr differences in other

sound. For, a different combination of the points of matter induces totally different sums properties.

for those distances up to which the curve of forces still twists about the axis ; where a very

slight change in the distances changes attractive forces into repulsive, & substitutes, vice versa,

differences for sums. Whereas, at those distances for which gravity obeys the laws we

have stated very approximately, the curve has its ordinates all in the same direction &,

even if the distance is slightly altered, practically unaltered in length. This of necessity Neariv everything

produces a huge difference in the former case, & a very great uniformity in the latter. depending on gray-

403. The distinction between gravitation (which is proportional to the mass on which |nyaOT^ement°with
it acts, directly, & as the square of the distance, inversely) & weight (which is, in addition, the usual theory :
proportional to the mass causing the gravitation) is just the same in my Theory as in that Oj

of Newton & all mechanicians. The former gives the accelerating force, the latter the motive easier in mine.

u

290

PHILOSOPHIC NATURALIS THEORIA

motricem, cum ilia determinet vim puncti gravitantis cujusvis, a qua pendet celeritas
massae ; [184] hoc summam virium ad omnia ejusmodi puncta pertinentium. Pariter
communia mihi sunt, quaecunque pertinet ad gravium motus a Galilaeo, & Hugenio definitos,
nisi quod gravitatis resolutionem in descensu per plana inclinata, & in gravibus sustentatis
per bina obliqua plana, vel obliqua fila, reducam ad compositionem juxta num. 284, & 286,
& centrum oscillationis, una cum centro Eequilibrii, & vecte, & libra, & machinarum principiis
deducam e consideratione systematis trium massarum in se mutuo agentium, ac potissimum
a simplici theoremate ad id pertinente, quae fuse persecutus sum a num. 307. Communia
pariter mihi sunt, quaecunque habentur in caelesti Newtoniana Mechanica jam ubique
recepta de planetarum, & cometarum motibus, de perturbationibus motuum potissimum
Jovis, & Saturni in distantiis minoribus a se invicem, de aberrationibus Lunae, de maris
aestu, de figura Telluris, de praecessione aequinoctiorum, & nutatione axis ; quin immo
ad hasc postrema problemata rite solvenda, multo tutior, & expeditior mihi panditur via,
quae me eo deducet post considerationem systematis massarum quatuor jacentium etiam
non in eodem piano communi, & connexarum invicem per vires mutuas, uti ad centrum
oscillationis etiam in latus in eodem piano, & ad centrum percussionis in eadem recta tarn
facile me deduxit consideratio systematis massarum trium.

cetur.

immobiiitas fix a- 404. Illud mihi prseterea non est commune, quod pertinet ad immobilitatem stellarum
iia fixarum, quam contra generalem Newtoni gravitatem vulgo solent objicere, quae nimirum
debeant ea attractione mutua ad se invicem accedere, & in unicam demum coire massam.
Respondent alii, Mundum in infmitum protendi, & proinde quamvis fixam aeque in omnes
partes trahi. Sed in actu existentibus infmitum absolutum, ego quidem censeo, haberi
omnino non posse. Recurrent alii ad immensam distantiam, quae non sinat motum in
fixis oriundum a vi gravitatis, n-e post immanem quidem saeculorum seriem sensu percipi.
li in eo verum omnino affirmant ; si enim concipiamus fixas Soli nostro aequales & similes,
vel saltern rationem luminum, quae emittunt, non multum discedere a ratione massarum ;
quoniam & vis ipsis massis proportionalis est, ac praeterea tarn vis, quam lumen decrescit
in ratione reciproca duplicata distantiarum ; erit vis gravitatis nostri Solaris systematis
in omnes Stellas, ad vim gravitatis nostrae in Solem, quae multis vicibus est minor, quam
vis gravitatis nostrorum gravium in Terram, ut est tota lux, quae provenit a fixis omnibus,
ad lucem, quae provenit a Sole, quae ratio est eadem, ac ratio noctis ad diem in genere lucis.
Quam exiguus motus inde consequi possit eo tempore, cujus temporis ad nos devenire
potuit notitia, nemo non videt. Si fixae omnes ad eandem etiam jaceant plagam, is motus
omnino haberi posset pro nullo.

Difficuitas residua 405. Adhuc tamen, quoniam nostra vita, & memoria respectu immensi fortasse subse-
subiata ab hac cuturi sevi est itidem fere nihil ; [185] si gravitas generalis in infmitum protendatur cum
eadem ilia lege, & eodem asymptotico crure, utique non solum hoc systema nostrum solare,
sed universa corporea natura ita, paullatim utique, sed tamen perpetuo ab eo statu recederet,
in quo est condita, & universa ad interitum necessario rueret, ac omnis materia deberet
demum in unicam informem massam conglobari, cum fixarum gravitas in se invicem, nullo
obliquo, & curvilineo motu elidatur. Id quidem haud ita se habere, demonstrari omnino
non potest ; adhuc tamen Divinae Providentise videtur melius consulere Theoria, quae
ejus etiam ruinse universalis evitandae viam aperiat, ut aperit sane mea. Fieri enim potest,
uti notavimus n. 170, ut postremus ille curvae meae arcus, qui exhibet gravitatem, posteaquam
recesserit ad distantias majores, quam sint cometarum omnium ad nostrum solare systema
pertinentium distantiae maximae a Sole, incipiat recedere plurimum ab hyperbola habente
ordinatas reciprocas quadratorum distantiae, ac iterum axem secet, & contorqueatur. Eo
pacto posset totum aggregatum fixarum cum Sole esse unica particula ordinis superioris
ad eas, quae hoc ipsum systema componunt, & pertinere ad systema adhuc in immensum
majus & fieri posset ut plurimi sint ejus generis ordines particularum ejusmodi etiam,
ut ejusdem ordinis particulae sint penitus a se invicem segregatae sine ullo possibili
commeatu ex una in aliam per asymptoticos arcus plures meae curvae juxta ea, quae
exposui a num. 171.

Cohiesio : expiicatio .Qg pjoc pacto difficultas qu33 a necessario fixarum accessu repetebatur contra

per quietem, vel _T T , Jf . . T. . . . . .

motus conspirantes. Newtomanam T heonam, in mea penitus evanescit ac simul a gravitate jam gradum fecimus
ad cohaesionem, quam ex generalibus materiae proprietatibus posueram postremo loco.

A THEORY OF NATURAL PHILOSOPHY 291

force ; since the former gives the force of any gravitating point, upon which depends
the velocity of the mass, & the latter the sum of all the forces pertaining to all such points.
Similarly, the agreement is the same in my Theory with regard to anything relating to
the motions of heavy bodies stated by Galileo & Huygens ; except that, in descent along
inclined planes, or bodies supported by two inclined planes or inclined strings, I substitute
for their resolution of gravity the principle of composition, as in Art. 284, 286 ; & I deduce
the centre of oscillation, as well as the centre of equilibrium, the lever, the balance & the
principles of machines from a consideration of three masses acting mutually upon one another ;
& this more especially by means of a simple theorem depending on that consideration,
which I investigated fully in Art. 307. The agreement is just as close in my Theory with
regard to anything occurring in the celestial mechanics of Newton, now universally accepted,
with regard to the motions of planets & comets, particularly the perturbations of the motions
of Jupiter & Saturn when at less than the average distances from one another, the aberrations
of the Moon, the flow of the tides, the figure of the Earth, the precession of the
equinoxes, & the nutation of the axis. Finally, for the correct solution of these latter
problems, a much safer & more expeditious path is opened to me, such as will lead me
to it after an investigation of the system of four masses, not even lying in the same common
plane, connected together by mutual forces ; just as the consideration of a system of three
masses led me with such ease to the centre of oscillation even to one side in the same
plane, & to the centre of percussion in the same straight line.

404. In addition to these, there is one thing in which I do not agree, namely, in that The manner in
which relates to the immobility of the fixed stars ; it is usually objected to the universal which the immo-
gravitation of Newton, that in accordance with it the fixed stars should by their mutual starJwas explained
attraction approach one another, & in time all cohere into one mass. Others reply to this, by Newton,
that the universe is indefinitely extended, & therefore that any one fixed star is equally

drawn in all directions. But in things that actually exist, I consider that it is totally impossible
that there can be any absolute infinity. Others fall back on the immense distance, which
they say will not permit the motion arising in the fixed stars from the force of gravity to
be perceived by the senses, even after an immense number of ages. In this they assert
nothing but the truth ; for if we consider the fixed stars equal & similar to our sun, or
at any rate the amounts of light that they emit, as not being far different from the ratio
of their masses ; then since also the force is proportional to the masses, & in addition both
force & light decrease in the inverse ratio of the squares of the distances, it must be that
the force of gravity of our solar system on all the stars is to the force of our gravity on the
Sun, which latter is many times less than the force of gravity of our heavy bodies on the
Earth, as the total light which comes from all the stars is to the light which comes from
the Sun ; & this ratio is the same as the ratio of night to day in respect of light. How
slight is the motion that can arise from this in the time (the comparatively short time
available for observation) nobody can fail to see. Even if all the fixed stars were
situated in the same direction, the motion could be considered as absolutely nothing.

405. However, since our period of life & memory, in comparison with the immense The remaining diffi-
number of ages perchance to follow, is almost as nothing, if universal gravitation th'is^heory"™7 "
extends indefinitely with the same law, & the same asymptotic branch, not only this solar

system of ours indeed, but the universe of corporeal nature, would, little by little in truth,
but still continuously, recede from the state in which it was established, & the universe
would necessarily fall to destruction ; all matter would in time be conglomerated into one
shapeless mass, since the gravity of the fixed stars on one another will not be cancelled by
any oblique or curvilinear motion. That this is not the case cannot be absolutely proved ;
& yet a Theory which opens up a possible way to avoid this universal ruin, in the way that
my Theory does, would seem to be more in agreement with the idea of Divine Providence.
For it may be that, as I remarked in Art. 170, the last arc of my curve, which represents
gravity, after it has reached distances greater than the greatest distances from the Sun of
all the comets that belong to our solar system, will depart very considerably from the hyperbola
having its ordinates the reciprocals of the squares of the distances, & once more will cut
the axis & be twined about it. In this way, it may be that the whole aggregate of the
fixed stars, together with the Sun, is a single particle of an order higher than those of
which the system is composed ; & that it belongs to a system immensely greater still. It
may even be the case that there are very many such orders of particles, of such a kind that
particles of the same class are completely separated from one another without any
possible means of getting from one to the other, owing to several asymptotic arcs to my
curve, as I explained in Art. 171.

406. In this way, the difficulty, which has been repeatedlybrought against the Newtonian Cohesion ; expiana-
theory on account of this necessary mutual approach of the fixed stars, disappears altogether ^"orof motions ?n
in my Theory. At the same time, we have now passed on from gravity to cohesion, which the same direction.

292 PHILOSOPHISE NATURALIS THEORIA

Cohsesionem explicuerunt aliqui per puram quietam ut Cartesian! alii per motus conspir-
antes, ut Joannes Bernoullius, ac Leibnitius, quam explicationem illustrarunt exemplo
illius veli aquse, quod in fontibus quibusdam cernimus, quod velum sit tantummodo ex
conspirante motu guttularum tenuissimarum, & tamen si quis digito velit perrumpere, eo
majorem resistentiam sentit, quo velocitas aquae effluentis est major, ut idcirco multo
adhuc major conspirantis motus velocitas videatur nostrorum cohsesionem corporum
exhibere, quae non nisi immani vi confringimus, ac in partes dividimus. Utraque explicandi
ratio eodem redit, si quietis nomine intelligatur non utique absoluta quies, quse translata
Tellure a Cartesianis nequaquam admittebatur, sed respectiva : nam etiam conspirantes
motus nihil sunt aliud, nisi quies respectiva illarum partium, quse conspirant in motibus.

nias exponere 407. At neutra earn explicat, quam cohsesionem reipsa dicimus, sed cohsesionis quendam

velut effectum. Ea, quse cohserent, utique respective quiescunt, sive motus conspirantes
habent, & id quidem ipsum in hac mea Theoria accidit [186] itidem, in qua cum singula
puncta materiae suam pergant semper eandem continuam curvam describere, ea, quse
cohserent inter se, toto eo tempore, quo cohserent, arcus habent curvarum suarum inter
se proximos, & in arcubus ipsis conspirantes motus. Sed in iis, quse cohaerent, id ipsum,
quod motus ibi sint conspirantes, non est sine causa pendente a mutuis eorum viribus, quse
causa impediat separationem alterius ab altero, ac in ea ipsa causa stat discrimen cohaeren-
tium a contiguis. Si duo lapides in piano horizontali jaceant, utique habent motum
conspirantem, quern circa Solem habet Tellus ; sed si tertius lapis in alterutrum incurrit,
vel ego ipsum submoveo manu, statim sine ulla vi mutua, quae separationem impediat,
dividuntur, & motus desinit esse conspirans. Hanc ipsam quasrimus causam, dum in
cohaesionem inquirimus. Nee velocitas motus, & exemplum veli aquse rem conficit.
Motus conspirans duorum lapidum contiguorum cum tota Tellure est utique multo velocior,
quam motus particularum aquse proveniens a gravitate in illo velo, & tamen sine ullo,
difficultate separantur. In aqua experimur difficultatem perrumpendi velum, quia ilia
motus conspirans non est communis etiam nobis & Telluri, ut est motus illorum lapidum ;
unde fit, ut vis, quam pro separatione applicamus singulis particulis, perquam exiguo
tempore possit agere, & ejus effectus citissime cesset, iis decidentibus, & supervenientibus
semper novis particulis, quse cum tota sua ingenti respectiva velocitate incurrunt in digitum.
At in corporibus, in quibus partes cohaerentes cernimus, eae partes nullam habent veloci-
tatem respectivam respectu nostri, nee alise succedunt aliis fugientibus. Quamobrem
longe aliter in iis se res habet, & oportet invenire causam longe aliam, prseter ipsum solum
conspirantem motum, ut explicetur difficultas, quam experimur in iis separandis, & in
inducendo motu non conspirante.

Expiicatio petita 408. Sunt, qui adducant pressionem fluidi cujuspiam tenuissimi, uti pressio atmo-

cur^aThtoeri^non sphseras extracto acre ex hemisphaeriis etiam vacuis ipsorum separationem impedit vi
possit. respondente ponderi ipsius atmosphaerse, quse vis cum in vulgaribus cohassionibus, & vero

etiam in hemisphaeriis bene ad se invicem adductis, sit multis vicibus major, quam pondus
atmosphserse ipsius, quod se prodit in suspensione mercurii in barometris ; aliud auxilio
advocant tenuius fluidum. At inprimis ejus fluidi hypothesis precaria est ; deinde hue
illud redit, quod supra etiam monui, ubi de gravitatis causa egimus, quod nimirum meo
quidem judicio explicari nullo modo possit, cur illud fluidum, quod sola pressione tantum
potest, nihil omnino ad sensum possit incursu suo contra celerrimos planetarum, &
cometarum motus. Accedit etiam, quod distractio & compressio fibrarum, quae habetur
ante fractionem solidorum corporum, ubi franguntur appenso inferne, vel superne imposito
[187] pondere ingenti, non ita bene cum ea sententia conciliari posse videatur.

ExpUcatio New- 409. Newtonus adhibuit ad earn rem attractionem diversam ab attractione gravitatis,

iiTmtnimU^istM6 quanquam is quidem videtur earn repetere itidem a tenuissimo aliquo fluido comprimente ;

tils : cur admitti repetit certe sub finem Opticse a spiritu quodam intimas corporum substantias penetrante,

cujus spiritus nomine quid intellexerit, ego quidem nunquam satis assequi potui ; cujus

A THEORY OF NATURAL PHILOSOPHY 293

I had put in the last place amongst the general properties of matter. Some have explained
cohesion from the idea of absolute rest, for instance, the Cartesians ; others, like Johann
Bernoulli, & Leibniz, by means of equal motions in the same direction. They illustrate
the explanation by means of the film of water, which we see in certain fountains ; this
film is formed merely from the equal motions in the same direction of the tiniest little
drops, & yet, if anyone tries to break it with his finger, he feels a resistance that is the
greater, the greater the velocity of the effluent water ; so that from this illustration it
would seem that a far greater velocity of equal motion in the same direction would account for
the cohesion of the bodies around us, which we cannot fracture & divide up into parts
unless we use a huge force. Either of these methods of explaining the matter reduces to
the same thing, if by the term ' rest ' we understand not only absolute rest which, since
the Earth is in motion, has in no sense been admitted by the Cartesians, but also relative
rest. For, equal motions in the same direction are nothing else but the relative rest of the
parts that have equal motions in the same direction.

407. Neither of these ideas explains that which we call cohesion in a real sense, but ^ut these methods
only an effect of cohesion. Things which cohere are certainly relatively at rest ; or they effect 6&P an"t the
have equal motions in the same direction. This is exactly what happens in my Theory also ; cause of cohesion,
for, in it, since each point of matter always keeps on describing the same continuous curve

which is peculiar to itself, those points that cohere to one another, during the whole of the
time in which they cohere, have the arcs of their respective curves very near to one another,
& the motions in those arcs equal & in the same direction. But in points that cohere,
the fact that their motions are then equal & in the same direction is not without a cause ;
& this depends on their mutual forces, which prevent separation of one point from another ;
& in this cause is involved the difference between cohering & contiguous points. If two
stones lie in the same horizontal plane, they will in all cases have equal motions in the same
direction as the Earth has round the Sun ; but if a third stone strikes against either of them, or
if I move this third stone up to the others with my hand, immediately, without any mutual
force preventing separation, the two are divided, & the equal motion in the same direction
comes to an end. This cause of cohesion is just what we want to find, when we seek to investi-
gate cohesion ; & velocity of motion, or the example of the film of water will not effect the
solution. The equal motions in the same direction as the whole Earth, possessed by the two
contiguous stones, is certainly much greater than the motions of the particles of water
produced by gravity in the film ; & yet the two stones can be separated without any difficulty.
In the water we encounter a difficulty in breaking the film, because the equal motion in the
one direction is not common to us & the Earth,^as the motion of the stones is. Hence it
comes about that the force, which we apply to separate the several particles, can only act for an
exceedingly small interval of time ; & the effect of this force ceases very quickly, as those
particles continually fall away & fresh ones come up ; & these strike the finger with the whole
of their relatively huge velocity. But, in bodies in which we perceive coherent parts, those
parts have no relative velocity with regard to ourselves, nor as one part flies off does
another take its place. Therefore the matter has to be explained in a totally different
manner ; & we must find a totally different cause to the idea of mere equality of motion in
the same direction, in order to solve the difficulty that is experienced in separating the
parts & inducing in them motions that are not equal & in the same direction.

408. There are some who bring forward the pressure of some fluid of very small density Explanation sought

i . i , ,., 11 -II j fr°m flul<i pressure ;

as an explanation. Just as the pressure of the atmosphere, when the air has been abstracted why it is impossible
from a pair of hollow hemispheres, prevents them from being separated with a force that this should be

r , . • i f i • i • <• • i • i- the case.

corresponding to the weight of the atmosphere ; &, since this force in ordinary cohesions,
& indeed also in the case of two hemispheres that fit one another very well, becomes many
times greater than the weight of the atmosphere, as shown in the suspension of mercury
in the barometer, they invoke the aid of another fluid of less density. But, first of all,
the hypothesis of such a fluid is uncertain ; next, there here arises the same objection that
I remarked upon above, when discussing the cause of gravity. Namely, that, in my opinion
no manner of reason could be given as to why this fluid, which by its mere pressure could
produce so great an effect, had as far as observation could discern absolutely no effect on
the swiftest motions of planets & comets, owing to impact with them. Also there is this
point in addition, that the extension & compression of fibres, which takes place before
fracture in solid bodies, when they are broken by hanging a weight beneath or by setting
a weight on top of them, does not seem to be in much conformity with this idea.

409. Newton derived an explanation of the matter from an attraction of a different The reason why it
kind to gravitation ; although he indeed seems to seek to obtain this attraction from some j^m}™]i£|s^pia*a°
compressing fluid of very small density. In fact, he seeks to obtain it, at the end of his tion from attraction
Optics, from a ' spirit ' permeating the inmost substances of bodies ; but I never was able tances^a

to grasp clearly what he intended by the term ' spirit ' ; & even he confessed that the Newton.

294 PHILOSOPHIC NATURALIS THEORIA

quidem agendi modum & sibi incognitum esse profitetur. Is posuit ejusmodi attractionem
imminutis distantiis crescentem ita, ut in contactu sit admodum ingens, & ubi primigeniae
particulae se in planis continuis, adeoque in punctis numero infinitis contingant, sit infinities
major, quam ubi particulae primigeniae particulas primigenias in certis punctis numero
finitis contingant, ac eo minor sit, quo pauciores contactus sunt respectu numeri particu-
larum primigeniarum, quibus constant particulae majores, quae se contingunt, quorum
contactuum numerus cum eo sit minor, quo altius ascenditur in ordine particularum a
minoribus particulis compositarum, donee deveniatur ad hsec nostra corpora ; inde ipse
deducit, particulas ordinum altiorum minus itidem tenaces esse, & minime omnium haec
ipsa corpora, quae malleis, & cuneis dividimus. At mihi positiva argumenta sunt contra
vires attractivas crescentes in infinitum, ubi in infinitum decrescant distantise, de quibus
mentionem feci num. 126; & ipsa meae Theoriae probatio evincit, in minimis distantiis
vires repulsivas esse, non attractivas, ac omnem immediatum contactum excludit : quam-
obrem alibi ego quidem cohaesionis rationem invenio, quam mea mihi Theoria sponte
propemodum subministrat.

Cohaesionem repe- 410. Cohaesio mihi est igitur iuxta num. 165 in iis virium limitibus, in quibus transit ur

tendam a limitibus i • • • • « j- •• j -L o i_ -j

virium. a vl repulsiva in minoribus distantiis, ad attractivam in majonbus ; & haec quidem est

cohaesio inter duo puncta, qua fit, ut repulsio diminutionem distantise impediat, attractio
incrementum, & puncta ipsa distantiam, quam habent, tueantur. At pro punctis pluribus
cohassio haberi potest, turn ubi singula binaria punctorum sunt inter se in distantiis limitum
cohaesionum, turn ubi vires oppositae eliduntur, cujusmodi exemplum dedi num. 223.

Cohassio duorum 411. Porro quod ad ejusmodi cohaesionem pertinet, multa ibi sunt notatu digna.

fes^o^aesionTs Inprimis ubi agitur de binis punctis, tot diversae haberi possunt distantiae cum cohaesione,
posse esse quot- quot exprimit numerus intersectionum curvae virium cum axe unitate auctus, si forte sit
fnrtp*6' mm™™!!!! impar, ac divisus per duo. Nam primus quidem limes, in quo curva ab arcu asymptotico

ceb, quo inque ... * . . r , . .. r. * .... i -i -i • i •

ordine positos. mo pnmo, sive a repulsionibus impenetrabilitatem exmbentibus transit ad primum
attractivum arcum, est limes cohaesionis, & deinde alterni intersectionum limites sunt non
cohaesionis, & coha2-[i88]sionis, juxta num. 179; unde fit, ut si intersectionum se conse-
quentium assumatur numerus par ; dimidium sit limitum cohaesionis. Hinc quoniam
in solutione problematis expositi num. 117 ostensum est, curvam simplicem illam meam
habere posse quemcunque demum intersectionum numerum ; poterit utique etiam pro
duobus tantummodo punctis haberi quicunque numerus distantiarum differentium a se
invicem cum cohaesione. Poterunt autem ejusmodi cohaesiones ipsae esse diversissimae
a se invicem soliditatis, ac nexus, limitibus vel validissimis, vel languidissimis utcunque,
prout nimirum ibi curva secuerit axem fere ad perpendiculum, & longissime abierit, vel
potius ad ilium inclinetur plurimum, & parum admodum discedat ; nam in priore eorum
casuum vires repulsivae imminutis, & attractivae auctis utcunque parum distantiis, ingentes
erunt ; in posteriore plurimum immutatis, perquam exiguae. Poterunt autem etiam e
remotioribus limitibus aliqui esse multo languidiores, & alii multo validiores aliquibus e
propioribus ; ut idcirco cohaesionis vis nihil omnino pendeat a densitate, sed cohaesio
possit in densioribus corporibus esse vel multo magis, vel multo minus valida, quam in
rarioribus, & id in ratione quacunque.

in massis numerus 412. Quae de binis punctis sunt dicta, multo magis de massis continentibus plurima,

ma^oVtU%robiema puncta, dicenda sunt. In iis numerus limitum est adhuc major in immensum, & discrimen

pro Us inveniendis utique majus. Inventio omnium positionum pro dato punctorum numero, in quibus tota

quomodo solve n- massa haberet limitem quendam virium, esset problema molestum, & calculus ad id

solvendum necessarius in immensum excresceret, existente aliquo majore punctorum

numero. Sed tamen data virium lege solvi utique posset. Satis esset assumere positiones

omnium punctorum respectu cujusdam puncti in quadam arbitraria recta ad arbitrium

collocati, & substitutis singulorum binariorum distantiis a se invicem in aequatione curvae

primae pro abscissa, ac valoribus itidem assumptis pro viribus singulorum punctorum pro

ordinatis, eruere totidem aequationes, turn reducere vires singulas singulorum punctorum

ad tres datas directiones, & summam omnium eandem directionem habentium in

quovis puncto ponere = o : orirentur aequationes, quae paullatim eliminatis valoribus

incognitis assumptis, demum ad aequationes perducerent definientes punctorum distantias

necessarias ad aequilibrium, & respectivam quietem, quae altissimae essent, & plurimas

A THEORY OF NATURAL PHILOSOPHY 295

mode of action was unknown to him. He supposed that there was such an attraction,
which, as the distances were diminished, increased in such a manner that at contact it
was exceedingly great ; & when the primary particles touched one another along continuous
planes, & thus in an infinite number of points, this attraction became infinitely greater
than when primary particles touched primary particles in a definite finite number of points ;
& the less the number of contacts compared with the number of primary particles forming
the larger particles which touch one another, the less the attraction becomes ; & since
the number of these contacts becomes smaller the higher we go in the orders of particles
formed from smaller particles, he deduces from this that particles of higher orders are
also of less tenacity, & the least tenacious of all are those bodies that we can divide with
mallet & wedge. But in my opinion there are positive arguments against attractive forces
increasing indefinitely, when the distances decrease indefinitely, as I remarked in Art. 126 ;
the very demonstration of my Theory gives convincing proof that the forces at very small
distances are repulsive, not attractive, & ezcludes all immediate contact. So that I find
the cause of cohesion from other sources ; & my Theory supplies me with this cause almost
spontaneously.

410. Cohesion, then, in rny opinion is, as I have said in Art. 165, to be ascribed to Cohesion is to be
the limit-points on the curve of forces, where there is a passage from a repulsive force at umit^oints^oiAhe
a smaller distance to an attractive force at a greater distance ; that is to say, this is the curve of forces,
cause of cohesion between two points, for here a repulsion prevents decrease, & attraction

increase, of distance ; & so the points preserve the distance at which they are. Cohesion
for more than two points can be obtained, both when each of the pairs of points is at a
distance corresponding to a limit-point of cohesion, & also when the opposite forces
cancel one another, an example of which I gave in Art. 223.

411. Further, with regard to such cohesion, there are many points that are worthy Cohesion of two
of remark. First of all, in connection with two points, we can have as many different ^ints^f' cohesion
distances corresponding with cohesion as is represented by the number of intersections can be anything
of the curve of forces with the axis (increased by one if perchance the number is odd) divided numte?*steJnjthd&
by two. For the first limit-point, at which the curve passes from the first asymptotic arc, orderof occurrence,
i.e., from repulsions that represent impenetrability, to the first attractive arc, is a limit-
point of cohesion ; & after that the points of intersection are alternately limit-points of
non-cohesion & cohesion, as was shown in Art. 179. Hence it comes about that, if the

number of intersections following one after the other are assumed to be even, half of them are
limit-points of cohesion. Hence, since, in the solution of the problem given in Art. 117,
it was shown that that simple curve of mine could have any number of intersections, it
will be possible for two points only to have any number of different distances from one
another that would correspond to limit-points of cohesion. Moreover these cohesions
could be of very different kinds, as regards solidity & connection, the limit-points being
either very strong or very weak ; that is to say, according as the curve at these points was
nearly perpendicular to the axis & departed far from it, or on the other hand was much
inclined to the perpendicular & only went away from the axis by a very small amount.
For, in the first case, the repulsive forces on diminishing the distances, or the attractive
forces on increasing the distances, ever so slightly, will be very great ; in the second
case, even when the distances are altered a good deal, the forces are very slight. Again
also, it is possible that some of the more remote limit-points would be much weaker, &
others much stronger, than some of the nearer limit-points. Thus, with me, the force of
cohesion is altogether independent of density ; the strength of cohesion, in denser bodies,
can be either much greater or much less than that in less dense bodies, & the ratio can be
anything whatever.

412. What has been said concerning two points applies also & in a far greater degree in masses the
to masses made up of a large number of points. In masses, the number of limit-points poTn/s is m'vufh
is immensely greater still, & the difference between them is greater in every case. The greater; how the
finding of all the positions for a given number of points, at which the whole mass has a t^enTis
limit-point of forces, would be a troublesome undertaking ; & the calculation necessary for its solved,
solution would increase immensely in proportion to the greater number of points taken.

However, it can certainly be solved, if the law of forces is given. It would be sufficient to
assume the positions of all the points with respect to any one point in any arbitrary straight
line in any arbitrary way, & having substituted the distances for each pair from one another
for the abscissa in the equation of the primary curve, & taking the values of the forces for
each of the points as ordinates, to make out as many equations ; then to resolve each of
the forces into three chosen directions, & to put the sum of all those in the same direction
for any point equal to zero. We shall thus obtain equations which, as the unknown assumed
values are one by one eliminated, will finally lead to equations determining the distances of
the points necessary for equilibrium, & relative rest ; but these would be of very high

296 PHILOSOPHISE NATURALIS THEORIA

haberent radices ; nam sequationes, quo altiores sunt, eo plures radices habere possunt, ac
singulis radicibus singuli limites exhiberentur, vel singulae positiones exhibentes vim nullam.
Inter ejusmodi positiones illse, in quibus repulsioni in minoribus distantiis habitae, succe-
derent attractiones in majoribus, exhiberent limites cohaesionis, qui adhuc essent quam
plurimi, & inter se magis diversi, quam limites ad duo tantummodo pun-[i89]cta pertin-
entes ; cum in compositione plurium semper utique crescat multitude, & diversitas casuum.
Sed haec innuisse sit satis.

Cur partes solid! AI*. Ubi confringitur massa aliqua, & dividitur in duas partes, quae prius tenacissime

fracti ad se invicem • 11 • • -n 11 i • i • •

appressae non acqui- inter se conserebant, si iterum mee partes adducantur ad se invicem ; conaesio prior non
rant cohaeskmem redit, utcunque apprimantur. Eius rei ratio apud Newtonianos est, quod in ilia divisione

priorem, ratio in j- 11 • i i -j • • j

Theoria Newton- non seque divellantur simul omnes particulse, ut textus remaneat idem, qui prius : sed
prominentibus jam multis, harum in restitutione contactus impediat, ne ad contactum
deveniant tarn multae particular, quam multae prius se mutuo contingebant, & quam multis
opus esset ad hoc, ut cohaesio fieret iterum satis firma : at ubi satis lse.viga.tze. binae superficies
ad se invicem apprimantur, sentiri primo resistentiam ingentem dicunt, donee apprimuntur ;
sed ubi semel satis appressae sint, cohasrere multis vicibus majore vi, quam sit pondus aeris
comprimentis ; quia antequam deveniatur ad eos contactus, haberi debet repulsiva vis
ingens, quam in majoribus distantiis, sed adhuc exiguis, agnovit Newtonus ipse, cui cum
deinde succedat in minoribus vis attractiva, quas in contactu evadat maxima, & in laevigato
marmore satis multi contactus obtineantur simul ; idcirco deinde satis validam cohcesionem
consequi. .

Ejusdem ratio in AIA. Quidquid ipsi de contactibus dicunt, id in mea Theoria dicitur aeque de satis

mea Theoria. ,. ,7 i • • T • -i r . . . . ,

vaiidis cohaesionis limitibus. In scabra superncie satis multae prommentes particulae
progressae ultra limites, in quibus ante sibi cohaerebant, repulsionem habent ejusmodi,
quae impediat accessum reliquarum ad limites illos ipsos, in quibus fuerant ante divulsionem.
Inde fit, ut ibi nimis paucae simul reduci possint ad cohaesionem particulse, dum in
laevigatis corporibus adducuntur simul satis multae. Ubi autem duo marmora, vel duo
quaecunque satis solida corpora, bene complanata, & laevigata sola appressione cohaeserunt
invicem, ilia quidem admodum facile divelluntur ; si una superficies per alteram excurrat
motu ipsis superficiebus parallelo ; licet motu ad ipsas superficies perpendiculari usque
adeo difficulter distrahi possint : quia particulae eo motu parallelo delatae, quae adhuc
sunt procul a marginibus partium congruentium, vires sentiunt hinc, & inde a particulis
lateralibus, a quibus fere aequidistant, fere aequales, adeoque sentitur resistentia earum
attractionum tantummodo, quas in se invicem exercent marginales particulae, dum augent
distantias limitum : nam mihi citra limitem quenvis cohaesionis est repulsio, ultra vero
attractio ; licet ipsi deinde adhuc aliae & attractiones, & repulsiones possint succedere.
Ubi autem 'perpendiculariter distrahuntur, debet omnium simul limitum resistentia
vinci.

Discrimen mass* 415. Nee vero idem accidit, ubi marmora integra, & nunquam adhuc divisa, inter se
fruSiretS'nf tevi- cohaerent ; turn enim fibrae possunt esse multae, quarum particulae adhuc in minori-[i9o]bus
gatisadse invicem distantiis, & multo validioribus limitibus inter se cohaereant, ad quos sensim devenerint
alise post alias iis viribus, quibus marmor induruit, ad quos nunc iterum reduci nequeant
omnes simul, dum marmora apprimuntur, quae ulteriorum limitum minus adhuc validorum,
sed validorum satis repulsivas vires simul sentiunt, ob quas non possunt denticuli, qui
adhuc supersunt perquam exigui post quamvis laevigationem, in foveolas se immittere, &
ad ulteriores limites validiores devenire ; praeterquam quod attritione, & Isevigatione ilia
plurimarum particularum ordinis proximi massis nobis sensibilibus inducitur discrimen
satis amplum inter massam solidam primigeniam, & binas massas complanatas, laevigatasque
ad se invicem appressas.

Distractio, & com- 4J6- Inde autem in mea Theoria satis commode explicatur & distractio, & compressio

pressio nbrarum fibrarum ante fractionem ; cum nimirum nihil apud me pendeat ab immediato contactu,
10 sed a limitibus, quorum distantia mutatur vi utcunque exigua : sed si satis validi sint, ad

A THEORY OF NATURAL PHILOSOPHY 297

degree & would have very many roots. For, the higher the degree, the more the roots
given by the equations ; & for each of the roots there would be a corresponding limit-point,
or a position representing zero force. Amongst such positions, those, in which we have
repulsion at a less distance followed by attraction at a greater distance, would yield limit-points
of cohesion ; & these would be as great in number & as different from one another as were
the limit-points pertaining to two points only ; for in a composition of several things there
certainly is always an increasing multitude & diversity of cases. But let it suffice that I
have called attention to these matters.

413. When a mass is broken, & divided into two parts, which originally cohered most xhe reason given
tenaciously, if the parts are again brought into contact with one another, the previous in the Newtonian

< ' f , j , rri, j- i • theory to account

cohesion does not return, however much they are pressed together. Ine reason of this, for the fact that the
according to the followers of Newton, is that in the division all the particles are not equally Parts of a broken

, ... , i_ r i e t solid, when brought

torn apart simultaneously, leaving the texture the same as before ; but as many of them closely together, do
now iut out bevond the rest, the contact between these in restitution prevents as many not attain their

. J, J . , i . i_ • • if i • i_ . J former cohesion.

particles coming into contact as there were touching one another originally, which number
is necessary for the purpose of again establishing a sufficiently strong cohesion. But when
two surfaces that are sufficiently well polished are brought closely together, they say that at
first there is felt a resistance of very great amount, until they are pressed into contact ;
but when once the surfaces are pressed together sufficiently closely, they cohere with a
force that is many times greater than that due to the weight of the air pressing upon them.
The reason they give is that, before actual contact is reached, there must be obtained a
very great repulsive force, such as Newton himself recognized as existing at comparatively
large, but actually very small, distances ; & after that, there followed an attractive force
at still smaller distances, which became exceedingly great when contact was reached. Thus,
in polished marble, a sufficiently great number of contacts was obtained simultaneously ;
& in consequence a comparatively great cohesion was obtained.

414. All that the Newtonians say with regard to contacts applies in my Theory equally The reason for the
well with regard to sufficiently strong limit-points of cohesion. In a rough surface, a ?amf thing accord-

,. .... -111 TIT T lng to my Theory.

sufficient number of jutting particles, pushed out beyond the distances corresponding to
those of the limit-points, at which they previously cohered, give rise to a repulsion of such
sort as prevents the other particles from approaching to the distances of the limit-points,
at which they were before being torn apart. Thus it comes about that in this case too
few of the particles can be brought into a state of cohesion ; whilst in the case of polished
bodies we have a sufficient number of particles brought together simultaneously. Moreover,
when two pieces of marble, or any two bodies of comparatively great solidity, after being
well smoothed & polished, cohere when they are merely pressed together, they can be forced
apart perfectly easily. If, for instance, one surface traverses the other with a motion parallel
to the surfaces ; although they can with difficulty be torn apart with a motion perpendicular
to the surfaces. For, particles carried along by this parallel motion, such as are still far
from the marginal surfaces of the parts in contact, feel the effects of forces on one side &
on the other, due to laterally situated particles from which they are nearly equidistant,
that are nearly equal to one another ; & thus resistance is only experienced from the
attractions which the particles in the marginal surfaces exert upon one another, whilst
they increase the distances of the limit-points. The reason is that with me there is repulsion
on the near side of any limit-point of cohesion, & attraction on the far side ; although
thereafter still other attractions & repulsions may follow. But when the bodies are
drawn apart perpendicularly, the resistance due to every limit-point must be overcome
simultaneously.

415. The same arguments do not apply to the case of whole pieces of marble that Distinction be-
have not as yet been broken at any time, when they cohere. For, in that case, there may *we^itive mass °&
be many filaments, the particles of which hitherto have been cohering at less distances & two pieces that
in much stronger limit-points ; these limit-points they would gradually reach one after h*ve be<;",,b5oke£

• i i c 11 • i i • i i 11 i , o«. polished &

the other with the forces that have given the marble its hardness ; but they cannot be reduced pressed together.

to them once more all at once, whilst the pieces of marble are being pressed together. At

the same time they feel the effect of the repulsive forces due to further limit-points still

less strong, but yet fairly powerful ; & on account of these, the little teeth which still are

left, though very small, after any polishing, cannot insert themselves into the little hollows,

& so reach the strong limit-points beyond. Besides, by this attrition & polishing of the

greater number of the particles of an order next to such masses as are sensible to us there

is induced a sufficiently wide distinction between a primitive solid mass & two masses that

have been smoothed & polished & then pressed together. Distension & com-

416. Hence also, in my Theory, we can give a fairly satisfactory explanation of the pression of fibres
distension & compression of fibres that precedes fracture ; for, with me, everything depends ^gooVwroianatton
not on immediate contact, but on the limit-points, the distance of which is changed by from my Theory.

298 PHILOSOPHIC NATURALIS THEORIA

vincendam satis magno accessu omnem repulsionem, vel recessu attractionem, requiritur
satis magna vis : quae quidem repulsio, & attractio in aliis limitibus longe mihi alia est, tarn
si vis ipsa consideretur quam si consideretur spatii, per quod ea agit, magnitude, quae omnia
pendent a forma, & amplitudine arcuum, quibus hinc, & inde circa axem contorquetur
mea virium curva. Hinc in aliis corporibus ante fractionem compressiones, & distractiones
esse possunt longe majores, vel minores, & longe major, vel minor vis requiri potest ad
fractionem ipsam, quae vis, ubi distantiis immutatis, superaverit maximam arcus ulterioris
repulsivam vim in recessu, superatis multo magis reliquis omnibus posterioribus viribus
repulsivis ope celeritatis quoque jam acquisitae per ipsam vincentem vim, & per attractivas
intermixtas vires, quae ipsam juvant, defert particulas massam cqnstituentes ad illas distantias,
in quibus jam nulla vis habetur sensibilis, sed ad tenuissimum gravitatis arcum acceditur.

Hinc cur sol id a 417. Hinc autem etiam illud in mea Theoria commodius accidit, quam in communi,

der^pressaconfrin" quod in mea statim apparet, cur pila quaecunque utcunque solid! corporis post certa imposita

gantur. pondera confringatur, & confringatur etiam solidus globus utriaque compressus ; cum

multo magis appareat, quo pacto textus, & dispositio particularum necessaria ad summam

virium satis validam mutari possit, ubi omnia puncta a se irwicem distant in vacuo libero,

quam ubi continuae compactae partes se contingant, nee ulla mihi est possibilis solida pila,

quae Mundum totum, si vi gravitatis in certain plagam feratur totus, sustineat, ut in sententia

de continua extensione materiae pila perfecte solida utcunque tenuis ad earn rem abunde

sufficeret.

Communia esse 418. Hisce omnibus jam accurate expositis, communia mihi sunt ea omnia, qua;
'

it ™ pertinent ad methodos explorandi per [191] experimenta diversam diversorum corporum
qua? pertinent ad cohaesionis vim, quod argumentum diligenter, ut solet, excoluit Musschenbroekius, &

:^ comparand! resistentiam ad fractionem, ubi divisio fieri debeat divulsione perpendicular!
resist entiam ad ad superficies divellendas, ut ubi trabi vertical! ingens pondus appenditur inferne, cum
v^rsU°positionibds" resistentia, quas habetur, ubi circa latus suum aliquod gyrare debeat superficies, quae
divellitur, quod accidit, ubi extremae parti trabis horizontalis pondus appenditur ; quam
perquisitionem a Galileo inchoatam, sed sine ulla consideratione flexionis & compressionis
fibrarum, quae habetur in ima parte, alii plures excoluerunt post ipsum ; & in quibus omnibus
discrimina inveniuntur quamplurima. Illud unum hie addam : posse cohaesionem
ingentem acquiri ab iis, quae per se nullum haberent, nova materia interposita, ut ubi
cineres, qui oleis actione ignis avolantibus inter se inertes remanserunt, oleis novis in massam
cohaerentem rediguntur iterum, ac in aliis ejusmodi casibus ; sed id jam pendet a discrimine
inter diversas particulas, & massas, ac pertinet ad soliditatem explicandam inprimis, non
generaliter ad cohaesionem, de quibus jam agam gradu facto a generalibus corporum
proprietatibus ad multiplicem varietatem Naturae, & proprietates corporum particulars.

Discrimen inter 419. Et primo quidem se hie mihi offert ingens illud plurium generum discrimen,

particulas diversas, i 1*1 • . . • . t* . __•__ ' ^*^ ^ j:

dpuncto- <luod haberi potest inter diversas punctorum congeries, quae constituunt diversa genera
rum, a mole, a particularum corpora constituentium. Primum discrimen, quod se objicit, repeti potest
quaeSltapotettfigesse ^ '1PSO numero punctorum constituentium particular^, qui potest esse sub eadem etiam

cum quavis mole admodum diversus. Deinde moles ipsa diversa itidem esse potest, ac diversa densitas,
damd eam ret ut nimirum duae particular i>ec massam habeant, nee molem, n.ec densitatem aequalem.
Deinde data etiam & massa, & mole, adeoque data densitate media particulae ; potest
haberi ingens discrimen in ipsa figura, sive in superficie omnia includente puncta & eorum
sequente ductum. Possunt enim in una particula disponi puncta in sphaeram, in alia in
pyramidem, vel quadratum, vel triangulare prisma. Sumatur figura quaecunque, & in
eam disponantur puncta utcunque : tot erunt ibi distantiae, quot erunt punctorum binaria,
qui numerus utique finitus erit. Curva virium potest habere limites cohaesionis quot-
cunque, & ubicunque. Fieri igitur potest, ut limites iis ipsis distantiis respondeant, &
turn eam ipsam formam habebit particula, & ejus formse poterit esse admodum tenax.
Quin immo per unicam etiam distantiam cum repagulo infinitae resistentiae, orto a binis
asymptotis parallelis, & sibi proximis, cum area hinc attractiva, & inde repulsiva infinita,

A THEORY OF NATURAL PHILOSOPHY 299

any force, however small this force may be. If these are sufficiently strong, then, to overcome
all repulsion by a sufficient great approach, or all attraction by a similar recession, there will
be required a force that is sufficiently great for the purpose. This repulsion & attraction,
with me, varies considerably for different limit-points, both when the force itself is considered,
& when the magnitude of the space through which it acts is taken into account ; & all of
these things depend on the form & size of the arcs with which my curve of forces is twined
round the axis, first on one side & then on the other. Hence, in different bodies, there may
occur, before fracture takes place, compressions & distensions that are far greater or far
less, & a force may be required for that fracture that is far greater or far less ; & this force,
when the distances are changed, having overcome the maximum repulsive force of the
further arc as it recedes, would (all the rest of the repulsive forces due to the first arcs
having been overcome all the more by the help of the velocity already acquired through
the overcoming force, assisted by the attractive forces that come in between) carry off the
particles forming the mass to those distances, at which there is no sensible force, but the
arc of exceedingly small amplitude corresponding to gravity is reached.

417. Hence, more easily in my Theory than in the common theory, because in mine Hence the reason

<• A • j- -i £ i • i_ i"ii \- wny solid bodies

it follows immediately, we have an explanation as to the reason why any pillar whatever, win be broken

made of a solid body, is broken when certain weights are imposed upon it ; & also why under the pressure

a solid sphere is crushed when compressed on both sides. For, it is much clearer how the weight."

texture & disposition of the particles, necessary to give such a comparatively great sum of

forces, can be changed, if all the points lie apart from one another in a free vacuum,

than if we suppose continuous compact parts that touch one another ; nor can I imagine

as possible any solid pillar that would sustain the whole Universe, if by the force of gravity

the whole of it were borne in a given direction ; & yet in the common idea of continuous

extension of matter a pillar that was perfectly solid, of no matter what thinness, would be

quite sufficient to do this.

418. These matters having now been accurately explained, I proceed in the ordinary There are many
manner in all things that relate to methods of experimental investigation of the different betwee^my^heory
force of cohesion in different bodies, a mode of demonstration that Mussenbroeck assiduously & the usual one,
practised with his usual care ; & methods of comparing the resistance to fracture in the ^fvestiga^io^of the
case when division must take place by a fracture perpendicular to the surfaces to be broken, forces of cohesion
such as occur when a great weight is hung beneath a vertical beam, with the resistance ture^in^difie^ent
that is obtained in the case when the surface has to rotate about one of its sides, which is positions.

torn off, as happens when a weight is hung at the end of a horizontal beam. This
investigation, first started by Galileo, but without considering bending or the compression
of the fibres that takes place on the under side of the beam, was carried on by several others
after him ; & in all cases of these there are very great differences to be found. I will here
add but this one thing ; it is possible for a very great cohesion to be acquired by things,
which of themselves have no cohesion, by the interposition of fresh matter. For instance
in the case of ashes, which, after the oily constituents have been driven off by the action
of fire, remained inert of themselves ; but, as soon as fresh oily constituents have been
added, become once more a coherent mass ; & in other cases of like nature. But this really
depends on the distinction between different kinds of particles & masses, & refers to the
explanation of solidity in particular, & not to cohesion in general. With such things I
will now deal, passing on from general properties of bodies to the multiplicity & variety
of Nature, & to particular properties of bodies.

419. The first thing that presents itself is the huge difference, of many kinds, which Distinction be-
there can be amongst different groups of points such as form the different kinds of particles kinds6 'of particles
of which bodies are formed. The first difference that calls our attention can be derived arising from the
from the number of points that form the particle ; this number can be quite different "he^their^cTume!
within the same volume. Then the volume itself may be different, as also may the their density, their
density ; for, of course, two particles need not have either equal masses, equal volumes, fatter Anything is
or equal densities. Then, even if the mass & the volume be given, that is to say, possible, & any
the mean density of the particle is given, there may be a huge difference in shape, «"eteTadnfor°the
that is to say, in the surface enclosing all the points, & conforming with them. For, purpose of main-
the points in one particle may be disposed in a sphere, in another in a pyramid, or a *

square or triangular prism. Take any such figure, & suppose the points are disposed in
any particular manner whatever ; then there will be as many distances as there are pairs
of points, & their number will be finite in every case. The curve of forces can have any
number of limit-points of cohesion, & these can occur anywhere along it. Therefore it
must be the case that limit-points can be found to correspond to those distances, & on
account of these the particle will have that particular form, & can be extremely tenacious
in keeping that form. Indeed, through a single distance, with a restraint of infinite resistance,
arising from a pair of parallel asymptotes close to one another, having the area on one side

300

PHILOSOPHIC NATURALIS THEORIA

potest haberi in quavis massa cujuscunque figurae soliditas etiam infinita, sive vis, quse
impediret dispositionis mutationem non minorem data quacunque. Nam intra illam
figuram [192] posset inscribi continuata series pyramidum juxta num. 363 habentium
pro lateribus illas distantias nunquam mutandas magis, quam pro distantia binarum illarum
asymptotorum, & positis punctis ad singulos angulos, haberetur massa punctorum, quorum
nullum jaceret extra ejusmodi figuram, nee ullum adesset intra illam figuram, vel in ejus
superficie spatii punctum, a quo ad distantiam minorem ilia distantia data non haberetur
punctum materiae aliquod. Possent autem intra massam haberi hiatus ubicunque, &
quotcunque prorsus vacui, inscriptis in solo residue spatio pyramidibus illis, & in angulis
quibusvis posset haberi quivis numerus punctorum distantium a se invicem minus, quam
distent illae binae asymptoti, & quivis eorum numerus collocari posset inter latera, & facies
pyramidum. Quare posset variari densitas ad libitum. Sed absque eo, quod singulis
distantiis respondeant in curva primigenia singuli limites, vel singula asymptotorum binaria,
vel ullae sint ejusmodi asymptoti praeter illam primam, innumera sunt sane figurarum
genera, in quibus pro dato punctorum numero haberi potest aequilibrium, & cohaesionis
limes per elisionem contrariarum virium, ex solutione problematis indicati num. 412.
Hoc discrimen est maxima notatu dignum.

Discnmen in punc- ^20. Data etiam figura potest adhuc in diversis particulis haberi discrimen maximum

torum distribu- 11' T >i • • n- • i i

tione per figuram ob diversam distributionem punctorum ipsorum. oic in eadem sphsera possunt puncta
eandem. esse admodum inaequaliter distributa ita, ut etiam paribus distantiis ex altera parte sint

plurima, ex altera paucissima, vel in diversis locis superficiei ejusdem concentricae esse
congeries plurimae punctorum conglobatorum, in aliis eorum raritas ingens, & haec ipsa
loca possunt in diversis a centro distantiis jacere ad plagas admodum diver'sas in eadem
etiam particula, & in eadem a centro distantia esse in diversis particulis admodum diversis
modis distributa. Verum etiam si particulse habeant eandem figuram, ut sphaericam, &
in singulis circumquaque in eadem a centro distantia puncta aequaliter distributa sint ;
ingens adhuc discrimen esse poterit in densitate diversis a centro distantiis respondente.
Possunt enim in altera esse fere omnia versus centrum, in altera versus medium, in altera
versus superficiem extimam : & in hisce ipsis discrimina, tarn quod pertinet ad loca densi-
tatum earundem, quam quod pertinet ad rationem inter diversas densitates, possunt in
infinitum variari.

Discrimen in vi, qua 421. Haec omnia discrimina pertinent ad numerum, & distributionem punctorum in

rJtinere : posselsse diversis particulis : sed ex iis oriuntur alia discrimina praecipua, quae maximam corporum,
taiem, ut null a & phaenomenorum varietatem inducunt, quae nimirum pertinent ad vires, quibus puncta
l particulam constituentia agunt inter se, vel quibus tota una particula agit in totam alteram.
Possunt inprimis, & in tanta dispositionum varietate debent, [193] puncta constituentia
eandem particulam habere vires cohaesionis admodum inter se diversas, ut aliae multo
facilius, alias multo difficilius dispositionem mutent mutatione, quae aliquam non ita parvam
rationem habeat ad totum. Est autem casus, in quo possint puncta particulae cohserere
inter se ita, ut nulla finita vi nexus dissolvi possit, ut ubi adsint asymptotici arcus in curva
primitiva, juxta ea, quae persecutus sum num. 362.

alia se ,22- Discrimina autem virium, quas una particula exercet in aliam, debent esse adhuc

attranentes, ana ,'_.. •*•-. • i ju

repeiientes, alia plura. Inprimis ex num. 222 patet, fieri posse, ut una particula constans etiam duobus
inertes inter so. punctis tertium punctum in iisdem distantiis collocatum ab earum medio attrahat per
totum quoddam intervallum, vel repellat per idem intervallum totum, vel nee usquam in
eo repellat, nee attrahat, conspirantibus in primo casu binis attractionibus, in secundo binis
repulsionibus itidem conspirantibus, & in tertio attractione, & repulsione aequalibus se
mutuo elidentibus. Multo autem magis summa virium totius cujusdam ^particulae in
aliam totam in eadem etiam distantia sitam, si medium utriusque spectetur, erit pro diversa
dispositione punctorum admodum inter se diversa, ut nimirum in una attractiones praeva-
leant, in alia repulsiones, in alia vires oppositae se mutuo elidant. Inde habebuntur,
particulae in se invicem agentes viribus admodum diversis, pro diversa sua constitutione
& particulae ad sensum inertes inter se, quae quidem persecutus sum ipso num. 222.

A THEORY OF NATURAL PHILOSOPHY 301

attractive & on the other side repulsive, there can be obtained in any mass of any form
whatever a solidity that is also infinite, or a force that would prevent any change of disposition
of the particles equal to or greater than any given change. For within that form there
could be inscribed a continued series of pyramids, after the manner of Art. 363, having
for sides those distances which are never to be altered by more than that corresponding to the
distance between the pair of asymptotes. If the points are placed one at each of the angles,
there would be obtained a mass consisting of points no one of which would lie outside a
figure of this sort ; & no other point could get within that figure or occupy a point of space
on its surface, from which there would not be some point oi matter at a less distance than the
given distance. Further, within the figure, there may be any kind & any number of gaps
quite empty of points, the pyramids being described only in the remainder of the space ;
& at the angles there may be any number of points distant from one another less than the
distance between the asymptotes ; & there may be any number of them situated along the
sides & faces of the pyramids. Hence, the density can be varied to any extent. But, apart
from the fact that to each distance there corresponds a limit-point in the primary curve,
or that there are pairs of asymptotes, or any other asymptotes of the sort except the first,
there are really an innumerable number of kinds of figures, in which with a given number
of points there can be equilibrium, & a limit-point of cohesion due to the cancelling of
equal & opposite forces, as can be seen from the solution of the problem indicated in Art.
412. The following distinction is especially worth remark.

420. Even if the figure is given, there can still be obtained a great difference between Difference in the
different particles on account of the different disposition of the points that form it. Thus, p^tf ^thin^ne
in the same sphere, the points may be quite unequally distributed, in such a way that, same figure,
even at equal distances, there may be very many in one part & very few in another ; or

in different places on the same concentric surface there may be very many groups of points
condensed together, whilst in others there are very few of them ; these very places may
be at quite different distances in different places even within the same particle, & in different
particles at the same distance from the centre they may be distributed in ways that are
altogether different. Further, even if particles have the same figure, say spherical, & in
each of them, round about, & at the same distance from, the centre the points are distributed
uniformly ; yet even then there may be a huge difference in the density corresponding to
different distances from the centre. For, in the one, they may be all grouped near the
centre, in another towards the middle surface, & in a third close to the outer surface. In
these the differences, both as regards the positions of equal density, & also as regards the
ratio of the different densities, can be varied indefinitely.

421. All such differences pertain to the number & distribution of points in the different Difference in the
particles. From them arise the principal differences that are left for consideration ; these partidea try "to
lead to the greatest variety in bodies & in phenomena. Such as those that relate to the forces conserve their
with which the points forming a particle act upon one another, or the forces with which beT'suc'h that 'the
the whole of one particle acts upon the whole of another particle. First of all, the points particle can be
forming the same particle may, & in such a great variety of distribution must, have forces of fin^force!
cohesion that are quite different one from the other ; so that some of them much more

easily, & others with much more difficulty, change this distribution with a change that bears
a ratio to the whole that is not altogether small. There is also the case, in which the
points of a particle can cohere so strongly together that the connection between them
cannot be broken by any finite force ; this happens when we have asymptotic arcs in
the primary curve, as I showed in Art. 362.

422. Moreover we may have still more differences between the forces which one Some particles
particle exerts upon another particle. First of all, it is evident from Art. 222, that it may oneraanothrerJP&
happen that a particle consisting of even two points may attract a third point situated at s?me have no ac-
the same distances from the middle point of the distance between the two points throughout other.0"

the whole of a certain interval of space, or they may repel it throughout the whole of the
same interval, or neither repel or attract it anywhere ; in the first case we have a
pair of attractions that are equal & in the same direction, in the second case a pair of
repulsions that are also equal & in the same direction, & in the third case an attraction
& a repulsion that are equal to one another cancelling one another. Also, to a far greater
degree, the sum of the forces for the whole of any particle upon the whole of another particle
even when situated at this same distance, if the mean for each is considered, will be altogether
different from one another for a different distribution of the points. Thus, in one particle
attractions will prevail, in another repulsions, & in a third equal & opposite forces will
cancel one another. Hence there will be particles acting upon one another with forces
that are altogether different, according to the different constitutions of the particles ; &
there will be particles that are approximately without any action upon one another, such
as I investigated also in the above-mentioned Art. 222,

302 PHILOSOPHIC NATURALIS THEORIA

Particula: qua: in 423. Aliud discrimen admodum notabile inter ejusmodi particularum vires est illud,

repe'iiantPUTn1Saiiis <luod eadem particula ex altera parte poterit datam aliam particulam attrahere, ex altera

attrahant : quse rcpellere ; quin immo possunt esse loca quotcunque in superficie particulae etiam sphaericae,

quaTTir^umqua^e <luae alteram particulam in eadem a centre distantia sitam attrahant, quae repellant, quas

eandem vim exer- nihil agant ; cum nimirum in iis locis possint vel plura, vel pauciora esse puncta, quam in

aliis, & ea ad diversas a centre, & a se invicem distantias collocata. Inde autem & illud

fieri poterit, ut, quemadmodum in iis, quae vidimus a num. 231, unum punctum a duorum

aliorum altero attractum, ab altero repulsum, vi composita urgetur in latus, ita etiam una

particula ab una alterius parte attracta, & repulsa ab altera in altera directione sita, urgeatur

itidem in latus, & certam assecuta positionem respectu ipsius, ad earn tuendam determinetur,

nee consistere possit, nisi in ea unica positione respectu ipsius, vel in quibusdam determinatis

positionibus, ad quas trudatur ab aliis rejecta. Quod si particula sphasrica sit, & in omnibus

concentricis superficiebus puncta aequaliter distributa sint, ad distantias a se invicem perquam

exiguas ; turn ejus, & alterius ejus similis particulae vires mutuas dirigentur ad sensum

ad earum centra, & fieri poterit, ut in quibusdam distantiis se repellant mutuo, in aliis se

attrahant, quo casu habebitur quidem diffi-[i94]cultas in avellenda altera ab altera, sed

nulla difficultas habebitur in altera circa alteram circumducenda in gyrum, sicut si Terrae

superficies horizontalis ubique sit, & egregie laevigata ; globus ponderis cujuscunque

posset quavis minima vi rotari per superficiem ipsam, elevari non posset sine vi, quae totum

ipsius pondus excedat.

Quo minojes par- ,2, jn ^ac act}0ne unius particulae in aliam generaliter, quo particulas ipsas minorem

ticulae, eo difficihus . . ~ T . r . .. T i • ... ..

dissoiubiies. habuerint molem, eo minus cetens paribus perturbabitur earum respectiva positio ab alia

particula in data quavis distantia sita : nam diversitas directionis & intensitatis, quam
habent vires agentes in diversas ejus partes, quae sola positionem turbare nititur, viribus
asqualibus & parallelis nullam mutuae positionis mutationem inducentibus, eo erit minor,
quo distantiarum, & directionum discrimen minus erit : atque idcirco, quemadmodum
jam "exposui num. 239, inferiorum ordinum particulae difficilius dissolvi possunt, quam
particulae ordinum superiorum.

Dart!cuiasa oriri* ex 425' ^•sec quidem praecipue notatu digna mihi sunt visa inter particularum ex homo-

punctorum vicima ; geneis etiam punctis compositarum discrimina, quae tamen, quod ad vires pertinet, intra

beant°diflferre cor- admodum exiguos distantiarum limites sistunt : nam pro majoribus distantiis particularum

pora, qua: ex iis omnium vires sunt prorsus uniformes, uti ostensum jam est num. 212, nimirum attractivae

in ratione reciproca duplicata distantiarum ad sensum. Porro hinc illud admodum

evidenter consequitur, massas majores ex adeo diversis particulis compositas, nimirum

haec ipsa nostra majora corpora, quas sub sensum cadunt, debere esse adhuc multo magis

diversa inter sc in iis, quae ad eorum nexum pertinent, & ad phaenomena exhibita a viribus

se extendentibus ad distantias illas exiguas, licet omnia in lege gravitatis generalis, quae

ad illas pertinet majores distantias, conformia sint penitus, quod etiam supra num. 402

notandum proposui. De hoc autem discrimine, & de particularibus diversorum corporum

proprietatibus ad diversas pertinentium classes jam agere incipiam.

Quae natura soiido- ^26. Prima se mihi offerunt solida, & fluida, quorum discrimina quas sint, & quomodo

q'u'i'd* fn^soHdis a mea Theoria ortum ducant, est exponendum. Solida ita inter se connexa sunt, ut quem-

rigida. quid virgae Hbet aliquot particularum motum sequantur reliquae : promotae, si illae promoventur :

qu?dICviscosa, quid retractae, si illae retrahuntur : conversae in latus, si linea, in qua ipsae jacent, directionem

humida. mutet : & in eo soliditas est sita : porro ea dicuntur rigida ; si ingenti etiam adhibita vi

positio, quam habet recta ducta per duas quasvis particulas massas, respectu rectas, quae

jungit alias quascunque, mutari ad sensum non possit, sed ad inclinandam unam partem

oporteat inclinare totam massam, & basim, & quanvis ejus rectam eodem angulo ; nam

in iis, quas flexilia sunt, ut elasticas virgae, pars una directionem positionis mutat, & [195]

inclinatur, altera priorem positionem servante : & priora ilia franguntur, alia majore,

alia minore vi adhibita ; haec posteriora se restituunt. Fluida autem passim non

utique carent vi mutua inter particulas, immo pleraque exercent, & aliqua satis magnam,

repulsivam vim, ut aer, qui ad expansionem semper tendit, aliqua attractivam, & vel non

exiguam, ut aqua, vel etiam admodum ingentem, ut mercurius, quorum liquorum particulae

se in globum etiam conformant mutua particularum suarum attractione, & tamen separantur

admodum facile a se invicem majores eorum massae, ac aliquot partibus motus facile ita

imprimitur : ut eodem tempore ad remotas satis sensibilis non protendatur ; unde fit

A THEORY OF NATURAL PHILOSOPHY 303

423. There is another difference that is well worth while mentioning amongst forces Particles which at
of this sort, namely, that the same particle in one part may exert attraction on another &eataothersnatSt"racte-
particle, & repulsion from another part ; indeed, there may be any number of places in some which urge
the surface of even a spherical particle, which attract another particle placed at the same °^ ^^ic^exert
distance from the centre, whilst others repel, & others have no action at all. For, at these the 'same force to
places there may be a greater or less number of points than in other places, & these may Produce rotation.
be situated at different distances from the centre & from one another. Thus, just as we

saw for the cases considered in Art. 231, that it may happen that a point is attracted
by one of two points & repelled by the other, & be urged to one side by the force that is
the resultant of these two, so also one particle may be attracted by one part of another
particle, & repelled by another part situated in another direction, & also be urged to one
side ; & having gained a certain position with respect to it, is inclined to preserve that
position ; nor can it stay in any position with regard to the other except the one, or perhaps
in several definite positions, to which it is forced when driven out from others. But if
the particle is spherical, & the points are equally distributed in all concentric surfaces, at
very small distances from one another ; then the mutual forces of it & another similar
particle are directed approximately to their centres ; & it may happen that at certain
distances they repel one another, & at other distances attract one another ; & in the latter
case there will be some difficulty in tearing them apart, but none in making them rotate
round one another. Just as, if the Earth's surface was everywhere horizontal, & perfectly
smooth, a ball of any weight whatever could be made to rotate along that surface by using
any very small force, whereas it could not be lifted except by using a force which exceeded
its own weight.

424. In general, in this action of one particle on another, the smaller volume the The smaller the
particles have, the less, other things being equal, is their relative position affected by another ^ffficuity therels in
particle situated at any given distance from it. For the differences in the directions & breaking it up.
intensities of the forces acting on different parts of it (which alone try to alter their positions,

since equal & parallel forces induce no alteration of mutual position) will be the less, the
less the difference in the distances & directions. Hence, just as I explained in Art.. 239,
particles of lower orders will be broken Up with more difficulty than particles of higher
orders.

425. The things given above seemed to me to be those especially worthy of remark Differences arise
amongst the differences between particles formed from even homogeneous points, which pf iwfaita "^"one
yet remained, as far as forces are concerned, within certain very narrow limits. For, as another ; how much
regards greater distances, the forces of all the particles are quite uniform ; that is to say, formed "from the'm
they are attractive forces varying approximately as the inverse square of the distances, differ from one
Further, from them it follows perfectly clearly that greater masses, formed from these a

already composite particles of different sorts, that is to say, the bodies that lie about us of
considerable size, such as come within the scope of our senses, must be still much more
different from one another in matters that have to do with the ties between them, & with
the phenomena exhibited by forces extending over very small distances ; although all of
them are quite uniform as regards the law of universal gravitation, which pertains to greater
distances, a point to which I also called attention in Art. 402. But I will now start to consider
this difference & the particular properties of different bodies belonging to different classes.

426. The first matters that offer themselves to me for explanation are the differences ^hefl^re °hSoli^
that exist between solids & fluids & how these arise according to my Theory. Solids are so\i^1 are rigfd. &
so connected together that the motion of any number of the particles is followed by the what elastic rods ;

.. . H ...... -•'•, 111 • r i_ j what in fluids are

remaining particles; it the former move forward, so do the latter; it they are retracted, viSCOUSj &whatare

so are the rest ; if a line in which they lie changes its direction, they are moved to one watery.

side ; & in these facts solidity is defined.. Further, solids are said to be rigid, if the position

of a straight line drawn through any two particles of the mass cannot be sensibly changed

with regard to the straight line joining any other pair of particles by using even a very

large force ; but in order to incline any one part of the mass it is necessary to incline the

whole mass, the base, & any straight line in the mass at the same angle. For, in those that

are flexible, such as elastic rods, one part may change the direction of its position & be

inclined, whilst the rest maintains its original position. The first are broken by using in

some cases a greater, & in others a less, force ; whereas the latter recover their form. Now

fluids in every case do not lack mutual force between their particles throughout ; indeed

very many of them exert, & some of them a fairly great, repulsive force, such as air,

which always tends to expand ; whilst others exert an attractive force, that is either

not very small, as in the case of water, or may even be very great, as in the case of mercury.

Of these liquids, the particles even form themselves into balls by the mutual attraction

of the particles forming them ; & yet larger masses of them are quite easily separated, &

motion is easily given to any number of parts in such a manner that the motion does not

304 PHILOSOPHIC NATURALIS THEORIA

ut fluida cedant vi cuicunque impressae, ac cedendo facile moveantur, solida vero nonnisi
tota simul mover! possint, & viribus impressis idcirco resistant magis : quae autem resistunt
quidem multum, sed non ita multum, ut solida, dicuntur viscosa. Ipsa vero fluida
dicuntur humida, si solido admoto adhaerescant, & sicca, si non adhsereant.

Unde fluiditas : tria 427. Haec omnia phaenomena praestari possunt per ilia sola discrimina, quas in diverse

fluidorum genera. • i r • , . TT • a -j- ... . -1

particularum textu consideravimus. Ut emm a fluiditate incipiamus, mpnmis in ipsis
fluidis omnes, particulae in aequilibrio esse debent, dum quiescunt, & si nulla externa vi
comprimantur, vel in certam dirigantur plagam ; id aequilibrium debebit haberi a solis
mutuis actionibus : sed ejusmodi casum non habemus hie in nostris fluidis, quae incumbentis
massse premuntur pondere, & aliqua. ut aer, etiam continentis vasis parietibus comprimuntur,
in quibus idcirco omnibus aliqua haberi debet repulsiva vis inter particulas proximas, licet
inter remotiores haberi possit attractio, ut jam constabit. Tria autem genera fluidorum
considerari poterunt : illud, in quo in majoribus ejus massulis nulla se prodit mutua
particularum vis : illud, in quo se prodit vis repulsiva : illud, in quo vis attractiva se prodit.
Primi generis fere sunt pulveres, & arenulae, ut illae, ex quibus etiam horologia clepsydris
veterum similia construuntur, & ad fluidorum naturam accedunt maxime, si satis laevigatam
habeant superficiem, quod in quibusdam granulis cernimus, ut in milio : nam plerumque
scabritiem habent aliquam & inaequalitates, quse motum difnciliorem reddunt. Secundi
generis sunt fluida elastica, ut aer : tertii vero generis liquores, ut aqua, & mercurius.
Porro in primis ostensum est num. 222, & 422, posse binas particulas eodem etiam punctorum
numero constantes, sed diverse modo dispositas, ita diversas habere virium summas in
iisdem etiam centrorum distantiis, ut aliae se attrahant, aliae se repellant, aliae nihil in se
invicem agant. Quamobrem ejusmodi discrimina exhibet abunde Theoria. Verum
multa in singulis diligenter notanda sunt ; nam ibi etiam, ubi nulla se prodit vis attractiva,
habetur inter proximas particulas repulsio, ut innui paullo ante, & jam patebit.

Unde faciiis motus [196! 428. Porro in primo casu statim apparet, unde facilis ille habeatur motus.

in fluidis primi/s- v *•• n ...... rr -, , ,

generis. (Juoniam, aucta distantia, nulla sensibili vi se attrahunt particular ; altera non sequetur

motum alterius ; nisi ubi ilia versus hanc promota ita accesserit, ut vi repulsiva mutua,
quemadmodum in corporum collisionibus accidit, cogatur illi loco cedere, quas cessio, si
satis laevigatae superficies fuerint, ut prominentes monticuli in exiguos hiatus ingressi motum
non impediant, & sit locus aliquis, versus quern possint vel in gyrum actae particulae, vel
elevatae, vel per apertum foramen erumpentes, loco cedere ; facile net, nee alia requiretur
vis ad eum motum, nisi quae ad inertiae vim vincendam requiritur, vel si graves particulae
sint versus externam massam, ut hie versus Tellurem, & fluidum motu impresso debeat
ascendere, vis, quae requiritur ad vincendam gravitatem ipsam : verum ad vincendam
solam vim inertiae, satis est quascunque activa vis utcunque exigua, & ad vincendam gravi-
tatem, in hoc fluidorum genere, si perfecta sit laevigatio ; satis est vis utcunque paullo major
pondere massae fluidae ascendentis : quanquam nisi excessus fuerit major ; lentissimus
erit motus ; ipsum autem pondus coget particulas ad se invicem accedere nonnihil, donee
obtineatur vis repulsiva ipsum elidens, uti supra ostendimus num. 348 ; adeoque in statu
aequilibrii se particulae, in hoc -etiam casu, repellent, sed erunt citra, & propre ejusmodi
limites, ultra quos vis attractiva sit ad sensum nulla. Quod si figura particularum praeterea
fuerit sphaerica, multo facilior habebitur motus in omnem plagam ob ipsam circumquaque
uniformem figuram.

Eadem ratio, & in 429. In. secundo, ac tertio genere motus itidem habebitur facilis, si particulae sphaericae
inter* ipsa?~ sint, & paribus a centre distantiis homogeneae, ut nimirum vires dirigantur ad centra. In
ejusmodi enim particulis motus quidem unius particulae circa aliam omni difficultate carebit,
& vires mutuae solum accessum vel recessum impedient. Hinc impresso motu particulis
aliquot, poterunt ipsae mover! in gyrum alias circa alias, & alia succedere poterit loco ab
alia relicto, quin partes remotiores motum ejusmodi sentiant : quanquam fere semper
fortuita quaedam particularum dispositio hiatus, qui necessario relinqui debent inter globos,
& directio impressionis varia inducent etiam accessus & recessus aliquos, quibus fiet, ut

A THEORY OF NATURAL PHILOSOPHY 305

spread simultaneously in any sensible degree to parts further off. Hence it comes about that
fluids yield to any impressed force whatever, &, in doing so, are easily moved ; but solids
cannot be moved except all together as a whole, & thus offer greater resistance to an impressed
force. Those fluids which offer a considerable resistance, but one that is not so great as it
is in the case of solids, are called viscous ; again, fluids are said to be moist when they
adhere to a solid that is moved away from them, & dry if they do not do so.

427. All these phenomena can be presented by means of the single difference, which The origin of fluid-
I have already considered in the different texture of particles. For, to begin with fluidity, jp. ' three kinds °*
we have first of all that in fluids all the particles must be in equilibrium, whilst they are

at rest ; &, if they are not under the action of an external force, or driven in a certain
direction, that equilibrium must be due to the mutual actions alone. But we do not have
this sort of case here, when considering the fluids about us, which are under the action of
the weight of a superincumbent mass, & some of them, like air, are also acted upon by the
walls of the vessel in which they are enclosed ; hence, in all of these, there must be some
repulsive force between the particles next to one another, although, as will now be evident,
there may also be an attraction between more remote particles. Now, three kinds of
fluids can be considered ; one kind, in which, amongst its greater parts, no mutual force
between its particles is shown ; another kind, in which a repulsive force appears ; & a third
kind, in which there is an attractive force. Of the first kind are nearly all powders & sands,
such as those, from which are constructed clocks similar to the clepsydras of the ancients ;
& these approximate very closely to the nature of fluids, if they have sufficiently polished
surfaces, such as we see in some grains, like millet ; for, the greater part of them have some
roughness, & inequalities, which render motion more difficult. To the second class belong
the elastic fluids, such as the air ; & of the third kind are such liquids as water & mercury.
Further, it has been shown particularly in Art. 222, 422, that it is possible for two particles,
made up even of the same number of points, though differently distributed, to have the sums
of the forces corresponding to them so different, even at the same distances from the centre,
that some of them attract, some repel, & some have no action at all upon one another :
hence, my Theory furnishes such differences in abundance. However, there are many
things to be carefully noted in each case ; for even when no attractive force is in evidence,
there is a repulsive force between adjacent particles, as I mentioned just above ; & this
will be evident without saying anything further.

428. Moreover, in the first case it is at once apparent why there is easy movement of The source of the
the particles. For, since when the distance is increased the particles do not attract one particie'°of "fluids of
another with any sensible force, the one does not follow the motion of the other ; except the first kind,
when the former moves towards the latter & approaches it to such an extent that, just as

happens in the cases of impact of bodies, it is forced to give way to it by a mutual repulsive
force ; & this giving way would easily take place, if the surfaces were sufficiently smooth,
so that the projecting hillocks of one did not hinder the motion by sticking into the tiny
gaps of another ; & if there were some place, to which the particles could be forced in a
curved path, or elevated, or could break through an orifice opened to them, they might
give way. This may easily happen ; no other force would be required for the motion except
that necessary to overcome the force of inertia ; or, if heavy particles are attracted towards
an external mass, as with us towards the Earth, & the fluid has to ascend, then no other force
is required save that necessary to overcome gravity. But to overcome the force of inertia
alone any active force, however small, is sufficient ; & to overcome gravity, in this kind of
fluids, if there is perfect smoothness, any force that is a little greater than the weight of the
ascending part of the fluid will suffice ; although, unless the excess were considerable, the
motion would be very slow. Moreover, the weight of the fluid will force the particles
somewhat closer together, until a mutual repulsive force is produced which will cancel it, as
I showed above in Art. 348. Thus, when in a state of equilibrium the particles, even in
this case, will repel one another ; but they will lie on the near side of, & close to such
limit-points as have the attractive force on the far side of them practically zero. But if,
in addition the shape of the particles should be spherical, there would be much easier
movement in all directions due to the uniformity of shape all round.

429. In the second & third classes of fluids there is also easy movsment, if the particles The same argument
are spherical, & homogeneous at equal distances from their centres, that is to say, so that the other°tw°o
the forces are directed towards their centres. For, in the case of such particles, the motion kinds ; differences
of one particle round another lacks difficulty of any sort, & the mutual forces prevent

approach or recession only. Hence, if a motion be impressed on any number of particles,
they could move in curved paths round one another, & some could take the place left free
by others, without the parts further off feeling the effects of such motion ; although nearly
always the accidental arrangement of the gaps empty of particles, which must of necessity
be left between the spheres, & the varied direction of the pressures will lead also to approach

306 PHILOSOPHIC NATURALIS THEORIA

motus ad remotiores etiam particulas deveniat, sed eo minor, quo major fuerit earum
distantia. Verum hie notandum erit discrimen ingenis inter duos casus, in quibus partes
fluidi se repellunt, & casus, in quibus se attrahunt.

in elasticis fluidis 430. In primo casu particulae proximae debebunt se omnino repellere, & vis ex parte

particulas esse ex- i. v j • i *• , •* ,. ,., f

traiimitessub altera elidet vim ex altera ; sed si repente relmquatur libertas ex parte quavis, sine ulla
arcubus repuisivis externa vi, sed sola ilia particularum actione mutua, recedent reipsa particulse a se invicem,
& fluidum dilatabitur ; quin [197] immo externa vi opus est, ad continendam in eo statu
massam ejusmodi, uti aerem gravitas superioris atmosphaerae continet, vel in vase occluso
vasis ipsius parietes ; & aucta ilia externa vi comprimente augeri poterit compressio,
imminuta imminui. Particulae illse inter se non erunt in limitibus quibusdam cohaesionis,
sed erunt sub repulsivo arcu curvae exprimentis vires compositas particularum ipsarum.

in fluidis humidis 4.31. At in tertio genere particulas quidem proximae se mutuo repellent, repulsione
fore sequali illi vi, quae necessaria est ad elidendam vim externam, & ad elidendam pressionem,

proximum, & si quae oritur a remotiorum attractionibus : verum si fluidum est parum admodum compres-

debere haberTprope sibile, vel etiam nihil ad sensum, ut aqua ; debent esse citra, & admodum prope limitem,

vaiidissimum arcum ultra quern vel immediate, vel potius, si id fluidum neque distrahitur (ut nimirum durante

repuisivum. gua forma nequeat acquirere spatium multo majus, quod itidem in aqua accidit) habeat

post limites alios satis inter se proximos arcum attractivum ad distantias aliquanto majores

protensum, a quo attractio ilia prodeat, quae se in ejusmodi fluidorum massulis prodit ;

licet si iterum id fluidum ma j ore vi abire possit in elasticos vapor es, ut ipsa aqua post eum

attractivum arcum ; arcus repulsivus debeat succedere satis amplus, juxta ea, quse diximus

num. 195.

Motus non obstante ^32. In hoc fluidi genere illud mirum videri potest, quod ilia attractiva vis, quae in
quad ad moTum niajoribus succedit distantiis, & ille validus cohaesionis limes, qui & compressionem &
aliquot particu- rarcfactionem impedit, non impediat divisionem massae, & separationem unius partis massae

larum non debeant iv A. j • j *_ *i .e • •!• • o •• TJ- L-T. i

mover! remotae a" aua- At quomodo id facile fieri ibi possit, & non possit in solidis, patebit hoc exemplo.
simul ut in solidis. Concipiatur Terrae superficies sphaerica accurate, & bene laevigata, ac gravitas sit eiusmodi,

Exemplumin qua- • j- ' c -LT • • • j- •

dam hypothesi gio- ut ln distantia perquam exigua hat jam msensibilis, ut vis magnetica in exigua distantia
borum gravium. sensum jam effugit. Sint autem globi multi itidem laeves mutua attractiva vi praediti,
quae vim in totam Terram superet. Si quis unum ejusmodi globum apprehendat, & attollat ;
secundus ipsi adhaerebit relicta Terra, & post ipsum ascendet, reliquis per superficiem
Terrae progredientibus, donee alii post alios eleventur, vi in globum jam elevatum superante
vim in Terram. Is, qui primum manu teneret globum, sentiret, & deberet vincere vim
unius tantummodo globi in Terram, quern separat, cum nulla sit difficultas in progressu
reliquorum per superficiem Terrae, quo distantia non augetur, & globorum jam altiorum
vis in Terram ponatur insensibilis. Vinceret igitur aliorum vim post vim aliorum, & vis
ab eo adhibita major tantummodo vi globi unici requireretur ad rem praestandam. At si
illi globi deberent elevari simul, ut si simul omnes colligati essent per virgas rigidas ; deberent
utique omnes illae vires omnium in Terram simul superari, & requireretur vis major omnibus
simul. Res eodem redit, ac ubi fasciculus virgarum [198] debeat totus frangi simul, vel
potius debeant aliae post alias frangi virgae.

Appiicatio exempli 433. Jd ipsum est discrimen inter fluida huius generis, & solida. In his motus parti-

ad fluida, & sohda : , ~JJ . r ..... ... J. ° . . .. *

successiva particu- cularum circa particuias liber ob earum unitormitatem permittit, ut separentur aliae post
larum separatio in aljas • dum in solidis vis in latus, de qua egimus iam in pluribus locis, & anguli prominentes,

fluidis. r • 1 • • j- • i* 11 • r •

ac ngurarum irregulantas, impediunt ejusmodi liberum motum, qui fiat sine mutatione
distantiarum, & cogunt divulsionem plurimarum particularum simul : unde oritur
difficultas ilia ingens dividend! a se invicem particulas solidas, quae in divisione fluidorum
est adeo tenuis, ac ad sensum nulla.

a

A THEORY OF NATURAL PHILOSOPHY 307

& recession of some kind ; & through these it will come about that the effect of the motion
will reach the particles further off, although this will be the less, the greater the distance
they are away. But here we have to notice the great difference between the two cases,
the one, in which the parts of the fluid repel, & the other, in which they attract, one
another.

4.30. In the first case adjacent particles must repel one another, in every instance, In e!aftic fluids the

o i r i- 11/-1- -HT TII particles are out-

& the force from one part must cancel the force from another part. Moreover, if all at side the limit-

once freedom of movement is left in any one part, without any external force to prevent it, P?^nts- & under

then by the mutual action of the particles alone, these particles will of themselves recede

from one another & the fluid will expand. Indeed, what is more, there is need of an external

force to maintain a mass of this kind in its original state, just as the gravity of the upper

atmosphere constrains the air, or the walls of a vessel the air contained within it. When

this compressing external force is increased the compression can be increased, & if diminished

diminished. The particles themselves will not be at distances from one another corresponding

to limit-points of cohesion of any sort ; but these will correspond to a repulsive arc of the

curve that represents the resultant forces of the particles.

431. Again, in the third kind, adjacent particles must indeed repel one another, the in watery fluids the
repulsion being equal to that force that is necessary to cancel the external force, & also must be™ ve'ry
the pressure which arises from the attractions of points further off. But, if the fluid is strong one, of co-
only very slightly compressible, or not to any appreciable extent (like water, for example), fluid goes off as a
then the particles must be on the near side, & quite close to, a limit-point ; & on the far vapour, there must
side of this limit-point, either there must follow immediately a comparatively ample attractive sterongSrep°uisiveVarc!
arc ; or, more strictly speaking, if the fluid does not expand (that is to say, whilst it maintains

its form, it cannot acquire much more space, which is also the case with water), then it
has, after several other limit-points fairly close to one another, an attractive arc extending
to somewhat greater distances, to which is due that attraction which is seen in small
globules of fluids ; but if, with a greater force applied, the fluid can after that go off
to still further distances in the form of elastic vapours (as water does), then, after the
attractive arc we must have the above-mentioned comparatively ample repulsive arc ;
as was shown in Art. 195.

432. In this kind of fluid it may appear strange that the attractive force which follows Mutual force not
at greater distances, or the strong limit-point of cohesion, which prevents both compression „!" n ""fs "Vasy"
& rarefaction, does not, either of them, prevent division of the mass or the separation of because particles
one part of it from the other. But the reason why this can take place here, & not in the not move^at "the
case of solids, will become evident on considering the following example. Suppose the same time, when
surface of the Earth to be perfectly spherical, & quite smooth ; & suppose gravity to be al^^are moved;
such, that when the distance becomes very small it becomes insensible, just as magnetic as is the case for
force practically vanishes at a very small distance. Then, suppose we have a number of ^el hypothesis6 of
smooth spheres endowed with an attractive force for one another, which exceeds the force heavy spheres.
each has for the whole Earth. If one of these spheres is taken & lifted, a second one will

adhere to it & leave the ground, & ascend after it ; the rest will move along the surface
of the Earth, until one after the other they are also lifted up, the attraction towards the
sphere just lifted exceeding the attraction towards the Earth. The person, who took hold
of the first sphere, would feel & would have to overcome the force of only the one sphere
towards the Earth, namely, that of the one he takes away ; for there is no difficulty about
the progress of the rest of the spheres along the surface of the Earth, supposing that the dis-
tance is not increased, & assuming that the force towards the Earth of spheres already lifted
is quite insensible. Hence the force of one after that of another would be overcome, &
the whole business would be accomplished by his using a force that was just greater than
the force due to a single sphere. But if all the spheres had to be raised at once, as if they
were all bound together by rigid rods, it would be necessary to overcome at one time all
the forces of all the spheres upon the Earth, & there would be required a force greater
than all these put together. It is just the same sort of thing as when a whole bundle of rods
has to be broken at the same time, or rather the rods have to be broken one after another.

433. This is exactly what causes the difference between fluids of this kind & solids. Application of the

TT7. *• , , , ' , . , , . , !_• example to the case

With the former, the free motion of the particles about one another, due to their of flui<is & solids ;
uniformity, allows them to be separated one after the other. Whilst, with solids, lateral successive separa-

, .<'.., 1111. 11 •• in- i • • tion of the particles

force, with which we have already dealt m several places, projecting angles & irregularities m the case of fluids.

of shape, prevent such freedom of motion, as (with fluids) takes place without any change

in the mutual distances ; & they compel us to tear away a very great number of particles

all at once. This is the cause of the very great difficulty in the way of dividing the

particles of solids from one another ; & is the reason why the difficulty is very slight, or

practically nothing, when dividing fluids.

308 PHILOSOPHISE NATURALIS THEORIA

Exempium ipsius 434. Successivam hujusmodi separationem particularum aliarum post alias videmus

tiamqin' fluidisfTd utique in ipsis aquae guttis pendentibus, quae ubi ita excreverunt : ut pondus totius guttae

separationem fieri superet vim attractivam mutuam partium ipsius ; non divellitur tota simul ingens cjus

soiidis,6 si ' veiocitas aliqua massa, sed a superiore parte, utut brevissimo tempore, attenuatur per gradus ;. donee

debeat esse ingens. illud veluti filum jam tenuissimum penitus superetur. Fuerunt prius mille particulae

in superficie, quae guttam pendentem connectebant cum superiore parte aquas, quas

relinquitur adhaerens corpori, ex quo pendebat gutta, fiunt paullo post ibi 900, 800, 700 :

& ita porro imminuto earum numero per gradus, dum laterales accedunt ad se invicem,

& attenuatur figura : quarum idcirco resistentia facile vincitur, ut ubi in illo virgarum

fascicule frangantur aliae post alias. At ubi celerrimo motu in fluidum ejusmodi incurritur

ita ; ut non possint tarn brevi tempore aliae aliis particulae locum dare, & in gyrum agi ;

turn vero fluida resistunt, ut solida. Id experimur in globis tormentariis, qui ex aqua

resiliunt, in earn satis oblique projecti, ut manente satis magna horizontali velocitate

collisio in perpendicular! fiat more solidorum : ac eandem quoque resistentiam in aqua

scindenda experiuntur, qui se ex editiore loco in earn demittunt.

Soiiditatis causa in 4-7 r. Hinc autem pronum est videre, unde soliditatis phaenomena ortum ducant.

vi, & motu in latus : •»-,-. . , . • t c j- i i_ • i j- -i

exempium in parai- JNimirum ubi particularum ngura recedit piurjmum a sphaenca, vei distnbutio punctorum
leiepipedis. intra particulam inaequalis est, ibi nee habetur libertas ilia motus circularis, & omnia, quae

ad soliditatem pertinent, consequi debent ex vi in latus. Cum enim una particula respectu
alterius non distantiam tantummodo, sed & positionem servare debeat ; non solum, ea
promota, vel retracta, alteram quoque promoveri, vel retrahi necesse est ; sed praeterea,
ea circa axem quencunque conversa, oportet & illam aliam loco cedere, ac eo abire, ubi
positionem priorem respectivam acquirat ; quod cum & tertia respectu secundas prsestare
debeat, & omnes reliquae circunquaque circa illam positae ; patet utique, non posse motum
in eo casu imprimi parti cuipiam systematis ; quin & totius systematis motus consequatur
respectivam po-[i99]-sitionem servantis, quae est ipsa superius indicata solidorum natura.
Res autem multo adhuc magis manifesta fit, ubi figura multum abludat a sphserica, ut si
sint bina parallelepipeda inter se constituta in quodam cohaesionis limite, alterum ex adverso
alterius. Alterum ex iis moveri non poterit, nisi vel utrinque a lateribus accedat ad alterum,
vel utrinque recedat, vel ex altero latere accedat, & recedat ex altero. In primo casu
imminuta distantia habetur repulsiva vis, & illud alterum progreditur : in secundo, eadem
aucta, habetur attractio, & illud secundum ad prioris motum consequitur ; in tertio casu,
qui haberi non potest, nisi per inclinationem prioris parallelepipedi, altero latere attracto,
& altero repulso inclinari necesse est etiam secundum ; quo pacto si ejusmodi parallelepi-
pedorum sit series quaedam continua, quse fibram longiorem, vel virgam constituat ; inclinata
basi, inclinatur illico series tota : & si ex ejusmodi particulis massa constet ; tota moveri
debet ac inclinari, inclinato latere quocunque.

om n?ushUund 43 ^' Quod de parallelepipedis est dictum, id ipsum ad figuras quascunque transferri
discrimen inte potest inaequales utcunque, quae ex altero latere possint accedere ad aliam particulam, ex
flexiiia, & rigida. altero recedere : habebitur semper motus in latus, & habebuntur soliditatis phasnomena,
nisi paribus a centro distantiis homogeneae, & sphaericas formae particulae sint. Verum
ingens in eo motu discrimen erit inter diversa corpora. Si nimirum vires illae hinc, &
inde a limite, in quo particulae constitutae sunt, sint admodum validae ; motus in latus
fiet celerrime, & nulla flexio in virga, aut in m'assa apparebit ; quanquam erit utique semper
aliqua. Si minores sint vires ; longiore tempore opus erit ad motum, & ad positionem
debitam acquirendam, quo casu, inclinata parte ima virgas, nondum pars summa obtinere
potest positionem jacentem in directum cum ipsa, adeoque habebitur inflexio, quae quidem
eo erit major, quo major fuerit celeritas conversionis ipsius virgae, uti omnino per experi-
menta deprehendimus.

Discrimen inter 437. Nee vero minus facile intelligitur illud, quid intersit inter flexiiia solida corpora,

unde!a> & fraglha & fragilia. Si nimirum vires hinc, & inde ab illo limite, in quo sunt particular, extenduntur

ad satis magnas distantias eaedem, arcu utroque habente amplitudinem non ita exiguam ;

A THEORY OF NATURAL PHILOSOPHY 309

434. We certainly see an example of this kind of successive separation of particles, one Example of this in
after the other, in the case of drops of water hanging suspended ; here, as soon as they *£° e^jdrtaace*to
have increased up to a point where the weight of the whole drop becomes greater than separation in fluids
the mutual attractive force of its parts, any great part is not torn away as a whole ; but {^° hfL^idf^fthe
by degrees, though in a time that is exceedingly short, the drop is attenuated at its upper velocity has 'to be
part, until the neck, which has by now become exceedingly narrow, is finally broken altogether. very srea*-
There were, say, initially, a thousand particles in the surface connecting the hanging drop

to the upper part of the water which is left adhering to the body from which the drop
was suspended ; these a little afterwards became 900, then 800, then 700, & so on, their
number being gradually diminished as the sides of the neck approach one another, & its
figure is narrowed. Hence, their resistance is easily overcome, just as when, in the bundle
of rods, the rods are broken one after the other. But, when it is a case of an onset with
high speed, so that the time is too short to allow the particles to give way one after the
other, & move in curved paths round one another ; then, indeed, fluids resist in just the
same way as solids. This is to be observed in the case of cannon-balls, which rebound
from the surface of water, when projected at sufficiently small inclination to it ; so that,
whilst the horizontal velocity remains sufficiently great, the vertical impact takes place
in the manner of that between solids. Also, those who dive into water from a fairly great
height will experience the same resistance in cle'aving the surface.

435. Further, from what has been said, it can be se£n without difficulty whence the The cause ol solid-
phenomena of solidity defrive their origin. For instance, when the shape of the particles f0rce 1CS& motion ;
is very far from being spherical, or the distribution of the points within the particle is not example ?*this in
uniform, then there is not that freedom of circular motion ; & all things that pertain to pai

solidity must follow from the presence of lateral force. For, since one particle must preserve
not only its distance, but also its position with regard to another ; not only, when the
one is driven forwards or backwards, must the other also be driven forwards or backwards,
but also if the one is turned about any axis, it is necessary that the other should give way
& move off to the place in which it will acquire its original relative position. Since also
the third must do the same thing with respect to the second, & all the rest of the particles
round it in all directions, it is quite clear that in this case motion cannot be imparted
to any part of the system, without a motion of the whole system following it, in which the
mutual position is preserved ; & this is the very nature of solids that was mentioned above.
Moreover, the matter becomes even still more evident, when the shape differs considerably
from the spherical ; for instance, if we have a pair of parallelepipeds situated with regard
to one another at a distance corresponding to a limit-point of cohesion, opposite one another.
It will not be possible for one of them to be moved, unless either it approaches the other
laterally at both ends, or recedes at both ends, or else approaches at one end & recedes at
the other. In the first case, the distance being diminished, we have a repulsive force, &
the second particle will move away ; in the second case, the distance being increased, there
will be an attraction, & the second particle will follow the motion of the first. In the
third case, which cannot take place unless there is an inclination of the first parallelepiped,
one end of the second being attracted, & the other repelled, it is necessary that the second
particle should also be inclined. In this way, if there is a continuous series of such
parallelepipeds, forming a fairly long fibre or rod, then, when the base is inclined, the
whole rod must be inclined along with it ; & if a mass is formed from such particles, then
if any side of the mass is inclined, the whole of the mass must move along with it & be also
inclined.

436. What has been said with regard to parallelepipeds can be said also about any The same thing for
figures whatever which are at all irregular, if they can approach another particle at one the ^tfleTence^be6
side & recede from it on the other side ; there will in every case be motion to one side, tween flexible &
& the phenomena of solidity will be obtained, unless the particles are homogeneous at n§

equal distances from the centre & spherical in form. But in this motion there is a very
great difference among different bodies. If, for instance, the forces on either side of the
limit-point, in which the particles are situated, are quite strong, the lateral motion will
be very swift, & no bending will be observed in the rod or in the mass ; although there
certainly will be some taking place. If the forces are not so great, there will be need of
a longer time for it to acquire motion & the proper position ; & in this case, if the bottom
part of the rod is inclined, the top part of the rod cannot for a little while attain to a position
lying in a straight line with the base, & thus there will be bending ; & this indeed will
be all the greater, the greater the speed with which the rod is turned ; as is proved by
experiment to be always the case.

437. Nor will it be less easy to understand the reason why there is a difference between The reason of the
flexible solids & fragile bodies. For instance, if the forces on each side of the limit-point, at fl^We"
which the particles are, are extended unaltered over sufficiently great distances from it, & the bodies.

310 PHILOSOPHIC NATURALIS THEORIA

turn vero, vi externa adhibita utrique extreme, vel majore velocitate impressa alteri,
incurvabitur virga, atque inflectetur, sed sibi relicta ad positionem abibit suam, & in illo
inflexionis violento statu vim exercebit perpetuam ad regressum, quod in elasticis virgis
accidit. Si vires illae non diu durent hinc, & inde eaedem, vel per satis magnum intervallum
sit ingens frequentia limitum ; turn quidem inflexio habebitur sine conatu ad se restitu-
endam, & sine fractione, tarn vi adhibita utrique extremo, quam ingenti velocitate impressa
alteri, ut videmus accidere in maxime ductilibus, [200] velut in plumbo. Si demum
vires hinc, & inde per exiguum intervallum durent, post quod nulla sit actio, vel ingens
repulsivus arcus consequatur, qui sequentes attractivos superet ; habebitur virga rigida,
& fractio, ac eo major erit soliditas, & ilia, quae vulgo appellatur durities, quo vires illse
hinc & inde statim post limites fuerint majores.

Quid, & unde vis- 438. Atque hie quidem jam etiam ad discrimen devenimus inter elastica, & mollia ;
verum antequam ad ea faciamus gradum, adnotabo non nulla, quae adhuc pertinent ad
solidorum, & fluidorum naturam, ac proprietates. Inprimis media inter solida, & fluida,
sunt viscosa corpora, in quibus est aliqua vis in latus, sed exigua. Ea resistunt mutation!
figurae, sed eo majore, vel minore vi, quo majus, vel minus est in diversis particularum
punctis virium discrimen, a quo oritur vis in latus. Viscosa autem praeter tenacitatem,
quam habent inter se, habent etiam vim, qua adhaerent externis corporibus, sed non
omnibus, in quo ad humidos liquores referuntur. Humiditas enim est itidem respectiva.
Aqua, quae digitis nostris adhaeret illico, & per vitrum, ac lignum diffunditur admodum
facile, oleaginosa, & resinosa corpora non humectat, in foliis herbarum pinguibus extat in
guttulas eminens, & avium plurium plumas non inficit. Id pendet a vi inter particulas
fluidi, & particulas extern! corporis ; & jam vidimus pro diversa punctorum distributione
particulas easdem respectu aliarum debere habere in eadem directione vim attractivam,
respectu aliarum repulsivam vim & respectu aliarum nullam.

Organicorum cor- 439. In particulis illis, quae ad soliditatem requiruntur, invenitur admodum expedita

porum eSormatio • I • j rj • • j TVL j • •

per vires in latus ratio phaenomeni ad solida corpora pertmentis, quod Physicos in summam admirationem
versus certa super- rapit, nimirum dispositio quaedam in peculiares quasdam figuras, quae in salibus inprimis

ficiei puncta. , . ," . . ^ . . & > . ,

apparent admodum constantes, in glacie, & in mvium stellulis potissimum adeo sunt
elegantes etiam, & ad certas quasdam leges accedunt, quas itidem cum constanti admodum
figurarum forma in gemmarum succis simplicibus observamus, quae vero nusquam magis
se produnt, quam in organicis vegetabilium, & animalium corporibus. In hac mea Theoria
in promptu est ratio. Si enim particulae in certis suae superficiei partibus quasdam alias
particulas attrahunt, in aliis repellunt ; facile concipitur, cur non nisi certo ordine sibi
adhaereant, in illis nimirum locis tantummodo, in quibus se attrahunt, & satis firmos limites
nancisci possunt, adeoque non nisi in certas tantummodo figuras possint coalescere.
Quoniam vero praeterea eadem particula, eadem sui parte, qua alteram attrahit, alteram
pro ejus varia dispositione repellit ; dum massa plurium particularum temere agitata
prastervolat ; eae tantummodo sistentur, quae attrahuntur, & ad ea se applicabunt puncta,
ad quae maxime attrahuntur, ac in illis haebebunt, in quibus post accessum maxime tenaces
limites [201] nanciscentur ; unde & secretionis, & nutritionis, vegetationis, & certarum
figurarum patet ratio admodum manifesta. Et haec quidem ad nutritionem, & ad certas
figuras pertinentia jam innueram num. 222, & 423.

Atomistarum sys- 440. Quoniam ostensum est, qui fieri possit, ut certam figuram acquirant certa

to'tum^x ^a" particularum genera, cujus admodum tenacia sint, si quis omnem veterum corpuscularium
Theoria, & cum sententiam, quam Gassendus, ac e recentioribus alii secuti sunt, adhibentes variarum
expiicata ^netefrea figurarum atomos, ut ad cohaesionem uncinatas, ab hac eadem Theoria velit deducere,

.
cohaesione partium is sane poterit, ut patet, & ejusmodi atomos adhibere ad explicationem eorum omnium

phaenomenorum, quae pendent a sola cohaesione, & inertia, quae tamen non ita multa sunt :
poterunt autem haberi ejusmodi atomi cum infinita figurae suae tenacitate, & cohaesione
mutua suarum partium per solas etiam binas asymptotes illas, de quibus num. 419, inter
se satis proximas. Et si curva virium habeat tantummodo in minimis distantiis duas
ejusmodi asymptotes, turn post crus repulsivum ulterioris statim consequatur arcus attrac-
tivus, primo quidem plurimum recedens ab axe cum exiguo recessu ab asymptote, turn

A THEORY OF NATURAL PHILOSOPHY 311

arc on either side of it has an amplitude that is not altogether small ; then, if an external force
is applied at both ends of the rod, or a fairly great velocity is impressed upon one of the
two ends, the rod will be curved, & bent ; but if it is left to itself it will return to its original
position ; & whilst in that violent state of inflection, it will continuously exert a force of
restoration, such as occurs in elastic rods. If the forces do not continue the same for such a
distance on each side of the limit-point, or if in a sufficiently large interval there exist a con-
siderable number of limit-points, then there will be bending without any endeavour towards
restoration, & without fracture, both when we apply a force to each end, & when a great
velocity is impressed upon one of them ; we see this happen in solids that are extremely
ductile, like lead. Finally, if the forces on either side of the limit-point only continue for
a very short space, after which there is no action at all, or if a large repulsive arc follows,
such as overcomes the attractive arcs that follow it ; then the rod will be rigid, & there
will be fracture ; & the solidity, & what is commonly called the hardness, will be the greater
the greater the forces on each side of the limit-points, & following immediately after them.

438. And now we come to the difference between elastic & soft bodies. But, before The nature &
we pass on to them, I will mention a few matters that have to do with the nature & properties source of viscosity,
of solids & fluids. First of all, intermediate between solids & fluids come viscous bodies ;

in these there is indeed some force to one side, but it is very slight. They resist a change
of shape ; but, the force of resistance is the greater or the less, the greater or the less the
difference of the forces on different points of the particles, from which arises the force to
one side. Viscous bodies, in addition to the tenacity which they have within their own
parts, have also another force with which they adhere to outside bodies, but not to all ;
& in this they are related to watery liquids. For humidity is also itself but relative. Water,
which adheres immediately to our fingers, & is quite easily diffused over glass or wood,
will not wet oily or resinous bodies ; on the greasy leaves of plants it stands up in little
droplets ; nor does it make its way through the feathers of the greater number of the birds.
This depends on the force between the particles of the fluid, & those of the external body ;
& we have already seen that, for a different distribution of their points, the same particles
may have with respect to some, in the same direction, an attractive force, with respect
to others a repulsive force, & with respect to others again no force at all.

439. In particles, such as are necessary for solidity, there is found quite easily the reason The formation of
for a phenomenon pertaining to solid bodies, which is a source of the greatest wonder to "fan^oUransverse
physicists. That is, a disposition in certain special shapes, which in salts especially seem forces directed to-
to be quite constant ; in ice, & the star-like flakes of snow more especially, they are wonderfully ^fa^ s^face150"113
beautiful ; & they observe certain definite laws, such as we also see, together with a constant

shape of figure, in the simple constituents of crystals. But these are nowhere to be found
so frequently as in the organic bodies of the vegetable & animal kingdoms. The reason
for this comes out directly in this Theory of mine. For, if particles, at certain parts of their
surfaces, attract other particles, & at other parts repel other particles, it can easily be
understood why they should adhere to one another only in a certain manner of arrangement ;
that is to say, in such places only as there is attraction, & where there can be produced
limit-points of sufficient strength ; & thus, they can only group themselves together in
figures of certain shapes. But since, in addition to this, the same particle, at the same
part of its surface, with which it attracts one particle, will repel another particle situated
differently with respect to it ; whilst the mass of the great number of particles, set in
motion at random, will slip by, those only will stay, which are attracted ; & they will attach
themselves to the points to which they are most attracted, & will adhere to those points
in which, after approach, limit-points of the greatest tenacity are produced. From this
the reason for secretion, nutrition, the growth of plants, & fixity of shape, is perfectly evident.
I have indeed already remarked on these matters, as far as they pertain to nutrition & fixity
of shape, in Arts. 222 & 423.

440. Since it has been shown how it may be possible for certain kinds of particles to The whole of the
acquire certain definite shapes, of which they are quite tenacious ; if anyone should wish to bySttthe f°™omists
derive from this same theory the whole idea of the ancient corpuscularians, such asGassendi can be derived
& others of the more modern philosophers have followed, employing atoms of various shapes, wTTh^whi^h0?!
hooked together for cohesion ; he will certainly be able, as is evident, to use atoms of this sort agrees very well ;
to explain all these phenomena that depend upon cohesion alone, & inertia ; but the number cohesfon^o"' the
of these is not very great. Moreover, atoms of this sort can be had with an infinite tenacity parts of their
of shape, & mutual cohesion of their parts, by even the sole assumption of those pairs of **°™s IS exPlamed
asymptotes sufficiently close to one another, of which I spoke in Art. 419. Even if the
curve of forces should have at very small distances two such asymptotes only, & then
immediately after the repulsive arc of the far one of these there should follow an attractive
arc, such as first of all recedes a great distance from the axis whilst it recedes only slightly
from the asymptote, & then returns towards the axis & approximates immediately to the

3i2 PHILOSOPHISE NATURALIS THEORIA

ad axem regrediens, & accedens statim ad formam gravitati exhibendae debitam ; haberentur
per ejusmodi curvam atomi habentes impenetrabilitatem, gravitatem, & figurae suse
tenacitatem ejusmodi, ut ab ea discedere non possent discessu quantum libuerit parvo ;
cum enim possint illse duae asymptoti sibi invicem esse proximo intervallo utcunque parvo,
posset utique ita contrahi intervallum istud, ut figurae mutatio aequalis datae cuicunque
utcunque parvae mutationi eviteatur. Ubi enim cuicunque figurae inscripta est series
continua cubulorum, & puncta in singulis angulis posita sunt, mutari non potest figura
externorum punctorum ductum sequens mutatione quadam data, per quam quaedam
puncta discedant a locis prioribus per quaedam intervalla data, manentibus quibusdam,
ut manente basi, nisi per quaedam data intervalla a se invicem recedant, vel ad se invicem
accedant saltern aliqua puncta, cum, data distantia puncti a tribus aliis, detur etiam ejus
positio respectu illorum, quae mutari non potest, nisi aliqua ex iisdem tribus distantiis
mutetur, unde fit, ut possit data quaevis positionis mutatio impediri, impedita mutatione
distantiae per intervallum ad earn mutationem necessarium. Quod si illae binae asymptoti
essent tantillo remotiores a se invicem, turn vero & mutatio distantiae haberi posset tantillo
major, & idcirco singulis distantiis illata vi aliqua posset figura non nihil mutari, & quidem
exigua mutatione distantiarum singularum posset in ingenti serie punctorum haberi inflexio
figurae satis magna orta ex pluribus exiguis flexibus. Sic & spirales atomi efformari possent,
quarum spiris per vim contractis sentiretur ingens elastica vis, sive determinatio ad
expansionem, ac per hujusmodi atomos possent iti-[202J-dem plurima explicari phsenomena,
ut & nexus massarum per uncos uncis, vel spiris insertos, quo pacto explicari itidem posset
etiam illud, quomodo in duabus particulis, quarum altera ad alteram cum ingenti velocitate
accesserit, oriatur ingens nexus novus, nimirum sine regressu a se invicem, unco nimirum
alterius in alterius foramen injecto, & intra illud converso per virium inaequalitatem in
diversas unci partes agentium, ut jam prodire non possit ; nam unci cavitas, & foramen,
seu porus alterius particulae, posset esse multo amplior, quam pro exigua ilia distantia
insuperabili, ut idcirco inseri posset sine impedimento orto a viribus agentibus in minore
distantia. Eaedem autem atomi haberi possunt, etiam si curva habeat reliquos omnes
flexus, quos habet mea, quo pacto ad alia multo plura, ut ad fermentationes inprimis, ac
vaporum, & luminis emissionem multo aptiores erunt ; & sine asymptoticis arcubus, qui
vires exhibeant extra originem abscissarum in infinitum excrescentes, idem obtineri poterit
per solos limites cohaesionis admodum validos cum tenacitate figurae non quidem infinita,
sed tamen maxima, ubi, quod illi veteres non explicarunt, cohaesio partium atomorum
inter se, adeoque atomorum soliditas, ut & continuata impenetrabilitatis resistentia, &
gravitas, ex eodem general! derivaretur principio, ex quo & reliqua universa Natura. Illud
unum hie notandum superest, ejusmodi atomos habituras necessario ubique distantiam
a se invicem majorem, quam pro ilia insuperabili distantia, ad quam externa puncta devenire
ibi non possunt.

Cur non omnia 441. Hue etiam pertinet solutio hujusmodi difficultatis, quae sponte se objicit : si

lice^omnra* puncta omnia niateriae puncta simplicia sunt, & vires in quavis directione circumquaque exercent

sint circumquaque easdem ; omnia corpora ex iis utique composita erunt fluida multo potiore jure, quam

ejusdem vis. fluida esse debeant, quae globulis constent easdem in omni circum directione vires exercen-

tibus. Huic difficultati hie facile occurritur : si particularum puncta possent vi adhibita

mutare aliquanto magis distantias inter se, nam aliqua etiam ad circulationem exigua

mutatio requiritur ; posset autem imprimi exiguo numero punctorum constituentium

unam e particulis primorum ordinum, quin imprimatur simul omnibus ejusmodi punctis,

vel satis magno eorum numero, motus ad sensum idem ; turn utique haberetur idem,

quod habetur in fluidis, & separates aliis punctis post alia, motus facilis per omnes omnium

corporum massas obtineretur. At particulae primi ordinis ab indivisibilibus punctis ortae,

ut & proximorum ordinum particulae ortae ab iis, sua ipsa parvitate molis tueri possunt

juxta num. 424 formam suam, & positionem punctorum : nam differentia virium exercit-

arum in diversa earum puncta potest esse perquam exigua, summa virium prohibente

tantum accessum unius particulae ad alteram, quo tamen accessu inaequalitas virium, &

A THEORY OF NATURAL PHILOSOPHY 313

form proper to represent gravitation ; by such a curve we should get atoms having
impenetrability, gravitation, & tenacity of shape of such a kind that they would not be
able to depart from this shape by any small amount we wish to assign. For, since the
two asymptotes can be very close together, distant from one another by any interval no matter
how small, this interval can in every case be contracted to such an extent, that the change
of shape will be just less than any given change no matter how small. For, if within any
figure there is inscribed a continuous series of little cubes, & points are situated at each
of their corners, the figure cannot be changed, following the lead of external points, by
any given change through which certain points depart from their original positions through
certain given intervals, whilst others stay where they are, i.e., whilst the base, say, stays
where it was ; unless they recede from one another through a certain given interval, or
approach one another, or some of the points do so at least. For, if the distances of a point
from three other points are given, its position with regard to them is also given ; & this
cannot be changed without altering some one of the three distances ; hence, any change
of position can be prevented by preventing the change of distance through any interval
that is necessary to such a change of position. But if the pair of asymptotes were just a
little further away from one another, then in truth there would be possibility of getting a
change of distance that was also just a little greater; & thus, a force being produced at each
distance, the figure might suffer some change ; & by a very slight change of each of the
distances in a very long series of points there might be obtained a bending of the figure of com-
paratively large amount, due to a large number of these slight bendings. In such a way atoms
might be formed like spirals ; &, if these spirals were compressed by a force, there would be
experienced a very great elastic force or propensity for expansion ; also by means of atoms
of this nature an explanation could be given of a very large number of phenomena, such
as the connection of masses by means of hooks inserted into hooks or coils ; & in this way
also an explanation could be given of the reason why, in the case of two particles of which
one has approached the other with a very great velocity, there arises a fresh connection
of great strength, that is, one so strong that there is no rebound of the particles from one
another. For instance, it may be said that the hook of the one is introduced into an opening
in the other, & twisted within it by the inequality of the forces acting on different parts
of the hook, so that it cannot get out again. For the concavity of the hook, & the opening
or pore of the second particle, may be much wider than that corresponding to that very -
slight distance limiting nearer approach ; & thus the hook can be inserted without hindrance
due to forces acting at those very small distances. These same atoms might be obtained,
even if the curve had all the inflected arcs that are present in mine ; & then such atoms
would be much more suitable to explain fermentations especially, as well as the emission
of vapours & of light. If there were no asymptotic arcs representing indefinitely increasing
forces beyond the origin of abscissae, the same result could be obtained by means of limit-
points of cohesion alone ; with tenacity of figure, not indeed infinite, but still very great if
these were very powerful. In this case, there could be derived from the same general
principle, from which is derived the whole of Nature in general, an explanation of the
cohesion of the parts of the atoms (which the ancients did not explain), & therefore of their
solidity ; & also the continued resistance of impenetrability, & gravitation too. There
remains but one thing for me to mention ; namely, that atoms of this kind will necessarily
keep to a greater distance from one another than that corresponding to the distance limiting
further approach, beyond which external points cannot come.

441. Here also is the place to solve a difficulty that spontaneously presents itself. If The reason why all
all points of matter are simple, & if they exert the same forces in all directions round bodies are not fluid,

, . . , , J 11 i T i r i although all points

themselves ; then it is far more natural to expect that all bodies that are composed of such in ail directions
points would be fluid than that those, which consist of little spheres exerting the same !?und arf under

f n T i i n • i r™ i • i-rr- i • tne same force-

forces in all directions around, are bound to be fluid. The answer to this difficulty is
easily given ; if the points of particles can, by application of force, increase their mutual
distances by a fair amount (for some slight change is necessary even for circulation), and if
further it were possible to impress a practically equal motion on a very small number of
points forming one of the particles of the first order, without at the same time giving this
motion to all such points, or even to any considerable number of them ; in that case we
certainly should obtain the same effect as is obtained in the case of fluids ; & the points being
separated one after the other, an easy movement would be obtained throughout all masses
of all bodies. But, particles of the first order, formed from indivisible points, as also those
of the next orders formed from the first, can, owing to their very smallness of volume,
preserve their form & the mutual arrangement of their points, as was shown in Art. 424.
For, the difference between the forces acting on different points of them may- be extremely
small, since the sum of the forces prevents too close an approach of one particle to the other ;
& yet by this approach an inequality in the forces & an obliquity in their directions is obtained,

3H PHILOSOPHIC NATURALIS THEORIA

obliquitas directionum ha-[203]-beatur adhuc satis magna ad vincendas vires mutuas,
mutandam positionem, qua positione manente, manet injequalitas virium, quas diversa
puncta ejus particulae exercent in aliam particulam. Ea inaequalitas itidem potest non
esse satis magna, ut possit illius mutuas vires vincere, & textum dissolvere, sed esse tanta,
ut motum inducat in latus, ac ejus motus obliquitas, & virium inaequalitas eo deinde erit
major, quo ad altiores ascenditur particularum ordines, donee deveniatur ad corpora,
quae a nobis sentiuntur.
Difficuitas deter- Ai2. Solida externum corpus ad ea delatum intra suam massam non recipiunt, ut

mmandi resisten- • T n • i v i • • • i • • T> • •

tiam fluidorum : vidimus : at liuida solidum intra se moven permittunt, sed. resistunt motui. Kesistentiam
method; indirectae eiusmodi accurate comparare, & eius leges accurate definire, est res admodum ardua.

idprsestandi eaedem fi . ,r. , J , ° . „ ,. . .

in hac Theoria ac Oporteret nosse ipsamvinum legem determinate, & numerum, & dispositionem punctorum,
m communi. ac habere satis promotam Geometriam, & Analysin ad rem praestandam. Sed in tanta

particularum, & virium multitudine, quam debeat esse res ardua, & humano captu superior
determinatio omnium motuum, satis constat ex ipso problemate trium corporum in se
mutuo agentium, quod num. 204 diximus nondum satis generaliter solutum esse. Hinc
alii ad alias hypotheses confugiunt, ut rem perficiant, & omnes ejusmodi methodi asque
cum mea, ac cum communi Theoria, consentire possunt.

fontes &r!rtriusltue 443' ^ tamen aliquid innuam etiam de eo argumento, duplex est resistentiae fons
lex. in fluidis ; primo quidem oritur resistentia ex motu impresso particulis fluidi ; nam juxta

leges collisionis corporum, corpus imprimens motum alteri, tantundem amittit de suo.
Deinde oritur resistentia a viribus, quas particulae exercent, dum alias in alias incurrunt,
quae earum motum impediunt, quo casu comprimuntur non nihil particulae ipsae etiam
in fluidis non elasticis egressae e limitibus, & aequilibrio : acquirunt autem motus admodum
diversos, gyrant, & alias impellunt, quae a tergo urgent non nihil corpus progrediens,
quod potissimum a fluidis elasticis a tergo impellitur, dilatato ibi fluido, dum a fronte a
fluido ibi compresso impeditur : sed ea omnia, uti diximus, accurate comparare non licet.
Illud generaliter notari potest : resistentia, quae provenit a motu communicate particulis
fluidi, & quae dicitur orta ab inertia ipsius fluidi, est ut ejus densitas, & ut quadratum
velocitatis conjunctim : ut densitas quia pari velocitate eo pluribus dato tempore particulis
motus idem imprimitur, quo densitas est major, nimirum quo plures in dato spatio
occurrunt particulae : ut quadratum velocitatis, quia pari densitate eo plures particulas
dato tempore loco movendae sunt, quo major est velocitas, nimirum quo plus spatii percur-
ritur, & eo major singulis imprimitur motus, quo itidem velocitas est major. Resistentia
autem, quae oritur a viribus, quas in se exercent particulae, si vis ea esset eadem in singulis,
quacunque velocitate [204] moveatur corpus progrediens, esset in ratione temporis, sive
constans : nam plures quidem eodem tempore particulae earn vim exercent, sed breviore
tempore durat singularum actio, adeoque summa evadit constans. Verum si velocitas
corporis progredientis sit major ; particulae magis compinguntur, & ad se invicem accedunt
magis, adeoque major est itidem vis. Quare ejusmodi resistentia est partim constans,
sive, ut vocant, in ratione momentorum temporis, & partim in aliqua ratione itidem
velocitatis.

Quam legem vide- AAA_ Porro ex expenmentis nonnulhs videtur erui, resistentiam in nonnullis fluidis

antur innuere ex- ' . • j v i • ... .,...

perimenta: in vis- esse partim m ratione duplicate velocitatum, partim in ratione earum simplici, & partim
co sis resistentiam constantem, sive in ratione momentorum temporis, quanvis ubi velocitas est ingens,
deprehendatur major : & ubi fluiditas est ingens, ut in aqua, ut secundum resistentiae
genus, quod est magis irregulare, & incertum, fit respectu prioris exiguum, satis accedit
resistentia ad rationem duplicatam velocitatum. Sed & illud cum Theoria conspirat,
quod viscosa fluida multo magis resistunt, quam pro ratione suae densitatis, & velocitate
corporis progredientis : nam in ejusmodi fluidis, quae ad solida accedunt, illud secundum
resistentiae genus est multo majus, quod quidem in solidis usque adeo crescit : quanquam
& in iis intrudi per ingentem vim intra massam potest corpus extraneum, ut clavus in murum,
vel in metallum, quae tamen, si fragilia sunt, & sensibilem compressionem non admittant,
diffringuntur.

Probiemata alia ad 44.5. Jam vero quaecunque a Newtono primum, turn ab aliis demonstrata sunt de

nTn^fa^Hid^m rnotu corporum, quibus resistitur in variis rationibus velocitatum, ea omnia consentiunt
communia huic itidem cum mea Theoria, & hujus sunt loci, ac ad illam pertinent Mechanicae partem,

Q motu solidorum per fluida. Sic etiam determinatio figurae, cui minimum

A THEORY OF NATURAL PHILOSOPHY

which is sufficiently great to overcome the mutual forces & to alter their position ; & when
this position stays as it was, so also does the inequality between the forces, which the different
points of the particle exert upon another particle. Again, this inequality may not be great
enough to overcome the mutual forces of that particle, & break up its formation ; but
yet great enough to induce lateral motion ; the obliquity of this motion, & the inequality
of forces will therefore be so much the greater, the further we ascend in the orders of the
particles, until we finally reach such bodies as affect our senses.

442. As we see, solids do not receive within their mass an external body that is brought
close up to them ; but fluids allow a solid to be moved within their mass, resisting however
the motion. To find such resistance accurately, & to make out the laws which govern
it, is a matter of great difficulty. It would be necessary to know the law of forces exactly,
the number & arrangement of the points, & to be in possession of fairly advanced geometry
& analysis to accomplish a solution. But, when dealing with such a great number of points
& forces, how difficult the matter is bound to be can be fairly seen by reference to that
problem of the three bodies acting upon one another, which I said, in Art. 204, had not
yet been solved at all generally. Hence, others resort to other hypotheses for their purposes ;
all such methods can be reconciled as well with my theory as with the common one.

443. So that I may not leave the point altogether untouched, I will just remark that
the source of resistance in fluids is twofold. First, we have resistance due to the motion
impressed on the particles of the fluid ; for, according to the laws of the impact of bodies,
the body which impresses the motion on the other will lose just as much of its own motion.
Secondly, there is resistance due to the forces exerted by the particles, as they approach
one another, which hinders their motion ; & in this case, the particles themselves are
compressed to some extent, even in non-elastic fluids, as they go beyond the limit-points
& equilibrium. Moreover they acquire different motions, they gyrate & drive off others
that are driving the moving body to some extent from the back ; & especially in the case
of elastic fluids we have this force at the back of the body, owing to the fluid being there
dilated, whilst at the same time there is a hindering force at the front due to the fluid being
compressed there. But all these things, as I have said, cannot be accurately determined.
It can, however, be in general noted that the resistance due to the motion communicated
to the particles of a fluid, which is said to arise from the inertia of the fluid, varies as its
density & the squares of the velocities j ointly. As the density, because in the same time,
for equal velocities, the same motion is impressed upon a number of particles which is the
greater, the greater the density, i.e., the greater the number of particles occupying the
same space. As the squares of the velocities, because in the same time, for equal densities,
the number of particles to be moved in position is the greater, the greater the velocity,
that is to say, the greater the space to be traversed ; & the motion that is impressed on
each point is the greater, the greater the velocity. Again, the resistance that is due to
the forces which the particles exert on one another, if the force is the same for each of them,
with whatever velocity the moving body proceeds, would be in proportion to the time,
or constant. For, it is true that a large number of particles exert this force in the same
time, but the action of each only lasts for a quite short time ; & thus the sum turns out
to be constant. If the velocity of the moving body is greater, the particles are driven
together more closely, & approach one another more nearly, & so also the force is greater.
Hence this kind of resistance is partly constant, or, as it is usually termed, proportional
to instants of time, & partly in some way proportional to the velocity as well.

444. Further the results of some experiments seem to indicate that the resistance
in some fluids is partly as the squares of the velocities, partly as the velocities simply, & partly
constant, or as the instants of time, although where the velocity is very great, it is found
to be greater. Also when the fluidity is great, as in the case of water, the second kind of
resistance, which is the more irregular & uncertain of the two, becomes exceedingly small
compared with that of the first kind, & the total resistance approaches fairly closely to a
variation as the squares of the velocities. It is also in agreement with the Theory that
the resistance for viscous fluids is much greater than that corresponding to the ratio of
densities & the velocities of the moving bodies. For, in such fluids, which are a near
approach to solids, the second kind of resistance is by far the greater, & indeed increases to
so great an extent as in solids. Although, in solids also, an extraneous body can be introduced
within their mass by means of a very great force, just as a nail may be driven into a wall, or
into metal ; yet if these are fragile & do not admit of sensible compression, they are broken.

445. But there are several other things, first demonstrated by Newton, & afterwards
by others, concerning the motion of bodies, under a resistance varying as different powers
of the velocity ; & all of these are also in agreement with my Theory, & come in in this
connection ; they belong also to that part of Mechanics which deals with the motion of
solids through fluids. So also the determination of the figure of least resistance, the

The difficulty of
determining the re-
sistance of fluids ;
the indirec t
methods for accom-
plishing this are tho
same in my Theory
as in the usual one.

Two sources of
resistance, & the
laws of each.

The law that ex-
periments seem to
indicate : the resist-
ance is greater in
viscous fluids.

Other problems
relating to resist-
ance that are
common also to this
Theory.

3i6 PHILOSOPHIC NATURALIS THEORIA

resistitur, determinatio vis fluid! solidum impellentis directionibus quibuscunque, mensura
velocitatis inde oriundae per corporum objectorum resistentiam observatione definitam,
innatatio solidorum in fluidis, ac alia ejusmodi, & mihi communia sunt : sed oportet
rite distinguere, quae sunt hypothetica tantummodo, ab iis, quas habentur reapse in
Natura.

Alia pertinentia 446. Ad fluida & solida pertinent itidem, quaecunque in parte secunda demonstrata
in" %rtePCsrcunda^ sunt de pressione fluidorum, & velocitate in efHuxu, quaecumque de aequilibrio solidorum,
discrimen inter de vecte, de centro oscillationis, & percussionis, quas quidem in Mechanica pertractari
eiastica, & moiha. solent. Illud unum addo, ex motu facili particularum fluidi aliarum circa alias, & irregulari
earum congestione, facile deduci, debere pressionem propagari quaquaversus. Sed de
his jam satis, quas ad soliditatem, & fluiditatem pertinent : illud vero, quod pertinet ad
discrimen inter eiastica, & mollia, brevi expediam. Eiastica sunt, quae post mutationem
[205] figurae redeunt ad formam priorem ; mollia, quae in nova positione perseverant.
Id discrimen Theoria exhibet per distantiam, vel propinquitatem limitum, juxta ea, quae dicta
num. 199. Si limites proximi illi, in quo particular coherent, hinc, & inde plurimum ab
eo distant, imminuta multum distantia, perstat semper repulsiva vis ; aucta distantia, perstat
vis attractiva. Quare sive comprimatur plus aequo, sive plus aequo distrahatur massa, ad
figuram veterem redit ; ubi rediit, excurrit ulterius, donee contraria vi elidatur velocitas
concepta, ac oritur tremor, & oscillatio, quae paullatim minuitur, & extinguitur demum,
partim actione externorum corporum, ut per aeris resistentiam sistitur paullatim motus
penduli, partim actione particularum minus elasticarum, quae admiscentur, & quae possunt
tremorem ilium paullatim interrumpere frictione, ac contrariis motibus, & sublapsu, quo
suam ipsam dispositionem nonnihil immutent. Si autem limites sint satis proximi ; causa
externa, quae massam comprimit, vel distrahit, posteaquam adduxit particulas ab uno
cohaasionis limite ad alium, ibi eas itidem cogit subsistere, quae ibidem semel constitutae
itidem in aequilibrio sunt, & habetur massa mollis.

Fluida eiastica, 447. Quaedam eiastica fluida non habent particulas positas inter se in limitibus cohae-

suTT\nPTimitibus sionis, sed in distantiis repulsionum, & quidem ingentium, ut aer : sed vel incumbente
cohaesionis ; omnia pondere, vel parietibus quibusdam impeditur recessus ille, & sunt quodammodo ibidem
& solida, & fluida jn statu violento ; licet semper puncta singula in aequilibrio sint, oppositis repulsionibus

eiastica esse, ocu * i „ i Pi n • \ • i • •

non dici, quia sensi- se mutuo elidentibus. Omnia autem & solida, & fluida, quae videntur nee comprimi, nee
r u^as nabere vires mutuas inter particulas, sed in limitibus esse, adhuc eiastica sunt, sive
vim repulsivam exercent inter particulas proximas, saltern quse sensibili gravitate sunt
prasdita, quae nimirum* vis repulsiva vim gravitatis elidat. Verum ea distantias parum
admodum mutant, mutatione, quae idcirco sensum omnem effugiat ; quod accidit in
aqua, quae in fundo putei, & prope superficiem supremam habet eandem ad sensum densi-
tatem, & in metallis, & in marmoribus, & in solidis corporibus passim, quaa pondere majore
imposito nihil ad sensum comprimuntur. Sed ea idcirco appellari non solent eiastica, &
ad ejusmodi appellationem non sufficit vis repulsiva etiam ingens inter particulas proximas :
sed etiam requiritur mutatio sensibilis distantiae respectu distantiae totalis respondens
sensibili mutationi virium.

Dura nuiia esse : 448. Dura corpora in eo sensu, in quo a Physicis duritiei nomen accipitur, ut nimirum

unde Ira^uitas" r& figuram nihil prorsus immutent, nulla sunt in mea Theoria, ut & nulla compacta penitus,

ductmtas. ac plane solida, quemadmodum diximus etiam num. 266 ; sed dura vocat vulgus, quae

satis magnam exercent vim, ne figuram mutent, sive eiastica sint, sive fragilia, sive mollia.

Fragilitas, unde ortum ducat, expositum est paullo su- [206] -perius num. 437, & inde etiam

quid ductilitas, ac malleabilitas sit, facile intelligitur. Ductilia nimirum a mollibus non

differunt, nisi in majore, vel minore yi, qua figuram tuentur suam : ut enim mollia pressione

tenui, & ipsis digitis comprimuntur, vel saltern figuram mutant, sed mutatam retinent,

ita ductilia ictu validiore mallei mutant itidem figuram suam veterem, & retinent novam,

quam acquirunt.

Superiora omnia 449. Atque hoc demum pacto quaecunque pertinent ad fluidorum, & solidorum diversa

Theori" ^ejus foe* genera, nam & eiastica, mollia, ductilia, fragilia eodem referuntur, invenimus omnia in

cunditas : ilia omnia illo particularum discrimine orto ex sola diversa combinatione punctorum, quam nobis

ajJensitate non pen- fheoria rite applicata exhibuit, in quibus omnibus immensa varietas itidem haberi poterit,

A THEORY OF NATURAL PHILOSOPHY 317

determination of the force of a fluid driving a solid in any directions, the measurement of
the velocity arising thence by means of the observed resistance of bodies placed in the way,
the flotation of bodies in fluids, & other things of the same kind, are all common to my
Theory. But it is necessary to distinguish which of them are only hypothetical & which
of them really occur In Nature.

446. To fluids & solids are to be referred all those matters, which in the second part other matters that
were demonstrated with regard to pressure of fluids, & velocity of efflux ; & all matters ^eere SecoCndSedpart
relating to equilibrium of solids, the lever, the centre of oscillation, & the centre of percussion ; really pertain to
all of which indeed are usually considered in connection with Mechanics. I will but add distincton^tween
that, from the ease of movement of the particles of a fluid about one another, & from their elastic & soft
irregular grouping, it readily follows that in them pressure must be propagated in every

direction. But I have now said enough about those matters that refer to solidity & fluidity ;
however, I will make a few remarks on matters that relate to the distinction between elastic
& soft bodies. Those bodies are elastic, which after change of shape return to their original
form ; & those are soft, which remain in their new state. This distinction my Theory
shows to be consequent upon the distance or closeness of the limit-points ; as I said in
Art. 199. If the limit-points, that are next to the one in which the particles cohere, are
far distant from it on either side, then, when the distance is much diminished, there
will still be a repulsive force all the time ; & if the distance is increased there will be a
similar attractive force. Hence, whether the mass is compressed more than is natural, or
expanded more than is natural, it will return to its original form. When it has returned to
its original form, it will go beyond it, until the velocity attained is cancelled by the opposite
force ; and a tremor, or oscillation, will be produced, which will be gradually diminished and
ultimately destroyed, partly by the action of external bodies, just as the motion of a pendulum
is stopped by the resistance of the air, & partly by the action of less elastic particles which
are interspersed, which can gradually break down the oscillation by their friction, & also
by contrary motions, & a relapse by which they change their own distribution somewhat.
But if these limit-points are fairly close, the external cause, which compresses or expands
the mass, after that it has brought the particles from one limit-point of cohesion to another,
will force them also to stay at the latter ; & these, when once grouped in this position, will
also be in equilibrium, & a soft mass will be the result.

447. The particles of some elastic fluids are not at limit-points of cohesion with respect Elastic fluids

r , ,. ,. -\-O-L whose particles are

to one another, but are at distances corresponding to repulsions, & these too very great ; not at limit-points
for instance, air. But recession is prevented either by superincumbent weight, or by of cohesion. AH

, . . <• • i j. • i j- 111 solids & fluids are

enclosing walls ; these are in some sort of violent condition at these distances, although really elastic, but
each point is always in equilibrium, due to the opposite repulsions cancelling one another. are not ,called so-

,, ,, ,.', „ n^. , 1-1 • i a • i because they do not

Moreover, all solids & fluids, which appear neither to suffer compression, nor to have any suffer sensible

mutual forces between their particles, but to be at limit-points, are however elastic ; that compression.

is to say, they exert a repulsive force between their adjacent particles ; at least those do

which are possessed of sensible gravitation, for it is this repulsive force that cancels the force

of gravity. The distances are in fact changed very slightly, the change being therefore

one that is beyond the scope of our senses. This is the case for water ; with it, the density is

practically the same at the bottom of a well as it is at the upper surface ; the same thing

happens in the case of metals & marbles & in all solid bodies, in which if a fairly large

weight is superimposed there is no sensible compression. But such things are not usually

termed elastic, for the reason that a repulsive force between adjacent particles, even if it

is very great, is not sufficient for such an appellation ; in addition, there is required to be a

sensible change of distance, compared with the whole distance, to correspond with a sensible

change in the forces.

448. There are in my Theory none of those bodies, that are hard in the sense in which Then5 are no hard
hardness is accepted by Physicists, namely such as do not suffer the slightest change of shape ; bodies are called
& also there are none that are perfectly compact, or quite solid, as I said in Art. 266. But haFd '• henc.e. fra-
those are usually termed hard, which exert a fairly great force to prevent change of form ; ^

they may be either elastic, fragile or soft. The source of fragility has been explained just
above, in Art. 437 ; & from this also the nature of ductility & malleability can be easily
understood. For instance, ductile & malleable solids only differ from one another in the
greater or less strength with which they preserve their form ; for, just as soft bodies under
slight pressure, even of the fingers, are compressed, or change their form, but retain the form
thus changed ; so ductile bodies under the stronger force of a blow with a 'mallet also
change their original shape, & retain the new form that they acquire.

449. Finally, in this way, whatever properties there may be relating to different kinds A11 *he above pro-
of fluids & solids (for elastic, soft, ductile & fragile bodies all come to the same thing), we fro'm^in^Tifeory ;
have made them all out from the difference between particles that is produced by the a11 of them do n°t
difference in the combination of the points alone ; this will be shown by my Theory if sity6"'

3i8 PHILOSOPHIC NATURALIS THEORIA

& debebit ; si curva primigenia ingentem habeat numerum intersectionum cum axe, &
particulse primi ordinis, ac reliquas ordinum superiorum dispositiones, quae in infinitum
variari possunt, habuerint plurimas, & admodum diversas inter se, ac eas inprimis, quae ad
haec ipsa figurarum, & virium discrimina requiruntur. Illud unum hie diligenter notandum
est, quod ipsam Theoriam itidem commendat plurimum, hasce proprietates omnes a densitate
nihil omnino pendere. Fieri enim potest, uti num. 183 notavimus, ut curva virium
primigenia limites, & arcus habeat quocunque ordine in diversis distantiis permixtos
quocunque numero, ut validiores, & minus validi, ac ampliores, & minus ampli commis-
ceantur inter se utcunque, adeoque phenomena eadem figurarum, & virium aeque inveniri
possint, ubi multo plura, & ubi multo pauciora puncta massam constituunt.

Communia quatuor 450. Jam vero ilia, qua; vulgo elcmenta appellari solent, Terra, Aqua, Aer, Ignis,

eiementa quid smt. njj1jj aliud mihi sunt, nisi diversa solida, & fluida, ex iisdem homogeneis punctis composita
diversimode dispositis, ex quibus deinde permixtis alia adhuc magis composita corpora
oriuntur. Et quidem Terra ex particulis constat inter se nulla vi conjunctis, quae solidi-
tatem aliarum admixtione particularum acquirunt, ut cineres oleorum ope, vel etiam
aliqua mutatione dispositionis internas, ut in vitrificatione evenit, quae transformationes
quo pacto accidant, dicemus postremo loco. Aqua est fluidum liquidum elasticitate carens
cadente sub sensum per compressionem sensibilem, licet ingentem exerceant repulsivam
vim ejus particulae, sustinentes velexternae vis, vel sui ipsius ponderis pressionem sine sensibili
distantiarum imminutione. Aer est fluidum elasticum, quern admodum probabile est
constare particulis plurimorum generum, cum e plurimis etiam fixis corporibus generetur
admodum diversis, ut videbimus, ubi de transformationibus agendum erit, ac propterea
continet vapores, & exhalationes plurimas, & heterogenea corpuscula, quae in eo innatant :
sed ejus particulae satis magna vi se repellunt, [207] & ea repulsiva particularum vis
imminutis distantiis diu perdurat, ac pertinet ad spatium, quod habet ingentem rationem
ad earn tanto minorem distantiam, ad quam compressione reduci potest, & in qua adhuc
ipsa vis crescit, arcu curvae adhuc recedente ab axe : is vero arcus ad axem ipsum deinde
debet ruere prasceps, ut circa proximum limitem adhuc ingentes in eo residue spatio
variationes in arcubus, & limitibus haberi possint. Porro extensionem tantam arcus repulsivi
evincit ipsa immanis compressio, ad quam ingenti vi aer compellitur, qui ut habeat com-
pressiones viribus prementibus proportionales, debet, ut monuimus num. 352, habere
vires repulsivas reciproce proportionales distantiis particularum a se invicem. Is autem
etiam in fixum corpus, & solidum transire potest, quod qua ratione fieri possit, dicam itidem,
ubi de transfoimationibus agemus in fine. Ignis etiam est fluidum maxime elasticum,
quod violentissimo intestine motu agitatur, ac fermentationem excitat, vel etiam in ipsa
fermentatione consistit, emittit vero lucem, de quo pariter agemus paullo inferius, ubi de
fermentatione, & emissione vaporum egerimus inter ea, quae ad Chemicas operationes
pertinet, ad quas jam progredior.

Chemicarum opera- 4.151. Chemicarum operationum principia ex eodem deducuntur fonte, nimirum ex

ducTTacffe^ex uio ^° particularum discrimine, quarum aliae inter se, & cum quibusdam aliis inertes, alias

particularum dis- ad se attrahunt, alias repellunt constanter per satis magnum intervallum, ubi attractio

ium'efiectuunfcau- ipsa cum aliis est major, cum aliis minor, aucta vero satis distantia, evadit ad sensum nulla ;

sas singuiares non quarum itidem aliae respectu aliarum habent ingentem virium alternationem, quam mutato

mente humana" * nonnihil textu suo, vel conjunctae, & permixtae cum aliis mutare possunt, succedente

pro particulis compositis alia virium lege ab ea, quae in simplicibus observabatur. Hasc

omnia si habeantur ob oculos ; mihi sane persuasum est, facile inveniri posse in hac ipsa

Theoria rationem generalem omnium Chemicarum operationum : nam singuiares deter-

minationes effectuum, qui a singulis permixtionibus diversorum corporum, per quas unice

omnia prasstantur in Chemia, sive resolvantur corpora, sive componantur, requirerent

intimam cognitionem textus particularum singularum, & dispositionis, quam habent in

massis singulis, ac prasterea Geometriae, & Analyseos vim, quae humanae mentis captum

excedit longissime. Verum illud in genere omnino patet, nullam esse Chemiae partem,

in qua praeter inertiam massae, & specificam gravitatem, alia virium mutuarum genera

inter particulas non ubique se prodant, & vel invitis incurrant in oculos, quod quidem

vel in sola postrema quaestione Opticae Newtoni abunde patet, ubi tam multa & attractionum,

A THEORY OF NATURAL PHILOSOPHY 319

properly applied, & in all such things also an immense variety can & must be produced.
Provided that the primary curve has a number of intersections with the axis, & provided
that particles of the first order, & the rest of the higher orders, have arrangements (which
indeed can be infinitely varied) that are great in number & all different from one another ;
& those especially that are required for these differences in shape & forces. Now, one
thing is at this point to be noted carefully, one that also supports the Theory itself very
strongly, namely, that all these properties are totally independent of density. For it is
possible that, as I mentioned in Art. 183, the primary curve of forces may have limit-points
& arcs mixed together in any order at different distances, and there may be any number
of either ; so that stronger & weaker limit-points, more & less ample arcs may be intermingled
in any manner amongst themselves ; & thus the same phenomena of shapes & forces can be
met with when the number of points constituting a mass is much larger or much smaller.

450. Now those things, which are commonly called the Elements, Earth, Water, Air The nature of the
& Fire, are nothing else in my Theory but different solids & fluids, formed of the same
homogeneous points differently arranged ; & from the admixture of these with others, called.

other still more compound bodies are produced. Indeed Earth consists of particles that
are not connected together by any force ; & these particles acquire solidity when mixed
with other particles, as ashes when mixed with oils ; or even by some change in their internal
arrangement, such as comes about in vitrification ; we will leave the discussion of the
manner in which these transformations take place till the end. Water is a liquid fluid devoid
of elasticity such as comes within the scope of the senses through a sensible compression ;
although there is a strong repulsive force exerted between its particles, which is sufficient
to sustain the pressure of an external force or of its own weight without sensible diminution
of the distances. Air is an elastic fluid, which in all probability consists of particles of
very many different sorts ; for it is generated from very many totally different fixed bodies,
as we shall see when we discuss transformations. For that reason, it contains a very large
number of vapours & exhalations, & heterogeneous corpuscles that float in it. Its particles,
however, repel one another with a fairly large force ; & this repulsive force of the particles
lasts for a long while as the distances are diminished, & pertains to a space that bears a very
large ratio to the so much smaller distance, to which it can be reduced by compression ;
& at this distance too the force still increases, the arc of the curve corresponding to it still
receding from the axis. But after that, the curve must return very steeply, so that in
the neighbourhood of the next limit-point there may yet be had in the space that remains
great variations in the arcs & the limit-points. Further such great extension of the repulsive
arc is indicated by the great compression induced by the pressure due to a large force ;
& this, in order that the compression may be proportional to the impressed force, shows,
as we pointed out in Art. 352, that there must be repulsive forces inversely proportional
to the distances of the particles from one another. Moreover it can pass into & through
a fixed & solid body ; & the reason of this also I will state when I deal with transformations
towards the end. Fire is also a highly elastic fluid, which is agitated by the most vigorous
internal motions ; it excites fermentations, or even consists of this very fermentation ; it
emits light, with which also we will deal a little later, when we discuss fermentation &
emission of vapours amongst other things referring to chemical operations ; to these we
will now pass on.

451. The principles of chemical operations are derived from the same source, namely, Th« different kinds
from the distinctions between particles ; some of these being inert with regard to themselves tions6™!^ readily
& in combination with certain others, some attract others to themselves, some repel others derived from the
continuously through a fairly great interval ; & the attraction itself with some is greater, p^tTcTe sTTh e
& with others is less, until when the distance is sufficiently increased it becomes practically special causes of
nothing. Further, some of them with respect to others have a very great alternation of

forces ; & this can vary if the structure is changed slightly, or if the particles are grouped intelligence of the

& intermingled with others ; in this case there follows another law of forces for the compound h

particles, which is different to that which we saw obeyed by the simple particles. If all

these things are kept carefully in view, I really think that there can be found in this

Theory the general theory for all chemical operations. For the special determination of

effects that arise from each of the different mixtures of the different bodies, through

which alone all effects in chemistry are produced, whether the bodies are resolved or

compounded, would require an intimate knowledge of the structure of each kind of particle,

& the arrangement of these in each of the masses ; &, in addition, the whole power of

geometry & analysis, such as exceeds by far the capacity of the human mind. But in

general it is quite evident that there is no part of chemistry, in which, in addition to

inertia of mass, & specific density, there are not everywhere produced other kinds of mutual

forces between the particles ; & these will meet our eyes without our looking for them,

as is indeed abundantly evident in the single question that comes last at the end of Newton's

320 PHILOSOPHISE NATURALIS THEORIA

& vero etiam repulsionum indicia, atque argumenta proferuntur. Omnia etiam genera
eorum, quae ad Chemiam pertinent, singillatim persequi, infinitum essct : evolvam
speciminis loco praecipua quaedam.

Quid sint : dissolu- [208] 45 2. Primo loco se mihi offerunt dissolutio, & ipsi contraria praecipitatio. Immissis in
pnma quomodo fiati qusedam fluida quibusdam solidis, cernimus, mutuum nexum, qui habebatur inter eorum
& quae sit ejus particulas, dissolvi ita, ut ipsa jam nusquam appareant, qua; tamen ibidem adhuc
manere in particulas perquam exiguas redacta, & dispersa, ostendit praecipitatio. Nam
immisso alio corpore quodam, decidit ad fundum pulvisculus tenuissimus ejus
substantiae, & quodammodo depluit. Sic metalla in suis quasque menstruis dissolvuntur,
turn ope aliarum substantiarum praecipitantur : aurum dissolvit aqua regia, quod immisso
etiam communi sale praecipitatur. Rei ideam est admodum facile sibi efformare satis
distinctam. Si particulae solidi, quod dissolvitur, majorem habent attractionem cum
particulis aquae, quam inter se ; utique avellentur a massa sua, & singulae circumquaque
acquirent, fluidas particulas, quae illas ambiant, uti limatura ferri adhaeret magnetibus,
ac fient quidam veluti globuli similes illi, quern referret Terra ; si ei tanta aquarum copia
affunderetur, ut posset totam alte submergere, vel illi, quern refert Terra submersa in
acre versus earn gravitante. Si, ut reipsa debet accidere, ilia vis attractiva in distantiis
paullo majoribus sit insensibilis ; ubi jam erit ad illam distantiam saturata eo fluido particula
solidi, ulterius fluidum non attrahet, quod idcirco aliis eodem pacto particulis solidi immersi
affundetur. Quare dissolvetur solidum ipsum, ac quidam veluti globuli terrulas suas
cum ingenti affusa marium vi exhibebunt, quae terrulae ob exiguam molem effugient nostros
sensus, nee vero decident sustentatae a vi, quae illas cum circumfuso mari conjungit : sed
globuli illi ipsi constituent quandam veluti continui fluidi massam. Ea est dissolutionis
idea.

Quomodo fiat pr«- 453. Quod si jam in ejusmodi fluidum immittatur alia substantia, cujus particulae

qUa l particulas fluidi ad se magis attrahant, & fortasse ad majores etiam distantias, quam
attrahuntur ab illis ; dissolvetur utique hasc secunda substantia, & circa ipsius particulas
affundentur particulae fluidi, quae prioris solidi particulis adhaerebant, ab illis avulsae, &
ipsis ereptae. Illae igitur nativo pondere intra fluidum specifice levius depluent, & habebitur
praecipitatio. Pulvisculus autem ille veterem particularum suarum nexum non acquiret
ibi per sese, vel quia & gluten fortasse aliquod admodum tenue, quo connectebantur invicem,
dissolutum simul jam deest in superficiebus illis, quarum separatio est facta, vel potius
quia, ut ubi per limam, per tunsionem, vel aliis similibus modis solidum in pulverem
redactum est, vel utcunque confractum, juxta ea, quae diximus num. 413, non potest iterum
solo accessu, & appressione deveniri ad illos eosdem limites, qui prius habebantur.

piuviam fortasse ACA Hoc pacto dissolutionis, & praecipitationis acquiritur idea admodum distincta ;

esse quoddam pra- „ r -i jj • : . ,

cipitationis genus : & fortassc etiam pluvia est quoddam praecipitationis genus, nee provenit e sola unione par-
mira phenomena f20Ql-ticularum aquas, quae prius tantummodo dispersae temere fuerint, & ob solam

commixtionum * • f. • • A -i • •

quomodo expiicen- tenuitatem suam sustentatae ac suspenses innatavermt. Apparet ibi etiam, qua ratione
tur- binae substantiae commisceri possint, & in unam massam coalescere. Id quidem in fluido

commixto cum solido patet ex ipso superiore exemplo solutionis. In binis fluidis facile
admodum fit, & si sint ejusdem ad sensum specificae gravitatis, solo motu, & agitatione
impressa fieri quotidie cernimus, ut in aqua, & vino, sed etiam si sint gravitatum admodum
diversarum, attractione particularum unius in particulas alterius fieri potest unius dissolutio
in altero, & commixtio. Fieri autem potest, ut ejusmodi commixtione e binis etiam fluidis
oriatur solidum, cujusmodi exempla in coagulis cernimus : & in Physica illud quoque
observatur quandoque, binas substantias commixtas coalescere in massam unicam minorem
mole, quam fuerit prius, cujus phaenomeni prima fronte admodum miri in promptu est
causa in mea Theoria : cum particulae, quae nimirum se immediate non contingcbant,
aliis interpositis possint accedere ad se magis, quam prius accesserint. Sic si haberetur
massa ingens elastrorum e ferro distractorum, quorum singulis inter cuspides adjungerentur
globuli magnetici ; hac nova accessione materiae minueretur moles, victa repulsione mutua

A THEORY OF NATURAL PHILOSOPHY 321

Optics, where there are many indications of both attractions & repulsions as well, & arguments
are brought forward with regard to them. Further, to investigate separately all matters
that relate to chemistry would be an endless task ; so I will discuss certain of the more
important, by way of example.

452. In the first place there occur to me solution & its converse, precipitation. When The nature of soiu-

TJ ,. , . r, . , • n • i i i • i'ii tlon & precipita-

certam solids are mixed with certain nuids, we see that the mutual connection which there tion ; how the first

used to be between the particles of each is dissolved in such a way that the solids are no comes about, & its

longer visible ; & yet that they are still there, reduced to extremely small particles &

dispersed, is shown by precipitation. For, if a certain other body is introduced, there falls

to the bottom an extremely fine powder of the original solid, as if it rained down. So metals,

each in its own solvent, dissolve, & with the help of other substances are precipitated.

"Aqua regia " dissolves gold; & this, on the addition of common salt, is precipitated. It

is quite easy to get a clear idea of the matter. Suppose that the particles of the solid have

a greater attraction for the particles of the water than for one another ; then they will

certainly be torn away from their own mass, & each of them will gather round itself fluid

particles, which will surround it, in the same manner as iron filings adhere to a magnet ; &

each would become something in the nature of little spheres similar to what the Earth

would resemble, if a sufficiency of water were to be poured over it to submerge it deeply,

or to what the Earth does resemble, submerged as it is in the air gravitating towards it.

If, as is bound to happen, the attractive force becomes insensible at distances a little greater,

then, when a particle of a solid has become saturated to that distance with the fluid, it

will no longer attract the fluid ; & therefore the latter will surround other particles of

the immersed solid in the same manner. Hence the solid will be dissolved, & each of

the little spheres, so to speak, would represent a little earth with its great abundance of sea

surrounding it ; & these little earths, on account of their exceedingly small volume will

escape our notice ; & they cannot fall, sustained as they are by the force that connects

them with the sea which surrounds them. Now these little globes themselves form a

certain mass of as it were continuous fluid ; hence we get an idea of the nature of solution.

453. If now another substance is introduced into a fluid of this kind, the particles The manner in
of which attract the particles of the fluid to themselves with a stronger force, & perhaps occurs ; sfits cause.
too at greater distances, than they are attracted by the particles of the first solid ; then

this second solid will be dissolved in every case, & its particles will be surrounded by the
particles of the fluid, which formerly adhered to the particles of the first solid, being torn
away from the latter & seized by the particles of the second solid. The particles of the
first solid will then rain down on account of their own weight within the fluid which
is specifically lighter, & there will be precipitation. Further, the fine powder will not
of itself then acquire the former connection between its particles ; this may be because
a sort of very thin cement, by which the particles were connected together, has perhaps
been at the same time dissolved, & this is now absent from the surfaces which have been
separated ; but more probably it is because, just as when, by means of a file or a hammer
or the like, a solid has been reduced to powder, or broken up in any manner, it cannot by
mere approach & pressing together get back once more to the same limit-points as before,
as I said in Art. 413.

454. In this way a perfectly clear idea of solution & precipitation is acquired. Perhaps Perhaps rain is

, TJ.T . f f • i . r , ^ . , . ,r some sort of preci-

also ram is some sort of precipitation, & does not merely come Irom the union oi particles pitation; how
of water which previously had been only dispersed at random, & had floated, sustained & certain wonderful

, , . . r . J. . . J r . , A i i phenomena in con-

suspended in the air, owing to their extreme tenuity alone. Also, we can now see how nection with mix-
two substances can be mixed together to coalesce into a single mass. This indeed, in the ^ures are exPlained-
case of a fluid mixed with a solid, is evident from the example of solution given above.
It takes place quite easily in the case of two fluids, &, if they are practically of the same
specific gravity, we see it happening every day by mere motion & the agitation impressed ;
as in the case of water & wine. But even if their specific gravities are quite different,
by the attraction of the particles of the one upon the particles of the other, there may
be solution of the one in the other, & thus a mixture of the two. Further, it may happen
that from a mixture of this kind, even of two fluids, there may be produced a solid ; we
see examples of such a thing in rennet. In Physics also, it is observed sometimes that two
substances mixed together coalesce into a single mass having a smaller volume than before ;
the cause of this phenomenon, which at first sight appears wonderful, is to be found
immediately with my Theory. For, the particles, which originally did not immediately
touch one another, when others are interposed, may approach nearer to one another than
they did before. Thus, if we have a large heap of springs made of iron, & to them we
add a number of little magnetic spheres, placing one between the tips of each spring ;
then, with this fresh addition of matter, the whole volume is diminished, the mutual

322 PHILOSOPHIC NATURALIS THEORIA

per attractionem magneticam, qua cuspides elastrorum ad se invicem accederent.
Cur ad commix- 455. Ubi solidum cum solido commiscendum est, ut fiat unica massa, ibi quidem
requiratur80 'cent™ oportet solida ipsa prius contundere, vel etiam dissolvere, ut nimirum exiguae particulae
sio : quid ad earn seorsim possint ad exiguas alterius solidi accedere, & cum iis conjungi. Id autem fit
"e- potissimum per ignem, cujus vehementi agitatione, & vero etiam fortasse actione ingenti
mutua inter ejus particulas, & inter quaedam peculiaria substantiarum genera, ut olea,
& sulphur, quas ut gluten quoddam conjungebant inter se vel inertes particulas, vel etiam
mutua repulsione prasditas, dissolvit omnium corporum nexus mutuos, & massas omnes
demum, si satis validus sit, cogit liquari, & ad naturam fluidorum accedere. Dissolutarum,
ac liquescentium massarum particulae commiscentur, & in unam massam coalescunt : ubi
autem sic coaluerunt, possunt iterum saepe dissimiles separari eadem actione ignis, qui
aliquas prius, alias posterius, cogit minore vi abire per evaporationem, & maxime fixas
majore vi reddit volatiles. Inaequalibus ejusmodi diversarum substantiarum attractionibus,
& inaequalibus adhaesionibus inter earum particulas, omnis fere nititur ars separandi metalla
a terris, cum quibus in fodinis commixta sunt, & alia aliorum ope prius uniendi, turn etiam
a se invicem separandi, quas omnia singillatim persequi infmitum foret. Generalis omnium
explicatio facile repetitur ab ilia, quam exposui, particularum diversa constitutione, quarum
alias respectu aliarum inertes sunt, respectu aliarum activitatem habent, sed admodum
diversam, turn [210] quod pertinet ad directionem, turn quod ad intensitatem virium.

TO?ataVza!tionem 45^ ^e Liquatione, & volatilizatione dicam illud tantummodo, eas fieri posse etiam
fieri posse per agita- sola ingenti agitatione particularum fluidi cujuspiam tenuissimi, cujus particulae ad solidi,
p'ar ticuTfr'um1 ^ ^x* corPoris particulas accedant satis, & inter ipsarum etiam intervalla irrumpant ; qui
Prima quomodo fiat] motus intestinus, unde haberi possit, jam exponam, ubi de fermentatione egero, & effer-
vescentia. Nam inprimis ea intestina agitatione induci potest in particulas corporis solidi,
& fixi motus quidam circa axes quosdam, qui ubi semel inductus est, jam illae particulas
vim exercent circunquaque circa ilium axem ad sensum eandem, succedentibus sibi invicem
celerrime punctis, & directionibus, in quibus diversae vires exercentur, qui etiam axes si
celerrime mutentur, irregulari nimirum impulsu, habebitur in iis particulis id, quod
asquivaleat sphaericitati & homogeneitati particularum, ex qua fluiditatem supra repetivimus,
atque hujus ipsius rei exemplum habuimus num. 237 in motu puncti per peripheriam
ellipseos, cujus focos bina alia puncta occupent. Haec fluiditas erit violenta, & desinente
tanta ilia agitatione, ac cessante vi, quae agitationem inducebat, cessabit, ac fluidum etiam
sine admixtione novas substantiae poterit evadere solidum. Poterit autem paullatim
cessare motus ille rotationis tarn per inasqualitatem exiguam, quae semper remanet inter
vires in diversis locis particulas diversas, & obsistit semper nonnihil rotationi, quam per
ipsam expulsionem illius agitatae substantiae, ut igneas, & per resistentiam circumjacentium.

Aiialiquationis .57. Deinde haberi etiam poterit liquatio per subtractionem heterogenearum, &

ratio per separa- , . „ T-> '. . , * . , / , , . . . ' ,

tionem partium dirtormium particularum, quae magis nomogeneas, & ad spnasricitatem accedentes particulas
heterogenearum. alligabant quodammodo impedito motu in gyrum. Id sane videtur accidere in pluribus
substantiis, quae quo magis depurantur, & ad homogeneitatem reducuntur, eo minus
tenaces evadunt, & viscosae. Sic viscositas est minima in petroleo, major in naphtha, &
adhuc major in asphalto, aut bitumine, in quibus substantiis Chemia ostendit, eo majorem
haberi viscositatem, quo habetur major compositip.

taizatic?" &Kitio0l& ^' Quoc^ s^ Pri°re modo liquatio accidat, & in eo motu particulae a limitibus
voiatiiizatio aeri's. cohaesionis, in quibus erant, abeant ad distantias paullo majores, in quibus habeatur ingens
repulsivus arcus, se repente fugient, quo pacto corpus fixum evadet volatile. Eandem
autem volatilitatem acquiret ; si particulae quae fixum corpus componebant, erant quidem
inter se in distantiis repulsionum validissimarum, sed per interjacentes particulas alterius
substantias cohibebatur ilia repulsiva vis superata ab attractione, quam exercebat
in eas nova intrusa particula : si enim haec agitatione ilia excutiatur, vel ab alia, quas
ipsam attrahat magis, praetervolante ad exiguam distantiam abripia-[2ii]-tur ; turn
vero repulsiva vis particularum prioris substantiae reviviscit quodammodo, & agit,
ac ipsa substantia evadit volatilis, quae iterum nova earundem particularum intrusione
figitur. Id sane videtur accidere in acre, qui potest ad fixum redigi corpus, & Halesius

A THEORY OF NATURAL PHILOSOPHY 323

repulsion being overcome by the magnetic attraction, with which the tips of the springs
would approach one another.

455. When a solid has to be mixed with a solid to form a single mass, it is necessary Why crushing is
to first of all crush the solids, or even to dissolve them, so that the exceedingly small particles fixture 'o^soHds*
of the one can separately approach those of the other solid, & combine with them. Now the effect of fire in
this especially takes place in the case of fire ; by its vigorous internal movement, & perhaps thefartof separating
too through a very great mutual attraction between its particles & those of certain particular metals,
kinds of substance, like oils & sulphur, these two causes acting as a sort of cement to join
together either inert particles, or even particles possessed of a mutual repulsion, fire dissolves
the mutual connections of all bodies & finally forces, if it is sufficiently powerful, all masses
to melt, & to approach fluids in their natures. The particles of the masses thus dissolved
& in a molten condition mingle together & coalesce into one single mass. Moreover, after
they have thus coalesced, the dissimilar substances can once more be separated by the
same action of fire, which forces, some at first & others later, the particles to go off, with
a smaller force through evaporation, & renders volatile the most refractory particles when
the intensity is greater. Upon the unequal attractions of different substances of this kind,
& upon the unequal adhesions between their particles, depends almost entirely the art
of separating metals from the earths with which they are mixed in the ores ; & some metals
from others, by means of first uniting them & then separating them once more ; but to
investigate all these matters singly would be an endless task. The general explanation
of them all is easily derived from that diverse constitution of the particles that I have
expounded ; namely, that some particles are inert with respect to others, & have activity
with respect to yet others ; where this activity is altogether varied, both as regards the
directions, & as regards the intensities, of the forces.

4156. With regard to liquefaction & volatilization, I will only say this : that these Liquefaction & voi-

1 • i . i i • i • . t n • J atilization can take

phenomena can take place simply through a violent agitation of some very tenuous fluid, piace owjng to a

whose particles approach sufficiently close to the particles of the solid fixed body, & push very great agitation

into the intervals between them. How this internal motion can happen I will explain, manner3 in6 which

when I discuss fermentation & effervescence. First of all, owing to the internal agitation, the first happens.

there can be induced in the particles of the solid fixed body motions about certain axes ;

& when these motions have once been set up, the particles will exert a rotary force about

the axis which is practically uniform, the points following one another extremely quickly,

& also the directions in which the different forces are exerted ; & if these axes are also

changed very rapidly, due, say to an irregular impulse, we shall have in the particles what

is equivalent to the sphericity & homogeneity of particles, from which we have derived

fluidity in a preceding article ; we had also an example of this kind of thing, in Art. 237,

in the motion of a point along the perimeter of an ellipse, of which two other points occupied

the foci. This fluidity will be very violent, &, as soon as the great agitation ends & the

force which caused the agitation ceases, the agitation will cease as well, & the fluid will

be able to become solid once more, without the admixture of any fresh substance. Further,

this motion of rotation may gradually cease, owing not only to the slight inequality that

will always remain between the different forces at different places of a particle, ever tending

to hinder the rotation to some extent, but also to the expulsion of the substance in agitation

(fire, say), & through the resistance of the particles lying in the neighbourhood.

4157. Secondly, there may be liquefaction through the subtraction of heterogeneous Another reason

10 ' . ,. ' ' . , i • i i ^ i i 6 i • i i • i_f°r liquefaction is

& non-uniform particles, which bound together the more homogeneous particles which through the separa-
approximate to sphericity, in such a way as to hinder their rotary motion. This is in tlon of heterogene-

i • i i'ii i • - • • • i_ ous parts.

fact seen to happen m several substances, which become less tenacious & viscous, the
more they are purified & reduced to homogeneity. Thus the viscosity is very small in
rock-oil, greater in naphtha, still greater in asphalt or bitumen ; &, in these substances,
chemistry shows that the viscosity is the greater, the more compound the substance.

458. But if liquation should take place in the first manner, & due to the motion the How volatilization

.', i i i n- r ii-'' i'ii T vi takes place ; fixa-

particles should go on from the limit-points at which they were to distances a little greater, tjon & volatilization

& if for these distances there should be a very large repulsive arc, then the particles will fly off °J air-

with great speed ; & in this way a fixed body will become volatile. Moreover it will acquire

the same volatility, if the particles which form the body were at such distances from one

another as correspond to very strong repulsions, but are held together by intervening

particles of another substance, the repulsive force being overcome by the attractions exerted

upon them by the new particles that have been introduced between them. For, if these are

displaced by the agitation, or are seized by others, which attract them more strongly, as

they fly past at a slight distance, then the repulsive force of the first substance will revive,

as it were, & come into action ; & the substance will become volatile, & will once again

become fixed on a fresh introduction of the same intervening particles. This in fact is

seen to happen in the case of air, which can be reduced to a fixed body. Hales has proved

324 PHILOSOPHIC NATURALIS THEORIA

demonstravit per experimenta, partem ingentem lapidum, qui in vesica oriuntur, &
calculorum in renibus constare puro acre ad fixitatem reducto, qui deinde potest iterum
statum volatilem recuperare : ac halitus inprimis sulphurei, & ipsa respiratio animalium
ingentem aeris copiam transf ert a statu volatili ad fixum. Ibi non habetur aeris compressio
sola facta per cellularum parietes ipsum concludentes ; ii enim disrumperentur penitus, cum aer
in ejusmodi fixis corporibus reducatur ad molem etiam millecuplo minorem, in quo statu, si in-
tegras haberet elasticas vires, omnia sane repagula ilia diffringeret. Halesius putat, eum in illo
statu amittere elasticitatem suam, quod fieret utique, si particulse ipsius ad earn inter se dis-
tantiam devenirent, in qua jam vis repulsiva nulla sit, sed potius attractiva succedat : sed fieri
itidem potest, ut vim quidem repulsivam adhuc ingentem habeant illae particulae, sed
ab interposita sulphurei halitus particula attrahantur magis, ut paullo ante vidimus in
elastris a globulo magnetico cohibitis, & constrictis. Turn quidem elasticitas in aere ad
fixitatem redacto maneret tola, sed ejus effectus impediretur a prasvalente vi. Atque id
quidem animadverti, & monui ante aliquot annos in dissertatione De Turbine, in qua omnia
turbinis ipsius phenomena ab hac aeris fixatione repetii.

459 P°rro agitatio ilia particularum in igne, ac in fermentationibus, & effervescentiis,
igne, fermenta- unde oriatur, facile itidem est in mea Theoria exponere. Ut primum crus meae curvse
^ impenetrabilitatem exhibuit, postremum gravitatem, intersectiones autem varia
contorsione curvae cohaesionum genera ; ita alternatio arcuum jam repulsivorum, jam attractivorum,
fermentationes exhibet, & evaporationes variorum generum, ac subitas etiam deflagrationes,
& explosiones, illas, quae occurrunt in Chemia passim, & quam in pulvere pyrio quotidie
intuemur. Quas autem hue ex Mechanica pertinet, jam vidimus num. 199. Dum ad
se invicem accedunt puncta cum velocitate aliqua, sub omni arcu attractive velocitatem
augent, sub omni repulsive minuunt : contra vero dum a se invicem recedunt, sub omni
repulsive augent, sub omni attractive minuunt, donee in accessu inveniant arcum
repulsivum, vel in recessu attractivum satis validum ad omnem velocitatem extinguendam.
Ubi eum invenerint, retro cursum reflectunt, & oscillant hinc, & inde, in quo itu, & reditu
perturbato, ac celeri, fermentationis habemus ideam satis distinctam.

cr 46°- Et in accessu quidem semper devenitur ad arcum repulsivum aliquem parem

semper .• . .J. r . >j.j

sisti a primo crure extinguendas velocitati cuinbet utcun-[2i2j-que magnae ; devenitur enim saltern ad

cessuS1Vbinipr<casus" Primum asymptoticum crus, quod in infinitum protenditur : at pro recessu duo hie casus

in primo cruris occurrunt potissimum considerandi. Vel enim etiam in recessu devenitur ad aliquod

UcTsemperasStiPre- crus asymptoticum attractivum cum area infinita, de cujusmodi casibus egimus jam num.

cessum etiam. 195, vel devenitur ad arcum attractivum recedentem longissime, & continentem aream

admodum ingentem, sed finitam. In utroque casu actio punctorum, quae extra massam

sunt sita, aliorum punctorum massse intestine illo motu agitatae oscillationem augebit

aliorum imminuet, & puncta alia post alia procurrent ulterius versus asymptotum, vel limitem

terminantem attractivas vires : quin etiam actiones mutuae punctorum non in directum

jacentium in massa multis punctis constante, mutabunt sane singulorum punctorum

maximos excursus hinc, & inde, & variabunt plurimum accessus mutuos, ac recessus, qui

in duobus punctis solis motum habentibus in recta, quae ilia conjungit, deberent, uti

monuimus num. 192, sine externis actionibus esse constantis semper magnitudinis. In

accessu tamen in utroque casu ad compenetrationem sane nunquam deveniretur : in

recessu vero in primo casu cruris asymptotici, & attractionis in infinitum crescentis cum

area curvae in infinitum aucta, itidem nunquam deveniretur ad distantiam illius asymptoti.

Quare in eo primo casu utcunque vehemens esset interna massae fermentatio, utcunque

magnis viribus, ab externis punctis in majore distantia sitis perturbaretur eadem massa,

ipsius dissolutio per nullam finitam vim, aut velocitatem alteri parti impressam haberi

unquam posset.

in secundo casu tfa At in secundo casu, in quo arcus attractivus ille ultimus ems spatii ingens esset,

arcus attractivi , ,7 . . ..,,.. J r.

ingentis, sed finiti sea nnitus, posset utique quorundam punctorum in ilia agitatione augen excursus usque

egressus partis ac[ limitem, post quern limitem succedente repulsione, iam illud punctum a massa ilia
punctorum excus- ,r . . 11 n- «• •

sorum e fine oscil- quodammodo velut avulsum avolaret, & motu accelerate recederet. 01 post eum limitem

latioms sine re- summa arearum repulsivarum esset maior, quam summa attractivarum, donee deveniatur
gressu. , ... r. . J . ?• .

ad arcum mum, qui gravitatem exprirmt, in quo vis jam est perquam exigua, & area
asymptotica ulterior in infinitum etiam producta, est finita, & exigua ; turn vero puncti"
elapsi recessus ab ilia massa nunquam cessaret actione massse ipsius, sed ipsum punctum
pergeret recedere, donee aliorum punctorum ad illam massam non pertinentium viribus
sisteretur, vel detorqueretur utcunque. In fortuita autem agitatione interna, ut & in

A THEORY OF NATURAL PHILOSOPHY

325

by means of experiments that the great part of stones, that are produced in the bladder,
& of the small ones in the kidneys, consists of pure air reduced to fixation ; & that this
can once again recover its volatile state. In this case the compression of the air is not
obtained simply by the boundaries that enclose it ; for these would be completely broken
down, since the air in such fixed solids is reduced to a volume that is even a thousand times
less ; & in this state, if the elastic forces still were unimpaired, all restraints would be easily
overcome. Hales thought that, when in this state, it loses its elasticity ; & this would
indeed happen if its particles attained that distance from one another, in which there is no
repulsive force, but rather an attractive force succeeds the repulsive force. It might also
happen that these particles still possess a very large repulsive force, but by the interposition
of particles of a sulphurous vapour they are attracted to a greater extent than they are
repelled ; as just above we saw was the case for springs restrained & constricted by little
magnetic spheres. Then, indeed, the elasticity in air reduced to fixity would remain
unaltered, but its effect would be prevented by a superior force. I considered this point
of view & mentioned it some years ago in my dissertation De Turbine, in which all the
phenomena of the whirlwind are derived from this fixation of the air.

459. Further, the source of the agitation of the particles in fire, fermentation, & Cause of the agita-
effervescence is also easily explained by my Theory. Just as the first branch of my curve *ion °f th<? particles

, .,. J / ', . 7 . J „ , . .,','. m fire, fermenta-

gives me impenetrability, & the last branch gravitation, & the intersections with the axis tions, & efferves-
the various kinds of cohesions ; so also the alternation of the arcs, now repulsive, now fence- derived from

.. . ,....' t, ' * the contortions of

attractive, represent fermentations & evaporations of various kinds, as well as sudden the curve round

conflagrations & explosions ; such things as occur everywhere in chemistry, & what we see the axls-

every day in the case of gunpowder. Those things from Mechanics that belong here

we have already seen in Art. 199. So long as points approach one another with any velocity,

they increase the velocity under every attractive arc, & diminish it under every repulsive

arc. On the other hand, so long as they recede from one another, they increase the velocity

under every repulsive arc & increase it under every attractive arc ; until, in approach,

they come to a repulsive arc, or in recession, to an attractive arc, which is sufficiently strong

to destroy the whole of the velocity. When they have reached this, they retrace their

paths, & oscillate backwards & forwards ; & in this, the backward & forward motion being

perturbed & rapid, we have a sufficiently clear notion of what fermentation is.

460. Now, on approach, there is always reached some repulsive arc or other, which Oscillations on ap-
is capable of destroying any velocity however great ; for at least finally the first asymptotic stopped aib y^hat
branch, which goes off to infinity, is reached. But on recession, there are two cases met first repulsive
with, which have to be considered in this connection. For, on recession, either there is cession' ^here^are
reached an asymptotic attractive branch having an infinite area, cases of which kind I two cases, in the
dealt with in Art. 194 ; or else we come to an attractive arc receding very far from the an^^nrptot^at-
axis, & containing an exceedingly great but finite area. In either case, the action of points tractive branch,
situated outside the mass will increase the oscillation of some of the points of the mass aiw^TsWped 1S
that is agitated by the internal motion, &. will diminish that of other points ; & one point

after another will go off beyond the mass towards the asymptote, or the limit-point bounding
the attractive forces. Moreover, the mutual actions, of points not lying in the same straight
line in a mass consisting of many points, will change considerably the largest oscillations
of each of the points ; especially will they alter their mutual approach & recession, which
for two points only, having a motion in the straight line joining them, must be, except
for external action, always of constant magnitude, as I remarked in Art. 192. On approach,
however, in either case, the position corresponding to compenetration can never really
be reached. But, on recession, in the first case, where there is an asymptotic branch, & an
attraction indefinitely increased along with an area of the curve also increasing indefinitely,
in this case also it can never attain the distance of that asymptote. Hence, in the first
case, however fierce the internal fermentation of the mass may be, no matter with how
great forces from external points situated further off the mass may be affected, its dissolution
can never be effected by any finite force, or velocity impressed on any one part of it.

461. Now, in the second case, in which the attractive arc at the end of the space is in the second case,
very large, but finite, it will indeed be possible for the motion of some points in the veiy great ebutfiriite
agitation to be increased right up to the limit-point ; &, as repulsion follows the limit-point attractive arc, there
that point of the mass will now be as it were torn off, & it will fly away & leave the mass Wf som^T^'he
with accelerated motion. If after the limit-point, the sum of the repulsive areas should points at the end
be greater than the sum of those that are attractive, that is, until that arc is reached which these" °wui atflyn'ofE
represents gravity, where the force then becomes exceedingly small, & the asymptotic area, without returning,
when produced still further, is finite & very small ; then indeed the recession of the point

that has left the mass will never cease owing to any action of the mass itself, but the point
will go on receding, until it is stopped by the forces from other points not belonging to that
mass, or its path is contorted in some manner. Moreover, in irregular internal agitation,

326 PHILOSOPHISE NATURALIS THEORIA

externa perturbatione fortuita, illud accidet, quod in omnibus fortuitis combinationibus
accidit, ut numerus casuum cujusdam dati generis in dato ingenti numero casuum aeque
possibilium dato tempore recurrat ad sensum idem, adeoque effluxus eorum punctorum,
si massa perseveret ad sensum eadem, erit dato tempore ad sensum idem, vel, massa multum
imminuta, imminuetur in aliqua ratione [213] massae, cum a multitudine punctorum
pendeat etiam casuum possibilium multitudo.

Jam P^UI"ima considerari possent. & casuum differentium, ac combinationum
evaporatio lenta. numerus in immensum excrescit ; sed pauca quaedam adnotabimus. Ubi intervallum,
quod massam claudit inter limites accessus, & recessus, est aliquanto majus, & posteriorum
arearum repulsivarum summa non multum excedit summam attractivarum, fiet paullatim
lenta quasdam evaporatio : puncta quae in fortuita agitatione ad eum finem deveniunt,
erunt pauca respectu totius massas, quae tamen in ingenti massa, & eodem fermentationis
statu erunt eodem tempore ad sensum aequali numero, ac, massa imminuta, imminuetur
& is numerus, massa autem diu perseverabit ad sensum nihil mutata. Habebitur ibi quaedam
velut ebullitio, & vaporum quantitas, ac vis in egressu in diversis substantiis variari plurimum
poterit, cum pendeat a situ, in quo ilia puncta collocata sint intra curvam : nam possunt
in aliis substantiis esse citra alios ingentes arcus attractivos, quorum posteriores vel sint
prioribus minus validi, vel arcus repulsivos se subsequentes minus validos habeant.

Vel subita explosio, ,g, gecj sj intervallum, quod massam claudit inter limites accessus, & recessus, sit

& deflagratio ; ac T J * . . . '

transformationes perquam exiguum, arcus attractivus postremus non sit ita validus, & succedat arcus repui-
vajiz>. avolante sivus validissimus ; fieri utique poterit, ut massa, quae respective quiescebat, adveniente,
exiguo motu a particulis externis satis proxime accedentibus, ut possint inaequalem motum
imprimere punctis particularum massse, agitatio ejusmodi in ipsa massa oriatur, qua
brevissimo tempore puncta omnia transcendant limitem, & cum ingenti repulsiva vi, ac
velocitate a se invicem discedant. Id videtur accidere in explosione subita pulveris pyrii,
qui plerumque non accenditur contusione sola ; sed exigua scintilla accedente dissilit fere
momento temporis, & tanta vi repulsiva globum e tormento ejicit. Idem apparet in iis
phosphoris, quae deflagrant solo aeris contactu : ac nemo non videt, quanta in iis omnibus
haberi possunt discrimina. Possunt nimirum alia facilius, alia difficilius deflagrare, alia
serius, alia citius : potest sine lenta evaporatione solvi tota massa tempore brevissimo ;
potest, ubi massa fuerit heterogenea, avolare unum substantiae genus aliis remanentibus.
& interea possunt ex iis, quae- remanent, fieri alia mixta admodum diversa a praecedentibus,
mutato etiam textu particularum altiorum ordinum per id, quod plures particulas ordinum
inferiorum, quas pertinebant ad diversas particulas superiorum, coalescant in particulam
ordinis superioris novi generis : hinc tarn multae compositiones, & transformationes in
Natura, & in Chemia inprimis : hinc tarn multa, tarn diversa vaporum genera, & in aere
elastico a tam diversis corporibus fixis genito tantum discrimen. Patet ubique immensus
excursui campus : sed eo relicto [214] progredior ad alia nonnulla, quae ad fermentationes,
& evaporationes itidem pertinent.

4^4" Substantia, quae fuerat dissoluta, non solum per praecipitationem colligitur iterum,
figur» residui, ut in ut ubi metalla cadunt suo pondere in tenuem pulvisculum redacta ; sed etiam per evapor-
sahbus. ationem, ut diximus, in salibus, qui evaporato illo fluido, in quo fuerant dissoluti, remanent

in fundo. Et quidem sales non remanent sub forma tenuis pulvisculi, particulis minutissimis
prorsus inertibus, sed colliguntur in massulas grandiusculas habentes certas figuras quae
in aliis salibus aliae sunt, & angulosas in omnibus, ac in maxime corrosivis horrendum in
modum cuspidatae, ac serratse, unde & sapores salium acutiores, & aliquorum ex iis, quas
corrosiva sunt, fibrillarum tenuium in animantibus proscissio, ac destructio organorum
necessariorum ad vitam. Quo autem pacto eas potissimum figuras induere possint, id
patet ex num. 439, ut & figuras crystallorum & succorum, ex quibus gemmae, & duri lapides
fiunt ubi simplices sunt, & suam quique figuram affectant, ac aliorum ejusmodi, quae post
evaporationem concrescunt, haberi utique possunt, ut ibidem ostensum est, per hoc, quod
in certis tantummodo lateribus, & punctis particulae alias particulas positas ad certas
distantias attrahant, adeoque sibi adjungant certo illo ordine, qui respondet illis punctis,
vel lateribus.

A THEORY OF NATURAL PHILOSOPHY 327

just as also in irregular external perturbation, the same thing happens, as always does happen
in irregular combinations ; namely, out of a given very large number of cases of a given
kind, all equally possible, the same number of cases will recur in any given interval of time.
Hence, so long as the mass remains practically the same, there will be the same number
of points going off ; £ when the mass is much diminished this number will also be diminished
in some way proportional to the mass ; for on the number of points depends also the number
of possible cases.

462. We may now consider a very large number of matters; & indeed the number Hence from a differ
of different cases & combinations increases immensely ; but we will only mention just a come

few of them. When the interval, which encloses the mass between limits of approach ation.
& recession, is somewhat large, & the sum of the later repulsive areas does not greatly exceed
that of the attraction, then a slow evaporation will take place. Points which, in the irregular
agitation, arrive at the outside, will be few in comparison with the whole mass ; & yet
these, in a very large mass, in the same state of fermentation, will be practically of
the same number in the same time ; & this number will be diminished if the mass is
diminished, but the mass itself will remain for a long time practically unaltered. Then
there will be a sort of ebullition ; & the amount of the vapour, & the force on egress
may be very different in different substances ; for it will depend on the position at which
the points are situated within the curve. In some substances they may be on the near
side of some, & in others of other, very great attractive arcs ; & of these the later arcs may
be either less powerful than those in front, or they may have less powerful repulsive arcs
following them.

463. But if the interval, which encloses the mass between limits of approach & recession °r there may be
should be exceedingly small, the last attractive arc may not be so very strong, & a very ^ deflagration^*
strong repulsive arc may follow it. Then indeed, it may happen that, as the mass, which various transfor-
was in a state of relative rest, coming up to the limit with but a slight motion due to oTthTmLxLre flies
external points approaching close enough to it to be capable of impressing a non-uniform off-
motion on the points of the particles, an agitation within the mass will be produced of

such a kind that owing to it all the points in an extremely short time will cross the limit,
& then they will fly off from one another with a huge repulsive force & a high velocity.
This kind of thing is seen to take place in the sudden explosion of gunpowder, which
commonly is not set on fire by a blow alone ; but on contact with the smallest spark goes
off almost at once, & with a very great repulsive force drives out the ball from the cannon.
The same thing is seen in phosphorous substances, which go on fire merely on contact with the
air ; & nobody can fail to see the differences that may exist in all these things. Thus,
some of them go on fire comparatively easily, others with greater difficulty, some slowly &
others more suddenly ; the whole of the mass may be broken up without any slow evapora-
tion in an exceedingly short time. If the mass was originally heterogeneous, one part may
fly off while the rest remains ; & while this happens, the parts that remain may form fresh
mixtures altogether different from the original, the structure of the particles of the higher
orders even being altered ; owing to the fact that several particles of lower orders, which
originally belonged to different particles of higher orders, now coalesce into a particle of a
higher order of a fresh kind. From this we get such a large number of compositions &
transformations in Nature, & more especially in chemistry ; hence we get such a large
number of different kinds of vapours, & the great differences in elastic air, which is formed
from such different fixed bodies. An immense field for inquiry is laid open ; but I must
leave it & go on to some other matters, which also refer to fermentations & evaporations.

464. A substance, which has been dissolved, can be once more obtained, not only by Concretions, after
precipitation, as when metals fall by their own weight reduced to the form of an impalpable ^^^"de finite
powder, but also by evaporation, as we have said, in the case of salts, which, on the fluid shapes in the resi-
in which they were dissolved being evaporated, remain behind at the bottom. Nor fnu^'It^formstance
indeed do salts remain behind in the form of a fine powder, with their minutest particles

quite inert ; but they are grouped together in fairly large masses having definite shapes,
which differ for different salts ; these are angular in all salts, & fearfully pointed & jagged
in those salts of a particularly corrosive nature. In consequence, the salts are rather
sharp to the taste ; & with some of them, which are corrosive, there is a power of cutting
the slender fibres of living things, & of destroying the organs that are necessary to life.
The manner in which they can acquire these shapes especially is clear from Art. 439 ; as
also the shapes of crystals & those jellies from which are formed gems & hard stones, when
they are simple, & each adheres to its own shape ; & also of some of the same kind, which
take form after evaporation ; & in every case this possibility is explained, as was also shown
in the same article, from the fact that particles attract other particles situated at certain
distances only at certain of their sides & points ; & thus they will only attach them to
themselves in a certain definite manner that corresponds to the particular points, or sides.

328 PHILOSOPHIC NATURALIS THEORIA

Quomodo possit ^gr Fermentatio paullatim minuitur, & demum cessat, cuius imminuti motus causas

fermentatio cessar. . 7 J. .. . ' . ... J. . ... .

attigi plunbus locis, ut num. 197. Modern autem pertmet illud etiam, quod mnui num.
440. Irregularitas particularum, ex quibus corpora constant, & inaequalitas virium,
plurimum confert ad imminuendum, & demum sistendum motum. Ubi nimirum aliquae
particulae, vel totse irruerunt in majorum cavitates, vel ubi suos uncos quosdam aliarum
uncis, vel foraminibus inseruerunt, explicari non possunt, & sublapsus quidam, & compres-
siones particularum accidunt in massa temere agitata, quse motum imminuunt & ad
sensum extinguunt, quo & in mollibus sisti motus potest post amissam figuram. Multum
itidem potest ad minuendum, ac demum sistendum motum sola asperitas ipsa particularum,
ut motus in scabro corpore sistitur per frictionem ; multum incursus in externa puncta,
ut aer pendulum sistit : multum particulae, quae emittuntur in omnes plagas, ut in
evaporatione, vel ubi corpus refrigescit, excussis pluribus igneis particulis, qua; dum evolant
actione paticularum massae, ipsis massae particulis procurrentibus motum in partes contrarias
imprimunt, & dum illae, quas oscillationem auxerant, aliae post alias aufugiunt, illae, quae
remanent, sunt, quae oscillationes ipsas internis, & externis actionibus minuebant.

Cur quaedam sub- 466. Porro non omnes substantiae cum omnibus fermentant, sed cum quibusdam

cifnf1qeuibeusdamfI& tantummodo : acidacum alcalinis ; & [215] quod quibusdam videtur mirum, sunt quaedam,

non cum aliis ; cur qU3e apparent acida respectu unius substantiae, & alcalina respectu alterius. Ea omnia

mentent' "debeant 'in mea Theoria facilem admodum explicationem habent : nam vidimus, particulas quasdam

contundi. respectu quarundam inertes esse, cum quibus commixtae idcirco non fermentant, respectu

aliarum exercere vires varias : adeoque si respectu quarundam habeant pro variis distantiis

diversas vires, & alternationem satis magnam attractionum, ac repulsionum ; statim, ac

satis prope ad ipsas accesserint, fermentant. Sic si limatura ferri cum sulphure commisce-

atur, & inspergatur aqua, oritur aliquanto post ingens fermentatio, quae & inflammationem

parit, ac terraemotuum exhibet imaginem quandam, & vulcanorum. Oportuit ferrum in

tenues particulas discerpere, ac ad majorem mixtionem adhuc adhibere aquam.

ignem esse fennen- 467. Ignem ego itidem arbitror esse quoddam fermentatioms genus, quod acquirat

tationis genus: , h '. . 6 111 • t

quomodo excitetur ve* potissimum, vel etiam sola sulphurea substantia, cum qua iermentat materia lucis
tanta fermentatio vehementissime, si in satis magna copia collecta sit. Ignem autem voco eum, qui non

ab exieua scintilla r • i « i r • « i • i i •

tantum rareiacit motu suo, sed & caleiacit, & meet, quae omma habentur, quando matena
ilia sulphurea satis fermentescit. Porro ignis comburit, quia in substantiis combustibilibus
multum adest substantiae cujusdam, quae sulphure abundat plurimum, & quae idcirco
sulphurea appellari potest, quas vel per lucem in satis magna copia collectam, vel per ipsam
jam fermentescentem sulphuream substantiam satis praegnantem ipsa lucida materia sibi
admotam fermentescit itidem, & dissolvitur, ac avolat. Is ingens motus intestinus
particularum excurrentium fit utique per vires mutuas inter particulas, quae erant in
aequilibrio : sed mutatis parum admodum distantiis exigui etiam punctorum numeri per
exiguum unius scintillae, vel tenuissimorum radiorum accessum, jam aliae vires succedunt,
& per earum reciprocationem perturbatur punctorum motus, qui cito per totam massam
propagatur.

Exempium avicuiae 468. Imaginem rei admodum vividam habere possumus in sola etiam gravitate.

djmota arenula m -,-, ° . . . , . ... ,.r.. r j •

summo monte de- Emergat e man satis editus mons, per cujus latera dispositae smt versus iundum mgentes
jicientis lapiiios, lapidum praegrandium moles, turn quo magis ascenditur, eo minores ; donee versus apicem

saxa, rupes, & , r.1v . r ,° . ^ , °. . , . ' .,., . \

excitantis in man lapilli smt, & in summo monte arenulae: smt autem omnia tere in asquiliDno pendentia
subjecto undas Jta, ut vi respectu molis exigua devolvi possint. Si avicula in summo monte commoveat
arenulam pede ; haec decidit, & lapiiios secum dejicit, qui, dum ruunt, majores lapides
secum trahunt, & hi demum ingentes illas moles : fit ruina immanis, & ingens motus, qui,
decidentibus in mare omnibus, mare ipsum commovet, ac in eo agitationem ingentem, &
undas immanes ciet, motu aquarum vehementissimo diutissime perdurante. Avi-[2l6]-cula
aequilibrium arenulae sustulit vi perquam exigua : reliquos motus gra vitas edidit, quae
occasionem agendi est nacta ex illo exiguo motu avicuiae. Haec imago quaedam est virium
intestinarum agentium, ubi cum vires crescere possint in immensum, mutata utcunque
parum distantia ; multo adhuc major effectus haberi potest, quam in casu gravitatis, quae

A THEORY OF NATURAL PHILOSOPHY 329

465. The fermentation diminishes gradually, & at length ceases ; I have touched The manner in
upon the causes of this diminished motion in several places, for instance, in Art. 197. The may cease!16
remarks I made in Art. 440 also refer to the same thing. The irregularity of the particles,

from which the bodies are formed, & the inequality of the forces, especially contribute
to the diminution & final stoppage of the motion. Thus, when certain particles, or the
whole of them enter cavities in larger particles, or when they insert their hooks into the
hooks or openings of others, these cannot be disentangled, & certain relapses & compressions
of the particles happen in a mass irregularly agitated, which diminish the motion &
practically destroy it altogether ; & due to this the motion even in soft bodies can be
stopped after a loss of shape. Also the roughness of the particles alone may do much toward
diminishing & finally stopping the motion ; just as motion in a rough body is stopped by
friction. Impact with external bodies has a great effect, e.g., the air stops a pendulum.
Much may be due to the emission of particles in all directions, as in evaporation ; or when
a body freezes, many igneous particles being driven off in the process ; & as these particles
fly off by the action of the particles of the mass, impress a motion in the opposite direction
on those particles as they move ; & while those that had increased the oscillation, one
after the other fly off, those that are left are such as were diminishing these oscillations
by internal & external actions.

466. Further, all substances do not ferment with every substance, but with some of The reason why
them only. Thus, acids ferment-only with alkalies ; &, what to some seems to be wonderful, £5°™^ S^J^^.S
there are some substances that appear to be acid with respect to one substance, & alkaline tain substances &
with respect to another. Now, all these things have a perfectly easy explanation in my ^ s^g ^ust fbe
Theory. For, we have seen that certain particles are inert with regard to certain other powdered before
particles, & therefore when these are mixed together there will be no fermentation. With they wU1 ferment-
regard to ethers, again, they exert various forces ; hence, if with respect to certain of them

they have different forces for different distances, & a sufficiently great alternation of
attractions & repulsions, they will immediately ferment on being brought into sufficiently
close contact with them. Thus, if iron-filings are mixed with sulphur, & moistened with
water, there will be produced in a little time a great fermentation ; & this also produces
inflammation, & exhibits phenomena akin to earthquakes & volcanoes. It is necessary,
however, that the iron should be powdered very finely, & that water should be used to
give a still closer mingling of the particles.

467. I believe also that fire itself is some kind of fermentation, which is acquired, either Fire is some sort of
more especially, or even solely by some sulphurous substance, with which the matter mlmner^'wh'ich*^
forming light ferments very vigorously, if it is concentrated in sufficiently great amount, great a fermenta-
Moreover I apply the term fire to that which not only rarefies through its own motion, ^"the^Ugntest^of
but also produces heat & light ; & all these conditions are present when the sulphurous sparks,
substance ferments sufficiently. Further, fire burns, because in combustible substances

there is present much of a substance largely consisting of something like sulphur, for which
reason it may be termed a sulphurous substance. Such a substance, either by contact with
light concentrated in sufficiently great amount, or by contact with the already fermenting
sulphurous substance which is charged with the matter of light to a sufficient degree,
will also ferment, & be broken up, & fly off. The very great internal motion of the particles
flying off is in every case due to the mutual forces between the particles, which originally
were in equilibrium ; but, the distances of even a very small number of points being changed
ever so little, by the slightest accession of a spark, or of its feeblest rays, other forces then
take their place, the motion of the points is also. disturbed by their oscillations, & this is
quickly propagated throughout the whole of the mass.

468. We can obtain a really vivid picture of the matter, even in the case of gravity As example, in the
alone. Suppose that from the sea there rises a mountain of considerable height, & that t^movfng^sin^e
along the sides of it there lie immense masses of huge stones, & the higher one goes, grain of sand on
the smaller the stones are ; until towards the top the stones are quite small, & at the very *h? toP °.f. a ™oun-

. , , 11 i M>I • tain, hurling down

summit they are mere grains of sand. Also suppose that all of these are just in equilibrium, stones, rocks,
so that they can be rolled down by a very slight force compared with their whole volume, j^e^aws^^the
If, now, a little bird on the top of the mountain moves with his foot just one grain of the sand, sea that lies at the
this will fall, & bring down with it the small stones ; these, as they fall, will drag with *°9* of the moun'
them the larger stones, & these in their turn will move the huge boulders. There will
be an immense collapse & a huge motion ; &, as all these stones fall into the sea, the motion
will communicate itself to the sea & cause in it a huge agitation & immense waves, & this
vigorous motion of the water will last for a very considerable time. The little bird disturbed
the equilibrium of the grain of sand with a very slight force ; gravity produced the remaining
motions, & it obtained its opportunity for acting through the slight motion of the little
bird. This is a kind of picture of the internal forces that act, when, owing to the possibility
of the forces increasing indefinitely, on the distance being changed ever so slightly, a much

330 PHILOSOPHISE NATURALIS THEORIA

quidem perseverat eadem, aucta tantummodo velocitate descensus per novas accelera-
tiones.

?uSX mateHa PSU!- 4^9* Quo& s'1 ^8™ excitatur tantummodo per sulphureae substantias fermentationem ;

phura:, ab igne ubi nihil adsit ejus substantiae, nullus crit metus ab igne. Videmus utique, quo minus

Sine fortasse inlpso ejusm°di substantiae corpora habeant, eo minus igni obnoxia esse, ut ex amianto & telas

Sole posse manere fiant, quas igne moderate purgantur, non comburuntur. Censeo autem idcirco nostras

substantias Utesas. nasce terrestres substantias ab igne satis intense dissolvi omnes, & inflammari, quod omnes

ejusmodi substantias aliquid admixtum habeant, quod nectat etiam inter se plurimas inertes

particulas. At si corpora haberentur aliqua, quae nihil ex ejusmodi substantia haberent

admixtum ; ea in medio igne vehementissimo illaesa perstarent, nee ullum motum acqui-

rerent, quern nimirum nostra haec corpora acquirunt ab igne non per incursum, sed per

fermentationem ab internis viribus excitatam. Hinc in ipso Sole, & fixis, ubi nostra corpora

momento fere temporis conflagrarent, & in vapores abirent tenuissimos, possunt esse corpora

ea substantia destituta, quae vegetent, & vivant sine ulla organici sui textus laesione minima.

Videmus certe maculas superficiei Solis proximas durantes aliquando per menses etiam

plures, ubi nostrae nubes, quibus eae videntur satis analogae, brevissimo tempore dissiparentur.

Exempium fermen- 470. Id mirum videbitur hommi prasjudicns praeoccupato ; nee mtelliget, qui fieri

tationis, quam .T/ • v -j • c i • • • i i • • ' i •

cum aceto habent possit, ut vivat aliquid in bole ipso, in quo tanto major esse debet vis ustoria, dum me
aiiquae terrae, aiiis exiguus radiorum solarium numerus majoribus cavis speculis, vel lentibus collectus dissolvit
omnia. At ut evidenter pateat, cujusmodi praejudicium id sit : fingamus nostra corpora
compacta esse ex illis terris, quas bolos vocant, quae a diversis aquis mineralibus deponuntur,
quas cum acidis fermentant, ac omnia corpora, quas habemus prae manibus, vel ex eadem
esse terra, vel plurimum ex ea habere admixtum. Acetum nobis haberetur loco ignis :
quascunque corpora in acetum deciderent, ingenti motu excitato dissolverentur citissime,
& si manum immitteremus in acetum : ea ipsa per fermentationem exortam amissa,
protinus horrore concuteremur ad solam aceti viciniam, & eodem modo videretur nobis
absurdum quoddam, ubi audiremus, esse substantias, quae acetum non metuant, & in eo
diu perstare possint sine minimo motu, atque sui textus laesione, quo vulgus rem prorsus
absurdam censebit, si audiat, in medio igne, in ipso Sole, posse haberi corpora, quas [217]
nullam inde laesionem accipiant, sed pacatissime quiescant, & vegetent, ac vivant.

Deiumme. senten- ^yj. Hasc quidem de igne; jam aliquid de luce, quam ignis emittit, & quas satis

tiam de emissione „ T' . . i ca j j • •

luminis prseferen- collecta ipsum excitat. Ipsa lux potest esse emuvium quoddam tenuissimum, & quasi

dam omnino undis vapor fcrmentatione ignea vehementi excussus. Et sane validissima, meo quidem iudicio,
fluidi elastici. 1-11 j i

argumenta sunt, contra omnes alias hypotheses, ut contra undas, per quas onm pnasnomena
lucis explicare conatus est Hugenius, quam sententiam diu consepultam iterum excitare
conati sunt nuper summi nostri asvi Geometras, sed meo quidem judicio sine successu (r) :
nam explicarunt illi quidem, & satis aegre, paucas admodum luminis proprietates, aliis
intactis prorsus, quas sane per earn hypothesim nullo pacto explicari posse censeo, & quarum
aliquas ipsi arbitror omnino opponi : sed earn sententiam impugnare non est hujus loci,
quod quidem alibi jam prasstiti non semel. Mirum sane, quam egregie in effluviorum
emanantium sententia ex mea Theoria profluant omnes tarn variae lucis proprietates, quam
explicationem fuse persecutus sum in secunda parte dissertationis De Lumine : prascipua
capita hie attingam ; interea illud innuam, videri admodum rationi consentaneam
ejusmodi sententiam materiae effluentis, vel ex eo, quod in Ingenti agitatione, quam habet
ignis, debet utique juxta id, quod vidimus num. 195, evolare copia quasdam particularum,
ut in ebullitionibus, effervescentiis, fermentationibus passim evaporationes habentur.

6

Proprietates lumi- 4.72. Praecipuas proprietates luminis sunt ejus emissio constans, & ab aequali massa,

ut ab eodem Sole, ab ejusdem candelae flamma, ad sensum eadem intensitate : immanis
velocitas, nam semidiametrorum terrestrium 20 millia, quanta est circiter Solis a Terra

(r) Cum htec scriberem, nondum •prodierant Opera Taurinensis Academies ; nee vero hue usque, dum hoc Opus
reimprimitur, adhuc videre potui, quie Geometra maximus La Grange hoc in genere protulit,

A THEORY OF NATURAL PHILOSOPHY 331

greater effect can be obtained, than is the case for gravity ; for, this remains the same,
the velocity of descent being only increased by fresh accelerations.

469. But if fire is excited only by the fermentation of sulphurous matter ; then, when Substances, that
none of this matter is present, there will be no danger from fire. We see indeed, the less sulphurous mattel*
of this substance the bodies have, the less liable they are to be injured by fire ; thus, a material are not necessarily
is woven from asbestos, & this is only purified, but not burned, by moderate fire. Further, henc^perhaps^lii
I consider that all our earthy substances are broken up by fire, provided it is sufficiently the Sun itself there
intense, & are set on fire, just because all substances of this kind have something mixed

with them, which connects a large number of inert particles together. However, if there
were any bodies which had nothing at all of such a substance mixed with them, these would
be unaltered in the heart of the most vigorous fire, & would not acquire any motion, that
is to say, such motion as the bodies about us acquire from fire, not through the entrance
of fiery particles, but through fermentation excited by internal forces. Hence, in the Sun
itself, & in the stars, in which our terrestrial bodies would burn up in an instant of time
& go off into the thinnest of vapours, there may exist bodies altogether lacking in such
a substance ; & these may grow & live without the slightest injury of any kind to their
organic structure. Indeed we see spots very close to the Sun lasting sometimes for several
months even ; whereas our clouds, to which these spots seem to bear a considerable analogy,
would be dissipated in a very short time.

470. Now this will appear wonderful to a man who is obsessed by prejudices ; nor Example, in the
will he be able to understand why it is that anything can live in the Sun, in which ^ which^some
there is bound to be ever so much greater burning force, while on earth an exceedingly earths have with
small number of solar rays, collected by fairly large concave mirrors or by lenses, will break areeunaffectcdthers
up all substances. However, in order to make plain how such a prejudice arises, let us

suppose that our substances are formed from those earths, which are termed boluses, such
as are deposited by certain minerals of different kinds & ferment with acids ; & that all
bodies around us either are formed out of this earth or are largely impregnated with it.
Let vinegar be taken to represent fire ; then if any of these bodies fell into the vinegar,
they would be very quickly broken up by the huge motion induced ; & if we placed our
hands in the vinegar, they too being lost by the fermentation produced, we should be
forthwith struck with horror at the mere vicinity of vinegar. It would seem to us that
it was something ridiculous if we were told that there were substances which were in no
fear of vinegar, but could last in it for a long time without slightest motion or injury to
their structure ; in exactly the same way as an ordinary man would think it ridiculous,
if he were told that in the heart of fire, or in the Sun itself, there might exist bodies which
received no injury from it, but remained at rest in the most calm fashion, & grew & lived.

471. So much on the subject of fire; now I will make a few remarks about light, Light; the theory
which is given off by fire, & which, when present in sufficient quantity, excites fire. It is to^bT^referred
possible that light may be a sort of very tenuous effluvium, or a kind of vapour forced altogether before
out by the vigorous igneous fermentation. Indeed, in my judgment, there are very strong elastic* fluid68 m *"
arguments in favour of this hypothesis, as opposed to all other hypotheses, such as that

of waves. On the hypothesis of waves, Huygens once tried to explain all the phenomena
of light ; & the most noted of the geometers of our age have tried to revive this theory,
which had been buried with Huygens; but, as I think, unsuccessfully (r). For, they have
explained, & even then poorly enough, a very few of the properties of light, leaving the
rest untouched ; & indeed I consider that such properties can not be explained in any way
by this hypothesis of waves, & my opinion is that some of them are altogether contrary to it.
But this is not the right place to impugn this theory ; indeed I have already, more than
once, presented my view in other places. It is really marvellous how excellently, on the
hypothesis of emanating effluvia, all the different properties of light are derived from my
Theory in a straightforward way. I gave a very full explanation of this in the second part of
my dissertation, De Lumine ; & the principal points of this work I will touch upon here.
Meanwhile, I will just mention that the idea of effluent matter seems to be altogether
reasonable; more especially from the fact that, in a very great agitation amongst particles,
such as there is in the case of fire, there is always bound to be, in accordance with what we
have seen in Art. 195, an abundance of particles flying off, just as we have evaporations
in ebullition, effervescence & fermentation.

472. The principal properties of light are : — its constant emission, & the fact that Those properties of

.1 • ' • • r 1 ,r f . , V, , , light for which we

the intensity is always the same from the same mass, such as from the Sun, or from the have to find the
flame of the same candle ; its huge velocity, for it traverses a distance equal to twenty reason,
thousand times the semidiameter of the Earth, which is about the distance of the Sun

(r) When I wrote this, the Transactions of the Academy of Turin had not been published ; and even now, at the time
of this reprint of my work, I have so far been unable to see what that excellent geometer La Grange has published on
the subject.

332 PHILOSOPHIC NATURALIS THEORIA

distantia, percurrit semiquadrante horae ; velocitatum discrimcn cxiguum in diversis
radiis, nam celeritatis discrimen in radiis homogeneis vix ullum esse, si quod est, colligitur
pluribus indiciis : propagatio rectilinea per medium diaphanum ejusdem densitatis ubique
cum impedimento progressus per media opaca, sine ullo impedimento sensibili ex impactu
in se invicem radiorum tot diversas directiones habentium, aut in partes internas diaphan-
orum corporum utcunque densorum : reflexio partis luminis ad angulos asquales in
mutatione medii, parte, quae reflectitur, eo majore respectu luminis, quo obliquitas
incidentiae est major ; refractio alterius partis eadem mutatione cum lege constantis rationis
inter sinum incidentiae, & sinum anguli refracti ; quae ratio [218] in diversis coloratis
radiis diversa est, in quo stat diversa diversorum coloratorum radiorum refrangibilitas :
dispersio & in reflexione, & in refractione exiguse partis luminis cum directionibus quibus-
cunque quaquaversus : alternatio binarum dispositionum in quovis radio, in quarum
altera facilius reflectatur, & in altera facilius transmittatur lux delata ad superficiem
dirimentem duo media heterogenea, quas Newtonus vocat vices facilioris reflexionis, &
facilioris transmissus, cum intervallis vicium, post quae nimirum dispositiones maxime
faventes reflexioni, vel refraction! redeunt, aequabilis in eodem radio ingresso in idem
medium, & diversis coloratis radiis, in diversis mediorum densitatibus, & in diversis
inclinationibus, in quibus radius ingreditur, ex quibus vicibus, & earum intervallis diversis
in diversis coloratis radiis pendent omnia phenomena laminarum tenuium, & naturalium
colorum tarn permanentium, quam variabilium, uti & crassarum laminarum colores, quae
omnia satis luculenter exposuit in celebri dissertatione De Lumine P. Carolus Benvenuti e
Soc. nostra Scriptor accuratissimus : ac demum ilia, quam vocant diffractionem, qua
radii in transitu prope corporum acies inflectuntur, & qui diversum colorem, ac diversam
refrangibilitatem habent, in angulis diversis.

Emissio quomodo 473. Quod pertinet ad emissionem jam est expositum num. 199, & num 461 ; ubi
si"mui etiam ostensum est illud, manente eadem massa quae emittit effluvia, ipsorum multitudinem

citissime d is sol- dato tempore esse ad sensum eandem. Porro fieri potest, ut massa, quae lumen emittit,

emittunt,UIut "ignis Penitus dissolvatur, ut in ignibus subitis accidit, & fieri potest, ut perseveret diutissime,

subitus, quaedam, Id potissimum pendet a magnitudine intervalli, in quo fit oscillatio fermentationis, & a

persVste'nt^slne natura arcus attractivi terminantis id intervallum juxta num. 195. Quin immo si Auctor

sensibili jactura. Naturae voluit massam vehementissima etiam fermentatione agitatam prorsus indissolubilem

quacunque finita velocitate, potuit facile id praestare juxta num. 460 per alios asymptoticos

arcus cum areis infinitis, intra quorum limites sit massa fermentescens ; quorum ope ea

colligari potest ita, ut dissolvi omnino nequeat, ponendo deinde materiam luminis emittendi,

ultra intervallum earum asymptotorum respectu particularum ejus massae, & citra arcum

attractivum ingentis areae, sed non infinitae, ex quo aliae lucidae particulae evolare possint

post alias. Nee illud, quod vulgo objici solet, tanta luminis effusione debere multum

imminui massam Solis, habet ullam difficultatem, posita ilia componibilitate in infinitum

& ilia solutione problematis quae habetur num. 395. Potest enim in spatiolo utcunque

exiguo haberi numerus utcunque ingens punctorum, & omnis massa luminis, quas diffusa

tarn immanem molem occupat, potest in Sole, vel prope Solem occupavisse spatiolum,

quantum libuerit, parvum, ut idcirco Sol post quotcunque sae-[2i9]-culorum millia ne

latum quidem unguem decrescat. Id pendet a ratione densitatis luminis ad densitatem

Solis, quae ratio potest esse utcunque parva ; & quidem pro immensa luminis tenuitate

sunt argumenta admodum valida, quorum aliqua proferam infra.

Unde tanta veloci- 474. Celeritas utcunque magna haberi potest ab arcubus repulsivis satis validis, qui

tfsS 'discrimln0<exi- occurrant post extremum limitem oscillationis terminatae ab arcu ingenti attractive juxta

guum, & in radiis num. 194 : nam si inde evadat particula cum velocitate nulla ; quadratum velocitatis

homogeneis multo totjus definitur ab excessu arearum omnium repulsivarum supra omnes attractivas juxta

num. 178, qui excessus cum possit esse utcunque magnus ; ejusmodi celeritas potest itidem

esse utcunque magna. Verum celeritatis discrimen in particulis homogeneis erit prorsus

insensibile, qui a particulae luminis ejusdem generis ad finem oscillationis advenient cum

velocitatibus fere nullis : nam eae, quae juxta Theoriam expositam num. 195, paullatim

augent oscillationem suam, demum adveniunt ad limitem cohibentem massam, & avolant ;

A THEORY OF NATURAL PHILOSOPHY 333

from the Earth, in an eighth of an hour ; the slight differences of velocity that exist in
different rays, for it is proved from several indications that there is scarcely any difference
for homogeneous light, if there is any at all ; its rectilinear propagation through a transparent
medium everywhere equally dense, along with hindrance to progression through opaque
media ; & this without any sensible hindrance due to impact with one another of rays
having so many different directions, or any that prevents passage into the inner parts of
transparent bodies, no matter how dense they may be ; reflection of part of the light at
equal angles" at the surface of separation of two media, the part that is reflected being greater
with regard to the whole amount of light, according as the obliquity of incidence is greater ;
refraction of the other part at the same surface of separation, with the law of a constant
ratio between the sines of the angle of incidence & the angle of refraction, the ratio being
different for differently coloured rays, upon which depends the different refrangibility of
the differently coloured rays ; dispersion, both in reflection & in refraction, of a very small
part of the light in directions of every description whatever ; the alternation of propensity
in any one ray, in one of which the light falling upon the surface of separation between
two media of different nature is the more easily reflected & in the other is the more easily
transmitted, which Newton calls ' fits ' of easier reflection & easier transmission, with
intervals between these fits, after which the propensities mostly favouring reflection or
refraction return, these intervals being equal in the same ray entering the same medium,
& different for differently coloured rays, for different densities of the medium, £ for
the different inclinations at which the ray enters the medium ; upon these fits & the
different intervals between them for differently coloured rays depend all the phenomena
of thin plates, & of natural colours, both variable & permanent, as well as the colours
of thick plates, all of which have been discussed with considerable clearness by Fr. C.
Benvenuti, a most careful writer of our Society, in his well-known dissertation, De
Lumine. Last of all, we have that property, which is called diffraction, in which rays,
passing near the edge of a body, are bent inwards, having a different colour & different
refrangibility for different angles.

473. What pertains to emission has been already explained in Art. 199 & Art. 461 ; How emission takes
there also it was shown that, if the mass emitting the effluvia remained the same, then p^nT'that* *lome
the amount emitted is practically the same in any given time. Further, it may bodies are very
happen that the mass emitting the light is completely broken up, as takes place in ^"the tkne^they
sudden flashes of fire ; or it may be that this mass persists for a very long time. This emit light, like a
to a very great extent depends on the size of the interval in which the oscillation due to wmfe^hers?* Rke
fermentation takes place, & on the nature of the attractive arc at the end of that interval, the Sun persist for
by Art. 195. Nay, if the Author of Nature had wished that a mass, agitated by the most ^^t Jnylppar-
vigorous fermentation even, should be quite irreducible by any finite force whatever, he ent loss.
could easily have accomplished this, as shown in Art. 460, by other asymptotic arcs with
infinite areas, between the confines of which the fermenting mass would be situated. By
the aid of these arcs the mass could be so bound together, that it would not admit of the
slightest dissolution ; & then by placing the material for emitting light further from
the particles of the mass than the interval between those asymptotes, & within the distance
corresponding to an attractive arc of huge but finite area ; from which we should have
particles, one after the other, of light flying off. Nor is there any difficulty from the
usual argument that is raised in objection to this, that the mass of the Sun must be much
diminished by such a large emission of light ; if we suppose indefinitely great componibility,
& the solution of the problem, given in Art. 395. For in any exceedingly small space
there may be any huge number of points whatever ; & the whole mass of the light, which
is diffused throughout & occupies such an immense volume, may, in the Sun or near the
Sun, have occupied a space as small as ever one likes to assign ; so that the Sun, after the
lapse of any number of thousands of centuries, will not therefore have decreased by even
a finger's breadth. It all depends on the ratio of the density of light to the density of the
Sun, & this ratio can be any small ratio whatever. Indeed there are perfectly valid arguments
for the immense tenuity of light, some of which I will give below.

474. Any velocity, no matter how great, can be obtained from sufficiently powerful Whence comes the
repulsive arcs, if these occur after the last limit of oscillation within the confines of a very withstanding7' the
great attractive arc, as shown in Art. 104. For if a particle goes off from here with no slight differences in

3 , . r , i • • i r i i j- 11 i i • velocity, & the still

velocity, the square of the whole velocity is defined by the excess of all the repulsive areas iess differences in

over all the attractive, as was shown in Art. 178 ; &, as this excess can be of any amount homogeneous rays.

whatever, the velocity can also be of any magnitude whatever. Again, the difference of

velocity for homogeneous particles is quite insensible, because particles of light of the same

kind come to the end of their oscillation with velocities that are almost zero ; for those

which, according to the Theory set forth in Art. 195, increase their oscillation gradually,

arrive at the boundary limiting the mass at last, & then fly off. Now, if, at the time they

334 PHILOSOPHIC NATURALIS THEORIA

quo si turn, cum avolant, advenirent cum ingenti velocitate, advenissent utique eodem, &
effugissent in oscillatione praecedenti. Demonstravimus autem ibidem, exiguum discrimen
velocitatis in ingressu spatii, in quo datae vires perpetuo accelerant motum, & generant
velocitatem ingentem, inducere discrimen velocitatis genitae perquam exiguum etiam
respectu illius exigui discriminis velocitatis initialis, quod demonstravimus ibi ratione petita a
natura quadrati quantitatis ingentis conjunct! cum quadrate quantitatis multo minoris, quod
quantitatem exhibet a priore ilia differentem multo minus, quam sit quantitas ilia parva,
cujus quadratum conjungitur. Discrimen aliquod sensibile haberi poterit ; siqua effugiunt,
non sint puncta simplicia, sed particulae non nihil inter se diversse : nam curva virium,
qua massa tota agit in ejusmodi particulas, potest esse nonnihil diversa pro illis diversis
particulis, adeoque excessus summae arearum repulsivarum supra summam attractivarum
potest esse nonnihil diversus & quadratum velocitatis ipsi respondens nonnihil itidem
diversum. Hoc pacto particulae luminis homogeneas habebunt velocitatem ad sensum
prorsus asqualem ; particulas heterogeneae poterunt habere nonnihil diversa m, uti ex
observatione phsenomenorum videtur omnino colligi. Illud unum hac in re notandum
superest, quod curva virium, qua massa tota agit in particulam positam jam ultra terminum
oscillationum, mutatis per oscillationem ipsam punctis massae, mutabitur nonnihil : sed
quoniam in fortuita ingenti agitatione massae totius celerrime succedunt omnes diversse
positiones punctorum ; summa omnium erit ad sensum eadem, potissimum pro particula
diutius hasrente in illo initio suae fugae, ad quod advenit, uti diximus, cum velocitate
perquam exigua, ut idcirco homogenearum velocitas, [220] ubi jam deventum fuerit ad
arcum gravitatis, & vires exiguas, debeat esse ad sensum eadem, & discrimen aliquod haberi
possit tantummodo in heterogeneis particulis a diverse earum textu. Patet igitur, unde
celeritas ingens provenire possit, & si quod est celeritatis discrimen exiguum.

Unde propagatio 475. Quod pertmet ad propagationem rectilmeam per medium homogeneum diaph-

rectilinea : incur- a j TL • 11 j- v • • i • • i V

sum immediatum anum, & ad. motum nberum sine ullo impedimento a particulis ipsius luminis, vel medii
punctorum lucis, diaphani, id in mea Theoria admodum facile exponitur, quod in aliis ingentem difficultatem

in puncta medii in. -j j j • j- • • • IT i i

nullum haberi: pant. Et quidem quod pertmet ad impedimenta, si curva virium nullum habeat arcum
virium in medio asymptoticum perpendicularem axi praeter primum : ostensum est num. 362. sola satis

homogeneo exi- ' i • t^j • • j i •

guam inaequaiita- ™agna velocitate obtinen posse apparentem compenetrationem duarum substantiarum,
tem eludi a tenui- quam tenuitas, & homogeneitas spatii, per quod transitur, plurimum iuvat. Quoniam

tate, & celeritate * • '.r ,.?.,.,. . . ' </,...-.

luminis. respectu punctorum matense prorsus mdivisibihum, & mextensorum infinities mnnita

sunt puncta spatii existentia in eodem piano ; infinities infinite est improbabilis pro quovis
momento temporis directio motus puncti materiae cujus vis accurate versus aliud punctum
materiae, ac improbabilitas pro summa momentorum omnium contentorum dato quovis
tempore utcunque longo evadit adhuc infinita. Ingens quidem est numerus punctorum
lucis, & propemodum immensus, sed in mea Theoria utique finitus. Ea puncta quovis
momento temporis directiones motuum habent numero propemodum immenso, sed in
mea Theoria finite. Verum quidem est, ubicunque oculus collocetur in immensa
propemodum superficie sphserse circa unam fixam remotissimam descripta, immo intra
ipsam sphaeram, videri fixam, & proinde aliquam luminis particulam afncere nostrum
oculum : sed id fit in mea Theoria non quia accurate in omnibus absolute infinitis
directionibus adveniant radii, sed quod pupilla, & fibrae oculorum non unicum punctum
£unt, & vires punctorum particulae luminis agunt ad aliquod intervallum. Hinc quovis
utcunque longo tempore nullus debet accidere casus in mea Theoria, in quo punctum
aliquod luminis directe tendat contra aliquod aliud punctum vel luminis, vel substantiae
cujusvis, ut in ipsum debeat incurrere. Quamobrem per incursum, & immediatum
impactum nullum punctum luminis aut sistet motum suum, aut deflectet.

si satis magnam 476. Id quidem commune est omnibus corporibus, quae corpora inter se congrediuntur.

antTq^evL, soiida Ea nullum habent in mea Theoria punctum immediatum incurrens in aliud punctum ;

etiam, transitura quam ob causam & illud ibidem dixi, si nullae vires mutuae adessent, debere utique haberi

sine3 umT motuum apparentem quandam compenetrationem omnium massarum ; sed adhuc vel ex hoc solo

perturbatione. capite veram compenetrationem haberi nunquam omnino posse. Vires igitur quae ad

aliquam distantiam protenduntur, im- [221] -pediunt progressum. Eae vires si circumquaque

essent semper aequales ; nullum impedimcntum haberet motus, qui vi inertiae deberet

A THEORY OF NATURAL PHILOSOPHY 335

fly off, they should reach this boundary with a very great velocity, then it is certain that
they would have reached it & flown off in a previous oscillation. Further, in the same
article, we have proved that a slight difference of velocity on entering a space, in which
given forces continually accelerate the motion & generate a huge velocity, also induces a
difference in the velocity generated that is very small even when compared with the slight
difference in the initial velocity. This we there prove from an argument derived from
the nature of the square of a very large quantity compounded with the square of a quantity
much less than it ; this gives a quantity differing from the first quantity by something
much less than the small quantity of which the square was added. A sensible difference
may be obtained, if what fly off are not simple points, but particles somewhat different
from one another. For the curve of forces, with which the mass acts upon such particles,
can be somewhat different for those different particles ; & thus, the excess of the sum of
the repulsive areas over the sum of the attractive may be somewhat different, & therefore
the square of the velocity corresponding to this excess may be somewhat different. In
this way particles of homogeneous light will have velocities that are practically equal ;
but particles of heterogeneous light may have velocities that are somewhat different ; as
seems to be conclusively shown from observations of phenomena. One thing remains to
be noted in this connection, namely, that the curve of forces, with which the whole mass
acts upon a particle placed already beyond the limit of the oscillation, when the points
of the mass are changed on account of the oscillation, will be somewhat altered. But since
in a very large irregular agitation of the entire mass all the different positions of the points
follow on after one another very quickly, the sum of all the forces will be practically the
same, especially in the case of a particle stopping for some time at the beginning of its ,
flight ; which point it has reached, as we have said, with a velocity that is exceedingly
small. Thus, the velocity of homogeneous particles must on that account be practically
the same, when they have reached the arc representing gravitation ; & a difference can
only be obtained in heterogeneous particles owing to their structure. It is therefore
clear from what source the very great velocity can come, & also the slight differences, if
there are any.

475. That which relates to the rectilinear propagation through a transparent The reason for
homogeneous medium, & the free motion, without hindrance, by particles either of the light eation^there^Tan
or of the transparent medium, is quite easily explained in my Theory, whereas in other be no' immediate
theories it begets a very great difficulty. Also as regards hindrance to this motion, so ^'pcTnt^oV^ight
long as the curve of forces has no asymptotic arc perpendicular to the axis besides the first, & the points of the
it has been shown, in Art. 362, that merely with a sufficiently great velocity there can be ^equ'aiit' of1 the
obtained an apparent compenetration of two substances ; & tenuity & homogeneity of forces in a homo-
space traversed will assist this to a very great extent. Now, since, compared with perfectly f re ^"uded "tf^the
indivisible & non-extended points of matter, there are an infinitely infinite number of points tenuity & the veio-
of space existing in the same plane, there is an infinitely infinite improbability that, for "ty0*1^*-

any instant of time chosen, the direction of motion of any one point of matter should be
accurately directed towards any other point of matter ; & this improbability, when we
consider the sum of all the instants contained in any given time, however long, still comes
out simply infinite. The number of points of light is indeed very large, not to say
enormous, but in my Theory it is at least finite. These points at any chosen instant of
time have an almost immeasurable number of directions of motion, but this number is
finite in my Theory. It is indeed true that, no matter where an eye is situated upon the
well-nigh immeasurable surface of a sphere described about one of the remotest stars as
centre, nay, or within that sphere, the star will be seen ; & thus, it is true that some particle
of light must affect our eye. But in my Theory, that does not come about because rays
of light come to it accurately in every one of an absolute infinity of directions ; but because
the pupil & the nerves of the eye do not form a single point, & the forces due to the
points of a particle of light act at some distance away. Hence, in any chosen time, no
matter how long, there need not happen in my Theory any case, in which any point of
light is directed exactly towards any other point either of light, or of any substance, so
that it is bound to collide with it. Hence, no point of light stays its motion, or deflects
it, through collision or immediate impact.

476. This is indeed a common property of all bodies, that is, of bodies that approach If they possess
one another. In my Theory, they have no point directly colliding with any other point, fity, anybodies.
For this reason I also stated, in the above-mentioned article, that, if no mutual forces were even solids, win
present, there is always bound to be an apparent compenetration of all bodies. Yet, from solids, without°any
this article alone, it is utterly impossible that there ever can be real compenetration. Hence, disturbance of their
forces extending over some distance will hinder the progressive motion. If these forces

are always equal in all directions, there would be no impediment to the motion, & it would
necessarily be rectilinear owing to the force of inertia. Hence, nothing but a difference in the

336 PHILOSOPHIC NATURALIS THEORIA

esse rectilineus. Quare sola differentia virium agentium in punctum mobile obstare potest.
At si nulla occurrat infinita vis arcus asymptotici cujuspiam post primum ; vires omnes
finitae sunt, adeoque & differentia virium secundum diversas directiones agentium finita
est semper. Igitur utcunque ea sit magna, ipsarri finita quaedam velocitas elidere potest,
quin permittat ullam retardationem, accelerationem, deviationem, quas ad datam quampiam
utcunque parvam magnitudinem assurgat : nam vires indigent tempore ad producendam
novam velocitatem, quae semper proportionalis est tempori, & vi. Hinc si satis magna
velocitas haberetur ; quaevis substantia trans aliam quanvis libere permearet sine ullo
sensibili obstaculo, & sine ulla sensibili mutatione dispositionis propriorum punctorum,
& sine ulla jactura nexus mutui inter ipsa puncta, & cohaesionis, quod ibidem illustravi
exemplo ferrei globuli inter magnetes disperses cum satis magna velocitate libere permeantis,
ubi etiam illud vidimus, in hoc casu virium ubique finitarum impenetrabilitatis ideam,
quam habemus, nos debere soli mediocritati nostrarum velocitatum, & virium, quarum
ope non possumus imprimere satis magnam velocitatem, & libere trans murorum septa, &
trans occlusas portas pervadere.
si per asymptoticos 477. Id quidem ita se habet, si nullas praeter primam asymptoti habeantur. quae vires

arcus particulae i i • r • • i • • f- } • ' • i r

essent prorsus im- absolute intimtas mducant : nam si per ejusmodi asymptoticos arcus particulse nant &
permeabiies, tum indissolubiles, & prorsus impenetrabiles iuxta num. 362 : turn vero nulla utcunque magna

recurrendum ad,. • i i i „ • i •

moiem imminutam velocitate posset una particula alteram transvolare, & res eodem recideret, quo in communi
quantum oportet. sententia de continua extensione materiae. Tum nimirum oporteret lucis particulas
minuere, non quidem in infinitum (quod ego absolute impossibile arbitror, quemadmodum
& quantitates, quas revera infinite parvae sint in se ipsis tales, ac independenter ab omni
nostro cogitandi modo determinatae : nee vero earum usquam habetur necessitas in Natura)
sed ita, ut adhuc incursus unius particulae in aliam pro quovis finite tempore sit, quantum
libuerit, improbabilis, quod per finitas utique magnitudines prsestari potest. Si enim
concipiatur planum per lucis particulam quancunque ductum, & cum ea progrediens ;
eorum planorum numerus dato quovis finito tempore utcunque longo erit utique finitus ;
si particulas inter se distent quovis utcunque exiguo intervallo, quarum idcirco finito quovis
tempore non nisi finitum numerum emittet massa utcunque lucida. Porro quodvis ex
ejusmodi planis ad medias, qua latissimae sunt, alias particulas luminis inter se distantes
finito numero vicium appellet utique intra finitum quodvis tempus, cum id per intervalla
finita tantummodo debeat accidere, [222] £ summa ejusmodi accessuum pertinentium
ad omnia plana particularum numero finitarum finita erit itidem, utcunque magna.
Licebit autem ita particularum diametros maximas imminuere, ut spatium plani ad datam
quamvis distantiam protensi circunquaque etiam exiguam, habeat ad sectionem maximam
particulae rationem, quantum libuerit, majorem ilia, quam exprimit ille ingens, sed finitus
accessuum numerus : ac idcirco numerus directionum, per quas possint transire omnia
ilia plana ad omnes particulas pertinentia sine incursu in ullam particulam, erit numero
earum, per quas fieri possit incursus, major in ratione ingenti, quantum libuerit ; etiam
si cum ea lege progredi deberent, ut altera non deberet transire in majore distantia ab
altera, quam sit intervallum illud determinans exiguum illud spatium, ad quod assumpta
est particularum sectio minor in ratione, quantum libuerit, magna. Infinite nusquam
opus erit in Natura, & series finitorum, quae in infinitum progreditur, semper aliquod
finitum nobis offert ita magnum, vel parvum, ut ad physicos usus quoscunque sufficiat.

Asymptoticis iis 478. Quod de particulis inter se collatis est dictum, idem locum habet & in particulis

ess" opus "ealu™ resPectu corporum quoruncunque, potissimum si corpora juxta meam Theoriam constituta

tius exciudenda : sint particulis distantibus a se invicem, & non continue nexu colligatis, sive extensionis

expUcen^ur2 Tin'e verc continu£e ilHus veli, aut muri continuam infinitam objicientis resistentiam, de quo

ipsis. egimus num. 362, & 363. Verum ejusmodi asymptoticorum arcuum nulla mini est

necessitas in mea Theoria, & hie itidem per nexus, ac vires limitum ingentis, quantum

libuerit, quanquam non etiam infiniti valoris, omnia prasstari possunt in Natura : & si

principio inductionis inhasrere libeat ; debemus potius arbitrari, nullos esse alios ejusmodi

asymptoticos arcus in curva, quam Natura adhibet : cum in ingenti intervallo a fixis ad

particulas minimas, quas intueri per microscopia possumus, nullus ejusmodi nexus occurrat,

quod indicat motus continuus particularum luminis per omnes ejusmodi tractus ; nisi forte

primus ille repulsivus, & postremus ejus naturae arcus, ad gravitatem pertinens, indicio sint,

esse & alios alibi in distantiis, quae citra microscopiorum, vel ultra telescopiorum potestatem

A THEORY OF NATURAL PHILOSOPHY 337

forces acting on a moving point can hinder it. But if no infinite force occurs corresponding
to any asymptotic arc after the first, all the forces are finite ; & so also the difference between
the forces acting in different directions will be always finite. Therefore, no matter how
great the force may be, there is some finite velocity capable of overcoming it, without
suffering any retardation, acceleration, or deviation amounting to any given magnitude,
no matter how small. For, the forces require time to produce a new velocity, this being
always proportional to the force & the time. Hence, if there were a sufficiently great velocity,
any substance would pass freely through any other substance, without any sensible hindrance,
& without any sensible change in the situation of the points belonging to either substance,
& without any destruction of the mutual connection between the points, or of cohesion.
There also I gave an illustration of an iron ball making its way freely through a group of
magnets with a sufficiently great velocity ; & here also we saw that we owe what idea
we have of impenetrability, in the case of forces that are everywhere finite, merely to
the moderate nature of our velocities & forces ; for by their help alone we cannot impress
a sufficiently great velocity, & freely pass through barrier- walls, or shut doors.

477. Now, this is the case, so long as there are no asymptotic arcs besides the first, n. owing to the
to induce absolutely infinite forces ; but if, owing to such asymptotic arcs, the particles totfce"rcs?fpartSes
become incapable both of dissolution & penetration, as in Art. 362, then indeed by no become 'imperme-
velocity, however great, could one particle pass through another ; & the matter would be Ihoaid. ha^e^ofau
reduced to the same idea, as is held generally about the continuous extension of matter, back upon diminu-
Thus, in that case it would be necessary to diminish the size of the particles of light ; not faraswas'necessary!
indeed infinitely — for I consider that that would be altogether impossible, just as also I

think that there are no quantities infinitely small in themselves, and so determined without
reference to any process of human thought ; nor is there anywhere in Nature any
necessity for such quantities. But they must be so diminished that the direct collision
of one particle with another in any chosen finite time will still be improbable, to any extent
desired ; & this can be secured in every case by finite magnitudes. For suppose a
plane area circumscribing each particle of light, & that this plane moves with the particle ;
then the number of these planes in any given finite time, however long, will in every case
be. finite, so long as the particles are distant from one another by any interval at all, no
matter how small ; & thus, in any given finite time the mass, however luminous, can only
emit a finite number of these particles. Further, any one of these planes will impinge, at their
broadest parts, upon the middle of other particles of light distant from one another by a
finite number of fits, in every case in a finite time ; for, this can only take place through
a finite interval. The sum of such approaches pertaining to all the planes of the particles,
finite in number, will also be finite, no matter how great the number may be. But
we may so diminish the greatest diameters of the particles that the area of the plane,
extended in all directions round to any given distance, however small, may bear to the
greatest section of the particle a ratio greater, to any arbitrary extent, than that which
is expressed by the huge but finite number of the approaches. Hence, the number of
directions, by which all the planes pertaining to all the particles may pass without colliding
with any particle, will be greater than the number of directions in which there may be
collision, the ratio being one that is as immense as we please. And this will even be the case,
if they should have to move in accordance with the law that one must not pass at a greater
distance from the other than that interval which determines the very small space, to
which it is supposed that the section of the particle bears a ratio of less inequality, no matter
what the magnitude. There will nowhere be any need of the infinite in Nature ; a series
of finites, extended indefinitely, will always give us something finite, which is large enough
or small enough to satisfy any physical needs.

478. All that has been said with regard to particles referred to one another, the same There k no ne^
will hold good for particles in reference to any bodies ; & especially if the bodies are formed, branch^s^^ather,
in accordance with my Theory, of particles distant from one another, & not bound together they should be ex-

J . J . . . ?. eluded ; how well

by a continuous connection, or possessing the truly continuous extension of the skin or an things can be
wall offering a continuous infinite resistance, with which we dealt in Art. 362, 363. But explained without
really there is no necessity for such asymptotic arcs in my Theory ; in it also, by means of
connections & forces of limits of any value however great, though not actually infinite,
everything in Nature can be accomplished. If we are to adhere to the principle of induction,
we are bound rather to think that there are no other asymptotic arcs in the curve which
Nature follows. For, in the mighty interval between the stars & the smallest particles
that are visible under the microscope, no connections of this kind occur, as is indicated
by the continuous motion of the particles of light throughout the whole of these regions.
Unless, perhaps, that first repulsive branch, & that last arc of the nature that pertains
to gravity, are to be taken as a sign that there are also somewhere others like them, at
distances which are less than microscopical, or greater than those within the range of the
* z

338 PHILOSOPHIC NATURALIS THEORIA

contrahuntur, vel protenduntur. Ceterum si vires omnes finitae sint, & puncta materiae
juxta meam Theoriam simplicia penitus, & inextensa : multo sane facilius concipitur,
qui fiat, ut habeatur haec apparens compenetratio sine ullo incursu, & sine ulla dissolutione
particularum cum transitu aliarum per alias.

Quomodo rem con- 4.79. Porro duo sunt, quorum singula rem praestare possunt, velocitas satis magna,

satis11 ma^na" && 1U2E nimirum utcunque magnam virium inaequalitatem potest eludere, & virium circum-

aequaiitas sensibilis quaque positarum aequalitas, quae differentiam relinquat omnino nullam. Differentia

quaque!" Quomodo nunquam sane habebitur omnino nulla, ubi [223] punctum materias praetervolet per

haec in homogeneo quandam punctorum veluti silvam, quorum alia ab aliis distent : necessario enim mutabit

;ur' distantiam ab iis, a quibus minimum distat, jam accedens nonnihil, jam recedens. Verum

ubi distributio particularum ad aequalitatem quandam multum accesserit, inaequalitas

virium erit perquam exigua ; si omnium virium habeatur ratio, quas exercent omnia puncta

disposita circa id punctum ad intervallum, ad quod satis sensibiles meae curvae vires

protenduntur. Concipiamus enim sphaeram quandam, quae habeat pro semidiametro

illam distantiam, ad quam protenduntur flexus curvae virium primigenise, sive ad quam

vires singulorum punctorum satis sensibiles pertingunt. Si medium satis ad homogenei-

tatem accedat ; secta ilia sphaera in duas partes utcunque per centrum, in utraque numerus

punctorum materiae erit quamproxime idem, & summa virium quam proxime eadem, se

compensantibus omnibus exiguis inaequalitatibus in tanta multitudine, quod in omnibus

fit satis numerosis fortuitis combinationibus : adeoque sine ullo sensibili impedimento,

sine ingenti flexione progredietur punctum quodcunque motu vel rectilineo, vel tremulo

quidem nonnihil, sed parum admodum, & ad sensum aeque in omnem plagam.

reioctafs0 exi8uam 4^°' Qu°d s^ accedat ingens velocitas ; multo adhuc minor erit inaequalitatum

inzquaiitatem eiu- effectus, turn quod multo minus habebunt temporis vires ut agant, turn quod in ipso

turbineXeiSneomnon contmuato progressu inasqualitates jam in unam plagam prasvalebunt, jam in aliam, quibus

cadente. sibi mutuo celerrime succedentibus, magis adhuc uniformis, & rectilineus erit progressus.

Sic ubi turbo ligneus gyrat celerrime circa verticalem axem cuspide tenuissima innixum

solo, stat utique, inaequalitate ponderis, quas ad casum determinat, jam ad aliam plagam

jacente, & totam inclinante molem, jam ad aliam, qui, celeritate motus circularis imminuta,

decidit inclinatus, quo exigit prasponderantia.

QUOCJ autem homogeneitas medii, & velocitas praestant simul, id adhuc auget
quid is multo magis is nexus, qui est inter materiae puncta particulam componentia, & aequali ad
prastet. sensum velocitate delata, qui mutuis viribus cum accessum ad se invicem punctorum

particulam componentium, & recessum impediat, cogit totam particulam simul trepidare
eo solo motu, quern inducit summa inaequalitatum pertinentium ad puncta omnia, quae
summa adhuc magis ad aequalitatem accedit : nam in fortuitis, & temere hac, iliac dispersis,
vel concurrentibus casu circumstantiis, quo major numerus accipitur, eo inaequalitatum
irregularium summa decrescit magis.

Rantatem plun- ,g2_ Demum raritas medii ad id ipsum confert adhuc magis : quo enim major est

mum prodesse : . ' . .... ^ J .

omnes eas quatuor raritas, eo minor occurnt punctorum numerus mtra illam sphaeram, adeoque eo minor
causas habere v}rium componendarum multitudo, & inaequalitas adhuc multo mi-r224l-nor. Porro

locum in lumme . v • , . . ,.. L 7i . v

non turbato a omnes hae quatuor causae aequalitatis concurrunt, ubi agitur de radus collatis cum alns
radns alia du-ectione rac|iis : homogeneitas, nam lumen a dato puncto progrediens suam densitatem imminuit
sum: priores tres in ratione reciproca duplicata distantiarum a puncto radiante, adeoque in tarn exiguo
' circuncluaclue circa, quodvis punctum intervallo, quantum est id, ad quod virium actio
sensibilis protenditur, ad homogeneitatem accedit in immensum : celeritas, quae tanta
est, ut singulis arteriae pulsibus quaevis luminis particula fere bis centum millia Romanorum
milliariorum percurrat : nexus particularum mutuus, nam ipsae luminis particulae ad
diversos coloratos radios pertinentes habent perennes proprietates suas, quas constanter
servant, ut certum refrangibilitatis gradum, & potentiam certo impulsu agitandi oculorum
fibras, per quam certain certi coloris sensationem eliciant : ac demum tenuitas immanis,
qua opus est ad tantam diffusionem, & tarn perennem efHuxum sine ulla sensibili imminu-
tione Solaris massas, & cujus indicium aliquod proferam paullo inferius. Ubi vero agitur
de lumine comparato cum substantiis pellucidis, per quas pervadit, priora ilia tria
tantummodo locum habent respectu particularum luminis, & omnia quatuor respectu
particularum pellucidi corporis, quarum nexus non dissolvitur, nee positio turbatur
quidquam ab intervolantibus radiorum particulis. Quamobrem errat qui putat, mea

A THEORY OF NATURAL PHILOSOPHY 339

telescope. Besides, if all the forces are finite, and points of matter, in accordance with, my
Theory, are perfectly simple & non-extended, it is far more easily understood why there
can be this apparent compenetration, without any collision, & without any dissolution of
the particles as they pass through one another.

479. Further, there are two things, each of which can accomplish the matter ; namely, How the matter
a sufficiently great velocity, such as will foil the inequality of the forces, however great uti&ed b ^suffi-
that may be ; & an equality of the forces in all directions, such as will leave the difference ciently large veio-
absolutely zero. Now the difference can never really be altogether zero, when a point Clty ^-ta ^"l1?6
of matter passes through, so to speak, a forest of points, which are separated from one forces in all direc-
another. For, of necessity, it will change its distance from those points, from which it is ti°ns- ?°w. ^^
least distant, at one time approaching & at another receding. But when the distribution a homogeneous
of the particles approaches very closely to an equality, the inequality of the forces will be medium-
exceedingly small, so long as account is taken of all the forces exerted by all the points

situated about that point at an interval equal to that over which the forces of my curve
extend while still fairly sensible. For, imagine a sphere, that has for its semi-diameter
the distance over which the windings of the primary curve extend, that is, the distance up
to which the forces of each of the points are fairly sensible. If the medium approximates
sufficiently closely to homogeneity, & the sphere is divided into any two parts by a plane
through the centre, the number of points of matter in each part will be nearly the same ;
& the sum of the forces will be very approximately the same, as the slight differences taken
as a whole compensate one another in so great a multitude ; for this is always the case in
sufficiently numerous fortuitous combinations. Thus, without any impediment, without
any very great flexure, any point will proceed with a motion that is rectilinear, or maybe
somewhat but very slightly wavy, & practically uniformly so in every direction.

480. But if the velocity is very great, the effect of inequalities will be still less ; both How a very great
because the forces will have much less time in which to act, & because in such a continued slight y inequality*
progress the inequalities will prevail first on one side & then on the other ; & as these follow example, from a
one another very quickly, the progress will be still more uniform & nearer to rectilinear ^p not faUhig.1118
motion. Thus, when a wooden spinning-top spins very quickly about a vertical axis with

a very fine point resting on the ground, it stays perfectly upright ; for, the inequality of
its weight, which disposes it to fall, lies first on one side & inclines the whole mass that way,
& then on the other side ; while, as soon as the circular motion decreases, it becomes inclined
to the side to which the preponderance forces it.

481. Again the effect produced by the homogeneity of the medium & the great velocity ?n addition there
together is still further increased by the connection that exists between the points of matter between the point"
forming; the particle & moving together with practically the same velocity. This connection, °* a particle ; the

1 , • i i i • r i • effect of this.

since, through the mutual forces, it prevents the mutual approach or recession ot the points
forming the particle, will force the entire particle to move as a-whole with the single motion
that is induced by the sum of the inequalities pertaining to all its points ; & this sum will
still further approximate to equality. For, in circumstances that are fortuitous, distributed
here & there at random, or concurring by chance, the greater the number taken, the
more the sum of the irregular inequalities decreases.

482. Lastly, rarity of the medium is of still further assistance ; for, the greater the Great effect of
rarity, the smaller the number of points that occur within the sphere imagined above, aif'four of^these
& therefore the smaller the number of forces to be compounded, & much smaller still the causes hold for
inequality. Further, all four of these causes of inequality occur together, when we are bj^rays^roceedSg
dealing with rays of light in regard to other rays. Homogeneity we have, because light in any other direc-
proceeding from a given point diminishes its density in the inverse ratio of the squares of threeoAhemin^he
the distances from the radiant point ; & thus, in the exceedingly small interval round about more dense trans-
any point, whatever the distance may be over which a sensible action of the forces extend, P31611* media-

the approach to homogeneity is exceedingly great. Velocity also we have, so great that
in a single beat of the pulse a particle of light travels a distance of nearly two hundred
thousand Roman miles. Mutual connection of the particles also, for the particles of light
pertaining to differently coloured rays have all their special lasting properties, which they
keep to unaltered, such as a definite refrangibility & the power of affecting the nerves of
the eye with a definite impulse, through which they give it a definite sensation of a definite
colour. Lastly, an extremely great tenuity, such as is necessitated by the greatness of
the diffusion & the endurance of the efflux without sensible diminution of mass in the case
of the Sun ; & of this I will bring forward some evidence a little further on. But when
we are dealing with light in regard to transparent substances, through which the light
passes, the first three only hold good with regard to the particles of light, but all four with
regard to the particles of the transparent body ; the connections between the particles of
the body are not broken, nor is their relative position affected to any extent by the particles
of the rays of light passing through them. Therefore he will be mistaken, who thinks

340 PHILOSOPHIC NATURALIS THEORIA

indivisibilia puncta prasdita insuperabili potentia repulsiva pertlngente ad finitam distantiam
esse tarn subjecta collisionibus, quam sunt particulae finitae magnitudinis, & idcirco nulli
adminiculo esse pro comprehendenda mutua lucis penetratione ; nam sine cruribus illis
asymptoticis posterioribus meae vires repulsivae non sunt insuperabiles, nisi ubi puncta
congredi debeant in recta, quae ilia jungit, qui casus in Natura nusquam occurrit.

Pelluciditatem oriri 483. Et vero sola homogeneitas pelluciditatem parit, uti jam olim notavit Newtonus,

a sola homogenei- • ... . vioij-

tate:soiam hetero- nec opacitas ontur ab impactu in partes corporum solidas, & a defectu pororum jacentmm
geneitatem impe- in directum, uti alii ante ipsum plures censuerant, sed ab inaequali textu particularum

dire posse progres- i £ «.F . , . , . *. . r ,. ..

sum per insquaii- neterogenearum, quarum alise alns minus densis, vel etiam pemtus vacuis amphoribus
tatem virium. spatiolis intermixtae satis magnam inducunt inaequalitatem virium, qua lumen in omnes
partes detorquent, ac distrahunt, flexu multiplici, & ambagibus per internes meatus
continuis, quibus fit, ut si paullo crassior occurrat massa corporis ex heterogeneis particulis
coalescentis, nullus radius rectilineo motu totam pervadat massam ipsam, quod nimirum
ad pelluciditatem requiritur. Indicia rei habemus quamplurima prseter ipsam omnem
superiorem Theoriam, quae rem sola evinceret ; cum nimirum sine inasqualitate virium
nullum haberi possit libero rectilineo progressui impedimentum. Id sane colligitur ex
eo, quod omnium corporum tenuiores laminae pellucidae sunt, uti norunt, qui microscopiis
tractandis assueverunt : id [225] evincunt illae substantiae, quae aliarum poris injectae
easdem ex opacis pellucidas reddunt, ut charta oleo imbuta fit pellucida, supplente aerem
ipso oleo, cum quo multo minus inaequaliter in lumen agunt particulae chartae, quam
agerent soli aeri, vel vacuo spatio intermixtae. Rem autem oculis subjicit vitrum contusum
in minores particulas, quod sola irregularitate figuras particularum temere ex contusione
nascentium, & aeris intermixti inaequalitate fit opacum per multiplicationem reflexionum,
& refractionum irregularium : nec aliam ob causam aqua in glaciem bullis continuis
interruptam abiens pelluciditatem amittit, ut & alia corpora sane multa, quae, dura
concrescunt vacuolis interrupta, illico opaca fiunt.

Reflexionem non .$* Quamobrem nec reflexio inde ortum ducit, sed habetur etiam in pellucidis

onri ab impactu, ' .7 .... ,, ... ,-J .

sed ab inzequaiitate corponbus ex inaequalitate virium seu repellentium, seu attrahentium, uti in Optica sua
vinum m mutatione Newtonus tarn multis notissimis argumentis demonstravit, quorum unum est illud ipsum

medu ; ubi pro re- . - . . . . i . . .r

fractions expiica- ex aspentate supernciei cujuscunque cujusvis corporis, utcunque nobis, nudo potissimum
pramissa inspectantibus oculo, laevis appareat, & perpolita, quod num. 299 exposuimus ; & ex eadem
causa oritur etiam refractio. Si velocitas luminis esset satis magna ; impediret etiam
hujusce inaequalitatis effectum, qui provenit a diversa mediorum constitutione : sed ex
ipsis reflexionibus, & refractionibus in mutatione medii, conjunctis cum propagatione
rectilinea per medium homogeneum, patet, celeritatem illam tantam luminis satis esse
magnam ad eludendam illam inaequalitatem tanto minorem, quae habetur in mediis homo-
geneis, non illam tanto majorem, quae oritur a mediorum discrimine. Quod vero ad
refractionis explicationem ex Mechanica requiritur, exposuimus a num. 302, ubi adhibuimus
principium illud virium inter duo plana parallela agentium aeque in distantiis aequalibus
ab eorum utroque, cujus explicationem ad luminis particulas jam expediemus.

ad "uam 485- Concipiatur (/) ilia sphaerula, cujus semidiameter^ [226] aequatur distantiae illi,
extenditur vis ad quam agunt actione satis sensibili particulae corporum in lucis particulam, quae cum

sensibilis agens in _~ ^ _____

lumen : inde vis

inter bina plana (f) Refert MN in fig. 70 superficiem dirimentem duo media, GE viam radii advenientis, H particulam luminis ;

parallela superficiei jjj? celeritatem, efus absolutam, HS parallelam, SE perpendicularem, quis est eo minor, quo radius incidit magis obliquus :

interfuse vis^ari? ' abc est sph<era, intra quam habetur actio sensibilis in particulam H, quie est adhuc tota in priore media : X, X', X" sunt
loca plura particulis progredientis inter plana AB, CD parallela superficiei MN, sita ad distantiam ab ea aqualem
semidiametro sphtera He. Particula sita inter ilia plana ubicunque, ut in X, ea sphtera habebit suum segmentum FRL
ultra superficiem MN : sit efus axis RT, ^ eodem axe segmentum QTZ priori tequale, ac mn planum per centrum par-
allelum MN. Segmenta mFLn, mQZn ejusdem medii agent izqualiter. Segmenta FRL, QTZ incequaliter, sed eorum
vires dirigentur per axem TR in alteram e binis plagis oppositis : adeoque i3 differentia virium dirigetur per eundem, qui
quidem perpendicularis est utique plants AB, CD. Ea actione viaiticurva radii sinuatur per XX'X". Prout vis dirigetur
versus CD, vel versus AB, curva erit cava versus easdem, y in mutatione directionis vis ipsius mutabitur ftexus curvtr.
Si autem curva evaserit alicubi parallela piano AB ; flectet cursum retro ; nisi id accidat accurate in situ vis = o, qui

A THEORY OF NATURAL PHILOSOPHY

34'

FIG. 70.

342

PHILOSOPHISE NATURALIS THEORIA

FIG. 70.

A THEORY OF NATURAL PHILOSOPHY 343

that my indivisible points, endowed with an insuperable repulsive force extending to a
finite distance, are just as subject to collisions as particles of finite magnitude ; & therefore
that there is no assistance to be derived from them in understanding the mutual penetration
of light ; for, unless there are those asymptotic branches after the first, my repulsive forces
are not insuperable, except when points are bound to move together in one straight line
joining them, a circumstance which never occurs in Nature.

483. Indeed homogeneity by itself creates transparence, as was long ago stated by

Newton ; & opacity does not arise from impact with the solid parts of bodies, or through alone ; & hetero-

a lack of pores lying in a straight line, as many others before Newton thought, but from geneity al°ne 1S

the unequal structure of heterogeneous particles ; of which some are interspersed amongst venting progressive

others of less density, or even in perfectly empty little spaces, of considerable size, motion through m-

and thus induce an inequality great enough to distort the light in all directions, & to

harass it with manifold windings & continuous meandering through internal channels ;

from which it comes about that, if a somewhat thick mass occurs of a body formed

from heterogeneous particles, no ray with rectilinear motion will pass through the

whole of that mass ; which is the requirement for transparence. We have very many

pieces of evidence on the subject, in addition to the whole of the Theory given above,

which of itself is sufficient to prove it. For, indeed, without inequality of forces there

can be no impediment to free rectilinear progressive motion. This can truly be deduced

from the fact that fairly thin plates of all bodies are transparent, as is known to those who

have been accustomed to microscopical work. Evidence is also afforded by such substances

as, on injection into the pores of other substances, turn the latter from opaque to transparent ;

thus, paper soaked with oil becomes transparent, the oil taking the place of the air ; for,

with it the particles of paper act far less unequally upon the light than they would act,

if merely air, or an empty space were interspersed. Moreover, glass broken up into fine

particles brings the matter right before our eyes ; for, from the mere irregularity of the

shape of the particles randomly produced by the powdering, & the inequality of the

interspersed air, it becomes opaque on account of the multiplication of reflections &

refractions occurring irregularly. From no other cause does water, turning into ice interrupted

by continuous bubbles, lose its transparence ; it is just the same also with many other

bodies, which, as they grow, are interspersed with little empty spaces, & from this cause

alone become opaque.

484. Therefore also reflection does not arise from impact ; but it is even found in Reflection does not

,j.j i . ...... ,. . ,-,.,. . take place through

transparent bodies due to the inequality of forces, whether repulsive or attractive. This impact, but owing
was proved by Newton in his Optics by a large number of arguments that are well known ; *° inequality of

r f , '. , i ' ° i • A i • i f i forces on the

one of these is that very reason that was stated in Art. 299, derived from the roughness medium being
of any surface of any body, no matter how smooth & polished it appears to us, especially changed : where

i • • » • i *i *i j T* r • • f -rr t the principles for

when viewed with the naked eye. Refraction also arises from the same cause. If the the explanation of
velocity of light were great enough, it would prevent even the effect of this inequality refraction have

J. f ' i T/r ° ' . • i * 11 T> r i r i been premised.

that arises from the different constitution of the media, .out, from the fact that there are
these reflections & refractions on a change of medium, taken in conjunction with the fact
of rectilinear propagation through a homogeneous medium, it is clear that the great velocity
of light is enough to foil the comparatively small inequality that is found in homogeneous
media, but is not enough for the comparatively greater inequality that arises from a difference
in the media traversed. But that which is necessary for the mechanical explanation of
refraction has been stated in Art. 302 onwards ; where we employed the idea of forces
acting between two parallel planes, the forces being equal for equal distances from either
of the planes ; we will now apply this idea to particles of light.

48;. Imagine (/) a sphere, of which the semidiameter is equal to the distance up to Consideration of

i'ii • i r i i • i /• i- i • i -1 r • i -11 • » the sphere whose

which the particles 01 a body act upon a particle or light with a fairly sensible action ; & radius is the dis-

_ tance to which the

(f) In Fig. 70, MN is the surface of separation between the two media, GE the path of an approaching ray, H a particle light extends-

of light, HE its absolute velocity, HS the parallel, SE the perpendicular component, which latter is the less, the more thence the force

oblique the incidence of the ray. abc is the small sphere, within which there is sensible action on the particle H, which between two planes

it as yet altogether in the first medium. X,X',X" are positions of the particle as it passes between the planes AB, CD, parallel to the

parallel to the surface MN, and situated at a distance from it equal to the semidiameter of the sphere He. // the particle tio^of the media,

is situated anywhere between the two planes, as at X, the sphere will have its segment FRL on the far side of the surface between which the

MN. Let the axis of the segment be RT, and let QTZ be a segment having the same axis and equal to the former force acts.

segment, and let mn be a plane through the centre parallel to MN. Then the segments mFLn, mQZn, lying in the

same medium, will act equally ; but the segments FRL, QTZ will act unequally ; yet their forces will be directed

along the axis TR in one or other of the two opposite directions, and thus also the difference between these forces will act

along the same straight line, which is perpendicular to the planes AB, CD in every case. Owing to this action the

curved path of the ray will wind along through X,X',X". According as the force is directed towards CD or towards

AB, the curve will be concave with respect to these same planes, and when the force changes its direction the flexure of

the curve will also change. Moreover, if the curve should anywhere happen to become parallel to the plane AB, the

path will be reflected ; unless it should fall out that exactly in that position the force was zero, a case that is infinitely

344 PHILOSOPHIC NATURALIS THEORIA

lucis particula progrediatur simnl. Donee ipsa sphaerula est in aliquo homogenco medio
tota,. vires in particulam circunquaque sequales erunt ad sensum, & cum nullus habeatur
immediatus incursus, motus inertiae vi factus erit ad sensum rectilineus, & uniformis.
Ubi ilia sphaerula aliquod aliud ingressa fuerit diversae naturae medium, cujus eadem moles
exerceat in particulas luminis vim diversam a prioris medii vi ; jam ilia pars novi medii,
quae intra sphserulam immersa erit, non exercebit in ipsam particulam vim aequalem illi,
quam exeret pars sphaerulae ipsi rcspondens ex altera centri-parte, & facile patet, differentiam
virium debere dirigi per axem perpendicularem illis segmentis sphaarulae, per quern singulae
utriusque segmenti vires diriguntur, nimirum perpendiculariter ad superficiem dirimentem
duo media, quae illud prius segmentum terminal : & quoniam ubicunque particula sit in
aequali distantia a superficie, illud segmentum erit magnitudinis ejusdem ; vis motum
perturbans in iisdem a superficie ilia distantiis eadem erit. Durabit autem ejusmodi vis,
donee ipsa sphserula tota intra novum medium immergatur. Incipiet autem immergi
ipsa sphaerula in novum medium, ubi particula advenerit ad distantiam ab ipsius superficie
sequalem radio sphasrulas, & immergetur tota, ubi ipsa particula jam immersa fuerit, ac
ad distantiam eandem processerit. Quare si concipiantur duo plana parallela ipsi superficiei
dirimenti media, quae superficies in exiguo tractu habetur pro plana, ad distantias citra,
& ultra ipsam sequales radio illius sphaerulae, sive intervallo actionis sensibilis ; particula
constituta inter ilia plana habebit vim secundum directionem perpendicularem ipsis planis,
quae in data distantia ab eorum altero utrovis aequalis erit.

Tres casus, qui ex- 486. Porro id ipsum est id, quod assumpsimus num. 302, & unde derivavimus reflexionis,

vel refractio°neIm ac refractionis legem : nimirum si concipiatur ejusmodi vis resoluta in duas, alteram

cum recessu a per- parallelam iis planis, alteram perpendicularem : ilia vis pot-[227]-est perpendicularem

sam 1Crefractionem velocitatem vel extinguere totam ante, quam deveniatur ad planum ulterius, vel imminuere,

cum accessu. vel augere. In primo casu debet particula retro regredi, & describere curvam similem illi,

quam descripsit usque ad ejusmodi extinctionem, recuperando iisdem viribus in regressu,

quod amiserat in progressu, adeoque debet egredi in angulo reflexionis aequali angulo

incidentiae : in secundo casu habetur refractio cum recessu a perpendiculo, in tertio

refractio cum accessu ad ipsum, & in utrolibet casu, quaecunque fuerit inclinatio in ingressu,

debet differentia quadratorum velocitatis perpendicularis in ingressu, & egressu esse

constantis cujusdam magnitudinis ex principio mechanico demonstrato num. 176 in adn.

& inde num. 305 est erutum illud, sinum anguli incidentiae ad sinum anguli refracti debere

esse in constanti ratione, quae est celeberrima lucis proprietas, cui tota innititur Dioptrica

& prasterea illud num. 306 velocitatem in medio prascedente ad velocitatem in medio

sequente esse in ratione reciproca sinuum eorundem.

Lumen debere in 487. Hoc pacto ex uniformi Theoria deductae sunt notissimae, ac vulgares leges

corpora reagere reflexionis, ac refractionis, ex quibus plura consectaria deduci possunt. Imprimis quoniam

sequahter : hinc , , .' * , , r. a

immensa lucis ten- debet actio semper esse mutua, dum corpora agunt in lumen ipsum renectenuo, oc reirm-
uitas : qui effectus gendo ; debet ipsum lumen agere in corpora, ac debet esse velocitas amissa a lumine ad

ipsi f also tribuantur ° . . ' r. . .r . ,

a nonnuiiis. velocitatem acquisitam a centre gravitatis corporis sistentis lumen, ut est massa corporis

ad massam luminis. Inde deducitur immensa luminis tenuitas : nam massa tenuissima
levissimae plumulae suspensae filo tenui, si impetatur a radio repente immisso, nullum
progressivum acquirit motum, qui sensu percipi possit. Cum tarn immanis sit velocitas
amissa a lumine ; facile patet, quam immensa sit tenuitas luminis. Newtonus etiam
radiorum impulsioni tribuit progressum vaporum cometicorum in caudam ; sed earn
ego sententiam satis valido, ut arbitror, argumento rejeci in mea dissertatione De Cometis.
Sunt, qui auroras boreales tribuant halitibus tenuissimis impulsis a radiis solaribus, quod
miror fieri etiam ab aliquo, qui radios putat esse undas tantummodo, nam undae progressivum

Cams est in infinitum improbabilis. Id accidet in aliis radiis citius, in aliis radiis serins, •pro diversa absoluta celeritate
radii, pro diversa indinatione incidentitz, W pro diversa natura, vel constitutione particula, abeuntibus aliis particulis
per QXIK, aliis per QXX'I'K', aliis per QXX'X'T'K". Porro perquam exiguum discrimen in vi, vel celeritate, potest
curvam uno aliquo in loco a positione proxima parallelismo ad ipsum parallelismum traducere, quo loco superato adhuc summa
actionum usque ad O potest esse ad sensum eadem. Reliqua sunt Me, ut num. 306.

ex-

A THEORY OF NATURAL PHILOSOPHY 345

suppose that this sphere moves along with the light particle. So long as the little sphere
is altogether In a homogeneous medium, the forces on the particle all round it are practically
equal ; &, since no immediate impact can take place, the motion will be kept practically
rectilinear & uniform by the force of inertia. When the little sphere enters some other
medium of a different nature, the same volume of which exerts on the particles of light
a force different from the force due to the first medium, then, that part of the new medium
which is intercepted within the little sphere will not exert on the particle a force equal
to that which the corresponding part on the other side of the centre exerts ; & it is easily
seen that the difference of the forces must be directed along the axis perpendicular to these
segments of the sphere, for the forces due to each segment separately are so directed ; that
is to say, perpendicular to the surface of separation between the two media, which is the
bounding surface of the first of the two segments. Now, since that segment will be of
the same magnitude whenever the distance of the particle from the surface of separation
is the same, the force determining the change of motion will be the same at equal distances
from that surface. Further, such force will continue unchanged so long as the little sphere
is altogether immersed in the new medium. Now, the little sphere will commence to be
immersed in the new medium as soon as the particle reaches a distance from the surface
of separation equal to the radius of the little sphere ; & it will become altogether immersed
in it as soon as the particle itself, after entering it, has gone forward a further distance
equal to the radius. Hence, if two planes are imagined to be drawn parallel to the surface
of separation of the media, & this surface is supposed to be plane, for the very small region
extending on every side to a distance equal to the radius of the little sphere, or the interval
corresponding to sensible action ; then, a particle situated between those planes will be
under the influence of a force in the direction perpendicular to the planes, which will be
the same for equal distances from either of them.

486. Now, this reduces to that very same supposition that we made in Art. 302, from J.1?1?? cases<
which we derived the laws of reflection & refraction. Thus, if such a force is supposed iy reflection, refrac-
to be resolved into two parts, one parallel & the other perpendicular to the planes, the *lon with recession

• i i t i i r ^ T i i • i r i r i from the normal,

latter force may either destroy the whole ot the perpendicular velocity before the further & refraction with

plane is reached, or it may reduce it, or it may increase it. In the first case the particle approach to the

must turn back in its path & describe a curve similar to that which it has already described

up to the point at which its perpendicular velocity was described ; & on its return it will

recover the velocity it lost during its advance, with the same forces ; & thus, it must leave

the second medium with an angle of reflection equal to its angle of incidence. In the

second case there will be refraction with recession from the normal ; & in the third case,

refraction with approach to the normal. In either of these cases, whatever the inclination

was on entering the second medium, the difference between the squares of the velocities

on entering & leaving must be of some constant magnitude, from the mechanical principle

demonstrated in the note to Art. 176. From which, in Art. 305, 1 have deduced that the sine

of the angle of incidence must bear a constant ratio to the sine of the angle of refraction ;

& this is the very well known property of light, upon which is established the whole theory of

dioptrics. Also, in addition, in Art. 306, I deduced that the velocity in the first medium

is to the velocity in the second in the inverse ratio of the sines of these angles.

487. In this way, from a uniform theory, all the principal well-known laws of reflection Lisht ^"^J^™
& refraction have been derived ; & from these a large number of corollaries can be deduced. On the bodies;
First of all, because the action must always be mutual, so long as bodies act upon light, hence the extreme

n • f • • i - i« i ' ill- » i I'lii tenuity of light ;

reflecting or refracting it, the light must react on the bodies ; & the velocity lost by the these effects are
light must bear a ratio to the velocity gained by the centre of gravity of the body resisting *als??y . attributed

r i !• i i • i • ' i 1 • r i r i i i i e to light itself by

the motion of the light, which is equal to the ratio of the mass of the body to the mass of some people,
the light. From this we deduce the extreme tenuity of light. For, the tiniest mass of
the lightest feather suspended by the finest of strings, if it should be struck by a ray of light
suddenly falling upon it, still would acquire no progressive motion, such as could be perceived.
Since the velocity lost by the light is so huge, it can be clearly seen how exceedingly small must
be the density of light. Newton even attributed to the impact of light rays the progressive
motion tail first of the vapours of comets ; but I overthrew this idea, by an argument
which I consider to be perfectly sound, in my dissertation De Cometis. Some people
attribute the aurora borealis to exhalations of extremely small density impelled by solar
light-rays ; & I am astonished that this should be put forward by anyone who considers

improbable. This reflection will take place sooner in some rays than in others, according to different velocities of the rays,
different angles of incidence, different natures and constitutions of the particle ; some of the particles will pass along a
path QXIK, others along QXX'I'K', and others again along QXX'X'T'K". Further, a very slight difference in the
force or velocity will be enough to turn the curve in some one position of the particle from being very nearly parallel
to being exactly parallel ; if this position is once passed, the sum of the actions thereafter as far as O may be practically
the same. The rest is now similar to that which has been stated in Art. 306.

346 PHILOSOPHIC NATURALIS THEORIA

motum per se se non imprimunt : qui autem censent, & fluvios retardari orient! Soli
contraries, & Terrae motus fieri ex impulsu radiorum Solis, ii sane nunquam per legitima
Mechanics principia inquisiverunt in luminis tenuitatem.

4

Tenuissimum mo- 488. Solis particulis tenuissimis corporum imprimunt motum radii, ex quo per internas

iuminelprartic1ulis Y"cs aucto oritur calor, & quidem in opacis corporibus multo facilius, ubi tantse sunt

corporum: cal- reflexionum, & refractionum internae vicissitudines : exiguo motu impresso paucis particulis,

provenire ab^arum re^^lua internae mutuae vires agunt juxta ea, quae diximus num. 467. Sic ubi radiis solaribus

viribus internis, speculo collectis comburuntur aliqua, alia calcinantur [228] etiam ; omnes illi motus ab

t'u°hicPSUm proba" internis utique viribus oriuntur, non ab impulsione radiorum. Regulus antimonii ita

calcinatus auget aliquando pondus decima sui parte. Sunt, qui id tribuant massae radiorum

ibi collects. Si ad ita esset ; debuisset citissime abire ilia substantia cum parte decima

velocitatis amissse a lumine, sive citius, quam binis arteriae pulsibus ultra Lunam fugere.

Quamobrem alia debet esse ejus phaenomeni causa, qua de re fusius egi in mea dissertatione

De Luminis Tenuitate.

Densiora agere in 489. Quoniam lumen in sulphuris particulas agit validissime, nam sulphurosae, &
l?,^.tn,,foftlu»: Sied oleosae substantias facillime accenduntur : eae contra in lumen validissime agunt. Sub-

Suipnurosu., cc OIG~ .• _ , . ._ _. . *

osa pan densitate stantise gencraliter eo magis agunt in lumen, quo densiores sunt, & attractionum summa
plus : cur id ipsum. pr3evalet, ubi radius utrumque illud planum transgressus refringitur : & idcirco generaliter
ubi sit transitus a medio rariore ad densius, refractio fit per accessum ad perpendiculum,
& ubi a medio densiore ad rarius, per recessum. Sed sulphurosa, & oleosa corpora multo
plus agunt in lucem, quam pro ratione suae densitatis. Ego sane arbitror, uti monui num.
467, ipsum ignem nihil esse aliud, nisi fermentationem ingentem lucis cum sulphurea
substantia.

Lumen in progressu 490. Lumen per media homogenea progredi motu liberrimo, & sine ulla resistentia

entiam 'po's it f ve me^"> Per quod propagetur, eruitur etiam ex illo, quod velocitas parallela maneat constans,

probatiir. uti assumpsimus num. 302, quod assumptum si non sit verum, manentibus ceteris ; ratio

sinus incidentiae ad sinum anguli refracti non esset constans : sed idem eruitur etiam ex

eo, quod ubi radius ex acre abivit in vitrum, turn e vitro in aerem progressus est, si iterum

ad vitrum deveniat ; eandem habeat refractionem, quam habuit prima vice. Porro si

resistentiam aliquam pateretur, ubi secundo advenit ad vitrum ; haberet refractionem

major em : nam velocitatem haberet minorem, quas semel amissa non recuperatur per

hoc, quod resistentia minuatur, & eadem vis mobile minori velocitate motum magis

detorquet a directione sui motus.

Unde lux in phos- 491. Posteaquam lux intra opaca corpora tarn multis, tarn variis erravit ambagibus
Lm' aliqua saltern sui parte deveniet iterum ad superficiales particulas, & avolabit. Inde omnino
ortum habebit lux ilia tarn multorum phosphororum, quse deprehendimus, e Sole retracta
in tenebras lucere per aliquot secunda, & a numero secundorum licet conjicere longitudinem
itineris confecti per tot itus, ac reditus intra meatus internos. Sed progrediamur jam ad
reliqua, quae num. 472 proposuimus.

Cur in majore obii- ^Q2. Prime quidem illud facile perspicitur, ex Theoria, quam exposuimus, cur, ubi

quitate plus lum- v T • • •,• • . ,. . r ., . '.*..* ' ,,

inis reflectatur. raaius inciait cum majore inchnatione ad supernciem, major luminis pars renectatur.
Et quidem In dissertatione, quam superiore anno die 12 Novembris legit [229] Bouguerius
in Academiae Parisiensis conventu publico, uti habetur in Mercurio Gallico hujus anni ad
mensem Januarii, profitetur, se invenisse in aqua in inclinatione admodum ingenti reflex-
ionem esse aeque fortem, ac in Mercurio ut nimirum reflectantur duo trientes, dum in
incidentia perpendicular! vix quinquagesima quinta pars reflectatur. Porro ratio in
promptu est. Quo magis inclinatur radius incidens ad superficiem novi medii, eo minor
est perpendicularis velocitas, uti patet : quare vires, quae agunt intra ilia duo plana, eo
facilius, & in pluribus particulis totam velocitatem perpendicularem elident, & reflex-
ionem determinabunt.

Diversam refrangi- .,.,. . . .. ..

biiitatem non pen- 493. Verum id quidem jam suppomt, non in omnes lucis particulas eandem exercen

ceterftatea articu* v*m' sec^ *n "s ^iscrimen haberi aliquod. Ejusmodi discrimina diligenter evolvam.
larum luminis, sed Inprimis discrimen aliquod haberi debet ex ipso textu particularum luminis, ex quo pendeat
earum textu'Vnduc" constans discrimen proprietatum quarundam, ut illud imprimis diversae radiorum refran-
ente vim diversam. gibilitatis. Quod idem radius refringatur ab una substantia magis, ab alia minus in eadem

A THEORY OF NATURAL PHILOSOPHY 347

that light-rays are only waves ; for, waves do not give any progressive motion of themselves.
Further there are some who consider that rivers running in a direction opposite to the rising
Sun are retarded, & that the motion of the Earth is due to impulse of solar rays ; but really
such people can never have investigated the tenuity of light by means of legitimate mechanical
principles.

488. The rays of the Sun impress a motion on the exceedingly small particles of bodies ; There is a very
& from this, when increased by internal forces, arises heat, & this all the more easily in the to^the^artlcifs^'o'f
case of opaque bodies, where there are such a number of internal alternations of reflections bodies by light;
& refractions. If a slight motion is impressed on but a few particles, the internal mutual arise &fro>mbUtheS
forces do all the rest, as we stated in Art. 467. Thus, when some substances are set on internal forces, as
fire by solar rays collected by a mirror, while some are even reduced to powder, all the B here proved-
motions arise in every case from internal forces, & not from the impulse of the light-rays.

Regulus of antimony (stibnite), thus calcined, will sometimes increase its weight by a tenth
part of it ; & there are some who attribute this fact to the mass of the rays so collected.
But if this were the case, the substance would have to fly off very quickly with a velocity
equal to a tenth part of the velocity lost by the light, or more quickly than would be
necessary to get beyond the Moon in two beats of the pulse. Hence there must be other
causes to account for this phenomenon, with which I have dealt fairly fully in my
dissertation De Luminis Tenuitate.

489. Since light acts very strongly on the particles of sulphur, for sulphurous & oily Denser substances
substances are very easily set on fire, these on the other hand act very strongly on light. ^ j^f! bS'"^
In general, substances have the greater action on light, the denser they are; & the sum phurous & oily
of the attractions will be stronger when the ray is refracted as it passes through each of substances more

„.. .o, 1 1*1 ° so than others of

the planes. Tor this reason, in general, when a ray passes from a less dense to a more dense equal density; the
medium, refraction takes place with approach to the normal, & when from a more dense reason for thls-
to a less dense, with recession from the normal. But sulphurous & oily bodies act much
more vigorously upon light than in proportion to their density. I am firmly convinced
that fire is nothing else but an exceedingly great fermentation of light with some
sulphurous substance, as I stated in Art. 467.

490. That light progresses through homogeneous media with a perfectly free motion, Positive demon-
without suffering any resistance from the medium through which it is propagated, is proved ^oTs'not ^er 'an*
by the fact that the parallel component of the velocity remains unaltered. We made this resistance in its pro-
assumption in Art. 302; & if the assumption is not true, other things being unaltered, gressive motion. ^
the ratio of the sine of incidence to the sine of refraction cannot be constant. Now the

same thing is also proved by the fact that when a light-ray goes from air into glass, & then
proceeds from the glass into air, then, if once more it should come to glass, it will have
the same refraction as it had in the first instance. Moreover, if it suffered any resistance,
when for the second time it came to glass, it would have a greater refraction ; for, the
velocity would be less, & once having lost this velocity, the particle could not regain it
simply because the resistance was diminished ; & the same force will cause a body moving
with a smaller velocity to deviate from the direction of its motion to a greater degree.

491. After light has wandered through so many & various paths within opaque bodies, The source of the
at some part at least it will once more arrive at the superficial particles of the bodies & fly hf>ht ln ff1*1111

n- rm > i -11 • • i i« i i • • i r i IT phosphorous bodies.

otr. i his alone will give rise to the light that we perceive with so many phosphorous bodies,
which on being withdrawn from the Sun into the shade shine for some seconds ; & from
the number of seconds one may conjecture the length of the path described by so many
backward & forward journeys within the internal channels. But let us now go on to the rest
of those things that we set forth in Art. 472.

492. In the first place, then, it is easily seen from the Theory which I have expounded, why at greater
why the proportion of light reflected is greater, when the ray falls on the surface with greater Jnore^oT thought
inclination to it. Indeed, in a dissertation, read on November I2th of last year by Bouguer reflected,
before a public convention of the Paris Academy, as is reported in the French Mercury

for January of this year the author professed to have found for water at a very great inclination

a reflection equal to that with mercury ; that is to say, two-thirds of the light was reflected,

while at perpendicular incidence barely a fifty-fifth part is reflected. Now, the reason

for this is not far to seek. The more inclined the incident ray is to the surface of the

new medium, the less is its perpendicular velocity, as is quite clear ; hence, the forces that

act between the two planes will the more easily, & for a larger number of particles, destroy Different refrangi-

the whole of the perpendicular velocity, & thus determine reflection. biiity does not

T> i • i i <• • i 11 -1 r v i_ L j. depend on different

493. Jout this supposes that the same force is not exerted on all particles or light, but velocities of the
that even for them there is some difference. I will carefully discuss these differences. First particles of light
of all, there is bound to be some difference owing to the structure of the particles of light ; their different
& upon this will depend a constant difference in some of its properties, such as that of the structure which
different refrangibilities of rays, in particular. The fact that the same ray is refracted by 3

348 PHILOSOPHIC NATURALIS THEORIA

etiam inclinatione incidentise, id quidem provenit a diversa natura substantiae refringentis,
uti vidimus : ac eodem pacto e contrario, quod e diversis radiis ab eodem medio, & cum
eadem inclinatione, alius refringatur magis, alius minus, id provenire debet a diversa
constitutione particularum pertinentium ad illos radios. Debet autem id provenire vel
a diversa celeritate in particulis radiorum, vel a diversa vi. Porro demonstrari potest, a
sola diversitate celeritatis non pro,venire, atque id prasstiti in secunda parte meae dissertationis
De Lumine : quanquam etiam radii diversae refrangibilitatis debeant habere omnino
diversam quoque celeritatem ; nam si ante ingressum in medium refringens habuissent
aequalem ; jam in illo inasqualem haberent, cum velocitas praecedens ad velocitatem
sequentem sit in ratione reciproca sinus incidentiae ad sinum anguli refracti : & hsec ratio
in radiis diverse refrangibilitatis sit omnino diversa. Quare provenit etiam a vi diversa,
quae cum constanter diversa sit, ob constantem in eodem radio, utcunque reflexo, &
refracto, refrangibilitatis gradum, debet oriri a diversa constitutione particularum, ex
qua sola potest provenire diversa summa virium pertinentium ad omnia puncta. Cum vero
diversa constanter sit harum particularum constitutio : nihil mirum, si diversam in
oculo impressionem faciant, & diversam ideam excitent.

EX eadem refrao 494. At quoniam experiments constat, radios ejusdem colons eandem refractionem
denf Icrforismem!s- Pat* eodem corpore, sive a stellis fixis provenerint, sive a Sole, sive a nostris ignibus, sive
sorum ab omnibus etiam a naturalibus, vel artificialibus phosphoris, nam ea omnia eodem telescopio aeque
evhfcf eandemlbin distincta videntur : manifesto patet, omnes radios ejusdem coloris pertinentes ad omnia
iis celeritatem, & ejusmodi lucida corpora eadem velocitate esse praeditos, & eadem [230] dispositione punc-
textum. torum : neque enim probabile est, (& fortasse nee fieri id potest), celeritatem diversam

a diversa vi compensari ubique accurate ita, ut semper eadem habeatur refractio per

ejusmodi compensationem.

vices facilioris re- 495. Sed oportet invenire aliud discrimen inter diversas constitutiones particularum
flexionis &c., oriri pertinentium ad radios eiusdem refrangibilitatis ad explicandas vices faciliores reflexionis, &

a contractione, & f . . J . ° . . r , ,.

expansione particu- facilions transmissus ; ac mde mini prodibit etiam ratio phsenomem radiorum, qui in rerlex-
larum in progressu jone & refractione irregularitur disperguntur, & ratio discriminis inter eos, qui reflectuntur

inducente dis- r .° ,, . . ,. . i*

crimen. potius, quam refnngantur, ex quo etiam fit, ut in majore inclinatione renectantur plures.

Newtonus plures innuit in Optica sua hypotheses ad rem utcunque adumbrandam, quarum
tamen nullam absolute amplectitur : ego utar hie causa, quam adhibui in ilia dissertatione
De Lumine parte secunda, quae causa & existit & rei explicandae est idonea : quamobrem
admitti debet juxta legem communem philosophandi. Ubi particula luminis a corpore
lucido excutitur fieri utique non potest, ut omnia ejus puncta eandem acquisierint veloci-
tatem, cum a punctis repellentibus diversas distantias habuerint. Debuerunt igitur aliqua
celerius progredi, quae sociis relictis processissent, nisi mutuae vires, acceleratis lentioribus,
ea retardassent, unde necessario oriri debuit particulae progredientis oscillatio quaedam,
in qua oscillatione particula ipsa debuit jam produci non nihil, jam contrahi : & quoniam
dum per medium homogeneum particula progreditur, inaequalitas summae actionum in
punctis singulis debet esse ad sensum nulla ; durabit eadem per ipsum medium homogeneum
reciprocatio contractionis, ac productionis particulae, quae quidem productio, & contractio
poterit esse satis exigua ; si nimirum nexus punctorum sit satis validus : sed semper erit
aliqua, & potest itidem esse non ita parva, nee vero debet esse eadem in particulis diversi
textus.

in Hmitibus ejus 496. Porro in ea reciprocatione figure habebuntur limites quidam productionis maxi-

dfutiu°SC^>erstare m3e> & maximae contractionis, in quibus juxta communem admodum indolem maximorum,

formam : in diver- & minimorum diutissime perdurabitur, motu reliquo, ubi jam inde discessum fuerit ad

vtriunTs^ m m^m distantiam sensibilem cum ingenti celeritate peracto, uti in pendulorum oscillationibus

esse diversam. videmus, pondus in extremis oscillationum Hmitibus quasi haerere diutius, in reliquis vero

locis celerrime praetervolare : ac in alio virium genere diverse a gravitate constanti, ilia

mora in extremis limitibus potest esse adhuc multo diuturnior, & excursus in distantiis

sensibilibus ab utrovis maximo multo magis celer. Deveniet autem particula ad medium

extremarum illarum duarum dispositionum diutius perseverantium post aequalia temporum

intervalla, ut aequales pendulorum oscillationes sunt aeque diuturnae, ac idcirco dum

particula progreditur per medium homogeneum, recurrent illae ipsae binae dispositiones

post aequa-[23l]-lia intervalla spatiorum pendentia a constanti velocitate particulae, &

A THEORY OF NATURAL PHILOSOPHY 349

one substance more, & by another substance less, even for the same inclination of incidence,
is due to the different nature of the refracting substance, as we have seen ; & in the same
way, on the other hand, the fact that, of different rays, & with the same inclination, one ray
is refracted & another less, by the same medium, is due to the different constitution of
the particles pertaining to those rays. Further, it is bound to be due either to a different
velocity in the particles of the rays, or to a different force. Lastly, it can be proved that
it is not due to the difference of velocity alone ; & this I showed in the second part of my
dissertation De Lumine ; although indeed rays of different refrangibilities are bound to
have altogether different velocities also. For, if before entering the refracting substance
they had equal velocities, then after entering they would have unequal velocities ; since
the first velocity is to the second in the inverse ratio of the sines of the angles of incidence
& refraction ; & this ratio for rays of different refrangibilities is altogether different. Hence
it must also be due to a difference of force ; & since this must be constantly different, on
account of the constant degree of refrangibility in the same ray, however it may be reflected
or refracted, it must be due to a difference in the constitution of the particles, from which
alone there can arise a difference in the sum of the forces pertaining to all points forming
them. Now, since the constitution of these particles is constantly different, it is no wonder
that they make a different impression on the eye, & incite a different sensation.

494. Now, since it is proved by experiment that rays of the same colour suffer the same From the equality
refraction by the same body, whether they come from the fixed stars, or from the pL" of ^he 'colour
Sun, or from our fires, or even from natural or artificial phosphorous substances, for they coming from ail
all appear equally distinct when viewed with the same telescope ; it is clearly evident that {j^r^i^'roved
all rays of the same colour pertaining to such light -giving bodies are endowed with the that for such rays
same velocities, & the same distribution of their points. For, it is very improbable, not to Velocity *&e struck
say impossible, that a difference in velocity should be everywhere exactly balanced by a ture.
difference in force to such a degree that by means of such a balance there should always

be the same refraction obtained.

495. But another difference must be found amongst the different constitutions of the Fits. of easier re-
particles belonging to rays of the same refrangibility, to account for the fits of easier reflection duetto' contraction
& easier transmission. From it I shall obtain also the reason for the phenomenon of rays that & expansion of the
are irregularly scattered in reflection & refraction ; & the reason for the difference between mduce^a^itoence
those that are reflected in preference to being refracted, from which also it comes about in the progressive
that the greater the angle the more numerous the rays reflected. Newton suggests several
hypotheses, in his Optics, to give a rough idea of the matter ; but he does not adhere

absolutely to any one of them. I will use in this connection the reason that I employed in
the dissertation De Lumine, in the second part ; this reason both really exists & is fitted for
explaining the matter ; & therefore, according to the usual rule in philosophizing, this
reason should be admitted. When a particle of light is driven off from a light-giving body,
it cannot in any case happen that all the points forming it have acquired the same velocity ;
for, they will have been at different distances from the repelling points of the body. Therefore
some of them are bound to progress more quickly than others, & the former would have left
their fellows behind in their advance, unless the mutual forces had retarded them, while the
slower ones were accelerated. Owing to this, there must necessarily have arisen a certain
oscillation of the particle as it goes along, & due to this oscillation the particle itself must have
been alternately extended & contracted to some extent. Now, since during the progress
of a particle through a homogeneous medium inequality of the sum of the actions at all
points of it must be practically zero, the same alternation of extension & contraction of
the particle will continue right through the homogeneous medium, although the
contraction & expansion will indeed be but slight, if the connections between the points
are fairly strong. But there will always be some oscillation, & it may also not be so very
small, nor need it be the same for particles of different structure.

496. Further, in this alternation of figure there will be certain bounding forms, At the boundaries
corresponding to maximum extension & maximum contraction ; & in these, according to ^ ^ Sp^atl°he
a universal property of all maxima & minima, there will be quite a long pause ; whereas, particle will' pre-
the rest of the motion, after a departure from them has taken place to a sensible distance, longer •**& the sum
is accomplished with a great velocity. Thus, we see in the oscillations of pendulums that of the forces at
the weight at the extreme ends of the oscillations seems to pause for a considerable time,

whereas in other positions it flies past very quickly. In another kind of forces different
from constant gravitation, this delay at the extreme ends may be still more prolonged, &
the motion at sensible distances from either maximum much more swift. Moreover the
particle will reach the mean, between the two extreme dispositions that last for some
considerable time, after equal intervals of time ; just as equal oscillations of pendulums
are of equal duration. Hence, as a particle proceeds through a homogeneous medium,
those two dispositions recur after equal intervals of space, depending on the constant velocity

350 PHILOSOPHIC NATURALIS THEORIA

a constant! tempore, quo particular cujusvis oscillatio durat. Demum summa virium,
quam novum medium, ad quod accedit particula, exercet in omnia particulae puncta,
non erit sane eadem in diversis illis oscillantis particulae dispositionibus.

inde binae disposi- 497. Hisce omnibus rite consideratis, concipiatur jam ille fere continuus affluxus
vkium in°maxima particularum etiam homogenearum ad superficiem duo heterogenea media dirimentem.
particuiarum parte Multo maximus numerus adveniet in altera ex binis illis oppositis dispositionibus, non
uSltibust kTpart'e quidem in medio ipsius, sed prope ipsam, & admodum exiguus erit numerus earum. quse
exigua appeiiente adveniunt cum dispositione satis remota ab illis extremis. Quae in hisce intermediis
inter eos dispersio. adveniunt, mutabunt utique dispositiones suas in progressu inter ilia duo plana, inter
quas agit vis motum particulae perturbans, ita, ut in datis ab utrovis piano distantiis vires
ad diversas particulas pertinentes, sint admodum diversae inter se. Quare illse, quae retro
regredientur, non eandem ad sensum recuperabunt in regressu velocitatem perpendicu-
larem, quam habuerunt in accessu, adeoque non reflectentur in angulo reflexionis aequali
ad sensum angulo incidentise, & illae, quae superabunt intervallum illud omne, in appulsu
ad planum ulterius, aliae aliam summam virium expertae, habebunt admodum diversa
inter se incrementa, vel decrementa velocitatum perpendicularium, & proinde in admodum
diversis angulis egredientur disperses. At quae advenient cum binis illis dispositionibus
contrariis, habebunt duo genera virium, quarum singula pertinebunt constanter ad classes
singulas, cum quarum uno idcirco facilius in illo continue curvaturae flexu devenietur
ad positionem illis planis parallelam, sive ad extinctionem velocitatis perpendicularis
cum altero difficilius : adeoque habebuntur in binis illis dispositionibus oppositis binae
vices, altera facilioris, altera difficilioris reflexionis, adeoque facilioris transitus, quae quidem
regredientur post aequalia spatiorum intervalla, quanquam ita, ut summa facilitas in media
dispositione sita sit, a qua quae minus, vel magis in appulsu discedunt, magis e contrario,
vel minus de ilia facilitate participent. Is ipse accessus major, vel minor ad summam
illam facilitatem in media dispositione sitam in Benvenutiana dissertatione superius
memorata exhibetur per curvam quandam continuam hinc, & inde aeque inflexam circa
suum axem, & inde reliqua omnia, quas ad vices, & earum consectaria pertinent, luculen-
tissime explicantur.

Unde discrimen 498. Porro hinc & illud patet, qui fieri possit, ut e radiis homogeneis ad eandem
reVex^'ad ^trans- superficiem advenientibus alii transmittantur, & alii reflectantur, prout nimirum advenerint
missum. in altera e binis dispositionibus : & quoniam non omnes, qui cum altera ex extremis illis

dispositionibus adveniunt, adve-[232J-niunt prorsus in media dispositione, fieri utique
poterit, ut ratio reflexorum ad transmissos sit admodum diversa in diversis circumstantiis,
nimirum diversi mediorum discriminis, vel diversas inclinationis in accessu : ubi enim
inaequalitas virium est minor, vel major perpendicularis velocitas per illam extinguenda
ad habendam reflexionem, non reflectentur, nisi illae particulas, quse advenerint in dispositione
illi medias quamproxima, adeoque multo pauciores quam ubi vel insequalitas virium est
major, vel velocitas perpendicularis est minor, unde fiet, ut quemadmodum experimur,
quo minus est mediorum discrimen, vel major incidentiae angulus, eo minor radiorum
copia reflectetur : ubi & illud notandum maxime, quod ubi in continue flexo curvaturae
viae particulae cujusvis, quae via jam in alteram plagam est cava, jam in alteram, prout
prasvalent attractiones densioris medii, vel repulsiones, devenitur identidem ad positionem
fere parallelam superficiei dirimenti media, velocitate perpendiculari fere extincta, exiguum
discrimen virium potest determinare parallelismum ipsum, sive illius perpendicularis
velocitatis extinctionem totalem : quanquam eo veluti anfractu superato, ubi demum
reditur ad planum citerius in reflexione, vel ulterius in refractione, summa omnium actionum
quae determinat velocitatem perpendicularem totalem, debeat esse ad sensum eadem,
nimirum nihil mutata ad sensum ab exigua ilia differentia virium, quam peperit exiguum
dispositionis discrimen a media dispositione.

Unde discrimen in 40.9. Atque hoc pacto satis luculenter jam explicatum est discrimen inter binas vices,

mtervalhs viaum. se(j SUperest exponendum, unde discrimen intervalli vicium, quod proposuimus nurn. 472.

Quod diversi colorati radii diversa habeant intervalla, nil mirum est : nam & diversas

A THEORY OF NATURAL PHILOSOPHY 351

of the particle, & on the constant time for which any oscillation of the particle lasts. Lastly,
the sum of the forces, which the new medium, approached by the particle, exerts upon
all the points of the particle, will not really be the same for the different dispositions of
the oscillating particle.

497. All such things being duly considered, a conception can be now formed of the almost Hence, we have the
continuous flow of even homogeneous particles towards the surface of separation of two positions5 fielding
unlike media. By far the greater number of them will arrive at the surface in one or other fits, with the greater
of those two opposite dispositions ; not indeed exactly so, but very nearly so. A very few tic°es° which Pare
of them will reach the surface with a disposition considerably removed from those extremes, striking in those
Those that do arrive in these intermediate states, will in all cases change their dispositions f™1 ^ SfewSthat
in their passage between the two planes, between which the force disturbing the motion strike in states
of the particle acts ; & in such a manner that at any given distance from either plane the twee^^them bwe
forces pertaining to different particles will be altogether different. Therefore, those which have dispersion,
return on their path, will not recover a velocity on the return, that is practically equal

to that perpendicular velocity that it had on approach ; & thus, it will not be reflected
at an angle of reflection practically equal to the angle of incidence. Those, which manage
to pass over the whole of the interval between the two planes, on moving away from the
further plane, will, under the influence of different sums of forces for different particles,
have quite different increments or decrements of the perpendicular velocities ; & they
will emerge at quite different angles from one another, in all directions. But, those that
reach the surface with either of those two opposite dispositions will have but two kinds
of forces ; & each of these will remain constant for its corresponding class of particles.
Hence, with one of these classes there will be more easy approach in its continually curving
path to a position parallel to the planes, corresponding to the extinction of the perpendicular
velocity ; & with the other, this will be more difficult. Therefore there will be produced,
in consequence of the two opposite dispositions, two fits, the one of more easy, & the other
of more difficult reflection, or more easy transmission ; these fits recur at equal intervals
of space. However, these will take place in such a manner that the greatest facility of
reflection will correspond to the mean disposition ; & the less or more the particles depart
from this mean on striking the surface, the more or the less, respectively, will they participate
in that facility. This greater or less approach to the maximum facility, corresponding
to the mean disposition, has been represented in the dissertation by Benvenuti mentioned
above by a continuous curve, which is equally inflected on each side of its axis ; & from
this curve all the other points that relate to fits & their consequences are explained in a
most excellent manner.

498. Further, from this also it is clear how it comes about that, out of a number of The cause of the
homogeneous rays reaching the same surface, some are transmitted & others are reflected, dlfference in tne

he amount
according as they reach it in one or other of two dispositions. Since, of those particles of light reflected to

which do [not] reach the surface with one of the two extreme dispositions, not all reach it ^*e^hlch 1S trans"

in the mean disposition exactly ; it may happen that the ratio of reflections to transmissions

will be altogether different in different circumstances of, say, various differences between

the media, or different inclinations of approach. For when the inequality of the forces

is less or the perpendicular velocity, which has to be destroyed by the inequality to produce

reflection, is greater, only those particles are reflected which reach the surface in dispositions

very near to that mean disposition ; & so, much fewer are reflected than is the case when

the inequality of forces is greater or the perpendicular velocity is less. Hence, it comes

about that the less the difference between the media, or the greater the angle of incidence,

the smaller the proportion of rays reflected ; which is in agreement with experience. In

this connection also it is especially to be observed that when in the continuous winding

of the curved path of any particle, the path being at one time concave on one side & at

another time on the other, according as the attractions or the repulsions of the denser

medium are more powerful, a position nearly parallel to the surface of separation between

the media is attained several times in succession, as the perpendicular velocity is nearly

destroyed, a very slight difference of the forces will be sufficient to produce exact

parallelism, or the total extinction of that perpendicular velocity. Although, when these,

so to speak, tortuosities are ended as the particle at length reaches the nearer plane in reflection

& the further plane in refraction, the sum of all the actions, which determines the total

perpendicular velocity, must be practically the same ; that is to say, in no wise changed

to any sensible extent by the slight difference of forces, such as produced the slight difference

of disposition from the mean disposition.

499. In this way we have a sufficient explanation of the difference between the two J*Le cause of the

- ~? ' <,, . , i- • i-rr i it i difference in the

fits ; but we have still to explain the source of the difference in the intervals between the intervals between
fits, which we propounded in Art. 472. There is nothing wonderful in the fact that successive fits,
differently coloured rays should have different intervals. For, different velocities require

352 PHILOSOPHIC NATURALIS THEORIA

velocitates diversa requirunt intervalla spatii inter vices oppositas, quando etiam eas vices
redeant aequalibiw temporis intervallis, & diversus particularum heterogenearum textus
requirit diversa oscillationum tempora. Quod in diversis mediis particulae ejusdem generis
habeant diversa intervalla, itidem facile colligitur ex diversa velocitate, quam in iis haberi
post refractionem ostendimus num. 493 ; sed praaterea in ipsa mediorum mutatione
inaequalis actio inter puncta particulam componentia potest utique, & vero videtur etiam
debere oscillationis magnitudinem, & fortasse etiam ordinem mutare, adeoque celeritatem
oscillationis ipsius. Demum ejusmodi mutatio pro diversa inclinatione vias particular
advenientis ad superficiem, diversa utique esse debet, ob diversam positionem motuum
punctorum ad superficiem ipsam, & ad massam agentem in ipsa puncta. Quamobrem
patet, eas omnes tres causas debere discrimen aliquod exhibere inter diversa intervalla,
uti reapse ex observatione colligitur.

Discnmen id non coo. Si possemus nosse peculiares constitutiones particula-tessl-rum ad diversos

posse definin, nisi ,3 r . . L JOJ _

per observationes : coloratos radios pertmentium, ordinem, & numerum, ac vires, & velocitates punctorum
vef ^ntdere a sola singulorum ; turn mediorum constitutionem suam in singulis, ac satis Geometrias, satis
imaginationis haberemus, & mentis ad omnia ejusmodi solvenda problemata ; liceret a
priori determinare intervallorum longitudines varias, & eorundem mutationes pro tribus
illis diversis circumstantiis exhibere. Sed quoniam longe citra eum locum consistimus
debemus illas tantummodo colligere per observationes, quod summa dexteritate Newtonus,
praestitit, qui determinatis per observationem singulis, mira inde consectaria deduxit,
& Naturae phenomena explicavit, uti multo luculentius videre est in ilia ipsa Benvenutiana
dissertatione. Illud unum ex proportionibus a Newtono inventis haud difficulter colligitur,
ea discrimina non pendere a sola particularum celeritate, nam celeritatum proportiones,
novimus per sinuum rationem : & facile itidem deducitur ex Theoria, quod etiam multo
facilius infertur partim ex Theoria, & partim ex observatione, radium, qui post quotcunque
vel reflexiones, vel refractiones regulares devenit ad idem medium, eandem in eo velocitatem
habere semper ; nam velocitates in reflexione manent, & in mutatione mediorum sunt in
ratione reciproca sinus incidentiae ad sinum anguli refracti : ac tarn Theoria, quam observatio
facile ostendit, ubi planis parallelis dirimantur media quotcunque, & radius in data
inclinatione ingressus e primo abeat ad ultimum, eundem fore refractionis angulum in
ultimo medio, qui esset, si a primo immediate in ultimum transivisset. Sed haec innuisse
sit satis.

Quod de crystalio 501. Illud etiam innuam tantummodo, quod Newtonus in Opticis Quaestionibus

isiandica Newtonus exponit esse miram quandam crystalli Islandicae proprietatem, quae radium quemvis, dum

prodidit, id in nac F , , ,, . . *• , .. • . i r • • !• • • J i • o

Theoria nuiiam refrmgit, discerpit in duos, & ahum usitato modo retrmgit, ahum musitato quodam, ubi &
habere difficuita- certa2 qusedam observantur leges, quarum explicationes ipse ibidem insinuat haberi posse
per vires diversas in diversis lateribus particularum luminis, ac solum adnotabo illud, ex
num. 423 patere, in mea Theoria nullam esse difficultatem agnoscendi in diversis lateribus
ejusdem particulae diversas dispositiones punctorum, & vires, qua ipsa diversitate usi sumus
superius ad explicandam solidorum cohassionem, & organicam formam, ac certas figuras
tot corporum, quse illas vel affectant constanter, vel etiam acquirunt.

piffractionem esse 502. Remanet demum diffractio luminis explicanda, quam itidem num. 472 proposue-
inchoatam reflexi- ramus< j?a est qusedam velut inchoata reflexio, & refractio. Dum radius advenit ad earn

oncm, vci rciitiCtiQ" ' - »• i i i * •«• -t •

nem. distantiam a corpore diversas naturae ab eo, per quod progreditur, quae vinum maequahtatem

inducit, incurvat viam vel accedendo, vel recedendo, & directionem mutat. Si corporis
superficies ibi esset satis ampla, vel reflecteretur ad angulos asquales, vel immergeretur
intra novum illud medium, & refrin-[234"|-geretur ; at quoniam acies ibidem progressum
superficiei interrumpit ; progreditur quidem radius aciem ipsam evitans & circa illam
praetervolat ; sed egressus ex ilia distantia directionem conservat postremo loco acquisitam,
& cum ea, diversa utique a priore, moveri pergit : ut adeo tota luminis Theoria sibi ubique
admodum conformis sit, & cum generali Theoria mea apprime consentiens, cujus rami
quidam sunt bina Newtoni praeclarissima comperta virium, quibus caslestia corpora motus
peragunt suos & quibus particulae luminis reflectuntur, refringuntur, diffringuntur. Sed
de luce, & coloribus jam satis.

De sapore, & odore : 503. Post ipsam lucem, quae oculos percellit, & visionem parit, ac ideam colorum

ratione^densrtat'is excitat, pronum est delabi ad sensus ceteros, in quibus multo minus immorabimur, cum

odoris propagati. circa eos multo minora habeamus comperta, quae determinatam physicam explicationem

ferant. Saporis sensus excitatur in palato a salibus. De angulosa illorum forma jam

°n

A THEORY OF NATURAL PHILOSOPHY 353

different intervals of space between opposite fits, when those fits recur also at equal intervals
of time ; & a difference in the structure of heterogeneous particles requires a difference
in the periods of oscillation. It is also easily seen that particles of the same kind have
different intervals in different media, owing to that difference in velocity, which, in
Art. 493, was proved to exist after refraction. But, in addition, on changing the
medium, an unequal action between the points composing the particle certainly can
and, apparently indeed, is bound to alter the magnitude of the oscillation also, & perhaps
even the order ; & thus the velocity of that oscillation must alter. Further, such a change,
for a difference in the inclination of the path of the particle approaching the surface,
is in every case bound to be different, on account of the difference in situation of
the motions of the points with respect to the surface & the mass acting upon the points.
Hence, it is clear that all three of these causes must stand for some difference between
diverse intervals ; & indeed we can deduce as much from observation.

500. If we could know the particular constitutions of particles for differently coloured This difference can-
rays, the order, number, forces & velocities of each point, & the constitution of each medium n°ven ^ unless ^b7
for each ray, and if we had a sufficiency of geometry, imagination & intelligence to solve observation; it
all problems of this kind, we could determine from first principles the various lengths

of the intervals, & could give the changes due to each of the three different circumstances.
But since this is far beyond us, we are bound to deduce them from observation alone. This
Newton accomplished with the greatest dexterity ; having determined each by observation,
he deduced from them wonderful consequences ; & explained the phenomena of Nature ;
as also it is to be seen much better in the dissertation by Benvenuti. There is one thing
that can be without much difficulty derived from the proportions discovered by Newton,
namely, that the differences do not solely depend upon the velocities of the particles ; for
we know the proportions of the velocities by the ratio of the sines. It can also easily be
deduced from the Theory, & indeed much more easily can it be inferred partly from the
Theory & partly from observation, that a ray which, after any number of regular reflections
& refractions, comes to the same medium will always have the same velocity in it as at first.
For the velocities remain unaltered in reflection, & on a change of medium they are in
the inverse ratio of the sine of the angle of incidence to the sine of the angle of refraction.
Both the Theory, & observation, clearly show that, when any number of media are separated
by parallel planes, & a ray, entering at a given inclination, leaves the first & reaches the
last, there will be the same angle of refraction in the last medium as there would have been,
if it had passed directly from the first medium into the last. But a mere mention of these
things is enough.

501. I will also merely mention that, as was stated by Newton in his Questions at the That which Newton
end of his Optics, there is a wonderful property of Iceland Spar ; namely, that when it inland "spar™^
refracts a ray of light it divides it into two, refracting one part according to the normal ?ents no difficulty
manner, & the other in an unusual way ; with the latter also definite laws are observed.

Newton himself suggested that the explanation of these laws could be attributed to different
forces on different sides of the particles of light ; & I will only remark that, according to
Art. 423, it is evident that in my Theory there is no difficulty over admitting for different
sides of the same particle different dispositions of the points, & different forces ; we have
already employed this sort of difference to explain cohesion of solids, & organic form, &
all those shapes of bodies, such as they always endeavour to acquire, & indeed do acquire.

502. Finally, we have to explain diffraction, which we also enunciated in Art. 472. Diffraction is in-
This is, so to speak, an incomplete reflection or refraction. When a ray of light attains complete reflection
the distance, from a body of a different nature from one through which it passes, which

induces an inequality of forces, its path becomes curved, either by approach or recession,
& the direction is altered. If the surface of the body at the point in question is sufficiently
wide, the ray will either be reflected at equal angles, or it will enter the new medium &
be reflected. But when a sharp edge terminates the run of the surface, the ray will pass
on, slipping by the edge, & flying past & round it. But, on emergence from that distance,
the ray will preserve the direction acquired in the last position, & with this direction, which
will be altogether different from that which it had originally, it will continue its motion.
Thus the whole theory of light will be quite consistent, & in close agreement with my Theory.
Of this Theory, the two most noted discoveries of Newton with respect to forces are just
branches ; namely, the forces with which the heavenly bodies keep up their motions, &
those by which particles of light are reflected, refracted & diffracted. But I have now
said sufficient about light & colour.

503. After light, which affects the eyes, begets vision, & excites the idea of colours, we Concerning taste &
naturally come to the other senses ; over these I will spend far less time, since we have ^ny' p^opJiTwith
far less knowledge of them, such as will help us to give a definite physical explanation, regard to the ratio
The sense of taste is excited in the palate by salts. I have already spoken of the

A A

354 PHILOSOPHIC NATURALIS THEORIA

diximus num. 464, quae ad diversum excitandum motum in papillis palati abunde sufficit ;
licet etiam dum dissolvuntur, vires varias pro varia punctorum dispositione exercere debeant,
quae saporum discrimen inducant. Odor est quidam tenuis vapor ex odoriferis corporibus
emissus, cujus rei indicia sunt sane multa, nee omnino assentiri possum illi, qui odorem
etiam, ut sonum, in tremore medii cujusdam interpositi censet consistere. Porro quae
evaporationum sit causa, explicavimus abunde num. 462. Illud unum hie innuam, errare
illos, uti pluribus ostendi in prima parte meae dissertationis De Lumine, qui multi sane
sunt, & praestantes Physici, qui odoribus etiam tribuunt proprietatem lumini debitam,
ut nimirum eorum densitas minuatur in ratione reciproca duplicata distantiarum a corpore
odorifero. Ea proprietas non convenit omnibus iis, quae a dato puncto diffunduntur in
sphaeram, sed quae diffunduntur cum uniformi celeritate, ut lumen. Si enim concipiantur
orbes concentrici tenuissimi datae crassitudinis ; ii erunt ut superficies, adeoque ut quadrata
distantiarum a communi centre, ac densitas materiae erit in ratione ipsorum reciproca :
si massa sit eadem : ut ea in ulterioribus orbibus sit eadem, ac in citerioribus ; oportet
sane, tota materia, quae erat in citerioribus ipsis, progrediatur ad ulteriores orbes motu
uniformi, quo fiet, ut, appellente ad citeriorem superficiem orbis ulterioris particula, quae
ad citeriorem citerioris appulerat, appellat simul ad ulteriorem ulterioris quae appulerat
simul ad ulteriorem citerioris, materia tota ex orbe citeriore in ulteriorem accurate translata :
quod nisi fiat, vel nisi loco uniformis progressus habeatur accurata compensatio velocitatis
imminutae, & impeditae a progressu partis vaporum, quae compensatio accurata est admodum
improbabilis ; non habebitur densitas reciproce proportionalis orbibus, sive eorum super-
ficiebus, vel distantiarum quadratis.

De sono difficult^ [235] 504. Sonus geometricas determinationes admittit plures, & quod pertinet ad
undis Ixcitetis,11 in vibrationes chordae elasticas, vel campani aeris, vel motum impressum aeri per tibias, &
fluido elastico. tubas, id quidem in Mechanica locum habet, & mihi commune est cum communibus theoriis.
Quod autem pertinet ad progressum soni per aerem usque ad aures, ubi delatus ad tympanum
excitat eum motum, a quo ad cerebrum propagate idea soni excitatur, res est multo opero-
sior, & pendet plurimum ab ipsa medii constitutione : ac si accurate solvi debeat problema,
quo quaeratur ex data medii fluidi elasticitate propagatio undarum, & ratio inter oscillationum
celeritates, a qua multipliciter variata pendent omnes toni, & consonantiae, ac dissonantiae,
& omnis ars musica, ac tempus, quo unda ex dato loco ad datam distantiam propagatur ;
res est admodum ardua ; si sine subsidiariis principiis, & gratuitis hypothesibus tractari
debeat, & determination! resistentiae fluidorum est admodum affinis, cum qua motum in
fluido propagatum communem habet. Exhibebo hie tantummodo simplicissimi casus
undas, ut appareat, qua via ineundam censeam in mea Theoria ejusmodi investigationem.

QUO pacto onantur cOr Sit in recta linea disposita series punctorum ad data intervalla aequalia a se invicem

undae in serie con- J •' , . .. . r. ,, . .. ...

tinua punctorum se distantium, quorum bma quaeque sibi proxima se repellant vinbus, quae crescant immmutis
invicem repeiien- distantiis, & dentur ipsae. Concipiatur autem ea series utraque parte in infinitum producta,

tium. . ' . . . ... ^ . r . r.

& uni ex ejus punctis concipiatur externa vi celernme agente in ipsum multo magis, quam
agant puncta in se invicem, brevissimo tempusculo impressa velocitas quaedam finita in
ejusdem rectae directione versus alteram plagam, ut dexteram, ac reliquorum punctorum
motus consideretur. Utcunque exiguum accipiatur tempusculum post primam systematis
perturbationem, debent illo tempusculo habuisse motum omnia puncta. Nam in momento
quovis ejus tempusculi punctum illud debet accessisse ad punctum secundum post se
dexterum, & recessisse a sinistro, velocitate nimirum in eo genita majore, quam generent
vires mutuae, quae statim agent in utrumque proximum punctum, aucta distantia a sinistro, &
imminuta a dextero, qua fiet, ut sinistrum urgeatur minus ab ipso, quam a sibi proximo
secundo ex ilia par^e, & dexterum ab ipso magis, quam a posteriore ipsi proximo, &
differentia virium producet illico motum aliquem, qui quidem initio, ob differentiam
virium tempusculo infinitesimo infinitesimam, erit infinities minor motu puncti impulsi,
sed erit aliquis : eodem pacto tertium punctum utraque ex parte debet illo tempusculo
infinitesimo habere motum aliquem, qui erit infinitesimus respectu secundi, & ita porro.

A THEORY OF NATURAL PHILOSOPHY 355

angular forms of salts, in Art. 464 ; these are quite sufficient for the excitement of different
motions in the papillae of the palate ; although, even when they are dissolved, they must
exert different forces for different dispositions of the points, which induce differences in
taste. Smell is a sort of tenuous vapour emitted by odoriferous bodies ; of this there are
really many points in evidence. I cannot agree altogether with one who thinks that smell,
like sound, consists of a sort of vibration of some intervening medium. Moreover, I have
fully explained, in Art. 462, what is the cause of evaporations. I will but mention here this
one thing, namely, that, as I showed in several places in the first part of my dissertation
De Lumine, those many and distinguished physicists are mistaken who attribute to smell
the same property as that proper to light, namely, that the density diminishes in the inverse
ratio of the squares of the distances from the odoriferous body. That is a property that
does not apply to all things that are diffused throughout a sphere from a given point ; but
only with those that are thus diffused with uniform velocity, as light is. For if we imagine
a set of concentric spherical shells of given very small thickness, they will be like surfaces.
Hence, they will be in the same ratio as the squares of the distances from the common
centre ; &, the density of matter will be inversely proportional to them, if the mass is the
same. Now, in order that it may be the same in the outer shells as it is in the inner, it is
necessary that the whole of the matter which was in the inner shells should proceed to the
outer shells with a uniform motion ; then, it would come about that two particles, which
have reached simultaneously the inner & outer surfaces of the inner shell respectively,
will reach simultaneously the inner & outer surfaces of the outer shell ; & the whole of
the matter will be transferred accurately from the inner shell to the outer. If this is not
the case, or, failing uniform progression, if instead there is not an accurate compensation of
the velocity thus diminished & hindered by the advance of part of the vapours (& such an
accurate compensation is in the highest degree improbable), then the density cannot be
inversely proportional to the shells, i.e., to their surfaces, or the squares of the distances.

504. Sound admits of several geometrical determinations ; & matters pertaining to Sound ; difficulty
vibrations of an elastic cord or bell-metal, or the motion given to the air by flutes & wavese^itedln^n
trumpets, all belong to the science of Mechanics ; & for them my Theory is in agreement elastic fluid,
with the ordinary theories. But, with respect to the progression of sound through the air

to the ears, where it is carried to the ear-drum & excites the motion by means of which,
when propagated to the brain, the idea of sound is produced, the matter is much more
laborious, & depends to a very large extent on the constitution of the medium itself. If
it is necessary to solve the problem, in which it is desired to find the propagation of waves
from a given elasticity of a fluid medium, & the ratio between the velocities of the oscillations
upon which, in its manifold variations, depend all musical sounds, harmonious or discordant,
the whole art of music, & the time in which a wave is propagated from a given point to a
given distance ; then, the matter is very hard, especially if it has to be treated without
the help of subsidiary principles or unfounded hypotheses. It is closely allied to the
determination of the resistance of fluids, with which subject it has common ground in the
motion propagated in a fluid. I will explain here merely waves of the very simplest kind ;
so that the manner in which I consider in my Theory such an investigation should be
undertaken will be seen.

505. Suppose we have a series of points situated in one straight line at given equal The manner in
intervals of distance from one another ; & of these let any two consecutive points repel one ^^ i^^continu-
another with forces, which increase as the distance decreases, & suppose that the magnitudes ous series of points
of these forces are also given. Also suppose that this series is continued on either side to ^her

infinity ; & suppose that, by means of an external force acting very quickly on one of the
points of the series much more than the points act upon one another, there is impressed
upon it in a very short time a certain finite velocity in the direction of the straight line
towards either end of it, say towards the right ; then we have to consider the motion of
all the other points. No matter how small the interval of time taken, after the initial
disturbance of the system, in that interval all points must have had motion. For, in any
instant of that interval of time, that point must have approached the next point to it on
the right, & have receded from the one on the left ; a velocity being generated in it greater
than that which the mutual forces would give. These forces immediately act on the points
next to it on either side, the distance on the left being increased, & on the right diminished.
Thus, the point on the left will be impelled by that point less than by the next one to it
on its left, & the one on the right more than by the next one to the right of it. The difference
of forces will immediately produce some motion ; this motion indeed at first, owing to
the difference of forces in an infinitesimal time being itself infinitesimal, will be infinitely
less than the motion of the point under the action of the external force ; but there will
be some motion. In the same way, a third point on either side must in that infinitesimally
small time have some motion, which will be infinitesimal with respect to that of the second ;

356 PHILOSOPHIC NATURALIS THEORIA

Post tempusculum utcunque exiguum omnia puncta aequilibrium amittent, & motum
habebunt aliquem. Interea cessante actione vis impellentis punctum primum incipiet
ipsum retar-[236]-dari vi repulsiva secundi dexteri praevalente supra vim secundi sinistri,
sed adhuc progredietur, & accedet ad secundum, ac ipsum accelerabit : verum post aliquod
tempus retardatio continua puncti impulsi, & acceleratio secundi reducent ilia ad veloci-
tatem eandem : turn vero non ultra accedent ad se invicem, sed recedent, quo recessu
incipiet retardari etiam punctum primum dexterum, ac paullo post extinguetur tota
velocitas puncti impulsi, quod incipiet regredi : aliquanto post incipiet regredi & punctum
secundum dexterum, & aliquanto post tertium, ac ita porro aliud. Sed interea punctum
impulsum, dum regreditur, incipiet urgeri magis a primo sinistro, & acceleratio minuetur :
turn habebitur retardatio, turn motus iterum reflexus. Dum id punctum iterum incipit
regredi versus dexteram, erit aliquod e dexteris, quod tune primo incipiet regredi versus
sinistram, & dum per easdem vices punctum impulsum iterum reflexit motum versus
sinistram, aliud dexterum remotius incipiet regredi versus ipsam sinistram, ac ita porro
motus semper progreditur ad dexteram major, & incipient regredi nova puncta alia post
alia. Undae amplitudinem determinabit distantia duorum punctorum, quae simul eunt
& simul redeunt, ac celeritatem propagationis soni tempus, quod requiritur ad unam
oscillationem puncti impulsi, & distantia a se invicem punctorum, quas simul cum eo eunt,
& redeunt ; & quod ad dexteram accidit ad sinistram. Sed & ea perquisitio est longe
altioris indaginis, quam ut hie institui debeat ; & ad veras soni undas elasticas referendas
non sufficit una series punctorum jacentium in directum, sed congeries punctorum, vel
particularum circumquaque dispersarum, & se repellentium.

Solutio difficuitatis 506. Interea illud unum adjiciam, in mea Theoria admodum facile solvi difficultatem,

pa^atione'm ^ectiii- quam Eulerus objecit Mairanio, explicanti propagationem diversorum sonorum, a quibus
neam diversorum diversi toni pendent, per diversa genera particularum elasticarum, quae habentur in acre,
f a^cTus "fn^h'ac quorurn singula singulis sonis inserviant, ut diversi sunt colorati radii cum diverse constant!
Theoria. refrangibilitatis gradu, & colore. Eulerus illud objicit, uti tarn multa sunt sonorum

genera, quae ad nostras, & aliorum aures simul possint deferri, ita debere haberi continuam
seriem particularum omnium generum ad ea deferenda, quod haberi omnino non possit,
cum circa globum quenvis in eodem piano non nisi sex tantummodo alii globi in gyrum
possint consistere. Difficultas in mea Theoria nulla est, cum particulas aliae in alias non
agant per immediatum contactum, sed in aliqua distantia, quae diametro globorum potest
esse major in ratione quacunque utcunque magna. Cum igitur certi globuli in iisdem
distantiis possint esse inertes respectu certorum, & activi respectu aliorum ; patet, posse
multos diversorum generum globulos esse permixtos ita, ut actionem aliorum sentiant
alii. Quin [237] immo licet activi sint globuli, fieri debet, ut alii habeant motus conformes
turn eos, qui pendent a viribus mutuis inter duos globulos, a quibus proveniunt undae,
turn eos qui pendent ab interna distributione punctorum, a qua proveniunt singularum
particularum interni vibratorii motus, & qui itidem ad diversum sonorum genus plurimum
conferre possint, & dissimilium globorum oscillationes se mutuo turbent, similium perpetuo
post primas actiones actionibus aliis conformibus augeantur, quemadmodum in consonantibus
instrumentorum chordis cernimus, quarum una percussa sonant & reliquae. LJbique
libertas motuum, & dispositionis, qua? sublato immediate impulsu, & accurata continuitate
in corporum textu, acquiritur ad explicandam naturam, est perquam idonea, & opportuna.

De caiore & frigore . coy. Quod pertinet ad tactiles propnetates, quid sit solidum, fluidum, ngidum, molle

materiae cientis , • n -i r -i i v • -j i • -j

caiorem expansio elasticum, flexile, fragile, grave, abunde explicavimus : quid laevigatum, quid asperum,

orta abeiasticitate: per se patet. Caloris causam repono in motu vehementi intestine particularum igneae,

vek>citasJU ut tor- vel sulphureae substantias fermentescentis potissimum cum particulis luminis, & qua ratione

rentis cujusdam. j^ fierj possit, exposuimus. Frigus haberi potest per ipsum defectum ejusmodi substantiae,

vel defectum motus in ipsa. Haberi possunt etiam particulae, quae frigus cieant actione

sua, ut nitrosas, per hoc, quod ejusmodi particularum motum sistant, & eas, attractione

A THEORY OF NATURAL PHILOSOPHY 357

& so on. Thus, after the lapse of any short interval of time, however small, all points will
lose their equilibrium & have some motion. Further, the action of the force acting upon
the first point will itself begin to be retarded by the repulsive force of the next point on
the right prevailing over the force from the next on the left ; but it will still progress,
approach the second & accelerate it. However, after some time, the continuous retardation
of the first point, & the acceleration of the second, will reduce them to the same velocity ;
& then they will no longer approach one another, but will recede from one another. When
this recession starts, the first point on the right will also begin to be retarded, & a little
while afterwards the whole of the velocity of the point impelled by the external force will
be destroyed, & it will commence to go backwards ; shortly afterwards, the second point
on the right will also commence to go backwards ; shortly after that, the third point ;
& so on, one after the other. But meanwhile, as it returns, the point, that was impelled
by the external force, will be more under the action of the first point on the left, &
its acceleration will be diminished ; there will follow first a retardation, & then once more
a reversal of motion. When the point once more begins to move towards the right, there
will be some one of the points on the right, which then for the first time is beginning to
move backwards to the left ; & when, after the same changes, the point impelled once
more reverses its motion & moves towards the left, there will be another point on the right,
further off, which will begin to move backwards towards the left. In this way, the motion
will always proceed further to the right, & fresh points, one after the other, will begin to
reverse their motion. The distance between two points, which go forward & backward
simultaneously, will determine the amplitude of the wave ; the velocity of propagation of
sound will be found from the time that is required for one oscillation of the impelled point,
& the distance between points, whose motion backwards & forwards is simultaneous ; &
what happens on the right will also happen on the left. But the investigation is one of
far too great difficulty to be properly treated here ; to render an account of the true elastic
waves of sound, one series of points lying in a straight line is insufficient ; we must have
groups of points or of particles, scattered in all directions round about, & repelling one
another.

506. I will add just one other thing ; in my Theory, it is quite easy to give a solution The solution of the
of the difficulty, which Euler brought forward in opposition to Mairan ; the latter tried spfct^thTrectifi-
to explain the propagation of the different sounds, upon which different musical tones near propagation of
depend, by the presence of different kinds of elastic particles in the air ; each kind of co^es "quhe0 easfiy
particle was of service to the corresponding sound, just as there are differently coloured from my Theory,
rays of light, having a constant different degree of refrangibility, & a different colour. Euler's

objection was that there are so many kinds of sounds, which can be borne simultaneously
to our ears & to those of others, that there must be a continuous series of particles of all
the different kinds to carry these sounds ; & that this was quite impossible, since only six
spheres could lie in a circle in the same plane round a sphere. There is no such difficulty
in my Theory, since particles do not act upon one another by immediate contact, but at
some distance, such as can bear to the diameter of the spheres any ratio whatever, however
large. Since, then, certain little spheres can be Inert, when placed at the same distances,
with regard to some & active with regard to others, it is clear that a large number of little
spheres of different kinds can be so intermingled that some of them feel the action of others.
Nay indeed, even if the little spheres are active, there are bound to be some that have
congruent motions ; not only those motions which depend upon the mutual forces between
two little spheres by which waves are produced, but also those which depend on the internal
distribution of the points forming them from which arise the internal vibratory motions
of the several particles. These, too, may contribute towards a different class of sounds to
a very great extent ; & they will disturb the mutual oscillations of unlike spheres, &, after
the first actions, the oscillations of like spheres will be increased by congruent actions ;
just as in the consonant strings of instruments we see that, when one of them is struck,
all the others sound as well. The freedom of motion everywhere, & of arrangement, which
is acquired by the removal of the ideas of immediate impact & accurate continuity in the
structure of bodies, is most suitable & convenient for the purpose of explaining the nature
of sound.

507. With respect to tactile properties, we have had full explanations of solid, fluid. Heat & cold ; the
rigid, soft, elastic, flexible, fragile & heavy bodies ; what a smooth, or a rough, body is, ^atter^producing
is self-evident. I consider the cause of heat to consist of a vigorous internal motion heat arises from
of the particles of fire, or of a sulphurous substance fermenting more especially with of^th^Lmer3*0"
particles of light ; & I have shown the mode in which this may take place. Cold may velocity as ' of a
be produced by a lack of this substance, or by a lack of motion in it. Also there may be torrent-
particles which produce cold by their own action, such as nitrous substances, through

something which stops the motion of such particles, &, as their attraction overcomes their

358 PHILOSOPHIC NATURALIS THEORIA

mutuas ipsarum vires vincente, ad se rapiant, ac sibi affundant quodammodo, veluti alligatas.
Potest autem generari frigus admodum intensum in corpore calido per solum etiam accessum
corporis frigefacti ob solum ejusmodi substantiae defectum. Ea enim, dum fermentat,
& in suo naturali volatilizationis statu permanet, nititur elasticitate sua ipsa ad expansionem,
per quam, si in aliquo medio conclusa sit, utcunque inerte respectu ipsius, ad aequalitatem
per ipsum diffunditur, unde fit, ut si uno in loco dematur aliqua ejus pars, statim illuc-
ex aliis tantum devolet, quantum ad illam aequalitatem requiritur. Hinc nimirum, si
in acre libero cesset fermentantis ejusmodi substantias quantitas, vel per imminutam con-
tinuationem impulsuum ad continuandum motum, ut imminuta radiorum Solis copia per
hyemem, ac in locis remotioribus ab yEquatore, vel per accessum ingentis copise particularum
sistentium ejusdem substantias motum, unde fit, ut in climatis etiam non multum ab
^Equatore distantibus ingentia pluribus in locis habeantur frigora, & glacies per nitrosorum,
efHuviorum copiam ; e corporibus omnibus expositis aeri perpetuo crumpet magna copia
ejusdem fermentescentis ibi adhuc, & elasticae materiae igneas ; & ea corpora remanebunt
admodum frigida per solam imminutionem ejus materiae, quibus si manum admoveamus,
ingens illico ex ipsa manu particularum earundem multitude avolabit transfusa illuc, ut
res ad aequalitatem redu-[238]-catur, & tarn ipsa cessatio illius intestini motus, qua immuta-
bitur status fibrarum organici corporis, quam ipse rapidus ejus substantiae in aliam
irrumpentis torrens, earn poterit, quam adeo molestam experimur, frigoris sensationem,
excitare.

tione,°&naffluxu.Xa" 5°8- Torrentis ejusmodi ideamhabemus in ipso velocissimo aeris motu, qui si in aliqua

spatii parte repente ad fixitatem reducatur in magna copia, ex aliis omnibus advolat
celerrime, & horrendos aliquando celeritate sua effectus parit. Sic ubi turbo vorticosus,
& aerem inferne exsugens prope domum conclusam transeat, aer internus expansiva sua
vi omnia evertit : avolant tecta, diffringuntur fenestras, & tabulate, ac omnes portae, quae
cubiculorum mutuam communicationem impediunt, repente dissiliunt, & ipsi parietes
nonnunquam evertuntur, ac corruunt, quemadmodum Romae ante aliquot observavimus
annos, & in dissertatione De Turbine superius memorata, quam turn edidi, pluribus exposui.

Attractio, quae po- 509. Verum haec sola substantiae hujusce fermentantis expansiva vis non est satis ad

nfotum 'sistere"1* rem explicandam, sed requiritur etiam certa vis mutua, qua ejusmodi substantia in
fixare : communi- alias quasdam attrahatur magis, in alias minus, quod qui fieri possit, vidimus, ubi de
»tu^tatenfqpao<st dissolutione, & praecipitatione egimus : & ejusmodi attractio potest esse ita valida, ut
partem fixatam : motum ipsum intestinum prorsus impediat appressione ipsa, ac fixationem ejus substantiae
varia inducat, quae si minor sit, permittet quidem motus fermentatorii continuationem, sed a
se totam massam divelli non permittet, nisi accedente corpore, quod majorem exerceat
vim, & ipsam sibi rapiat. Hie autem raptus fieri potest ob duplicem causam ; primo
quidem, quod alia substantia majorem absolutam vim habeat in ejusmodi substantiam
igneam, quam alia, pari etiam particularum numero : deinde, quod licet ea aeque, vel
etiam minus trahat, adhuc tamen cum utraque in minoribus distantiis trahat plus, in
majoribus minus, ilia habeat ejus substantiae multo minus etiam pro ratione attractionis
suae, quam altera ; nam in hoc secundo casu, adhuc ab hac posteriore avellerentur particulae
affusae ipsius particulis ad distantias aliquanto majores, & affunderentur particulis prioris
substantiae, donee in utravis substantia haberetur aequalis saturitas, si ejus partes inter
se conferantur, & asqualis itidem attractiva vis particularum substantiae igneae maxime
remotarum a particulis utriusque substantiae, quibus ea affunditur : sed copia ipsius
substantiae igneae possit adhuc esse in iis binis substantiis in quacunque ratione diversa inter
se ; cum possit in altera ob vim longius pertinentem certa vis haberi in distantia majore,
quam in altera, adeoque altitude ejusmodi veluti marium in altera esse major, minor in
altera, & in iisdem distantiis possit in altera haberi ob vim majorem densitas major sub-
stantias ipsius igneae affusae, quam in altera. Ex hisce quidem principiis, ac diversis
combinationibus, mirum sa-[239]-ne, quam multa deduci possint ad explicationem Naturae
per quam idoneis.

Quae a diffuskme 510. Sic etiam ex hac diffusione ad ejusmodi aequalitatem eandem inter diversas

consequanturpo^ ejusdem substantias partes, sed admodum diversam inter substantias diversas, facile intelli-

simum respectu re- gitur, qui fiat, ut manus in hyeme exposita libero aeri minus sentiat frigoris, quam solido

&rrongfcfciationis.S' cuipiam satis denso corpori, quod ante ipsi aeri frigido diu fuerit expositum, ut marmori,

& inter ipsa corpora solida, multo majus frigus ab altero sentiat, quam ab altero, ac ab

acre humido multo plus, quam a sicco, rapta nimirum in diversis ejusmodi circumstantiis

A THEORY OF NATURAL PHILOSOPHY 359

mutual forces, these substances draw these particles towards themselves & surround themselves
with them as if the particles were bound to them. Moreover, a very intense cold can be
produced in a warm body merely by the approach of a body made cold by a mere defect
of such a substance. For, the substance, while it ferments, & remains in its natural state
of volatilization, avails itself of its own elasticity to expand ; & thereby, if it is enclosed in any
medium, however inert it may be with respect to the medium, the substance diffuses through
the medium equally. Hence, it comes about that, if from any one place there is taken away
some part of the substance, immediately there flies to it from other places just that quantity
which is required for equality. Thus, for instance, if in the open air a quantity of such fer-
menting substance is lacking, whether through a diminution in the continued impulses neces-
sary for the continued motion, such as the diminished supply of rays from the Sun in winter,
or in places more remote from the equator, or whether through the presence of a large
supply of particles that stop such motion of the substance, due to which there is, even in
regions not far distant from the equator, great coldness in several places, & ice, through
an abundance of nitrous exhalations ; then, from all bodies exposed to such air there will
rush forth a great abundance of the substance still fermenting in them, & of the elastic
matter of fire. The bodies themselves will remain quite cold, merely by the diminution
of this matter ; & if we touch them with the hand, immediately a large number of these
particles will fly out of the hand & be transfused into the bodies, so as to bring about equality ;
& not only the cessation of that internal motion by which the state of the nerves of the
organic body is altered, but also the rapid rush of the substance entering into the other,
will give rise to that feeling of cold which we experience so keenly.

508. We have an idea of such a rush in the very swift motion of the air ; if the air in An illustration
some part of space is suddenly reduced to fixation in large quantities, air will rush in violently ^°."^ thef fi*atlon
from all other places, & sometimes produces dreadful effects by its velocity. Thus, when

a whirlwind, sucking out the air below, passes near to a house that is shut up, the air inside
the house overcomes everything by its expansive force ; roofs fly off, windows are broken,
the floors & all the doors that prevent mutual communication between the rooms are
suddenly burst apart, & the very walls are sometimes overthrown & fall down ; just as
was seen at Rome some years ago, & as I fully explained in the dissertation De Turbine
already mentioned, which I published at the time.

509. But the mere expansive force of such a fermenting substance is insufficient to The at fraction

explain thoroughly what happens ; we require also a certain mutual force, due to which yhl.cli can. st°2 &
.° ' , IT . , . , fix internal motion ;

the substance is attracted more by some bodies & less by others ; & the manner in which motion shared so

this can happen was explained when we dealt with solution & precipitation. Such an aj ^t^tio^after

attraction may be so powerful as to prevent that internal motion altogether by its pressure, a part is fixed ;

& lead to fixation of the substance ; but if this is fairly small, it will indeed allow some d^e2tnkillds of

fermentatory motion to go on, but will not allow the whole mass to be broken up, unless

a body approaches which exerts a greater force & draws the substance to itself. Now

this attraction can take place in two ways. In the first, because one substance has a greater

absolute force on this fiery substance than another, for the same number of particles ; in

the second, because although the one attracts the substance equally or even less than the

other, yet, since either of them attracts it more at smaller distances & less at greater distances,

the one has much less of the substance in proportion to its attraction than the other.

In this second case, particles will still be torn away from the latter body, intermingled

with particles of the substance, to distances somewhat greater, & will be surrounded with

particles of the former, until in both there will be an equal saturation when parts of it are

compared with one another ; & also an equal attractive force for particles of the fiery substance

that are remote from particles of either of the substances by which it is surrounded. But

there still may be an abundance of the fiery substance in each of the two substances, in

any ratio, different for each. For, in the one, due to a more extended continuation of

the force, there may be had a given force at a greater distance than in the other ; & thus

the depth, so to speak, of the oceans surrounding the one may be greater than for the other ;

& for the same distances, for the one there may be, on account of the greater force, a greater

density of the affused fiery substance, than for the other. From these principles, & different

combinations of them, it is truly wonderful how many things can be derived extremely

suitable to explain the phenomena of Nature.

510. Thus, from the principle of such diffusion tending to establish the same equality The consequences

i vrr r i ° T i • T<T r of this diffusion

between different parts of the same substance, but an equality that is quite different for ten(iing to estab-
different substances, it is easily seen how It comes about that in winter the hand when exposed Hsh equality;
to the open air, feels the cold less than when exposed to a solid body of sufficient density, Batter of refrigera-
such as marble, which has previously been exposed to the same cold air for a long time; tion & congelation.
& amongst solids, feels far more cold from some than from others, from damp air much
more than from dry. For, in different circumstances of the same kind, in the same time,

360 PHILOSOPHIC NATURALIS THEORIA

eodem tempore admodum diversa copia igneas substantial, quae calorem in manu fovebat.
Atque hie quidem & analogiae sunt quaedam cum iis, quae de refractione diximus : nam
plerumque corpora, quae plus habent materias, nisi oleosa, & sulphorosa sint, majorem
habent vim refractivam, pro ratione densitatis suae, & corpora itidem communiter, quo
densiora sunt, eo citius manum admotam calore spoliant, quae idcirco si lineam telam
libero expositam aeri contingat in hyeme, multo minus frigescit, quam si lignum, si marmora
si metalla. Fieri itidem potest, ut aliqua substantia ejusmodi substantiam igneam repellat
etiam, sed ob aliam substantiam admixtam sibi magis attrahentem, adhuc aliquid surripiat
magis, vel minus, prout ejus admixtae substantiae plus habet, vel minus. Sic fieri posset,
ut aer ejusmodi substantiam igneam respueret, sed ob heterogenea corpora, quae sustinet, inter
quae inprimis est aqua in vapores elevata, surripiat nonnihil ; ubi autem in ipso volitantes
particulae, quae ad fixitatem adducunt, vel expellunt ejusmodi substantiam igneam, accedant
ad alias, ut aqueas, fieri potest, ut repente habeantur & concretiones, atque congelationes,
ac inde nives, & grandines. A diffusione vero ad aequalitatem intra idem corpus fieri utique
debet, ut ubi altius infra Terrae superficiem descensum sit, permanens habeatur caloris
gradus, ut in fodinis, ad exiguam profunditatem pertinente effectu vicissitudinum, quas
habemus in superficie ex tot substantiarum permixtionibus continuis, & accessu, ac recessu
solarium radiorum, quae omnia se mutuo compensant saltern intra annum, antequam
sensibilis differentia haberi possit in profundioribus locis : ac ex diversa vi, quam diversas
substantiae exercent in ejusmodi substantiam igneam, provenire debet & illud, quod
experimenta evincunt, ut nimirum nee eodem tempore aeque frigescant diversae substantiae
aeri libero expositae, nee caloris imminutio certam densitatum rationem sectetur, sed
varietur admodum independenter ab ipsa. Eodem autem pacto & alia innumera ex iisdem
principiis, ubique sane conformibus admodum facile explicantur.

Eodem pacto expli- 511. Patet autem ex iisdem principiis repeti posse explica-[24o]-tionem etiam praeci-

tem :&priencTpia Puorum omnium ex Electricitatis phaenomenis, quorum Theoriam a Franklino mira sane
Frank linianae sagacitate inventam in America & exornavit plurimum, & confirmavit, ac promovit Taurini
tads"* EIectnci" P. Beccaria vir doctissimus opere egregio ea de re edito ante hos aliquot annos. Juxta
ejusmodi Theoriam hue omnia reducuntur : esse quoddam fluidum electricum, quod in
aliis substantiis & per superficiem, & per interna ipsarum viscera possit pervadere, per
alias motum non habeat, licet saltern harum aliquae ingentem contineant ejusdem substantiae
copiam sibi firmissime adhaerentem, nee sine frictione, & motu intestine effundendam,
quarum priora sint per communicationem electrica, posteriora vero electrica natura sua :
in prioribus illis diffundi statim id fluidum ad aequalitatem in singulis ; licet alia majorem,
alia minorem ceteris paribus copiam ejusdem poscant ad quandam sibi veluti connaturalem
saturitatem : hinc e duobus ejusmodi corporibus, quse respectu naturae suse non eundem
habeant saturitatis gradum, esse alterum respectu alterius electricum per excessum, &
alterum per defectum, quae ubi admoveantur ad earn distantiam, in qua particulae circa
ipsa corpora diffusae, & iis utcunque adhaerentes ad modum atmosphaerarum quarundam,
possint agere aliae in alias, e corpore electrico per excessum fluere illico ejusmodi fluidum
in corpus electricum per defectum, donee ad respectivam aequalitatem deventum sit, in
quo effluxu & substantiae ipsae, quae fluidum dant, & recipiunt, simul ad se invicem accedant,
si satis leves sint, vel libere pendeant, & si motus coacervatae materiae sit vehemens, explo-
siones habeantur, & scintillae, & vero etiam fulgurationes, tonitrua, & fulmina. Hinc
nimirum facile repetuntur omnia consueta electricitatis phaenomena, praeter Batavicum
experimentum phialae, quod multo generalius est, & in Frankliniano piano aeque habet
locum. Id enim phasnomenum ad aliud principium reducitur : nimirum ubi corpora
natura sua electrica exiguam habent crassitudinem, ut tenuis vitrea lamella, posse in altera
superficie congeri multo majorem ejus fluidi copiam, dummodo ex altera ipsi ex adverse
respondente aequalis copia fluidi ejusdem extrahatur recepta in alterum corpus per com-
municationem electricum, quod ut per satis amplam superficiei partem fieri possit, non
excurrente fluido per ejusmodi superficies ; aqua affunditur superficiei alteri, & ad alteram
manus tota apprimitur, vel auro inducitur superficies utraque, quod sit tanquam vehiculum,
per quod ipsum fluidum possit inferri, & efferri, quod tamen non debet usque ad marginem
deduci, ut citerior inauratio cum ulteriore conjungatur, vel ad illam satis accedat : si enim
id fiat, transfuse statim fluido ex altera superficie in alteram, obtinetur aequalitas, & omnia
cessant electrica signa.

A THEORY OF NATURAL PHILOSOPHY 361

a different quantity of the fiery substance is seized, & this originally kept the hand warm.
Here, too, there are certain analogies with what we have said about refraction. For, very
many bodies possessing a considerable amount of matter, unless they are oily or sulphurous,
have a greater refractive force in proportion to their density ; & commonly, too, the denser
they are, the more quickly they withdraw heat from the hand that touches them ; & thus,
if the hand touches a linen cloth exposed to the open air in winter, it is made cold to a far
less degree than it would be in the case of wood, marble, or metal. Further it may be
that some substance of this sort even repels the fiery substance ; but, owing to the fact
that another substance mixed with it has a stronger attraction, it will still carry off some
of the fiery substance, more or less in amount according as there is more or less of the second
substance mixed with it. Thus, it might be the case that air would reject a fiery substance
of this sort ; but, owing to the presence of heterogeneous bodies in it, amongst which
there is in particular water uplifted in the form of vapour, it seizes some portion of it. Also,
when particles hovering in it, which either induce fixity, or repel such fiery substance, approach
others, like those of water -vapour, it may happen that sudden concretions & congelations take
place ; & thus cause snow & hail. But from a diffusion tending to produce equality within
the same body it must come about that, when one goes deeper down beneath the surface
of the Earth, there is a permanent degree of warmth. Thus, in mines, the effect of the
vicissitudes which take place on the surface owing to the continual mingling of so many
substances, & the accession & recession of the solar rays, only continues for a very small
depth ; for these all compensate one another in the course of a year at any rate, before
any sensible difference can be produced in places of fair depth. Because of this, and also
on account of the different force exerted by different substances on this fiery substance,
it must come about, as is proved experimentally, that different bodies are not cooled equally
in the same time when exposed to the open air, nor is the diminution of heat in a fixed
ratio to the density, but varies altogether independently of it. In the same way, innumerable
other things can be quite readily derived from these same principles, which agree with one
another perfectly.

511. Further, it is clear that from these principles there can be derived an explanation Electricity can also
of all the chief phenomena in electricity ; the theory of these, discovered by Franklin in ^m^wa'y^Frlnk-
America with truly marvellous sagacity, has been greatly embellished & confirmed, & even lin's principles of
further developed at Turin by Fr. Beccaria, a most learned man, in his excellent work tricky C0ry °* el&
on this subject, published some years ago. According to such theory, all things reduce
to this ; there is a certain electric fluid, which can in some substances move along the surface
& also through their inward parts ; but has no motion through others, although some of
these at any rate hold an abundance of the substance very firmly adherent to themselves,
& not to be loosened without friction & internal motion. Of these, the former are electric
by communication, the latter electric by nature. In the former, the fluid is immediately
diffused to produce equality on each of them ; although some of them require more, others
less, of the fluid to produce, so to speak, an intrinsic saturation, other things being the same.
Thus, of two of these bodies, of which the saturation corresponding to their natures is
not the same, one will be electric by excess, & the other by defect, with respect to one
another. If these bodies approach one another to within that distance, for which the
particles surrounding the bodies, & adhering to them like atmospheres, can act upon one
another ; then, from the body that is electric by excess this fluid will immediately flow
towards the one that is electric by defect, until equality is reached. During this flow,
the substances which respectively yield & receive the fluid will simultaneously approach
one another, if they are light enough, or if they are freely suspended ; & if the motion of
the concentrated matter is vigorous, there will be explosions, & sparks, & even lightning,
thunder, & thunderbolts. Hence, forsooth, can be derived all the customary phenomena
of electricity, besides the experiment of the Leyden Jar, which is much more general, &
the same holds equally good for Franklin's plate. For this phenomenon reduces to another
principle ; namely, that when bodies that are naturally electric have a very small thickness,
such as a thin glass plate, there can be collected on one of the surfaces a much greater amount
of the fluid, & at the same time from the other surface exactly opposite to it there can be
withdrawn an equal amount of the fluid, & this may be passed into another body by electric
communication. In order that this can take place over a sufficiently ample part of the
surface, as the fluid does not run away from such surfaces, water is brought into contact
with one surface, & the other is pressed with the whole hand ; or each of the surfaces is
overlaid with gold, which forms as it were a medium through which the fluid can be borne
either in or out. The gold, however, must not be brought right up to the edge, so that
the inner gilding touches the outer, or even approaches it too closely ; for if this happens,
the fluid is immediately transfused from one surface to the other, equality is obtained,
& all signs of electricity cease.

362 PHILOSOPHISE NATURALIS THEORIA

Eorum expiicatio tjI2. Hujusmodi Theoriae ea pars, quae continet respectivam [241] illam saturitatem,

conspirat cum iis, quse diximus de ignea substantia, ubi ipsam respectivam saturitatem
abunde explicavimus. Dum autem fluidum vi mutua agente abit ex altera substantia
in alteram : facile patet, debere ipsa etiam ea corpora, quorum particulse ipsum fluidum,
quanquam viribus inaequalibus, ad se trahunt, ad se invicem accedere, ac facile itidem
patet, cur aer humidus, in quo ob admixtas aquae particulas vidimus citius manum frigescere,
electricis phasnomenis contrarius sit, vaporibus abripientibus illico, quod in catena a globi
sibi proximi frictione in ipso excitatum, & avulsum congeritur. Secunda pars, ex qua
Batavicum experimentum pendet, & successus plani Frankliniani, aliquanto difricilior,
explicatione tamen sua non caret. Fieri utique potest, ut in certis corporibus ingens sit
ejus substantise copia ob attractionem ingentem, & ad exiguas distantias pertinentem,
congesta, quae in aliquanto majore distantia in repulsionem transeat, sed attraction! non
praevalentem. Haec repulsio cum ilia copia materiae potest esse in causa, ne per ejusmodi
substantias transire possit is vapor, & ne per ipsam superficiem excurrat, nee vero ad earn
accedat satis ; nisi alterius substantiae adjunctae actio simul superveniat, & adjuvet. Turn
vero ubi lamina sit tenuis, potest repulsio, quam exercent particulae fluidi prope alteram
superficiem siti, agere in particulas sitas circa superficiem alteram : sed adhuc fieri potest,
ut ea non possit satis ad vincendam attractionem, qua haerent particulis sibi proximis :
verum si ea adjuvetur ex una parte ab attractione corporis admoti per communicationem
electrici, & ex altera crescat accessu novi fluidi advecti ad superficiem oppositam, quod
vim ipsam repulsivam intendat : turn vero ipsa praevaleat. Ipsa autem praevalente,
effluet ex ulteriore superficie ejus fluidi pars novum illud corpus admotum ingressa, ac
ex ejus partis remotione, cessante parte vis repulsivae, quam nimirum id, quod effluit,
exercebat in particulas citerioris superficiei, ipsi citeriori superficiei adhaereat jam idcirco
major copia fluidi electrici admota per aquam, vel aurum, donee tamen, communicatione
extrorsum restituta per seriem corporum sola communicatione electricorum, defluxus ex
altera superficie pateat ad alteram. Porro explicationem hujusmodi & illud confirmat,
quod experimentum in lamina nimis crassa non succedit. Quod autem per substantiam
natura sua electricam non permeet, ut aequalitatem acquirat, id ipsum provenire posset
ab exigua distantia, ad quam extendatur ingens ejus attractiva vis in illam substantiam
fluidam, & aliquanto majore distantia suarum particularum a se invicem : nam in eo casu
altera particula substantiae per se electricae, utut spoliata magna parte sui fluidi, non poterit
rapere partem satis magnam fluidi alteri parti affusi, & appressi.

Quod videatur esse 513. Haec quidem an eo modo se habeant, defmire non licet [242] nisi & illud ostendatur

discrimen inter sjmui rem aliter se habere non posse. Sed illud jam patet, Theoriam meam, servato semper

materiam elec- . * . . J. •*,...

tricam, & igneam. eodem agendi modo, suggerere ideam earum etiam dispositionum materiae, quae possint
maxime omnium ardua, & composita explicare Naturae phaenomena, ac corporum discrimina.
Illud unum hie addam ; quoniam & ingens inter igneam substantiam, & electricum fluidum
analogia deprehenditur, & habetur itidem discrimen aliquod ; fieri etiam posse, ut inter
se in eo tantummodo discrepent, quod altera sit cum actuali fermentatione, & intestine
motu, quamobrem etiam comburat, & calefaciat, & dilatet, ac rarefaciat substantias, altera
ad fermentescendum apta sit, sed sine ulla, saltern tanta agitatione, quantam fermentatio,
inducit orta ex collisione ingenti mutua, vel ex aliarum admixtione substantiarum, quse
sint ad fermentandum idoneae.

r»e magnetica yi : 51^. Quod ad magneticam vim pertinet, adnotabo illud tantummodo, ejus phaenomena

variationem "Ven- omnia reduci ad solam attractionem certarum substantiarum ad se invicem. Nam directio,

dere ab attractione, ad quam & inclinatio, & declinatio reducitur, repeti utique potest ab attractione ipsa sola.

ar^m^ingeSfum Videmus acum magneticam inclinari statim prope fodinas ferri, intra quas idcirco nullus

attrahentium. est pyxidis magneticae usus. Si ingens adesset in ipsis polis, & in iis solis, massa ferrea ;

omnes acus magneticae dirigerentur ad polos ipsos : sed quoniam ubique terrarum fodinae

ferreae habentur, si circa polos eaedem sint in multo majore copia, quam alibi ; dirigentur

utique acus polos versus, sed cum aliqua deviatione in reliquas massas per totam Tellurem

dispersas, quae nunquam poterit certum superare graduum numerum ; nisi plus sequo ad

fodinam aliquam accedatur. Declinatio ejusmodi diversa erit in diversis locis, ob diversam

A THEORY OF NATURAL PHILOSOPHY 363

512. That part of this theory, which deals with the relative saturation, agrees with Explanation of
what we have said with respect to the fiery substance, when we gave a full explanation of n^Ttoory.

its relative saturation. Moreover, when the fluid, under the action of a mutual force,
passes from one substance to another, it is readily seen that those bodies, of which the
particles attract the fluids to themselves although with unequal forces, must also attract
one another. It is also quite clear why moist air, in which, on account of the admixture
of water particles, we see that the hand is cooled more rapidly, works in an exactly opposite
manner with electric phenomena, the vapour immediately carrying off the fluid, that is
accumulated in a chain, after it has been excited in a sphere very close to it by friction &
expelled from it into the chain. The second part, upon which the Leyden jar experiment
depends, as also the Franklin plate, is somewhat more difficult, yet does not altogether
lack an explanation. For, it may indeed be the case that in certain bodies there may be
concentrated a huge amount of the substance, due to a huge attraction, which however
only lasts for exceedingly small distances ; & this attraction for a somewhat greater distance
may pass into a repulsion, without however overcoming the attraction. This repulsion
taken in conjunction with the large amount of matter may be for the purpose of preventing
the possibility of this vapour from passing through such bodies, or of running along its
surface, or even of approaching very near to it ; unless the action of some other substance
adjoined simultaneously supervenes & assists it. Then, indeed, when the plate is thin,
there can be a repulsion, exerted by the particles of the fluid situated on one of the surfaces,
acting on particles situated near the other surface. Still, it may be that this is not sufficient
to overcome the attraction by which the particles adhere to those that are next to them.
But, If this is assisted on the one side by the attraction of a body, which is electric by
communication, moving towards it, & on the other side it is increased by a fresh accession
of fluid brought up to the opposite surface, because this will augment the repulsive force
also ; then, the repulsive force will overcome the attraction. Now, when this is the case,
part of the fluid will flow off from the further surface & enter the new body that has been
brought close to It ; & since part of the repulsive force ceases owing to the removal of this
part of the fluid (namely, that repulsive force that was exerted on the particles of the nearer
surface by the part of the fluid that flowed off), in consequence, there will adhere to the
nearer surface a greater amount of the electric fluid brought to it by the water or the gold ;
until, however, communication being restored from without by means of a series of bodies
that are merely electric by communication, the flow of the fluid from one surface to the
other will be unhindered. Moreover, this explanation is confirmed by the fact that, if
the experiment is tried with a plate that is too thick, it will not succeed. Further, the
fact that the fluid will not pass through a substance that is naturally electric, so that equality
is produced, can be produced by the very small distance over which the huge attractive
force on the fluid substance extends, & the somewhat greater distance of its particles from
one another. For, in this case, one particle of the naturally electric substance, when it
has lost the greater part of its fluid, will not seize upon any great part of the fluid surrounding
another part, & in close contact with it.

513. Whether these things are indeed as stated cannot be determined, unless it can The manner in
be shown at the same time that it is impossible for them to be otherwise. But this fact which electric mat-

, , m, i . \ . , if- i i ter seems to difier

is clear, that my Theory, always maintaining the same mode of action, suggests also the from fire,
idea of these dispositions of matter, such as are most of all capable of explaining the difficult
& compound phenomena of Nature, & the differences between bodies. I will add but one
thing further : since we can detect a very great analogy between the fiery substance &
the electric fluid, & also some difference, it may possibly be that they only differ from one
another in the fact that the one occurs in conjunction with actual fermentation & internal
motion, due to which it burns, heats, dilates & rarefies substances ; while the other is suitable
to the setting up of fermentation, but without that agitation, or at least without an agitation
so great as that produced by fermentation arising from a very great mutual collision, or
from admixture of other substances that are liable to fermentation.

514. With regard to magnetic force, I will make but the one observation, that all Magnetic force ; its
phenomena with regard to it reduce to a mere attraction of certain substances for one tion° defends V^01n
another. For direction, to which both inclination & declination can be reduced, can always the attraction &
be derived from attraction alone. We notice that a magnetic needle is immediately inclined fa ^'"iLsses* at-
near iron mines ; & therefore within these a magnetic compass-box is of no service. If tracting.

there were present at the poles, & there only, immense masses of iron, every magnetic needle
would be directed towards those poles. But, since there are iron mines in all lands, if
about the poles there were the same in much greater abundance than in other places, then,
in every case needles would be directed towards the poles, but with some deviation towards
the other masses scattered over the whole Earth ; this deviation could never exceed a
certain number of degrees, unless it was taken too near some one mine. Declination of

364 PHILOSOPHISE NATURALIS THEORIA

eorum locorum positionem ad omnes ejusmodi massas, & vero etiam variabitur, cum fodinae
ferri & destruantur in dies novae, & generentur, ac augeantur, & minuantur in horas.
Variatio intra unum diem exigua erit, cum eae mutationes in fodinis intra unum diem
exiguae sint : procedente tempore evadet major, eritque omnino irregularis ; si mutationes,
quae in fodinis accidunt, sint etiam ipsae irregulares.

Attract jo n e m, & rjr QUod autem ad attractionem pertinet earn in particulis haberi posse patet, &

polos cohaerere cum , J J , . , f ...

hac Theoria : diffi- an earum textu aebere penaere : plunma autem sunt magnetismi phaenomena, quae
cuitas de distantia ostendant, mutata dispositione particularum generari magneticam vim, vel destrui, &

ad quam visea , , . . r , . , r . . . °. , ° .

extenditur: conjee- multo irequentius intendi, vel remitti, cujus rei exempla passim occurrunt apud eos, qm
tura de soiutione de magneticis agunt. Poli autem ex altera parte attractivi, ex altera repulsivi, qui haben-
tur in magnetismo itidem, cohasrent cum Theoria ; cum virium summa ex altera parte possit
esse major, quam ex altera. Difficultatem aliquam majorem parit distantia ingens, ad
quam ejusmodi vis extenditur : at fieri utique id ipsum potest per aliquod efnuviorum
intermedium genus, quod tenui-[243]-tate sua efTugerit hue usque observantium oculos,
& quod per intermedias vires suas connectat etiam massas remotas, si forte ex sola diversa
combinatione punctorum habentium vires ab eadem ilia mea curva expressas id etiam
phenomenon provenire non possit. Sed ad haec omnia rite evolvenda, & illustranda singu-
lares tractatus, & longae perquisitiones requirerentur ; hie mihi satis est indicasse
ingentem Theoriae meae foecunditatem, & usum in difficillimis quibuscunque Physicas etiam
particularis partibus pertractandis.

Superest, ut postremo loco dicamus hie aliquid de alterationibus, & transforma-
tria diversa prin- tionibus corporum. Pro materia mihi sunt puncta indivisibilia, inextensa, praedita vi
Crove!nire%siu1ntlS i061^33? & viribus mutuis expressis per simplicem continuam curvam habentem deter-
minatas illas proprietates, quas expressi a num. 117, & quae per aequationem quoque
algebraicam definiri potest. An haec virium lex sit intrinseca, & essentialis ipsis indivisi-
bilibus punctis ; an sit quiddam substantiate, vel accidentale ipsis superadditum,
quemadmodum sunt Peripateticorum formae substantiales, vel accidentales ; an sit libera
lex Auctoris Naturae, qui motus ipsos secundum legem a se pro arbitrio constitutam dirigat :
illud non quaero, nee vero inveniri potest per phaenomena, quae eadem sunt in omnibus
iis sententiis. Tertia est causarum occasionalium ad gustum Cartesianorum, secunda
Peripateticis inservire potest, qui in quovis puncto possunt agnoscere materiam, turn formam
substantialem exigentem accidens, quod sit formalis lex virium, ut etiam, si velint, destructa
substantia, remanere eadem accidentia in individuo, possint conservare individuum istud
accidens, unde sensibilitas remanebit prorsus eadem, & quas pro diversa combinatione
ejusmodi accidentium pertinentium ad diversa puncta, erit diversa. Prima sententia
videtur esse plurimorum e Recentioribus, qui impenetrabilitatem, & activas vires, quas
admittunt Leibnitiani, & Newtonian! passim, videntur agnoscere pro primariis materiae
proprietatibus in ipsa ejus essentia sitis. Potest utique hasc mea Theoria adhiberi in
omnibus hisce philosophandi generibus, & suo cujusque peculiari cogitandi modo aptari
potest.

Homogeneitas ele- 517. Hsec materia mihi est prorsus homogenea, quod pertinet ad legem virium, &

no^admittatuT argumenta, quae habeo pro homogeneitate, exposui num. 92. Siqua occurrent Naturae
quanto piures com- phaenomena, quae per unicum materiae genus explicari non possint ; poterunt adhiberi
versas°ieges virium- plura genera punctorum cum pluribus legibus inter se diversis, atque id ita, ut tot leges
formam substantia- sint, quot sunt binaria generum, & praeterea, quot sunt ipsa genera, ut illarum singulae
p^se Pen^atftiicos! exprimant vires mutuas inter puncta pertinentia ad bina singulorum binariorum genera,
si yeiint, agnoscere & harum singulae vires mutuas inter puncta pertinentia ad idem genus, singulae pro generibus
pui singulis. Porro inde mirum sane, quanto major [244] combinationum numerus oriretur,

& quanto facilius explicarentur omnia phaenomena. Possent autem illse leges exponi
per curvas quasdam, quarum aliquae haberent aliquid commune, ut asymptoticum impene-
trabilitatis arcum, & arcum gravitatis, ac aliae ab aliis possent distare magis, ut habeantur
quaedam genera, & quaedam differentiae, quae corporum elementa in certas classes
distribuerent ; & hie Peripateticis, si velint, occasio daretur admittendi materiam ubique
homogeneam, ac formas substantiales diversas, quae accidentalem virium formam diversam
exigant, & vero etiam piures accidentales formas, quae diversas determinent vires, ex quibus
componatur vis totalis unius elementi respectu sui similium, vel respectu aliorum.

A THEORY OF NATURAL PHILOSOPHY 365

this kind will be different in different places, on account of the different situation of these
places with respect to all such masses ; & it will vary, since mines of iron are destroyed &
generated every day, & are increased & diminished hourly. The variation within a day
will be very slight, since the daily change in mines is very small ; as time goes on it becomes
greater, & it will be quite irregular, if the changes that take place in mines are themselves
also irregular.

515. With regard to attraction, it is clear that this can be had in the particles, & that Attraction, & the
it must depend upon their structure. Moreover, there are very "many phenomena of terft ^v it h°n^y
magnetism, which will show that magnetic force is generated by changing the disposition Theory; difficulty
of the particles, or is destroyed, or more frequently is augmented or abated ; examples tcTwhich6 the^force
of this everywhere come under the observation of those who study magnets. Moreover, extends ; conjee-
poles that are attractive on one side & repulsive on the other, which are also had in magnetism, tionoTthS problem!
agree with my Theory ; for, the sum of the forces on one side may be greater than the

sum of the forces on the other. A somewhat greater difficulty arises from the huge distance
to which this kind of force extends. But even this can take place through some intermediate
kind of exhalation, which owing to its extreme tenuity has hitherto escaped the notice
of observers, & such as by means of intermediate forces of its own connects also remote
masses ; if perchance this phenomenon cannot be derived from merely a different combination
of points having forces represented by that same curve of mine. But to explain all these
things properly, & to furnish them with illustrations would require separate treatment
& long investigations. It is enough for me that- 1 have pointed out the extreme fertility
of my Theory, & its use in any of the most difficult & special problems of
physics.

516. It remains for me here to say a few words finally about alterations & trans- The nature of mat-
formations of bodies. To me, matter is nothing but indivisible points, that are non-extended, *®J' forces ^"Tkree
endowed with a force of inertia, & also mutual forces represented by a simple continuous different principles
curve having those definite properties which I stated in Art. 117 ; these can also be defined m^a^e1011 they
by an algebraical equation. Whether this law of forces is an intrinsic property of indivisible

points ; whether it is something substantial or accidental superadded to them, like the
substantial or accidental shapes of the Peripatetics ; whether it is an arbitrary law of the
Author of Nature, who directs those motions by a law made according to His Will ; this
I do not seek to find, nor indeed can it be found from the phenomena, which are the same
in all these theories. The third is that of occasional causes, suited to the taste of followers
of Descartes ; the second will serve the Peripatetics, who can thus admit the existence
of matter at any point ; & then a substantial form producing a circumstance (accidens)
which becomes a formal law of forces ; so that, if they wish, having destroyed the substance,
that the same circumstances shall remain in the individual, they can preserve that individual
circumstance. Hence the sensibility will remain the same exactly, & such as will be different
for a different combination of such circumstances pertaining to different points. The
first theory seems to be that of most of the modern philosophers, who seem to admit
impenetrability & active forces, such as the followers of Leibniz & Newton all admit, as
the primary properties of matter founded on its very essence. This Theory of mine can
indeed be used in all these kinds of philosophising, & can be adapted to the mode of thought
peculiar to any one of them.

517. Matter, in my opinion, is perfectly homogeneous ; what pertains to the law of Homogeneity of the
forces, & the arguments which I have in favour of homogeneity, I have stated in Art. 92. ^Ie™e0ntts' admitted!
If there are any phenomena of Nature, which cannot be explained by a single kind of matter, there will be ail the
then we should have to make use of many different kinds of points, with many laws that

differ from one another ; & this, too, in such a manner that there are as many laws as there laws of forces ;

are pairs of kinds of points ; &, in addition, as many more as there are kinds of points. For,

each of the former express the mutual forces between the points belonging to two kinds substantial form &

of each pair, & each of the latter the mutual forces between points belonging to the same theseprintT3 mt°

kind, one for each kind. Further, from this it is truly marvellous how much greater the

number of combinations will become, & how much more easily all phenomena can be

explained. Moreover, the laws can be expressed by curves, some of which would have

something in common, such as the asymptotic arc of impenetrability, or the arc of gravitation ;

while some might be considerably different from others, so that certain classes &' certain

differences could be obtained, such as would distribute the elements of bodies into certain

classes. This would give the Peripatetics an opportunity, if they so wished, of admitting

matter that was everywhere homogeneous, as well as substantial forms of different kinds

such as would necessitate a different accidental form of forces ; & also many accidental

forms, which determine different laws, from which is compounded the total force of one

element upon others similar to it, or upon others that are not.

366 PHILOSOPHIC NATURALIS THEORIA

Mira yariatas con- 518. Posset autem admitti vis in quibusdam generibus nulla. & tune substantia unius

sectariorum : possi- •• -i TI • ,.,..,,

biiitas quotiibuerit ex lls genenbus iibernme permearet per substantiam altenus sine ullo occursu, qui m
Mundorum in eo- numero finite punctorum indivisibilium nullus haberetur, adeoque transiret cum impene-

dem spatiocum ap- L-T v o • n- r

parent! compene- trabilitate reah, & compenetratione apparente : ac posset unum genus esse colligatum
tratione, sine uiia cum alio per kgem virium, quam habeant cum tertio, sine ulla lege virium mutua inter

notitia unius cuius- • i j 1111 11 •

vis in aiiis. JPsa> vel possent ea duo genera nullum habere nexum cum ullo tertio : atque in hoc

posteriore casu haberi possent plurimi Mundi materiales, & sensibiles in eodem spatio ita
inter se disparati, ut nullum alter cum altero haberet commercium, nee alter ullam alterius
notitiam posset unquam acquirere. Mirum sane, quam multae aliae in casibus illius nexus
cujuspiam duorum generum cum tertio combinationes haberi possint ad explicanda Naturae
phenomena : sed argumenta, quae pro homogeneitate protuli, locum habent pro omnibus
punctis, cum quibus nos commercium aliquod habere possumus, pro quibus solis inductio
locum habere potest. An autem sint alia punctorum genera vel hie in nostro spatio, vel
alibi in distantia quavis, vel si id ipsum non repugnat, in aliquo alio spatii genere, quod
nullam habeat relationem cum nostro spatio, in quo possint esse puncta sine ulla relatione
distantias a punctis in nostro existentibus, nos prorsus ignoramus, nihil enim eo pertinens
omnino ex Naturae phaenomenis colligere possumus, & nimis est audax, qui eorum omnium,
quae condidit Divinus Naturae Fabricator limitem ponat suam sentiendi, & vero etiam
cogitandi vim.

ffjmfm- m homo~ 519. Sed redeundo ad meam homogeneorum elementorum Theoriam, singulares

gGIlCllcitlS SUppOSl~ r i • • i 11 i • •

tioneessenumerum, corporum formae erunt combmatio punctorum homogeneorum, quae habetur a distantus
pun^torumSltl0nua3 ^ positionibus, ac praeter solam combinationem velocitas, & directio motus punctorum
sunt radix 'omnium singulorum ; pro individuis vero corporum massis accedit punctorum numerus. Dato
did^ossint^onnae numero & dispositione punctorum in data massa, datur radix omnium proprietatum, quas
specifics : unde habet eadem massa in se, & omnium relationum, [245] quas eadem habere debet cum
transform t^n S & a^*s massis> quas nimirum determinabunt numeri, & combinationes, ac motus earum,
& datur radix omnium mutationum, quae ipsi possunt accidere. Quoniam vero sunt
qusedam combinationes peculiares, quae exhibent quasdam peculiares proprietates con-
stantes, quas determinavimus, & exposuimus, nimirum suse pro cohaesione, & variis solidi-
tatum gradibus, suse pro fluiditate, suae pro elasticitate, suae pro mollitie, suae pro certis
acquirendis figuris, suae pro certis habendis oscillationibus, quae & per se, & per vires sibi
affixas diversos sapores pariant, & diversos ordores, & colorum diversas constantes proprie-
tates exhibeant, sunt autem aliae combinationes, quae inducunt motus, & mutationes non
permanentes, uti est omne fermentationum genus ; possunt a primis illis constantium
proprietatum combinationibus desumi specificae corporum formas, & differentiae, & per
hasce posteriores habebuntur alterationes, & transformationes.

Discrimen inter T -n • i« • j

transformation em, 52°- inter illas autem proprietates constantes possunt seligi quaedam, quae magis

& aiterationem. constantes sint, & quae non pendeant a permixtione aliarum particularum, vel etiam, quae
si amittantur, facile, & prompte acquirantur, & illas haberi pro essentialibus illi speciei,
quibus constanter mutatis habeatur transformatio, iisdem vero manentibus, habeatur
tantummodo alteratio. Sic si fluidi particulae alligentur per alias, ut motum circa se
invicem habere non possint, sed illarum textus, & virium genus maneat idem ; conglaciatum
illud fluidum dicetur tantummodo alteratum, non vero etiam mutatum specifice. Ita
alterabitur etiam, & non specifice mutabitur corpus, aucta quantitate materia igneae, quam
in poris continet, vel aucta quantitate materias igneas, quam in poris continet, vel aucto
motu ejusdem, vel etiam aucta aliqua suarum partium oscillatione, ac dicetur calefactione
nova alteratum tantummodo : & aquae massa, quas post ebullitionem redit ad priorem
formam, erit per ipsam ebullitionem alterata, non transformata : figurae itidem mutatio,
ubi ex cera, vel metallo diversa fiunt opera, aiterationem quandam inducet. At ubi mutatur
ille textus, qui habebatur in particulis, atque id mutatione constanti, & quae longe alia
phaenomena praebeat ; turn vero dicetur corrumpi, & transformari corpus. Sic ubi e
solidis corporibus generetur permanens aer elasticus, & vapores elastici ex aqua, ubi aqua
in terram concrescat, ubi commixtis substantiis pluribus arete inter se cohasreant novo
nexu earum particulae, & novum mixtum efforment, ubi mixti particulas separatae per
solutionem nexus ipsius, quod accidit in putrefactione, & in fermentationibus plurimis,
novam singulas constitutionem acquirant, habebitur transformatio.

A THEORY OF NATURAL PHILOSOPHY 367

518. Also, in some of these classes, the absence of any force may be admitted ; & then Wonderful variety
the substance of one of these classes will pas° perfectly freely through the substance of the po"sfbiiityCeSof
another without any collisions ; for, with a finite number of indivisible points, there would anv number of
not be any ; & thus the substance would pass through with real impenetrability & apparent ing^h^sam^space
compenetration. Also it would be possible for one kind to be bound up with another by with apparent cpm-
means of a law of forces, which they have with a third, without any mutual law of forces out^any'^'dication
between themselves, or these two kinds might have no connection with any third. In this of the presence of
latter case there might be a large number of material & sensible universes existing in the same thJ others. *
space, separated one from the other in such a way that one was perfectly independent of

the other, & the one could never acquire any indication of the existence of the other. It
is truly wonderful how many other combination- in cases of any such connection of two
kinds with a third could be obtained for the purpose of explaining the phenomena of Nature.
But the arguments, which I brought forward in favour of homogeneity, hold good for all
points, with which we can have any relation ; & for these alone the principle of induction
can hold good. Further, whether there may be other kinds of points, either here in the space
around us, or somewhere else at a distance from us, or, if the idea of such a thing is not
opposed to our reason, in some other kind of space having no relation with our space, in
which there may be points that have no distance-relation with points existing in our space ;
of this we can know nothing. For, nothing relating to it in the slightest degree can be

fathered from the phenomena of Nature ; & it would be great presumption for any one to
x as a limit his own power of perception, or even of imagination, of all the things that
the Divine Author of Nature has founded.

519. But, to return to my Theory of homogeneous elements, the several forms of Form in the hyp°-
bodies will consist of a combination of homogeneous points, which comes from their distances ity uf the number &
& positions, &, in addition to combination alone, the velocity & direction of the motion disposition of the
of each of the points ; also for individual masses of bodies there is to be added the number Ftitute'the basis of
of points that form them. Given the number & disposition of the points in a given mass, ail properties ; what

i_ v • r 11 • • i_-i_ • i_ • i •• OI-L r 11 may be sald about

the basis or all its properties, which are inherent in the mass, is given ; & also that ot all specific form ; hence,

the relations that the same mass must have with other masses ; that is to say, those determined alterations & trans-

by their numbers, combinations & motions ; moreover, the basis of all changes that can

happen to it is also given. Now, since there are certain special combinations, representing

certain special constant properties, which we have determined & explained, namely, those

corresponding to cohesion, & various degrees of solidity, those for fluidity, for elasticity,

for softness, for the acquisition of certain shapes, for the existence of certain oscillations, which

combinations, both of themselves & through forces connected with them, produce different

tastes & different smells, & exhibit the different constant properties of colours ; & also there

are other combinations which induce motions & changes that are not permanent, like all

sorts of fermentations ; there can be derived from the primary combinations of constant

properties the specific forms of bodies & their differences, & from the latter also can be

obtained alterations & transformations in these forms.

520. Now, amongst these constant properties there may be chosen, some that are more Distinction between
constant than others ; such as do not depend upon admixture with other particles, &. also ^e^°ion?'tl0n &
such as, if they should be lost, would be easily & quickly acquired. These propertiep could

be considered to be essential to the specie'? ; & if such properties suffered a permanent change,
we should have a transformation ; whereas, if they persisted, there would only be an alteration.
Thus, if the particles of a fluid were bound together by other particles, so that they could
have no motion about one another, but their structure & the kind of forces corresponding
to them remained the same, the fluid thus congealed would be said to have been merely
altered, & not to have been specifically changed as well. Thus also, a body would be said
to be altered, but not specifically changed, if the quantity of fiery matter which it contains
in its pores is increased ; or if there is an increase in its motion, or even in some oscillation
of its parts ; similarly, it would be said to be merely altered by a fresh accession of heat.
A mass of water, which after ebullition returns to its original form, will be altered by that
ebullition, but not transformed ; & a change of shape, as when different things are made
from wax & metal, gives some sort of alteration. But when the structure in the particles
is changed, & the change is such as will give far different phenomena, then the body would
be said to have been broken down & transformed. Thus, when from solid bodies there is
generated a permanent elastic gas, & elastic vapour from water, when water is congealed
into earth, when several substances are intimately mixed with one another & in consequence
adhere with some fresh connection between their particles, & form a new mixture, when
the mixed particles, separated by the breaking of this connection, as happens in the case
of putrefaction & in most fermentations, severally acquire fresh constitutions ; then a
transformation takes place.

368 PHILOSOPHIC NATURALIS THEORIA

Quid requireretur 521. Si possemus inspicere intimam particularum constitutionem, & textum, ac

formamnSPint!mam) distinguere a se invicem particulas ordinum gradatim altiorum a punctis elementaribus

unde liceret a priori ad haec nostra corpora ; fortasse inveniremus aliqua particularum genera [246] ita suae

genera"5 ^specLsl f°rmse tenacia, ut in omnibus permutationibus ea nunquam corrumpantur, sed mutentur

quid praestandum, quorundam altiorum ordinum particulse per solam mutationem compositionis, quam

ia ' habent a diversa dispositione particularum constantium ordinis inferioris ; liceret multo

certius dividere corpora in suas species, & distinguere elementa quaedam, quse haberi

possent pro simplicibus, & inalterabilibus vi Naturae, turn compositiones mixtorum

specifkas, & essentiales ab accidentalibus proprietatibus discernere. Sed quoniam in

intimum ejusmodi textum penetrare nondum licet ; eas proprietates debemus diligenter

notare, quae ab illo intimo textu proveniunt, & nostris sensibus sunt perviae, quae quidem

omnes consistunt in viribus, motu, & mutatione dispositionis massularum grandiuscularum,

quae sensibus se nostris objiciunt, & constanter habitas, vel facile, & brevi recuperatas

distinguere a transitoriis, vel facile, & constanter amissas, & ex illarum aggregate distinguere

species, hasce vero habere pro accidentalibus.

Videri, nos nun- ^22. Verum quod ad omne hoc areumentum pertinet, non erit abs re, si postremo

quam posse devenire , J. , ^ ,, °. r, . . . r . ..

a d cognoscendam loco hue transferam ex Stayana Recentiore Philosophia, ac meis in earn adnotatiombus,
intimam substan- iiluj quod habeo ad versum 547 libri i : " Quamvis intrinsecam corporum naturam intueri

tiam, & essentiam, ' T. .... ,*T» J* . r

acdiscrimina speci- non liceat, non esse adjiciendum, amrmat, .Naturae investigandae studium : posse ex exterms
fica- illis proprietatibus plures detegi in dies : ad ipsum summae laudi esse : ideam sane, quam

habemus confusam substantiae eas habentis proprietates, proprietatibus ipsis auctis exten-
dimus. Rem illustrat aptissimo exemplo ejus substantiae, quam aurum appellamus, ac
seriem proprietatum eo ordine proponit, quo ipsas detectas esse verosimiliter arbitratur ;
colorem fulvum, pondus gravissimum, ductilitatem, fusilitatem, quod in fusione nihil
amittat, quod rubiginem non contrahat. Diu his tantummodo proprietatibus auri sub-
stantiam contineri est creditum, sero additum, solvi per illam, quam dicunt aquam regiam,
& praecipitari immisso sale. Porro & aliae supererunt plurimae ejusmodi proprietates olim
fortasse detegendae : quo plures detegimus eo plus ad confusam illam naturae auri
cognitionem accedimus : a clara, atque intima ipsius naturae contemplatione adhuc absumus.
Idem, quod in hoc vidimus peculiar! corpore, de corporis in genere natura affirmat.
Investigandas proprietates, quibus detectis ilium intimum proprietatum fontem attingi
nunquam posse : nil nisi inania proferri vocabula, ubi intimae proprietates investigantur."

Quid tamen praes- 523. Haec ego quidem ex illo : turn meam hanc ipsam Theoriam respiciens, quam

generaies)SSpropri^ & ipse Hbro io exposuit nondum edito, sic persequor : " Quid autem, si partim observatione

tates, & generaiia partim ratiocinatione adhibita, constaret demum, materiam homogeneam esse, ac omne

hic^pKestitum. e & discrimen inter corpora prove-[247]-nire a forma, nexu, viribus, & motibus particularum,

quae sint intima origo sensibilium omnium proprietatum. Ea nostros sensus non alia

effugiunt ratione, nisi ob nimis exiguam particularum molem : nee nostrae mentis vim,

nisi ob ingentem ipsarum multitudinem, & sublimissimam, utut communem, virium legem,

quibus fit, ut ad intimam singularum specierum compositionem cognoscendam aspirare

non possimus. At generalium corporis proprietatum, & generalium discriminum explica-

tionem libro io ex intimis iis principiis petitam, exhibebimus fortasse non infeliciter :

peculiarium corporum textum olim cognosci, difficillimum quidem esse, arbitror, prorsus

impossibile, affirmare non ausim."

QUO pacto interea 524. Demum ibidem illud addo, quod pertinet ad genera, & species : " Interea specificas

species tmgua- naturas sestimamus, & distinguimus a collectione ilia externarum proprietatum, in quo
plurimum confert ordo, quo deteguntur. Si quaedam collectio, quae sola innotuerat,
inveniatur simul cum nova quadam proprietate conjuncta, in aliis fere aequali numero
cum alia diversa ; earn, quam pro specie infima habebamus, pro genere quodam habemus
continente sub se illas species, & nomen, quod prius habuerant, pro utraque retinemus.
Si diu invenimus cunjunctam ubique cum aliqua nova, deinde vero alicubi multo posterius
inveniatur sine ilia nova : turn, nova ilia jam in naturae ideam admissa, hanc substantiam
ea carentem ab ejusmodi natura arcemus, nee ipsi id nomen tribuimus. Si nunc inveniretur
massa, quae ceteras omnes enumeratas auri proprietates haberet, sed aqua regia non solveretur,

A THEORY OF NATURAL PHILOSOPHY 369

521. If we could inspect the innermost constitution of particles & their structure, What is required
& distinguish particles from one another & separate them into classes, step by step of hieher *° enaj)le ?s to look

, j? *- . 11- i i *• \ i\ r~ i ° m"> the innermost

orders, from elementary points up to our own bodies ; then, perhaps, we should find some constitution,™
classes of particles to be so tenacious of their form that in all changes they would never order .tl£lt j^0"^,1*
be broken down ; but the particles of higher orders would be changed by mere change fronTfirst principles
of the composition that they have owing to a different disposition of the particles of a lower *? reduc« matter to
order from which they are formed. It would then be possible to divide with far greater what is to be done!
certainty bodies into their species, & to distinguish certain elements which could be taken since such a thing is
as the simple elements, unalterable by any force in Nature ; & then to distinguish
the specific & essential compositions of mixtures from accidental properties. But, since
we cannot as yet penetrate into the innermost structure of this sort, we must carefully
observe those properties, that arise from this innermost structure, & are accessible to our
senses ; these indeed all consist of the forces, motion & change of disposition of those
comparatively large, though really small, masses that meet our senses ; & we must distinguish
between those properties that are constantly possessed, or easily & quickly recovered, &
those that are transitory, or easily lost & lost for good ; & from the aggregate of the former
to distinguish the species, while considering the latter as accidental properties.

522. But, with respect to all this argument, it will not be out of place if, in the last it is thus to be seen
place, I here quote from Stay's Recentior Pbilosopbia, & my notes thereon, that which I tha:t we J*11 ne7U

f -r, t -T tc AII i • . arrive at a full

have written on verse 547 of Book 1. Although we cannot peer into the intrinsic nature knowledge of the
of bodies, the endeavour to investigate Nature, he states, must not be abandoned. Many "™ermost & essen-

, . i i -i r i mi . . • , ' tial substance, or

things can be detected daily from those external properties. This is worthy of all praise ; the distinction be-

we truly extend the idea, which we have in a confused form of a substance possessing these tween sPecies-

properties, if the properties are increased. He illustrates the matter with a very fitting

example of the substance, which we call gold, & enunciates the series of properties in the

order in which he considers that in all probability they were detected : — yellow colour,

very heavy weight, ductility, fusibility, that nothing is lost in fusion, that it does not rust.

For a long time it was believed that the substance of gold was comprised in these properties

only ; later, there was added, that it was dissolved by what Is called aqua regia, & precipitated

from the solution by salt. Moreover, there will be in addition very many other properties

of this kind, perhaps to be detected in the future ; & the more of these we find out, the

nearer we shall approach to that hazy knowledge of the nature of gold ; but we are still

far from obtaining a clear & intimate view of this nature. He asserts the same thing about

the nature of a body In general, as we have seen in the case of this particular body. He*

states that the properties should be investigated, although from their detection the inmost

source of the properties can never be reached ; that nothing except empty words can be

produced, when fundamental properties are investigated."

523. These were my words in that book ; then considering my own Theory, which What may, how-
he also explained in Book 10, not yet published, I went on thus : — " But what if, partly euShed^thareC'ard
by observation & partly by using deduction, it should finally be established that matter to general properties
is homogeneous, & that all distinction between bodies comes from form, connection, forces, * iefen<has been
& motions of the particles, such as may be the fundamental origin of all sensible properties ? done 'in this work.
These escape our senses for no other reason than the exceedingly small volume of the

particles ; nor are they beyond the powers of our intelligence, except on account of
their huge number, & the very complicated, though general, law of forces. Owing
to these, we cannot hope to obtain an intimate knowledge of the composition of each
species. But we will present, perhaps not unsuccessfully, in book 10, an explanation
of the general properties of a body & the general distinctions between them, derived from
such fundamental principles. I consider that the attainment of a knowledge of the structure
of particular bodies in the future will be very difficult ; that it will be altogether impossible,
I will not dare to assert."

1524. Lastly, I add this in the same connection, relating to classes & species : — " Amongst T n e manner,

11- ' . •/- „ v ' -i i e i 11 • t 3, amongst other

other things, we estimate specific natures, & distinguish them from the collection ot external things, in which we
properties ; & In this the order in which they are detected is of special assistance. If any shall distinguish
collection, which had alone been observed, should be discovered conjoined with some fresh
property, & in others of nearly equal number conjoined with something different ; then
that, which we had considered as a fundamental species, we should now consider as a class
containing within it both these species ; & the name that they had originally, we should
retain for both species. If for some time we found it conjoined with some fresh property,
& then at another time much later it is found without that fresh property ; then, this
fresh property being admitted into the idea of nature, we should exclude the substance
lacking this property from a nature of this kind, & should not give it that name. If now
a mass should be found, which had all the other enumerated properties of gold, but was
not dissolved by aqua rcgia, we should say that it was not gold. If at the beginning it was

P P

370 PHILOSOPHIC NATURALIS THEORIA

earn non esse aurum diceremus. Si initio compertum esset, alias ejusmodi massas solvi,
alias non solvi per aquam regiam, sed per alium liquorem, & utrumque in sequali fere earum
massarum numero notatum esset, putatum fuisset, binas esse auri species, quarum altera
alterius liquoris ope solveretur."

Haec ego ibi ; unde adhuc magis patet, quid specifics formae sint, & inde, quid sit
transformatio. Sed de his omnibus jam satis.

A THEORY OF NATURAL PHILOSOPHY 371

discovered that certain masses of the same sort were dissolved by aqua regia, but that others
were not, but were dissolved by another liquid ; & each of the two phenomena was
observed in an approximately equal number of masses ; then, it would be considered that
there were two sorts of gold, & that one sort was dissolved by one liquid, & the other by
the other."

Those are my words ; & from them it can be easily seen what specific forms are ;
& from that, what transformation is. But I have now said sufficient on the point.

[248] APPENDIX

AD META PHYSIC AM PERTINENS

DE ANIMA ET DEO

525' Quae Pertment a& discrimen animae a materia, & ad modum, quo anima in corput
sit addita.' 3git, rejecta Leibnitianorum harmonia praestabilita, persecutus jam sum in parte prima

a num. 153. Hie primum & id ipsum discrimen evolvam magis, & addam de ipsius animae,
& ejus actuum vi, ac natura, nonnulla, quae cum eodem operis argumento arctissime con-
nectuntur : turn ad eum colligendum, qui semper maximus esse debet omnium philosophic-
arum meditationum fructus, nimirum ad ipsum potentissimum, ac sapientissium Auctorem
Naturae conscendam.

Discrimen inter am- r26. Imprimis hie iterum patet, quantum discrimen sit inter corpus. & animam, ac

mam & corpus :in.-' . * .. . A . ....-? . , '

hoc omnia peragi inter ea, quae corporeae matenae tnbuimus, & quae in nostra spintuali substantia expenmur.
per distantias lo- j^j omnia perfecimus tantummodo per distantias locales, & motus, ac per vires, quae nihil

cales, motus, ac r . . , . i i i • i t

vires inducentes ahud sunt, nisi determmationes ad motus locales, sive ad mutandas, vel conservandas locales
motum locaiem. distantias certa lege necessaria, & a nulla materias ipsius libera determinatione pendentes.
Nee vero ullas ego repraesentativas vires in ipsa materia agnosco, quarum nomine haud
scio, an ii ipsi, qui utuntur, satis norint, quid intelligant, nee ullum aliud genus virium,
aut actionum ipsi tribuo, praster illud unum, quod respicit locaiem motum, & accessus
mutuos, ac recessus.

in anima nos 527. At in ea nostra substantia, qua vivimus, nos quidem intimo sensu, & reflexione,

expend sensationes, i i t» j • • o • i j« •

& cogitationes, ac duplex aliud operationum genus expenmur, & agnoscimus, quarum alterum dicimus
voiitiones: vim sensationem, alterum coeitationem, & volitionem. Profecto idea, quam de illis habemus

esse in nobis inna- .. - • i t T i • i 11 11*

tam, qua videamus mtimam, & prorsus cxperimcntalem, est longe diversa ab idea, quam habemus, locahs
nanim discrimina, & distantiae, & motus. Et quidem illud mihi, ut in prima parte innui, omnino persuasum

relationem quam . . . j • • j o -11 11

habent ad sutetan- est, inesse animis nostns vim quandam, qua ipsas nostras ideas, & illos, non locales, sed
tias, a quibus animasticos motus, quos in nobis ipsis inspicimus, intime cognoscamus, & non solum similes

procedunt essential- •,• • *vi • j« • r .....

fter diversas. • a dissimihbus possimus discerncrc, quod omnino facimus, cum post equi visi ideam, se
nobis idea piscis objicit, & hunc dicimus non esse equum ; vel cum in [249] primis prin-
cipiis ideas conformes affirmando conjungimus, difformes vero separamus negando ; verum
etiam ipsorum non localium motuum, & idearum naturam immediate videamus, atque
originem ; ut idcirco nobis evidenter constet per sese, alias oriri^in nobis a substantia aliqua
externa ipsi animo, & admodum discrepante ab ipso, utut etiam ipsi conjuncta, quam
corpus dicimus, alias earum occasione in ipso animo exurgere, atque enasci per longe aliam
vim : ac primi generis esse sensationes ipsas, & directas ideas, posterioris autem omne
reflexionum genus, judicia, discursus, ac voluntatis actus tam varies : qua interna evidentia,
& conscientia sua illi etiam, qui de corporum, de aliorum extra se objectorum existentia
dubitare vellent, ac idealismum, & egoismum affectant, coguntur vel inviti internum
ejusmodi ineptissimis dubitationibus assensum negare, & quotiescunque directe, & vero
etiam reflexe, ac serio cogitant, & loquuntur, aut agunt, ita agere, loqui, cogitare, ut alia
etiam extra se posita sibi similia, & spiritualia, & materialia entia agnoscant : neque enim
libros conscriberent, & ederent, & suam rationibus confirmare sententiam niterentur ;
nisi illis omnino persuasum esset, existere extra ipsos, qui, quae scripserint, & typis
vulgaverint, perlegant, qui eorum rationes voce expressas aure excipiant, & victi demum
se dedant.

Duo genera actuum

no^^pWs^icfmus! 52^' ^ vero ex motibus quibusdam localibus in nostro corpore factis per impulsum
sensationes, & cogi- ab externis corporibus, vel per se etiam eo modo, quo ab externis fierent, ac delatis ad
c^^as^ss^mus cerebrum (in eo enim alicubi videtur debere esse saltern praecipua sedes animae, ad quam
etiam sine corpore nimirum tot nervorum fibrae pertingunt idcirco, ut impulsiones propagatae, vel per succurn

37?

APPEND I X

RELATING TO METAPHYSICS
THE MIND AND GOD

525. What relates to the distinction between the mind & matter, & the manner in The theme of this
which the mind acts on the body, I have already investigated in the First part, from Art. r^o'rf'for adding
153 on, after rejecting the pre-established harmony of the followers of Leibniz. Here I will it.

first of all consider more fully this distinction ; & I will add something with regard to the
mind itself, the force of its actions, & its nature ; these are closely connected with the very
theme of this work. After that, I will proceed to consider that which always ought to be
the most profitable of all philosophical meditations, namely, the power & wisdom of the
Author of Nature.

526. Here, in the first place, it is clear how great a distinction there is between the Distinction be-
body & the mind, & between those things that we term corporeal matter & those which ^f^ty*. lathis
we feel in our spiritual substance. In Art. 153, we did everything by the sole means of everything is ac-
local distances & motions, & by forces that are nothing else but propensities to local motions, ^°™a3n1cse^ed b& 'mo-
or propensities to change, or preserve, local distances in accordance with a certain necessary tions, & forces mdu-
law ; & these do not depend on any free determination of the matter jtself. But I do not cins local motlons-
recognize any representative forces in matter itself — I do not know whether those, who

use the term, are really sure of what they mean by it — nor do I attribute to it any other
type of forces or actions besides that one which has to do with local motions & mutual
approach & recession.

527. But in this substance of ours, by which we live, we feel & recognize, by an inner in the mind we
sense & thought, another twofold class of operations ; one of which we call sensation, & *£ei ^sensations,
the other thought or will. Without any doubt, the idea which we have within us, which pose ; force is in-
is altogether the result of experience, of the former, is far different to that which we have ^which'we'see the
of local distance & motion. Indeed I am quite of the opinion, as I remarked in the First differences between
Part, that there is in our minds a certain force, by means of which we obtain full cognition ^^o^tha't^tey
of our very ideas & those non-local, but mental, motions that we observe in our own selves ; bear to essentially
& we can distinguish between like & unlike, as we assuredly do, when after the idea gta^ge^m

of a horse that has been seen there presents itself the idea of a fish, & we say that this is they proceed.

not a horse ; or when, in elementary principles, we join together affirmatively like ideas,

& separate unlike ideas with a negation. Indeed, we also see immediately the nature &

origin of these non-local motions & ideas. Hence, it is self-evident to us that some of them

arise through a substance external to the mind, & altogether different from it, but yet in

connection with it, which we call the body ; & that others take rise from direct encounter

with the mind itself, & spring from a far different force. We see that to the first class

belong sensations & direct ideas, & to the second all kinds of reflections, decisions, trains

of reasoning, & the numerous different acts of the will. By this internal evidence, & their own

consciousness, even those, who would like to doubt the existence of bodies, & other objects

external to themselves, & affect idealism & egoism, are forced to refuse, though unwillingly,

their inward assent to such very absurd doubts. As often as directly, or even reflectively

& seriously, they think, speak, or act, they are forced so to act, speak, or think, that they

recognize other entities situated external to themselves, which are like to themselves, both

spiritual & material. For, they would not write & publish books, or try to corroborate their

theory with arguments ; unless they were fully persuaded that, external to themselves,

there exist those who will read what they have written & published in printed form, &

those who will hear the reasons they have spoken, & at length acknowledge themselves

convinced.

528. Now, certain local motions in our body are engendered by impulse from external J^^"^ ^e ~..T.
bodies, or even self-produced by the manner in which they come from without, & these ceive in ourselves,
are carried to the brain. For in the brain, somewhere, it seems that the seat of the mind ^fought* 10orS 'wilt
must be situated ; & that is why so many nerve-fibres extend to it, so that the impulses which we can exer-
can be carried to it, propagated either by a volatile juice or by rigid fibres in all directions, ?^e even wlthou

373

374 PHILOSOPHIC NATURALIS THEORIA

volatilem, vel per rigidas fibras quaquaversus deferri possint, & inde imperium in universum
exerceri corpus) exurgunt motus quidam non locales in animo, nee vero liberi, & ideae
coloris, saporis, odoris, soni, & vero etiam doloris, qui oriuntur quidem ex motibus illis
localibus ; sed intima conscientia teste, qua ipsorum naturam, & originem intuemur, longe
aliud sunt, quam motus ipsi locales : sunt nimirum vitales actus, utut non liberi. Praeter
hos autem in nobis ipsis illud aliud etiam operationum genus perspicimus cogitandi, ac
volendi, quod alii & brutis itidem attribuunt, cum quibus illud primum operationum
genus commune nobis esse censent jam omnes, praeter Cartesianos paucos, Philosophi :
nam & Leibnitiani brutis ipsis animam tribuunt, quanquam non immediate agentem in
corpus : sed ex iis, qui ipsam cogitandi, & volendi vim brutis attribuunt, in iis agnoscunt
passim omnes, qui sapiunt, nostra inferiorem longe, & ita a materia pendentem, ut sine
ilia nee vivere possint, nee agere ; dum nostras animas etiam a corpore separatas credimus
posse eosdem seque cogitationis, & volitionis actus exercere.

Si ea brutis con- [250] 529. Porro ex his, qui cogitationem, & voluntatem brutis attribuunt, alii

hnj^rfectiora1tnltiis utli°Lue gcneri applicant nomen spiritus, sed distinguunt diversa spirituum genera, alii

essedebeant, & quid vocem spiritualis substantise tribuunt illis solis, quae cogitare, & velle possint etiam sine

"e s^rttus ullo nexu cum corpore & sine ulla materiae organica dispositione, & motu, qui necessarius

est brutis, ut vivant. Atque id quidem admodum facile revocari potest ad litem de nomine,

& ad ideam, quae affigatur huic voci spiritus, vel spiritualis, cujus vocis latina vis originaria

non nisi tenuem flatum significat : nee magna erit in vocum usurpatione difficultas ;

dummodo bene distinguantur a se invicem materia expers omni & sentiendi, & cogitandi,

ac volendi vi, a viventibus sensu praeditis ; & in viventibus ipsis anima immortalis, ac per

se ipsam etiam extra omne organicum corpus capax cogitationis, & voluntatis, a brutis

longe imperfectioribus, vel quia solum sentiendi vim habeant omnis cogitationis, &

voluntatis expertia, vel quia, si cogitent, & velint, longe imperfectiores habeant ejusmodi

operationes, ac dis^oluto per organici corporis corruptionem nexu cum ipso corpore,

prorsus dispereant.

inter c-0> Ceterum longe aliud profecto est & tenuitas lamellae, quae determinat hunc

motus, a quibus . JJ ... P r,. , n '. * .

idea excitatur, & potms, quam ilium coloratum radium ad renexionem, ut ad oculos nostros devemat, in
ideam ipsam: quo sensu adhibet coloris nomen vulgus, & opifices ; & dispositio punctorum componentium

quatuor acceptiones ^ . . . . .° . ' ff.. ' .£.. . r r .

vocis color. particulam lumims, quae certum ipsi conciliat reirangibilitatis gradum, certum in certis

circumstantiis intervallum vicium facilioris reflexionis, & facilioris transmissus, unde fit,
ut certam in oculi fibris impressionem faciat, in quo sensu nomen coloris adhibent Optici ;
& impressio ipsa facta in oculo, & propagata ad cerebrum, in quo sensu coloris nomen
Anatomici usurpare possunt ; & longe aliud quid, & diversum ab iis omnibus, ac ne analogum
quidem illis, saltern satis arcto analogiae, & omnimodae similitudinis genere, est idea ilia,
quae nobis excitatur in animo, & quam demum a prioribus illis localibus motibus determi-
natam intuemur in nobis ipsis, ac intima nostra conscientia, & animi vis, de cujus vera in
nobis ipsis existentia dubitare omnino non possumus, evidentissima voce admonent ea
de re, & certos nos reddunt.

Commercium ani- r^ porro commercium illud inter animam, & corpus, quod unionem appellamus,

mae cum corpore . fj. . ,. , . ,. , r

continere triaiegum tria habet inter se diversa legum genera, quarum bina sunt prorsus diversa ab ea etiam,
genera: quae sint quae habetur inter materiae puncta, tertium in aliquo genere convenit cum ipsa, sed ita

pnora duo. J1 ........ ,.r . v ? • • •+. T> •

longe in alns plunmis ab ea distat, ut a material! mechamsmo pemtus remotum sit. rnores
sunt in ordine ad motus locales organici nostri corporis, vel potius ejus partis, sive ea sit
fluidum quoddam tenuissimum, sive sint solidae fibrae ; & ad motus non locales, sed
animasticos nostri a-[25i]-nimi, nimirum ad excitationem idearum, & ad voluntatis actus.
Utroque legum genere ad quosdam motus corporis excitantur quidam animi actus, &
vice versa, & utrumque requirit inter cetera positionem certam in partibus corporis ad
se invicem, & certam animae positionem ad ipsas : ubi enim laesione quadam satis magna
organici corporis ea mutua positio partium turbatur, ejusmodi legum observantia cessat :
nee vero ea locum habere potest, si anima procul distet a corpore extra ipsum sita.

in aitero ex iis 532. Sunt autem ejusmodi legum duo genera : alterum genus est illud, cujus nexus

A^co^uTnecSsS- est necessarius, alterum, cujus nexus est liber : habemus enim & liberos, & necessaries
ius, in aitero liber : motus, & saepe fit, ut aliquis apoplexia ictus amittat omnem, saltern respectu aliquorum
exponuntur ambo. membrorum, facultatem liberi motus ; at necessarios, non eos tantum, qui ad nutritipnem
pertinent, & a sola machina pendent, sed & eos, quibus excitantur sensationes, retineat.

APPENDIX 375

& from it control can be exercised over the whole body. From these local motions there
arise certain non-local motions in the mind, that are not indeed free motions, such as the
ideas of colour, taste, smell, sound, & even grief, all of which indeed arise from such local
motions. But, on the evidence of our inner consciousness, by means of which we observe
their nature & origin, they are something far different to these local motions ; that is to
say, they are vital actions, although not voluntary. Besides these we also perceive in our
own selves that other kind of operations, those of thinking & willing. This kind some
people also attribute to brutes as well ; & all philosophers, except a few of the Cartesians,
already believe that the first kind of operations is common to the brutes & ourselves. The
followers of Leibniz attribute a mind even to the brutes, although one that does not act
directly on the body. But of those who attribute to the brutes the power of thinking &
willing, all those that have any understanding admit that in the brutes it is far inferior to
our own ; & so dependent on matter, that without it they cannot live or act ; while they
believe that our minds, even if separated from the body, are capable of exercising the same
acts of thought & will just as well.

529. Again, of those who attribute to brutes the power of thought & will, some apply if these powers are
to either class the term " spirit," but distinguish between two different kinds of spirits ; ^e ^rStls^they
others attribute the name of spiritual substance to those only that can think & will without must be much more
any connection with the body, & without any organic disposition of matter, & the motion

that is necessary to the brutes in order that they may live. This may quite easily be reduced
to a quarrel over a mere term, & the idea that is assigned to the word spirit, or spiritual,
of which the original Latin signification is merely " a tenuous breath." There will not be
any great difficulty over the use of the terms, so long as matter (which is devoid of all power
of feeling, thinking & willing) & living things possessed of feeling are carefully distinguished
from one another ; & also amongst living things, the immortal mind, &, on account of it,
in addition also every organic body capable of thinking & willing, from the far more imperfect
brutes ; either, because they have the power of feeling only, & are unable to think or will ;
or because, if they do think & will, they have these powers far more imperfectly, &, if the
connection with the body is destroyed by some corruption of the organic body, they perish
altogether.

530. Besides, there is certainly a very great difference between thinness of the plate, Distinction be-
which determines one coloured ray of light rather than another to be reflected, so that it *ween, .the m°tjon

,.,',. ° , , „ by which an idea

comes to the eyes, in which sense ordinary people & craftsmen use the term colour; & is excited & the idea
the disposition of the points forming a particle of light, to which corresponds a definite itself; four accepca-

" , .. ..... r .. . ° . r ° , '- . . . -r i r r tions of the term

degree of refrangibility, & in certain circumstances a definite interval between the fits of colour.
easier reflection & easier transmission, whence there arises the fact that it makes a definite
impression upon the nerves of the eyes, in which sense the term colour is used by investigators
in Optics ; & the impression itself that is made upon the eyes, & propagated to the brain,
in which sense anatomists may employ the term ; & something far different, & of a diverse
nature to all the foregoing, being not even analogous to them, or only with a kind of analogy,
& total similitude that is sufficiently close, is the idea itself, which is excited in our minds,
& which, determined at length by the former local motions, we perceive within ourselves ;
& our inner consciousness, & the force of the mind, concerning the existence of which within
us there cannot be the slightest doubt, warn us with no uncertain voice about the matter,
& make us acquainted with it.

;-?i. Now, the intercourse between the mind & the body, which we term union, has The intercourse of

i !• i c i i-n- r i „ f i i • V/T the mind with the

three kinds of laws different from one another ; & of these, two are also quite different body contains
also from that which obtains between points of matter ; while the third in some sort agrees three kinds of laws ;

..... -,.._ - . \ , .... , the nature of the

with it, but is so far different from it m very many other ways that it is altogether remote first two.
from any material mechanism. The two former are especially applicable to local motions,
of our organic bodies, or rather of part of them, whether that part consists of a very tenuous
fluid, or of solid fibres ; & to motions that are not local motions, but to mental motions
of our minds, such as the excitation of ideas, & acts of the will. According to each of these
laws, certain acts of the mind are transmitted to certain motions of the body, & vice versa ;
& each kind demands, amongst other things, a certain relative situation of parts of the body,
& a certain situation of the mind with regard to these parts. For, when this mutual
situation between the parts is sufficiently disturbed by a sufficiently great lesion of the
organic body, observance of these laws ceases ; nor indeed does it hold, if the mind is far
away from the body situated outside it. in one of these, the

532. Moreover, of such laws there are two kinds ; the one kind is that in which the connection between
connection is necessary, while in the other the connection is free. For, we have both j^dy is Of a
necessary & free motions ; & it often happens that one who is stricken with apoplexy loses necessary nature,
all power of free motions, at least with respect to some of his limbs ; while he retains the £ee ;° explanation
necessary motions, not only those which relate to nutrition, & depend solely upon a mechanism, °f each of them.

3/6 PHILOSOPHIC NATURALIS THEORIA

Unde apparet & illud, diversa esse instrumenta, quibus ad ea duo diversa motuum genera
utimur. Quanquam & in hoc secundo legum genere fieri posset, ut nexus ibi quidem
aliquis necessarius habeatur, sed non mutuus. Ut nimirum tota libertas nostra consistat
in excitandis actibus voluntatis, & eorum ope etiam ideis mentis, quibus semel libero
animastico motu intrinseco excitatis, per legem hujus secundi generis debeant illico certi
locales motus exoriri in ea corporis nostri parte, quae est primum instrumentum liberorum
motuum, nulli autem sint motus locales partis ullius nostri corporis, nullae ideae nostrae
mentis, qux animum certa lege determinent ad hunc potius, quam ilium voluntatis liberum
actum ; licet fieri possit, ut certa lege ad id inclinent, & actus alios aliis faciliores reddant,
manente tamen semper in animo, in ipsa ilia ejus facultate, quam dicimus voluntatem,
potestate liberrima eligendi illud etiam, contra quod inclinatur, & efficiendi, ut ex mera
sua determinatione praeponderet etiam illud, quod independenter ab ea minorem habet
vim. In eodem autem genere nexus quidam necessarii erunt itidem inter motus locales
corporis, ac ideas mentis, cum quibusdam indeliberatis animi affectionibus, quae leges,
quam multas sint, quam variae, & an singula genera ad unicam aliquam satis generalem
reduci possint, id vero nobis quidem saltern hue usque est penitus inaccessum.

533- Tertium legum genus convenit cum lege mutua punctorum in hoc genere, quod
nexu mutuo inter ad motum localem pertinet animae ipsius, ac certam ejus positionem ad corpus, & ad certam
mUIquo a^ecT^i'ui^ organorum dispositionerri. Durante nimirum dispositione, a qua pendet vita, anima
mum difierat. necessario debet mutare locum, dum locum mutat corpus, atque id ipsum quodam necessario

nexu, non libero : si enim praeceps gravitate sua corpus ruit, si ab alio repente impellitur,
si vehitur navi, si ex ipsius ani-[252]-mae voluntate progreditur, moveri utique cum ipso
debet necessario & anima, ac illam eandem respectivam sedem tenere, & corpus comitari
ubique. Dissolute autem eo nexu organicorum instrumentorum, abit illico, & a corpore,
jam suis inepto usibus, discedit. At in eo haec virium lex localem motum animae respiciens
plurimum differt a viribus materiae, quod nee in infinitum protenditur, sed ad certam
quandam satis exiguam distantiam, nee illam habet tantam reciprocationem determinationis
ad accessum, & recessum cum tot illis limitibus, vel saltern nullum earum rerum habemus
indicium. Fortasse nee in minimis distantiis a quovis materiae puncto determinationem
ullam habet ad recessum, cum potius ipsa compenetrari cum materia posse videatur : nam
ex phaenomenis nee illud certo colligi posse arbitror, an cum ullo materiae puncto com-
penetretur. Deinde nee hujusmodi vires habet perennes, & immutabiles, pereunt enim
destructa organizatione corporis, nee eas habet, cum suis similibus, nimirum cum aliis
animabus, cum quibus idcirco nee impenetrabilitatem habet, nee illos nexus cohaesionum,
ex quibus materiae sensibilitas oritur. Atque ex iis tarn multis discriminibus, & tarn
insignibus, satis luculenter patet, quam longe haec etiam lex pertinens ad unionem animae
cum corpore a materiali mechanismo distet, & penitus remota sit.

Ubisit sedesanimsE, 53^. Ut>i sit animae sedes, ex puris pbcenomenis certo nosse omnino non possumus : an

" nimirum ea sit praesens certo cuidam punctorum numero, & toti spatio intermedio habens
virtualem illam extensionem, quam num. 84 in primis materiae elementis rejecimus, an
compenetretur cum uno aliquo puncto materiae, cui unita secum ferat & necessaries illos,
& liberos nexus, ut vel illud punctum cum aliis etiam legibus agat in alia puncta quaedam,
vel ut, enatis certis quibusdam in eo motibus, caetera fiant per virium legem toti materiae
communem ; an ipsa existat in unico puncto spatii, quod a nullo materiae puncto occupetur,
& inde nexum habeat cum certis punctis, respectu quorum habeat omnes illas motuum
localium, & animasticorum leges, quas diximus ; id sane ex puris Nature phanomenis, &
vero etiam, ut arbitror, ex reflexione, & meditatione quavis, quae fiat circa ipsa phenomena,
nunquam nobis innotescet.

Demonstratur id
ipsum producendo,
quid oporteret nosse . . . . , .

ad resoivendam 535. Nam ad id determmandum ex phaenomenis utcunque consideratis, oporteret

- nosse, an ea phaenomena possint haberi eadem quovis ex iis modis, an potius requiratur
aliquis ex iis determinatus ut conjunctio, localis etiam, animae cum magna corporis parte,

APPENDIX 377

but also those by which sensations are excited. From which it is also clear that the
instruments which we employ to produce the two different kinds of motions must be
different. Also, although in the second kind of these laws it may happen that there is,
even in it, some sort of necessary connection, yet it is not a mutual connection. Thus,
the whole of our power of free action consists of the excitation of acts of the will, & by
means of these of ideas of the mind also ; once these have been excited by a free & intrinsic
motion of the mind, owing to a law of this second kind there must immediately arise certain
local motions in that part of the body which is the prime instrument of free motions ;
but there may be no motions of any part of the body, no motions of the mind, which
determine the mind to this rather than to that free act of the will. It may happen, possibly,
that by a certain law there is an inclination to one thing & that the motions produce some
acts more easily than others ; & yet, because there always remains in the mind & that
faculty of it which we call the will a perfectly free power of choosing even that thing against
which it is naturally inclined, there will even be a power of bringing it about that, due
merely to its own determination, the thing, which independently of this determination
would have the less force, will preponderate. However, in this same kind of law, there
will be also certain connections of the necessary type between the local motions of the body
& the ideas of the mind, together with some involuntary affections of the mind ; & how
many of these laws there may be, & how different they may be, & whether all the several
kinds can be reduced to a single law of fair generality, is indeed, at least up till now, quite
impossible to determine.

533. The third kind of law agrees with the mutual law of points in the fact that it The points in
pertains to local motion of the mind itself, to a definite position which it has with regard %££hot ^ a^£
to the body, & to the definite arrangement of the organs. Thus, while the arrangement with the mutual
persists, upon which life depends, the mind must of necessity change its position, as the tweenCtl<points b of
body changes its position, & that on account of some connection of the necessary type, matter: and those
& not a free connection. For, if the body rushes headlong through its own gravity, or UJ^I* £0£ most
is vigorously impelled by another, or if it is borne on a ship, or if it progresses through the

will of the mind itself, in every case the mind also must necessarily move along with the
body, & keep to its seat with respect to the body, & accompany the body everywhere. But
if this connection of the organic instruments is dissolved, straightway it goes off & leaves
the body which is now useless for its purposes. But this law of forces governing the
local motion of the mind differs greatly from the law of forces between points of matter
in this, that it does not extend to infinity, but only to a fairly small distance, & that it does
not contain that great alternation of propensity for approach & recession, going with
as many limit-points ; or at least we have no indication of these things. Perhaps too, even
at very small distances from any point of matter, it has no propensity for recession, since
it seems rather to have a power of compenetration with matter. For, I do not think that
it can with certainty be decided from phenomena, whether there is compenetration with
any point of matter or not. Secondly, it has no lasting & unvarying forces of this kind ;
for they are destroyed as soon as the organization of the body is destroyed ; nor are there
forces with things like itself, that is to say other minds, & so there can be no impenetrability
existing between them ; nor can there be those connections of cohesion from which the
sensibility of matter arises. From the number of these differences & special characteristics,
it is fairly evident how far even this law pertaining to the union of the mind with the body
differs from a material mechanism, & that it is something of quite a different nature.

534. We are quite unable to ascertain with any certainty from phenomena alone the it is not possible
position of the seat of the mind. That is to say, we cannot ascertain whether it is present f*™e toPdetemSn8
in any definite number of points, & has such a virtual extension through the whole of the the position of the
intermediate space, as, in Art. 84, we rejected in the case of the primary elements of sea °

matter. It cannot be ascertained whether it has compenetration with some one point of

matter, &, united with this, bears along with itself those necessary & free motions, so that

either this point acts on certain other points with even other laws, or so that, certain definite

motions being produced in this point, others take place on account of the law of forces

that is common to the whole of matter. It cannot be ascertained whether it exists in a

single point of space, which is unoccupied by any point of matter, & on that account has

a connection with certain definite points, with respect to which it has all those laws of

local & mental motions, of which we have spoken. We can never become acquainted

with any of these points from the phenomena of Nature alone certainly, & indeed, as I think,

neither can we by reflection or any consideration whatever, that may be made with regard xhis is proved by

to these phenomena. setting forth what

535. For, in order to determine it from any consideration of phenomena in any way. Wnown mV0rder to
it would be necessary to know whether these phenomena could happen in any of these obtain a solution of
ways, or rather some particular one of them is required, determined as a conjunction, also phenomena!"

378 PHILOSOPHIC NATURALIS THEORIA

vel etiam cum toto corpora. Ad id autem cognoscendum oporteret distinctam habere
notitiam earum legum, quas secum trahit conjunctio animae cum corpora, & totius
dispositionis punctorum omnium, quae corpus constituunt, ac legis virium mutuarum
inter materias puncta, turn etiam ha-[253]-bere tantam Geometriae vim, quanta opus est
ad determinancies omnes motus, qui ex sola mechanica distributione eorundem punctorum
oriri possint. lis omnibus opus esset ad videndum, an ex motibus, quos anima imperio
suae voluntatis, vel necessitate suae naturae induceret in unicum punctum, vel in aliqua
determinata puncta, consequi deinde possent per solam legem virium communem punctis
materiae omnes reliqui spirituum, & nervorum motus, qui habentur in motibus nostris
spontaneis, & omnes motus tot particularum corporis, ex quibus pendent secretiones,
nutritio, respiratio, ac alii nostri motus non liberi. At ilk omnia nobis incognita sunt,
nee ad illud adeo sublime Geometrise genus adspirare nobis licet, qui nondum penitus
determinare potuimus motus omnes trium etiam massularum, quae certis viribus in se
invicem agant.

•

Faisitas plurium 536. Fuerunt, qui animam concluserint intra certam aliquam exiguam corporis nostri

opmionum de ejus particulam, ut Cartesius intra glandulam pinealem : at deinde compertum est, ea parte

sede : non proban, ± '.. ° f . . . . . ^ ...... .

earn non extendi sola non contmeri : nam ea parte dempta, vita superfuit : sic sine pineali glandula aliquando
per totum corpus. vitam perdurasse, compertum jam est, ut animalia aliqua etiam sine cerebro vitam produx-
erunt. Alii diffusionem animse per totum corpus impugnant ex eo, quod aliquando
homines, rescissa etiam manu, dixerint, se digitorum dolorem sentire, tanquam si adhuc
haberent digitos ; qui dolor cum sentiatur absque eo quod anima ibi digitis sit praesens :
inde inferri posse arbitrantur, quotiescumque digitorum sentimus dolorem, illam sentiri
sine praesentia animas in digitis. At ea ratio nihil evincit : fieri enim posset, ut ad habendum
prima vice sensum, quern in digitorum dolore experimur, requireretur praesentia animae
in ipsis digitis, sine qua ejus doloris idea primo excitari non possit, possit autem efformata
semel per ejusmodi praesentiam excitari iterum sine ipsa per eos motus nervorum, qui
cum motu fibrarum digiti in primo illo sensu conjunct! fuerant : praeterquam quod adhuc
remanet definiendum illud, an ad nutritionem requiratur praesentis animae impulsus aliquis,
an ea per solum mechanismum obtineri possit tota sine ulla animae operatione.

Conclusio pro ignor- 537. Haec omnia abunde ostendunt, phaenomenis rite consultis nihil satis certo definiri

rn^topcwJit Sse.U°" posse circa animae sedem, nee ejus diffusionem per magnam aliquam corporis partem, vel
etiam per totum corpus excludi. Quod si vel per ingentem partem, vel etiam per totum
corpus protendatur, id ipsum etiam cum mea theoria optime conciliabitur. Poterit enim
anima per illam virtualem extensionem, de qua egimus a num. 83, existere in toto spatio,
quo continentur omnia puncta constituentia illam partem, vel totum corpus : atque eo
pacto adhuc magis in mea theoria differet anima a materia ; cum simplicia materiae elementa
non nisi in singulis spatii punctis existant singula singulis momentis temporis, anima autem
licet itidem sim-[2S4]-plex, adhuc tamen simul existet in punctis spatii infinitis^conjungens
cum unico momento temporis seriem continuam punctorum spatii, cui toti simul erit
praesens per illam extensionem virtualem, ut & Deus per infinitam Immensitatem suam
praesens est punctis infinitis spatii (& ille quidem omnibus omnino), sive in iis materia
sit, sive sint vacua.

Nunquamproduciab 538. Et haec quidem de sede animae : illud autem postremo loco addendum hie censeo

z™SerT°ir?mpartes & legibus omnibus constituentibus ejus conjunctionem cum corpore, quod est observa-

oppositas: quid inde tionibus conforme, quod diximus num. 74, & 387, nunquam ab anima produci motum in

consequatur. unQ j^gj.^ puncto, quin in alio aliquo sequalis motus in partem contrariam producatur,

unde fit, ut nee liberi, nee necessarii materiae motus ab animabus nostris orti perturbent

actionis, & reactionis aequalitatem, conservationem ejusdem status centri communis

gravitatis, & conservationem ejusdem quantitatis motus in Mundo in eandem plagam

computari.

Transitus ad Auc- 53^ Haec quidem de anima : jam quod pertinet ad ipsum Divinum Naturae Opificem,

cujusp^rfectionesTn in hac Theoria elucet maxime & necessitas ipsum omnino admittendi, & summa ipsius,
hac Theoria elucent atque infinita Potentia, Sapientia, Providentia, quae venerationem a nobis demississimam,

m^v-imp * * *

maxime.

APPENDIX 379

local, of the mind with a great part of the body, or even with the whole of the body. But
to know this, it would be necessary to have a clear knowledge of their laws, which conjunction
of the mind with the body necessitates , & also a knowledge of the entire disposition of
all the points constituting the body, & the laws for the mutual forces between points of
matter. In addition, there would be the necessity for as great geometrical powers, as
would be enough to determine all the motions, which might be produced merely on account
of the mechanical distribution of these points. All of these would be needed for
perceiving whether, from the motions, which the mind could induce, by the power of its
own will or the necessity of its nature, on a single point, or on certain given points, by
means of the single law of forces common to points of matter, there could follow all the other
motions of the spirits & nerves, such as take place in our voluntary motions ; as well as
all those different motions of particles of the body upon which depend secretions, nutrition,
respiration, & other motions of ours that are not voluntary. But all these are unknown
to us ; nor may we aspire to such a sublime kind of geometry, for as yet we cannot altogether
determine all the motions of even three little masses, which act upon one another with
forces that are known.

536. There have been some who would confine the mind to some very small portion of Fa-}s&y °f several
the body ; for instance, Descartes suggested the pineal gland. But, later, it was discovered seat'of^the mind":
that it could not be contained in that part alone ; for, if that part were removed, life still it cannot be proved
went on. It has been already discovered that life endured for some time without the extend throughout
pineal gland, just as some animals produced life even without a brain. Others argued *h^ whole °f the
against the diffusion of the mind throughout the whole of the body, from the fact that

sometimes men, after the hand had been cut off, said that they could still feel the pain in
the fingers, as if they still had fingers ; & smce this pain is felt, although in this case there is
not the fact that the mind is present in the fingers, they thought that it could be inferred
that, as often as we feel a pain in the fingers, we feel it without the presence of the
mind in the fingers. But such argument proves nothing at all ; for it might happen that,
in order that there should be in the first place that feeling, which we experience of pain
in the fingers, there were required the presence of the mind in the fingers, without which
it would be impossible that an idea of the pain could be excited in the first place ; but,
once this idea had been formed, it might be possible that it could once more be excited,
without the presence of the mind in the fingers, by the motions of the nerves, which had
been conjoined with a motion of the fibres of the finger when the pain was first felt. Be- .
sides, it still remains to be decided whether any impulse of a present mind is required
for nutrition, or whether this can be obtained wholly without any operation of the mind,
by means of a mere mechanism alone.

537. All these things show fully that nothing certain can be stated with regard to the Conclusion that the
seat of the mind from a due consideration of phenomena ; nor that its diffusion through- Unknown •* "he r'e
out any great part of the body, or even throughout the whole body, is excluded. But if & in what manner
it should extend throughout a great part, or even the whole, of the body, that also would

fit in excellently with my Theory. For, by means of such virtual extension as we discussed
in Art. 83, the mind might exist in the whole of the space containing all the points which
form that part of the body, or that form the whole body. With this idea, in my Theory,
the mind will differ still more from matter ; for the simple elements of matter cannot exist
except in single points of space at single instants of time, each to each, while the mind can
also be one-fold, & yet exist at one & the same time in an infinite number of points of space,
conjoining with a single instant of time a continuous series of points of space ; & to the
whole of this series it will at one & the same time be present owing to the virtual extension
it possesses ; just as God also, by means of His own infinite Immensity, is present in an
infinite number of points of space (& He indeed in His entirety in every single one),
whether they are occupied by matter, or whether they are empty.

538. These things indeed relate to the seat of the mind; but I think there should Motion can never
be added here in the last place, concerning all the laws governing its conjunction with the nTinPdr,°tnkssbyitthis
body, that which is in conformity with the observations that I made in Art. 74 & Art. equal in opposite
387 ; namely, that motion can never be produced by the mind in a point of matter, without

producing an equal motion in some other point in the opposite direction. Whence it comes
about that neither the necessary nor the free motions of matter produced by our minds can
disturb the equality of action & reaction, the conservation of the same state of the centre
of gravity, & the conservation of the same quantity of motion in the Universe, reckoned
in the same direction.

539. So much for the mind ; now, as regards the Divine Founder of Nature Himself, m
there shines forth very clearly in my Theory, not only the necessity of admitting His existence the perfections of
in every way, but also His excellent & infinite Power, Wisdom, & Foresight ; which demand ^hoic1le

from us the most humble veneration, along with a grateful heart, & loving affection. The Theory.

380

PHILOSOPHISE NATURALIS THEORIA

vanam sine re.

& simul gratum animum, atque amorem exposcant : ac vanissima illorum somnia corruunt
penitus, qui Mundum vel casu quodam fortuito putant, vel fatal! quadam necessitate
potuisse condi, vel per se ipsum existere ab aeterno suis necessariis legibus consistentem.
Error tnbuentium r ^O- j?t primo quidem quod ad casum pertinet, sic ratiocinantur : finiti terminorum

Munch onginem JT . -T . ^ n ~ . i • • • r- •

casui fortuito: numeri combmationes numero nmtas habent, combmationes autem per totam innmtam
casum esse vocem aeternitatem debent extitisse numero infinitae : etiamsi nomine combinationum assumamus

vanam cin*» m *

totam seriem pertinentem ad quotcunque millenos annos. Quamobrem in fortuita
atomorum agitatione, si omnia se aequaliter habuerint, ut in longa fortuitorum serie semper
accidit, debuit quaevis ex ipsis redire infinitis vicibus, adeoque infinities major est prob-
abilitas pro reditu hujus individuae combinationis, quam habemus, quocunque finite
numero vicium redeuntis mero casu, quam pro non reditu. Hi quidem inprimis in eo
errant, quod putent esse aliquid, quod in se ipso revera fortuitum sit ; cum omnia deter-
minatas habeant in Natura causas, ex quibus profluunt, & idcirco a nobis fortuita dicantur
quaedam, quia causas, a quibus eorum existentia determinatur, ignoramus.

Numerurn cpmbina- r jj ge(j eo omisso, falsissimum est, numerum combinationum esse finitum in terminis

tionum in terminis JT ,... . . i n/r i- ••

etiam numero finitis numero nnitis : si omnia, quae ad Mundi constitutionem necessana sunt, perpendantur.
esse infinitum : si Est nuidem finitus numerus combinationum, si nomine combinationis assumatur tantum-

nte omnia expen- , ^ , . , ... . . .. ' . . . . ... , .

dantur. modo ordo quidam, quo alii termini post alios jacent : nine ultro agnosco mud : si omnes

litterae, quae [255] Virgilii poema componunt, versentur temere in sacco aliquo, turn
extrahantur, &ordinentur omnes litterae, aliae post alias, atque ejusmodi operatic continuetur
in infinitum, redituram & ipsam combinationem Virgilianam numero vicium quenvis
determinatum numerum superante. At ad Mundi constitutionem habetur inprimis
dispositio punctorum materiae in spatio patente in longum, latum, & profundum : porro
rectas in uno piano sunt infinitae, plana in spatio sunt infinita, & pro quavis recta in quovis
piano infinita sunt curvarum genera, quae cum eadem ex dato puncto directione oriantur,
in quarum singularum classibus infinities plures sunt, quae per datum punctorum numerum
non transeant. Quare ubi seligenda sit curva, quae transeat per omnia materiae puncta,
jam habemus infinitum saltern ordinis tertii. Praeterea, determinata ejusmodi curva,
potest variari in infinitum distantia puncti cujusvis a sibi proximo : quamobrem numerus
dispositionum possibilium pro quovis puncto materiae adhuc ceteris manentibus est infinitus,
adeoque is numerus ex omnium mutationibus possibilibus est infinitus ordinis expositi a
numero punctorum aucto saltern ternario. Iterum velocitas, quam habet dato tempore
punctum quodvis, potest variari in infinitum, & directio motus potest variari in infinitum
ordinis secundi ob directiones infinitas in eodem piano, & plana infinita in spatio.
Quare cum constitutio Mundi, & sequentium phaenomenorum series pendeat ab ipsa velo-
citate, & directione motus ; numerus, qui exprimit gradum infiniti, ad quern assurgit
numerus casuum diversorum, debet multiplicari ter per numerum punctorum materise.

aeternitate.

Cujus ordinis infini- 54.2. Est igitur numerus casuum diversorum non finitus, sed infinitus ordinis expositi

aWsshni, ^irTinv a quarta potentia numeri punctorum aucta saltern ternario, atque id etiam determinata
mensum aitioris curva virium, quae potest itidem infinitis modis variari. Quamobrem numerus combina-
tionum relativarum ad Mundi constitutionem non est finitus pro dato quovis momento
temporis, sed infinitus ordinis altissimi, respectu infiniti ejus generis, cujus generis est
infinitum numeri punctorum spatii in recta quapiam, quae concipiatur utrinque in infinitum
producta. At huic infinite est analogum infinitum momentorum temporis in tota utraque
aetemitate, cum unicam dimensionem habeat tempus. Igitur numerus combinationum
est infinitus ordinis in immensum aitioris ordine infiniti momentorum temporis, adeoque
non solum non omnes combinationes non debent redire infinities : sed ratio numeri earum,
quae non redeunt, est infinita ordinis altissimi, quam nimirum exponit quarta potentia
numeri punctorum aucta saltern binario, vel, si libeat variare virium leges, saltern ternario.
Quamobrem ruit futile ejusmodi, atque inane argumentum.

in ipso immense 54.3. Sed hide etiam illud eruitur, in immenso isto com-[2S6]-binationum numero

mero'S immensuni infinities esse plures pro quovis genere combinationes inordinatas, quae exhibeant incertum
plures esse combina- chaos, & massam temere volitantium punctorum, quam quae exhibeant Mundum ordinatum,
quam orduTata?*^' & certis constantem perpetuis legibus. Sic ex. gr. ad efformandas particulas, quae constanter
suam formam retineant, requiritur collocatio in punctis illis, in quibus sunt limites, &

APPENDIX 381

truly groundless dreams of those, who think that the Universe could have been founded
either by some fortuitous chance or some necessity of fate, or that it existed of itself from
all eternity dependent on necessary laws of its own, all these must altogether come to nothing.

540. Now first of all, the argument that it is due to chance is as follows. The The error made by
combinations of a finite number of terms are finite in number; but the combinations Jha^ the° Universe
throughout the whole of infinite eternity must have been infinite in number, even if we was produced by
assume that what is understood by the name of combinations is the whole series pertaining f^^nce -^u^an
to so many thousands of years. Hence, in a fortuitous agitation of the atoms, if all cases empty phrase with-
happen equally, as is always the ca^e in a long series of fortuitous things, one of them is out a ^iP8^*0 cor"
bound to recur an infinite number of times in turn. Thus, the probability of the recurrence

of this individual combination, which we have, is infinitely more probable, in any finite
number of succeeding returns by mere chance, than of its non-recurrence. Here, first of
all, they err in the fact that they consider that there is anything that is in itself truly
fortuitous ; for, all things have definite causes in Nature, from which they arise ; & there-
fore some things are called by us fortuitous, simply because we are ignorant of the causes by
which their existence is determined.

541. But, leaving that out of account, it is quite false to say that the number of The number of
combinations from a finite number of terms is finite, if all things that are necessary to amongst 'terms' that
the constitution of the Universe are considered. The number of combinations is indeed are even finite in
finite, if by the term combination there is implied merely a certain order, in which jf'th^a

some of the terms follow the others. I readily acknowledge this much ; that, if all ly considered.

the letters that go to form a poem of Virgil are shaken haphazard in a bag, & then

taken out of It, & all the letters are set in order, one after the other, & this operation is

carried on indefinitely, that combination which formed the poem of Virgil will return

after a number of times, if this number is greater than some definite number. But, for

the constitution of the Universe, we have first of all the arrangement of the points of matter,

in a space that extends in length, breadth & depth ; further, there are an infinite number

of straight lines m any one plane, an infinite number of planes in space, & for any straight

line in any plane there are an infinite number of classes of curves, which will start from

a given point in the same direction as the straight line ; & in every one of these classes

there are infinitely more which do not pass through a given number of points. Hence,

when a curve has to be selected which shall pass through all points of matter, we now have

an infinity of at least the third order. Besides, after any curve has been chosen, the distance

of each point from the one next to it can be varied indefinitely ; hence the number of

possible arrangements for any one point of matter, while the rest remain fixed, is infinite.

Therefore it follows that the number derived from the possible changes m all of these things

is infinite, of the order determined by the number of points increased at least three times.

Again, the velocity which any point has at a given time can be varied indefinitely ; & the

direction of motion can be varied to an infinity of the second order, on account of the

infinity of directions in the same plane & the infinity of planes in space. Hence, since the

constitution of the Universe, & the series of consequent phenomena, depend on the velocity

& the direction of motion ; the number, which expresses the degree of infinity to which

the number of different cases mounts up, must be multiplied three times by the number

of points of matter.

542. Therefore the number of cases is not finite, but infinite of the order expressed The order of the
by the fourth power of the number of points increased threefold at least ; & that is so, even ceedfng'iy* "igh"
if there is a definite curve of forces which also can be varied in an infinity of ways. Hence immensely higher
the number of relative combinations necessary to the formation of the Universe is not instants* of" timer in
finite for any given instant of time ; but it is infinite, of an exceedingly high order with the whole of eter-
respect to an infinity of the kind to which belongs the infinity of the number of points mty'

of space in any straight line, which is conceived to be produced to infinity in both directions.
To this infinity the infinity of the instants in the whole of eternity past & present is analogous ;
for time has but one dimension. Hence, the number of combinations is infinite of an order
that is immensely higher than the order of the Infinity of instants of time ; & thus, not
only does it follow that not all the combinations are not bound to return an infinite number
of times, but the ratio even of those that do not return is infinite, of a very high order,
namely that which is expressed by the fourth power of the number of points increased
twofold at least, or threefold at least if we choose to vary the laws of forces. Hence, the
arguments of this sort that are brought forward are futile & worthless.

543. Moreover from this it also follows that, in this immense number of combinations, in this immense
there will be, for any kind, infinitely more irregular combinations, such as represent indefinite atio^even^here
chaos & a mass of points flying about haphazard, than there are of those that exhibit the are immensely
regular combinations of the Universe, which follow definite & everlasting laws. For instance, """j,^

in order to form particles which continually maintain their form, there is required their there are regular.

382 PHILOSOPHIC NATURALIS THEORIA

quorum numerus debet esse infinities minor, quam numerus punctorum sitorum extra
ipsos : nam intersectiones curvae cum axe debent fieri in certis punctis, & inter ipsa debent
intercedere segmenta axis continua, habentia puncta spatii infinita. Quamobrem nisi sit
aliquis, qui ex omnibus seque per se possibilibus seligat unam ex ordinatis ; infinities
probabilius est, infinitate ordinis admodum elevati, obventuram inordinatam combinationum
seriem, & chaos, non ordinatam, & Mundum, quern cernimus, & admiramur. Atque ad
vincendam determinate earn infinitam improbabilitatem, requiritur infinita vis Conditoris
Supremi seligentis unam ex iis infinitis.

Non determinari ab r A* Nec vero illud obiici potest, etiam hominem, qui statuam aliquam effingat,

homine individuum: ., . -/'T .. .,, • j- -j t -IT i • • c • 11-

sed eo determinante finita vi eligere illam individuam formam, quam illi dat, inter mfimtas, quas naben possunt.
intraiimites,adquos Nam imprimis ille earn individuam non eligit, sed determinat modo admodum confuso
reifquamTn defer- figuram quandam, & individua ilia oritur ex Naturae legibus, & Mundi constitutione ilia
minationem yinci individua, quam naturae Opifex Infinitus infinitam indeterminationem superans deter-

ab Ente in infimtum . , . * , . .,,. . , F , .. . .

Ubero. mmavit, per quam ab ejus voluntatis actu onuntur mi certi motus in ejus brachns, & ab

hisce motus instrumentorum. Quin etiam in genere idcirco tarn multi Philosophi
determinationem ad individuum, & determinationem ad omnes illos gradus, ad quos
cognitio creati determinantis non pertingit, rejecerunt in Deum infinita cognoscendi, &
discernendi vi praeditum, necessaria ad determinandum unum individuum casum ex
infinitis ad idem genus pertinentibus ; cum creatae mentis cognitio ad finitum tantummodo
graduum diversorum numerum distincte percipiendum extendi possit : sine ullo autem
determinante ex casibus infinitis, & quidem tanto infinitatis gradu, individuus unus prae
aliis per se, aut per fortuitam eventualitatem prodire omnino non potest.

Hunc ordinem non CAC, Sed nee dici potest, hunc ipsum ordinem necessarium esse, & aeternum ac per

posse dici per se , . . j . • , , . .r

necessarium: prima se subsistere, casu quovis sequentc determinate a proxime praecedente, & a lege vinum
impugnatio a nuUo intrinseca, & necessaria iis individuis punctis, & non aliis. Nam contra hoc ipsum miserum

nexu, qui videtur a • i • T • • j j j-fc -i

haberi inter distan- sane ettugmm quamplunma sunt, quae opponi possunt. Inprimis admodum dirncile est,
tiam, & vim, quae ut homo sibi serio persuadeat, hanc unam virium legem, quam habet hoc individuum

idcirco liberum de- , • • j- -j • r • 'Li o • • •

terminantem requi- punctum respectu hujus individui puncti, fuisse possibiJem, & necessanam, ut nimirum
"jnt- in hac individua [257] distantia se potius attrahant, quam repellant, & se attrahant tanta

potius attractione, quam alia. Nulla appaiet sane connexio inter distantiam tantam,
& tantam talis speciei vim, ut ibi non potuerit esse alia quaevis, & ut hanc potius, quam
aliam pro hisce punctis non selegerit arbitrium entis habentis infinitam determinativam
potentiam, vel pro hisce punctis id, si libeat, ex natura sua petentibus, non posuerit alia
puncta illam aliam petentia ex sua itidem natura.

Secunda a numero 54.6. Praeterea cum & infinitum, & infinite parvum in se determinatum, & in se tale,

qu^determinantem m creatis sit impossibile (quod de infinite in extensione demonstravi (') pluribus in locis,

voluntatem poscit. nee una tantum demonstratione, ut in dissertatione De Natura, & usu infinitorum, &

infinite -parvorum, ac in dissertatione adjecta meis Sectionum Conicarum Elementis, Element.

torn. 3) ; finitus est numerus punctorum materiae, vel saltern in communi etiam sententia

finita est materiae existentis massa, quae finitum spatium occupare debet, & non in infinitum

(t) En unam ex ejusmodi demonstrationibus. Sit in fig. 71 spatium
a C versus AE infinitum, & in eo angulus rectilineus ACE bifariam
sectus per rectum CD. Sit autem GH parallela CA, ques occurrat CD
in H, ac producatur ita, ut HF fiat dupla GH, ducaturque CF, y omnes
CA, CB, CD, CE in infinitum producantur. Inprimis totum spatium
infinitum ECD debet esse eequale infinite ACD : nam ob angulum ACE
bifariam sectum sibi invicem congruerent. Deinde triangulum HCF est
duplum HCG, ob FH duplum HG. Eodem pacto ductis aliis ghf ipsi
parallelis, hCf erit duplam hCg, adeoque & area FHhf dupla HGgh.
Quare W summa omnium FHhf dupla summte omnium HGgh, nimirum
tota area infinita BCD dupla infinite DCE, adeoque dupla ACD, nimi-
rum pars dupla totius, quod est absurdum. Porro absurdum oritur ab
ipsa infinitate, si enim sint arcus circulares GMI, gmi centra C ; sector
ICM erit aqualis GCM, y triangulum FCH duplum GCH. Donee
sumus in quantitatibus finitis, res bene procedit, qui a FCH non est pars
ICM, sicut BCD est pars ACD, nee MCG, tf HCG sunt unum, W
idem, ut DCE est unicum infinitum absolutum contentum cruribus CD,
CE. Absurdum oritur tantummodo, ubi sublatis prorsus limitibus, a
quibus oriuntur discrimina spatiorum inclusorum iisdem angulis ad C, sit
suppositio infiniti absoluti, qute contradictionem involvit.

APPENDIX 383

grouping together in those points in which there are limit-points ; & of these the number
must be infinitely less than the number of points situated without them. For the
intersections of the curve with the axis must take place in certain points ; & between these
points there must lie continuous segments of the axis, having on them an infinite number
of points of space. Hence, unless there were One to select, from among all the combinations
that are equally possible in themselves, one of the regular combinations, it would be infinitely
more probable, the infinity being of a very high order, that there would happen an irregular
series of combinations & chaos, rather than one that was regular, & such an Universe as we
see & wonder at. Then, to overcome definitely this infinite improbability, there would
be required the infinite power of a Supreme Founder selecting one from among those
infinite combinations.

544. Nor can the argument be raised that even man, when he fashions a statue, with The individual is
but a finite force selects that individual form which he gives to it, from among an infinite m°antefri™whenbit
number which are possible. For, first of all, the man does not select that individual form ; has been determined
he determines in a very confused way a certain shape, & that individual form arises from which1 maV-s'know0
the laws of Nature, & from that individual constitution of the Universe which the ledge attains, the
Infinite Founder of Nature, overcoming the infinite lack of determination, has determined ; termined* S "over-
through which, by an act of his will, arise those definite motions in the arms of the man, come by a Being
& from these the motions of his tools. Moreover, in general, on this account, so many £e° infinitely
philosophers have thrown back individual determination, & a determination for all those

stages to which the knowledge of a determining created thing cannot attain, upon a God
endowed with an infinite power of knowing & distinguishing, such as is necessary for the
task of determining one individual case from among an infinite number pertaining to the
same class. For the knowledge of a created mind can only be extended to perceiving
distinctly a finite number of different stages. But, unless there is someone to determine
it, one individual cannot of itself, or through fortuitous happening, possibly come forth in
preference to others, from among an infinite number of cases, let alone from an infinity of
such a high degree.

545. No more can it be said that this very regularity is necessary, everlasting, & self- This regularity
sustained, any one case following the one next before it & determined by it, & by a law of b^^ecessa^ in
forces that is intrinsic & necessary to those individual points & to no others. For against this itself ; first, be-
really worthless subterfuge there are very many arguments that can be brought forward, parent °fabsenceaof
First of all, it is very difficult to see how a man can seriously persuade himself that one any connection
particular law of forces, which one particular point has with regard to another particular ^force" th^atter
point, should be possible & necessary, so that, for instance, at one particular distance the therefore requires
points should attract one another rather than repel one another, & attract one another lio^ee determma"
with an attraction that is so much greater than that with which they attract others. In

truth, there is apparently no connection between so great a distance & so great a force of
such a sort, that there could not be any other in the circumstances ; & that the will of a
Being having infinite determinative power should not select one in particular rather than
another for these points ; or should not substitute, for these points that from their very
nature, if you like to say so, require the first, other points that also from their nature require
that other connection.

546. Besides, the infinite & the infinitely small, self-determined & such of themselves, Second argument
is impossible in created things ; as I proved concerning the infinite in extension (*) in fofce^number *of
several places, & with more than one proof, for instance, in the dissertation De Natura, points, which re-
y usu infinitorum, £f? infinite parvorum, & in a dissertation added to my Sectionum Conicarum j^g"^/1 determln~
Elementa, Elem. Vol. 3. It therefore follows that the number of points of matter is finite ;

or at least, even in the commonly accepted opinion, the mass of existing matter is finite ;

(t) Here is one of these -proofs. In Fig. 71, let the space from C in the direction of A, E be infinite ; y in
this space, let the rectilineal angle ACE be bisected by the straight line CD. Also let GH be parallel to CA, meeting
CD in H ; y let it be produced so that HF is double GH ; join CF, fcf let all the straight lines CA, CB, CD,
CE be produced to infinity. Now, first of all, the whole of the infinite space ECD must be equal to the infinite space
ACD ,- for, on account of the bisection of the angle ACE, they will be congruent with one another. Secondly, the
triangle HCF is double the triangle HCG, sinee FH is double HG. In the same way, if other parallels like ghf are
drawn, hCf will be double hCg ,- y thus the area FHhf will be double HGgh. Hence, the sum of all such areas as FHhf
will be double the sum of all such as HGgh ; that is to say, the whole of the infinite area BCD will be double the
infinite area DCE, y there f 01 e double ACD ; the part double the whole, which is impossible. Further, the impossibility
springs from the supposition of infinity ; for, if GMI, gmi are circular arcs whose centre is C, the sector ICM will
be equal to GCM, y the triangle FCH will be double GCH. So long as we are dealing with finite quantities, the
matter goes on quite correctly, because FCH is not a part of ICM, as BCD is a part of ACD, nor are MCG y
HCG one y the same, as DCE is the unique infinite absolute content of the arms CD, DE. The impossibility only
arises when, all limits being taken away, from which arise the differences between the spaces included by the same
angles at C, the supposition is made of absolute infinity, which involves the contradiction.

PHILOSOPHIC NATURALIS THEORIA

protendi. Porro cur hie sit potius numerus punctorum, haec potius massse quantitas in
Natura, quam alia ; nulla sane ratio esse potest, nisi arbitrium entis infinita determinativa
potentia praediti, & nemo sanus sibi facile serio persuadebit, in quodam determinate numero
punctorum haberi necessitatem existentiae potius, quam in alio quovis.

Tertiaabaetemitate, 547. Accedit illud, quod si Mundus cum hisce legibus fuisset ab aeterno ; extitissent

motusT^um^iinea ']am motus seterni, & lineae a singulis punctis descriptae debuissent fuisse jam in infinitum
necessario infinita : productae : nam in se ipsas non redeunt sine arbitrio entis infinitam improbabilitatem vincentis,
cum demonstraverim supra pluribus in locis infinities improbabilius esse, [258] aliquod
punctum redire aliquando ad locum, quem alio temporis momento occupaverit, quam
nullum redire unquam. Porro infinitum in extensione impossibile prorsus esse, ego quidem
demonstravi, uti monui, & ilia impossibilitas pertinere debet ad omne genus linearum,
quae in infinitum productae sint. Potest utique motus continuari in infinitum per
aeternitatem futuram, quia si aliquando coepit, nunquam habebitur momentum temporis,
in quo jam fuerit existentia infinitae lineae : secus vero, si per aeternitatem praecedentem
jam extiterit : nee in eo futuram aeternitatem cum prseterita prorsus analogam esse
censeo, ut illud indefinitum futurae non sit verum quoddam infinitum praeteritae. Quod
si linea infinita non fuerit, & quies est infinities adhuc improbabilior, quam regressus pro
unico temporis momento ad idem spatii punctum, ac multo magis aeterna quies : utique
nee motum habuit aeternum materia, nee existere potuit ab seterno, cum sine & quiete, &
motu existere non potuerit, adeoque creatione omnino, & Creatore fuit opus, qui idcirco
infinitam haberet effectivam potentiam, ut omnem creare posset materiam, ac infinitam
determinativam vim, ut libero arbitrio suo utens ex omnibus infinitis possibilibus momentis
totius aeternitatis in utramque partem indefinitae illud posset seligere individuum momentum,
in quo materiam crearet, ac ex omnibus infinitis illis possibilibus statibus, & quidem tarn
sublimi infinitatis gradu, seligere ilium individuum statum, complectentem unam ex illis
curvis per omnia puncta dato ordine accepta transeuntibus, ac in ea determinatas illas
distantias, ac determinatas motuum velocitates, & directiones.

Validissima ab im-
possibilitate seriei
infinitae terminorum,
in qua alii ab aliis
determinentur ad
existendum sine ex-
trinseco determin-
ants ; ea hie demon-
stratur.

548. Verum hisce omnibus etiam omissis, est illud a determinatione itidem necessaria
repetitum, in qua vis Theoria validissimum, sed adhuc magis in mea, in qua omnia phaenomena
pendent a curva virium, & inertiae vi. Nimirum materia licet ponatur ejusmodi, ut habeat
necessariam, & sibi essentialem vim inertiae, & virium activarum legem ; adhuc ut quovis
dato tempore posteriore habeat determinatum statum, quem habet, debet determinari
ad ipsum a statu praecedenti, qui si fuisset diversus, diversus esset & subsequens ; neque
enim lapis, qui sequent! tempore est in Tellure, ibi esset ; si immediate antecedent! fuisset
in Luna. Quare status ille, qui habetur tempore sequent!, nee a se ipso, nee a materia,
nee ab ullo ente materiali turn existente, habet determinationem ad existendum, & pro-
prietates, quas habet materia perennes, indifferentiam per se continent, nee ullam
determinationem inducunt. Determinationem igitur, quam habet ille status ad existendum,
accipit a statu praecedenti. Porro status praecedens non potest determinare sequentem,
nisi quatenus ipse determinate existit. Ipse autem nullam itidem in se habet determin-
ationem ad existendum, sed illam accipit a praecedente. [259] Ergo nihil habemus adhuc
in ipso secundum se considerato determinationis ad existendum pro postremo illo statu.
Quod de secundo diximus, dicendum de tertio praecedente, qui determinationem debet
accipere a quarto, adeoque in se nullam habet determinationem pro existentia sui, nee
idcirco ullam pro existentia postremi. Verum eodem pacto progrediendo in infinitum,
habemus infinitam seriem statuum, in quorum singulis habemus merum nihil in ordine
ad determinatam existentiam postremi status. Summa autem omnium nihilorum utcunque
numero infinitorum est nihil ; jam diu enim constitit, ilium Guidonis Grandi, utut summi
Geometrae, paralogismum fuisse, quo ex expressione seriei parallelae ortae per divisionem

intulit summam infinitorum zero esse revera aequalem dimidio. Non potest igitur

ilia series per se determinare existentiam cujuscunque certi sui termini, adeoque nee tota
ipsa potest determinate existere, nisi ab ente extra ipsam posito determinetur.

In quo hoc argu-

communi adhibente 549- Hoc quidem argumento jam ab annis multis uti soleo, quod & cum aliis pluribus

impossibiiitatem se- communicavi, neque id ab usitato argumento, quo reiicitur series contingentium infinita

riei contingentium . . * , • • ••••vj-rr j

sine eote necessario. sine ente extrinseco dante existentiam seriei toti, in alio dittert, quam in eo, quod a

APPENDIX 385

& this must occupy finite space & cannot extend indefinitely. Now, there is truly no
possible reason why there should be this finite number of points, or this quantity of matter
in Nature, rather than that ; except the will of a Being possessed of infinite determinative
power. No one in his right senses will easily persuade himself seriously that there is any
necessity for existence in any one number of points, rather than m any other.

547. In addition, if the Universe had gone on with these laws from eternity, then A third argument
already there would have been eternal motions, & straight lines described by the several n^d^n^wMch
points would already have been produced to infinity. For they do not re-enter themselves, motions have last-

except by the will of a Being who overcomes the infinite improbability ; since I have necesSu36 infinite13

proved above in several places that it is infinitely more improbable that any point should the impossibility of

return at some time to the same place as it had occupied at some other instant, than that thls-

no point should ever return. Moreover, I have proved that infinity in extension is quite

impossible, as I have already observed ; & this impossibility must pertain to every kind

of lines that have been produced indefinitely. Anyhow, the motion can be continued

indefinitely throughout future eternity ; for, if it commenced at any one instant there never

would be an instant of time, in which there has already been the existence of an infinite

line ; but otherwise, if it has already existed throughout past eternity. However, in this

connection, I do not think that future eternity is quite analogous with past eternity ; so

that this indefinite of the future is not really the same thing as an infinite of the past.

But if there has not been an infinite line (& absolute rest is still more infinitely improbable

than a return for a single instant to the same point of space, & eternal rest is even more

improbable still), then it certainly follows that matter cannot have had eternal motion, nor

can it have existed from eternity. For, it could not have existed without both rest &

motion ; & thus, there was altogether a need for creation, & a Creator, & therefore He

would have an infinite effective power, so that He could create all matter, & an infinite

determinative force ; so that, out of all the possible instants, infinite in number, in the

whole of eternity indefinitely prolonged in either direction, He could choose of His Own

untrammelled will that particular instant in which to create matter ; & out of all the

infinite number of possible states, & this to such a high degree of infinity, He could select

that one particular state, which involves one of those curves passing through all the points

taken in a certain order ; & in it could choose those determinate distances, & the determinate

velocities & directions of the motions.

548. But, leaving all these things out of the question, there is a very strong argument A very strong
in any Theory, derived also from a necessity for determination ; but especially strong in "o^the impoS
my Theory, where all phenomena depend on a curve of forces, & the force of inertia. Thus, biiity of an infinite
although matter may be assumed to be of such a nature as to have a necessary & essential *£^ °theerideter°
force of inertia & a law of active forces ; yet, in order that at any subsequent time it may minatkm to exist
have the determinate state, which it actually has, it must be determined to that state, from that^o^ano^her
the state just preceding ; & if this preceding state had been different, the subsequent state without something
would also have been different. For a stone, which at a subsequent instant is on the Earth, enc^from'witho'ut1"
would not have been there at the instant, if at the instant immediately preceding it had been proof of the impos-
on the Moon. Hence the state which occur? at the subsequent instant, neither of itself, slbUltv here &lven-
nor from matter, nor from any material entity then existing, has any determination to exist ;

& the properties, which matter has unvarying, contain of themselves indifference nor do
they lead to any determination. The determination, then, which that state has to exist, is
derived from the state preceding it. Further, a preceding state cannot determine the one
which follows it, except in so far as it itself has existed determinately. Moreover, this
preceding state also has no determination in itself to exist, but derives it from one that
precedes it. Consequently, we have as yet nothing in this, considered by itself, yielding
determination to exist for that last state. What has been said with regard to this second state,
is to be said also about the third preceding state ; this must receive its determination from a
fourth, & so in itself has no determination for its own existence, nor on that account has it
any for the existence of the last state. Now, going on indefinitely in the same manner, we
have an infinite series of states, in each of which we have absolutely nothing for the purpose
of determining the existence of the last state. Moreover, the sum of all these nothings, no
matter how infinite the number of them, is nothing also. For, it has been long ago made
clear that Guide Grandi, although a very eminent geometer, enunciated a fallacy when, from
an expression of a parallel series derived by division of I by (i + i), he deduced that the sum
of an infinite number of zeros was really equal to -J-. Therefore, that series of states T

, •! f • • i . .In what this

cannot determine the existence of any particular one of its terms, & so neither can the whole argument differs
of it exist determinately, unless it be determined by a Being situated without itself. *rom tj1.6 usual °°e'

T , iii- /• , T i • i • depending upon the

549. 1 have employed this argument for many years past, & I have communicated it impossibility of a
to several others ; it does not differ from the usual argument employed, which denies se.ri^s of events

.. ... . . ~ . . - . . . . , _r . - ' . . . without a necessary

the possibility of an innnite series of contingents without an outside Being giving existence being.

c c

386 PHILOSOPHIC NATURALIS THEORIA

contingentia res ad determinationem est translata, & a defectu determinationis pro sua
cujusque existentia res est translata ad defectum determinationis pro existentia unius
determinati status assumpti pro postremo ; id autem praestiti, ne eludatur argumentum
dicendo, in tota serie haberi determinationem ad ipsam totam, cum pro quovis termino
habeatur determinatio intra eandem seriem, nimirum in termino praecedente. Ilia
reductione ad vim determinativam existentiae postremi quaesitam per omnem seriem,
devenitur ad seriem nihilorum respectu ipsius quorum summa adhuc est nihilum.

m "habere 55°' Jam vero ^oc ens extrinsecum seriei ipsi, quod hanc seriem elegit prae seriebus

debet. aliis infinitis ejusdem generis, infinitam habere debet determinativam, & electivam vim,

ut unam illam ex infinitis seligat. Idem autem & cognitionem habere, debuit, &
sapientiam, ut hanc seriem ordinatam inter inordinatas selegerit : si enim sine cogni-
tione, & electione egisset, infinities probabilius fuisset, ab illo determinatum iri aliquam
ex inordinatis, quam unam ex ordinatis, ut hanc ; cum nimirum ratio inordinatarum ad
ordinatas sit infinita, & quidem ordinis altissimi, adeoque & excessus probabilitatis pro
cognitione, & sapientia, ac libera electione supra probabilitatem pro caeco agendi modo,
fatalismo, & necessitate, sit infinitus, qui idcirco certitudinem inducit.

brutas "Ser°hic 55 J< Atque n^c notandum & illud, pro quovis indivi-[26o]-duo statu respondente

occurrit,'a quo unp cuivis momento temporis, & multo magis pro quavis individua serie respondente cuivis

rum' !L°soio' infinite contmuo tempori, improbabilitas determinatae ipsius existentias est infinita, & nos

Ubero. deberemus esse certi de ejus non existentia, nisi determinaretur ab infinite determinantes

& nisi ejus determinationis notitiam nos haberemus. Sic si in urna sint nomina centum,

& unum, & agatur de uno determinate, an extractum inde prodierit, centuplo major est

improbabilitas ipsi contraria : si mille, & unum, millecupla : si numerus sit infinitus ;

improbabilitas erit infinita, quae in certitudinem transit : sed si quis viderit extrac-

tionem, & nobis nunciet ; tota improbabilitas ilia repente corruit. Verum & in hoc

exemplo individua ilia determinatio a create agente non habebitur inter infinitas possi-

biles, nisi ex legibus ab infinite determinante jam determinatis in Natura, & ab ejusdem

determinatione ad individuum, uti paullo ante dicebamus de individuae figurae electione

pro statua.

Quanta sapientia ^52. Porro qui aliquanto diligentius perpenderit vel ilia pauca, quae adnotavimus

opus fuerit ad -'-'.. ,. ~ ., . » , * ~ r IT r • i -1

seiigendum numer- necessaria in distnbutione punctoru-m ad efformanda diversa particularum genera, quae
um & ordinem exhibeant diversa corpora, videbit sane quanta sapientia, & potentia sit opus ad ea omnia

punctorum, & legem ..,.., r ,-> • i S « i • • T»II

virium. perspicienda, eligenda, praestanda. Quid vero, ubi cogitet, quanta altissimorum rroble-

matum indeterminatio occurrat in infinite illo combinationum possibilium numero, &
quanta cognitione opus fuerit ad eligendas illas potissimum, quae necessariae erant ad hanc
usque adeo inter se connexorum phaenomenorum seriem exhibendam ? Cogitet, quid
una lux praestare debeat, ut se propaget sine occursu, ut diversam pro diversis coloribus
refrangibilitatem habeat, & diversa vicium intervalla, ut calorem & igneas fermentationes
excitet : interea vero aptandus fuit corporum textus, & laminarum crassitudo ad ea
potissimum remittenda radiorum genera, quae illos determinates colores exhiberent sine
ceterarum & alterationum, & transformationum jactura, disponendse oculorum partes,
ut imago pingeretur in fundo, & propagaretur ad cerebrum, ac simul nutritioni daretur
locus, & alia ejusmodi praestanda sexcenta. Quid unus aer, qui simul pro sono, pro
respiratione, & vero etiam nutritione animalium, pro diurni caloris conservatione per
noctem, pro ventis ad navigationem, pro vaporibus continendis ad pluvias, pro innumeris
aliis usibus est conditus ? Quid gravitas, qua perennes fiunt planetarum motus, & comet-
arum, qua omnia compacta, & coadunata in ipsorum globis, qua una suis maria continentur
littoribus, & currunt fluvii, imber in terram decidit, & earn irrigat, ac foecundat, sua mole
aedificia consistunt, temporis mensuram exhibent pendulorum oscillationes ? [261] si ea
repente deficeret ; quo noster incessus, quo situs viscerum, quo aer ipse sua elasticitate
dissiliens ? Homo hominem arreptum a Tellure, & utcunque exigua impulsum vi, vel
uno etiam oris flatu impetitum, ab hominum omnium commercio in infinitum expelleret,
nunquam per totam aeternitatem rediturum.

Congeries e o r u m,
quae evincunt in eli-

fan^tiamentrovi- 553* ^ed <lu^ eg° nsec singularia persequor ? quanta Geometria opus fuit ad eas com-

dentiam, immensas,

APPENDIX 387

to the whole series, except in the detail that the matter is altered from a contingence to
a determination, £ from a lack of determination of the existence of any thing in itself the
question is transferred to a lack of determination for the existence of one determined state
assumed as the last of the series. But my argument is superior to the usual one, in that it
cannot be evaded by saying that there is in the whole series a determination to the series
as a whole ; since for any term there is a determination within the same series, namely
one derivable from the preceding term. By my reduction to a force determining existence
of the last term throughout the whole series, the result is a series of zeros with regard to
this last term, & the sum of these is still zero.

550. Now, the Being external to the series, which chooses this series in preference The necessary
to all others of the infinite number in the same class, must have infinite determinative & attributes of the
elective force, in order that He may select this one out of an infinite number. Also He

must have knowledge & wisdom, in order to select this regular series from among the
irregular series ; for, if He had acted without knowledge & selection, it would have been
infinitely more probable that there would have been a determination by Him of one of
the irregular series, than of one of the regular series, such as the one in question. For
the ratio of the number of irregular series to the number of regular serie? is infinite, & that
too of a very high order ; & thus, the excess of the probability in favour of knowledge,
wisdom, & arbitrary selection is infinitely greater than the probability in favour of blind
choice, fatalism. & necessity ; & this therefore leads to a certainty.

551. Here also it is to be observed that for any individual state corresponding: to any The sort of Being

J . t • o -L f • i f • who could over-

given instant 01 time, & much more tor any particular series corresponding to a given come the infinite

continuous time, the improbability of a self-determined existence is infinite ; & we ought improbability

i .,. A. ' , , -11 • « • . i • . which here occurs :

to be certain of its non-existence, unless it were determined by an infinite determmator, & it could be accom-

we had evidence of the determination. Thus, if in an urn there are a hundred & one names, P1131"^ OI^y . b/

& it is a question with regard to one determined name, whether it has been drawn from nitely free.

the urn, the improbability is a hundredfold to the contrary ; & if there were a thousand

& one names, a thousandfold ; if the number of names is infinite, the improbability

will be infinite ; & this passes into a certainty. But if anyone should have seen the drawing

& give us information, then the whole of the improbability would immediately be destroyed.

Again, in this example, the particular determination by a created agent will not be from

among an infinite number of possibles,except on account of laws already determined in Nature

by an infinite Determinator and from the determination to the individual by the same power ;

as I said, a little earlier, when speaking of the selection of a particular form for a statue.

552. Now, if anyone will consider a little more carefully even the few things I have H.°w great the

J.J , ' . , , i . , ', , ... °,.ff Wisdom would

mentioned as necessary in the arrangement of the points for the formation of the different have to be to select
kinds of particles, which different bodies exhibit, he must perceive how great the wisdom & the nuniber &
power must needs be, to comprehend, select & establish all these things. What then, pomts.'Tthe °aw of
when he considers how great an indeterminateness in problems of very high degree occurs forces-
through the infinite number of possible combinations ; & how great the knowledge would
have to be to select those of them especially, which were necessary to yield this series of
phenomena so far connected with one another ? Let him consider what properties the
single substance called light must exhibit, such that it is propagated without collision, that
it has different refrangibilities for different colours, & different intervals between its fits, that it
should excite heat, & fiery fermentations. At the same time the texture of bodies & the
thickness of plates had to be made suitable for the giving forth of those kinds of rays especially,
which were to exhibit determinate colours, without sacrificing other alterations and trans-
formations ; the arrangement of parts of the eyes, so that an image is depicted at the back
& propagated to the brain ; & at the same time place should be given to nutrition, & thou-
sands of other things of the same sort to be settled. What the properties of the single
substance called. air, which at one & the same time is suitable for sound, for breathing, even
for the nutrition of animals, for the preservation during the night of the heat received
during the day, for holding rain-clouds, & innumerable other uses. What those of gravity,
through which the motions of the planets & comets go on unchanged, through which all
things became compacted & united together within their spheres, through which each sea
is contained within its own bounds, & rivers flow, the rain falls upon the earth & irrigates
it, & fertilizes it, houses stand up owing to their own mass, & the oscillations of pendu-
lums yield the measure of time. Consider, if gravity were taken away suddenly, what would
become of our walking, of the arrangement of our viscera, of the air itself, which would fly
off in all directions through its own elasticity. A man could pick up another from the Groups of points.
Earth, & impel him with ever so slight a force, or even but blow upon him with his breath, which prove

o j • i • e • -1111 • •/-• conclusively the im-

& drive him from intercourse with all humanity away to infinity, nevermore to return measurabiiityofthe
throughout all eternity. power wisdom and

553. But why do I enumerate these separate things ? Consider how much geometry rmp

388 PHILSOPHI^ NATURALIS THEORIA

binationes inveniendas, quae tot organica nobis corpora exhiberent, tot arbores, & flores
educerent, tot brutis animantibus, & hominibus tarn multa vitae instrumenta submini-
starent ? Pro fronde unica efformanda quanta cognitione opus fuit, & providentia, ut
motus omnes per tot saecula perdurantes, & cum omnibus aliis motibus tarn arete connexi
illas individuas materiae particulas eo adducerent, ut illam demum, illo determinato tempore
frondem illius determinatae curvaturae producerent ? quid autem hoc ipsum respectu
eorum, ad quae nulli nostri sensus pervadunt, quae longissime supra telescopiorum, & infra
microscopiorum potestatem latent ? Quid respectu eorum, quae nulla possumus
contemplatione assequi, quorum nobis nullam omnino licet, ne levissimam quidem con-
jecturam adipisci, de quibus idcirco, ut phrasi utar, quam alibi ad aliquid ejusdem generis
exprimendum adhibui, de quibus inquam, hoc ipsum, ignorari ea a nobis, ignoramus ?
Ille profecto unus immensam Divini Creatoris potentiam, sapientiam, providentiam humanae
mentis captum omnem longissime superantes, ignorare potest, qui penitus mente cascutit,
vel sibi ipsi oculos eruit, & omnem mentis obtundit vim, qui Natura altissimis undique
inclamante vocibus aures occludit sibi, ne quid audiat, vel potius (nam occludere non est
satis) & cochleam, & tympanum, & quidquid ad auditum utcunque confert, ' proscindit,
dilacerat, eruit, ac a se longissime projectum amovet.

Quid prospiciendum CCA. Sed in hac tanta eligentis, ac omnia providentis Supremi Conditoris sapientia,

fuent pro nostra JJT • . j • • j i o • -11 f ji

existentia, & nostris atque exsequentis potentia, quam admirari debemus perpetuo, & veneran, mud adhuc
commodis: quantum magis cogitandum est nobis, quantum inde in nostros etiam usus promanarit, quos utique

ipsi inde simus ob- • -11 -j • o /* -i • • • • o

stricti. respexit me, qurvidet omnia, & fines sibi istos omnes constituit, qui per ea omnia & nostrae

ipsi existentiae viam stravit, ac nos prae infinitis aliis hominibus, qui existere utique poterant,
elegit ab ipso Mundi exordio, motus omnes, ad horum, quibus utimur, organorum forma-
tionem disposuit, praeter ea tarn multa quae ad tuendam, & conservandam hanc vitam,
ad tot commoda, & vero etiam voluptates conducerent. Nam illud omnino credendum
firmissime, non solum ea omnia vidisse unico intuitu Auctorem Naturae, sed omnes eos
animo sibi constitutes habuisse fines, ad quos conducunt media, quae videmus adhibita.

Mundum non esse err. Haud ego quidem Leibnitianis, & aliis quibuscunque [262] Optimismi defen-

possibilium optim- •, • •**» j i • • • o •

um, cum in possibi- sonbus assentior, qui Mundum nunc, in quo vivimus, & cujus pars sumus, omnium
litms nuiius terminus perfectissimum esse arbitrantur, ac Deum faciunt natura sua determinatum ad id creandum
officere sapiential qu°d perfectissimum sit, ac eo ordine, qui perfectissimus sit. Id sane nee fieri posse arbitror :
ac bonitati infinite, cum nimirum in quovis possibilium genere seriem agnoscam finitorum tantummodo,

quod non fecerit, nee . • ~ . r r, .. ^ • j* ^ ._•' j

potentia, quod non quanquam in mfimtum productam, ut num. 90 exposui, in qua, ut in distantns duorum
potuent id facere. punctorum nulla est minima, nulla maxima ; ita ibidem nulla sit perfectionis maximse,
nulla minimae, sed quavis finita perfectione utcunque magna, vel parva, sit alia perfectio
major, vel minor : unde fit, ut quancunque seligat Naturae Auctor, necessario debeat
alias majores omittere : nee vero ejus potentiae illud ofHcit, quod creare nonpossit optimum,
aut maximum, ut nee officit, quod non possit simul creare totum, quodcunque creare
potest : nam id eo evadit, ut non possit se in eum statum redigere, in quo nihil melius,
aut majus, vel absolute nihil aliud creare possit : nee officit aut sapientiae, aut bonitati
infinitae, quod optimum non seligat, ubi optimum est nullum.

Quam multa pessima cc6. £x ajia parte determinatio ilia ad optimum, & libertatem Divinam tollit, &

consectaria secum.. • • • r

trahat sententia contmgentiam rerum omnium, cum, quae existunt, necessana riant, quae non existunt,
Mundi perfectissimi. evadant impossibilia ; ac praeterea nobis quodammodo in ilia hypothesi debemus, quod
existimus, non illi. Qui enim potuit non existere id, quod habuit pro sua existentia
rationem praevalentem, quam Naturae Auctor cum viderit, non potuerit non sequi, nee
vero potuerit non videre ? Qui existere potuit id, quod eandem habuit non existendi
necessitatem ? Quid vero illi pro nostra existentia debeamus, qui nos condidit idcirco,
quia in nobis invenit meritum majus, quam in iis, quos omisit, & a sua ipsius natura necessario
determinatus fuit, & adactus ad obsequendum ipsi huic nostro intrinseco, & essential!
merito praevalenti ? Distinguendum est inter hsec duo : unum esse alio melius, & esse
melius creare potius unum, quam aliud. Illud primum habetur ubique, hoc secundum
nusquam, sed aeque bonum est creare, vel non creare quodcunque, quod physicam bonitatem
quancunque habeat, utcunque majorem, vel minorem alio quovis omisso : solum enim

APPENDIX 389

was needed to discover those combinations which were to display to us so many
organic bodies, produce so many trees & flowers, & supply so many instruments
of life to living brutes & men. For the formation of a single leaf, how great
was the need for knowledge & foresight, in order that all those motions, lasting
for so many ages, & so closely connected with all other motions, should so bring
together those particular particles of matter, that at length, at a certain deter-
minate time, they should produce that leaf with that determinate curvature. What
is this in comparison with those things to which none of our senses can penetrate, things
that lie hidden far & away beyond the power of telescopes, & too small lor the
microscope ? What of those which we can never understand no matter how hard we think
about them, of which we can never attain not even the slightest idea ; concerning which
therefore, to use a phrase I have elsewhere employed to express something of the same sort,
of which I say this : — " We do not know the very fact of our ignorance. " Undoubtedly
he alone can be ignorant of the immeasurable power, wisdom & foresight of the Divine
Creator, far surpassing all comprehension of the human intellect, whose mind is
altogether blind, or who tears out his eyes, & dulls every mental power, who shuts
his ears to Nature, so that he shall not hear her as she proclaims in accents loud
on every side, or rather (for to shut them is not enough) cuts away, tears up & destroys,
& hurls far from him the cochlea & the tympanum & anything else that helps him to hear.

554. But, in this great wisdom of selection & universal foresight on the part of the HOW our existence
Supreme Founder, & the power of carrying it out, there i« still another thing for us to and our c°nveni-

r . , , , , / 111-1 i ences would have

consider ; namely, how much proceeds from it to meet the needs of us, who are all under to be provided
the care of Him Who sees all things, & has imposed on Himself the accomplishment of all for; what a debt

\Tn i ill i r » -11 11 n r i we are therefore

those purposes ; Who has smoothed the path of our existence with them all, & from the under to Him.
commencement of the Universe has chosen us in preference to an infinite number of other
human beings that might have existed ; Who has planned all the motions nece>sary for the
formation of the organs we employ, besides all the many things that should conduce towards
the protection & preservation of this life, to its many conveniences, nay, even to its pleasures.
For, it cannot be but a matter of the firmeat belief, not only that the Author of Nature saw
all these things with a single intuition, but also that He had settled in his mind all
those purposes, to which the means that we see employed conduce.

555. I do not indeed agree with the followers of Leibniz, or with any of the upholders The Universe is not
of Optimism, who consider that this Universe, in which we live & of which we are part, 'the . best toi a11
is the most perfect of all ; & who thus make God determined by His own nature for the amongst e possibles
creation of that which is the most perfect, & in that order which is the most perfect. In there is no . last
truth, I think that such a thing would be impossible ; for, I recognize, in any kind of argument* Eigauist
possibles, a series of finites only, although prolonged to infinity, as I explained in Art. 90 ; infinite wisdom &
& in this series, just as in the case of the distances between two points, there is no greatest He'dkTn'ot mTue^t
or least, here also there is no case of greatest or of least perfection ; but, for any finite so : nor against His
perfection, however great or small, there is another perfection that is greater or smaller. HeWwas unaWe^to
Hence it comes about that, whatever the Author of Nature should select, He would have make i<: so-

to omit some that were of greater perfection. But, neither is it an argument against His
power, that He cannot create the best or the greatest ; nor similarly is it an argument
against His power that, whatever He could create, He could not create it as a whole at one &
the same time. For, it would come to this, that He would put Himself in the position
where He could create nothing better, nothing greater, or absolutely nothing else. Similarly,
it is no argument against His infinite wisdom & goodness, that He did not select the
best, when there is no best.

556. On the other hand, that determination for the best takes away altogether the The number of
freedom of God, & the contingency of all things ; for, those things which exist become Fosi? imperfections

0 i , -, i .. , ¥» « i i , ° i . involved in the idea

necessary, & those that do not are impossible. Besides, on that hypothesis, we should be Of a most perfect
under some sort of obligation to ourselves, & not to Him, for the fact that we exist. For Universe-
how was it possible that a thing should not exist, which had a powerful reason for its exist-
ence ; for, when the Author of Nature saw this reason, He could not fail to follow it,
nor indeed could He fail to see it ? How could a thing exist which had a like need for non-
existence ? For what should we have to thank Him, if He had created us for the simple
reason that in us He found a greater merit than in those whom He omitted, if He was
necessarily determined by His own nature, & driven by it to submit to our mere
intrinsic & essential overpowering merit ? We must mark the distinction between the
two dictums : — (i) this thing is better than that, (2) it would be better to create this
thing than to create that. There is a possibility of the first in all cases, but never any of
the second. It is an equally good thing to create or not to create anything whatever,
which has any physical goodness, however much greater or less than anything else which
has been omitted. The exercise of Divine freedom alone is infinitely more perfect than

390 PHILOSOPHIC NATURALIS THEORIA

Divinse libertatis exercitium infinities perfectius est quavis perfectione creata, quae idcirco
nullum potest offerre Divinae libertati meritum determinativum ad se creandum.

Media tamen idonea 557. Cum ea infinita libertate Divina componitur tamen illud, quod ad sapientiam

- pertinet, ut ad eos fines, quos sibi pro liberrimo suo arbitrio praefixit Deus, media semper
ad fines sibi apta debeat seligerc, quae finem propositum frustrari non sinant. Porro haec media etiam
1" m nostrum bonum selegit plurima, dum totam Naturam conderet, quod quern a nobis
exigat beneficiorum memorem, & gratum animum, quern etiam tan-[263]-tae beneficentiae
respondentem amorem cum ingenti ilia admiratione, & veneratione conjunctum, nemo
non videt.
Deduci nos inde ad rrg> Superest & illud innuendum. neminem sanae mentis hominem dubitare posse,

revelationem, quae . JJ . r . ,. . , . ,. . r . .

tamen hue non quin, qui tantam in ordmanda JNatura providentiam ostendit, tantam erga nos in nobis
pertineat, ad opus seligendis, in consulendo nostris & indigentiis. & commodis beneficentiam, illud etiam

mmirum pure philo- , . ...... °. , .

sophicum. prasstare voluerit, ut cum adeo imbecilla sit, & nebes mens nostra, & ad ipsius cogmtionem

per sese vix quidquam possit, se ipse nobis per aliquam revelationem voluerit multo uberius
praebere cognoscendum, colendum, amandum ; quo ubi devenerimus, quae inter tarn
multas falso jactatas absurdissimas revelationes unica vera sit perspiciemus utique admodum
facile. Sed ea jam Philosophise Naturalis fines excedunt, cujus in hoc opere Theoriam
meam exposui, & ex qua uberes hosce, & solidos demum fructus percepi.

APPENDIX 391

any perfection created ; & the latter can therefore offer no determinative merit to the
freedom of God in favour of its own creation.

557. With this infinite Divine liberty is bound up all that relates to wisdom; for, Fit means, however,
God, to those purposes which he of His own unfettered will had designed, was always be^ei^cted^^the
bound to select suitable means, such as would not allow these purposes to be frustrated. Author of Nature
Further, He has selected these means for the most part suitable for our welfare, whilst he ^J^the purposes
founded the whole of Nature ; & this demands from us a remembrance of His favours & a He has designed
thankful heart, nay, even a love that shall correspond to such great beneficence together

with a mighty wonder & admiration, as every one will see. Him.

558. It now remains but to mention that there is no man of sound mind who could we are thus led to
possibly doubt that One, Who has shown such great foresight in the arrangement of Nature, revelation, which

, ' . . , « i i • r i i i however does not

such great beneficence towards us in selecting us, & in looking after both our needs & our come within the
comforts, would not also wish to accomplish this also ; namely that, since our mind is so scop.e °.f, .such. *

',,,,. , , . f. /. _.. work as this, which

weak & dull that it can scarcely of itself arrive at any sort of knowledge about Him, He is purely phiio-

would have wished to present Himself to us through some kind of revelation much more fully

to be known, honoured & loved. This being done, we should indeed quite easily perceive

which was the only true one, from amongst so many of those absurdities falsely brought

forward as revelations. But such things as this already exceed the scope of a Natural

Philosophy, of which in this work I have explained my Theory, & from which I have finally

gathered such ripe & solid fruit.

[264] SUPPLEMENT A

§1
De Spatio, ac Tempore

Argumentum: quae i. Ego materiae extensionem prorsus continuam non admitto, sed earn constituo

spatu attributa. punctis prorsus indivisibilibus, & inextensis a se invicem disjunctis aliquo intervallo, &
connexis per vires quasdam jam attractivas, jam repulsivas pendentes a mutuis ipsorum
distantiis. Videndum hie, quid mihi sit in hac sententia spatium, ac tempus, quomodo
utrumque dici possit continuum, divisibile in infinitum, aeternum, immensum, immobile,
necessarium, licet neutrum, ut in ipsa adnotatione ostendi, suam habeat naturam realem
ejusmodi proprietatibus prasditam.

2- InPrimis i^ud mmi videtur evidens, tarn eos, qui spatium admittunt absolutum,
reaies modos exist- natura sua reali, continuum, sternum, immensum, tarn eos, qui cum Leibnitianis, &
tem" Cartesianis ponunt spatium ipsum in ordine, quern habent inter se res, quae existunt, praeter
ipsas res, quae existunt, debere admittere modum aliquem non pure imaginarium, sed realem
existendi, per quem ibi sint, ubi sunt, & qui existat turn, cum ibi sunt, pereat cum ibi esse
desierint, ubi erant. Nam admissso etiam in prima sententia spatio illo, si hoc, quod est
esse rem aliquam in ea parte spatii, haberetur tantummodo per rem, & spatium ; quoties-
cunque existeret res, & spatium, haberetur hoc, quod est rem illam in ea spatii parte collocari.
Rursus si in posteriore sententia ordo ille, qui locum constituit, haberetur per ipsas tantum-
modo res, quae ordinem ilium habent, quotiescunque res illae existerent, eodem semper
existerent ordine illo, nee proinde unquam locum mutarent. Atque id, quod de loco
dixi, dicendum pariter de tempore.

Quocunque is modus 3. Necessario igitur admittendus est realis aliquis existendi modus, per quem res est

nomine appeiietur. jj^ u^- es^ £ tunij cum esti gjve js mO(jus dicatur res, sive modus rei, sive aliquid, sive

nonnihil ; is extra nostram imaginationem esse debet, & res ipsum mutare potest, habens
jam alium ejusmodi existendi modum, jam alium.

Modi reaies, qui sint ^ ]?gO igitur pro singulis materise punctis, ut de his [265] loquar, e quibus ad res etiam

tempus? a m' immateriales eadem omnia facile transferri possunt, admitto bina realia modorum existendi

genera, quorum alii ad locum pertineant, alii ad tempus, & illi locales, hi dicantur tem-

porarii. Quodlibet punctum habet modum realem existendi, per quem est ibi, ubi est,

& alium, per quem est turn, cum est. Hi reaies existendi modi sunt mihi reale tempus,

& spatium : horum possibilitas a nobis indefinite cognita est mihi spatium vacuum, &

tempus itidem, ut ita dicam, vacuum, sive etiam spatium imaginarium, & tempus

imaginarium.

Eorum natura, & c Modi illi reaies singuli & oriuntur, ac pereunt, & indivisibiles prorsus mihi sunt, ac

relationes. .•>... °. ,. . T. „ , «•

inextensi, & immobiles, ac in suo ordine immutabiles. Ii & sua ipsorum loca sunt realia,
ac tempora, & punctorum, ad quae pertinent. Fundamentum praebent realis relationis
distantiae, sive localis inter duo puncta, sive temporariae inter duos eventus. Nee aliud
est in se, quod illam determinatam distantiam habeant ilia duo materiae puncta, quam quod
illos determinatos habeant existendi modos, quos necessario mutent, ubi earn mutent
distantiam. Eos modos, qui in ordine ad locum sunt, dico puncta loci realia, qui in ordine
ad tempus, momenta, quae partibus carent singula, ac omni ilia quidem extensione, haec
duratione, utraque divisibilitate destituuntur.

Contiguitas puncto- 6. pOrro punctum materiae prorsus indivisibile, & inextensum, alteri puncto materiae

Spa " contiguum esse non potest : si nullam habent distantiam ; prorsus coeunt : si non coeunt

penitus ; distantiam aliquam habent. Neque enim, cum nullum habeant partium genus,

(a) Hie, fcsf sequens paragraphus habentur in Suppkmtntis tomi I. Philosophic Retentions Benedicti Stay, § 6, W 7.

392

SUPPLEMENTS

§1

Of Space and Time (a)

1. I do not admit perfectly continuous extension of matter ; I consider it to be made The theme; what
up of perfectly indivisible points, which are non-extended, set apart from one another by oiespa«: ?attrlbutes
a certain interval, & connected together by certain forces that are at one time attractive &

at another time repulsive, depending on their mutual distances. Here it is to be seen,
with this theory, what is my idea of space, & of time, how each of them may be said to be
continuous, infinitely divisible, eternal, immense, immovable, necessary, although neither
of them, as I have shown in a note, have a real nature of their own that is possessed of these
properties.

2. First of all it seems clear to me that not only those who admit absolute space, which Real local and
is of its own real nature continuous, eternal & immense, but also those who, following Leibniz existence "m^tf °f
& Descartes, consider space itself to be the relative arrangement which exists amongst necessity be ad-
things that exist, over and above these existent things ; it seems to me, I say, that all must ^"ed by eVWy
admit some mode of existence that is real & not purely imaginary ; through which they are

where they are, & this mode exists when they are there, & perishes when they cease to be
where they were. For, such a space being admitted in the first theory, if the fact that
there Is some thing in that part of space depends on the thing & space alone ; then, as often
as the thing existed, & space, we should have the fact that that thing was situated in that
part of space. Again, if, in the second theory, the arrangement, which constitutes position,
depended only on the things themselves that have that arrangement ; then, as often as
these things should exist, they would exist in the same arrangement, & could never change
their position. What I have said with regard to space applies equally to time.

3. Therefore it needs must be admitted that there is some real mode of existence, due The name by which
to which a thing is where it is, & exists then, when it does exist. Whether this mode is is immaterial,
called the thing, or the mode of the thing, or something or nothing, it is bound to be

beyond our imagination ; & the thing may change this kind of mode, having one mode at
one time & another at another time.

4. Hence, for each of the points of matter (to consider these, from which all I say Real modes ; what
can be easily transferred to immaterial things), I admit two real kinds of modes of existence, [fme ^ay be.

of which some pertain to space & others to time ; & these will be called local & temporal
modes respectively. Any point has a real mode of existence, through which it is where it
is ; & another, due to which it exists at the time when it does exist. These real modes
of existence are to me real time & space ; the possibility of these modes, hazily apprehended
by us, is, to my mind, empty space & again empty time, so to speak ; in other words, imagin-
ary space & imaginary time.

5. These several real modes are produced & perish, and are in my opinion quite Their nature &
indivisible, non-extended, immovable & unvarying in their order. They, as well as the relatlons-
positions & times of them, & of the points to which they belong, are real. They

afford the foundation of a real relation of distance, which is either a local relation between
two points, or a temporal relation between two events. Nor is the fact that those two
points of matter have that determinated distance anything essentially different from the
fact that they have those determinated modes of existence, which necessarily alter when
they change the distance. Those mode? which are descriptive of position I call real
points of position ; & those that are descriptive of time I call instants ; & they are without
parts, & the former lack any kind of extension, while the latter lack duration ; both are
indivisible.

6. Further, a point of matter that is perfectly indivisible & non-extended cannot be Contiguity of
contiguous to any other point of matter ; if they have no distance from one another, they

coincide completely ; if they do not coincide completely, they have some distance between

(a) This W the fallowing section are to be found in the Philosophise Recentior, by Benedict Stay, Vol. I, § 6, 7.

393

394 PHILOSOPHIC NATURALIS THEORIA

possunt ex parte coire tantummodo, & ex parte altera se contingere, ex altera mutuo
aversari. Praejudicium est quoddam ab infantia, & ideis ortum per sensus acquisitis, a
debita reflexione destitutis, qui nimirum nobis massas semper ex partibus a se invicem
distantibus compositas exhibuerunt, cum videmur nobis puncta etiam invisibilia, & inextensa
posse punctis adjungere ita, ut se contingant, & oblongam quandam seriem constituant.
Globules re ipsa nobis confingimus, nee abstrahimus animum ab extensione ilia, & partibus,
quas voce, & ore secludimus.

Posse binis punctis 7. Porro ubi bina materiae puncta a se invicem distant, semper aliud materiae punctum

aaai alia in directum n • • j- i i IT-

ad distantias squa- potest collocan m dnectum ultra utrumque ad eandem distantiam, & alterum ultra hoc,
^ & *ta Porro> ut Patet> sme ullo fine. Potest itidem inter utrumque collocari in medio aliud
punctum, quod neutrum continget : si enim alterum contigeret, utrumque contingeret,
adeoque cum utroque congrueret, & ilia etiam congruerent, non distarent, contra hypo-
thesim. Dividi igitur poterit illud intervallum in partes duas, ac eodem argumento ilia
itidem duo in alias quatuor, & ita porro sine ullo fine. Quamobrem, utcunque ingens
fuerit binorum punctorum intervallum, semper [266] aliud haberi poterit majus, utcunque
id fuerit parvum, semper aliud haberi poterit minus, sine ullo limite, & fine.

8> Hmc u^tra' & inter kma *oc* Puncta realia quaecunque alia loci puncta realia possibilia
finita. numero, & in sunt, quas ab iis recedant, vel ad ipsa accedant sine ullo limite determinato, & divisibilitas realis
pOTsibii^u^nuUum ^nterva^i inter duo puncta in infinitum est, ut ita dicam, interseribilitas punctorum realium
finem. sine ullo fine. Quotiescunque ilia puncta loci realia inteiposita fuerint, interpositis punctis

materiae realibus, finitus erit eorum numerus, finitus intervalloium numerus illo priore
interceptorum, & ipsi simul aequalium : at numerus ejusmodi partium possibilium finem
habebit nullum. Illorum singulorum magnitudo certa erit, ac finita : horum magnitude
minuetur ultra quoscunque limites, sine ullo determinato hiatu, qui adjectis novis inter-
mediis punctis imminui adhuc non possit; licet nee possit actuali divisione, sive inter-
positione exhauriri.

con- 9' ^mc vero dum concipimus possibilia haec loci puncta, spatii infinitatem, &
tinuum.necessarium continuitatem habemus, cum divisibilitate in infinitum. In existentibus limes est semper
sternum^iminobite certus, certus punctorum numerus, certus intervallorum : in possibilibus nullus est finis.
praecisivam. Possibilium abstracta cognitio, excludens limitem a possibili augmento intervalli, &

diminutione, ac hiatu, infinitatem lineae imaginarae, & continuitatem constituit, quae
partes actu existentes non habet, sed tantummodo possibiles. Cumque ea possibilitas &
aeterna sit, & necessaria, ab seterno enim, & necessario verum fuit, posse ilia puncta cum
illis modis existere ; spatium hujusmodi imaginarium continuum, infinitum, simul etiam
aeternum fuit, & necessariurn, sed non est aliquid existens, sed aliquid tantummodo potens
existere, & a nobis indefinite conceptum : immobilitas autem ipsius spatii a singulorum
punctorum immobilitate orietur.

in momentis eadem, IQ. Atque haec omnia, quae hucusque de loci punctis sunt dicta, ad temporis momenta

quae in punctis : j iiir-i • • 111 i •

post primum nullum e°dem modo admodum facile transferuntur, inter quae ingens qusedam habetur analogia.
secundum, aut ulti- Nam & punctum a puncto, & momentum a momento quo vis determinato certain distantiam

mum: sed in tern- i i • • • •««••«* ......

pore unica dimensio, habet, nisi coeunt, qua major, & minor haben alia potest sine ullo limite. In quo vis
in spatio triplex. intervallo spatii imaginarii, ac temporis adest primum punctum, vel momentum, & ultimum,
secundum vero, & penultimum habetur nullum : quovis enim assumpto pro secundo, vel
penultimo, cum non coeat cum primo, vel ultimo, debet ab eo distare, & in eo intervallo
alia itidem possibilia puncta vel momenta interjacent. Nee punctum continuae lineae,
nee momentum continui temporis, pars est, sed limes & terminus. Linea continua, &
tempus continuum generari intelligentur non repetitione puncti, vel momenti, sed ductu
continue, in quo intervalla alia aliorum sint partes, non ipsa puncta, vel momenta, quae
continue ducuntur. Illud unicum erit [267] discrimen, quod hie ductus in spatio fieri
poterit, non in unica directione tantum per lineam, sed in infinitis per planum, quod
concipietur ductu continue in latus lineae jam conceptae, & iterum in infinitis per solidum,
quod concipietur ductu continue plani jam concept!, in tempore autem unicus ductus
durationis habebitur, quod idcirco soli lineae erit analogum, & dum spatii imaginarii extensio

SUPPLEMENT I 395

them. For, since they have no kind of parts, they cannot coincide partly only ; that is,
they cannot touch one another on one side, & on the other side be separated. It is but
a prejudice acquired from infancy, & born of ideas obtained through the senses, which
have not been considered with proper care ; £ these ideas picture masses to us as always
being composed of parts at a distance from one another. It is owing to this prejudice
that we seem to ourselves to be able to bring even indivisible and non-extended points so
close to other points that they touch them & constitute a sort of lengthy series. We imagine
a series of little spheres, in fact ; & we do not put out of mind that extension, & the parts,
which we verbally exclude.

7. Again, where two points of matter are at a distance from one another, another Given two points,
point of matter can always be placed in the same straight line with them, on the far side ^the^Tn^tne^ame
of either, at an equal distance ; & another beyond that, & so on without end, as is evident, straight line at
Also another point can be pkced halfway between the two points, so as to touch neither apart* *• &lStitnC<is
of them ; for, if it touched either of them it would touch them both, & thus would coincide possible to insert
with both ; hence the two points would coincide with one another & could not be separate them" t^Tny
points, which is contrary to the hypothesis. Therefore that interval can be divided into extent ' in either
two parts ; & therefore, by the same argument, those two can be divided into four others, case'

& so on without any end. Hence it follows that, however great the interval between two
points, we could always obtain another that is greater ; &, however small the interval
might be, we could always obtain another that is smaller ; '&, in either case, without any
limit or end.

8. Hence beyond & between two real points of position of any sort there are other The number of
real points of position possible ; & these recede from them & approach them respectively, ^^ wUi^aiways
without any determinate limit. There will be a real divisibility to an infinite extent of be finite, & the
the interval between two points, or, if I may call it so, an endless ' insertibility ' of real the^finite^there
points. However often such real points of position are interpolated, by real points of is no end to the
matter being interposed, their number will always be finite, the number of intervals possit
intercepted on the first interval, & at the same time constituting that interval, will be finite ;

but the number of possible parts of this sort will be endless. The magnitude of each
of the former will be definite & finite ; the magnitude of the latter will be diminished
without any limit whatever ; & there will be no gap that cannot be diminished by adding
fresh points in between ; although it cannot be completely removed either by division or
by interposition of points.

9. In this way, so long as we conceive as possibles these points of position, we have Hence, the manner
infinity of space, & continuity, together with infinite divisibility. With existing things a° ^ace^hat^is
there is always a definite limit, a definite number of points, a definite number of intervals ; finite, continuous,
with possibles, there is none that is finite. The abstract concept of possibles, excluding ^immovabi^"^
as it does a limit due to a possible increase of the interval, a decrease or a gap, gives us the means of an abstract
infinity of an imaginary line, & continuity ; such a line has not actually any existing parts, concePt-

but only possible ones. Also, since this possibility is eternal, in that it was true from
eternity & of necessity that such points might exist in conjunction with such modes, space
of this kind, imaginary, continuous & infinite, was also at the same time eternal & necessary ;
but it is not anything that exists, but something that is merely capable of existing, & an
indefinite concept of our minds. Moreover, immobility of this space will come from
immobility of the several points of position.

10. Everything, that has so far been said with regard to points of position, can quite The same things
easily in the same way be applied to instants of time ; & indeed there is a very great analogy j^ ;f°L7or points*
of a sort between the two. For, a point from a given point, or an instant from a given after' the first
instant, has a definite distance, unless they coincide ; & another distance can be found or^as't ; n<inSetimed
either greater or less than the first, without any limit whatever. In any interval of imaginary however, there is
space or time, there is a first point or instant, & a last ; but there is no second, or last but ^iie "In

one. For, if any particular one is supposed to be the second, then, since it does not coincide there are three,
with the first, it must be at some distance from it ; & in the interval between, other possible
points or instants intervene. Again, a point is not a part of a continuous line, or an instant
a part of a continuous time ; but a limit & a boundary. A continuous line, or a continuous
time is understood to be generated, not by repetition of points or instants, but by a continuous
progressive motion, in which some intervals are parts of other intervals ; the points them-
selves, or the instants, which are continually progressing, are not parts of the intervals.
There is but one difference, namely, that this progressive motion can be accomplished
in space, not only in a single direction along a line, but in infinite directions over a plane
which is conceived from the continuous motion of the line already conceived in the direction
of its breadth ; & further, in infinite directions throughout a solid, which is conceived
from the continuous motion of the plane already conceived. Whereas, in time there will
be had but one progressive motion, that of duration ; & therefore this will be analogous

396 PHILOSOPHIC NATURALIS THEORIA

habetur triplex in longum, latum, & protundum, temporis habetur unica in longum, vel
diuturnum tantummodo. In triplici tamen spatii, & unico temporis geneie, punctum,
ac momentum erit principium quoddam, a quo ductu illo suo haec ipsa generata intelligentur.

Quodyis punctum n. Illud jam hie diligenter notandum : non solum ubi duo puncta materiae existunt,

tegnmf spatium, "ac" & aliquam distantiam habent, existere duos modos, qui relationis illius distantiae funda-
tempus imaginanum mentum prsebeant, & sint bina diversa puncta loci realia, quorum possibilitas a nobis

suum : quidsitcom- i_-i. u- ...*... j • £ • • •t-'VL

penetratio. concepta, exhibeat bma puncta spatii imaginarn, adeoque mfimtis numero possibmbus

materiae punctis respondere infinites numero possibiles existendi modos, sed cuivis puncto
materiae respondere itidem infinites possibiles existendi modos, qui sint omnia ipsius puncti
possibilia loca. Haec omnia satis sunt ad totum spatium imaginarium habendum, & quodvis
materiae punctum habet suum spatium imaginarium immobile, infinitum, continuum,
quae tamen ornnia spatia pertinentia ad omnia puncta sibi invicem congruunt, & habentur
pro unico. Nam si assumatur unum punctum reale loci ad unum materiae punctum
pertinens, & conferatur cum omnibus punctis realibus loci pertinentibus ad aliud punctum
materiae ; est unum inter haec posteriora, quod si cum illo priore coexistat, relationem
inducet distantiae nullius, quam compenetrationem appellamus. Unde patet punctorum,
quae existunt, distantiam nullam non esse nihil, sed relationem inductam a binis quibusdam
existendi modis. Reliquorum quivis cum illo eodem priore induceret relationem aliam,
quam dicimus cujusdam determinatae distantiae, & positionis. Porro ilia loci puncta, quae
nullius distantiae relationem inducunt, pro eodem accipimus, & quenvis ex infinitis hujus-
modi punctis ad infinita puncta materiae pertinentibus pro eodem accipimus, ac ejusdem
loci nomine intelligimus. Ea autem haberi debere pro quovis punctorum binario, sic
patet. Si tertium punctum ubicunque collocetur, habebit aliquam distantiam, & positionem
respectu primi. Summoto prime, poterit secundum collocari ita, ut habeat eandem illam
distantiam, & positionem, respectu tertii, quam habebat primum. Igitur modus hie, quo
existit, pro eodem habetur, ac modus, quo existebat illud primum, & si hi bini modi simul
existerent, nullius distantiae relationem inducerent inter primum, ac secundum : & haec
pariter, quae hie de spatii punctis dicta sunt, aeque temporis momentis conveniunt.

piura momenta [268] 12. An autem possint simul existere, id vero pertinet ad relationem, quam
n nabent puncta loci cum momentis temporis, sive spectetur unicum materiae punctum,
sive plura. Inprimis plura momenta ejusdem puncti materiae coexistere non possunt, sed
alia necessario post alia, sic itidem bina puncta localia ejusdem puncti materiae conjungi
non possunt, sed alia jacere debent extra alia, atque id ipsum ex eorum natura, & ut
ajunt, essentia.

J3' Deinde considerentur conjunctiones variae punctorum loci, & momentorum.
poris pro unico Quodvis punctum materiae, si existit, conjungit aliquod punctum spatii cum aliquo
Puatuor "ro* binis momento temporis. Nam necessario alicubi existit, & aliquando existit ; ac si solum
notabiies : existat, semper suum habet, & localem, & temporarium existendi modum, per quod, si
5 osrtf a^uc* °iuodpiam existat, quod suos itidem habebit modus, distantiae & localis, & temporis
relationem ad ipsum acquiret. Id saltern omnino accidet, si omnium, quae existunt, vel
existere possunt, commune est spatium, ut puncta localia unius, punctis localibus alterius
perfecte congruant, singula singulis. Quid enim, si alia sunt rerum genera, vel a nostris
dissimilium, vel nostris etiam prorsus similium, quae aliud, ut ita dicam, infinitum spatium
habeant, quod a nostro itidem infinite non per intervallum quoddam finitum, vel infinitum
distet, sed ita alienum sit, ita, ut ita dicam, alibi positum, ut nullum cum hoc nostro com-
mercium habeat, nullam relationem distantiae inducat. Atque id ipsum de tempore etiam
dici posset extra omne nostrum aeternum tempus collocate. At id menti, ipsum conanti
concipere, vim summam infert, ac a cogitatione directa admitti vel nullo modo potest,
vel saltern vix potest. Quamobrem iis rebus, vel rerum spatiis, & temporibus, quae ad nos
nihil pertinere possent, prorsus omissis, agamus de notris hisce. Si igitur primo idem
punctum materiae conjungat idem punctum spatii, cum pluribus momentis temporis aliquo
a se invicem intervallo disjunctis ; habebitur regressus ad eundem locum. Si secundo id
conjungat cum serie continua momentorum temporis continui ; habebitur quies, quae
requirit tempus aliquod continuum cum eodem loci puncto, sine qua conjunctione habetur
continuus motus, succedentibus sibi aliis, atque aliis loci punctis, pro aliis, atque aliis

SUPPLEMENT I 397

to a single line. Thus, while for imaginary space there is extension in three dimensions,
length, breadth & depth, there is only one for time, namely length or duration only.
Nevertheless, in the threefold class of space, & in the onefold class of time, the point &
the instant will be respectively the element, from which, by its progression, motion, space
& time will be understood to be generated.

11. Now here there is one thing that must be carefully noted. Not only when two Every point of
points of matter exist, & have a distance from one another, do two modes exist which give ^fhe *t h°o lefTi
the foundation of the relation of this distance ; & there are two different real points of imaginary space,
position, the possibility of which, as conceived by us, will yield two points of imaginary naturelmof; con>
space ; & thus, to the infinite number of possible points of matter there will correspond penetration.

an infinite number of possible modes of existence. But also to any one point of matter
there will correspond the infinite possible modes of existing, which are all the possible
positions of that point. All of these taken together are sufficient for the possession of
the whole of imaginary space ; & any point of matter has its own imaginary space, immovable,
infinite & continuous ; nevertheless, all these spaces, belonging to all points coincide with
one another, & are considered to be one & the same. For if we take one real point of
position belonging to one point of matter, & associate it with all the real points of position
belonging to another point of matter, there is one among the latter, which, if it coexist
with the former, will induce a relation of no-distance, which we call compenetration.
From this it is clear that, for points which exist, no-distance is not nothing, but a relation
induced by some two modes of existence. Any of the others would induce, with that
same former point of position, another relation of some determinate distance & position,
as we say. Further, those points of position, which induce a relation of no-distance, we
consider to be the same ; & we consider any of the infinite number of such points belonging
to the infinite number of points of matter to be the same ; & mean them when we speak
of the ' same position.' Moreover this is evidently bound to be true for any pair of points.
If now a third point is situated anywhere, it will have some distance & position with respect
to the first. If the first is removed, the second can be so situated that it has the same
distance & position with respect to the third as the first had. Hence the mode, in which
it exists, will be taken to be the same in this case as the mode in which the first point was
existing ; & if these two modes were existing together, they would induce a relation of
no-distance between the first point & the second. All that has been said above with regard
to points of space applies equally well to instants of time.

12. Now, whether they can coexist is a question that pertains to the relation between Several instants
points of position & instants of time, whether we consider a single point of matter or ^"gp"fnt Cannot
several of them. In the first place, several instants of time belonging to the same point coexist.

of matter cannot coexist ; but they must necessarily come one after .the other ; & similarly,
two points of position belonging to the same point of matter cannot be conjoined, but
must lie one outside the other ; & this comes from the nature of points of this kind, &
is essential to them, to use a common phrase.

13. Next, we have to consider the different kinds of combinations of points of space Four combinations
& instants of time. Any point of matter, if it exists, connects together some point of ^ sTn^ie VoSt * of
space & some instant of time ; for it is bound to exist somewhere & sometime. Even if matter ; four
it exists alone, it always has its own mode of existence, both local & temporal ; & by this fo^twtr'pofa^
fact, if any other point of matter exists, having its own modes also, it will acquire a relation extraordinary idea
of distance, both local & temporal, with respect to the first. This at least will certainly s

be the case, if the space belonging to all that exist, or can possibly exist, is common ; so
that the points of position belonging to the one coincide perfectly with those belonging
to the other, each to each. But, what if there are other kinds of things, either different from
those about us, or even exactly similar to ours, which have, so to speak, another infinite
space, which is distant from this our infinite space by no interval either finite or infinite,
but is so foreign to it, situated, so to speak, elsewhere in such a way that it has no com-
munication with this space of ours ; & thus will induce no relation of distance. The
same remark can be made with regard to a time situated outside the whole of our eternity.
But such an idea requires an intellect of the greatest power to try to grasp it ; & it cannot
be admitted by direct consideration, in any way, or at least with difficulty. Hence,
omitting altogether such things, or the spaces & times of such things which are no concern
of ours, let us consider the things that have to do with us. If therefore, firstly, the same,
point of matter connects the same point of space with several instants of time separated
from one another by any interval, there will be return to the same place. If, secondly,
it connects the point of space to a continuous series of instants of continuous time, there
will be rest, which requires a certain continuous time to be connected with the same point
of position ; without this connection there will be continuous motion, points of position
succeeding one another corresponding to instants of time, one after the other. Thirdly,

398 PHILOSOPHIC NATURALIS THEORIA

momentis temporis. Si tertio idem punctum materiae conjungat idem momentum tem-
poris cum pluribus punctis loci a se invicem distantibus aliquo intervallo ; habebitur ilia,
quam dicimus replicationem. Si quarto id conjungat cum serie continua punctorum loci
aliquo intervallo continue contentorum, habebitur quaedam quam plures Peripatetici
admiserunt, virtualem appellantes extensionem, qua indivisibilis, & partibus omnino
destituta materiae particula spatium divisibile occuparet. Sunt alias quatuor combina-
tiones, ubi plura materiae pun-[269J-cta considerentur. Nimirum quinto si conjungant
idem momentum temporis cum pluribus punctis loci, in quo sita est coexistentia. Sexto
si conjungant idem punctum spatii cum diversis momentis temporis, quod fieret in successive
appulsu diversorum punctorum materiae ad eundem locum. Septimo si conjungant idem
momentum temporis cum eodem puncto spatii, in quo sita esset compenetratio. Octavo
si nee momentum ullum, nee punctum spatii commune habeant, quod haberetur, si nee
coexisterent, nee ea loca occuparent, quae ab aliis occupata fuissent aliquando.

Relation** earum j, gx }j,isce octo casibus primo respondet tertius, secundo quartus, quinto sextus,

ad se invicem : quae, , T __,. ..*«.. T-

& quomodo possi- septimo octavus. Tertium casum, mmirum replicationem, commumtur censent naturahter
haberi non posse. Quartum censent multi habere animam rationalem, quam putant esse
in spatio aliquo divisibili, ut plures Peripatetici in toto corpore, alii Philosophi in quadam
cerebri parte, vel in aliquo nervorum succo ita, ut cum indivisibilis sit, tota sit in toto
spatio, & tota in quavis spatii parte, quemadmodum eadem indivisibilis Divina Natura
est in toto spatio, & tota in qualibet spatii parte, ubique necessario praesens, & omnibus
creatarum rerum realibus locis coexistens, ac adstans. Eundem alii casum in materia
admittunt, cujus particulas eodem pacto extendi putant, ut diximus ; licet simplices sint,
licet partibus expertes, non modo actu separatis, sed etiam distinctis, ac tantummodo
separabilibus. Earn sententiam amplectendam esse non censeo idcirco, quod ubicunque
materiam loca distincta occupantem sensu percipimus, separabilem etiam, ingenti saltern
adhibita vi, videmus ; sejunctis partibus, quae distabant : nee vero alio ullo argumento
excludimus a Natura replicationem, nisi quia nullam materiae partem, quantum sensu
percipere possumus, videmus, bina simul occupare loca. Virtualis ilia extensio materiae
infinities ulterius progreditur ultra simplicem replicationem.

Quietem, & regres- jr Si secundus casus quietis, & primus casus regressus ad eundem locum naturaliter

sum ad eundem , , . .A . , , ... A

locum in Natura haberi possent, esset is quidem defectus quidam analogiae inter spatium, & tempus. At
esse in mfinitum j^j^j v}deor probare illud posse, neutrum unquam in Natura contingere, adeoque naturaliter

improbabiles, &,,. . , c.

indeingensanaiogia. haberi non posse. Id autem evinco hoc argumento. bit punctum matenas quodam
momento in quodam spatii puncto, & pro quovis alio momento ignorantes, ubi sit, quaeramus,
quanto probabilius sit, ipsum alibi esses, quam ibidem. Tanto erit probabilius illud,
quam hoc, quanto plura sunt alia spatii puncta, quam illud unicum. Hsec in quavis linea
sunt infinita, infinitus in quovis piano linearum numerus, infinitus in toto spatio planorum
numerus. Quare numerus aliorum punctorum est infinitus tertii generis, adeoque ilia
probabilitas major infinities tertii generis infinitate, ubi de quovis alio determinato momento
agitur. Agatur jam inde-[269]-finite de omnibus momentis temporis infiniti, decrescet
prior probabilitas in ea ratione, qua momenta crescunt, in quorum aliquo saltern posset
ibidem esse punctum. Sunt autem momenta numero infinita infinitate ejusdem generis,
cujus puncta possibilia in linea infinita. Igitur adhuc agendo de omnibus momentis
infiniti temporis indefinite, est infinities infinite improbabilius, quod punctum in eodem
illo priore sit loco, quam quod sit alibi. Consideretur jam non unicum punctum loci
determinato unico momento occupatum, sed quodvis punctum loci, quovis indefinite
momento occupatum, & adhuc probabilitas regressus ad aliquod ex iis crescet, ut crescit
horum loci punctorum numerus, qui infinite etiam tempore est infinitus ejusdem ordinis,
cujus est numerus linearum, in quovis piano. Quare improbabilitas casus, quo determin-
atum quodpiam materiae punctum redeat, quovis indefinite momento temporis, ad quodvis
indefinite punctum loci, in quo alio quovis fuit momento temporis indefinite sumpto,
remanet infinita primi ordinis. Eadem autem pro omnibus materiae punctis, quae numero
finita sunt, decrescit in ratione finita ejus numeri ad unitatem (quod secus accidit in com-
muni sententia, in qua punctorum materise numerus est infinitus ordinis tertii). Quare

SUPPLEMENT I 399

if the same point of matter connects the same instant of time with several points of position
distant from one another by some interval, then we shall have replication. Fourthly, if
it connects the instant with a continuous series of points of position contained within some
continuous interval, we shall have something which several of the Peripatetics admitted,
calling it virtual extension ; by virtue of which an indivisible particle of matter, quite
without parts, could occupy divisible space. There are four other combinations, when
several points are considered. That is to say, fifthly, if several points connect the same
instant of time with several points of position ; in this is involved coexistence. Sixthly,
if they connect the same point of space with several instants of time ; as would be the
case when different points of matter were forced successively into the same position.
Seventhly, if they connect the same point of space with the same instant of time ; in
this is involved compenetration. Eighthly, if they have no instant of time, & no point
of space, common to them ; as would be the case, if they did not coexist, nor, any of them,
occupied the positions that had been occupied by any of the others at any time.

14. Out of these eight cases, the third corresponds to the first, the fourth to the second, The relations of
the sixth to the fifth, the eighth to the seventh. The third case, namely replication, is anotherT^Mch0""
usually considered to be naturally impossible. Many think that the fourth case holds them are possible,
good for the rational soul, which they consider to have its seat in some divisible space ;

for instance, the Peripatetics think that it pervades the whole of the body, other philosophers
think it is situated in a certain part of the brain, or in some juice of the nerves ; so that,
since it is indivisible, the whole of it must be in the whole of the space, & the whole of it in
any part of the space. Just in the same way as the same indivisible Divine Nature is as
a whole in the whole of space, & as a whole in any part of space, being necessarily present
everywhere, & coexisting with & accompanying created things wherever created things are.
Others admit this same case for matter, & consider that particles of matter can be extended in
a similar manner, as we have said ; although they are simple, & although they are devoid
of parts, not only parts that are really separated, but also such as are distinct & only -separable.
I do not consider that this supposition can be entertained, for the reason that, whenever
we perceive with our senses matter occupying positions distinct from one another, we see
that it is also separable, although we may have to use a very great force ; here, parts are
separated which were at a distance from one another. Indeed, by no other argument
can we exclude replication from Nature, than that we never see any portion of
matter, as far as can be perceived by the senses, occupying two positions at the same
time. The idea of Virtual extension of matter goes infinitely further beyond the idea
of simple replication.

15. If the second case of rest, & the first case of return to the same position could ^est & "turn to
be obtained naturally, then indeed there would be a certain defect in the analogy between

space & time. But it seems to me that I can prove that neither ever happens in Nature ; £able in Nature ;

& so they cannot be obtained naturally ; this is my argument. If a point of matter at great6 analogy

any instant of time is at a certain point of space, & we do not know where it is at some tween them-

other instant, let us inquire how much more probable it is that it should be somewhere

else than at the same point as before. The former will be more probable than the latter

in the proportion of the number of all the other points of space to that single point. There

are an infinite number of these points in any straight line, the number of lines in any plane

is infinite, & the number of planes in the whole of space is infinite. Hence, the number

of other points of space is an infinity of the third order ; & thus the probability is infinitely

greater with an infinity of the third order, when we are concerned with any other particular

instant of time. Now let us deal indefinitely with all the instants of infinite time ; then

the first probability will decrease in proportion as the number of instants increases, at any

of which it might at least be possible that the point was in the same place as before. Moreover,

there are an infinite number of instants, the infinity being of the same order as that of the

number of possible points in an infinite line. Hence, still considering indefinitely all the instants

of infinite time, it is infinitely more improbable that the point should be in the same position

as before, than that it should be somewhere else. Now consider, not a single point of position

occupied at a single particular instant, but any point of position occupied at any indefinite

instant ; then still the probability of return to any one of these points of position will

increase as the number of them increases ; & this number, in a time that is also infinite,

is an infinity of the same order as the number of lines in any plane. Hence the improbability

of this case, in which any particular point of matter returns at some indefinite instant of

time to some indefinite point of position, in which it was assumed to be at some other

indefinite instant of time, remains an infinity of the first order. Moreover, this, for all

points of matter, which are finite in number, will decrease in the finite ratio of this number

to infinity (which would not be the case with the usual theory, in which the number of

points of matter is taken to be an infinity of the third order). Hence we are still left with

400 PHILOSOPHIC NATURALIS THEORIA

adhuc remanet infinita improbabilitas regressus puncti materiae cujusvis indefinite, ad
punctum loci quodvis, occupatum quovis momento praecedenti indefinite, regressus inquam,
habendi quovis indefinite momento sequent! temporis, qui regressus idcirco sine ullo erroris
metu debet excludi, cum infinitam improbabilitatem in relativam quandam impossibilitatem
migrare censendum sit : quae quidem Theoria communi sententiae applicari non potest.
Quamobrem eo pacto, patet, in mea materiae punctorum Theoria e Natura tolli & quietem,
quam etiam supra exclusimus, & vero etiam regressum ad idem loci punctum, in quo
semel ipsum punctum materias extitit : unde fit, ut omnes illi primi quatuor casus exclud-
antur ex Natura, & in iis accurata temporis, & spatii servetur analogia.

Nuiium punctum jg. Quin imo si quaeratur, an aliquod materiae punctum occupare debeat quopiam

materue advenire , -1 , ,. ^ ,. ,. ,r . . T."wr*'

ad uiium punctum momento punctum loci, quod alio momento aliquo aliud materiae punctum occupavit ; adhuc
spatii in quo improbabilitas erit infinities infinita. Nam numerus punctorum materiae existentium est

aliquando fuerit c f . . . . , r . .

aliud punctum fimtus, adeoque si pro regressu puncti cujusvis ad puncta loci a se occupata adhibeatur
quodvis. in sola regressus ad puncta occupata a quovis alio, numerus casuum crescit in ratione unitatis ad

coexistentia respon- cr. * • . . . . r . TT.

dente huic adventui numerum punctorum nnitum utique, nimirum in ratione finita tantummodo. Hmc
analogiam. improbabilitas appulsus alicujus puncti materiae indefinite sumpti ad punctum spatii
aliquando ab alio quovis puncto occupati adhuc est infinita, & ipse appulsus habendus pro
impossibili, quo quidem pacto excluditur & sextus casus, qui in eo ipso situs erat regressu,
& multo magis septimus, qui binorum punctorum mate-[27i]-riae simultaneum appulsum
continet ad idem aliquod loci punctum, sive compenetrationem. Octavus autem pro
materia excluditur, cum tota simul creata perpetuo duret tota, adeoque semper idem
momentum habeat commune.^) Solus quintus casus, quo plura materiae puncta idem
momentum temporis cum diversis punctis loci conjungant, non modo possibilis est, sed
etiam necessarius pro omnibus materiae punctis, coexistentibus nimirum : fieri enim non
potest, ut septimus, & octavus excludantur ; nisi continue ob id ipsum includatur quintus
ille, ut consideranti patebit facile. Quamobrem in eo analogia deficit, quod possint plura
materiae puncta conjungere diversa puncta spatii cum eodem momento temporis, qui est
hie casus quintus, non autem possit idem punctum spatii, cum pluribus momentis temporis,
qui est casus tertius, quern defectum necessario inducit exclusio septimi, & octavi, quorum
altero incluso, excludi posset hie quintus, ut si possent materiae puncta, quae simul creata
sunt, nee pereunt, non coexistere, turn enim idem momentum cum diversis loci punctis
nequaquam conjungeretur.

^"' ibilUS Si"r 17. Ex illis 7 casibus videntur omnino 6 per Divinam Omnipotentiam possibiles,

Divinam Omnipo- dempta nimirum virtuali ilia materiae extensione, de qua dubium esse poterit, quia deberet
tentiam: usus simui existere numerus absolute infinitus punctorum illorum loci realium, quod impossibile

supenons theore- ..,,. . . ,.../•»«*

matis in impenetra- est ; si infimtum numero actu existens repugnat in modis ipsis. (juoniam autem possunt
bihtate. omnia existere alia post alia puncta loci in qua vis linea constituta, in motu nimirum con-

tinuo, & possunt itidem momenta omnia temporis continui, alia itidem post alia in rei
cujusvis duratione ; ambigi poterit, an possint & omnia simul ipsa loci puncta, quam quaes-
tionem definire non ausim. Illud unum moneo, sententiam hanc meam de spatii natura,
& continuitate praecipuas omnes difficultates, quibus premuntur reliquae, peni-[272]-tus
evitare, & ad omnia, quae hue pertinent, explicanda commodissimam esse. Turn illud
addo, excluso appulsu puncti cujusvis materiae ad punctum loci, ad quod punctum quodvis
materiae quovis momento appellit, & inde compenetratione, veram impenetrabilitatem
materiae necessario consequi, quod in decimo nobis libro (') plurimum proderit. Nimirum

(b) Hie casus nusquam itidem haberetur ; si duratio non esset quid continenter permanent, sed loco ipsius admit-
Uretur qutedam existentia, ut ita dicam, saltitans, nimirum si quodvis materits punctum (Eif idem potest transferri ad
qutsvis creata entia) existeret tantum in momentis indivisililibus a se invicem remotis, in omnibus vero intermediis
possibilibus omnino non existeret. Eo casu coexistentia esset infinite improbabilis eodem fere argumento, quo adventus
unius puncti materite ad punctum spatii, in quo aliud quodvis punctum unquam fuerit. In eodem nullum haberetur
reale continuum ne in motu quidem ; diversis celeritates multo melius explicarentur : multo magis pateret, quomodo
vita insecti brevissima possit tequivalere vitce cuivis longissimce, per eundem nimirum numerum existentiarum inter
extrema momenta. Verum y exclusio cujusvis coexistentid abriperet secum omnes prorsus influxus pbysicos immediatos,
ac determinations, y deberet haberi continua reproductio, immo creatio nova perpetua, y alia ejusmodi, quts admitti,
non possunt, haberentur.

(c) Stay ants nimirum philosophies, in quo Auctor elegantissimus, & doctissimus hanc meam Philosophiam exponit.
Hunc ejus theorematis fructum jam cepimus hie supra, ubi in ipso opere de impenetrabilitate egimus, W de apparenti
compenetratione, qute sine viribus mutuis haberetur a num. 360.

SUPPLEMENT 1 401

an infinite improbability of the return of any indefinitely chosen point of matter to any
point of position, occupied at any previous instant of time indefinitely, of a return, I say,
taking place at any indefinite instant of subsequent time ; hence, such a return must be
excluded, without any fear as to error, since it must be considered that an infinite
improbability merges into a sort of relative impossibility. This Theory indeed cannot
be applied to the ordinary view. Hence, in this way it is clear, in my Theory of points
of matter, there must be excluded from Nature both rest, which also we excluded above,
& even return to the same point of position in which that point of matter once was situated.
Therefore it comes about that all those first four cases will be excluded from Nature, &
in them the analogy of time & space will be preserved accurately.

1 6. Finally, if we seek to find whether any point of matter is bound to occupy at some NO point of matter
instant a point of position which was occupied by some other point of matter at some p^nH>TspTce that
other instant, still the improbability will be infinitely infinite. For the number of existing wa» once occupied
points of matter is finite ; & thus, if instead of the return of any point to points of position it^1^0 onfymin
occupied by itself we consider the return to points that have been occupied by another, coexistence, which
the number of cases increases in the ratio of unity to a number of points that is in every that^the "

case finite, that is to say, in a finite ratio only. Hence, the improbability of the arrival is broken,
of any point of matter indefinitely taken at a point of space that has been occupied at some
time by any other point is still infinite ; & this arrival must therefore be taken to be impossible.
In this way, indeed, the sixth case, which depended on this return, is excluded ; & much
more so the seventh case, which involves the simultaneous arrival of a pair of points of matter
at any the same point of position, that is to say, compenetration. The eighth case also
is excluded for matter ; for all things created together as a whole will continually last as
a whole, & so will always have a common instant of time.(*) Only the fifth case, in which
several points of matter connect the same instant of time with different points of position
remains ; & this is not only possible, but also necessary for all points of matter, seeing
that they coexist. For it cannot be the case that the seventh & the eighth are excluded,
unless straightway, on that very account, the fifth is included, as will be easily seen on
consideration. Therefore in this point the analogy fails, namely, in that several points
of matter can connect different points of space with the same instant of time, which is
the fifth case ; whereas it is impossible for the same point of space to be connected with
several instants of time, which is the third case. This defect is necessarily induced by
the exclusion of the seventh & eighth cases ; for if either of the latter is included, the
fifth might be excluded ; just as if it were possible for points of matter, which had been
created together, & do not perish, not to coexist ; for then the same instant of time would
in no way be connected with different points of position.

17. At least six of the seven cases seem to be possible through Divine Omnipotence, Which of the cases
that is to say, omitting the virtual extension of matter, about which there may be possibly through0 oYvine
some doubt ; for in this case there must exist at the same time an absolutely infinite number Omnipotence; use
of those real points of position ; & this is impossible, if an existing thing that is infinite above "
in number is contradictory in the modes. Moreover, since all points of position can exist trabiiity.

one after another, arranged along any line, for instance, in continuous motion, & so can
also all instants of continuous time, one after another in the duration of any thing, there
will be reason for doubt as to whether all those points of position can also exist at the same
time. This is a matter upon which I dare not make a definite statement. All I say is
that this theory of mine with regard to the nature of space & continuity completely avoids
all the chief difficulties that are obstacles in other theories ; & that it is very suitable for
the explanation of everything in connection with this matter. I will also add the remark
that, if the arrival of any point of matter at a point of position, at which any point of
matter has arrived at any instant, is excluded, & along with it compenetration is thus
excluded, then real impenetrability of matter must necessarily follow, which will be of great
service to us in our tenth book (^). That is, unless repulsive forces prevent such a thing, any

(b) This case also would never happen, if the duration were not something continuously permanent ; in -place of
it, we should have to admit a kind of, so to speak, skipping existence ; that is to say, as if any point of matter (and
the same thing applies to all created entities) existed only in indivisible instants remote from one another, and in all
intermediate instants possible did not exist at all. Coexistence, in this case, would be infinitely improbable, the argu-
ment being nearly the same, as in the case of the arrival of one point of matter at a point of space in which some other
point had once been. In this case too, there would be no real continuum even in motion ; different velocities could be
explained much more easily ; it would be much more evident in what way the very short life of an insect can be
equivalent to the longest of lives, by means of the same number of existences coming in between the first & last instants.
Indeed the exclusion of any coexistence would carry away with it all immediate physical influence altogether, 13 deter-
minations ; indeed, a continually fresh creation, 13 other inadmissible things of that sort, would be obtained.

(c) The reference is to Stay's " Philosophy," in which that most refined {3 learned author expounds my Philosophy.
On what I have said above, I have plucked the fruit of the theorem, in which, in Art. 360 of this work, I dealt with
impenetrability, 13 the apparent compenetration that would result, if there were no mutual forces.

DP

402 PHILOSOPHISE NATURALIS THEORIA

nisi vires repulsivse prohiberent ; liberrime massa qusevis per quamvis aliam massam
permearet, sine ullo periculo occursus ullius puncti cum alio quovis, ubi haberetur apparens
quaedam compenetratio similis penetrationi luminis per crystalla, olei per ligna, & marmora,
sine ulla reali compenetratione punctorum. In massis crassioribus, & minori celeritate
praeditis vires repulsivse motum ulteriorem plerumque impediunt sine ullo impactu, &
sensibilem etiam illam, ac apparentem compenetrationem excludunt : in tenuissimis, &
celerrimis, ut in luminis radiis per homogeneas substantias, vel per alios radios propagatis,
evitatur per celeritatem ipsam, actionum exigua insequalitas, ex circumjacentium punctorum
insequali distantia orta, ac liberrimus habetur progressus in omnes plagas sine ullo occursus
periculo, quod summam, & unicam difficultatem propagations luminis per substantiam
emissam, & progredientem, penitus amovet. Sed de his jam satis.

SUPPLEMENT I 403

perfectly free mass will permeate through any other mass, without there being any danger
of a collision of one point with another. Here there would be an apparent compenetration
similar to the penetration of light through crystals, oils through wood, & marble, without
any real compenetration of the points. In denser masses, & those endowed with a smaller
velocity, the repulsive forces for the most part prevent further motion without any impact ;
& this also excludes sensible as well as apparent compenetration. In very tenuous masses
moving with very great velocities, as rays of light propagated through homogeneous
substances, or through other rays, the very slight inequality of the actions, derived from
the unequal distances of the circumjacent points, will be prevented by the high velocity ;
& perfectly free progress will take place in all directions without any danger of collisions.
This removes altogether the greatest & only real difficulty in the idea of the propagation
of light by means of a substance that is emitted & travels forward. But I have now said
quite enough upon this matter.

D

[273] § II
Tempore, ut a nobis cognoscuntur

Nos nee modos ig. Diximus in superiore Supplemento de spatio, ac tempore, ut sunt in se ipsis :

posse60 absolute superest, ut illud attingam, quod pertinet ad ipsa, ut cognoscuntur. Nos nequaquam
cognoscere, nee immediate cognoscimus per sensus illos existendi modos reales, nee discernere possumus

absolute distantias T i !•• o • • i s» • • • i • • •

& magnitudines. a^los a® a^lls' Sentimus quidem a discrimme idearum, quae per sensus excitantur m ammo,
relationem determinatam distantiae, & positionis, quae e binis quibusque localibus existendi
modis exoritur, sed eadem idea oriri potest ex innumeris modorum, sive punctorum realium
loci binariis, quae inducant relationes aequalium distantiarum, & similium positionum
tarn inter se, quam ad nostra organa, & ad reliqua circumjacentia corpora. Nam bina
materiae puncta, quae alicubi datam habent distantiam, & positionem inductam a binis
quibusdam existendi modis, alibi possunt per alios binos existendi modos habere relationem
distantise aequalis, & positionis similis, distantiis nimirum ipsis existentibus parallelis.
Si ilia puncta, & nos, & omnia circumjacentia corpora mutent loca realia, ita tamen, ut
omnes distantise aequales maneant, & prioribus parallelae ; nos easdem prorsus habebimus
ideas, quin imo easdem ideas habebimus ; si manentibus distantiarum magnitudinibus,
directiones omnes in sequali angulo converterentur, adeoque aeque ad se invicem inclinarentur
ac prius. Et si minuerentur etiam distantiae illas omnes, manentibus angulis, & manente
illarum ratione ad se invicem, vires autem ex ea distantiarum mutatione non mutarentur,
rite mutata virium scala ilia, nimirum curva ilia linea, per cujus ordinatas ipsae vires
exprimuntur ; nullam nos in nostris ideis mutationem haberemus.

Motum communem 19. Hinc autem consequitur illud, si totus hie Mundus nobis conspicuus motu parallelo

non1Sposse a nobis promoveatur in plagam quamvis, & simul in quovis angulo convertatur, nos ilium motum,

cognosci, nee s i & conversionem sentire non posse. Sic si cubiculi, in quo sumus, & camporum, ac montium

ipse in quavis

ratione au

'geatur, tractus omnis motu aliquo Telluris communi ad sensum simul convertatur ; motum ejusmodi
vei minuatur totus. sentire non possumus : ideas enim eaedem ad sensum excitantur in animo. Fieri autem
posset, ut totus itidem Mundus nobis conspicuus in dies contraheretur, vel produceretur,
scala virium tantundem contracta, vel producta ; quod si fieret ; nulla in animo nostro
idearum mutatio haberetur, adeoque nullus ejusmodi mutationis sensus.

Mutata positione
nostra, & omnium,
quae videmus, non
mutari n o s t r a s
ideas, & idcirco nos
motum nee nobis
adscribere, nee
reliquis.

20. Ubi vel objecta externa, vel nostra organa mutant illos suos existendi modos ita,
ut prior ilia aequalitas, [274] vel similitudo non maneat, turn vero mutantur ideae, &
mutationis habetur sensus, sed ideas eaedem omnino sunt, sive objecta externa mutationem
subeant, sive nostra organa, sive utrumque inaequaliter. Semper ideae nostrae difrerentiam
novi status a priore referent, non absolutam mutationem, quae sub sensus non cadit. Sic
sive astra circa Terram moveantur, sive Terra motu contrario circa se ipsam nobiscum ;
eaedem sunt ideae, idem sensus. Mutationes absolutas nunquam sentire possumus, discrimen
a priori forma sentimus. Cum autem nihil adest, quod nos de nostrorum organorum
mutatione commoneat ; turn vero nos ipsos pro immotis habemus communi praejudicio
habendi pro nullis in se, quae nulla sunt in nostra mente, cum non cognoscantur, & muta-
tionem omnem objectis extra nos sitis tribuimus. Sic errat, qui in navi clausus se immotum
censet, littora autem, & monies, ac ipsam undam moveri arbitratur.

404

§ II

Of Space ^f Time, as we know them

1 8. We have spoken, in the preceding Supplement, of Space & Time, as they are in We cannot obtain
themselves ; it remains for us to say a few words on matters that pertain to them, in so fe^geof°iocai modes
far as they come within our knowledge. We can in no direct way obtain a knowledge of existence ; nor
through the senses of those real modes of existence, nor can we discern one of them from tances ^or^agni-
another. We do indeed perceive, by a difference of ideas excited in the mind by means tudes.

of the senses, a determinate relation of distance & position, such as arises from any two
local modes of existence ; but the same idea may be produced by innumerable pairs of
modes or real points of position ; these induce the relations of equal distances & like positions,
both amongst themselves & with regard to our organs, & to the rest of the circumjacent
bodies. For, two points of matter, which anywhere have a given distance & position induced
by some two modes of existence, may somewhere else on account. of two other modes of
existence have a relation of equal distance & like position, for instance if the distances exist
parallel to one another. If those points, we, & all the circumjacent bodies change their
real positions, & yet do so in such a manner that all the distances remain equal & parallel
to what they were at the start, we shall get exactly the same ideas. Nay, we shall get the
same ideas, if, while the magnitudes of the distances remain the same, all their directions
are turned through any the same angle, & thus make the same angles with one another as
before. Even if all these distances were diminished, while the angles remained constant,
& the ratio of the distances to one another also remained constant, but the forces did not
change owing to that change of distance ; then if the scale of forces is correctly altered,
that is to say, that curved line, whose ordinates express the forces ; then there would be
no change in our ideas.

19. Hence it follows that, if the whole Universe within our sight were moved by a The motion, if any,
parallel motion in any direction, & at the same time rotated through any angle, we could thTuniverse could
never be aware of the motion or the rotation. Similarly, if the whole region containing not come within
the room in which we are, the plains & the hills, were simultaneously turned round by some ncV^ould winnow
approximately common motion of the Earth, .we should not be aware of such a motion ; it. if it were in-
for practically the same ideas would be excited in the mind. Moreover, it might be the ratfofor diminished^
case that the whole Universe within our sight should daily contract or expand, while the as a whole,
scale of forces contracted or expanded in the same ratio ; if such a thing did happen, there

would be no change of ideas in our mind, & so we should have no feeling that such a change
was taking place.

20. When either objects external to us, or our organs change their modes of existence s.mce- '£ °!?r P°si-
in such a way that that first equality or similitude does not remain constant, then indeed everything we see
the ideas are altered, & there is a feeling of change ; but the ideas are the same exactly, is changed, our ideas
whether the external objects suffer the change, or our organs, or both of them unequally, therefore Cwen8can
In every case our ideas refer to the difference between the new state & the old, & not to ascribe no motion
the absolute change, which does not come within the scope of our senses. Thus, whether anything eise.°
the stars move round the Earth, or the Earth & ourselves 'move in the opposite direction

round them, the ideas are the same, & there is the same sensation. We can never perceive
absolute changes ; we can only perceive the difference from the former configuration that
has arisen. Further, when there is nothing at hand to warn us as to the change of our
organs, then indeed we shall count ourselves to have been unmoved, owing to a general
prejudice for counting as nothing those things that are nothing in our mind ; for we cannot
know of this change, & we attribute the whole of the change to objects situated outside
of ourselves. In such manner any one would be mistaken in thinking, when on board ship,
that he himself was motionless, while the shore, the hills & even the sea were in
motion.

405

406

PHILOSOPHISE NATURALIS THEORIA

Quomodo judice-
mus de aequalitate
duorum, ex aequal-
itate cum tertio :
nunquam h a b e r i
congrue n t i a m in
longitudine, ut nee
in tempore, sed in-
ferri a causis.

21. Illud autem notandum inprimis ex hoc principio immutabilitatis eorum, quorum
mutationem per sensum non cognoscimus, oriri etiam methodum, quam adhibemus in
comparandis intervallorum magnitudinibus inter se, ubi id, quod pro mensura assumimus,
habemus pro immutabili. Utimur autem hoc principio, quce sunt cequalia eidem, sunt
cequalia inter se, ex quo deducitur hoc aliud, ad ipsum pertinens, quce sunt ceque multipla,
vel submuti-pla alterius, sunt itidem inter se cequalia, & hoc alio, quce congruant, cequalia sunt.
Assumimus ligneam, vel ferream decempedam, quam uni intervallo semel, vel centies
applicatam si inveniamus congruentem, turn alteri intervallo applicatam itidem semel,
vel centies itidem congruentem, ilia intervalla aequalia dicimus. Porro illam ligneam, vel
ferream decempedam habemus pro eodem comparationis termino post translationem. Si
ea constaret ex materia prorsus continua, & solida, haberi posset pro eodem comparationis
termino ; at in mea punctorum a se invicem distantium sententia, omnia illius decempedse
puncta, dum transferuntur, perpetuo distantiam revera mutant. Distantia enim con-
stituitur per illos reales existendi modos, qui mutantur perpetuo. Si mutentur ita, ut
qui modi succedunt, fundent reales sequalium distantiarum relationes ; terminus compara-
tionis non erit idem, adhuc tamen Eequalis erit, & aequalitas mensuratorum intervallorum
rite colligetur. Longitudinem decempedae in priore situ per illos priores reales modos
constitute, cum longitudine in posteriore situ constituta per hosce posteriores, immediate
inter se conferre nihilo magis possumus, quam ilia ipsa intervalla, quae mensurando conferi-
mus. Sed quia nullam in translatione mutationem sentimus, quae longitudinis relationem
nobis ostendat, idcirco pro eadem habemus longitudinem ipsam. At ea revera semper
in ipsa translatione non nihil mutabitur. Fieri posset, ut ingentem etiam mutationem
aliquam subiret [275] & ipsa, & nostri sensus, quam nos non sentiremus, & ad priorem
restituta locum ad priori sequalem, vel similem statum rediret. Exigua tamen aliqua
mutatio habetur omnino idcirco, quod vires, quse ilia materiae puncta inter se nectunt,
mutata positione ad omnia reliquarum Mundi partium puncta, non nihil immutantur.
Idem autem & in communi sententia accidit. Nullum enim corpus spatiolis vacat inter jectis,
& omnis penitus compressionis, ac dilatationis est incapax, quae quidem dilatatio, &
compressio saltern exigua in omni translatione omnino habetur. Nos tamen mensuram
illam pro eadem habemus, cum, ut monui, nullam mutationem sentiamus.

Conciusio

discri- 22. Ex his omnibus consequitur, nos absolutas distantias nee immediate cognoscere

omnino posse, nee per terminum communem inter se comparare, sed sestimare magnitudines
ab ideis, per quas eas cognoscimus, & mensuras habere pro communibus terminis, in quibus
nullam mutationem factam esse vulgus censet. Philosophi autem mutationem quidem
debent agnoscere, sed cum nullam violatae notabili mutatione sequalitatis causam agnoscant,
mutationem ipsam pro aequaliter facta habent.

Licet translata de-
cempeda, mutentur
modi, qui intervalli
relationem consti-
tuunt ; tamen inter-
valla aequalia haberi
pro eodem ex
causis.

23. Porro licet, ubi puncta materiae locum mutant, ut in decempeda translata, mutetur
revera distantia, mutatis iis modis realibus, quae ipsam constituunt ; tamen si mutatio ita
fiat, ut posterior ilia distantia aequalis prorsus priori sit, ipsam appellabimus eandem, & nihil
mutatam ita, ut eorundem terminorum aequales distantiae dicantur distantia eadem, &
magnitude dicatur eadem, quae per eas aequales distantias definitur, ut itidem ejusdem
directionis nomine intelligantur binae etiam directiones paralleiae ; nee mutari distantiam,
vel directionem dicemus in sequentibus, nisi distantiae magnitudo, vel parallelismus mutetur.

Eadem ad tempus
transferenda, sed
in eo etiam vulgo
notum esse, inter-
vallum t e m p o r-
arium no n posse
transferri idem pro
comparatione duo-
rum : errari ab eo
circa spatium.

24. Quae de spatii mensura diximus, haud difficulter ad tempus transferentur, in quo
itidem nullam habemus certam, & constantem mensuram. Desuminus a motu illam,
quam possumus, sed nullum habemus motum prorsus aequabilem. Multa, quae hue perti-
nent, & quae ad idearum ipsarum naturam, & successionem spectant, diximus in notis.
Unum hie addo, in mensura temporis, ne vulgus quidem censere ab uno tempore ad aliud
tempus eandem temporis mensuram transierri. Videt aliam esse, sed aequalem supponit ob
motum suppositum aequalem. In mensura locali aeque in mea sententia, ac in mensura
temporaria impossibile est certam longitudinem, ut certam durationem e sua sede abducere
in alterius sedem, ut binorum comparatio habeatur per tertium. Utrobique alia longi-
tude, ut alia duratio substituitur, quae priori illi aequalis censetur, nimirum nova realia

SUPPLEMENT II 407

21. Again, it is to be observed first of all that from this principle of the unchangeability The man
of those things, of which we cannot perceive the change through our senses, there comes ^g^ofTh
forth the method that we use for comparing the magnitudes of intervals with one another ; ity of two things
here, that, which is taken as a measure, is assumed to be unchangeable. Also we make wrt^tnird^there
use of the axiom, things that are equal to the same thing are equal to one another ; & from this never can be con-
is deduced another one pertaining to the same thing, namely, things that are equal multiples, fJ^moreThanthe-e
or submulti-ples, of each, are also equal to one another; & also this, things that coincide are can be in time; the
equal. We take a wooden or iron ten-foot rod ; & if we find that this is congruent with j^f ^rr ed tfrom
one given interval when applied to it either once or a hundred times, & also congruent to causes,
another interval when applied to it either once or a hundred times, then we say that these

intervals are equal. Further, we consider the wooden or iron ten-foot rod to be the same
standard of comparison after translation. Now, if it consisted of perfectly continuous &
solid matter, we might hold it to be exactly the same standard of comparison ; but in
my theory of points at a distance from one another, all the points of the ten-foot rod, while
they are being transferred, really change the distance continually. For the distance is
constituted by those real modes of existence, & these are continually changing. But if they
are changed in such a manner that the modes which follow establish real relations of equal
distances, the standard of comparison will not be identically the same ; & yet it will still
be an equal one, & the equality of the measured intervals will be correctly determined.
We can no more transfer the length of the ten-foot rod, constituted in its first position
by the first real modes, to the place of the length constituted in its second position by the
second real modes, than we are able to do so for intervals themselves, which we compare
by measurement. But, because we perceive none of this change during the translation,
such as may demonstrate to us a relation of length, therefore we take that length to be
the same. But really in this translation it will always suffer some slight change. It might
happen that it underwent even some very great change, common to it & our senses, so that
we should not perceive the change ; & that, when restored to its former position, it would
return to a state equal & similar to that which it had at first. However, there always is
some slight change, owing to the fact that the forces which connect the points of matter,
will be changed to some slight extent, if its position is altered with respect to all the rest
of the Universe. Indeed, the same is the case in the ordinary theory. For no body is
quite without little spaces interspersed within it, altogether incapable of being compressed
or dilated ; & this dilatation & compression undoubtedly occurs in every case of translation,
at least to a slight extent. We, however, consider the measure to be the same so long
as we do not perceive any alteration, as I have already remarked.

22. The consequence of all this is that we are quite unable to obtain a direct knowledge Conclusion reached;
of absolute distances ; & we cannot compare them with one another by a common standard. *he difference. be-

•r . ,1-1 } • tween ordinary

We have to estimate magnitudes by the ideas through which we recognize them ; & to people & phiio-
take as common standards those measures which ordinary people think suffer no change, sophers in the

... . . ' r r . 6 matter of judgment.

.but philosophers should recognize that there is a change ; but, since they know of no case
in which the equality is destroyed by a perceptible change, they consider that the change
is made equally.

23. Further, although the distance is really changed when, as in the case of the translation Although, when
of the ten-foot rod, the position of the points of matter is altered, those real modes which the t^11:*00* r°d 1S

T v i • i i 1 -r i • moved in position,

constitute the distance being altered ; nevertheless it the change takes place in such a way those modes that
that the second distance is exactly equal to the first, we shall call it the same, & say that it is constitute thereia-

, . ' i1.. -11 i • i i tions of the interval

altered in no way, so that the equal distances between the same ends will be said to be the are also altered, yet
same distance & the magnitude will be said to be the same : & this is defined by means of eq"al intervals are

, ,. . ° HIT- -11 i i • i i reckoned as same

these equal distances, just as also two parallel directions will be also included under the name for the reasons
of the same direction. In what follows we shall say that the distance is not changed, or stated-
the direction, unless the magnitude of the distance, or the parallelism, is altered.

24. What has been said with regard to the measurement of space, without difficulty The same obser-
can be applied to time ; in this also we have no definite & constant measurement. We equally11 to ^TimJ;
obtain all that is possible from motion ; but we cannot get a motion that is perfectly uniform. but in it. * is wel1
We have remarked on many things that belong to this subject, & bear upon the nature & ordinary people!
succession of these ideas, in our notes. I will but add here, that, in the measurement of that the. sam<j
time, not even ordinary people think that the same standard measure of time can be translated c^m^bWansfated
from one time to another time. They see that it is another, consider that it is an equal, for the purpose of
on account of some assumed uniform motion. Just as with the measurement of time, m^en/a^V^is be-
so in my theory with the measurement of space it is impossible to transfer a fixed length cause of this that

f • ' i ,. ...r.11 . -,. i r • they fall into error

irom its place to some other, just as it is impossible to transfer a fixed interval ot time, ^th regard to
so that it can be used for the purpose of comparing two of them by means of a third. In space,
both cases, a second length, or a second duration is substituted, which is supposed to be
equal to the first ; that is to say, fresh real positions of the points of the same ten-foot

4o8 PHILOSOPHIC NATURALIS THEORIA

punctorum ejusdem decempedae loca novam distantiam constituentia, ut [276] novus
ejusdem styli circuitus, sive nova temporaria distantia inter bina initia, & binos fines. In
mea Theoria eadem prorsus utrobique habetur analogia spatii, & temporis. Vulgus tan-
tummodo in mensura locali eundem haberi putat comparationis terminum : Philosophi
ceteri fere omnes eundem saltern haberi posse per mensuram perfecte solidam, & continuam,
in tempore tantummodo aequalem : ego vero utrobique aequalem tantum agnosco, nuspiam
eandem.

SUPPLEMENT II 409

rod which constitute a new distance, such as a new circuit made by the same rod, or a
fresh temporal distance between two beginnings & two ends. In my Theory, there is in each
case exactly the same analogy between space & time. Ordinary people think that it is
only for measurement of space that the standard of measurement is the same ; almost all
other philosophers except myself hold that it can at least be considered to be the same
from the idea that the measure is perfectly solid & continuous, but that in time there is only
equality. But I, for my part, only admit in either case the equality, & never the identity.

[277] § HI

Solutio analytica 'Problematis determinants naturam Legis

Virium (d)

Denomin a t i o, ac 25. Ut hasce conditiones impleamus, formulam inveniemus algebraicam, quae ipsam

continebit legem nostram, sed hie elementa communia vulgaris Cartesianae algebras suppone-
mus ut nota, sine quibus res omnino confici nequaquam potest. Dicatur autem ordinata
y, abscissa x, ac ponatur xx = z. Capiantur omnium AE, AG, AI &c. valores cum signo
negative, & summa quadratorum omnium ejusmodi valorum dicatur a, summa productorum
e binis quibusque quadratis b, summa productorum e ternis c, & ita porro ; productum,
autem ex omnibus dicatur /. Numerus eorundem valorum dicatur m. His positis ponatur
zm + azm~* + bzmz + czm~3 &c ...+/= P. Si ponatur P = o, patet aequationis ejus
omnes radices fore reales, & positivas, nimirum sola ilia quadrata quantitatum AE, AG,
AI &c, qui erunt valores ipsius z ; adeoque cum ob xx = z, sit x = ± Vz, patet, valores
x fore tarn AE, AG, AI positivas, quam AE', AG', &c negativas.

Assumptio cujus- 26. Deinde sumatur qusecunque quantitas data per z, & constantes quomodocunque,

1 dummodo non habeat ullum divisorem communem cum P, ne evanescente z, eadem evan-
escat, ac facta x infinitesima ordinis primi, evadat infinitesima ordinis ejusdem, vel inferioris,
ut erit quaecunque formula z' + gz'"1 + hzr~z &c + /, quae posita = o habeat radices
quotcunque imaginarias, & quotcunque, & quascunque reales (dummodo earum nulla sit
ex iis AE, AG, AI &c, sive positiva, sive negativa), si deinde tota multiplicetur per z. Ea
dicatur Q.

Formula continens 2y. Si jam fiat P — Qy = o ; dico, hanc aequationem satisfacere reliquis omnibus

stain.10 hujus curvae conditionibus, & rite determinate valore Q, posse infmitis modis satisfied

etiam postremae condition! expositae sexto loco.
Aequationem fore [278] 28. Nam inprimis, quoniam valores P, & Q positi = o, nullam habent radicem

simplicem non re- *• ' J n * i i i T • TT- i '

soiubiiem in piures. communem, nullum habebunt divisorem communem. Hinc hsec aequatio non potest per
divisionem reduci ad binas, adeoque non est composita ex binis aequationibus, sed simplex,
& proinde simplicem quandam curvam continuam exhibet, quae ex aliis non componitur.
Quod erat primum.

Exhibituram da- 29. Deinde curva hujusmodi secabit axem C'AC in iis omnibus, & solis punctis, E,

ter^ctiomim^curvae] G, I, &c, E', G', &c. Nam ea secabit axem C'AC solum in iis punctis, in quibus y = o,

in datis punctis. ' & secabit in omnibus. Porro ubi fuerit y = o, erit & Qy = o, adeoque ob P — Qy = o ;

erit P = o. Id autem continget solum in iis punctis, in quibus z fuerit una e radicibus

aequationis P = o, nimirum, ut supra vidimus, in punctis E, G, I, vel E', G', &c. Quare

solum in his punctis evanescet y, & curva axem secabit. Secaturam autem in his omnibus

patet ex eo, quod in his omnibus punctis erit P = o. Quare erit etiam Q y = o. Non

erit autem Q = o ; cum nulla sit radix communis aequationum P = o, & Q = o. Quare

erit y = o, & curva axem secabit. Quod erat secundum.

p
Singuias ordinatas *o> Praeterea cum sit P — Qx = o, erit y = -^- ; determinata autem utcunque abscissa

responsuras singu- (J

x, habebitur determinata quaedam z, adeoque & P, Q erunt unicse, & determinatae. Erit
igitur etiam y unica, & determinata ; ac proinde respondebunt singulis abscissis z singulas
tantum ordinatae y. Quod erat tertium.

(d) Heec solutio excerpta est ex dissertatione De Lege Virium in Natura existentium. Acced.it iis, quce Me
sunt eruta, scholium 3 primo adjectum in hue editions Veneta prima. Ipsum problema hie solvendum habetur in ipso
hoc Optre parte i num. 117, ac ejus conditiones num. 118.

410

§111

Analytical Solution of the Problem to determine the nature of the

Law of Forces

25. To fulfil these conditions, we will find an algebraical formula, such as will represent statement, & pre.
our law ; to do so, we shall take it that the first principles of the ordinary Cartesian algebra Potion,
are known ; for, without that, the thing can in no way be accomplished. Suppose that
y is the ordinate, x the abscissa, & let x* = z. Take the values of AE, AG, AI, &c., all with
a negative sign, & let a be the sum of the squares of all such values, b the sum of the products
of all these squares two at a time, c the sum of the products three at a time, & so on ; &
let the product of them all together be called / ; suppose that the number of these values
is m. Then suppose P to stand for

If P is put equal to zero, it is plain that all the roots of this equation will be real & positive,
namely, only the squares of the quantities AE, AG, AI, &c. ; & these will be the values of z.
Hence, since x* = z, & therefore x = ± V z, it is evident that the values of x will be
AE, AG, AI, positive, & AE', AG', &c., negative. [See Fig. i.]

26. Next, assume some quantity that is given by z, & constants, in any manner, so Assumption of
long as it has not got any common measure with P, nor vanishes when z vanishes ; also, to'the^aSer11^'6
if x is made an infinitesimal of the first order, let the quantity become an infinitesimal of

the same order, or of a lower order. Such a formula will be any one such as

zf + gz'~~l -f- hz'~2 -\- -j- /

(if this is put equal to zero, it will have a number of imaginary, & a number of real roots of
ome kind ; but none of them will be equal to AE, AG, AI, &c., whether positive or negative)
-f we multiply the whole by z. Call the product Q.

27. If now we put P — Qy =• o, I say that this equation will satisfy all the remaining Formula containing
conditions of the curve ; & if Q is correctly determined, it can satisfy in an infinite number quired*1113*10

of ways the last condition also, given as sixthly.

28. For, first of all, since the values, P &Q, when separately put equal to zero, have no The equation will
common root, they cannot have a common divisor. Hence this equation cannot by division jj.e ^j^' ^l* re'-
be reduced to two ; & therefore it is not a composite equation formed from two equations, solved into several
but is simple. Hence, it will represent some simple continuous curve, which is not made c

up of others. This was the first condition.

29. Next, this curve will cut the axis C'AC in all those points, & in them only, such it will represent
as E, G, I, &c.,E', G', &c. For it will cut the axis C'AC in those points only, for which »f {££5ecto£bS
y = o, & it will cut it in all of them. Further, when y = O, we have also Qy = 0 ; & there- the curve at given
fore, since P — Qy = o, we have P = o. Now this happens only at those points for which P°mts-

z would be one of the roots of the equation P = o ; that is to say, as we saw above, at
the points E, G, I, &c., E', G', &c. Hence it is only at these points that y will vanish,
& the curve will cut the axis. It is clear that it will cut the axis at all these points, from
the fact that at all these points we have P = 0. Hence also Qy = o. But Q is not
equal to zero, since there is no root common to the equations P = o, Q = o. Hence
y = o, & the curve will cut the axis. This was the second condition.

30. Further, since P — Qy = o, it follows that y = P/Q ; hence, for any determinate To each. abscissa
abscissa x, there will be a determinate z ; & thus P & Q will be uniquely determinate, sporfd one ordinate
Therefore also y will be uniquely determinate ; hence, to each abscissa x there will correspond & °ne only.

one ordinate, y, & only one. This was the third condition.

(d) This solution is abstracted from my dissertation De Lege Virium in Natura existentium. In addition to
these things that have been taken from that dissertation, there has been added a third scholium, which appears for the
first time in this Venetian edition. The problem here set for solution will be found in Art. 117 of the first part of
this work, y the conditions in Art. 118.

411

412 PHILOSOPHIC NATURALIS THEORIA

Abscissis hinc inde 31. Rursus sive x assumatur positiva, sive negativa, dummodo ejusdem longitudinis

^raf aqualerord"*- s'lt> semPer val°r z = xx er^ idem ; ac proinde valores tarn P, quam Q erunt semper iidem.
natas. ' Quare semper eadem y. Sumptis igitur abscissis z aequalibus hinc, & inde ab A, altera

positiva, altera negativa, respondebunt ordinatse aequales. Quod erat quartum.

Primum arcum 32. Si autem x minuatur in infinitum, sive ea positiva sit, sive negativa; semper z

toticumUcumSyarea minuetur in infinitum, & evadet infinitesima ordinis secundi. Quare in valore P decrescent
infinita. in infinitum omnes termini praeter /, quia omnes praeter eum multiplicantur per z, adeoque

valor P erit adhuc finitus. Valor autem Q, qui habet formulam ductam in z totam,

p
minuetur in infinitum, eritque infinitesimus ordinis secundi. Igitur ^-= y augebitur in

infinitum ita, ut evadat infinita ordinis secundi. Quare curva habebit pro asymptoto
rectam AB, & area BAED excrescet in infinitum, & si ordinatae y assumantur ad partes AB,
& exprimant vires repulsivas, arcus asymptoticus ED jacebit ad partes ipsas AB. Quod
erat quintum.

Post eas condi- [279] 33. Patet igitur, utcunque assumpto Q cum datis conditionibus, satisfieri primis
indeterSmina«onem quinque conditionibus curvae. Jam vero potest valor Q variari infinitis modis ita, ut
parem cuicunque adhuc impleat semper conditiones, cum quibus assumptus est. Ac proinde arcus curvae
curva-Tin1 punctis mtercepti intersectionibus poterunt infinitis modis variari ita, ut primae quinque ipsius
datis quibusvis. curvae conditiones impleantur ; unde fit, ut possint etiam variari ita, ut sextam conditionem
impleant.

Quid requirature 34. Si enim dentur quotcunque, & quicunque arcus, quarumcunque curvarum, modo

s^nt ejusmodi, ut ab asymptoto AB perpetuo recedant, adeoque nulla recta ipsi asymptoto
parallela eos arcus secet in pluribus, quam in unico puncto, & in iis assumantur puncta
quotcunque, utcunque inter se proxima ; poterit admodum facile assumi valor P ita, ut
curva per omnia ejusmodi puncta transeat, & idem poterit infinitis modis variari ita, ut
adhuc semper curva transeat per eadem ilia puncta.

standum° "* pra" 35' ^ enim numerus punctorum assumptorum quicunque = r, & a singulis ejusmodi

punctis demittantur rectae parallelae AB usque ad axem C'AC, quae debent esse ordinatae
curvae quaesitae, & singulae abscissae ab A usque ad ejusmodi ordinatas dicantur Mi. Mz,
M3, &c, singulae autem ordinatae N'l, N'z, N'3, &c. Assumatur autem quaedam quantitas
Azr + Bz™ + Cz'"2 + .... -f" Gz, quae ponatur = R. Turn alia assumatur
quantitas T ejusmodi, ut evanescente z evanescat quivis ejus terminus, & ut nullus sit
divisor communis valoris P, & valoris R -f- T, quod facile fiet, cum innotescant omnes
divisores quantitatis P. Ponatur autem Q = R -f- T, & jam aequatio ad curvam erit
P — R y — Ty = o. Ponantur in hac sequatione successive Mi, M2, M3, &c, pro x, &
N'l, N'z, N'3, &c pro y. Habebuntur aequationes numero r, quae singulae continebunt
valores A, B, C, . . . G, unius tantum dimensionis singulos, numero pariter r, & praeterea
datos valores Mi, Mz, M3, &c, Ni, Nz, N3, &c, ac valores arbitrarios, qui in T sunt
coefficientes ipsius z.

Progressus ulterior. 36. per illas aequationes numero r admodum tacile determinabuntur illi valores A, B,

C, . . . G, qui sunt pariter numero r, assumendo in prima aequatione, juxta methodos
notissimas, & elementares valorem A, & eum substituendo in sequationibus omnibus sequen-
tibus, quo pacto habebuntur aequationes r — I. Hae autem ejecto valore B reducentur
ad r — z, & ita porro, donee ad unicam ventum fuerit, in qua determinato valore G, per
ipsum ordine retrograde determinabuntur valores omnes praecedentes, singuli in singulis
aequationibus.

Conciusio, & cohaer. 37. Determinatis hoc pacto valoribus A, B, C, . . . G [280] in aequatione P — Ry

pr^dTtirTslon! --Ty = o, siveP-Qy = o, patet positis successive pro x valorib™ Mi, Mz, M3, &c,

ditionibus. debere valores ordinatae y esse successive Ni, Nz, N3, &c ; ac proinde debere curvam

transire per data ilia puncta in datis illis curvis : & tamen valor Q adhuc habebit

omnes conditiones prsecedentes. Nam imminuta z ultra quoscunque limites, minuentur

singuli ejus termini ultra quoscunque limites, cum minuantur termini singuli valoris T,

qui ita assumpti sunt, & minuantur pariter termini valoris R, qui omnes sunt ducti in z,

& praeterea nullus erit communis divisor quantitatum P, & Q, cum nullus sit quantitatum

P, & R + T.

inde contactus, 38. Porro si bina proxima ex punctis assumptis in arcubus curvarum ad eandem axis

partem concipiantur accedere ad se invicem ultra quoscunque limites, & tandem congruere,
factis nimirum binis M aequalibus, & pariter asqualibus binis N ; jam curva quaesita ibidem

aCC< 5

SUPPLEMENT III 413

31. Again, whether x is taken positive or negative, so long as its length is the same, T? equal at>-

i i r -11 i i TT i i ' t i i T« a r\ -11 i scissae, therefore,

the value of z, or x*, will be the same. Hence the values of both r & Q will be the same, there will corre-

Hence y will always be the same for either. Hence, if equal abscissae x are taken one on sP°nd equal or-

either side of A, the one positive & the other negative, the corresponding ordinates will side1 of' the origin!
be equal. This was the fourth condition.

32. Now, if x is diminished indefinitely, whether it is positive or negative, z will be The first arc wil1
also diminished indefinitely, & will become an infinitesimal of the second order. Hence, branch aSwith>t°an
every term in the value of P, except /, will diminish indefinitely ; for each of them except infinite area,
this one has a factor z. Thus the value of P will remain finite. But the value of Q, in

which the whole expression was multiplied by z, will diminish indefinitely ; & it will become
an infinitesimal of the second order. Hence y, which is equal to P/Q, will be increased
indefinitely, so that it becomes an infinity of the second order. Therefore, the curve will
have the straight line AB as an asymptote, & the area BAED will become infinite ; also,
if AB is taken to be the positive direction for the ordinates y, these will represent repulsive
forces, & the asymptotic arc ED will fall in the direction given by AB. This was the fifth
condition.

33. Hence, it is clear that, however Q is chosen subject to the given conditions, the After these con-
first five conditions for our curve will be satisfied. Now, the value of Q can be varied ^ifMed htherebetm
in an infinite number of ways, such that it will still fulfil the conditions under which it remains an equal
was assumed. Then the arcs of the curves intercepted between the intersections with indeterminat1°n
the axis could be varied in an infinite number of ways, such that the first five conditions any given curves
for the curve are satisfied. Hence it follows that they can be varied also, in such a way at any given P°ints-
that the sixth condition is satisfied.

34. Now, if any number of arcs of any kind, belonging to any curves, are given; so The conditions
long as these are such that they continually recede from the asymptote AB, & therefore th^o'ug^g^vTn
such that no straight line parallel to this asymptote will cut any of them in more than one points of these
point ; & if in these arcs there are taken any number of points, no matter how close they c

are together, a value of P can be obtained quite easily, such that the curve will pass through
all these points. Moreover, this can be done in an infinite number of ways, such that
the curve will still pass through all these points in every case.

35. For, let the number of points taken be any number r. From each of these points, How this can be
let a straight line be drawn parallel to AB, to meet the axis C'AC ; these must be ordinates mana8ed-

of the curve required. Let the several abscissae measured from A to these ordinates be
MI, M», Ms, &c. ; & let the corresponding ordinates be NI, Nz, Na, &c. Then assume
some quantity Azf + B z^1 + Cz'1"2 + . . . + Gz, & suppose that this is R. Next,
take another quantity, T, of such a kind that, when z vanishes, each term of T vanishes,
& there is no common divisor of P & R + T. This can easily be done, since the divisors
of the quantity P are known. Now, suppose that Q = R+ T ; the equation to the curve,
will then be P — Ry -- Ty = O. In this equation, substitute in succession MI, Ma, M3
&c. for x, & NI, NS, Ns &c. for y. Then we shall have r equations, each of which will

contain the values A, B, C, , G, which are also r in number ; & these will all appear

linearly. The equations will also contain, in addition, the given values MI, M2, M3, &c.,
NI, Nj, NS, &c., & the arbitrary values which aopear as the coefficients of z in the expression
T.

36. From these equations, r in number, the values of A, B, C, . . . , G, which are also Further progress.
r in number, can quite easily be determined. Thus, from the first equation, according

to well-known elementary methods, obtain the value of A in terms of the rest, & substitute
this value in each of the other equations. In this way we shall obtain r — I equations.
Eliminating B from these, we shall get r — ^ equations ; & so on, until at last we shall come
to a single equation. Having determined from this the value of G, we can determine, by
retracing our steps, the preceding values in succession, one value from each set of equations.

37. The values of A, B, C, . . . . , G, in the equation P — Ry — Ty = o, or Conclusion ; agree-
P — Qy = o, having been thus found, it is clear that, if the values Ml5 M,, M3, &c., are ^nt preceding
substituted for x in succession, the values of y will be NI, N», N3, &c. Hence, the curve conditions,
must pass through the given points on the given arcs ; & still the value of Q will satisfy

all the preceding conditions. For, if z is diminished beyond all limits, each of its terms
will be diminished beyond all limits ; since each of the terms of the value of T, according
to the supposition made, will be so diminished, & likewise each of the terms of R, which
all contain a factor z. In addition, there will be no common divisor of P & Q, since there
is none for the quantities P & R + T.

38. Again, if two of the chosen points, next to one another in the arcs of the curves, Hence contacts,
are supposed to approach one another on the same side of the axis beyond all limits, & approach' °ofS' any
finally to coincide with one another, namely, by making two values of M equal to one

another, & therefore also the corresponding values of N, then also the required curve will

4H PHILOSOPHIC NATURALIS THEORIA

tanget arcum curvae datae : & si tria ejusmodi puncta congruant, earn osculabitur : quin
immo illud prasstari poterit, ut coeant quot libuerit puncta, ubi libuerit, & habeantur
oscula ordinis cujus libuerit, & ut libuerit sibi invicem proxima, arcu curvae datse accedente,
ut libuerit, & in quibus libuerit distantiis ad arcus, quos libuerit curvarum, quarum libuerit,
& tamen ipsa curva servante omnes illas sex conditiones requisitas ad exponendam legem
illam virium repulsivarum, ac attractivarum, & datos limites.

Adhuc indetermina- »Q Cum vero adhuc infinitis modis variari possit valor T; infinitis modis idem

tio relicta pro infmi- J* * . i • r- • • T • •• • • i i • v • -i

tis modis. praestan poterit : ac promde infinitis modis mvenin poterit curva simplex datis conditiombus

satisfaciens. Q.E.F.

Posse & axem ^O- Coroll. I. Curva poterit contingere axem C'AC in quot libuerit punctis, &

con mgere. os a.n, contmgere gimu^ ac secare in iisdem, ac proinde eum osculari quocunque osculi genere.
Nam si binse quaevis e distantiis limitum fiant aequales ; curva continget rectam C'A,
evanescente arcu inter binos limites ; ut si punctum I abiret in L, evanescente arcu IKL ;
haberetur contactus in L, repulsio per arcum HI perpetuo decresceret, & in ipso contactu
IL evanesceret, turn non transiret in attractionem, sed iterum cresceret repulsio ipsa per
arcum LM. Idem autem accideret attractioni, si coeuntibus punctis LN, evanesceret
arcus repulsivus LMN.

Posse contingere ^j. Si autem tria puncta coirent, ut LNP ; curva contingeret simul axem C'AC, &

ab eodem simul secaretur, ac promde haberet in eodem puncto contactus flexum
contrarium. Haberetur autem ibidem transitus ab attractione ad repulsionem, vel vice
versa, adeoque verus limes.

Quid congruentia 4.2. Eodem pacto possunt congruere puncta quatuor, quinque, quotcunque : & si

congruat numerus punctorum par; habebitur contactus : si impar ; contactus simul, &
sectio. Sed quo plura puncta coibunt ; eo magis curva accedet ad [281] axem C'AC in
ipso limite, eumque osculabitur osculo arctiore.

Posse axem secari 43. Coroll. 2. In iis limitibus, in quibus curva secat axem C'AC, potest ipsa curva

anguiis,U&aSCquavis secare eundem in quibuscunque angulis ita tamen, ut angulus, quern efficit ad partes A
g nit u d i n e arcus curvae in perpetuo recessu ab asymptote appellens ad axem C'AC non sit major recto,
& ibidem potest aut axem, aut rectam axi perpendicularem contingere, aut osculari, quo-
cunque contactus, aut osculi genere, nimirum habendo in utrolibet casu radium osculi
magnitudinis cujuscunque, & vel utcunque evanescentem, vel utcunque abeuntem in
innnitum.

Demonstratio : A A. Nam pro illis punctis datis in arcubus curvarum quarumcunque, quas curva

umitatio necessana. jnventa pOtest vel contingere, vel osculari quocunque osculi genere, ex quibus definitus
est valor R, possunt assumi arcus curvarum quarumcunque secantium axem C'AC, in
angulis quibuscunque : solum quoniam semper arcus curvae, ut *Ny debet ab asymptote
recedere, non poterit punctum ullum t praecedens limitem N jacere ultra rectam axi perpen-
dicularem erectam ex N, vel punctum y sequens ipsum N jacere citra ; ac proinde non
poterit angulus AN*, quern efficit ad partes A arcus *N in perpetuo recessu ab asymptote
appellens ad axem C'AC, esse major recto.

Quid possint arcus ^ Possunt autem arcus curvarum assumptarum in iisdem punctis aut axem, aut

taVum™ omTiPa rectam axi perpendicularem contingere, aut osculari, quocunque contactus, aut osculi

posse & inventam. genere, ut nimirum sit radius osculi magnitudinis cujuscunque, & vel utcunque evanescens,

vel utcunque abiens in innnitum. Quare idem accidere poterit ut innuimus, & arcui

curvae inventae, quae ad eos arcus potest accedere, quantum libuerit, & eos contingere, vel

osculari quocunque osculi genere in iis ipsis punctis.

Conditio necessana, ^g. Solum si curva inventa tetigerit in ipso limite rectam axi C'AC perpendicularem,

naturaUJUS CUrV8e debebit simul ibidem eandem secare ; cum debeat semper recedere ab asymptote, adeoque
debebit ibidem habere flexum contrarium.

Coroi. i includi in ^7. Scholium I. Corollarium I est casus particularis hujus corollarii secundi, ut patet :

sed libuit ipsum seorsum diversa methodo, & faciliore prius eruere.

Quid ubivis etiam ^g. Coroll. 3. Arcus curvae etiam extra limites potest habere tangentem in quovis

angulo inclinatam ad axem, vel ei parallelam, vel perpendicularem cum iisdem contactuum,

& osculorum conditionibus, quae habentur in corollario 2.
Demonstratio ea- ^ Demonstratio est prorsus eadem : nam arcus curvarum dati, ad quos arcus

curvse inventae potest accedere ubicunque, quantum libuerit, possunt habere ejusmodi

conditiones.

m a
arcuum.

SUPPLEMENT III 415

touch the arc of the given curve at this point. If three such points coincide with one
another, it will osculate the given curve. Indeed, it can be brought about that any number
of points desired shall coincide, & thus osculations of any order desired can be obtained.
These may be as close together as desired, the arc approaching the given curve to any desired
degree of closeness ; or they may be at any distances from any of the arcs of any of the
curves, as desired. Yet the curve will observe all those six conditions, which are required
for representing the law of repulsive & attractive forces, as well as the limit-points.

39. Now, since the value of T can still be varied in an infinite number of ways, this can There is stm left
be brought about in an infinite number of ways. Hence, in an infinite number of ways, coun^ie^wa10^ '"
a simple curve can be found satisfying the given conditions. Q . E . F .

40. Cor. i. The curve may touch the axis C'AC in any desired number of points ; it is possible also
or at the same time touch & cut it at the same points ; & hence it may osculate the axis touchhtheCaxIs o°
with any kind of osculation. For, if any two of the distances for the limit-points become to osculate it, etc.
equal, the curve will touch the straight line C'A, the arc between these two limit-points

vanishing. Thus, if the point I should go off to L, the arc IKL vanishing, we should have
contact at L, & repulsion would continually decrease along the arc HI, vanish at the point
of contact IL ; after that it would not become an attraction, but the repulsion would
continually increase along the arc LM. The same thing would also happen in the case
of attraction, if, owing to the points L,N coinciding, the repulsive arc LMN should vanish.

41. Again, if three points, say L,N,P, should coincide, the curve would at the same *t is possible that
time touch the axis C'AC & intersect it ; thus, at that point of contact there would be simultaneous7 con!
contrary flexure. Also, there would be there a passage from attraction to repulsion, or tact & section of
vice versa, & therefore a true limit-point.

42. In the same way, four points may coincide, or five, or any number. If the number The result of the
of points that coincide is even, there will be touching contact; if the number is odd, there several ninterse°c-
will be contact & intersection at the same time. The greater the number of the points tions.

that coincide, the more the curve will approach to coincidence with the axis C'AC at that
limit-point ; & thus the higher the order of the osculation.

43. Cor. 2. At these limit-points, where the curve cuts the axis C'AC, the curve The axis may be
may cut it at any angle ; but in such a way that the angle, which the arc of the curve, atlny^ngfe, C&rby
in its continuous recession from the asymptote, makes with the direction of A as it comes arcs of any size,
up to the axis C'AC, is not greater than a right angle ; & it may touch either the axis or

the straight line at right angles to the axis, or osculate the axis ; the contact or the osculation
being of any order. That is to say, it may have in either case a radius of osculation of
any magnitude whatever, either vanishing or becoming infinite, in any way whatever.

44. For, we may take as our chosen points in the arcs of any curves, which the curve Demonstration;
of forces is found to touch or to osculate with an osculation of any order, from which tion.SSa

the value of R is determined, arcs of any curves cutting the axis C'AC at any angles. Except
that, since the arc of the curve, such as tNy, must always recede from the asymptote, it
would not be possible for any point such as t, which precedes the limit-point N, to lie on
the far side of the straight line perpendicular to the axis erected at N ; or for the point y,
which follows N, to lie on the near side of this perpendicular. Thus, the angle AN/, which
it makes with the direction of A, as the arc tN continually recedes from the asymptote,
as it comes up to the axis C'AC, cannot be greater than a right angle.

45. Again, the arcs of the assumed curves may, at these points either touch the axis what the arcs of
or the straight line perpendicular to the axis, or they may osculate, the contact or the Say^^the^me
osculation being of any order ; that is, the radius of osculation may be of any magnitude properties may aii
whatever, either vanishing or becoming infinite, in any way. Hence, as I said, this may curve^oumi by the
also be the case for an arc of the curve that has been found ; for it can be made to approximate

as closely as desired to these curves, so as to touch them or osculate them, with any order
of osculation, at these points.

46. Except that, if the curve should touch at the limit-point the straight line Necessary condi-
perpendicular to the axis C'AC, it must at the same time cut it at that point ; for the the ' nature^f The
curve must always recede from the asymptote, & thus is bound to have contrary flexure curve-

at the point.

47. Scholium i. The first corollary is a particular case of the second, as is evident. Th<r *jrs,t corollary

•n e i ^ • c. • i_ • j j t L J-JT a • ls included in the

but 1 preferred to take it first, with an independent proof by a different & an easier second,
method.

48. Cor. 3. Even beyond the limit-points, the arc of the curve can have a tangent What happens also
inclined at any angle to the axis, or parallel to it, or perpendicular to it ; with the same the "mut-pointl°n
conditions as to contact or osculation as we had in the second corollary.

49. The proof is exactly the same as before ; for, the given arcs of the curves, to ^^ the same as
which the arc of the curve that is found can be made to approximate as closely as desired,

may have the conditions stated.

PHILOSOPHIC NATURALIS THEORIA

sssposs are 5?- Coroll. 5. Mutata abscissa per quodcunque intervallum datum, potest ordinata

ad mutationem or- mutari per aliud quodcunque datum utcunque minus, vel majus ipsa mutatione abscissae,
quan^nque^10 * & ut-[282]-cunque majus quantitate quacunque data ; ac si differentia abscissa; sit infini-
tesima, & dicatur ordinis primi ; poterit differentia ordinatae esse ordinis cujuscunque,
vel utcunque inferioris, vel intermedii, inter quantitates finitas, & quantitates ordinis
primi.

prO 51- Patet primum ex eo, quod, ubi determinatur valor R, potest curva transire per

quotcunque, & quaecunque puncta, adeoque per puncta, ex quibus ducts ordinals sint
utcunque inter se proximae, & utcunque insequales.

iSe s'mor°uvm ^2' 7*tQt secundum : quia in curvis, ad quas accedit arcus curvs invents vel quas

ordine. osculatur quocunque osculi genere, potest differentia abscissae ad differentiam ordinats

esse pro diversa curvarum natura in datis earum punctis in quavis ratione, quantitatis
infinitesimae ordinis cujuscunque ad infinitesimam cujuscunque alterius.

mottpdere a . 53> Scholi.um 2' Illud notandum, ubicunque fuerit tangens curvae invents inclinata
positione tangentis. 'm angulo finito ad axem, fore differentiam abscissae ejusdem ordinis, ac est differentia
ordinats : ubi tangens fuerit parallela axi, fore differentiam ordinats ordinis inferioris,
quam sit differentia abscissae, & vice versa, ubi tangens fuerit perpendicularis axi.

mnr™ 54- Prseterea notandum : si abscissa fuerit ipsa distantia limitis, quae vel augeatur, vel

termini, tiir in Jim* • t*^v * t* * • i • •

ite. mmuatur utcunque ; differentia ordinats erit ipsa ordinata Integra ; cum nimirum in

limite ordinata sit nihilo aequalis.
Posse arcus utcun. 55. Coroll. c. Arcus repulsionum, vel attractionum intercepti binis limitibus

que rccedcre ftD **_ i i i»i»i /•»

axe. quibuscunque, possunt recedere ab axe, quantum libuent, adeoque fieri potest, ut alii

propiores asymptote recedant minus, quam alii remotiores, vel ut quodam ordine eo minus
recedant ab axe, quo sunt remotiores ab asymptoto, vel ut post aliquot arcus minus
recedentes aliquis arcus longissime recedat.

Demonstrate. ^ Omnia manifesto consequuntur ex eo, quod curva possit transire per quaevis

data puncta.

m0uSmhcrusrirs°ySmp. 57' CorolL 6- Potest curva ipsum axem C'AC habere pro asymptoto ad partes C',

toticum.&aiiacrura & C ita, ut arcus asymptoticus sit vel repulsivus, vel attractivus ; & potest arcus quivis
asymptotica. bims limitibus quibuscunque interceptus abire in infinitum, ac habere pro asymptoto

rectam axi perpendicularem, utcunque proximam utrilibet limiti, vel ab eo remotam.

Ratio praestandi eg. Nam si concipiatur, binos postremos limites coire, abeuntibus binis intersectionibus

primum. * •••*«•• . . « . .

in contactum, turn concipiatur, ipsam distantiam contactus excrescere in infinitum ; jam
axis aequivalet rectae curvam tangenti in puncto infinite remoto, adeoque evadit asymptotus :
& si arcus evanescens inter postremos duos limites coeuntes fuerit arcus repulsionis ;
postremus arcus asymptoticus erit arcus attractionis. Contra vero, si arcus evanescens
fuerit arcus attractionis.

& [2^5l 59' Eodem pacto si concipiatur, quamvis ordinatam respondentem puncto cuilibet,
per quod debet transire curva, abire in infinitum ; jam arcus curvae abibit in infinitum, &
erit ejus asymptotus in ilia ipsa ordinata in infinitum excrescens.

exhfberiVpermfunc> ^°' ^c^°^um 3- Ope formulae exhibentis curvam propositam habetur lex virium

tionem distantiae, expressa per functionem quandam distantiae constantem plurimis terminis, immo per

^1°z™e1r°en<d1aem ^qu*11^0116111 commiscentem abscissam, & ordinatam, ac utriusque potentias inter se, &

unicam potentiam : cum rectis datis, non per solam ipsius distantiae potentiam. Sunt, qui censeant expres-

sionem per solam potentiam debere prseferri expressioni per functionem aliam, quia haec sit

simplicior, quam ilia, & quia in ilia praeter distantias debeant haberi aliquae aliae parametri,

quae non sint solae distantiae ; dum in formula - exprimente x distantias, distantiae solae

oc

rem confidant, videatur autem vis debere pendere a solis distantiis, potissimum si sit
quaedam essentialis proprietas materiae : praeterea addunt, nullam fore rationem suffi-
cientem, cur una potius, quam alia parameter expressionem virium deberet ingredi, si
parametri sint admiscendae.

Qua occasione haec 6i. Hsc agitata sunt potissimum ante hos aliquot annos in Academia Parisiensi,

a^tetaY^parfeiensi cum censeretur, motum Apogei Lunaris observatum non cohaerere cum gravitate decrescente
Academia. ^ ratione reciproca duplicata distantiarum, & ad ipsum exhibendum adhiberetur gravitas

SUPPLEMENT III 417

50. Cor. 4. If the abscissa is changed by any given interval, the ordinate can be The change of the
changed by any other given interval however much the latter may be smaller or greater an^^atio^to ^he
than the change of the abscissa, or however much greater than any given quantity it may change of the ordi-
be. Further, if the difference in the abscissa is infinitesimal, & we call it an infinitesimal nate'

of the first order, then the difference in the ordinate may be of any order, either of any
order below the first whatever, or intermediate between finite quantities & quantities of
this first order.

51. The first part is evident from the fact that, when the value of R is determined, P""?0* for fini*e
a curve can be made to pass through any number of points of any sort ; & thus, through

points, from which ordinates are drawn as close to one another as we please, & unequal
to one another in any way.

52. The second part is evident, because in the curves, to which the arcs of the curve The same for any
found approximates, or which it osculates with any order of osculation, the difference of ^fsr of mfimtesi-
the abscissa can bear any ratio to the difference of the ordinate for a different nature of

the curves at given points on them ; this ratio may be that of an infinitesimal quantity
of any order to an infinitesimal quantity of any other order.

53. Scholium 2. It is to be observed that, whenever the tangent to the curve that This relation de.
has been found is inclined at a finite angle to the axis, the difference of the abscissa is of tkm ^"t

the same order as the difference of the ordinate ; when the tangent is parallel to the axis, gent,
the difference of the ordinate will be of an inferior order to the difference of the abscissa ;
& the opposite is the case when the tangent is perpendicular to the axis.

54. In addition, it is to be observed that, if the abscissa corresponds to a limit-point, The. case
& this is either increased or diminished in any way, the difference of the ordinate will be mg^at
the whole ordinate itself, for at the limit-point itself the ordinate is indeed equal to zero, limit-points.

55. Cor. 5. The arcs of repulsion or attraction, which are intercepted between any The arcs may re.
pair of limit-points, may recede from the axis to any extent ; & thus, it may happen that ^ aeny gSent6 aX1S
some that are nearer to the asymptote may recede less than others that are more remote ;

or that, to any order, they may recede the less, the further they are from the asymptote ;
or that, after a number of arcs that recede less, there may be one which recedes by a very
large amount.

56. Everything clearly follows from the fact that the curve can be made to pass through Prooi ot this state-
any given points.

1:7. Cor. 6. The curve may have the axis C'AC as an asymptote in the directions ,The .curve can

, f^fi .", . , , J, . . . , t • • have lts last branch

or C & C, m such a manner that the asymptotic arc is either repulsive or attractive ; also any asymptotic, & also
arc intercepted between a pair of limit-points may go off to infinity, & have for an other asymptotic

• iv ,. , i , ' , v • t v • branches.

asymptote a straight line perpendicular to the axis, however near or tar irom either limit-
point.

58. For, if we suppose that the last two limit-points coincide, as the two intersections Jhe Proof of the

. • , „ i r • f • i i first part of this

coincide & become a point where the curve touches the axis ; & then suppose that the statement,
distance of this point of contact becomes infinite ; then the axis will become equivalent
to a straight line touching the curve at a point infinitely remote, & will thus be an asymptote.
If the vanishing arc that is intercepted between those two last coincident limit-points
should be an arc of repulsion, the last asymptotic arc will be an arc of attraction. But
the opposite would be the case if the vanishing arc should be an arc of attraction.

59. In the same way, if it is supposed that any ordinate corresponding to any point, The proof of the
through which the curve has to pass, should go off to infinity ; then the arc of the curve statement. °f the
will also go off to infinity, and that ordinate, as it increases indefinitely, will become an

asymptote of the curve.

60. Scholium 3. By the help of the formula corresponding to the proposed curve, Th?r law of foilce?

,, .... 9 . ] . . , ,-,.rr. ° . .. r r . , 'is here represented

the law of forces is obtained expressed as a definite function of the distance with many by a function of the
terms ; or rather, by means of an equation involving the abscissa & the ordinate, & powers distance; many

, . i ' . • i i • i • i f i i • Vni others think that a.

of these, along with given straight lines, & not by a single power of the distance. There single power of the
are some who think that representation by means of a single power is to be preferred to djstance .1S prefer-

. rr. , J , , . ° . r , , r .. able ; their reasons.

representation by another function ; because the latter is simpler than the former ; &
because in it, besides the distances, there are bound to be other parameters that are not
merely distances. Whereas, in the formula i/;e»», where x represents the distances, the
distances alone settle the matter ; & it is seen that the force must depend on the distance
alone, especially if it should be an essential property of matter. Besides, they add, there
is no sufficient reason why any one, rather than any other, parameter should enter the
expression for the forces, if parameters are to be admitted.

61. This question came in for a large amount of discussion a number of years ago in The occasion on

... r T> • T-I • , 111 • rit 11 which this question

the Academy of Pans. For, it was thought that the motion of the lunar apogee, as observed, was discussed in the
did not agree with the idea of gravity decreasing in the inverse duplicate ratio of the distances. Academy of Paris.
They considered that an expression for gravity should be employed, in which it was represented

EE

4i 8 PHILOSOPHIC NATURALIS THEORIA

, . . a b . . . ....

expressa per bmommm — -\ -- cujus pars prior in magnis, pars posterior in exiguis

X X

distantiis respectu socise partis evanesceret ad sensum, sed ilia prior in distantia Lunae a
Terra adhuc turbaret hanc posteriorem, quantum satis erat ad earn praestandam rem.
Atque earn ipsam binomii expressionem adhibuerant jam plures Physici ad deducendam
simul ex eadem formula gravitatem, & majores minimarum particularum attractiones, ac
multo validiorem cohaesionem, ut innuimus num. 121 : atque hae difficultates in Parisiensi
Encyclopaedia inculcantur ad vocem dttractio, Tomo I turn edito.

Occasionem substi- 62. Paullo post, correctis calculis innotuit, motum Apogei lunaris ea composita formula

tuendi turn functio- • j • . . ,

nem cessasse, sed non mdigere : at rationes contra id propositae, quae multo magis contra meam virium
rationes contra legem pugnarent, meo quidem iudicio nullam habent vim. Nam in primis quod ad

allatas nullam • v • • 1-111 • j- •• /•

habere vim : curvas simplicitatem pertmet, hie habent locum ea omnia, quas dicta sunt in ipso opere num. no
omnes unifprmis de simplicitate curvarum. Formula exprimens solam potentiam quandam distantiae
aqvuTsimpiices! Se designatae per abscissam exprimit ordinatam ad locum geometricum pertinentem ad
familiam, quam exhibet [284] y = x", qui quidem locus est Parabola quaedam ; si m
sit numerus positivus, nee sit unitas : recta ; si sit unitas, vel zero : quaedam Hyperbola ;
si sit numerus negativus : formula autem continens functionem aliam quamvis exprimit
ordinatam ad aliam curvam, quae erit continua, & simplex, si ilia formula per divisionem non
possit discerpi in alias plures. Omnes autem ejusmodi curvae sunt aeque simplices in se,
& alia; aliis sunt magis affines, aliae minus. Nobis hominibus recta est omnium simplicissima,
cum ejus naturam intueamur, & evidentissime perspiciamus, ad quam idcirco reducimus
alias curvas, & prout sunt ipsi magis, vel minus affines, habemus eas pro simplicioribus, vel
magis compositis ; cum tamen in se aeque simplices sint omnes illae, qua ductum uniformem
habent, & naturam ubique constantem.

Esse «eque simpli- 63. Hinc ipsa ordinata ad quamvis naturae uniformis curvam est quidam terminus

cem relationem • .. . . \ . . . , . , •,. j i_ • • *.

ordinatarum ad simplicissimae relatioms cujusdam, quam habet ordinata ad abscissam, cui termmo impositum
abscissas : termino- est generale nomen functionis continens sub se omnia functionum genera, ut etiam quam-

rum multitudinem , . „ • i i • j • j • j- • j •

pro ea exprimenda cunque solam potentiam, & si haberemus nomma ad ejusmodi functiones denommandas
oriri a nostro cog- singillatim ; haberet nomen suum quaevis ex ipsis, ut habet quadratum, cubus, potestas

noscendi mode. • c- • • j- i •

quaevis. Si omnia curvarum genera, omnes ejusmodi relationes nostra mens mtueretur

immediate in se ipsis ; nulla indigeremus terminorum farragine, nee multitudine signorum

ad cognoscendam, & enuntiandam ejusmodi functionem, vel ejus relationem ad abscissam.

Origo ejus modi ab 64. Verum nos, quibus uti monui recta linea est omnium locorum geometricorum

habe'muTnos loam- simplicissima, omnia referimus ad rectam, & idcirco etiam ad ea, quae oriuntur ex recta,

ines naturae soiius ut est quadratum, quod fit ducendo perpendiculariter rectam super aliam rectam aequalem,

omnw curvas refer* & cubus> qui fit ducendo quadratum eodem pacto per aliam rectam primae radici aequalem,

imus. quibus & sua signa dedimus ope exponentium, & universalizando exponentes efformavimus

nobis ideas jam non geometricas superiorum potentiarum, nee integrarum tantummodo,

& positivarum, sed etiam fractionariarum, & negativarum : & vero etiam, abstrahendo

semper magis, irrationalium. Ad hasce potentias, & ad producta, quae simili ductu conci-

piuntur genita, reducimus caeteras functiones omnes per relationem, quam habent ad

ejusmodi potentias, & producta earum cum rectis datis, ac ad earn reductionem, sive ad

expressionem illarum functionum per hasce potentias, & per haec producta, indigemus

terminis jam paucioribus, jam pluribus, & quandoque etiam, ut in functionibus transcendent-

alibus, serie terminorum infinita, quae ad valorem, vel naturam functionis propositae accedat

semper magis, utut in hisce casibus earn nunquam ac-[28s]-curate attingat : habemus

autem pro magis, vel minus compositis eas, quae pluribus, vel paucioribus terminis indigent,

sive quae ad solas potentias relationem habent propiorem.

Aiiud mentium 65. At si aliud mentium genus aliam curvam ita intime cognosceret, ut nos rectam;

haberet pro maxime simplici solam ejus functionem, & ad exprimendum quadratum, vel

potentiae necessario aliam potentiam, contemplaretur illam eandem relationem, sed inverse assumptam ita, ut
Tem! Ve^maljorem incipiendo a functione ipsa per earn, & per similes ejus functiones, ac functionum citeriorum
fan-aginem. functiones ulteriores, addendo, ac subtrahendo deveniret demum ad quaesitam. Relatio

potentiae ad functionem, & nexus mutuus compositionem habet, & multitudinem terminorum

inducit : uterque relationis terminus est in se aeque simplex.

Sola etiam poten-

in'cTudi etta^amTd6 *>6. Quod pertinet ad parametros, quas dicitur includere functio, non autem potentia

nos homines para- distantiae, non est verum id ipsum, quod potentia parametros non includat. Formula

metres plures: para- •. . . . , .

meter in unitate _ includit unitatem ipsam, quae non est aliquid in se determmatum, sed potest expnmere

arbitraria, & affix- xm

certame£Santiamd magnitudinem quamcunque. Et quidem ea species includit omnes species Hyperbolarum,

SUPPLEMENT III 419

by the formula of two terms, afxz + b/xz ; of this, the first part at large distances, & the
last part at very small distances, would practically become evanescent with respect to the
other part associated with it. But the first part, for the distance of the Moon from the
Earth, would still disturb the last part sufficiently to account for the observed inequality.
Already, several Physicists had employed such an expression with two terms to deduce
at the same time from the one formula both gravity & the greater attractions of very
small particles, & much more so the still stronger forces of cohesion, as I have mentioned
in Art. 121. These difficulties are included in the Encyclopedia Parisiensis under the
heading Attraction, in Vol. I published at that time.

62. Shortly afterwards, the calculations were corrected & it was found that the motion The reasons for
of the lunar apogee did not necessitate this compound formula. But the arguments brought formula" *) or *the
forward against it, which were still more in opposition to this Theory of mine with regard function, which
to the law of forces, have no weight, at any rate in my eyes. For, in the first place, as ceawd toexitt^
regards simplicity, all those things held good in this case, which I stated in this work, but the arguments
Art. 116, with regard to simplicity of curves. A formula in terms of a single power of agamst it

the distance represented by an abscissa expresses the ordinate of a geometrical locus belonging weight ; all curves

to the family, represented by y = xm ; & this locus is a Parabola, if m is any positive number

except unity ; a straight line, if m is unity or zero ; & a hyperbola, if m is a negative number, equally simple

But a formula containing some other function expresses the ordinate of some other curve ;

& this will be continuous & simple, if the formula cannot be separated by division into

several others. Further, all such curves are equally simple in themselves ; & some of

them are more, some less, of the same nature as others. To us men, a straight line is the

simplest of all ; for we observe its nature & understand it clearest of all. To it therefore

we refer all other curves ; & according as they are more or less like it in nature, we consider

them to be the more or less simple. However, in themselves, all curves, which are composed

of a continuous line & have a constant nature everywhere, are equally simple.

63. Hence, the ordinate to any curve of a uniform nature is some term of some very The relation be.
simple relation that the ordinate has to the abscissa. To this term there is given the ^the ^bscissT^s
general name, function ; this name includes every kind of function, for instance, even a equally simple ; the
single power. If we had names to denote such functions singly, each of them would have usedn't o°f express
its own name, just as a square, a cube, or any other power. If our minds were capable of this relation arises
viewing all kinds of curves, & all such relations in themselves, at a glance, then there would ^nowing^t^7
be no need of a medley of terms, & a multitude of signs in order to know & state such a

function or its relation to the abscissa.

64. But we, to whom, as I mentioned, the straight line is the simplest of all geometrical The origin of this

• • *« 11 • T T j i r i i • • f method comes from

loci, refer all curves to a straight line, and therefore also to all those things that arise from a the intuition which
straight line ; such as a square, which is formed by moving a straight line perpendicular we men have of.the

• iv >. , '. . ' V. , . ,° , , r . , nature of a straight

to another straight line which is equal to it ; & a cube, which is formed by moving the iine alone, to which

square in the same way all along another straight line equal to its prime root. To these we refer ail curves.

we have given their own signs by the help of exponents ; &, generalizing exponents, we

have formed for ourselves ideas, that are not now geometrical, of higher powers ; & these

not integral only, & positive, but also fractional, & negative ; & indeed, by continual

abstraction, ever more & more, ideas of irrational powers. To these powers, & to products

which may be considered to arise in a similar fashion, we reduce all other functions, by

means of the relation they bear to such powers & their products with given straight lines.

For this reduction, or expression of the functions by means of these powers & these products

we require sometimes more, sometimes less, terms ; even when, as in the case of transcendental

functions, we have to use an infinite series of terms, which approximates more & more

closely to the value & the nature of the given function, although in such cases it never

actually reaches this value. Moreover, we consider these to be more or less composite,

according as they require more or less terms, or have a nearer relation to single powers.

65. But if another type of mind knew another curve as intimately as we know the straight Another type of
line, it would consider a single function of that curve to be the most simple of all ; &, to ™en reiatkm^of CSa
express a square or another power, it would consider the self-same relation, inversely taken, power would neces.
so that, beginning with the function, through it & like functions of it, & of higher functions an^quafor greater
of these lower functions, by addition & subtraction, the mind would finally arrive at the medley of terms.
function required. The relation of a power to a function, & the mutual connection, has

a compositeness, & leads to a multitude of terms. Each term of the relation is in itself

equally simple. power, we men

66. As regards the introduction of parameters, which they say are included in a function ^'"a meters6 "a*
but not in a power of the distance, it is not true that a power does not include a parameter in the
parameter. The formula i/x™ includes unity itself ; & this is not something that is the'cTmbinatfoA ol
self-determinate, but something that can express any magnitude. Indeed, that a certain force with
species of formula includes all species of hyperbolas, &, if the exponent m is given

420 PHILOSOPHIC NATURALIS THEORIA

ac definite exponente m, exprimit unicam quidem earum speciem, sed quae continet infinitas
numero individuas Hyperbolas, quarum quselibet suam parametrum diversam habet pro
diversitate unitatis assumptae. Potest quidem quaevis ex iis Hyperbolis ad arbitrium
assumi ad exprimendam vim decrescentem in ea ratione reciproca ; sed adhuc in ipsa
expressione includitur quaedam parameter, quae determinet certam vim a certa ordinata
exprimendam, sive certam vim certae distantiae respondentem, qua semel determinata
remanent determinatae reliquse omnes, sed ipsa infinitis modis determinari potest, stante
expressione facta per ordinatas ejusdem curvae, sive per eandem potentise formulam.
Ejusmodi primus nexus a sola distantia utique non pendet.
Parameter in ex- 5^^ Accedit autem alia quasi parameter in exponente potentiae : illius numeri m

ponente potentiae. , ' . j 5- • j- • i- •

determmatio utique non pendet a distantia, nee distantiam aliquam exprimit.
Non SS'V1 68. Sed nee illud video, cur etiam si dicatur vis esse proprietas qusedam materise

sola distantia essentialis, ea debeat necessario pendere a solis distantiis. Si esset quaedam virtus, quae a

etiam, si ** *~ materiae puncto quovis egressa progrederetur motu uniformi, & rectilineo ad omnes circum

tas materiae. distantias : turn quidem diffusio ejus virtutis per orbes majores aeque crassos fieret in

ratione reciproca duplicata distantiarum, & a distantiis solis penderet ; quanquam ne turn

quidem ab iis penitus solis, sed ab iis, & exponente secundae potentiae, ac primo nexu cum

arbitraria [286] unitate. At cum nulla ejusmodi virtus debeat progredi, & in progressu

ipso ita attenuari ; nihil est, cur determinatio ad accessum debeat pendere a solis distantiis,

ac proinde solae distantiae ingredi formulam functionis exprimentis vim.

69. Verum admisso etiam, quod necessario vis debeat pendere a solis distantiis, nihil
distantiis, "ordlna- habetur contra expressioncm factam per functionem quandam. Nam ipsa functio per se
dataqcurvae endere immediate pendet a distantia, & est ordinata quaedam ad curvam quandam certae naturae,
a solis abscissis. respondens abscissae datae cuilibet sua. Parametri inducuntur ex eo, quod illius relationem
ad abscissam exprimere debeamus per potentias abscissae, & potentiarum producta cum
aliis rectis ; sed in se, uti supra diximus, ejusdem est naturae & ilia functio, ac potentia
quaevis, & ilia, ut haec, ordinatam immediate simplicem exhibet respondentem abscissae
ad curvam quandam uniformis, & in se simplicis curvae.

Parametros ipsas 70. Praeterea ipsae illae parametri, quae formulam functionis ingrediuntur, possunt

eTs dfuncticmem esse certae quaedam distantiae & assumi debere ad hoc, ut illis datis distantiis illae datae, &

esse ingressas, quod non aliae vires respondeant. Sic ubi quaesita est formula, quae exprimeret aequationem

debuerit haberrvls a& curvam quaesitam, assumpsimus quasdam distantias, in quibus curva secaret axem,

data, vei nulla. nimirum in quibus, evanescente vi haberentur limites, & earum distantiarum valores ingressi

sunt formulam inventam, ut quaedam parametri. Possunt igitur ipsae parametri esse

distantiae qusedam ; ac proinde posito, quod omnino debeat vis exprimi per solas distantias,

potest adhuc exprimi per functionem continentem quotcunque parametros, & non

exprimetur necessario per solam aliquam potentiam.

Argumentum con- 71. Reliquum est, ut dicamus aliquid de Ratione Sufficienti, quae dicitur parametros

rationis sufficientis! excludcre, cum non sit ratio, cur aliae prae aliis parametri seligantur.

si vis sit essentialis 72. Inprimis si vis est in ipsa natura materiae ; nulla ratio ulterior requiri potest

tTnum' "param™ praeter earn ipsam naturam, quae determinet hanc potius, quam aliam vim pro hac potius,

trorum esse ipsam quam pro ilia distantia, adeoque hanc potius, quam aliam parametrum. Quaeri ad summum

cu"hocgenuUsma!tei poterit, cur clegcrit Naturae Auctor earn potissimum materiam, quae earn legem virium

riae existat, ration- haberet cssentialem, quam aliam : ubi ego quidem, qui summam in Auctore Naturae

Creatoris T^idenT libertatem agnosco, censeo, ut in aliis omnibus, nihil aliud requiri pro ratione sufficient!

si ea non sit essen- electionis, quam ipsam liberam determinationem Divinae voluntatis, a cujus arbitrio

pendeat turn, quod hanc potius, quam aliam eligat rem, quam condat, turn quod ea re

hanc in se naturam habente, ubi jam condita fuerit, utatur ad hoc potius, quam ad illud

ex tarn multis, ad quae natura quaevis a tanti Artificis manu adhibita potest esse idonea.

Atque haec responsio [287] aeque valet, si vis non est ipsi materiae essentialis, sed libera

Auctoris lege sancita, quo casu ipse pro libero arbitrio suo hanc huic materiae potuit legem

dare prse aliis electam.

Praeter arbitrium
retorsio in poten-
tia : rationemutro- 73. At si ratio etiam exhiberi debeat, quae Auctorem Naturae potuerit impellere ad

quoTsibTTpse^ro8- seligendam materiam hac potissimum prasditam essentiali virium lege, vel ad seligendam
posuerit. qui pos- pro hac materia hanc legem virium ; quaeri primo potest, cur hunc potius exponentem
ignoti. e 3 potentiae elegerit, & hanc parametrum in unitate inclusam, sive in quadam determinata

SUPPLEMENT III 421

it represents one of these species ; & any one of these has its own different parameter for
a difference in the unity assumed. It is possible for any one of these hyperbolas to be
arbitrarily chosen to represent a force which decreases in that reciprocal ratio ; but still
there is included in the expression a certain parameter ; namely, one which determines a
certain force to be represented by a certain ordinate, or a certain force to correspond with
a certain distance ; when once this is determined, all the rest are at the same time determined.
But this can be done in an infinite number of ways, without altering the generation of the
expression from the ordinates of the self-same curve, or the same formula of a power. A
primary connection of this kind certainly does not depend on distance alone.

67. Besides there is another thing, that is very like a parameter, in the exponent of There is a para-
the power ; the determination of the number m at any rate does not depend on the distance, ™e^r ofntheepower
nor does it express any distance.

68. But, really, I do not see why, if it is said that force is some property essential to There is no reason
matter, it should of necessity depend on distances alone. If it were some virtue, which Thy l\ should

i 1 r . i , 1-1 T • • • i dependonthe

proceeded from any point of matter & progressed with uniform motion in a straight line distance alone, if
to all distances round ; then indeed the diffusion of this virtue through greater spheres force ls an e336"*^1

..... , r i T property of matter.

equally thick would be as the inverse squares of the distances ; & thus would depend on
distance alone. Although not even then would it depend altogether on distances alone ;
but on them & the exponent of the second power, in addition to the prime connection
with an arbitrary unity. But since no such virtue is bound to progress, & even in progression
to be so attenuated, there is no reason why determination for approach should depend
on distances alone ; & that therefore distances alone should enter the formula of the function
that expresses the force.

69. But even if it is admitted that force must necessarily depend on the distances Even if the force
alone ; still there is nothing against the expression being formed of some function. For *jld, dePend on the
the function in itself depends directly upon distance, & is an ordinate to some curve of the ordinates also!
known nature, corresponding to its own given abscissa, which may be anything you please. m themselves, de.

• l 1 l l r i i 11- r 6i *i. Pend on the ab-

Parameters are induced by the fact that we have to express the relation of the ordinate scissae alone, for
to the abscissa by means of powers of the abscissa, & the products of these powers with any siven curve-
other straight lines. But in themselves, as I said above, both the function & any power
are of the same nature ; & the former, like the latter, will give a perfectly simple ordinate
corresponding to the abscissa to any arc of a curve that is uniform & simple in itself.

70. Besides, these very parameters, which come into the formula, may be certain The parameters
known distances ; & they have to be assumed for the purpose of ensuring that to these tanc^e-lheyrehave
given distances those given forces, & not others, correspond. So, when we seek a formula come into the
to express the equation to the curve required, we assume certain distances in which the ^"riven distances
curve shall cut the axis ; that is to say, distances for which, as the force vanishes, we shall there must be a
obtain limit-points ; & the values of these distances have entered the formula we have j=jv|{| force or none
found, as certain parameters. Hence the parameters themselves may be distances.

Therefore, if it is stated that force is absolutely bound to depend on distances alone, it is
still possible to express the force by a function containing any number of parameters ;
& it is not necessarily expressed by some single power.

71. It only remains to say a few words with regard to Sufficient Reason ; this being The argument
said to exclude parameters, because there is no reason why some parameters should be jfiJ2?0f .rfictont
chosen in preference to others. reason.

72. First of all, if force is an essential property of matter, there is no need for any if force is an
other reason beside that of the very nature of matter, to determine that this, rather esse ntiai^ prope rt y
than another, force should correspond to this, rather than to another, distance ; & therefore reason for ' such
this parameter, rather than any other. It may be asked, & we can go no further, why parameters is the

r TVT i • • • i 1111 i • ' i very nature of

the Architect of Nature chose this matter m particular, such as should have this essential matter ; why such
law of forces, & no other. In that case, I, who believe in the supreme freedom of the Jjf^ ^Jm* Jf tjj*
Architect of Nature, think, as in all other things, that there is nothing else required for the creator; the same
sufficient reason for His choice beyond the free determination of the Divine will. Upon thins if f°™e w not
the free exercise of this depends not only the fact that He chose this thing rather than
another to create ; & also that, the thing having this nature in itself, when it was once
created, He should use it for this purpose rather than for any other of the very many purposes,
to which any nature employed by the hand of so mighty an Artificer may be suitable. This
reply applies just as well, even if the force is not an essential property of matter, but
established by the free law of the Author ; for, in that case, He, of his own free will, could

,..'.. ,. .. , 11 11 There is something

give this law to this matter, having chosen it m preference to all other laws. beyond win in the

73. Now, if we have also to give the reason which might have forced the Author of limitation of his
Nature to select in particular this matter possessed of this essential law of forces, or to bonfcases^s'Sfe'aim
select for this matter this law of forces especially ; it may first be asked why He should have *h.at He set before

- ... ,. i . ' • • i j j • i_ • Himself ; & this we

preference for this exponent of the power, this parameter that is included in the unity, may not know.

422 PHILOSOPHIC NATURALIS THEORIA

distantia quandam determinatam vim. Quod de iis dicitur, applicari poterit parametris
reliquis functionis cujusvis. Ut ille exponens, ilia unitas, ille nexus potuit habere aliquid,
quod caeteris praestaret ad eos obtinendos fines, quos sibi Naturae Auctor praescripsit ; sic
etiam aliquid ejusmodi habere poterant reliquse omnes quotcunque, & qualescunque
parametri.

Evoiutio finis ip- 74. Deinde rem ipsam diligenter consideranti facile patebit, ad obtinendos fines,

hatendi "nfac cluos s*ki Naturae Auctor debuit proponere, non fuisse aptam solam potentiam quandam

nexum ab Algebra distantiae pro lege virium, sed debuisse assumi functionem, quae ubi exprimi deberet per

primufiie'm1 nisi nostram humanum Algebram, alias quoque parametros admisceret. Si ex. gr. voluisset

per functionem, ad per eandem vim & motum Planetarum ad sensum ellipticum cum Kepleriano nexu inter

on'iTproWem^pro quadrata temporum periodicorum, & cubos distantiarum mediarum, & cohsesionem per

hac corporum con contactum, nulla sola potentia ad utrumque praestandum finem fuisset satis, quem finem

stitutionc, & mo- a L

tuum serie. obtinuisset ilia, formula — + — . At nee ea formula potuit ipsi sufEcere, si vera est Theoria

OC X

mea, cum ea formula nullam habeat in minimis distantiis vim contrariam vi in maximis,
sed in omnibus distantiis eandem, nimirum in minimis attractivam, ut in maximis.
Cohaesio punctorum se invicem repellentium in minimis distantiis, & attrahentium in
majoribus haberi non potuit sine intersectione curvae cum axe, quae intersectio sine para-
metro aliqua non obtinetur. Verum ad omnem hanc phaenomenorum seriem obtinendam
multo pluribus, uti ostensum est suo loco, intersectionibus curvae, & flexibus tarn variis
opus erat, quae sine plurimis parametris obtineri non poterant. Consideretur elevatissimum
inversum problema affine alteri, cujus mentio est facta num. 547, quo quaeratur numerus
punctorum, & lex virium mutuarum communis omnibus necessaria ab habendam ope
cujusdam primae combinationis, hanc omnem tam diuturnam, tarn variam phaenomenorum
seriem, cujus perquam exiguam particulam nos homines intuemur, & statim patebit eleva-
tissimum debere esse, & respectu habito ad nostros exprimendi modos complicatissimum
genus curvae ad ejusmodi problematis solutionem ne-[288]-cessarium ; quod tamen problema
certas quasdam parametros in singulis saltern solutionibus suis, quae numero fortasse infinite
sunt, involveret, sola unica potentia ad tanti problematis solutionem inepta.

id non potuisse 75.^Debuit igitur Naturae Auctor, qui hanc sibi potissimum Phaenomenorum seriem

potentiam: legeS proposuit, parametros quasdam seligere, & quidem plures, nee potuit solam unicam pro
quadrati distantia: lege virium exprimenda distantiae potentiam adhibere : ubi & illud praeterea ad rem eandem
confirmandam recolendum, quod a num. 124, dictum est de ratione reciproca duplicata
distantiarum, quam vidimus non esse omnium perfectissimam, nee omnino eligendam,
& illud, quod sequenti horum Supplementorum paragrapho exhibetur contra vires in
minimis distantiis attractivas & excrescentes in infinitum, ad quas sola potentia demum
deducit.

Conciusio contra 76. Atque hoc demum pacto, videtur mihi, dissoluta penitus omnis ilia difficultas,

necessitate1?, V^J quae proposita fuerat, nee ulla esse ratio, cur sola potentia qusedam distantiae anteferri

convementiam sol- > , f . r. ' . . ,.r , r . .

ius potentia:. debuent function! utcunque, si nostrum exprimendi modum spectemus, complicatissimae.

SUPPLEMENT III

423

or a certain determined force for a certain determined distance. Now, what is to be said
about these things, can be also applied to all the other parameters of any function. Namely,
that this exponent, this unity, this connection might have had something in them, which
was superior to all other things for the purpose of obtaining those aims which the Author
of Nature had set before Himself. Similarly, all the other parameters might have something
of the same sort, no matter how many or of what kind they are.

74. Next, it will easily be clear to anyone, who considers the matter with care, that,
for the purpose of obtaining the aims which the Author of Nature was bound to have set
Himself, any single power of the distance would not have been convenient for the law of
forces ; but a function would have had to be taken ; & this, as it was destined to be expressed
in our human algebra, would bring in other parameters also. If, for instance, He had
wished to make subject to the same force, both the practically elliptic motion of the planets,
with the Keplerian connection between the squares of the periodic times & the cubes of
the mean distances, & also cohesion by contact ; then no single power would have been
sufficient for the establishment of both aims ; this aim would have been met by the formula
a/x3 + b/xz. But this formula even would not have been sufficient, if my Theory is
true ; for it has not the force at very small distances in the opposite direction to the force
at very great distances ; but the same kind of force at all distances, that is, an attractive
force at very small distances, just as at very great distances. Now, the cohesion of points
that repel one another at very small distances, & attract one another at very large distances,
cannot be obtained without intersection of the curve & the axis ; & this intersection could
not be obtained without the introduction of some parameter. Indeed, to obtain the
whole series of phenomena, there was need, as has been shown in the proper place for each,
of far more intersections of the curve, & for flexures of such different sorts ; & these could
not be obtained without introducing a large number of parameters. Just consider for a
moment this most intricate problem, akin to another of which mention was made in
Art. 547 : — Required to find the number of points, & the law of mutual forces common
to all of them, which would be necessary to obtain, by the aid of a given initial combination,
the whole of this series of phenomena, of such duration & variety, of which we men behold
but the very smallest of small portions. Immediately it will be evident that it is bound
to be of the most intricate character, &, having regard to our methods of expressing things,
that the kind of curve necessary for the solution of such a problem must be very complicated.
This problem, however, would involve certain known parameters in each of its solutions
at least, & the number of these might perchance be infinite ; & a single power by itself
would be ill-suited for the solution of so great a problem.

75. Hence, the Author of Nature, who decided on this series of phenomena in particular,
must have selected certain parameters, & indeed a considerable number of them; nor
could He have used a single power of the distance by itself for expressing the law of forces,
In this connection also, we must recall to mind, for the confirmation of this matter, what,
from Art. 124 onwards, has been said with regard to the inverse ratio of the squares of the
distances. We saw that this ratio was not the most perfect of all, nor one to be chosen
in all circumstances. Also, we must look at that which is shown, in the next section of
these supplements, in opposition to forces that are attractive at very small distances,
increasing indefinitely, to which a single power reduces in the end.

76. Finally, in this way, it seems to me that the whole of the difficulty that was put
forward has been quite done away with ; there is no reason why any single power of the
distance should be preferred to a function, no matter how complicated it may be, if
regard is paid to our methods of expressing it.

The evolution

the necessity ' for
tnis connection

e^pr'essitfie "by
human algebra,

problems of crea-

t»tk>& of bodies8 &
series of motions.

it could not be

™^ ^tneSaw^f
the squares of the

Conclusion against

convenSnof °oftha
single power.

[289] § IV

Contra vires in minimis distantiis attractivas^ & excrescentes in

infinitum

, qduodCUubi 77' At praeterea contra solam attractionem plures habentur difficultates, quae per

conatus deberet gradus crescunt. Nam inprimis si eae imminutis utcunque distantiis agant, augent veloci-
apSmiisunaXindebeat tatem usque ad contactum, ad quern ubi deventum est, incrementum velocitatis ibi per
esse nuiius, vei saltum abrumpitur, & ubi maxima est, ibi perpetuo incassum nituntur partes ad ulteriorem
effectum habendum, & necessario irritos conatus edunt.

Secunda, si ratio yg. Quod si in infinitum imminuta distantia, crescant in aliqua ratione distantiarum

tantiae^aviabsoiute reciproca ; multae itidem difficultates habentur, quae nostrum oppositam sententiam
infinita, ad quam confirmant. Inprimis in ea hypothesi virium deveniri potest ad contactum. in quo vis,

deveniri deberet. , , • j- • j i_ •••£.•*. • • v i- •

sublata omni distantia, debet augen m innmtum magis, quam esset in aliqua distantia.
Porro nos putamus accurate demonstrari, nullas quantitates existere posse, quae in se
infinitae sint, aut infinite parvae. Hinc autem statim habemus absurdum, quod nimirum
si vires in aliqua distantia aliquid sunt, in contactu debeant esse absolute infinitae.

Tertia ex eo, 79. Augetur difficultas, si debeat ratio reciproca esse major, quam simplex (ut ad gravi-

™ai°r tatem requiritur reciproca duplicata, ad cohaesionem adhuc major) & ad bina puncta

quam simplex, ue- _ j. ± .1. . i i • i i • r • •

beat in contactu pertmeat. Nam ilk puncta in ipso congressu devement ad velocitatem absolute innmtam.

veioSutem^infini^ Velocitas autem absolute infinita est impossibilis, cum ea requirat spatium finitum percursum

tam. momento temporis, adeoque replicationem, sive extensionem simultaneam per spatium

finitum divisibile, & quovis finite tempore requirat spatium infinitum, quod cum inter
bina puncta interjacere non possit, requireret ex natura sua, ut punctum ejusmodi
velocitatem adeptum nusquam esset.

Alia absurda: si 80. Accedunt plurima absurda, ad quse ejusmodi leges nos deducunt. Tendat punctum

' aliquod in fig. ']^ in centrum F in ratione reciproca duplicata distantiarum, & ex A pro-

regressus

saitus ab acceiera. jiciatur directione AB perpendicular! ad AF, cum velocitate satis exigua : describet

nu\"amC7nCingressu Ellipsim ACDE, cujus focus erit F,& semper regredietur ad A. Decrescat velocitas AB per
insuperficiemsphae. gradus, donee demum evanescat. Semper magis arctatur Ellipsis, & vertex D accedit ad
focum F, in quern demum recidit abeunte Ellipsi in rectam AF. Videtur igitur id [290]
punctum sibi relictum debere descendere ad F, turn post acquisitam ibi infmitam veloci-
tatem, earn sine ulla contraria vi convertere in oppositam, & retro regredi. At si id punctum
tendat in omnia puncta superficiei sphericae, vel globi EGCH in eadem ilia ratione ;
demonstratum est a Newtono, debere per AG descendere motu accelerate eodem modo,
quo acceleraretur, si omnia ejusmodi puncta superficiei, vel sphaerae compenetrarentur in
F : abrupta vero lege accelerationis in G, debere per GH ferri motu sequabili, viribus
omnibus per contrarias actiones elisis, turn per HI tantundem procurrere motu retardato,
adeoque perpetuam oscillationem peragere, velocitatis mutatione bis in singulis oscillation-
ibus per saltum interrupta.

snu ^ocursus ^Ip In eo jam absurdum quoddam videtur esse : sed id quidem multo magis crescit ;

ultra' ad eandcm si consideretur, quid debeat accidere, ubi tota sphaerica superficies, vel tota sphaera abeat

distantiam, v e i • unicum punctum F. Turn itidem corpus sibi relictum, deveniet ad centrum cum

saltus in tanto . . J *. . . i T i • i • -r«iv • i.

procursu, sine infinita velocitate, sed procurret ulterms usque ad 1, dum pnus, ubi Ellipsis evanescebat,

praeviis minoribus. Jebebat redire retro. Nos quidem pluribus in locis alibi demonstravimus, in prima

(e) Hac excer-pta sunt ex eadem dissertatione De Lege Virium in Natura existentium a num. 59.

424

SUPPLEMENT IV

425

FIG. 72.

426

PHILOSOPHIC NATURALIS THEORIA

§IV

Arguments against forces that are attractive at very small
distances and increase indefinitely ("}

77. Besides, there are many difficulties in the way of attraction alone, which increase The first difficulty
by degrees. For, first of all, if these act at diminished distances of any sort, they will SiatT ^hen*5 *the
increase the velocity right up to the moment of contact : & when contact is attained, effort should be
the increment of the velocity will then be suddenly broken off ; & when this is greatest, ^oach^it Abound
the parts will continually strive in vain to produce a further effect, & the efforts will to be either nothing
necessarily turn out to be fruitless.

78. But if, when the distances are infinitely diminished, the forces increase according The second diffi.
to some ratio that is inversely as the distances, many difficulties will again be had, which culty arises fro™

• . /~VT_ i_ • r j -11 the fact that, if

confirm our opposite opinion. Un that hypothesis ot iorces especially, contact may be the ratio is in-
attained, in which, as all distance is taken away, the force is bound to be increased ^erseiy as the
infinitely more than it would be at a distance of some amount. Further, I think that it come to Wa force
is rigorously proved that no quantities can possibly exist, such as are infinite in themselves !hat. is absolutely
or infinitely small. Hence, we immediately have an absurdity ; namely, that if the forces
at any distance are anything, on contact they must be absolutely infinite.

79. The difficulty is increased, if the inverse ratio is greater than a simple ratio (as A third difficulty
for gravity we require the inverse square, & for cohesion one that is still greater) ; & it J °he Averse ratfo
has to do with a pair of points. For these points on collision will attain a velocity that is »? greater than a
absolutely infinite. But such an absolutely infinite velocity is impossible, since it requires bound a°so 'tolmve
that a finite space should be passed over in an instant of time, that is, replication, or simul- on contact an
taneous extension through finite divisible space ; & for any finite time it would require " ' velocity-
infinite space, which, since there cannot be such between the two points, would require of

its own nature that there should not be anywhere a point that has attained such a velocity.

80. There are many more absurdities, to which such laws of forces lead us. In Fig. other absurdities ;
72, let any point tend towards a centre F in the inverse ratio of the squares of the distances, square "of^th^ dis6
& suppose it to be projected from the point A in a direction, AB, perpendicular to AF, tance, there will be
with a fairly small velocity. Then it will describe the ellipse ACDE, of which F is the cenutrre; ^"suddtn
focus ; & it will always return to A. Now let the velocity AB decrease by degrees, until change from an
finally it vanishes. Then the ellipse will continually become more & more pointed, & toawSnjT'to^oM
the vertex D will approach the focus F, & will coincide with it when the ellipse becomes tnat is nothing on
the straight line AF. It seems therefore that the point, if left to itself would fall towards surfac"g * Sph<
the focus F, then, after acquiring an infinite velocity as it reaches F, it would convert it

into an equal velocity in the opposite direction without the assistance of any opposing force,
& return to its original position. But if that point tended towards all the points of a
spherical surface, or the sphere EGCH, in that same ratio, it was proved by Newton that
it would have to descend along AG with a motion accelerated in the same manner as it
would be if all such points of the surface, or the sphere, were condensed at F. Now the
law of acceleration being broken at G, it will have to go on along GH with uniform velocity,
all forces being counterbalanced by contrary reactions ; then it will have to travel along
HI for the same interval with retarded motion. Thus, there would be a continual
oscillation, with the change of velocity suddenly interrupted twice in each oscillation.

81. Here there is already seen to be considerable absurdity ; but there is still greater Simultaneous
to follow. For, let us consider what will necessarily happen when the whole of the spherical ^n"™ &onmotion
surface, or the whole of the sphere, becomes but a single point at F. Then indeed, the beyond it to an
body if left to itself would arrive at the centre with infinite velocity ; but it would pass a^udden^nge °n
through it & beyond as far as I, whereas in the former case when the ellipse vanished, it this great motion,
had to return to its original position. Indeed, in many places elsewhere, I have proved "
. - tions.

(e) These paragraphs are quoted from the same dissertation De Lege Virium in Natura existentium, starting with
Art. 59.

427

428 PHILOSOPHIC NATURALIS THEORIA

determinatione latere errorem, cum Ellipsi evanescente, nullae jam adsint omnes vires,
quae agunt per arcum situm ultra F ad partes D, quae priorem velocitatem debebant extin-
guere, & novam producere ipsi aequalem. Verum adhuc habetur saltus quidam, cui &
Natura, & Geometria ubique repugnat. Nam donee utcunque parva est velocitas, habetur
semper regressus ad A cum procursu FD eo minore, quo velocitas est minor : facta autem
velocitate nulla, procursus immediate evadit FI, quin ulli intermedii minores adfuerint.
Quod si quis ejus priorem determinationem tueri velit, ut punctum, quod agatur in centrum
vi, quae sit in ratione reciproca duplicata distantiarum, debeat e centre regredi retro ;
turn saltus habetur similis, ubi prius in sphaericam superfkiem vel sphaeram tendat, quae
paullatim abeat in centrum. Donee enim aderit superficies ilia, vel sphaera, habebitur
semper is procursus, qui abrumpetur in illo appulsu totius superficiei ad centrum, quin
habeantur prius minores procursus.

- quidem in ratione reciproca duplicata distantiarum : in reciproca triplicata

latio puncti in habentur etiam graviora. Nam si cum debita quadam velocitate projiciatur per rectam
trum'SU ad °en" •^••^ fy?" 73 c°ntinentem angulum acutum cum AP, mobile, quod urgeatur in P vi crescente
in ratione reciproca triplicata distantiarum ; demonstratur in Mechanica, ipsum debere
percurrere curvam ACDEFGH, quae vocatur spiralis logarithmica, quae hanc habet pro-
prietatem, ut quaevis recta, ut PF, ducta ad quodvis ejus punctum, contineat cum recta
ipsam ibidem tangente angulum aequalem angulo PAB, unde illud consequitur, ut ea quidem
ex una parte infmitis spiris cir-[29i]-cumvolvatur circa punctum P, nee tamen in ipsum
unquam desinat : si autem ducatur ex P recta perpendicularis ad AP, quae tangenti AB
occurrat in B, tota spiralis ACDEFGH in infinitum continuata ad mensuram longitudinis
AB accedat ultra quoscunque limites, nee unquam ei sequalis fiat : velocitas autem in
ejusmodi curva in continuo accessu ad centrum virium P perpetuo crescat. Quare finito
tempore, & sane breviore, quam sit illud, quo velocitate initiali percurreret AB, deberet
id mobile devenire ad centrum P, in quo bina gravissima absurda habentur. Primo
quidem, quod haberetur tota ilia spiralis, quae in centrum desineret, contra id, quod ex ejus
natura deducitur, cum nimirum in centrum cadere nequaquam possit : deinde vero, quod
elapso eo finito tempore mobile illud nusquam esse deberet. Nam ea curva, ubi etiam in
infinitum continuata intelligitur, nullum habet egressum e P. Et quidem formulas ana-
lyticae exhibent ejus locum post id tempus impossibilem, sive, ut dicimus, imaginarium ;
quo quidem argumento Eulerus in sua Mechanica afnrmavit illud, debere id mobile in
appulsu ad centrum virium annihilari. Quanto satius fuisset inferre, earn legem virium
impossibilem esse ?

Pejus in potentiis g-j. Quanto autem maiora absurda in ulterioribus potentiis, quibus vires alligatae

altioribus : prseca- • *o-ii r- A TIT? « • • VAT • •

ratio ad demon- sint, consequentur ? Sit globus in fig. 74 ABE, & intra ipsum alms Abe, qui priorem
strandum absur- contingat in A, ac in omnia utriusque puncta agant vires decrescentes in ratione reciproca
quadruplicata distantiarum, vel majore, & quaeratur ratio vis puncti constituti in concursu
A utriusque superficiei. Concipiatur uterque resolutus in pyramides infinite arctas, quae
prodeant ex communi puncto A, ut BAD, bAd. In singulis autem pyramidulis divisis
in partes totis proportionales sint particulas MN, mn similes, & similiter positae. Quantitas
materiae in MN, ad quantitatem in mn erit, ut massa totius globi majoris ad totum minorem,
nimirum, ut cubus radii majoris ad cubum minoris. Cum igitur vis, qua trahitur punctum
A, sit, ut quantitas materiae directe, & ut quarta potestas distantiarum reciproce, quae
itidem distantiae sunt, ut radii sphaerarum ; erit vis in partem MN, ad vim in partem mn
directe, ut tertia potestas radii majoris ad tertiam minoris, & reciproce, ut quarta potestas
ipsius. Quare manebit ratio simplex reciproca radiorum.

Partem fore majo- g. Minor erit igitur actio singularum particulaium homologarum MN, quam mn,

rem tolo. ~ ,. * \ , A -nT->

in ipsa ratione radiorum, adeoque punctum A minus trahetur a tota sphaera ABE, quam
a sphaera Abe, quod est absurdum, cum attractio in earn sphseram minorem debeat esse pars

SUPPLEMENT IV

429

FIG. 73.

FIG. 74.

430

PHILOSOPHIC NATURALIS THEORIA

FIG. 73.

FIG. 74.

SUPPLEMENT IV 431

that there is an error in the first determination ; for when the ellipse vanishes, there are
no longer present any of all these forces, which act on the body as it goes along the arc
situated beyond F in the direction of D ; & these were necessary to extinguish the former
velocity & to generate a new velocity equal to it. But still there is a sudden change, to
which both Nature & geometry are in all cases opposed. For, so long as there is a velocity,
no matter how small, we always have a return to A with a further motion beyond F, equal
to FD, which is correspondingly smaller as the velocity becomes smaller ; & yet, when the
velocity is made nothing at all, the further motion beyond F at once becomes FI, without
there being present any intermediate smaller motions. Now, if anyone would wish to
adhere to the first determination of the problem, so that a point, which is attracted towards
a centre by a force in the inverse ratio of the squares of the distances, is bound to return
from the centre to its original position ; then there too there is a sudden change of a like
nature to that which took place in the first case when it tended towards a spherical surface,
or a sphere, which gradually dwindled to a point at the centre. For, as long as the spherical
surface, or the sphere, is there, there will always be obtained that further motion ; but
this is suddenly stopped on the arrival of the whole of the spherical surface, or the whole
of the sphere, at the centre, without any previous smaller motions being had.

82. Such indeed are the results that we obtain for the inverse ratio of the squares of if. the ratio is the
the distances ; for the inverse ratio of the cubes, we have even more serious difficulties. wo^r-feT annihUa-
For, if a body is projected along AB, in Fig. 73, making an acute angle with AP, with a tion of the point on
certain suitable velocity, & it is attracted towards P with a force increasing in the inverse a*
ratio of the cubes of the distances ; in that case, it is proved in Mechanics that the motion

will be along a curve such as ACDEFGH, which is called the logarithmic spiral. This
curve has the property that any straight line, PF, drawn from P to any point F of the curve,
contains with the tangent to the curve at the point an angle equal to the angle PAB. Hence
it follows that, on the one hand indeed it will rotate through an infinite number of con-
volutions round the point P, but will never reach that point ; yet, on the other hand, if
a straight line is drawn through P perpendicular to AP, to meet the tangent AB in B, then
the whole length of the spiral ACDEFGH continued indefinitely will approximate to the
length of AB beyond all limits, & yet never be equal to it. Further the velocity in such
a curve, as it continually approaches the centre of forces P, continually increases. Hence
in a finite time, & that too one that is shorter than that in which it would pass over the
distance AB with the given initial velocity, the moving body would be bound to arrive
at the centre P ; & in this we have two very serious absurdities. The first is that the
whole of the spiral, which terminates in the centre, is obtained, in opposition to the principle
deduced from its nature, since truly it can never get to the centre ; & secondly, that
after that finite time has elapsed the moving body would have to be nowhere at all. For,
the curve, even when it is understood that it is continued to infinity, has no exit through &
past the point P. Indeed the analytical formulae represent its position after the lapse of
this time as impossible, or, as it is usually called imaginary. By this very argument, Euler,
in his Mechanics, asserts that the moving body on approaching the centre of forces is
annihilated. How much more reasonable would it be to infer that this law of forces is
an impossible one ?

83. How much greater absurdities are those that follow for higher powers, with which still worse for
the forces may be connected ! In Fig. 74, let ABE be a sphere, & within it let there be higher P°wers;

It i • i r ° T i r 11 • r i preparation for

another one Abe, touching the former at A ; & suppose that on all points of each of them demonstrating an

there act forces which decrease in the inverse ratio of the fourth powers of the distances, absurdlty-

or even greater ; & suppose that we require the ratio of the forces due to a point situated at

the point of contact A of the two surfaces. Imagine each of the spheres to be divided into

infinitely thin pyramids, proceeding from the common vertex A, such as BAD, bAd. In each

of these little pyramids, which are then divided into parts proportional to the wholes, let

MN & mn be particles that are similar & similarly situated. The quantity of matter in MN

will be to the quantity of matter in mn as the mass of the larger sphere to the mass of the

whole of the smaller ; i.e., as the cube of the radius of the larger to the cube of the radius

of the smaller. Hence, since the force exerted upon A varies as the quantity of matter

directly, & as the fourth power of the distance inversely, & these distances also vary as

the radii of the spheres. Therefore, the force on the part MN is to the force on the part

mn directly as the third power of the radius of the larger sphere to the third power of the

radius of the smaller, & inversely as the fourth powers of the same. That is, there results

the simple inverse ratio of the radii.

84. Hence the action of each of the homologous particles MN will be less than each T*16 Part greater
of the corresponding particles mn, in the ratio of the radii ; & thus the point A will be

attracted less by the whole sphere ABE than by the sphere Abe. This is absurd ; for, the
attraction on the smaller sphere must be a part of the attraction on the greater sphere

432 PHILOSOPHIC NATURALIS THEORIA

attractionis in sphaeram majorem, quae continet minorem, cum magna materias parte sita
extra ipsam usque ad superficiem sphaerae majoris, unde concluditur esse partem majorem
toto, maximum nimirum absurdum. Et qui-[292]-dem in altioribus potentiis multo major
est is error ; nam generaliter, si vis sit reciproce, ut Rm, posito R pro radio, & m pro quovis
numero ternarium superante, erit attractio sphserae eodem argumento reciproce, ut Rm~3,
quae eo majorem indicat vim in sphaeram minorem respectu majoris ipsam continentis,
quo numerus m est major.

Omnia absurda gij. Hoc quidem pacto inveniuntur plurima absurda in variis generibus attractionum

minimis distantiis quae si repulsiones, in minimis distantiis habeantur pares extinguendae velocitati cuilibet
habeatur repuisio, utcunque magnae, cessant illico omnia, cum eae repulsiones mutuum accessum usque ad
ilnpediat. appUlSUm concursum penitus impediant. Inde autem manifesto iterum consequitur, repulsiones

in minimis distantiis praeferendas potius esse attraction!, ex quarum variis generibus tarn

multa absurda consequuntur.

SUPPLEMENT IV 433

which contains the smaller one, together with a great part of the matter situated beyond
it as far as the surface of the greater sphere ; hence the conclusion is that the part is greater
than the whole, which is altogether impossible. Indeed, in still higher powers the error
is much greater ; for, in general, if the force varies inversely as R"*, where R is taken as
the radius, & m for some number greater than three, then the attraction of the sphere
will be inversely as Rw~3 ; & this points to a force that is the greater on a smaller sphere
compared with that on a larger sphere containing it, in proportion as the number m is
greater.

85. Thus we find very many absurdities in various kinds of attractions ; if there are All these absurd-
repulsions at very small distances, sufficiently great to destroy any velocity however large, there^tsT repulsion
all these absurdities would cease to be immediately, for these repulsions would prevent at very small
mutual approach up to the point of actual contact. Hence it once again manifestly follows p^e^ent*
that repulsions at very small distances are to be preferred before an attraction ; for from approach,
the various kinds of the latter so many absurdities follow.

near

FF

[293] § V

De ALquilibrio binarum massarum connexarum invicem per bina

alia puncta (/)

de squih- ^6. Continetur autem, quod pertinet ad momentum in vecte, & ad aequilibrium,

brio punctorum sequentis problematis solutione. Sit in fig. 75 quivis numerus punctorum materias in A,
Mnta°r'extreIma I1" dicatur A, in D quivis alius, qui dicatur D, & puncta ea omnia secundum directiones
habeant quas- AZ, DX parallelas rectse datae CF sollicitentur simul viribus, quae sint aequales inter omnia
v^ibusexternisTibi puncta sita in A, itidem inter omnia sita in D, licet vires in A sint utcunque diversae a
proportionaiitms, & viribus in D. Sint autem in C, & B bina puncta, quae in se invicem, & in ilia puncta sita
medus vim in A, & D mutuo agant, ac ejusmodi mutuis actionibus impediri debeat omnis actio virium
illarum in A, & D, & omnis motus puncti B : motus autem puncti C impediri debeat
actione contraria fulcri cujusdam, in quod ipsum agat secundum directionem compositam
ex actionibus omnium virium, quas habet : quaeritur ratio, quam habere debent summae
virium A, & D ad hoc, ut habeatur id aequilibrium, & quantitas, ac quaeritur directio vis,
qua fulcrum urgeri debet a puncto C.

tremis* in^aiterum
e mediis.

^7' Exprimant AZ, & DX vires illas parallelas singulorum punctorum positorum
in A, & D. Ut ipsae elidantur, debebunt in iis haberi vires AG, DK contrariae, & aequales
ipsis AZ, DK. Quoniam eae debent oriri a solis actionibus punctorum C, & B agentium
in A secundum rectas AC, AB, & in D secundum rectas DC, DB, ductis ex G rectis GI,
GH parallelis BA, AC usque ad rectas AC, BA, & ex K rectis KM, KL parallelis BD, DC,
usque ad rectas DC, BD ; patet, in A vim AG debere componi ex viribus AI, AH, quarum
prima quodvis punctum in A repellat a C, secunda attrahat ad B, & in D vim DK componi
itidem ex viribus DM, DL, quarum prima quodvis punctum situm in D repellat a C,
secunda attrahat ad B. Hinc ob actionem reactioni aequalem debebit punctum C repelli
a quo vis- puncto sito in A secundum directionem AC vi aequali IA, & a quovis puncto sito
in D secundum directio-[294]-nem DC vi aequali MD : punctum vero B debebit attrahi
a quovis puncto sito in A secundum directionem BA vi aequali HA,' & a quovis puncto sito
in D vi aequali LD. Habebit igitur punctum C ex actione punctorum in A, & D binas
vires, quarum altera aget secundum directionem AC, & erit aequalis IA ductae in A, altera
aget secundum directionem DC, & erit aequalis MD ductas in D. Punctum vero B itidem
binas, quarum altera aget secundum directionem BA, & erit aequalis HA ductae in A, altera
aget secundum directionem BD, & erit sequalis LD ductae in D.

vis, quam debet g8. Porro vis composita ex illis binis, quibus urgetur punctum B, elidi debet ab actione

mum"6 compositar e mutua inter ipsum, & C ; quare debebit habere directionem rectae BC in casu, quern exhibet

quatiior : enum- figura, in quo C jacet in angulo ABD : nam si angulus ABD hiatum obverteret ad partes

pertinentJm""^ oppositas, ut C jaceret extra angulum ; ea haberet directionem CB, &_ reliqua omnis

omnia puncta. demonstratio rediret eodem. Punctum autem C ob actionem, & reactionem aequales

debebit habere vim aequalem, & contrarium illi, quam exercet B, adeoque vim aequalem,

& ejusdem directionis cum vi, quam e prioribus illis binis compositam habet punctum B :

nempe debebit habere binas vires aequales, & directionis ejusdem cum viribus illam com-

ponentibus, nimirum vim secundum directionem parallelam BA aequalem ipsi HA ductae

in A, & vim secundum directionem parallelam BD aequalem ipsi LD ductae in D. Habebit

(f) Excerfta btsc sunt ex Synopsi Physics Generalis P. Caroli Benvenuti Soc. Jesu, num. 146, cui hanc solutionem
ibi imprimendam tradideram,

434

SUPPLEMENT V

435

Kir,. 75.

436

PHILOSOPHIC NATURALIS THEORIA

M

FIG. 75-

§v

Equilibrium of two masses connected together by two other

points (/)

86. All that pertains to moment in the lever, & to equilibrium is contained in ^"enf""^0' the
the solution of the following problem. In Fig. 75, let there be any number of points of equilibrium o f
matter at the point A, & let the number be called A ; similarly, any other number at D, ^™ch P^f :t^
called D ; & suppose that all these are at the same time under the action of forces along outside points have
the directions AZ, DX parallel to the given straight line CF, & that these forces are equal g"^^^63 fj^
to one another for all the points situated at A, & also for all the points situated at D, proportional to
although the forces at A may be altogether different from those at D. Also, at C & B, let ^™nne* £™is °s
there be two points, which act mutually upon one another & upon the points situated at subject to a force
A & D. Suppose that by such actions the whole of the action of the forces on A & D has from a fulcrum-
to be prevented, as well as any motion of the point B. Also suppose that the motion of

the point C is to be prevented by the contrary action of a fulcrum, upon which the point
C acts according to the direction compounded from all the forces that act upon it. It
is required to find- the ratio which there must be between the forces on A & D, for the
purpose of obtaining equilibrium ; also to find the quantity & direction of the force to
which the fulcrum must be subjected by the point C.

87. Let AZ & DX represent the parallel forces of each of the points situated at A & D T116 for<* from the

•-n 1*1 • i • r » /-i „ TN TT- two extremes on

respectively, lo cancel these, we must have acting at these points forces ACj & DK, either of the means,
which are equal and opposite to AZ, DX. Now, these must arise purely from the actions
of the points C & B, acting on A along the straight lines AC, AB, & on D along the straight
lines DC & DB. Hence, if we draw through G straight lines GI, GH, parallel to BA, AC,
to meet the straight lines AC, BA ; & through K, straight lines KM, KL, parallel to BD,
DC, to meet DC, BD ; then it is plain that the force AG on A must be compounded of
the forces AI, AH, of which the first will repel any one of the points at A away from C,
& the second will attract it towards B ; & similarly, the force DK on D must be compounded
of the two forces DM, DL, of which the first repels any one of the points situated at D
away from C, & the second attracts it towards B. Hence, on account of the equality of
action & reaction, the point C must be repelled by every point situated at A in the direction
AC by a force equal to IA, & by every point situated at D in the direction DC with a force
equal to MD. Also the point B will be attracted by every point situated at A in the
direction BA with a force equal to HA, & by every point at D with a force equal to LD.
Therefore, the point C will have, due to the actions of the points at A & D, two forces, of
which one will act in the direction AC, & be equal to IA multiplied by A, & the other will
act in the direction DC & be equal to MD multiplied by D. The point B will also be under
the action of two forces, one of which will act in the direction BA & be equal to HA mul-
tiplied by A, & the other will act in the direction BD & be equal to LD multiplied by D.

88. Further the force composed from the two forces, which act upon the point B, The force, which
must be cancelled by the mutual action between it & C ; hence, this must be in the ^ust^na^e11*' %
direction of the straight line BC, in the case given by the figure, where C lies within the composed out of
angle ABD ; for, if the angle ABD should turn its opening in the other direction, so that *°i"nrj£ ^forces
C should lie outside the angle, then the force would have the direction CB, & all the rest pertaining to all
of the proof would come to the same thing. Now, the point C, on account of the equality of the P°ints-
action & reaction, must have a force that is equal & opposite to that exerted by B ; & thus,

a force that is equal to, & in the same direction as, the force which B has, compounded of
those first two forces. That is to say, it must have two forces that are equal to, & in the
same direction as, the two forces that compose it ; namely, a force in a direction parallel
to BA & equal to HA multiplied by A, & a force in a direction parallel to BD & equal to

(f) These are quoted from the Synopsis Physicae Generalis of FT. Carolus Senvtnuto, S.J., Art. 146, to which
author I gave this solution to print in that work.

437

438 PHILOSOPHIC NATURALIS THEORIA

igitur quodvis punctum A binas vires AI, AH, quodvis punctum D binas vires DM, DL,
punctum B binas vires, quarum altera dirigetur ad A, & sequabitur HA ductae in A, altera
dirigetur ad D, & aequabitur LD ductae in D, ex quibus componi debet vis agens secundum
rectam BC : & demum habebit punctum C vires quatuor, quarum prima dirigetur ad
partes AC, & erit aequalis IA ductae in A, secunda ad partes DC, & erit aequalis MD ductae
in D, tertia habebit directionem parallelam BA, & erit aequalis HA ductse in A ; quarta
habebit directionem BD, & erit aequalis LD ductae in D : ac ipsum punctum C urgebit
fulcrum vi composita ex illis quatuor, quae omnia, si habeatur ratio directionis rectarum
secundum ordinem, quo enunciantur per literas, hue reducuntur :

Quodvis punctum A habebit vires binas .... AI, AH
Quodvis punctum D vires binas ..... DM, DL

Punctum B binas A X HA, D X DL

Punctum C quatuor . A X IA, D X MD, A X HA, D X LD

Constructioprapar- g^. Exprimat jam recta BC magnitudinem vis compositae e binis CN, CR parallelis

DB, AB ; expriment BN, BR magnitudinem virium illarum componentium, cum exprimant
[295] earum directiones, adeoque RC, NC ipsis aequales, & parallelae expriment vires illas
tertiam, & quartam puncti C. Producantur autem DC, AC donee occurrant in O, & T
rectis ex N, & R parallelis ipsi CF, sive ipsis GAZ, KDX, & demittantur AF, DE, NQ, RS
perpendicula in ipsam FC productam, qua opus est, quae occurrat rectis AB, DB in V, P.

vires sub nova 9°. Inprimis ob singula latera singulis lateribus parallela erunt similia triangula IAG,

expression inde CTR, & triangula MDK, CON. Quare erit ut IG, sive AH, ad CR, sive NB, vel A X AH,
nimirum ut I ad A, ita AG ad TR, & ita AI ad TC. Erit igitur TR aequalis GA, sive AZ
ductae in A, & CT aequalis IA ductae in A ; adeoque ilia exprimet summam omnium virium
AZ omnium punctorum in A, haec vim illam primam puncti C, nimirum A X IA. Eodem
prorsus argumento, cum sit MK, sive DL ad CN, sive RB, vel D X DL, nimirum I ad D,
ita DK ad ON, & ita DM ad OC ; erit NO aequalis KD, sive DX ductae in D, & OC
aequalis MD ductae in D, adeoque ilia exprimet summam omnium virium DX omnium
punctorum in D, haec vim illam secundam puncti C, nimirum D X DM. Quare jam
erunt

Summa virium parallelarum in A . . . . TR

Summa virium parallelarum in D . . . . NO

Binse vires in B BN, BR

Quatuor vires in C CT, OC, RC, NC

vis in fulcrum cui 91. Jam vero patet, ex tertia RC, & prima CT componi vim RT aequalem summae

aequalls' virium parallelarum A : & ex quarta NC, ac secunda OC componi vim NO aequalem

summae virium parallelarum in D. Quare patet, ab unico puncto C fulcrum urgeri vi,
quae eandem directionem habeat, quam habent viies parallelae in A, & D, & aequetur earum
summae, nimirum urgeri eodem modo, quo urgeretur, si omnia ilia puncta, quae sunt in
D, & A, cum his viribus essent in C, & fulcrum per se ipsa immediate urgerent.

Proportio quae g2 prseterea °b parallelismum itidem omnium laterum similia erunt triangula if

CNO, DPC : 2° CNQ, PDE : 3? CPR, VCN : 4? CRS, VNQ : 5° CVA, TCR : 6? VAF,
CRS. Ea exhibent sequentes sex proportiones, quarum binae singulis versibus continentur.

ON . CP : : NC . PD : : NQ . DE
CP . CV : : CR . NV : : RS . NQ
CV . RT : : VA . RC : : AF . RS

Porro ex iis componendo primas, & postremas, ac demendo in illis CP, CV ; in his QN,
RS communes tarn antecedentibus, quam consequentibus, fit ex aequalitate nimirum pertur-
bata ON . RT : : AF. DE. Nempe summa omnium virium parallelarum in D, cui
aequatur ON, ad summam om-[296]-nium in A, cui aequatur RT, ut e contrario distantia
harum perpendicularis AF a recta CF ducta per fulcrum directioni virium earumdem
parallela, ad illarum perpendicularem distantiam ab eadem. Quare habetur determinatio
eorum omnium quae quaerebantur (8).

(g) Porro applicatio ad vectem est similis illi, qute habetur hie post (equilibrium trium massarum num. 326.

SUPPLEMENT V

439

M

FIG. 75.

440

PHILOSOPHIC NATURALIS THEORIA

FIG. 75.

SUPPLEMENT V 441

LD multiplied by D. Hence, any point at A will have two forces, AI, AH ; any point at
D two forces DM, DL ; the point B two forces, of which one is directed towards A & is
equal to HA multiplied by A, & the other is directed towards D & is equal to LD multi-
plied by D ; & lastly, the point C will have four forces, of which the first is directed along
AC and is equal to IA multiplied by A, the second along DC and equal to MD multiplied by
D, the third has a direction parallel to BA and is equal to HA multiplied by A, and the fourth
has a direction parallel to BD and is equal to LD multiplied by D. The point C will exert on
the fulcrum a force compounded from all four forces ; and all of these, if the sense of the
direction of the straight lines is considered to be that given by the order of the letters by
which they are named, will be as follows : —

Any point at A will have two forces . . . AI, AH

Any point at D, two forces ..... DM, DL

The point B, two A X HA, D X LD

The point C, four . . A x IA, D x MD, A X HA, D x LD

89. Now let BC represent the magnitude of the force compounded from the two Construction neces
forces CN, CR, parallel to DB, AB : then BN, BR will represent the magnitude of the ^ £°
component forces, since they represent their directions, and thus RC, NC, which are equal

and parallel to them, will represent the third and fourth forces on the point C. Also
let DC & AC be produced, until they meet in O and T respectively the straight lines drawn
through N & R parallel to CF, i.e., to GAZ & KDX ; & let AF, DE, NQ, RS be drawn
perpendicular to CF, produced if necessary ; & let CF meet AB, DB in V & P.

90. First of all, on account of their corresponding sides being parallel, the triangles IAG, The forces that
CTR are similar, & so also are the triangles MDK, CON. Hence, as IG, or AH, is to CR, new* method ^of
or NB, i.e., A X AH, in other words, as i is to A, so is AG to TR, or as IA to TC. Hence, representation.
TR will be equal to GA or AZ multiplied by A, & CT will be equal to IA multiplied by

A. Therefore the former will represent the sum of all the forces AZ on all the points at A,
& the latter that first force on the point C, i.e., A X IA. With precisely the same argu-
ment, since MK, or DL, is to CN, or RB, i.e., D X DL, in other words, as I to D, so is
DK to ON, or DM to OC ; therefore NO will be equal to KD or DX multiplied by D,
& CO equal to MD multiplied by D ; & therefore the former will represent the sum of
all the forces DX for all the points at D, & the latter that second force on the point C
namely, D X DM. Hence, we now have : —

The sum of the parallel forces on A . . . . TR

The sum of the parallel forces on D . . . . NO

The two forces on B ... ... BN, BR

The four forces on C CT, OC, RC, NC

91. And now it is plain that, from the third force RC, & the first, CT, we have a The force on the
resultant force RT which is equal to the sum of the parallel forces at A ; & from the fourth, ["lc™"J : to what it:
NC, & the second, OC, we get a resultant force NO, which is equal to the sum of all the

parallel forces at D. Therefore, it is evident that the fulcrum at C is subject to but a
single force, which has the same direction as that of the parallel forces on the points at
A & D, & that its magnitude is equal to their sum. In other words, the force acting upon
it is exactly the same as if all those points which are at A & D were transferred together
with the forces acting upon them to the point C, & there acted upon the fulcrum directly.

92. In addition, on account of all sides being parallel, the following pairs of triangles The proportion
are similar :-(i) CNO, DPC ; (2) CNQ, PDE ; (3) CPR, VCN ; (4) CRS, VNQ ; (5)

CVA, TCR ; (6) VAF, CRS. These will give the following six proportions, two of which
are contained in each of the following lines : —

ON : CP = NC : PD = NQ : DE
CP : CV = CR : NV = RS : NQ
CV : RT = VA : RC = AF : RS.

Further, by compounding together the first & last of these, & removing from the
antecedents the ratio CP : CV, & from the consequents the ratio QN : RS, we are left
with the proportion, ON : RT = AF : DE. That is to say, the sum of all the parallel
forces on D, to which ON is equal, is to the sum of all those on A, to which RT is equal,
as the opposite perpendicular distance AF from the straight line CF drawn through the
fulcrum in a direction parallel to that of these forces, is to the perpendicular distance of
the former from the same straight line. Hence, we have obtained a solution of all that
was required (£).

(g) Moreover, the application to the lever u similar to that given in this work, after the equilibrium of three points
in Art. 326.

[297] §VI

EPISTOLA AUCTORIS AD P. CAROLUM SCHERFFER

SOCIETATIS JESU

Occasio, & argu- 93- 1° meo discessu Vienna reliqui apud Reverentiam Vestram imprimendum opus,
mentum epistoiae. cujus conscribendi occasionem praebuit Systema trium massarum, quarum vires mutuse
Theoremata exhibuerunt & elegantia, & foecunda, pertinentia tarn ad directionem, quam
ad rationem virium compositarum e binis in massis singulis. Ex iis Theorematis evolvi
nonnulla, quae in ipso prime inventionis aestu, & scriptionis fervore quodam, atque impetu
se se obtulerunt. Sunt autem & alia, potissimum nonnulla ad centrum percussionis
pertinentia ibi attactum potius, quam pertractatum, quae mihi deinde occurrerunt & in
itinere, & hie in Hetruria, ubi me negotia mihi commissa detinuerunt hucusque, qua
quidem ad Reverentiam Vestram transmittenda censui, ut si forte satis mature advenerint,
ad calcem operis addi possint ; pertinent enim ad complementum eorum, quae ibidem
exposui, & ad alias sublimiores, ac utilissimas perquisitiones viam sternunt.

<^ Inprimis ego quidem ibi consideravi directiones virium in eodem illo piano, in
a massis jacentibus quo jacent tres massae, & idcirco ubi Theoremata applicavi ad centrum aequilibrii, &
ad* iTbicuiT ueaDUI^' osc^ati°nis Pro phiribus etiam massis, restrinxi Theoriam ad casum, in quo omnes massae
tas affirmata ^L jaceant in eodem piano perpendiculari ad axem conversionis. In nonnullis Scholiis tantum-
°trandahlC demon" modo innui, posse rem transferri ad massas, utcunque dispersas, si eae reducantur ad id
planum per rectas perpendiculares piano eidem ; sed ejus applicationis per ejusmodi
reductionem nullam exhibui demonstrationem, & affirmavi, requiri systema quatuor
massarum ad rem generaliter pertractandam.

Viribus trium or. At admodum facile demonstratur eiusmodi reductionem rite fieri, & sine nova

massarum in eodem f • TL r i. U • --PL

piano, i n quo peculiari Theona massarum quatuor generalis habetur applicatio tenui extensione Theonae
^?eat' j trans'atis massarum trium. Nimirum si concipiatur planum quodvis, & vires singulae resolvantur

ad ahud, rem ob- • j , j- i • • i HI • v i

tinen. in duas, alteram perpendicularem piano ipsi, alteram paraiielam ; pnorum summa elidetur,

cum oriantur e viribus mutuis contrariis, & aequalibus, quae ad quamcunque datam
directionem redactae aequales itidem remanent, & con-[298]-trariae, evanescente (£) summa :
posteriores autem componentur eodem prorsus pacto, quo componerentur ; si massae per
illas perpendiculares vires reducerentur ad illud planum, & in eo essent, ibique vires haberent
aequales redactas ad directionem ejusdem plani, quarum oppositio & aequalitas redderet
eandem figuram, & eadem Theoremata, quae in opere demonstrata sunt pro viribus jacentibus
in eodem piano, in quo sunt massae. Porro haec consideratio extendet Theoriam aequilibrii,
& centri oscillationis ad omnes casus, in quibus systema quodvis concipitur connexum cum
unico puncto axis rotationis, ut ubi globus, vel systema quotcunque massarum invicem
connexarum oscillat suspensum per punctum unicum.

9^- Quod si sint quatuor massae, & concipiatur planum perpendiculare rectae transeunti
omnes ad planum per binas ex iis, ac fiat resolutio eadem, quae superius ; res iterum eodem recidet : nam
recta:0 Vungenti ^^ bhiae massae ita in illud planum projectae, coalescent in massam unicam, & vires ad

duas : inde transi-

tus ad massas

quotcunque.

(h) Htec turn quidem in bac epistola. Addi potest illud, ubi nulla externa vis in ea directions agens, & in
contraria applicetur diversis partibus ipsius systematis, dtbere vim hujusmodi in singulis etiam ipsius systematis punctis
esse nullam. Nam per mutuum nexum impeditur mutatio positionis mututs,que utique induceretur, si in aliquibus tantum-
modo ejus partibus remaneret vis externis viribus non impedita. Porro ubi agitur de centra oscillationis, & percussionis,
ac etiam de (equilibria, nulla supponitur vis, externa agens secundum directionem axis rotationis, sen conversionis. Quare
in iis casibus, pro quibus b<zc tbeoria hie extenditur, satis est considerare reliquas illas vires, qua agunt secundum
directionem plani perpendicularis eidem axi, quod hie pnsstatur in iis, qu<z consequuntur .

442

§ VI

transition to
theory of the
of oscilla-

A LETTER FROM THE AUTHOR

TO
FR. CAROLUS SCHERFFER, S.J.

03. When I departed from Vienna, I left with Your Reverence to be printed a work, The occasion for,

, . 7J-, . . , ' i • j • r r i i & the contents of,

which I had written as an outcome of the consideration of a system of three masses ; the the letter,
mutual forces between these brought out several theorems that were both elegant & fruitful,
with regard to the direction & the ratio of the forces on each of the masses compounded
from the other two. From these theorems I worked out certain results, which, in the
first surge of discovery, & a certain fervour & impetus of writing, had forced themselves
on my attention. But there are also other matters, especially some relating to the centre
of percussion that are in it merely touched upon rather than dealt with thoroughly ; these
came to me later, some during my journey, & some here in Tuscany, where the business
entrusted to me has kept me up till now. These matters I thought should be sent to Your
Reverence, so that, if perchance they should reach you soon enough, they might be added
at the end of the work ; for they deal with the further development of those things which
I have expounded therein, & open the road to more sublime & useful matters for inquiry.

94. First of all, I there indeed considered the directions of the forces in the same The
plane as that in which the masses were situated ; &, therefore, when I applied the theorems *h£t
to the centre of equilibrium & oscillation even for several masses, I restricted the Theory tion'from* the case
to the case in which all the masses were lying in the same plane, perpendicular to the axis ^ ^e^n^11 p^ne
of rotation. Only in some notes did I mention that the matter could be developed for to masses lying
masses that were disposed in any manner, if these were reduced to that plane by perpendi- any™here, . mer^
culars to the plane. But I gave no demonstration of this application by means of such a work itself, is here
reduction ; & I asserted that the consideration of a system of four masses would be necessary to be Proved-
before the matter could be dealt with thoroughly, & in general.

OS. But it is quite easily proved that such a reduction can be correctly made ; & a T.he forces . f°r

... . . ' ...... r ,. . , three masses m the

general application, without any special fresh theory for lour masses, can take place, with same plane as that
a very slight extension of the theory for three masses. Thus, if any plane is taken & each m . whlch they he,

• 11- e -L • i • j- i o i 11 i belnS transferred

force is resolved into two forces, of which one is perpendicular & the other parallel to the to another, the
plane ; then the sum of all the first will be eliminated, since they arise from mutual forces tnins ^ done-
that are equal & opposite to one another ; for, these when reduced to any given direction
whatever will still remain equal & opposite to one another, & their sum will vanish (£). Also
the latter will be compounded in exactly the same manner as they would have been com-
pounded, if the masses, by means of those perpendicular forces, had been reduced to that
plane, & were really in it, & had there equal forces reduced to the direction of that plane :
the equal & opposite nature of these forces would give the same figure, & the same theorems
as were proved in the work itself for forces in the same plane as that in which the masses
were lying. Further, this way of looking at the matter will extend the Theory of the
centre of equilibrium & oscillation to all cases, in which any system is supposed to be con-
nected with a single point on the axis of rotation, as when a sphere, or a system of any
number of masses connected together oscillates under suspension from a single point.

06. Now if there are four masses, & a plane is taken perpendicular to the straight If there are fou,^

, . 7 . . , . f . . r . r . . . °, masses, they are all

line joining any two of them, & the same resolution is made as in the preceding paragraph ; to be reduced to a
then, the matter will again come to the same thing. For, those two masses, being thus plane perpendicular

, ..... .° , . . 11. to the straight line

thrown into the same plane, will coalesce into a single mass ; & the forces belonging to joining two of

them ; hence the

transition to any

(h) This is what I said in the letter. To it may be added the point that, when no external force is applied acting number of masses.
in one direction on one part, y the opposite direction on another part, of the system, this kind of force must also he zero
for each of the points of the system. For, a change of mutual position is prevented by the mutual connection ; y at
any rate this would be induced, if in any of the parts of it there but remained a force that was not checked by external
forces. Further, when dealing with the centre of oscillation, & of percussion, W with equilibrium, no external force is
supposed to act in the direction of the axis of rotation or conversion. Hence, in these cases, for which the theory is
here extended, it is sufficient to consider these other forces, which act in the direction of the plane perpendicular to the
axis i £jf this is done in what follows.

443

444 PHILOSOPHIC NATURALTS THEORIA

reliquas binas massas pertinentes habebunt ad se invicem eas rationes, quae pro systemate
trium massarum deductae sunt. Hinc ubi systema massarum utcunque dispersarum
convert! debet circa axem aliquem, sive de aequilibrii centre agatur, sive de centra oscilla-
tionis^ sive de centre percussionis, licebit considerare massas singulas connexas cum binis
punctis utcunque assumptis in axe, & cum alio puncto, vel massa quavis utcunque assumpta,
vel concepta intra idem systema, & habebitur omnium massarum nexus mutuus, ac
applicatio ad omnia ejusmodi centra habebitur eadem, concipiendo tantummodo massas
singulas redactas ad planum perpendiculare per rectas ipsi axi parallelas.

Applicatio ad 97. Sic ex. gr. ubi agitur de centre oscillationis, quae pro massis existentibus in unico

centn oscillationis t j- i • i •

generaiem deter- piano perpendiculan ad axem rotationis proposui, ac demonstravi respectu puncti suspen-
minationem. sionis, centri gravitatis, traducentur ad massas quascunque, utcunque dispersas respectu

axis, & respectu rectae parallelae axi ductae per centrum gravitatis, quam rectam Hugenius
appellat axem gravitatis. Nimirum centrum oscillationis jacebit in recta perpendiculari
axi rotationis transeunte per centrum gravitatis, ac ad habendam ejus distantiam ab axe
eodem, si-[299]-ve longitudinem penduli isochroni, satis erit ducere massas singulas in
quadrata suarum distantiarum perpendicularium ab eodem axe, & productorum summam
dividere per factum ex summa massarum, & distantia perpendiculari centri gravitatis
communis ab ipso axe. Rectangulum autem sub binis distantiis centri gravitatis ab axe
conversionis, & a centre oscillationis erit aequale summse omnium productorum, quae
habentur, si massae singulae ducantur in quadrata suarum distantiarum perpendicularium
ab axe gravitatis, divisae per summam massarum. Si enim omnes massae reducantur ad
unicum planum perpendiculare axi conversionis, abit is totus axis in punctum suspensionis,
totus axis gravitatis in centrum gravitatis, & singulas distantiae perpendiculares ab iis
axibus evadunt distantiae ab iis punctis : unde patet generaiem Theoriam reddi omnem
per solam applicationem systematis massarum trium rite adhibitam.

Ahud utile coroi- ng. Quod ad centrum oscillationis pertmet, erui potest almd Corollanum, praeter ilia,

larmm pertmens ad • , . r • j • c • » •

centrum osciiia- q1136 proposui, quod summo saepe usui esse potest : est autem ejusmodi. 01 pLurium -partium
tionis- systematis compositarum ex massis quotcunque, utcunque disperses inventa fuerint seorsim

centra gravitatis, tff centra oscillationis respondentia data puncto suspensionis, vel dato axi
conversionis ; inveniri poterit centrum oscillationis commune, ducendo singularum partium
massas in distantias perpendiculares sui cujusque centri gravitatis ab axe conversionis, &
centri oscillationis cujusvis ab eodem, W dividendo productorum summam per massam totius
systematis ductam in distantiam centri gravitatis communis ab eodem axe. Hoc corollarium
deducitur ex formula generali eruta in ipso opere num. 334 pro centre oscillationis, quae
respondet figurae 63 exprimenti unicam massam A ex pluribus quotcunque, quae concipi
possint ubicunque : exprimit autem ibidem P punctum suspensionis, vel axem conversionis,
G centrum gravitatis, Q centrum oscillationis, M summam massarum A + B + C &c,

a t i r>r\ A X AP2 + B X BP2 + &c.
& formula est PQ = M X GP '

Ejus demonstrate. 99. Nam ex ejusmodi formula est M X GP X PQ = A X AP2 + B X BP2 &c.

Quare si singularum partium massae M ducantur in suas binas distantias GP, PQ ; habetur
in singulis summa omnium A X AP2 + B X BP2 &c. Summa autem omnium ejusmodi
summarum debet esse numerator pro formula pertinente ad totum systema, cum oporteat
singulas totius systematis massas ducere in sua cujusque quadrata distantiarum ab axe.
Igitur patet numeratorem ipsum rite haberi per summam productorum M X GP X PQ
pertinentium ad singulas systematis partes, uti in hoc novo Corollario enunciatur.

i

Usus pro longi- IOO. Usus hujus Corollarii facile patebit. Pendeat ex. gr. globus aliquis suspensus

com^osit^Uso-1 per filum quoddam. Pro globo jam constat centrum gravitatis esse in ipso centra globi,

chroni facilius & constat [soo] itidem, ac e superioribus etiam Theorematis facile deducitur, centrum

oscillationis jacere infra centrum globi, per f tertiae proportionalis post distantiam puncti

suspensionis a centra globi, & radium ; pro filo autem considerate ut recta quadam habetur

centrum gravitatis in medio ipso filo, & centrum oscillationis, suspensione facta per fili

extremum est in fine secundi trientis longitudinis ejusdem fili, quod itidem ex formula

LETTER TO FR. SCHERFFER 445

the other two masses will have to one another those ratios that have already been deter-
mined for a system of three masses. Hence, when a system of masses arranged in any manner
must rotate about some axis, whether it is a question of the centre of equilibrium, or of the
centre of oscillation, or of the centre of percussion, we may consider each of the masses
as being connected with a pair of points chosen anywhere on the axis, & with some other
point, whether this is some mass taken in any manner or assumed to be within the same
system ; & then, there will be a mutual connection between all the masses, & the same
application can be made to all such centres, by merely considering that each of the masses
is reduced to a perpendicular plane by means of straight lines parallel to the axis.

97. Thus, for example, when we are concerned with the centre of oscillation, the Application to the
results which I enunciated for masses existing in a single plane perpendicular to the axis f^6™,1

of rotation, and proved, with respect to the point of suspension & the centre of gravity, may of oscillation.

be applied to any masses, however disposed with respect to the axis, & with respect to a

straight line drawn parallel to the axis through the centre of gravity ; this straight line is

called the axis of gravity by Huyghens. That is to say, the centre of oscillation will lie

in a straight line perpendicular to the axis of rotation drawn through the centre of gravity ;

& to obtain the distance of this centre of oscillation from the axis, or the length of the iso-

chronous pendulum, it will be sufficient to multiply each of the masses by the square of

its distance measured perpendicular to the same axis, & to divide the sum of the products by

the product of the sum of the masses & the perpendicular distance of the common centre

of gravity from the axis. Also the rectangle contained by the two distances of the centre

of gravity from the axis of rotation & the centre of oscillation will be equal to the sum

of all the products, which are obtained by multiplying each of the masses by the square

of its perpendicular distance from the axis of gravity, divided by the sum of the masses.

For, if all the masses are reduced to a single plane perpendicular to the axis of rotation,

the whole axis merely becomes the point of suspension, the whole axis of gravity becomes

the centre of gravity, & each of the perpendicular distances from these axes becomes a

distance from these points. Thus, it will be clear that the whole of the general theory

is obtained by the application of the system of three masses alone, if this is correctly done.

98. As regards the centre of oscillation, there can be derived another corollary, besides Another useful
the one that I have enunciated ; & this has often been of great service to me ; it is as fol- 1°^°"*

lows. //, for two or more parts of a system composed of any number of masses, situated in any oscillation.
manner, the centres of gravity, & the centres of oscillation corresponding to a given point of
suspension, or a given axis of rotation, have been separately determined ; then, the common
centre of oscillation can be determined by multiplying the mass of each of the parts by the per-
pendicular distance of its centre of gravity from the axis of rotation, & the perpendicular distance
of the centre of oscillation from the same axis ; W dividing the sum of these products by the
mass of the whole system, y the distance of the common centre of gravity from the same axis.
This corollary is derived from the general formula derived in the work itself, Art. 334, for
the centre of oscillation, which corresponds to Fig. 63, showing a single mass A out of any
number whatever that might be conceived anywhere ; also in the same diagram, the
point P is the point of suspension, or the axis of rotation, G the centre of gravity, Q the
centre of oscillation, M the sum of the masses A + B + C, &c, and the formula is

po _ A x AP* -t- B x BP* + &c.
M x GP

99. Thus, from the formula given, we have Demonstration of

M x GP x PQ = A x AP» + B x BP* + &c.

Hence, if the mass, M, of each of the parts is multiplied each by its own two distances GP,
PQ, we have for each the total sum A X AP2 + B X BP2 -}- &c. But the sum of all such
sums as these must be the numerator belonging to the formula for the whole system, since
we have to multiply each of the masses of the whole system by the square of its distance
from the axis. Therefore, it is plain that the numerator can be correctly taken to be the
sum of the products M X GP X PQ belonging to the several parts of the system, as we have
stated in this new corollary.

100. The use of this corollary will be easily seen. For example, suppose we have a its use in providing

, , , , . /_ . •: . ., , r ' rr - . an easy determma-

sphere suspended by a thin rod. ror a sphere, it is well-known that the centre 01 gravity tion of the length
is at the centre of the sphere ; and it is also well-known, & indeed it can be easily deduced of a pendulum
from the theorems given above, that the centre of oscillation lies below the centre of the given composite
sphere, at a distance from it equal to two-fifths of the third proportional to the distance of pendulum.
the point of suspension from the centre & the radius. For the rod, considered as a straight
line, the centre of gravity is at the middle point of the rod ; & the centre of oscillation,
when the suspension is made from one end of the rod, is two-thirds of the length of the
rod from that end ; & this can also be deduced quite easily from the general formula. Hence

446 PHILOSOPHIC NATURALIS THEORIA

general! facillime deducitur. Inde centrum oscillationis commune globi, & fili nullo
negotio definietur per corollarium superius.

roenoo I01' ^ Longitude fili a, massa seu pondus b, radius globi r, massa seu pondus p : erit

pendentis e filo. distantia centri gravitatis fili ab axe conversionis erit %a, distantia centri oscillationis
ejusdem ftf. Quare productum illud pertinens ad filum erit J a*b. Pro globo erit
distantia centri gravitatis a + r, quas ponatur = m ; Distantia centri oscillationis erit

TT

m + f X — . Quare productum pertinens ad globum erit m?p + f rrp. Horum summa

est mzp + | rrp + J d*b. Porro cum centra gravitatis fili, & globi jaceant in directum
cum puncto suspensionis, ad habendam distantiam centri gravitatis communis ductam
in summam massarum satis erit ducere singularum partium massas in suorum centrorum
distantias, ac habebitur mp -f- i ab. Quare formula pro centro oscillationis utriusque
simul, erit

m*p -f f rrp -f %azb
mp + \ab

Non Hcere hie con- 102. Hie autem notandum illud, ad centrum oscillationis commune habendum non

uiaf6 uT^oDectf s" licere singularem partium massas concipere, ut collectas in suis singulas aut centris oscilla-

in suis centris oscii- tionis, aut centris gravitatis. In primo casu numerator colligeretur ex summa omnium

tati°nlS'aaut ^ailis Productorum, quae fierent ducendo singulas massas in quadrata distantiarum centri

inter'mediis docu- oscillationis sui ; in secundo in quadrata distantiarum sui centri gravitatis. In illo nimirum

haberetur plus justo, in hoc minus. Sed nee possunt concipi ut collects in aliquo puncto

intermedio, cujus distantia sit media continue proportionalis inter illas distantias ; nam

in eo casu numerator maneret idem, at denominator non esset idem, qui ut idem perseveraret,

oporteret concipere massas singulas collectas in suis centris gravitatis, non ultra ipsa. Inde

autem patet, non semper licere concipere massas ingentes in suo gravitatis centro, & idcirco,

ubi in Theoria centri oscillationis, vel percussionis dico massam existentem in quodam

puncto, intelligi debet, ut monui in ipso opere, tota massa ibi compenetrata vel concipi

massula extensionis infinitesimse ut massae compenetratae in unico suo puncto aequivaleat.

Transitus ad cen- [30 il jo1?. Quod atthiet ad centrum percussionis, id attigi tantummodo determinando

trum percussionis : "• J J J . . . r. p ... . .

ejusnotioneshaberi punctum systematis massarum jacentium in recta quadam, & hbere gyrantis, cujus puncti

posse piures. impedito motu sistitur motus totius systematis. Porro aeque facile determinatur centrum

. percussionis in eo sensu acceptum pro quovis systemate massarum utcunque dispositarum,

& res itidem facile perficitur, si aliae diversae etiam centri percussionis ideas adhibeantur.

Rem hie paullo diligentius persequar.

adhibita in 'o^ere'- I04" InPrimis ut agamus de eadem centri percussionis notione, moveatur libere

centri gravitatis systema quodcunque ita inter se connexum, ut ejus partes mutare non possint distantias a se

inamcrtu^beroVatUS mvicem- Centrum gravitatis totius systematis vel quiescet, vel movebitur uniformiter

in directum, cum per theorema inventum a Newtono, & a me demonstratum in ipso Opere

num. 250, actiones mutuae non turbent statum ipsius : systema autem totum sibi relictum

vel movebitur motu eodem parallelo, vel convertetur motu aequali circa axem datum

transeuntem per ipsum centrum gravitatis, & vel quiescentem cum ipso centro, vel ejusdem

uniformi motu parallelo delatum simul, quod itidem demonstrari potest haud difficulter.

temlte™ traLiato IO5' Inde autem colligitur illud, in motu totius systematis composite ex motu uniformi

cam nrtatione, few in directum, & ex rotatione circular! circa axem itidem translatum haberi semper rectam

rectam cum eo quandam pertinentem ad systema, nimirum cum eo connexam, pro quovis tempusculo

bneemam q u c" vTs suam, qu82 illo tempusculo maneat immota, & circa quam, ut circa quendam axem immotum

tempusculo suam ; COnvertatur eo tempusculo totum systema. Concipiatur enim planum quodvis transiens

poSit.a' ' per axem rotationis circularis, & in eo piano sit recta quaevis axi parallela ; ea convertetur

circa axem velocitate eo majore, quo magis ab ipso distat. Erit igitur aliqua distantia

ejus rectae ejusmodi, ut velocitas conversionis aequetur ibi velocitati, quam habet centrum

gravitatis cum axe translate ; & in altero e binis appulsibus ipsius rectae parallelae gyrantis

LETTER TO FR. SCHERFFER

447

the common centre of oscillation for the sphere & the rod together can with little difficulty
be determined from the corollary given above.

101. Let the length of the rod be a, its mass or weight b, the radius of the sphere r,
and p its mass or weight. The distance of the centre of gravity of the rod from the axis
of rotation will be %a, & the distance ot its centre of oscillation will be fa. Hence, the
product required in the case of the rod is ^a*b. For the sphere, the distance of the centre
of gravity will be a -f- r ; call this m. Then the distance of the centre of oscillation will be

r~
m + f X — Hence, the product for the sphere will be m2p -{- %r*p. The sum of these is

Calculation giving
the formula for a
pendulum formed
of a sphere hanging
at the end of a thin
rod.

m

m*P + 5 r*P ~t~ s^b. Further, since the centres of gravity of the rod & of the sphere lie
in a straight line through the point of suspension, to obtain the distance of the common
centre of gravity multiplied by the sum of the masses, it is enough to multiply the mass of
each part by the distance of its own centre ; in this way we obtain mp -f- i#b. Hence the
formula for the centre of oscillation for both together will be

m*P ~\~ 7irZP "4- 3- &*b.
mp + f ab

102. Now, here we have to observe that, in order to find the common centre of oscilla-
tion, it will not be permissible to suppose that the mass of each part is condensed at either
its centre of oscillation or its centre of gravity. In the first case, the numerator would
be formed of the sum of all the products, obtained by multiplying each mass by the square
of the distance of its centre of oscillation ; & in the second case, by multiplying by the
square of the distance of its centre of gravity. Thus, in the former, the numerator found
would be greater than it ought to be ; & in the latter, less. Further, the masses cannot
be considered to be condensed in any point intermediate to these centres, such that its
distance is some term of a continued proportion between their distances. For, in that
case, the numerator would remain the same when the denominator was not the same ; for,
in order that the latter should remain the same, it would be necessary to suppose that each
mass was condensed at its centre of gravity, & not beyond it. From this it is also evident
that it is not always permissible to suppose that huge masses can be at their centre of gravity ;
&, on this account, when in the theory of the centre of oscillation or percussion I say that
there is a mass at a certain point, it must be understood, as I mentioned in the work itself,
that the whole mass is compenetrated at the point, or supposed to be a small mass of
infinitesimal extension, so as to be equivalent to a mass compenetrated at a single point.

103. Now, as regards the centre of percussion, I merely touched upon this point, when
I determined its position for the case of a system of masses lying in a straight line & gyrating
freely ; using the idea that the point was such that, if its motion was prevented, the whole
system was brought to rest. Further, the centre of percussion is determined with equal
facility, when considered in this way, for any system of masses no matter how they are
arranged. The matter is also easily accomplished, even if diverse other ideas of the centre
of percussion are adopted. In what follows here, I will investigate the matter a little more
carefully.

104. First of all, to use the same notion of the centre of percussion as above, let the
system be in free motion of any sort so long as it is so self-connected that its parts cannot
change their distances from one another. Then, the centre of gravity of the whole system
will either be at rest, or will move uniformly in a straight line ; for, according to a theorem,
discovered by Newton, and demonstrated by myself in Art. 250 of the work, the mutual
actions will not disturb the state of the centre of gravity. Also the whole system, if left
to itself, will either move with the same parallel motion, or will rotate with uniform motion
about a given axis passing through the centre of gravity ; this axis either remains at rest
along with the centre of gravity, or moves together with it with the same parallel uniform
motion, as also can be proved without much difficulty.

105. Also from this it can be deduced that, in a motion of the whole system, com-
pounded of an uniform motion in a straight line and a circular motion about an axis that is
also translated, there will always be found a certain straight line belonging to the system, that
is to say, connected with it, corresponding to every small interval of time ; & this straight
line for that small interval of time remains motionless, and about it, as about an immovable
axis, the whole system is turned in that short interval of time. For, let any plane be taken
passing through the axis of circular motion, and in that plane take any straight line parallel
to the axis • then this straight line will be turned about the axis with a velocity that is greater
in proportion as its distance from the axis is increased. There will therefore be some
distance for such a straight line, such that in that position the velocity of turning will be
equal to that velocity of the centre of gravity & the axis carried along with it ; & in one or
other of the two positions of the parallel straight line, gyrating with the system, when it

We cannot in this
consider each mass
as being condensed
at either its centre
of oscillation or its
centre of gravity,
or other points
inter mediate; a
serviceable warn-
ing, to be taken
from the example
above.

Passing on to the
centre of per-
cussion ; several
different ideas of
this point are
possible.

We will start with
the same idea as
that used in the
work itself ; the
state of the centre
of gravity is con-
served in free
motion.

Hence we derive
the fact that, when
a system is trans-
lated with rotation,
there will be, corres-
ponding to any short
interval of time, a
certain straight line
connected with the
system, which is
motionless ; & this
straight line can
easily be deter-
mined.

448 PHILOSOPHISE NATURALIS THEORIA

cum systemate ad planum perpendiculare ei piano, quod axis uniformiter progrediens
describit, ejus rectae motus circularis net contrarius motui axis ipsius, adeoque motui, quo
ipsa axem comitatur, cui cum ibi & aequalis sit, motu altero per alterum eliso, ea recta
quiescet illo tempusculo, & systema totum motu composite gyrabit circa ipsam. Nee
erit difficile dato motu centri gravitatis, & binarum massarum non jacentium in eodem
piano transeunte per axem rotationis, invenire positionem axis, & hujus rectae immotae pro
quovis dato momento temporis.
Propositio proble- 106. Quaeratur jam in eiusmodi systemate punctum aliquod, cuius motus, si per aliquam

matis, & praaparatio • • j- j i • -i • • • r • -1 ,

ad soiutior.em. vim externam impediatur, debeat mutuis actiombus sisti motus totius systematis, quod
punctum, si uspiam fuerit, dicatur centrum percussionis. Concipiantur autem massas
omnes translatae per rectas parallelas rectae [302] illi manenti immotae tempusculo, quo
motus sistitur, quam rectam hie appellabimus axem rotationis, in planum ipsi perpendiculare
transiens per centrum gravitatis, & in figura 64 exprimatur id planum ipso piano schematis :
sit autem ibidem P centrum rotationis, per quod transeat axis ille, sit G centrum gravitatis,
& A una ex massis. Consideretur quoddam punctum Q assumptum in ipsa recta PG, &
aliud extra ipsam, ac singularum massarum motus concipiatur resolutus in duos, alterum
perpendicularem rectae PQ agentem directione Aa, alterum ipsi parallelum agentem
directione PG, ac velocitas absoluta puncta Q dicatur V.

PA v V

totfsnateoiuute°C& I07> Erit pQ • PA : : V • P , quae erit velocitas absoluta massae A. Erit autem

relativaru

vis massae.

relativarum cujus- p. p

o

pa . . x y _ x y s erjt velocjtas secundum directionem Aa, &

QA QA

PA Aa

PA . Aa : : ^- X V . =-^ X V, quae erit velocitas secundum directionem PG.

Nam in compositione, & resolutione motuum, si rectae perpendiculares directionibus
motus compositi, & binorum componentium constituant triangulum, sunt motus ipsi, ut
latera ejus trianguli ipsis respondentia, velocitas autem absoluta est perpendicularis ad
AP. Inde vero bini motus secundum eas duas directiones erunt

Evan esc entia lo$f jam Vero summa f^~X Ax V est zero, cum ob naturam centri gravitatis

summae determi- J PQ

nans problema. y

summa omnium Aa X A sit aequalis zero, & =^r sit quantitas data. Quare si per vim
externam applicatam cuidam puncto Q, & mutuas actiones sistatur summa omnium motuum
=^-X A X V, sistetur totus systematis motus, reliqua summa elisa per solas vires mutuas,

quarum nimirum summa est itidem zero.

inventio summae 109. Ut habeatur id ipsum punctum Q, concipiatur quaevis massa A connexa cum eo,

nihUo. a;quands & cum puncto P, vel cum massis ibidem conceptis, & summa omnium motuum, qui ex

nexu derivantur in Q, dum extinguitur is motus in omnibus A, debet elidi per vim externam,

summa vero omnium provenientium in P, ubi nulla vis externa agit, debet elidi per sese.

Haec igitur posterior summa erit investiganda, & ponenda = o.

Calculus, & formula [303] no. Porro posito radio = I, est ex Theoremate trium massarum ut P X PQ X i
derivata. ad A X AQ X sin QAa, sive ut P X PQ ad A X Qa, ita actio in A perpendicularis ad

PQ = -^ X V ad actionem in P secundum eandem directionem, quae evadit — = ^- — X V:

1\£ A X A \2

• i s~\ T*/~\ T» « • • T> ** X A vj X A# — A X JL & TT r*

nimirum ob Qa = PQ — Pa, erit actio in P = - P v PQ2

V

harum summa debat aequari zero demptis communibus = pnv sequabuntur positiva

-L />, Jt \^.

negativis, nimirum posita pro characteristica summae, habebitur/.AxPQxPtf =/.AxP02,

sive PQ = -f' A ^g~"> ve^ °b /- A X Pa — M X PG, posito ut prius M pro summa
j • AX ±a

massarum, fiet PQ = -^ ^ , qui valor datur ob datas omnes massas A, datas omnes

rectas Pa, datam PG. Q.E.F.

LETTER TO FR. SCHERFFER 449

arrives in a plane perpendicular to the plane which the uniformly progressing axis describes,
the circular motion of the straight line will be in the opposite direction to that of the axis
itself, and thus of the motion with which it accompanies the axis ; & since it is also equal
to it there, the one motion cancels the other, & the straight line will be at rest for the
small interval of time, & the whole system will gyrate about it with a compound motion.
Nor will it be difficult, given the motion of the centre of gravity, & of two masses not lying
in the same plane passing through the axis of rotation, to find the position of this axis &
that of the motionless straight line for any given instant of time.

106. Now let it be required to find in such a system a point, such that, if its motion Enunciation of a
is prevented by some external force, the motion of the whole system is thereby checked by problem & prepar-
mutual actions ; this point, if there is one, will be called the centre of percussion. Suppose solution*01 the
all the masses to be translated along straight lines parallel to the straight line that remains

motionless for the small interval of time in which the motion is checked ; this straight line
we will now call the axis of rotation ; & suppose that by this translation they are all brought
into a plane perpendicular to the axis of rotation & passing through the centre of gravity.
In Fig. 64, let this plane be represented by the plane of the diagram ; & there also let P
stand for the centre of rotation through which the axis passes ; let G be the centre of gra-
vity, & A one of the masses. Consider any point Q, taken in the straight line PG, & another
point that is not on this line ; & let the motion of each mass be resolved into two, of which
one is perpendicular to the straight line PG & acts in the direction Aa, & the other is
parallel to it & acts in the direction PG ; let the absolute velocity of the point Q be
called V.

107. If v is the absolute velocity of the mass A, we have PQ : PA = V : v ; therefore Determination of
v = V X PA/PQ. Similarly, since we have PA : Pa = V X PA/PQ : V X Pa/PQ ; teiocity.t l^Vhe
therefore V X Pa/PQ will be the velocity in the direction Aa. Also, since we have relative' velocities,
PA : A*=V X PA/PQ : V X Aa/PQ • hence, V X Aa/PQ will be the velocity in the of any mass"
direction PG. For, in composition and resolution of motion, if straight lines perpendicular

to the directions of the resultant motion & its two components form a triangle, then the
motions are proportional to the corresponding sides of the triangle ; & the absolute velocity
is perpendicular to AP. Hence, the two motions in these two directions will be

equal to *± X A X V, and X A X V.

108. Now, the sum of all such as =^- X A X V is equal to zero, since, on account Evanescence of this

JrQ sum which deter-

, . . , ... .. mines the problem.

ot the nature of the centre of gravity, the sum of all such as Aa X A is equal to zero, and
V/PQ is a given quantity. Hence, if by means of an external torce applied at any point Q,

& the mutual actions, the sum of all the motions X A X V is checked, then the

whole motion of the system is checked also ; for the remaining sum is cancelled by the
mutual forces only, of which indeed the sum is also zero.

109. In order to find the point Q, take any mass A connected with it & the point P, The determination
or with masses supposed to be situated at these points ; then the sum of all the motions, u^be eq^atld ' to
which are derived from the connection for Q, when this motion is destroyed for every A, zero-
must be cancelled by the external force ; but the sum of all these that arise for P, upon
which no external force acts, must cancel one another. Hence it is the latter sum that will
have to be investigated & put equal to zero.

no. Now, if the radius is made the unit, then, from the theorem for three masses, The calculation,
we have the ratio of P X PQ X I to A X AQ X sinQAa, or P X PQ to A X Qa, equal "

to the ratio of the action at A perpendicular to PQ (which is equal to ^ X V) to the action

at P in the same direction; & therefore the latter is equal to ^ pr» x V. that

X /\ X \£

is to say, since Qa = PQ — Pa, the action at P = Ax PQ X Pf — A X Pa* x v.

i X A \2

Since the sum of all of these has to be equated to zero, on cancelling the common factor
V/(P X PQ2), the positives will be equal to the negatives ; hence, using the symbol /
as the characteristic of a sum, we have /. A X PQ X Pa = /. A X Pa* ; that is,
PQ =/. A XPa*/(f.A X Pa). Now, if as before we put M for the sum of all the masses,
then/. A X Pa= M X PG, & we have PQ =/. A X Pa*/(M X PG). This value can be
determined ; for all the masses like A are given, also all the straight lines such as Pa
are given, & PG is given. Q.E.F.

GG

45° PHILOSOPHIC NATURALIS THEORIA

Thf°rmuia erutum ni- Corollarium I. Quoniam aP aequatur distantiae perpendicular! A a piano trans-

eunte per P perpendicular! ad rectam PG, habebitur hujusmodi Theorema. Distantia
centri percussionis ab axe rotationis in recta ipsi axi perpendiculari transeunte •per centrum
gravitatis habebitur, ducendo singulas massas in quadrata suarum distantiarum perpendicularium
a -piano perpendiculari eidem rectce transeunte per axem ipsum rotationis, ac dividendo summam
omnium ejusmodi productorum per factum ex summa massarum in distantiam perpendicularem
centri gravitatis communis ab eodem plano.ty

Deductio casus, [304] 112. Corollarium II. Si massae jaceant in eodem unico piano quovis transeunte
quo jaceant omnes per axem ; A, & a congruunt, adeoque distantiae Pa sunt ipsae distantise ab axe. Quamobrem

massae in eodem f , ', , °, .' . r . . , ,

piano. in hoc casu formula haec inventa pro centro percussionis congruit prorsus cum iormula

inventa pro centro oscillationis, & ea duo centra sunt idem punctum, si axis rotationis sit
idem, adeoque in eo casu transferenda sunt ad centrum percussionis, qucecunque pro centro
oscillationis sunt demonstrata.

Si qua massa sit j™. Corollarium III. Si aliqua massa iaceat extra eiusmodi planum pertinens ad aliam

extra : discnmen ? • -i • T> • T> A j • • j • * L •* Z.

centri oscillationis, quampiam ; erit ibi ra minor, quam r A, adeoque centrum percussionis aistabit minus ab

a centro percus- axe rotationis, quam distet centrum oscillationis.

Slonis- r AvPrf2

Formula deducts 114. Corollarium IV. In formula generali PG = V- *:: habetur Pa2 = PG2 +

pro pluribus aliis M X GP

theorematis. Gat _ 2pg x Q^ pQrro y_ ^ x aPQ X Ga evanescit ob evanescentem /. A X Ga, &

.

deducuntur sequentia Theoremata affinia similibus pertinentibus ad centrum oscillationis
deductis in ipso opere.

Theorema de posi- 115. Si impressio ad sistendum motum fiat in recta perpendiculari axi rotationis transeunte

tatis* centri gravi" per centrum gravitatis, centrum gravitatis jacet inter centrum percussionis, W axem rotationis.

Nam PQ evasit major quam PG.

Theorema de n 6. Productum sub binis distantiis illius ab his est constant, ubi axis rotationis sit in

an^ TOoductoantI~ fodem piano quovis transeunte per centrum gravitatis cum eadem directione in quacunque distantia

, ~,~ f.AxGa* . ^n ^, -or* J.Ax*Ga*
ab ipso centro gravitatis. Nam ob GQ = *-* - pp- erit GQ X PG =

deductum"1 inde HJ. In eo casu punctum axis pertinens ad id planum, & centrum percussionis recipro-

cantur ; cum nimirum productum sub binis eorum distantiis a constants centro gravitatis sit
constans.

Axe rotationis ab- 1 1 8. Abeunte axe rotat ionis in infinitum, ubi nimirum totum systema movetur tantummodo

eunte in infinitum. mgfu paraiiei0 centrum percussionis abit in centrum gravitatis. Nam altera e binis distantiis

centrum p crcussio~ / • • *• * •% * i T> • * j *

nis abire in cent- excrescente in infinitum, debet altera evanescere. rorro is casus accidit semper etiam,
rum gravitatis. ^ omnes massae abeunt in unum punctum, quod erit turn ipsum gravitatis centrum to-
[305]-tius systematis, & progredietur sine rotatione ante percussionem.

si axis rotationis nq Abeunte axe rotationis in centrum gravitatis, nimirum quiescente ipso gravitatis

transeat per cen- ........ •« • ?• •

trum gravitatis, centra, centrum percussionis abit in infinitum, nee ulla percussione appiicata unico puncto motus
motum sisti non s-st^ potestt Nam e COntrario altera distantia evanescente, altera abit in infinitum.

osse • *

posse

Centri percussionis I2O Corollarium V. Centrum percussionis debet jacere in recta perpendiculari ad axem

positio notabihs. t. .,...' ••> r^-,

rotationis transeunte per centrum gravitatis. Id evmcitur per quartum e supenonbus ineo-
rematis. Solutio problematis adhibita exhibet solam distantiam centri percussionis ab axe
illo rotationis. Nam demonstratio manet eadem, ad quodcunque planum perpendiculare

(i) Facile deducitur ex hoc primo corollario, ad habendum centrum percussionis massarum utcunque dispersarum satis
esse singulas massas reducere ad rectam transeuntem per centrum gravitatis, W perpendicularem axi rotationis per rectas
ipsi axi perpendiculares, W invenire massarum ita reductarum centrum oscillationis, habito puncto rotationis pro puncto
suspensionis ; id enim erit ipsum centrum percussionis qutesitum. Nam distantite ab ipso piano perpendiculari illi recite,
quarum distantiarum fit mentio in hoc corollario, manent eeelem in ejusmodi translatione massarum, W evadunt distantia
a puncto suspensionis. Theorema autem post substitutionem distantiarum a puncto suspensionis pro Us ipsis distantiis ab
illo piano exhibet ipsam formulam distantia: centri oscillationis a puncto suspensionis, qua habetur num. 334. Hinc autem
consequitur generalis reciprocatio puncti rotationis, W centri percussionis, ac alia plura in sequentibus deducta multo
immediatius deducuntur e proprietatibus centri oscillationis jam demonstratis.

LETTER TO FR. SCHERFFER 451

111. Collar ary I. Since aP is equal to the perpendicular distance of A from a plane A Theorem derived
passing through P perpendicular to the straight line PG, we have the following theorem. from this formula.
The distance of the centre of percussion from the axis of rotation in a straight line perpendicular

to it passing through the centre of gravity, will be obtained by multiplying each mass by the
square of its perpendicular distance from a plane passing through the axis of rotation, y perpen-
dicular to the straight line ; W then dividing the sum of all such products by the product of
the sum of all the masses multiplied by the perpendicular distance of the common centre of gravity
from the same plane. (')

112. Corollary II. If the masses lie in any the same single plane passing through the Dedwstfcmol «w

e ° a

axis, A & a coincide, & therefore the distances fa become the distances of the masses from

the axis. Hence, in this case, the formula here found for the centre of percussion agrees the same plane

in every way with the formula found for the centre of oscillation ; thus the two centres

are the same point, if the axis of rotation is the same. Hence, in this case, everything that

has been proved for the centre of oscillation, holds good for the centre of percussion.

113. Corollary III. If any mass lies outside the plane belonging to any other, then if any mass does
Pa will be less than PA ; hence, the centre of percussion will be at a less distance from the axis ** in

is a

of rotation than the centre of oscillation. distinction between

f A V P/72 tne centre of oscilla-

114. Corollary IF. In the general formula PQ = J', " we have P^ = PG2 tion & the centre of

M X GP percussion.

+ Ga* — 2 PQ X Ga. Also, the sum/.A XzPQxGa vanishes, since/.A X Ga vanishes 5

&/.A X PG2/(M X PG)= PG. Hence we have theorems.

PO pr i /• A x Ga2 . rf) /.A x Ga*
- M x PG ' * MxPG-

From this can be deduced the following theorems like to similar theorems pertaining to the
centre of oscillation deduced in the work itself.

115. If the impressed force applied for the purpose of checking motion is in a straight line Theorem

perpendicular to the axis of rotation & passing through the centre of gravity, the centre of gravity J£f th®e

will lie between the centre of percussion W the axis of rotation. For PQ is greater than PG. gravity.

1 1 6. The product of the two distances of the former from the two latter is constant, when Theorem

the axis of rotation is in any the same plane passing through the centre of gravity, the direction 0"8 ^

of measurement being the same for any distance from the centre of gravity. For, since tances.

CQ - -' <h«efore GQ x PG -

117. In that case, the point on the axis corresponding to the plane y the centre of Corollary derived

percussion will be interchangeable ; for, the product of their two distances from a constant centre from this'
of gravity is constant.

1 1 8. // the axis of rotation goes off to infinity, that is to say, when the whole system is trans- if the axis of

lated with simply a parallel motion, the centre of percussion will become coincident with the centre l^i°y, tTe^centre

of gravity. For, if one of the two distances increases indefinitely, the other must become of percussion win

evanescent. Also, this will always happen, when all the masses coincide at a single point ; wlthTh

this point will then be the centre of gravity of the whole system, & it will be moving without gravity.
rotation before percussion.

119. // the axis of rotation passes through the centre of gravity, the centre of percussion "^^

passes of to infinity, W" the motion cannot be checked by any blow applied at a single point, through the cente

For, on the contrary, when the finite distance vanishes, the other distance must become of , . s^'ty- *e

. ,. . J motion cannot be

innnite. checked.

120. Corollary F. The centre of percussion must lie in the straight line perpendicular A noteworthy
to the axis of rotation & passing through the centre of gravity. This is proved by the fourth centre^of °percuse
of the theorems given above. The method of solution of the problem that was employed sion.

shows the unique distance of the centre of percussion from the axis of rotation. For, the
demonstration remains the same, no matter to what plane perpendicular to the axis all the

(i) /* is easily deduced from this first corollary that, in order to obtain the centre of percussion of any masses
however arranged, it is sufficient to reduce each of the masses to a straight line •passing through the centre of gravity
y perpendicular to the axis of rotation, by means of straight lines perpendicular to the axis ; & then to find the centre
of oscillation of the masses thus reduced, the point of rotation being taken as the point of suspension. This will be the
centre of percussion required. For, the distances from the plane perpendicular to the straight line, such as are men-
tioned in this corollary, remain the same in this kind of translation of the masses y become the distances from the
point of suspension. Moreover, the theorem, after the substitution of the distances from the point of suspension for the
distances from the plane, gives the same formula for the distance of the centre of oscillation from the point of suspension,
which was obtained in Art. 334. From it also there follows the general reciprocity of the point of rotation y the centre
of percussion ; y many other things deduced in what follows can be more easily derived from the properties of the centre
of oscillation already proved.

45 1

PHILOSOPHIC NATURALIS THEORIA

axi reducantur per rectas ipsi axi parallelas & massae omnes, & ipsum centrum gravitatis
commune, adeoque inde non haberetur unicum centrum percussionis, sed series eorum
continua parallela axi ipsi, quae abeunte axe rotationis ejus directionis in infinitum, nimirum
cessante conversione respectu ejus directionis, transit per centrum gravitatis juxta id
Theorema. Porro si concipiatur planum quodvis perpendiculare axi rotationis, omnes
massae respectu rectarum perpendicularium axi priori in eo jacentium rotationem nullam
habent, cum distantiam ab eo piano non mutent, sed ferantur secundum ejus directionem,
adeoque respectu omnium directionum priori axi perpendicularium jacentium in eo piano
res eodem modo se habet, ac si axis rotationis cujusdam ipsas respicientis in infinitum distet
ab earum singulis, & proinde respectu ipsarum debet centrum percussionis abire ad distan-
tiam, in qua est centrum gravitatis, nimiium jacere in eo planorum parallelorum omnes
ejusmodi directiones continentium, quod transit per ipsum centrum gravitatis : adeoque
ad sistendum penitus omnem motum, & ne pars altera procurrat ultra alteram, & earn
vincat, debet centrum percussionis jacere in piano perpendiculari ad axem transeunte per
centrum gravitatis, & debent in solutione problematis omnes massae reduci ad id ipsum
planum, ut praestitimus, non ad aliud quodpiam ipsi parallelum : ac eo pacto habebitur
aequilibrium massarum, hinc & inde positarum, quarum ductarum in suas distantias ab
eodem piano summae hinc, & inde acceptae aequabuntur inter se. Porro eo piano ad solu-
tionem adhibito, patet ex ipsa solutione, centrum percussionis jacere in recta perpendiculari
axi ducta per centrum gravitatis : jacet enim in recta, quae a centro gravitatis ducitur ad
illud punctum in quo axis id planum secat, quae recta ipsi axi perpendicularis toti illi piano
perpendicularis esse debet.

impactus m cen- I2i. Corollarium VI. Impactus in centro percussionis in corpus externa vi eius motum

trum percussionis . ., . . . '. . . . '.....'

qui sit. sistens est idem, qui esset, si singulce masses incurrerent in ipsum cum suis velocitatibus respecti-

[$o6\-vis redactis ad directionem perpendicularem piano transeunti per axem rotationis, £ff
centrum gravitatis, sive si massarum summa in ipsum incurreret directione, & velocitate motus,
qua fertur centrum gravitatis.

Demonstratio pri- 122. Patet primum, quia debet in Q haberi vis contraria directioni illius motus

mae partis. perpendicularis piano transeunti per axem, & PG, par extinguendis omnibus omnium

massarum velocitatibus ad earn directionem redactis, quae vis itidem requireretur, si omnes

massse eo immediate devenirent cum ejusmodi velocitatibus.

p

patet secundum ex eo, quod velocitas ilia pro massa A sit pf) x V, adeoque

Demonstratio

motus

A x

X V, quorum motuum summa est

M x PG

X V.

PG

Est autem .g-^-J x V,

n — , - n

velocitas puncti G, quod punctum movetur solo motu perpendiculari ad PG, adeoqiie si
massa totalis M incurrat in Q cum directione, & celeritate, qua fertur centrum gravitatis G,
faciet impressionem eandem.

impressio ubi fieri 124. Corollarium VII. Potest motus sisti impressione facta etiam extra rectam PG,

trum percussionis seu extra planum transiens per axem rotationis, & centrum gravitatis, nimirum si impressio
cum eodem effectu. flat in quodvis punctum rectce eidem piano perpendicularis, & transeuntis per Q, directione

rectee ipsius. Nam per nexum inter id punctum, & Q statim impressio per earn rectam

transfertur ab eo puncto ad ipsum Q.

Motus communi-
qU1

quiescenti.

125. Corollarium VIII. Contra vero si imprimatur dato cuidam puncto systematis
sys1temati quiescentis vis qucedam matrix ; invenietur facile motus inde communicandus ipsi systemati.
Nam ejusmodi motus erit is, qui contrario aequali impactu sisteretur. Determinatio autem
regressu facto per ipsam problematis solutionem erit hujusmodi. Centrum gravitatis
commune movebitur directione, qua egit vis, & velocitate, quam ea potest imprimere massae
totius systematis, quae ad earn, quam potest imprimere massae cuivis, est ut haec posterior
massa ad illam priorem, & si vis ipsa applicata fuerit ad centrum gravitatis, vel immediate,
vel per rectam tendentem ad ipsum ; systema sine ulla rotatione movebitur eadem veloci-
tate : sin autem applicetur ad aliud punctum quodvis directione non tendente ad ipsum
centrum gravitatis, praeterea habebitur conversio, cujus axis, & celeritas sic invenietur.
Per centrum gravitatis G agatur planum perpendiculare rectae, secundum quam fit impactus,
& notetur punctum Q, in quo eidem piano occurrit eadem recta. Per ipsum punctum G
ducatur in eo piano recta perpendicularis ad QG, quae erit axis quaesitus. Per punctum Q
concipiatur alterum planum perpendiculare rectae GQ, ca-[307]-piantur omnes distantiae
perpendiculares omnium massarum A ab ejusmodi piano, aequales nimirum suis aQ :

LETTER TO FR. SCHERFFER 453

masses & their common centre of gravity are reduced by straight lines parallel to the axis.
Thus, from it, we should not obtain a single centre of percussion, but a continuous series
of them parallel to the axis ; & this, when the axis of rotation goes off to infinity for this
direction, that is, when turning ceases for this direction, will pass through the centre of
gravity, according to the theorem. Further, if any plane perpendicular to the axis of
rotation is taken, all the masses have no rotation with regard to straight lines perpendicular
to the former axis which lie in the plane ; for they will not change their distances from
that plane, but are carried in its direction. Hence, with regard to all directions perpendi-
cular to the former axis which lie in that plane, the matter comes out in the same way ; &,
if the axis of rotation for any one of the former is infinitely distant from each of the latter,
and therefore with respect to the former, the centre of percussion has to pass to that distance
at which is the centre of gravity, that is to say, has to lie in that one of the parallel planes con-
taining all such directions, which passes through the centre of gravity. Thus, to stop all
motion entirely, & to prevent one part outrunning another part & overcoming it, the centre
of percussion must lie in a plane perpendicular to the axis & passing through the centre of
gravity ; & in the solution of the problem, all the masses are bound to be reduced to that
plane, as we have shown, & not to any other that is parallel to it. In this way, we shall
obtain equilibrium of the masses, situated on either side of it ; & the sums of these multiplied
by their distances from this plane, taken together on one side & on the other, will be equal
to one another. Moreover, if this plane is used for the solution, it is clear from the solution
itself, that the centre of percussion lies in a straight line perpendicular to the axis, drawn
through the centre of gravity. For, it will lie in the straight line that is drawn from the
centre of gravity to that point in which the axis cuts the plane, & this straight line must
be perpendicular to the axis, since the axis is perpendicular to the whole of the plane.

121. Corollary VI. The impact at the centre of percussion on a body by an external The nature of the
force, which checks its motion, is the same as we should have, if each mass were to collide with

it with its velocity resolved in the direction perpendicular to the plane passing through the axis
of rotation & the centre of gravity ; or if the sum of the masses collided with it with the direction
6f velocity of motion, with which the centre of gravity is moving.

122. The first part is evident, because there must be at Q a force opposite in direction Proof of the first
to the motion perpendicular to the plane passing through the axis & PG, capable of destroying part'

all the velocities of all the masses resolved in that direction ; & this force would also be
required, if all the masses collided with it directly with such velocities.

123. The second part is evident from the fact that the velocity for the mass A is Proof of the second

part.

X V ; & thus, the motion is — pf\-~- X V ; & the sum of these motions is

1VT v Pf* Pf

- X V. But pp- X V is the velocity of the point G, & the sole motion of this

point is perpendicular to PG ; & thus, if the total mass M collided with Q with the
direction & speed with which the centre of gravity G moves, it would produce the same
effect.

1 24. Corollary VII. The motion may be checked even by a blow applied without the when the blow can
straight line PG, or without the plane passing through the axis of rotation W the centre of gravity ; {j^Sntre £f yperd
that is, if it is applied at any point of a straight line perpendicular to the same plane, & passing cussion with the
through Q, in the direction of this straight line. For, through the connection between that same effect'
point & Q, the blow is immediately transferred along the straight line from the point to Q
itself.

125. Corollary VIII. On the other hand, if any motive force is impressed upon any Motion communi-
given point of a system at rest, it is easy to find the motion thereby communicated to the system. System at rest. °
For such motion will be that which would be checked by an equal & opposite blow. The
determination of the motion, made by retracing our steps through the solution of that
problem, would proceed as follows. The common centre of gravity will be moved in the
direction in which the force acts, & with a velocity which it can give to the mass of the whole
system ; this velocity is to that which it could give to any mass as the latter mass is to the
former. If the force were applied at the centre of gravity, either directly, or along a straight
line tending to it, then the system, without any rotation, would move with the same velocity.
But if it were applied at any other point in a direction not tending towards the centre of
gravity, we should have in addition a rotation, of which the axis & the velocity will be found
thus. Let a plane be drawn through the centre of gravity G perpendicular to the straight
line along which the blow is impressed, & let the point in which the straight line meets this
plane be denoted as the point Q. Through G draw in this plane a straight line perpendicular
to QG ; this will be the axis required. Draw another plane through the point Q, per-
pendicular to the straight line QG ; take all the perpendicular distances of all the masses A

454 PHILOSOPHIC NATURALIS THEORIA

singularum quadrata ducantur in suas massas, & factorum summa dividatur per summam
massarum, turn in recta GQ producta capiantur GP aequalis ; ei quoto diviso per ipsam
QG, & celeritas puncti P revolventis circa axem inventum in circulo, cujus radius GP, erit
aequalis celeritati inventae centri gravitatis, directio autem motus contraria eidem. Unde
habetur directio, & celeritas motus punctorum reliquorum systematis.

Demonstrate. 126. Patet constructio ex eo, quod ita motu composito movebitur systema circa axem

immotum transeuntem per P, qui motus regressu facto a constructione tradita ad inventi-
onem praemissam centri percussionis sisteretur impressione contraria, & aequali impressioni
datae.

Aditus ad perquisi- 127- Scholium. Hoc postremo corollario definitur motus vi externa impressus

tiones uiteriores systemati quiescenti. Quod si jam systema habuerit aliquem motum progressivum, &

motu impresso < , •>..•',. ,,^- . . .

systemati moto. circularem, novus motus externa vi inductus juxta corollanum ipsum componendus erit
cum priore, quod, quo pacto fieri debeat, hie non inquiram, ubi centrum percussionis
persequor tantummodo. Ea perquisitio ex iisdem principiis perfici potest, & ejus ope
patet, aperiri aditum ad inquirendas etiam mutationes, quae ab inaequali actione Solis,
& Lunae in partes supra globi formam extantes inducuntur in diurnum motum, adeoque
ad definiendam ex genuinis principiis prascessionem aequinoctiorum, & nutationem axis :
sed ea investigatio peculiarem tractionem requirit.

Transitus ad aliam I28- Interea gradum hie faciam ad aliam notionem quandam centri percussionis,

notionem ejus cen- nihilo minus, imo etiam magis aptam ipsi nomini. Ad earn perquisitionem sic progrediar.

tri.

Probiema conti- 129. Problema. Si systema datum gyrans data velocitate circa axem datum externa vi

nens hanc ideam. immotum incurrat in dato suo -puncto in massam datam, delatam velocitate data in directions

motus •puncti ejusdem, quam massam debeat abripere secum ; queeritur velocitas, quam ei masses

im-primet, & ipsum systema retinebit post im-pactum.

Soiutio : formula 130 Concipiatur totum systema projectum in planum perpendiculare axi rotationis

continents motum transiens per centrum gravitatis G, in quo piano punctum conversionis sit P, massa autem

massse in quam . —t— . **.**?, ,?.r •"•• IT i • n

incidit, & suum in recta PG in Q. Velocitas puncti cujusvis systematis, quod distet ab axe per intervallum
reiiquum. _ I? ante incursum sit = a, velocitas ab eodem amissa sit = x, adeoque velocitas post

impactum = a — x, velocitas autem massae Q ante impactum sit = PQ X b. Erit ut I
ad AP, ita x ad velocitatem amissam a massa A, quae erit AP X x. Erit autem ut I ad
a — x ita PQ ad velocitatem residua m in puncto systematis Q, quas net PQ X (a — #),
& ea erit itidem velocitas massse Q post [308] impactum, adeoque massa Q acquiret veloci-
tatem PQ X (a — b — x), sive posito a — b=c, habebitur PQ X (c — x). Porro ex
mutuo nexu massae A cum P, & Q erit Q X PQ ad A X AP, ut effectus ad velocitatem

A X AP2

pertinens in A = AP X x ad effectum in Q = j~— \ np X x. Summa horum effec-

(j. X vj.r

tuum provenientium e massis omnibus erit aequalis velocitati acquisitae in Q. Nimirum

X c, *

x = _ Q X °-F* - x c . Dato autem x datur a — x, & is valor ductus in distantiam

/. A x AP2 + Q x QP2
puncti cujusvis systematis, vel etiam massae Q, exhibebit velocitatem quaesitam. Q.E.F.

Casus particulares, 1 31. Scholium. Formula habet locum etiam pro casu, quo massa Q quiescat, vel quo

ad quos appiicari feratur contra motum systematis, dummodo in primo casu fiat b = o, & c = a, ac in secundo

valor b mutetur in negativum, adeoque sit c = a + b. Posset etiam facile appiicari ad

casum, quo in conflictu ageret elasticas perfecta vel imperfecta. Determinatio tradita

exhiberet partem effectus in collisione facti tempore amissas figurae, ex quo effectus debitus

LETTER TO FR. SCHERFFER 455

from this plane, each equal to the corresponding aQ ; multiply the square of each of these
by the corresponding mass, & divide the sum of all the products by the sum of the masses.
Then in the straight line QG produced take GP equal to this quotient divided by QG.
The velocity of the point P rotating in a circle about the axis which has been found, of
which the radius is GP, will be equal to the velocity of the centre of gravity which has also
been found, but the direction of the motion will be in the opposite direction. From this,
we have the direction & the velocity for all the other points of the system.

126. The correctness of the construction is evident from the fact that in this way Demonstration.
the system will move with a compound motion in a circle about a motionless axis passing

through P ; & this motion, by retracing our steps from the construction for finding the
centre of percussion, already given, would be checked by a blow equal & opposite to the
given blow.

127. Scholium. In the last corollary the motion impressed by an external force on The way is open
a system at rest is determined. But if now the system should have some motion, progressive gationf w^e^mo-
& circular, the new motion induced by the external force in accordance with the corollary tion is impressed on
will have to be compounded with what it already has. I do not inquire here, how this will a movms system.
happen, for here I am only concerned with the centre of percussion. The investigation can

be carried out by means of the very same principles ; & by the help of this investigation,
it is clear that the door would be opened also for the investigation of the variations which are
induced in the daily motion by the unequal actions of the Sun, & of the Moon, on parts of
the Earth that jut out beyond the figure of the sphere ; & thus for determining from
real principles the precession of the equinoxes & the nutation of the axis. But this investiga-
tion requires a special treatise.

128. Meanwhile, I will now go on to another idea of the centre of percussion, which Passing on to an-
is no less, nay it is even more, fit to have that name given to it. To this investigation oth" ldea of thls

' . ' , .... . centre.

I proceed in the following manner.

129. Problem. // a given system, gyrating with given velocity about a given axis, not Problem embody-
acted upon by an external force, collides at a given point of itself with a given mass, which

is moving with a given velocity in the direction of the motion of this point, the mass being of
necessity borne along with the system ; it is required to find the velocity impressed on the mass,
y retained by the system after impact.

130. Suppose that the whole system is projected on a plane perpendicular to the axis Solution ; formulae
of rotation passing through the centre of gravity G ; in this plane let the point of rotation containing the

, .° , . . .. ° —. .^ XT f i • r • r i motion of the mass

be P, & let the mass be in the straight line PG at Q. Let the velocity of any point of the with which it col-
system, whose distance from the axis is unity, before the impact be a, & let the velocity Udes- a n d * n e

i i • i t • f • MI u AI i i i • r i motion left in it-

lost by it be x ; & thus, the velocity alter impact will be a— x. Also let the velocity of the seif.
mass at Q before impact be PQ X b. Then, as I is to AP so is x to the velocity lost by the
mass at A, which will therefore be AP X x. Also, as I is to a — x so is PQ to the velocity
that remains in the point Q of the system ; & therefore this is PQ X (a — x) ; this will
also be the velocity of the mass Q after impact. Hence, the mass Q will acquire a velocity
PQ X (a — b — x} ; or, if we put a — b = c , it will be PQ X (c — x). Further, from
the mutual connection between the mass A & P & Q, we shall have the ratio of Q X PQ to
A X AP equal to that of the effect pertaining to the velocity at A, which is equal to

A x AP*

AP X x, to the effect at Q, which is therefore equal to _ X x. The sum of

Q X Qir

these effects, arising from all the masses, will be equal to the velocity acquired at Q. That
is to say, we have

Q x QPZ
1 * = /.Ax AP' + QxQP2

But, if we are given x, we are also given a — x ; and this value, multiplied by the distance
of any point of the system, or also that of the mass Q, will give the velocity required. Q.E.F.

131. Scholium. The formula holds good even when the mass Q is at rest, or when Particular cases to
it moves in the opposite direction to the system ; so long as, in the first case, b is made ^hlj£d li can be
equal to zero, or c = a ; & in the second case, the value of b is changed from positive to
negative, so that c = a -j- b. It might also easily be applied to the case in which elasticity,
either perfect or imperfect, would take a part in the collision. The determination given
would represent that part of the effect of the collision which was produced during the
interval of time corresponding to loss of shape ; & from this the proper effect for the whole

456 PHILOSOPHIC NATURALIS THEORIA

tempori totus collisionis usque ad finem recuperatae figurae colligitur facile, duplicando
priorem, vel augendo in ratione data uti fit in colKsionibus.

ulterior 132. Itidem locum habet pro casu, quo massa nova non jaceat in Q in recta PG, sed

in quovis alio puncto plani perpendicularis axi transeuntis per G, ex quo si intelligatur
perpendiculum in PG ei occurrens in Q ; idem prorsus erit impactus ibi, qui esset in Q,
translata actione per illam systematis rectam. Qui imo si Q non jaceat in eo piano perpen-
diculari ad axem, quod transit per centrum gravitatis, sed ubivis extra, res eodem redit,
dummodo per id punctum concipiatur planum perpendiculare axi illi immoto per vim
externam ad quod planum reducatur centrum gravitatis, & qusevis massa A ; vel si ipsa
massa Q cum reliquis reducatur ad quodvis aliud planum perpendiculare axi. Omnia
eodem recidunt ob id ipsum, quod axis externa vi immotus sit. Sed jam ex generali
solutione problematis deducimus plura Corollaria.

Reiatio ad cen- 133. Corollarium I. Si distantia centri oscillationis totius systematis ab axe P dicatur

trum oscillationis. R} distantia centri gravitatis G, massa tota M, habebitur

_ _ Qx PQ2 v . ro, f_MxGxR ,

~ MxGxR + QxPQ2 Ct l3 9J x~ Q xPQ*

f A x AP2
Patet ex eo, quod ex natura centri oscillationis habetur R = -A^ - -=- , adeoque

M x G
/. A x AP' = M x G x R.

Expressio veioci- ,,. Corollarium II. Velocitas acquisita a massa Q erit J^ * G * R * PQ- X c.

tatis m massa sim- M X G X R+ Q X PQ2

phcior ope dims. Q pQl

Est enim ea velocitas PQ X (e - x), sive PQ (c -- M x G ^ R + Q x PQ« X f)» 1uod
reductum ad eundem denominatorem elisis terminis contrariis eo redit.

UbicoiHgendumes- j-jr Corollarium III. Si manente velocitate circular! systematis tota ejus massa

set totum systema . ,J .... . . i . „ ... . J .

ad eandem veioci- concipiatur collecta m unico puncto jacente inter centra gravitatis, & oscillationis, cujus
tatem impri- distantia a puncto conversionis sit media geometrice proportionalis inter distantias reli-

mendam massae. , . , ,. , . ,

quorum punctorum, vel m eadem distantia ex parte opposita ; velocitas eadem impnmeretur
novae massae in quovis puncto sitae. Tune enim abiret in illud punctum utrumque centrum,
& valor G X R esset idem, ac prius, nimirum aequalis quadrate ejus distantiae ab axe, quod
quadratum est positivum etiam, si distantia accepta ex parte opposita fiat negativa.

in quot, & quibus 136. CorollariumlV. Si capiatur hinc, vel inde in PG segmentum, quod ad distantiam

massa^eandem ^x ejus puncti ab axe sit in subduplicata ratione massae totius systematis ad massam Q ; ipsa
impactu veiotita- massa Q in -quatuor distantiis ab axe, binis hinc, & binis inde, quarum binarum producta
' aequentur singula quadrato ejus segmenti, acquiret velocitatem in omnibus eandem
magnitudine, licet in binis directionis contrariae, & ea net maxima, ubi ipsa massa sit in
fine ejus segmenti ex parte axis ultralibet. Erit enim velocitas acquisita directe ut

'• vel dividendo per const""em ~^ x '- *

M
endo illud segmentum = ± T, cujus quadratum T* debet esse = ^ X G X R, erit

PQ ^a

directe ut _I_PO«' adeoque reciproce ut ^^ + PQ. Is autem [310] valor manet

idem, si pro PQ ponantur bini valores, quorum productum aequatur T*, migrante tantum-
modo altera binomii parte in alteram Si enim alter valor sit m, erit alter — ; & posito

m

W

illo pro PQ : habetur — + m, posito hoc habetur -=— + — , sive m + — . Sed cum

m 1 m m

T2 ...

eae distantiae abeunt ad partes oppositas, fiunt — w, & — , migrante in negativum etiam

tn

LETTER TO FR. SCHERFFER 457

time of collision, up to the end of recovery of shape could be easily derived, by doubling
in the first case, & by increasing in a given ratio in the second case ; just as was done when
we considered collisions.

132. The formula also holds good for the case in which the new mass does not lie at S^ jde|*tei
the point Q in the straight line PG, but at some other point of a plane perpendicular to

the axis & passing through G ; if from this point a perpendicular is supposed to be drawn
to PG, meeting it in Q, then the effect will be exactly the same as if the impact had been
at Q, the action being transferred by this straight line of the system. Indeed, if Q does not
lie in the plane perpendicular to the axis, which passes through the centre of gravity, but
somewhere without it, it all comes to the same thing, so long as through that point a plane
is supposed to be drawn perpendicular to the axis that is unmoved by the external force,
and the centre of gravity is reduced to this plane, together with any mass A ; or if the
mass Q, together with the rest, is reduced to any plane perpendicular to the axis. It all
comes to the same thing, on account of the fact that there is an axis that is unmoved by the
external force. But now we will deduce several corollaries from the general solution of
the problem.

133. Corollary I. If the distance of the centre of oscillation of the whole system from Relation to the
the axis P is denoted by R, the distance of the centre of gravity by G, & the total mass by ""

M, then we have „ = M x ****Q x Qp. X , i & |= + «• It is

evident from the fact that, from the nature of the centre of oscillation, we have

R = • . & thus

1VL /\ \J

1 34. Corollary II. The velocity acquired by the mass Q will be A simpler expres-

sion for the velo-

MXGXRXPQ.. city in the mass

MXGXR + QXPQ«X by its help.

for, this is the velocity PQ (c - x), or PQ (c - ^--^^W^- p.Qi- x c) ;

and this, when reduced to the same denominator, comes to that which was given, after
cancelling terms of opposite sign.

135. Corollary III. If, while the circular velocity remained unaltered, the whole mass The point in which
of the system is supposed to be collected at a single point lying between the centres of ^ouid^veTo^
gravity & oscillation, the distance of which from the point of rotation is a geometrical mean collected in order
between the distances of the other points, or at the same distance on the other side of the ^meim^^St ^n
point of rotation ; then, the same velocity would be impressed on the new mass situated the mass.

at any point. For, in that case, each centre would coincide with that point, & the value
of G X R would be the same as before, namely, equal to the square of its distance from
the axis ; & this square is positive, even if the distance, when taken on the other side of the
point of rotation, is negative.

136. If, on one side or the other, in PG a segment is taken, which is to the distance Th.e number of

t .1 ' • r , ... i i v ' r i 11 r i i points, and their

ot the point from the axis in the subduphcate ratio of the whole mass of the system to the distances from the

mass Q ; then, the mass Q, if placed at one of four distances from the axis, two on one side axis- for wh^h the

& two on the other, so that the products for each pair should be equal to the square of the quire the same

segment, would at each distance acquire a velocity of the same magnitude although in yelocities from the

. , . . , . A i i • i • iii T impact ; where the

opposite directions tor the two pairs. Also this velocity would be greatest, when the mass velocity would be
was placed at the end of the segment on either side of the- axis. For, the velocity acquired greatest.

varies directly as M x Q x R . Q x a x c '> dividing this by the constant

- X c , and denoting the segment by ± T, of which the square, T1, must be

equal to ^- X G X R, the velocity will vary directly as _\_T>(\*> ^ therefore, inversely

T*

as 7=c + PQ- Now, this value remains the same, if for PQ we substitute either of the

pair of values whose product is T*, the first part of the binomial expression merely
interchanging with the second. For, if either value is denoted by m, the other will be
T2/« ; &, if the former is substituted for PQ, we get T*/m + m ; or, if the latter, we
have T*wi/T* -f- T*/m, i.e., m + T*/m. But, when these distances are taken on the
opposite side, they become — m & — T*/m, & the value also of the formula becomes nega-
tive ; this shows that the direction of the motion is opposite to what it was before ; in

458 PHILOSOPHIC NATURALIS THEORIA

valore formulae, quod ostendit directionem motus contrariam priori, systemate nimirum
hinc, & inde ab axe in partibus oppositis habente directiones motuum oppositas.

rpa

Demonstratio de- 137. Quoniam autem assumpto quovis valore finite pro PQ, formula ~>c+PQ est

terminationis max-

finita, & evadit infinita facto PQ tarn infinite, quam = o ; patet in hisce postremis duobus
casibus velocitatem e contrario evanescere, in reliquis esse finitam, adeoque alicubi debere
esse maximam. Non potest autem esse maxima, nisi ubi ad eandem magnitudinem redit,
quod accidit in transitu PQ per utrumvis valorem ± T, circa quern hinc & inde valores
aequales sunt. Ibi igitur id habetur maximum.

^g Scholium 2. Libuit sine calculo differential! invenire illud maximum, quod ope

difierentiaiem. C3L\C}3k ipsius admodum facile definitur. Ponantur T = *, & PQ = z. Fiet formula -+z,

z

& differentiando -- - + dz — o, sive — ** + z* — o, vel z* = /*, & z = ± t , sive
zz

PQ = ± T, ut in corollario 4 inventum est.
DUX aliae accep- j,g Ljcebit autem iam ex postremis duobus corollariis deducere alias duas notiones

tiones centn per .•>*.. J . ., „ . „ .

cussionis, & ejus centn percussioms, cum suis eorundem determinatiombus. rotest pnmo appellan centrum
determinatio ex percussionis illud punctum, in quo tota systematis massa collecta eandem velocitatem

superionbus. f r., . • *. ' , , 1

impnmeret massae eidem incurrendo m earn eodem suo puncto cum eadem velocitate, quae
videtur omnium aptissima centri percussionis notio. Centrum percussionis in ea acceptione
determinatur admodum eleganter ope corollarii 3 : jacet nimirum inter centrum gravitatis,
& centrum oscillationis ita, ut ejus distantia ab axe rotationis sit media geometrice pro-
portionalis inter illorum distantias, vel ubivis in recta axi parallela ducta per punctum ita
inventum. Potest secundo appellari centrum percussionis illud punctum, per quod si fiat
percussio, imprimitur velocitas omnium maxima massae, in [311] quam incurritur. In
hac acceptione centrum percussionis itidem eleganter determinatur per corollarium
quartum, mutando earn distantiam in ratione subduplicata massae, in quam incurritur,
ad massam totius systematis.

watum lta&0npro I4°- ^n ^oc secundo sensu acceptum, & investigatum esse centrum percussionis a

particular! casu summo Geometra Celeberrimo Pisano Professore Perrellio, nuper mihi significavit Vir

determmatum. itidem Doctissimus, & geometra insignis Eques Mozzius, qui & suam mihi ejus centri

determinationem exhibuit pro casu systematis continentis unicam massam in rectilinea

virga inflexili.

?u t ernedftermi if I4I> Libuit rem longe alia methodo hie erutam generaliter, & cum superioribus

atum ad foecundi- omnibus conspirantem, ac ex iis sponte propemodum profluentem proponere, ut innotescat
ta*tejn A Theoriae mira sane foecunditas Theorematis simplicissimi pertinentis ad rationem virium

ostendendam. . r „ . . .r. ...

compositarum in systemate massarum tnum. bed de his omnibus jam satis.
DABAM FLORENTIAE, 17 Junii, 1758.

FINIS.

LETTER TO FR. SCHERFFER

459

other words, the system has opposite directions for motions of opposite parts on either side
of the axis.

137. Now, since, for any assumed finite value of PQ, the formula T2/PQ -f PQ is
finite, & comes out infinite both when PQ is made infinite & when it is made zero, it is
clear that the velocity, which varies inversely as the formula, must vanish in these two
extreme cases, & be finite in all other cases ; hence, at some time there must be a maxi-
mum. But it cannot be a maximum, except when the two parts of the formula become
equal ; & this happens as PQ passes through either of the values ± T, about which, on
either side, the values are equal. Hence there is a maximum there.

138. Scholium ^. I have preferred to find this maximum without the help of the
differential calculus ; but with the help of the calculus, it can be determined very easily.
Put T = t, & PQ = z ; then the formula becomes t*/z + z. Differentiating, we have

- t dz/z* + dz = o, or - t* + z2 = o, or z2 = t* ; & z = ± /, or PQ = ± T, as
was found in corollary IV.

139. We may now, from the last two corollaries, deduce two other ideas of the centre
of percussion, together with the determination of each. In the first place, we may call
the centre of percussion that point which is such that if the whole mass of the system were
collected therein, it would impress the same velocity on the same mass by colliding with it
with this same point of itself with the same velocity ; & it seems that this is the most apt
idea of all for the centre of percussion. The centre of percussion, in this acceptation, is
determined in a very elegant manner by the aid of corollary III. Thus, it will lie between
the centre of gravity & the centre of oscillation, in such a manner that its distance from
the axis of rotation is a geometrical mean between those two distances, or anywhere in a
straight line parallel to the axis drawn through the point thus found. Again, the name
centre of percussion may be given to that point which is such that, if the blow is delivered
through it, it will give to the mass on which it falls the greatest possible velocity. In this
acceptation, the centre of percussion is also elegantly determined by the fourth corollary,
by changing the distance in the subduplicate ratio of the mass struck to the whole mass of
the system.

140. That learned man & fine geometer, Signer Mozzi, has but lately acquainted
me with the fact that the centre of percussion was taken, in this second sense, & investigated
by that excellent geometer, the well-known Professor at Pisa, Perrelli ; & Mozzi also showed
me his own determination for the case of a system consisting of a single mass in the form
of a rectilinear inflexible rod.

141. I have preferred to set forth the matter here derived in general in a far different
manner, agreeing as it does with all that has gone before, & arising from it almost automa-
tically, so as to make known the truly wonderful fertility of that very simple theorem
dealing with the ratio of the composite forces in a system of three masses. But now I
have said enough about all these things.

FLORENCE,

17 'th June, 1758.

Demonstration that
the maximum is
correctly given.

Determination of
the maximum by
means of the dif-
ferential calculus.

Two other accep-
tations of the term
centre of percus-
sion ; and its
determination by
means of what has
been given above.

By whom so con-
sidered, and deter-
mined in a parti-
cular case.

Here a more gen-
eral determination
from other prin-
ciples has been
given, in order to
show the fertility
of the theory.

THE END.

INDEX

PARS I

Pag. Num.

Introductio .............. i i

Expositio Theorise ............. 4 7

Occasio inveniendae, & ordo, ac analytica deductio invents Theoriae . . . .8 16

Lex continuitatis quid sit . . . . . . . . . . .13 32

Ejus probatio ab inductione ; vis inductionis ........ 16 39

Ejusdem probatio metaphysica ........... 22 48

Ejus applicatio ad excludendum immediatum contactum ...... 28 63

Deductio legis virium, & determinatio curvae earn exprimentis ..... 33 73

Primorum elementorum materiae indivisibilitas, & inextensio . . . . -37 81

Eorundem homogeneitas . . . . . . . . . . . -41 91

Objectiones contra vires in genere, & contra hanc virium legem . . . -45 100

Objectiones contra hanc constitutionem primorum elementorum materiae . . -59 I3l

PARS II

Applicatio Theories ad. Mechanicam

Argumentum hujus partis . . . . . . . . . . -77 166

Consideratio curvae virium . . . . . . . . . . . 77 167

De arcubus .............. 77 168

De areis ............... 79 172

De appulsibus ad axem, & recessibus in innnitum, ubi de limitibus virium . . . .82 179

De combinationibus punctorum, & primo quidem de systemate punctorum duorum . 86 189

De systemate punctorum trium . . . . . . . . . 92 204

De systemate punctorum quatuor . . . . . . . . . .no 238

De massis, & primo quidem de centro gravitatis, ubi etiam de viribus quotcunque generaliter

componendis . . . . . . . . .'••'. . . .111 240

De aequalitate actionis, ac reactionis . . . . . . . . . .124 265

De collisionibus corporum, & incursu in planum immobile . . . . . .125 266

Exclusio verae virium resolutionis ...... 132 279

De compositione, & imaginaria resolutione virium, ubi aliquid etiam de Viribus vivis . 136 289
De continuitate servata in variis motibus, ubi quaedam de collisionibus, de reflexionibus, &

refractionibus motuum .... • J39 297

De systemate trium massarum ..... . 143 3°7

Theoremata pertinentia ad directiones virium compositarum in singulis . . 143 308

Theoremata pertinentia ad ipsarum virium magnitudines . . . 145 313

Centrum asquilibrii, & vis in fulcrum inde. ..... . 148 321

Momenta pro machinis, & omnia vectium genera inde itidem . . .150 325

Centrum itidem oscillationis ..... • J52 32^

Centrum etiam percussionis ..... • JS7 344

Multa huic Theorize communia cum aliis hie tantummodo indicata . .158 347

De fluidorum pressione ...... • '59 34^

De velocitate fluidi erumpentis ..... • 162 354

PARS III

Applicatio Theories ad Physicam

Argumentum hujus partis .... • 164 358

Impenetrabilitas .... . 164 360

Extensio cujusmodi sit in hac Theoria, ubi de Geometria . . 169 371

Figurabilitas, ubi de mole, massa, densitate • I72 375

Mobilitas, & continuitas motuum . . • J75 3^3

jEqualitas actionis, & reactionis . . • 17% 3^8

Divisibilitas quae sit : componibilitas aequivalens divisibilitati in innnitum . 179 391

460

INDEX

PART I

Page Art.

Introduction .............. 35 I

Statement of the theory 37 7

What led to its discovery, the order, & the analytical deduction of the theory found . 45 16

What the Law of Continuity is 51 32

Proof of this law by induction ; the power of induction ...... 55 39

Metaphysical proof of the same thing ......... 63 48

Its application for the purpose of eliminating the idea of immediate contact . . 71 63

Deduction of the law of forces & the determination of the curve representing this law . 77 73

Indivisibility & non -extension of the primary elements of matter . . . 83 81

Their homogeneity . . . . . . . . . . . . 89 91

Objections against forces in general, & against this particular law of forces ... 95 100

Objections against this particular constitution of the primary elements of matter . in 131

PART II

Application of the Theory to Mechanics

The theme of the second part . . . . 135 166

Consideration of the curve of forces 135 167

The arcs 135 168

The areas . 139 172

Approach towards & recession to an infinite distance from the axis ; the limits of the forces 143 179

Combinations of points ; firstly, a system of two points 149 189

System of three points ............ 155 204

System of four points ............ 187 238

Masses ; firstly, the centre of gravity ;& the general composition of any number of forces . 189 240

Equality of action & reaction 203 265

Collision of solid bodies, & impact on a fixed plane 205 266

Exclusion of the idea of real resolution of forces .213 279

Composition, & hypothetical resolution, of forces ; remarks on " living forces." . . 223 289
Continuity observed in various motions ; also some remarks upon collisions, reflections &

refractions of motions 227 297

System of three masses 237 307

Theorems relating to the directions of the resultant forces on each of the systems . 237 308

Theorems relating to the magnitude of these forces ....... 241 313

The centre of equilibrium, & the force on the fulcrum derived from it ... 247 321

Theory of moments for machines, & hence all kinds of levers also . . . 249 325

The centre of oscillation . . . . . . . . . . . .251 328

The centre of percussion ............ 257 344

Many points of this Theory common to others, merely mentioned .... 259 347

Pressure of fluids .......... ... 259 348

Velocity of a fluid issuing from a vessel ...... ... 263 354

PART III

Application of the Theory to Physics

The theme of the third part 267 358

Impenetrability ............. 267 360

What kind of extension is admitted in this Theory ; geometry ..... 273 371

Figurability ; also volume, mass & density ......... 275 375

Mobility, & continuity of motions . . . . . . . . . 281 383

Equality of action & reaction 283 388

What divisibility is ; componibility equivalent to infinite divisibility .... 285 391

461

462 INDEX

Pag. Num.

Immutabihtas pnmorum matenae elementorum . . . . . . . .181 398

Gravitas 182 399

Cohssio . 185 406

Discrimen inter particulas. ........... 191 419

Soliditas, & fluiditas ............ 194 426

Virgx rigidae, flexiles, elasticae, fragiles ......... 199 436

Viscositas _ 200 438

Certas quorundam corporum figurae .......... 200 439

De fluidorum resistentia ............ 203 442

De elasticis, & mollibus . 204 446

Ductilitas, & Malleabilitas 205 448

Densitas indifferens ad omnes proprietates ........ 206 449

Vulgaria 4 elementa quid sint ........... 206 450

De operationibus chemicis singillatim ......... 207 451

De natura ignis ............. 215 467

De lumine, ubi de omnibus ejus proprietatibus, ac de Phosphoris .... 217 472

De sapore, & odore ............. 234 503

De sono . 235 504

De tactu, ubi de frigore, & calore .......... 237 507

De electricitate, ubi de analogia, & differentia materiae electricae, & ignese . . . 239 511

De Magnetismo ............. 242 514

Quid sit materia, forma, corruptio, alteratio 243 516

APPENDIX

Ad Metapbysicam pertinens

De Anima .............. 248 526

De DEO .. 254 539

SUPPLEMENTA

§ I De Spatio, & Tempore .......... 264 i

§ II De Spatio, ac Tempore, ut a nobis cognoscuntur 273 18

§ III Solutio analytica Problematis determinantis naturam legis virium . . . 277 25

§ IV Contra vires in minimis distantiis attractivas, & excrescentes in infinitum . . 289 77

§ V De Jiquilibrio binarum massarum connexarum invicem per bina alia puncta . 293 86

§ VI Epistola ad P. Scherffer 297 93

NOI RIFORMATORI

Dello Studio di Padova.

AVENDO veduto per la Fede di Revisione, ed Approvazione del P. F. Gio. Paolo Zapparella,
Inquisitor Generale del Santo Officio di Fenizia, del Libro intitolato Philosophic Naturalis Theoria

redacta ad unicam legem virium in natura existentium, Auctore P. Rogerio Josepho Boscovicb &c. non v'esser

cosa alcuna contro la Santa Fede Cattolica, e parimente per attestato del Segretario Nostro, niente contro
Principi, e buoni costumi concediamo licenza a Giambattista Remondini Stampator di Venezia, che possa
essere stampato, osservando gli ordini in materia di stampe, e presentando le solite Copie alk Publiche
Librerie di Venizia, e di Padova.

Dat. li 7. Settembre 1758.
(Gio. Emo, Procurator, Rif.
(Z. Alvise Mocenigo, Rif.

(
Registrato in Libro a carte 47. al num. 383.

Gio. Girolamo Zuccato, Segretario.
Adi 18 Settembre 1758.

Registrato nel Magistr. Eccellentiss. degli Esec. contro la Bestemmia.

Gio. Pietro Dolfin, Segretario.

INDEX 463

Page Art.

Immutability of the primary elements of matter 287 398

Gravity 287 399

Cohesion ............... 291 406

Distinction between particles 299 419

Solidity & fluidity 303 426

Rigid, flexible, elastic & fragile rods 309 436

Viscosity 311 438

Definite shapes of certain bodies . . . . . . . . . .311 439

Resistance of fluids 315 442

Elastic, & soft, bodies . . .317 446

Ductility & malleability 317 448

Density unrelated to all other properties . . . . . . . . .317 449

The so-called "four elements" . . . -3*9 45°

Chemical operations, each in turn . . . . . . . . . . 319 451

The nature of fire 329 467

Light, its properties, & light -giving bodies • 331 472

Taste and smell 353 503

Sound 355 504

Touch ; cold & heat 357 507

Electricity; the resemblances & differences between electric matter & fire . . . 361 511

Magnetism .............. 363 514

Matter, form, corruption, alteration . ........ . 365 516

APPENDIX

Relating to Metaphysics

The Soul .............. 373 526

GOD . 379 539

SUPPLEMENTS

I On Space & Time ............ 393 i

II On Space & Time, as we know them 405 18

III Analytical solution of the problem to determine the nature of the law of forces . 411 25

IV Arguments against forces that are attracti ve at very small distances, & increase indefinitely 427 77

V Equilibrium of two masses connected together by two other points . . . 437 86

VI A letter from the Author to Father Scherffer 443 93

WE, as Censors of the College of Padua, having seen, through trust in the revision & approval of
Father F. Gio. Paolo Zapparella, Inquisitor General of the Holy Office in Venice, that there
is nothing in the book, entitled Philosophic Naturalis Iheoria redacta ad unicam legem virium in natura
existentium, by P. Rogerius Josephus Boscovich, that is contrary to the Holy Catholic Faith ; & also,
on the testimony of our Secretary, that there is nothing contrary to our Rules, according to good usance,
give leave to Giambattista Remondinus, printer in Venice, to print the book ; provided that he observe
the regulations governing the press, & present the usual copies to the Public Libraries of Venice & Padua.
Given this 7th of September, 1758.

GIG. EMO, Procurator, Censor.
Z. ALVISE MOCENIGO, Censor.

??*
Registered in Book, p. 47, no. 383.

September iSth, 1758. Gio. Girolamo Zuccato, Secretary.

Registered in the High Court for the Prevention of Blasphemy.

Gio. Pietro Dolfin, Secretary.
* There is here a space for another name that was not filled in.

CATALOGUS OPERUM
P. ROGERII JOSEPHI BOSCOV1CH, S.J.

imfressorum usque ad initium anni 1763.

Annus prima
Opera, fcf opuscula justa molis. edition,

Sopra il Turbine, che la notte tra gli 1 1, e 12 Giugno del 1749 danneggio una gran parte di Roma. Dis- 1749
sertazione del P. Ruggiero Giuseppe Boscovich della Comp. di Gesii. In Roma appresso Nicolo,
e Marco Pagliarini, in 8.

Elementorum Matheseos tomi tres, in 4. Prodierunt anno 1752 sub titulo, Elementorum Matheseos ad 1752
usum studiosae juventutis, tomi primi pars prima complectens Geometriam planam, Arithmeticam
vulgarem, Geometriam Solidorum, & Trigonometriam cum planam, turn sphaericam. Pars altera,
in qua Algebrae finite elementa traduntur. Romae : excudebat Generosus Salomoni. Us binis tomis
sine nova eorum impressionf mutatus est titulus anno 1754 in hunc, Elementorum Universe Matheseos
Auctore P. Rogerio Josepho Boscovich Soc. Jesu Publico Matheseos Professore Tomus I continens
&c. Tomus II continens &c, y adjectus est sequens.

Tomus III continens Sectionum Conicarum Elementa nova quadam methodo concinnata, & Disserta- 1754
tionem de Transformatione locorum Geometricorum, ubi de Continuitatis lege, ac de quibusdam
Infiniti mysteriis : Typis iisdem ejusdem Generosi Salomoni omnes in 8. Extat eorundem impressio
Veneta anni 1758, sed typorum mendis deformatissima.

De Litteraria Expeditione per Pontificiam ditionem ad dimetiendos duos Meridian! gradus, & corrigendam 1755
mappam geographicam, jussu, & auspiciis Benedicti XIV. P.M. suscepta Patribus Soc. Jesu Chris-
tophoro Maire, & Rogerio Josepho Boscovich, Romae 1755. In Typographic Palladis : excudebant
Nicolaus, & Marcus Palearini, in 4. Quidquid eo volumine continetur, est Patris Boscovich prater bina
brevia opuscula Patris Maire, qua ipse P. Boscovich inseruit. Prostat etiam Mappa Geographica
ditionis Pontificia delineata P. Maire ex observationibus utrique communibus.

De Inasqualitatibus, quas Saturnus, & Jupiter sibi mutuo videntur inducere, praesertim circa tempus 1756
conjunctionis. Opusculum ad Parisiensem Academiam transmissum, & nunc primum editum.
Auctore P. Rogerio Josepho Boscovich Soc. Jesu ; Romae ; ex Typographia Generosi Salomoni, in 8.

Philosophic Naturalis Theoria redacta ad unicam legem virium in Natura existentium Auctore P. Rogerio I7*fi
Jos. Boscovich S.J. publico Matheseos Professore in Collegio Romano. Prostat Vienna Austria in
Officina libraria Kalivvodiana : in 4. In fine accedit Epistola ad P. Carolum Scherffer Soc. Jesu.
Habetur secunda editio Viennensis paullo posterior : tertia hie exhibetur : Epistola babetur in ejus
Supplements.

Adnotationes in aliorum Opera.

Caroli Noceti e Societate Jesu de Iride, & Aurora Boreali Carmina . . . cum notis Josephi Rogerii Bosco- X747
vich ex eadem Societate. Romae : excudebant Nicolaus, & Marcus Palearini, in 4. Perperam
nomen Josephi antepositum est ibi nomini Rogerii.

Philosophise Recentioris a Benedicto Stay in Romano Archigymnasio Publico Eloquentiae Professore . . . 1755
cum adnotationibus, & Supplementis P. Rogerii Joseph! Boscovich S.J. in Collegio Rom. Publici
Matheseos Professoris. Tomus I. Romae : Typis, & sumptibus Nicolai, & Marci Palearini, in 8.
Duce ejus editiones prodierunt simul.

Tomus II Romae : Typis, & sumptibus Nicolai, & Marci Palearini, in 8. In singulis hisce voluminibus J7°°
ea, qua ad P. Boscovich pertinent, efficerent per se ipsa justum volumen. In solis primi Stayani tomi
supplements occurrunt 39 ipsius Dissertationes de variis argumentis pertinentibus potissimum ad Meta-
physicam {3 Mechanicam.

Dissertationes impresses pro exercitationibus annuis, y publics propugnattz : omnes

in 4.

De Maculis Solaribus, Exercitatio Astronomica habita in Collegio Romano Soc. Jesu. Romae : ex Typo- J73^

graphia Komarek.
De Mercurii novissimo infra Solem transitu. Dissertatio habita in Seminario Romano. Romse, Typis J737

Antonii de Rubeis.
Constructio Geometrica Trigonometric sphaericae. Romae, ex Typographia Komarek. Hujus titulus vel

est hie ipse, vel parum ab hoc differt.
De Aurora Boreali Dissertatio habita in Seminario Romano. Romas : Typis Antonii de Rubeis. Eadem *738

eodem anno edita fuit etiam typis Komarek.
De Novo Telescopii usu ad objecta caelestia determinanda. Dissertatio habenda a PP. Soc. Jesu in Collegio J739

Romano. Romae, ex Typographia Komarek. Extat recusa sine ulla mutatione in Actis Lipsiensibus

ad annum 1740.

HH

Annus primee

edition. £)e Veterum argumentis pro Telluris sphaericitate. Dissertatio habita in Seminario Romano Soc. Jesu.

Romae : Typis Antonii de Rubeis.

Dissertatio de Telluris Figura habita in Seminario Romano Soc. Jesu. Romae : Typis Antonii de Rubeis.
Eadem prodiit in 8, anno 1744 in opere, cui titulus Memorie &c. In Lucca per li Salani, e Giuntini,
y in titulo additur : nunc primum aucta, & illustrata ab ipsomet Auctore ; "sed ea editio scatet typorum
erroribus, ut y reliqua inferius nominanda in eadem collectione inserta.

1740 De Circulis Osculatoribus. Dissertatio habenda a PP. Societatis Jesu in Collegio Romano. Romae :

ex Typographia Komarek.

De Motu corporum projectorum in spatio non resistente. Dissertatio habita in Seminario Romano Soc.
Jesu. Romae : Typis Antonii de Rubeis.

1741 De Natura, & usu infinitorum, & infinite parvorum. Dissertatio habita in Collegio Romano Soc. Jesu.

Romae : ex Typographia Komarek.

De Inaequalitate gravitatis in diversis Terrae locis. Dissertatio habita in Seminario Romano Soc. Jesu.
Romae : Typis Antonii de Rubeis.

1742 De Annuis Fixarum aberrationibus. Dissertatio habita in Collegio Romano Societatis Jesu. Romae :

ex Typographia Komarek.
De Observationibus Astronomicis, & quo pertingat earundem certitudo. Dissertatio habita in Seminario

Romano Soc. Jesu. Romae : Typis Antonii de Rubeis.
Disquisitio in Universam Astronomiam publicae Disputation! proposita in Collegio Romano Soc. Jesu.

Romae : ex Typographia Komarek.

1743 De Motu Corporis attracti in centrum immobile viribus decrescentibus in ratione distantiarum reciproca

duplicata in spatiis non resistentibus. Dissertatio habita in Collegio Romano. Romae : Typis
Komarek. Eadem prodiit anno 1747 sine ulla mutatione in Commentariis A cad. Bononiensis Tom. II.
•par. III.

1744 Nova methodus adhibendi phasium observationes in Eclipsibus Lunaribus ad exercendam Geometriam,

& promovendam Astronomiam. Dissertatio habita in Collegio Romano. Romae : ex Typographia
Komarek. Eadem prodiit in 8, anno 1747 cum exigua mutatione, vel additamento in Opere superius
memorato, cui titulus Memorie &c. In Lucca per li Salani, e Giuntini.

X745 De Viribus Vivis. Dissertatio habita in Collegio Romano Soc. Jesu. Romae : Typis Komarek. Eadem
prodiit anno 1747 sine ulla mutatione in Commentariis A cad. Bonon. To. II. par. Ill, y in Germania
pluribus vicibus est recusa.

174" De Cometis. Dissertatio habita a PP. Soc. Jesu in Collegio Rom. Romae : ex Typographia Komarek.

J747 De .<Estu Maris. Dissertatio habita a PP. Soc. Jesu in Collegio Romano. Romas : ex Typographia
Komarek. Ea est Dissertationis pars I ; secunda pars nunquam prodiit. Quee pro ilia fuerant des-
tinata, habentur in Opere De Expeditione Litteraria, y in supplements Philosophies Stayanes tomo II.

1748 Dissertationis de Lumine pars prima publice propugnata in Seminario Romano Soc. Jesu. Romae : Typis

Antonii de Rubeis.

Dissertationis de Lumine pars secunda publice propugnata a PP. Soc. Jesu in Collegio Romano. Romae :
ex Typographia Komarek.

1749 De Determinanda Orbita Planetae ope Catoptricae, ex datis vi, celeritate, & directione motus in dato

puncto. Exercitatio habita a PP. Soc. Jesu in Collegio Romano. Romae : ex Typographia Komarek.

'7S1 De Centre Gravitatis. Dissertatio habita in Collegio Romano Soc. Jesu. Romae: ex Typographia
Komarek. Eadem paullo post prodiit iterum cum sequenti titulo, y additamento. De Centre Gravitatis.
Dissertatio publice propugnata in Collegio Romano Soc. Jesu Auctore P. Rogerio Josepho Boscovich
Societatis ejusdem. Editio altera. Accedit Disquisitio in centrum Magnitudinis, qua quasdam in
ea Dissertatione proposita, atque alia iis affinia demonstrantur. Romae, Typis, & sumptibus Nicolai,
& Marci Palearini.

I7S3 De Lunae Atmosphaera. Dissertatio habita a PP. Soc. Jesu in Collegio Romano. Romae: ex Typographia
Generosi Salomoni. Multa eorundem typorum exemplaria prodierunt paullo post cum nomine Auctoris
in ipso titulo, y cum exigua unius loci mutatione.

X754 De Continuitatis Lege, & Consectariis pertinentibus ad prima materiae elementa, eorumque vires. Disser-
tatio habita a PP. Societatis Jesu in Collegio Romano. Romae : ex Typographia Generosi Salomoni.

J7SS De Lege virium in Natura existentium. Dissertatio habita a PP. Soc. Jesu in Collegio Romano. Romae :

Typis Generosi Salomoni.

De Lentibus, & Telescopiis dioptricis. Dissertatio habita in Seminario Romano. Romae : ex Typographia
Antonii de Rubeis.

Plures ex hisce Dissertationibus prodierunt etiam iisdem typis, sed cum alia titulo, habente non locum, ubi
sunt habitee, vel propugnatee, sed tantummodo nomen Auctoris. In hac postrema mutates sunt bines pagince,
posteaquam plurima exemplaria fuerant distracta. In prioribus tribus sunt pauca qucedam mutata, vel
addita a P. Horatio Burgundio adhuc Professore Matheseos in Collegio Romano, qui fuerat ejus Prce-
ceptor ; sed eo jam ad Dissertationes ejusmodi conscribendas utebatur.

Eee omnes, quce -pertinent ad. Seminarium Romanum, habent in ipso titulo adscripta nomina Nobilium Con-
victorum, qui illas propugnarunt, y sub eorum nomine referuntur plures ex iis in Actis Lipsiensibus.

Multa pertinentia ad ipsum P. Boscovich babentur in binis Dissertationibus, quarum tituli, Synopsis Physicae
Generalis, & De Lumine, quarum utraque est edita Romae anno 1754, Typis Antonii de Rubeis, in 4.
Id ibidem testatur earundem Auctor (is est P. Carolus Benvenutus Soc. ejusdem) affirmans, ea sibi ab
eodem P. Boscovich fuisse communicata.

Habetur etiam ampliatio solutionis cujusdam problematis pertinentis ad Auroram Borealem, soluti in adnota-
tionibus ad Carmen P. Noceti, inserta in quadam Dissertatione impressa Romcs circa annum 1756, y

Annus prinue

publice propugnata, cujus Auctor est P. Lunardi Soc . Jesu, qui affirmat ibidem, se ea.nd.em acceptam ab edition,
ipso P. Boscovich proponere ejusdem verbis.

Subjiciemus jam bina opuscula Italica, quee communi nomine PP.»m Le Seur, Jacquier, ac suo conscripsit
ipse P. Boscovich. Utrumque est sine loco impressionis, & nomine Typography ; impresserunt autem
Palearini Fratres Romce jussu Prasulis, qui turn curabat Fabricam 5. Petri, a quo & publice distributa
sunt per Urbem.

Parere di tre Matematici, sopra i danni, che si sono trovati nella Cupola di S. Pietro sul fine del 1742, dato 1742
per ordine di Nostro Signore Benedetto XIV, in 4. In fine opusculi habentur subscripta omnium trio,
nomina.

Riflessioni de' PP. Tomaso Le Seur, Francesco Jacquier dell' Ordine de' Minimi, e Ruggerio Giuseppe 1743
Boscovich della Comp. di Gesu sopra alcune difficolta spettanti i danni, e rifarcimenti della Cupola
di S. Pietro proposte nella Congregazione tenutasi nel Quirinale a' 20 Gennaro 1743, e sopra alcune
nuove Ispezioni fatte dopo la medesima Congregazione.

Habentur itidem Italico sermone bina ex Us, quas Itali vacant Scritture, -pro quadam lite Ecclesia S. Agnetis 1757
Romance, pertinentes ad aquarum cursum Romce editce anno 1757.

Inserta,

Nunc faciemus gradum ad inserta in Publicis Academiarum monumentis, in diariis, in collectionibus, y in
privatorum Auctorum Operibus.

In Monumentis Acad. Bononiensis.

Prater reimpressionem binarum Dissertationum in To. II, de quibus supra, habetur in To. IV De Litteraria 1 757
Expeditione per Pontificiam ditionem. Est Synopsis amplioris Operis, ac habentur plura ejus exemplaria
etiam seorsum impressa.

In Romano Litteratorum diario vulgo Giornale de' Letterati
appresso i Fratelli Pagliarini.

D'Un' antica villa scoperta sul dosso del Tuscolo : d'un antico Orologio a Sole, e di alcune altre rarita, 1746
che si sono tra le rovine della medesima ritrovate. Luogo di Vitruvio illustrate. Ibi ejus schedias-
matis Auctor profert, uti ipse profitetur, quee singillatim audierat ab ipso P. Boscovich.

Dimostrazione facile di una principale proprieta delle Sezioni Coniche, la quale non dipende da altri
Teoremi conici, e disegno di un nuovo metodo di trattare questa dottrina.

Dissertazione della Tenuita della Luca Solare, Del P. RuggieroGi us. Boscovich Matematico del Collegio 1747
Romano.

Dimostrazione di un passo spettante all' angolo massimo, e minimo dell' Iride, cavato dalla prop, ix
par. 2 del libro i dell' Ottica del Newton con altre riflessioni su quel capitolo. Del P. Ruggiero
Gius. Boscovich dell Comp. di Gesu.

Metodo di alzare un Infinitinomio a qualunque potenza. Del P. Ruggiero Gius. Boscovich.

Parte prima delle Riflessioni sul metodo di alzare un Infinitinomio a qualunque potenza. Del P. Ruggiero 1748
Gius. Boscovich della Comp. di Gesu.

Parte seconda &c.

Soluzione Geometrica di un Problema spettante 1'ora delle alte, e basse maree, e suo confronto con una
soluzione algebraica del medesimo data dal Sig. Daniele Bernoulli. Del P. Ruggiero Giuseppe
Boscovich della Compagnia di Gesu.

Dialogi Pastorali V sull' Aurora Boreale del P. Ruggiero Gius. Boscovich della Comp. di Gesu.

Dimostrazione di un metodo dato dall' Eulero per dividere una frazione razionale in piu frazioni piu semplici X749
con delle altre riflessioni sulla stessa materia.

Lettera del P. Ruggiero Gius. Boscovich della Comp. di Gesu al Sig. Ab. Angelo Bandini in risposta '75°
alia lettera del Sig. Ernesto Freeman sopra L'Obelisco d'Augusto. Nomen Freeman est fictitium,
Auctorem denotans Neapoli latentem, y aliis Operibus satis notum. Extat eadem etiam in folio.

Altera de eodem Obelisco ad.mod.um prolixa Epistola, Italice, W Latine scripta ad eundem Bandinium suo
nomine ab ipso P. Boscovich habetur in ejusdem Bandinii Opere, cui titulus, De Obelisco Caesaris Augusti
e Campi Martii ruderibus nuper eruto. Commentarius Auctore Angelo Maria Bandinio. Romas
apud Fratres Palearinos, in folio. Ibidem in fine habetur alia epistola itidem admodum prolixa de eodem
argumento nomine Stuarti, e cujus schedis relictis apud Cardinalem Falentium in ejus discessu ab Urbe
earn Epistolam conscripsit, ac ejus comperta illustravit, ac auxit ipse P. Boscovich.

Osservazioni dell' ultimo passaggio di Mercurio sotto il Sole seguito a' 6. di Maggio 1753, fatte in Roma, 1753
e raccolte dal P. Ruggiero Gius. Boscovich della Comp. di Gesu con alcune reflessioni sulle medesime.

In aliis monumentis.

In Collectione Opusculorum Lucensi cui titulus : Memorie sopra la Fisica, e Istoria naturale di diversi Valen-
tuomini. In Lucca per li Salani, e Giuntini, in 8, Prater binas dissertationes, de quibus supra, habetur.

Problema Mechanicum de solido maxima attractions solutum a P. Rogerio Josepho Boscovich Soc. Jesu
Publico Professore Matheseos in Collegio Romano : Tomo I.

De Materias divisibilitate, & Principiis corporum. Dissertatio conscripta jam ab anno 1748, & nunc J757
primum edita. Auctore P. Rogerio Jos. Boscovich Soc. Jesu, To. IV.

Omnium horum quatuor Opusculorum habentur etiam exemplaria seorsum impressa.

Annus primes

edition. jn eJitione Elementorum Geometries Patris Tacqueti facta Roma sumptibus Venantii Monaldini, Typis
1745 Hieronymi Mainardi, in 8. habetur Trigonometria sphaerica P. Rogerii Josephi Boscovich, qua: deinde

adhuc magis expolita prodiit Tomo I. ejus Elementorum Matheseos. Habetur praterea ibidem Trac-
tatus De Cycloide, & Logistica, qui etiam seorsum impressus est iisdem typis.

J7S2 In Opere Comitis Zoannis Baptists Soardi, cui titulus Nuovi instrument! &c. in Brescia dalle stampe di
Gio. Battista Rizzardi, in 4., habentur bina epistola Italica ipsius P. Boscovich de Curvis quibusdam,
cum figuris, & demonstrationibus.
!758 In Optica Abbatis De la Caille latine reddita a P. Carolo Scherffer Soc. Jesu, y impressa Vienna in Austria

habetur schediasma Patris Boscovich de Micrometre objective.

In postremo tomo Commentar. Academiae Parisiensis in Historia, & in uno e tomis Correspondent tium
ejusdem Academics, creditur esse breve aliquid pertinens ad ipsum P. Boscovich. Est aliquid etiam in
diario Gallico Journal des javans, y fortasse in Anglicanis Transactionibus, atque alibi insertum hisce
itinerum annis.

Poetica.

J7S3 P. Rogerii Josephi Boscovich Soc. Jesu inter Arcades Numenii Anigrei Ecloga recitata in publico Arcadum
consessu primo Ludorum Olympicorum die, quo die Michael Joseph Morejus Generalis Arcadias
Gustos illustrium Poetarum Arcadum effigies formandas jaculorum ludi substitnerat. Romae in 8.
Extat eadem iisdem Typis etiam in Collectione turn impressa omnium, qua ea occasione sunt recitata.
Stanislai Poloniae Regis, Lotharingiae, ac Barri Ducis, & inter Arcades Euthimii Aliphiraei, dum ejus effigies
in publico Arcadum Ccetu erigeretur, Apotheosis. Auctore P. Rogerio Josepho Boscovich Soc.
Jesu inter Arcades Numenio Anigreo. Romas ex Typographia Generosi Salomon!, in 8. Est
poema versu heroico. Idem autem recusum fuit Nancei cum versione Gallic a Domini Cogolin,

1757 Pro Benedicto XIV. P.M. Soteria. Est itidem poema Heroicum ejusdem P. Boscovich pertinens vel ad hunc,

vel ad superiorem annum : est autem impressum Romae in 4, apud Fratres Palearinos, occasione periculi
mortis imminentis, evitati a Pontif.ce convalescente.

1758 In Nuptiis Joannis Corrarii, & Andrianae Pisauriae e nobilissimis Venetae Reip. Senatoriis familiis. Carmen

P. -Rogerii Jos. Boscovich S.J. Public! in Romano Collegio Matheseos Professoris. Romae : ex

Typographic Palladis : excudebant Nicolaus, & Marcus Palearini, in 4.
1760 De Solis, ac Lunse defectibus libri V P. Rogerii Josephi Boscovich Societatis Jesu ad Regiam Societatem

Londinensem, Londini 1760. in 4. Nan habetur nomen Typographi, qui impressit, sed Bibliopolarum

quorum sumptibus est impressum : deest hie ejus editionis exemplar, ex quo ea nomina corrects describantur.

Idem recusum fuit anno 1761 Venetiis apud Zattam in 8°. cum exiguo additamento in fine, y cum hoc

catalogo, quern inde hue derivavimus. Habentur in adnotationibus bina Epigrammata cum versionibus

Italicis, sive Sonetti.
Est & aliud ejus poema Heroicum anno 1756 impressum Vienna in Austria in collectione carminum facta

occasione inaugurationis novarum Academic Viennensis cedium.
Sunt y epigrammata nonnulla in Collec tionibus Arcadum, inter qua unum pro recuperata valetudine Johannis

V Lusitania Regis, tf unum pro Rege turn utriusque Sicilia, y nunc Hispania, ac pro Regina ejus

conjuge.
Extant etiam pauca admodum exemplaria unius ex illis, quas in Italia appellamus Cantatine, impressa Viterbii

anno 1750 pro Visitatione B. Maria Virginis, in qua sex, quas dicimus Ariette, prof ana; ad sacrum

argumentum transferenda erant, manente Musica, y inter se connectenda.

ERRATA

p. 2, 1. II, for ac omnem read ad omnem

p. 3, 1. 5, for has been read should be

p. 4, 1. 18, for Venetisis read Venetiis

p. 6, 1. 9 from bottom, for exceres read cxerces

1. 4 from bottom, for eocatum read evocatum
p. 7, 1. 18 from bottom, after despatched add to the
Court of Spain

1. 13 from bottom, for befits read befit
p. 8, 1. i, for publico read publice

1. 13, for utique read ubique

1. 28, for infiliciter read infeliciter
p. 10, 1. 8, for opportunam read opportunum

1. 9, for mediocrum read mediocrium
p. 12, 1. 13, for aliquando read aliquanto

1. 10 from bottom, for repulsivis read repulsivas
p. 14,1. 13, for adhibitis read adhibitas

1. 24, for postremo read postrema
p. 1 8, 1. 2, for alter read altera
p. 22, 1. 15, after vero etiam insert leges
p. 28, 1. 17, for acquiretur read acquireretur

1. 28, for -menae read -mena
p. 40, 1. 22, for Naturam read Natura

1. 23, for quandem read quandam

1. 29, for recidit read recedit

1. 32, for postquam read post quam
p. 47, 1. 34, for many lead most
p. 48, 1. 1 8, for lins read lineae

1. 29, for genere read generis
p. 50, 1. 26, for deferendam read deserendam

1. 31, for viderimus read videremus

1. 46, for nominandi read nominando
p. 52, 11. 5, 6 of marginal note to § 7, for nihilmu read

nihilum.
p. 54, 1. i, for exhibit read exhibet

1. 3, for opposite read opposita

1. 12, for sit read fit
p. 55, 1. 4, after & add then
p. 56, 1. 3, for servat read servant
p. 58, 1. 32, for crederit read crederet
p. 60, 1. 3, marg. note, Art. 46, for sit read fit
p. 64, 1. 2, for terio read tertio
p. 65, 1. 57, for fact read by the fact
p. 66, 1. 9, for concipiantur read concipiatur

1. 16, for ordinate read ordinatae
p. 67, 1. 48, for before & read previously
p. 68, 1. II, for in GM' read in GM
p. 71, 1. 46, for and this read and that this is found

nowhere
p. 72, 1. i, for ejusmodi read hujusmodi

1. 4 from bottom, for potissimuim read

potissimum

p. 74, 1. 38, for illo read ilia

p. 76, 1. 3 from bottom, for devenirent read devenerint
p. 81, 1. 42, for is read ought to be
p. 82, 1. 5, marg. note, Art. 82, for se read sed
p. 86, Art. 89, in marg. note, for densitatis read densitas
p. 88, 1. II, for adi read ad

1. 1 6, for reliquent read relinquent
p. 90, 1. 30, for diversimodo read diversimode

1. 34, for distantia read distantiae

Art. 95, marg. note, for de- read dif-
p. 92, 1. 33, for apparent read apparerent
p. 94, 1. 22, for incurrant read incurrunt
p. 95, Art. 103, 1. I, for are read is ; and in marg. note

insert in between and and what
p. 96, 1. 8, for potissimo read potissimum

1. 16, for praecedentum read prascedentem

marg. note, Art. 105, for transire read transiri
p. 97, 1. 9 from bottom, for quite enough read better

p. 99, 1. 40, insert a comma after locus

p. loo, marg. note, Art. 112, for recte read rectae

1. 32, for ellipis read ellipsis
p. 106, marg. note, Art. 125, for perfectionum read

perfectiorum
p. 107, 1. 23, for off they are read away they go

1. 13 from bottom, for have read has
p. 109,' 1. 27, after tantummodo add admitto
p. no, 1. 17, for expandantur read expendantur

1. 24, for a read &

bottom line, for distinctis read distinctas
p. 112, 1. 27, for veteram read veterem
p. 113, 1. 5 from bottom, for because read that

1. 4 from bottom, after change add is excluded by
p. 115, 1. 33, for and read et

marg. note, Art. 139, add at end impugned
p. 1 1 8, 1. 7 from bottom for ali read alia
p. 122, 1. 26, for justmodi read ejusmodi
p. 125, 1. 29, for ignored read urged in reply
p. 128, 1. 31, for ea read eas
p. 129, 1. 16, for Principii read Principiis
p. 139, 1. 8, for arm E read arm ED

footnote, 1. 5, for DP read OP
p. 140, 1. 34, insert cum before directione

1. 4, footnote, for ut in read ut n
p. 148, 1. 10, for Expositas read Expositis, for curva read

curvam
p. 156, 1. I, for a que read atque

1. 7, for caculo read calculo

1. 39, for Tam read Turn

p. 158, Art. 209, marg. note, add at endLegum multitudo
& varietas

footnote, 1. 1 1 from bottom of page, for obvenerit

read obveniret
p. 160, footnote, 1. I, for sit read fit

1. 12, for ed read sed

p. 161, footnote, 1. 20, after segment add DR
p. 162, 1. 7 from bottom, for reflexionis read reflex-

iones

p. 167, 1. 40, for ae read da

p. 168, 1. 8, from bottom for zjC—'AC read 27 C'AC
p. 171, 1. 4 from bottom for GL, or LI read GI, or IL
p. 172, 1. 34, for compositas read compositis
p. 175, 1. 13, for 30 read 27

1. 39, insert a comma after approximately
p. 176, 1. 7, for delatam read delatum
p. 178, marg. note, Art. 230, 1. 5, for foco read focos
p. 188, 1. 31, for summa read summas
p. 195, 1. 19, insert a comma after point P
p. 197, 1. 35, for sum of the (at end of line) read sums of the
p. 198, Fig. 40, insert F where AE cuts CD
p. 199, 1. 35, for ceases read cease

1. 37, after all, & insert I assume
p. 202, 1. 6, for summa read summam

Art. 264, 1. 3, for quacunque read quancunque and
in marg. note for corallarium read corol-
larium

p. 205, 1. 21, for -recessions read recession
p. 206, 1. 3, for globis read globus

1. 2 from bottom, insert motu before quodam

p. 208, last line, for

m -)- n

read

m-(-n

in each case

n m

p. 209, 1. 11, for (2CQ — 2Cq) read (2CQ — 2cQ)
p. 210. 1. 8, for quiescat read quiescit
p. 211, 1. 25, the denominator (Q + q) should be (Q + q)2
p. 215, 1. 5, for BP read BO
p. 223, 1. 26, for 50,61,62 read 50,51,52
p. 227, 1. 21 from bottom, for to read of
p. 228, 1. 5, from bottom, for Angulum read Angulus

ERRATA

p. 233, 1. 8, for in volute read evolute

bottom line, after vary insert inversely

p. 241, marg. note, Art. 313, add at end This is very soon
proved

p. 242, last line, for denominator AD read BD

p. 247, 1. 15 from bottom, for A & B read B & C

p. 248, 1. 24, for conversione read conversionem

p. 250, 1. 39, for justa read jurta

p. 252, 1. 5 from bottom, for gravitas read gravitatis

p. 256, 1. lo, for quassitum read quaesitam

p. 270, marg. note, Art. 366, 1. 5, for magnas read magna

p. 278, 1. 7, for varior read rarior

p. 280, 1. 12 from bottom, for tranctanda read tractanda

p. 284, 1. 15, for sit read fit

p. 286, 1. 20, for multuplicetur read multiplicetur

p. 288, 1. 21, for Solum read Solem

p. 292, 1. i, for quietam read quietem
1. 3, for sit, read fit

1. 25, for ullo, read ulla deleting the comma
1. 26, for ilia read ille

p. 304, 1. 12 from bottom, for propre read prope

p. 310, 1. 15 from bottom, for hsebebunt read hzrebunt

p. 314, 1. 2, insert & before mutandam

p. 319, 1. 1 8 from bottom, for some repel read and repel

p. 324, 1. 22, for pertinet read pertinent

p. 329, 1. 25, for ethers read others

p. 332, 1. 16, for aequabilis read aequalibus

1. 21, for Benvenuti read Benvenutus
1. 3 from bottom, for qui a read quia

P- 336, 1. 35) insert utcunque afar circunquaque

p. 346, 1. 19, for sit read fit

p. 348, 1. 28, for irregularitur read irregulariter
p. 350, 1. 13 from bottom, for flexo read flexu
P- 355) !• H> /<"" with read to
p. 356, 1. 9, after porro aliud insert post aliud

1. 19, after accidit insert idem accidit
p. 366, 1. 32, for ordores read odores
p. 394, 1. 31, for imaginarae read imaginariae
p. 396, 1. 19 from bottom, after solum add etiam

1. 14 from bottom, for sunt read sint
p. 398, 1. 20, after est insert tota

1. 35, for esses read esse
p. 400, 1. 33, after omnino insert saltern
p. 406, 1. 6, for congruant read congruunt
p. 410, 1. 8, for bzma read bz™'3

Art 27, marg. note, for quaesitum read quaesitam
p. 412, 1. II, insert positives before assumantur
p. 422, 1. 22, for ab read ad
p. 434, Art 86, marg. note, for massa read massas

1. 5 from bottom, for contrarium read contrariam
p. 444, 1. n, insert & before centri
p. 448, 1. 17, for PQ read PG

Art. 107, marg. note, for absoluutae read absolutas

1. 4 from bottom, after posita insert J
p. 454, 1. 2 from bottom, for elasticas read elasticitas

>J.J rp2

p. 456, 1. 5 from bottom for — read—

p. 458,1. n, for ^ read

zz zz

p. 459, 1. 13, for — tdz/.z2 read — t2dz/z2
p. 464, 1. 4, for Discrimen read Discrimina

1. 10 from bottom, for Venizia read Venezia

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