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T H E O R. I. A
PHILOSOPHIAE NATURALIS
R E D A CT A A D UNICA M L E G E M W I R I UM
IN NATURA EX IST ENT I UM,
A v C 7" O R E
P. ROGERIO JOSEPHO BOSCOVICH
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NUNC AB I PSO PER POLITA, ET A U CTA,
Ac a plurimis pracedentium
mendis expurgata.
E D I T I O W, E N E T A P R I M A
IPSO vºſ UCTORE PRAES ENTE, ET CORRIG ENT E.
V E N E T I I S,
M D C C L X I I I.
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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

PRINTED IN GREAT BRITAIN
BY
BUTLER & TANNER, FROME, ENGLAND
Copyright
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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 Petronievič, 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
Petroniević 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 Theoria 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,
Boško, 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, Boško,
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 Boškovic', in Serbo-Croatian) was born at Ragusa
on September 18th, 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 Lunae 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 Philosophiae 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-
—re
* 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 Litteraria Expeditione per Pontificiam, &c.; this
was translated into French under the title of Voyage astronomique et géographique dans
l'État de l'Église, 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 Lunae 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
Poniatowski. 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
I773. 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 1785, 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 13th, 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, déjeúner.
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 Mémoirs de l'Académie 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 Stavans, Février, 1792, pp. 113-118.
INTRODUCTION
LTHOUGH 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 Virium 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
Yl

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 Viribus 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 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. I64, 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 A C E
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 B D F
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.
* 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 motrix (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 materiae puncta determinari aeque 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 motrix),
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 =ſid: F Jºd. =ſo.d.). See Note () to Art. 118, where it is
evident that the word wires, translated “forces,” strictly means “accelerations;” see also 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 f 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 f. Hence, the body A will give to each point of B, and
therefore to the whole of B, an acceleration equal to af. 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
s 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 " w”y= c, in
which & 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 xy
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
Wirium 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 forée 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 = 0, 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
XVI &
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 wires viva.
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 = 0, referred to above. Note.—In this connection
(p. 4Io, 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 () on p. 141. He assumes that
xviii INTRODUCTION
the equation of the curve is a "y" = 1, and obtains the integral -º- wy + A, where A is the
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 m3m. He says: “The area is infinite, when n = m, because
this makes the divisor zero; and thus the area becomes still more infinite if nºm.” Put
into symbols, the argument is : Since m-m-30,
"->"> Co. The historically interesting
71–771 O -
point about this is that it represents the persistance of an error originally made by Wallis
in his Arithmetica Infinitorum (it was Wallis who invented the sign oc to stand for “simple
infinity,” the value of 1/0, and hence of m/Q). 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
0. f; tº .- -
using the formula; tºy +A, instead of *-a” + A, for the area. From the latter
* 77—777 -
it is easily seen that since the initial area is zero, we must have A =#2”. If n is
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 1758, 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
p
p
:
27, l. 8, for in one plane read in the same direction
47, 1, 62, literally on which . . . is exerted
49, 1.33, for just as . . . is read so that . . . may be
53, 1. 9, after a line add but not parts of the line itself -
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; r
67, l. 5, for it is allowable for me read I am disposed; unless in the original libet is taken to be a misprint for licet
73, 1. 26, after nothing add in the strict meaning of the term
85, 1. 27, after conjunction add of the same point of space
. 91, l. 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
. II5, 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
. I 17, 1.41, for precisely read abstractly
. I25, l. 29, for ignored read urged in reply
. I26, 1.6 from bottom, it is possible that acquirere is intended for acquiescere, with a corresponding change in the
translation
. I29, Art. I62, marg. note, for on what they may be founded read in what it consists.
. I67, Art. 214, l. 2 of marg. note, transpoſe by and on
footnote, l. I, for be at read bisect it at
. I99, l. 24, for so that read just as
. 233, l. 4 from bottom, for base to the angle read base to the sine of the angle
last line, after vary insert inversely
. 307, l. 5 from end, for motion, as (with fluids) takes place read motion from taking place
. 323, 1.39, for the agitation will read the fluidity will
. 345, l. 32, for described read destroyed
. 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
PHILOSOPHIAE NATURALIS
TYPOGRAPHUS
VENETUS
LECTORI
§®§PUS, quod tibi offero, jam ab annis quinque Viennæ editum, quo plausu
V; exceptum sit per Europam, noveris sane, si Diaria publica perlegeris, inter
quæ si, ut omittam cætera, consulas ea, quæ in Bernensi pertinent ad
initium anni 1761 ; videbis sane quo id loco haberi debeat. Systema
continet Naturalis Philosophiæ omnino novum, quod jam ab ipso Auctore
suo vulgo Boscovichianum appellant. Id quidem in pluribus Academiis
jam passim publice traditur, nec 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 archetypo.
Verum qui omnem systematis compagem, arctissimum partium nexum mutuum, fœcun-
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.
Hæc 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 Europæ partibus, utut expetita, aut nuspiam
venalia prostant, aut vix uspiam : systema vero in Italia natum, ac ab Auctore suo pluribus
hic apud nos jam dissertationibus adumbratum, & casu quodam Viennæ, 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
tum 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, quæ 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 editioni præsentem haberem, & curantem omnia per sese. At id quidem per
hosce annos obtinere non licuit, eo universam fere Europam peragrante ; donec demum
ex tam longa peregrinatione redux huc nuper se contulit, & toto adstitit editionis tempore,
ac præter 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
humanæ mentis conditio, ut in eadem re diu satis intente defigi non possit.
Hæc idcirco ut prima quædam, atque originaria editio haberi debet, quam qui cum
Viennensi contulerit, videbit sane discrimen. E minoribus mutatiunculis multæ pertinent
ad expolienda, & declaranda plura loca ; sunt tamen etiam nonnulla potissimum in pagin-
arum fine exigua additamenta, vel mutatiunculæ exiguæ factæ post typographicam
constructionem idcirco tantummodo, ut lacunulæ implerentur quæ aliquando idcirco
supererant, quod plures phyliræ a diversis compositoribus simul adornabantur, & quatuor
simul præla sudabant; quod quidem ipso præsente fieri facile potuit, sine ulla pertur-
batione sententiarum, & ordinis.

THE PRINTER AT VENICE
TO
THE READER
2.”
º
ºf - rº
; -
** 3:
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.
3
35; § OU will be well aware, if you have read the public journals, with what applause

4 TYPOGRAPHUS VENETUS LECTORI
Inter mutationes occurret ordo numerorum mutatus in paragraphis : nam numerus 82
de novo accessit totus : deinde is, qui fuerat 26I discerptus est in 5 ; demum in Appendice
post num. 534 factæ sunt & mutatiunculæ nonnullæ, & additamenta plura in iis, quæ
pertinent ad sedem animæ.
Supplementorum ordo mutatus est itidem ; quæ 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 controversiæ occasionem exhibuit in Encyclopedico etiam dictionario attactum,
in qua dissertatiuncula demonstrat Auctor non esse, cur ad vim exprimendam potentia
quæpiam distantiæ adhibeatur potius, quam functio.
Accesserunt per totum Opus notulae marginales, in quibus eorum, quæ 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, quæ huc
usque ab ipso Auctore sunt edita, quorum collectionem omnem expolitam, & correctam,
ac eorum, quæ nondum absoluta sunt, continuationem meditatur, aggressurus illico post
suum regressum in Urbem Romam, quo properat. Hic catalogus impressus fuit Venetisis
ante hosce duos annos in reimpressione ejus poematis de Solis ac Lunæ defectibus.
Porro eam omnium suorum Operum Collectionem, ubi ipse adornaverit, typis ego meis
excudendam suscipiam, quam magnificentissime potero.
Hæc erant, quæ 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 Encyclopédique. 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
PRIMÆ EDITIONIS VIENNENSIS
AD CELSISSIMUM TUNC PRINCIPEM ARCHIEPISCOPUM
VIENNENSEM, NUNC PRÆTEREA ET CARDINALEM
EMINENTISSIMUM, ET EPISCOPUM VACCIENSEM
CHRISTOPHORUM E COMITATIBUS DE MIG AZZI
3ABIS veniam, Princeps Celsissime, si forte inter assiduas sacri regiminis curas
| importunus interpellator advenio, & libellum Tibi offero mole tenuem, nec
arcana Religionis mysteria, quam in isto tanto constitutus fastigio adminis-
tras, sed Naturalis Philosophiæ principia continentem. Novi ego quidem,
quam totus in eo sis, ut, quam geris, personam sustineas, ac vigilantissimi
sacrorum Antistitis partes agas. Videt utique Imperialis hæc 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 Romæ dantem operam
studiis cognoscerem, mihi fors obtigit, altissime jam insederat animo, id in omni
reliquo amplissimorum munerum Tibi commissorum cursu hæsit firmissime, atque idipsum
inprimis adjectum tam multis & dotibus, quas a Natura uberrime congestas habes, &
virtutibus, quas tute diuturna Tibi exercitatione, atque assiduo labore comparasti, sanc-
tissime observatum inter tam varias forenses, Aulicas, Sacerdotales occupationes, istos Tibi
tam celeres dignitatum gradus quodammodo veluti coacervavit, & omnium una tam
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 Romæ in celeberrimo illo,
quod Auditorum Rotæ appellant, collegio toti Christiano orbi jus diceres, accesserat
Hetrusca Imperialis Legatio apud Romanum Pontificem exercenda ; cum repente Mech-
liniensi Archiepiscopo in amplissima illa administranda Ecclesia Adjutor datus, & destinatus
Successor, possessione præstantissimi muneris vixdum capta, ad Hispanicum Regem ab
Augustissima Romanorum Imperatrice ad gravissima tractanda negotia Legatus es missus,
in quibus cum summa utriusque Aulæ 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, & singulari cura, ac diligentia Religionem administras, & sacrorum
exceres curam ; non ea tantum urbs, atque ditio, sed universum Hungariæ Regnum,
quanquam exterum hominem, non ut civem suum tantummodo, sed ut Parentem aman-
tissimum habuit, quem adhuc ereptum sibi dolet, & angitur ; dum scilicet minore, quam
unius anni intervallo ab Ipsa Augustissima Imperatrice ad Regalem hanc Urbem, tot
Imperatorum sedem, ac Austriacæ 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,
quæ ad sacra pertinent, curandis vel per Te ipsum usque adeo, ut sæpe, 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 WACZ
fºss-ZOU will pardon me, Most Noble Prince, if perchance I come to disturb at an
7A) 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
7
ºº:

8 EPISTOLA AUCTORIS DEDICATORIA PRIMÆ EDITIONIS VIENNENSIS
hæc 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 hæc omnia ; novi hanc indolem, hanc animi constitutionem ; nec
sum tamen inde absterritus, ne, inter gravissimas istas Tuas Sacerdotales curas, Philosophicas
hasce meditationes meas, Tibi sisterem, ac tantulæ 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 Naturæ contemplationem omnino
arbitror cum Sacerdotii sanctitate penitus consentire. Mirum enim, quam belle ab ipsa
consideratione Naturæ ad cælestium rerum contemplationem disponitur animus, & ad
ipsum Divinum tantæ molis Conditorem assurgit, infinitam ejus Potentiam Sapientiam,
Providentiam admiratus, quæ erumpunt undique, & utique se produnt.
Est autem & illud, quod ad supremi sacrorum Moderatoris curam pertinet providere,
ne in prima ingenuæ juventutis institutione, quæ semper a naturalibus studiis exordium
ducit, prava teneris mentibus irrepant, ac perniciosa principia, quæ sensim Religionem
corrumpant, & vero etiam evertant penitus, ac eruant a fundamentis ; quod quidem jam
dudum tristi quodam Europæ fato passim evenire cernimus, gliscente in dies malo, ut fucatis
quibusdam, profecto perniciosissimis, imbuti principiis juvenes, tum 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 Universæ Physicæ Theoriam, & novam potissimum
Theoriam sistat, rem is quidem præstet æquissimam, nec alienum quidpiam ab ejus munere
Sacerdotali offerat, sed cum eodem apprime consentiens.
Nec 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 collocavi, ubi si quid præstitero, si minus infiliciter me gessero, nemo sane me
impudentiæ arguat, quasi vilem aliquam, & tanto indignam fastigio rem offeram. Habetur
in eo novum quoddam Universæ Naturalis Philosophiæ genus a receptis huc usque, usi-
tatisque plurimam discreparis, quanquam etiam ex iis, quæ maxime omnium per hæc tempora
celebrantur, casu quodam præcipua quæque 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 quædam ad arbitrium selecta, quæ
utcumque inter se componerem, atque compaginarem : sed omni præjudicio seposito, a
principiis exorsus inconcussis, & vero etiam receptis communiter, legitima ratiocinatione
usus, & continuo conclusionum nexu deveni ad legem virium in Natura existentium unicam,
simplicem, continuam, quæ mihi & constitutionem elementorum materiæ, & Mechanicæ
leges, & generales materiæ ipsius proprietates, & præcipua 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, quæ ad diversa præstanda phænomena sunt adhibendæ, ad
quas combinationes Conditoris Supremi consilium, & immensa Mentis Divinæ vis ubique
requiritur, quæ infinitos casus perspiciat, & ad rem aptissimos seligat, ac in Naturam
inducat.
Id mihi quidem argumentum est operis, in quo Theoriam meam expono, comprobo,
vindico : tum 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 fictitious 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-
ro EPISTOLA AUCTORIS DEDICATORIA PRIMÆ EDITIONIS VIENNENSIS
communibus Philosophorum consentio placitis, & ubi ea, quæ habemus jam pro compertis,
ex meis etiam deductionibus sponte fluunt, quod usque adeo voluminis molem contraxit.
Dederam ego quidem dispersa dissertatiunculis variis Theoriæ meæ quædam velut specimina,
quæ 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, quem laborem postremo hoc mense, molestiori-
bus negotiis, quæ 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, quæ proposui, est duplex : primo quidem
ipsum argumenti genus, quod Christianum Antistitem non modo non dedecet, sed etiam
apprime decet: tum ipsius argumenti vis, atque dignitas, quæ 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 efflagitabat
tanta erga me humanitas Tua, qua jam olim immerentem complexus Romæ, hic etiam
fovere pergis, nec in tanto dedignatus fastigio, omni benevolentiæ 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 hic orbatam suo, in dies
nimirum discessuro, & quodammodo veluti posthumam post ipsum ejus discessum typis
impressam, & in publicum prodeuntem tueberis, fovebisque.
Hæc sunt, quæ meum Tibi consilium probent, Princeps Celsissime : Tu, qua soles
humanitate auctorem excipere, opus excipe, & si forte adhuc consilium ipsum Tibi visum
fuerit improbandum ; animum saltem æquus respice obsequentissimum Tibi, ac devinct-
issimum. Vale.
Dabam Viemmae im Collegio Academico Soc. jESU
Idibus Febr. MDCCLVIII.
AUTHOR'S EPISTLE DEDICATING THE FIRST VIENNA EDITION II
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
attalilmentS.
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 13th, 1758.
AD LECTOREM
EX EDITIONE VIENNENSI
$22AT$4 BES, amice Lector, Philosophiæ Naturalis Theoriam ex unica lege virium
JAISAG } deductam, quam & ubi jam olim adumbraverim, vel etiam ex parte explica-
verim, & qua occasione nunc uberius pertractandum, atque augendam etiam,
susceperim, invenies in ipso primae partis exordio. Libuit autem hoc opus
dividere in partes tres, quarum prima continet explicationem Theoriae ipsius,
ac ejus analyticam deductionem, & vindicationem : secunda applicationem
satis uberem ad Mechanicam ; tertia applicationem ad Physicam.
Porro illud imprimis curandum duxi, ut omnia, quam liceret, dilucide exponerentur, nec
sublimiore Geometria, aut Calculo indigerent. Et quidem in prima, ac tertia parte non tantum
nullae analyticae, sed nec geometricae demonstrationes occurrunt, paucissimis quibusdam, quibus
indigeo, rejectis in adnotatiunculas, quas in fine paginarum quarumdam invenies. Quaedam
autem admodum pauca, quae majorem Algebræ, & Geometriae cognitionem requirebant, vel erant
complicatiora aliquando, &$ alibi a me jam edita, in fine operis apposui, quae Supplementorum
appellavi nomine, ubi &5 ea addidi, quae sentio de spatio, ac tempore, Theoriæ meæ consentanea,
ac edita itidem jam alibi. In secunda parte, ubi ad Mechanicam applicatur Theoria, a geome-
tricis, & aliquando etiam ab algebraicis demonstrationibus abstinere omnino non potui ; sed
eae ejusmodi sunt, ut vix umquam requirant aliud, quam Euclideam Geometriam, & primas
Trigonometriae notiones maxime simplices, ac simplicem algorithmum.
In prima quidem parte occurrunt Figuræ geometricae complures, quae prima fronte vide-
buntur etiam complicatae rem ipsam intimius non perspectanti ; verum eae nihil aliud exhibent,
nisi imaginem quamdam rerum, quae ipsis oculis per ejusmodi figuras sistuntur contemplandae.
Ejusmodi est ip$a illa curva, quae legem virium exhibet. Invenio ego quidem inter omnia
materiae puncta vim quandam mutuam, quae a distantiis pendet, &5 mutatis distantiis mutatur
ita, ut in aliis attractiva sit, in aliis repulsiva, sed certa quadam, &5 continua lege. Leges
eju$modi variationis binarum quantitatum a se invicem pendentium, uti hic sunt distantia,
&5 vis, exprimi possunt vel per analyticam formulam, vel per geometricam curvam ; sed illa
prior expressio & multo plures cognitiones requirit ad Algebram pertinentes, & imaginationem
non ita adjuvat, ut haec posterior, qua idcirco sum usus in ipsa prima operis parte, rejecta in
Supplementa formula analytica, quae & curvam, &5 legem virium ab illa expressam exhibeat.
Porro huc res omnis reducitur. Habetur in recta indefinita, quae axis dicitur, punctum
quoddam, a quo abscissa ipsius rectae segmenta referunt distantias. Curva linea protenditur
secundum rectam ipsam, circa quam etiam serpit, &5 eamdem in pluribus secat punctis : rectæ
a fine segmentorum erectae perpendiculariter usque ad curvam, exprimunt vires, quae majores
sunt, vel minores, prout ejusmodi rectae sunt itidem majores, vel minores ; ac eaedem ex attrac-
tivis migrant in repulsivis, vel vice versa, ubi illæ ipsae perpendiculares rectæ directionem
mutant, curva ab altera axis indefiniti plaga migrante ad alteram. Id quidem nullas requirit
geometricas demonstrationes, sed meram cognitionem vocum quarumdam, quae vel ad prima per-
timent Geometriae elementa, &5 notissimæ sunt, vel ibi explicantur, ubi adhibentur. Notissima
' autem etiam est significatio vocis Asymptotus, unde &5 crus asymptoticum curvae appellatur ;
dicitur nimirum recta asymptotus cruris cujuspiam curvæ, cum ipsa recta in infinitum producta,
ita ad curvilineum arcum productum itidem in infinitum semper accedit magis, ut distantia
minuatur in infinitum, $ed nusquam penitus evanescat, illis idcirco numquam invicem con-
venientibus.
Consideratio porro attenta curvae propositæ in Fig. I, & rationis, qua per illam exprimitur

I2
THE PREFACE TO THE READER
THAT APPEARED IN THE VIENNA EDITION
FºEAR 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. e
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
I3

I4 AD LECTOREM EX EDITIONE VIENNENSI
nexus inter vires, & distantias, est utique admodum necessaria ad intelligendam Theoriam ipsam,
cujus ea est praecipua quaedam veluti clavis, sine qua omnimo incassum tentarentur cetera ; sed
& ejusmodi est, ut tironum, &5 same etiam mediocrium, immo etiam longe infra mediocritatem
collocatorum, captum mom excedat, potissimum si viva accedat Professoris vox mediocriter etiam
versati im Mechanica, cujus ope, pro certo habeo, rem ita patentem omnibus reddi posse, ut
ii etiam, qui Geometriae penitus ignari sunt, paucorum admodum explicatione vocabulorum
accidente, eam ipsis oculis intueantur omnimo perspicuam.
In tertia parte supponuntur utique nonnulla, quae demonstrantur in secunda ; sed ea ipsa
sunt admodum pauca, & iis, qui geometricas demonstrationes fastidiunt, facile admodum exponi
po$$unt res ipsae ita, ut penitus etiam sine ullo Geometriae adjumento percipiantur, quanquam
sine iis ipsa demonstratio haberi non poterit ; ut idcirco in eo differre debeat is, qui secundam
partem attente legerit, &5 Geometriam calleat, ab eo, qui eam omittat, quod ille primus veritates
in tertia parte adhibitis, ac ex $ecunda erutas, ad explicationem Physicae, intuebitur per evi-
dentiam ex ipsis demonstrationibus haustam, hic $ecundus easdem quodammodo per fidem Geo-
metris adhibitam credet. Hujusmodi imprimis est illud, particulam compositam ex punctis
etiam homogeneis, praeditis lege virium proposita, po$$e per solam diversam ipsorum punctorum
dispositionem aliam particulam per certum intervallum vel perpetuo attrahere, vel perpetuo
repellere, vel nihil in eam agere, atque id ipsum viribus admodum diversis, & quæ respectu diver-
sarum particularum diversae simt, &5 diversae respectu partium diversarum ejusdem particulae,
ac aliam particulam alicubi etiam urgeant in latus, unde plurium phaenomenorum explicatio in
Physica sponte fluit.
Verum qui omnem Theoriae, & deductionum compagem aliquanto altius inspexerit, ac
diligentius perpenderit, videbit, ut spero, me in hoc perquisitionis gemere multo ulterius
progressum esse, quam olim Newtonus ipse desideravit. Is enim in postremo Opticae questione
prolatis iis, quae per vim attractivam, &5 vim repulsivam, mutata distantia ipsi attractivae suc-
cedentem, explicari poterant, haec addidit : “ Atque haec quidem omnia si ita sint, jam Natura
universa valde erit simplex, & consimilis sui, perficiens nimirum magnos omnes corporum
caelestium motus attractione gravitatis, quae est mutua inter corpora illa omnia, &5 minores fere
omnes particularum suarum motus alia aliqua vi attrahente, &$ repellente, quæ est inter particulas
illas mutua.” Aliquanto autem inferius de primigeniis particulis agens sic habet : “ Porro
videntur mihi hae particulæ primigemiae non modo in se vim inertiae habere, motusque leges passivas
illas, quae ex vi ista necessario oriuntur ; verum etiam motum perpetuo accipere a certis principiis
actuosis, qualia nimirum sunt gravitas, &5 causa fermentationis, &5 cohaerentia corporum. Atque
haec quidem principia considero non ut occultas qualitates, quae ex specificis rerum formis oriri
fimgantur, sed ut universales Naturae leges, quibus res ipsae sunt formatae. Nam primcipia
quidem talia revera existere ostendunt phaenomena Naturae, licet ipsorum causae quae sint,
nondum fuerit explicatum. Affirmare, singulas rerum species specificis praeditas e$$e qualita-
tibus occultis, per quas eae vim certam in agendo habent, hoc utique est nihil dicere : at ex
phaenomenis Naturae duo, vel tria derivare generalia motus principia, &5 deinde explicare,
quemadmodum proprietates, & actiones rerum corporearum omnium ex istis principiis conse-
quantur, id vero magnus esset factus in Philosophia progressus, etiamsi principiorum istorum
causae nondum essent cognitae. Quare motus principia supradicta proponere non dubito, cum
per Naturam universam latissime pateant.”
Haec ibi Newtonus, ubi is quidem magnos im Philosophia progressus facturum arbitratus
est eum, qui ad duo, vel tria generalia motus principia ex Naturæ phænomenis derivata phaemo-
memorum explicationem reduxerit, & sua principia protulit, ex quibus inter se diversis eorum
aliqua tantummodo explicari posse censuit. Quid igitur, ubi &5 ea ipsa tria, &5 alia praecipua
quaeque, ut ipsa etiam impenetrabilitas, & impulsio reducantur ad principium unicum legitima
ratiocinatione deductum ? At id per meam unicam, & simplicem virium legem praestari, patebit
sane consideranti operis totius Synopsim quamdam, quam hic 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
far more clear to him who studies the whole work with some earnestness,
II
I6
I9
SYNOPSIS TOTIUS OPERIS
EX EDITIONE VIENNENSI
PARS I
Ε$&RIMIS sex numeris exhibeo, quando, & qua occasione Theoriam meam
j) j 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 præstet : addo, quid
alibi promiserim pertinens ad æquilibrium, & 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 justæ
molis evaserit tractatio. &
Tum usque ad num. II expono Theoriam ipsam : materiam constantem punctis
prorsus simplicibus, indivisibilibus, & inextensis, ac a se invicem distantibus, quæ puncta
habeant singula vim inertiæ, & præterea vim activam mutuam pendentem a distantiis, ut
nimirum, data distantia, detur & magnitudo, & directio vis ipsius, mutata autem distantia,
mutetur vis ipsa, quæ, imminuta distantia in infinitum, sit repulsiva, & quidem
excrescens in infinitum : aucta autem distantia, minuatur, evanescat, mutetur in attrac-
tivam crescentèm primo, tum decrescentem, evanescentem, abeuntem iterum in repul-
sivam, idque per multas vices, donec demum in majoribus distantiis abeat in attractivam
decrescentem ad sensum in ratione reciproca duplicata distantiarum ; quem 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 binæ elastri cuspides conantur ad
es invicem accedere, vel a se invicem recedere, prout sunt plus justo distractæ, vel con-
traCtæ.
Inde ad num. I6 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 curvæ ductum, & naturam expono,
ac ostendo, in quo differat ab hyperbola illa gradus tertii, quæ Newtonianum gravitatem
exprimit : ac demum ibidem & argumentum, & divisionem propono operis totius.
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 I9 illud, in collisione corporum debere vel
haberi compenetrationem, vel violari legem continuitatis, velocitate mutata per saltum, si
cum inæqualibus velocitatibus deveniant ad immediatum contactum, quæ continuitatis lex
cum (ut evinco) debeat omnino observari, illud infero, antequam ad contactum deveniant
corpora, debere mutari eorum velocitates per vim quandam, quæ sit par extinguendæ
velocitati, vel velocitatum differentiæ, cuivis utcunque magnæ. -
A num. I9 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, quæ sunt in textu.


I6
SYNOPSIS OF THE WHOLE WORK
(FROM THE VIENNA EDITION)
PART I
ºšN the first six articles, I state the time at which I evolved my Theory, what
§ 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,
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. I6 I show that it is not merely an aggregate of forces combined
haphazard, but that it is represented by a single continuous curve, by means of abscissae
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
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
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.
I7 & C
I+
II
I6
I9

I8 SYNOPSIS TOTIUS OPERIS
28
3o
32
39
45
52
73
74.
75
Sequentibus num. 28 & 29 binas alias responsiones rejicio aliorum, quarum altera, ut
mei argumenti vis elidatur, affirmat quispiam, prima materiæ elementa compenetrari, alter
dicuntur materiæ puncta adhuc moveri ad se invicem, ubi localiter omnino quiescunt, &
contra primum effugium evinco impenetrabilitatem ex inductione ; contra secundum
expono æquivocationem quandam in significatione vocis motus, cui æquivocationi totum
1nn1t1tur.
Hinc num. 3o, & 3I ostendo, in quo a Mac-Laurino dissentiam, qui considerata eadem,
quam ego contemplatus sum, collisione corporum, conclusit, continuitatis legem violari,
cum ego eandem illæsam esse debere ratus ad totam devenerim Theoriam meam.
Hic igitur, ut meæ deductionis vim exponam, in ipsam continuitatis legem inquiro, ac
a num. 32 ad 38 expono, quid ipsa sit, quid mutatio continua per gradus omnes intermedios,
quæ nimirum excludat omnem saltum ab una magnitudine ad aliam sine transitu per
intermedias, ac Geometriam etiam ad explicationem rei in subsidium advoco : tum eam
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 erutæ passim per inductionem, donec demum ejus vim
applicem ad legem continuitatis demonstrandam : ac sequentibus numeris casus evolvo
quosdam binarum classium, in quibus continuitatis lex videtur lædi nec tamen læditur.
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, quæ nimirum nec suo principio,
nec suo fine carere possunt. Ejus rationis vim ostendo in motu locali, & in Geometria
sequentibus duobus numeris : tum num. 52 expono difficultatem quandam, quæ petitur
ex eo, quod in momento temporis, in quo transitur a non esse ad e$$e, videatur juxta ejusmodi
Theoriam debere simul haberi ipsum esse, & non esse, quorum alterum ad finem præcedentis
seriei statuum pertinet, alterum ad sequentis initium, ac solutionem ipsius fuse evolvo,
Geometria etiam ad rem oculo ipsi sistendam vocata in auxilium. -
Num. 63, post epilogum eorum omnium, quæ de lege continuitatis sunt dicta, id
principium applico ad excludendum saltum immediatum ab una velocitate ad aliam, sine
transitu per intermedias, quod & inductionem læderet pro continuitate amplissimam, &
induceret pro ipso momento temporis, in quo fieret saltus, binas velocitates, ultimam
nimirum seriei præcedentis, & primam novæ, 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, quæ ad ipsarum naturam, ac mutationes pertinent, diligenter evolvo, nonnullis
etiam, quæ inde contra meæ Theoriæ probationem objici possunt, dissolutis.
His expositis concludo jam illud ex ipsa continuitate, ubi corpus quodpiam velocius
movetur post aliud lentius, ad contactum immediatum cum illa velocitatum inæqualitate
deveniri non posse, in quo scilicet contactu primo mutaretur vel utriusque velocitas, vel
alterius, per saltum, sed debere mutationem velocitatis incipere ante contactum ipsum.
Hinc num. 73 infero, debere haberi mutationis causam, quæ appelletur vis : tum num. 74
hanc vim debere esse mutuam, & agere in partes contrarias, quod per inductionem evinco,
& inde infero num. 75, appellari posse repulsivam ejusmodi vim mutuam, ac ejus legem
exquirendam propono. In ejusmodi autem perquisitione usque ad num. 8o invenio illud,
debere vim ipsam imminutis distantiis crescere in infinitum ita ut par sit extinguendæ
velocitati utcunque magnæ ; tum & 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 : tum sensim plures, ac etiam
plurimos ejusmodi limites invenio, sive transitus ab attractione ad repulsionem, & vice
versa, ac formam totius curvæ per ordinatas suas exprimentis virium legem determino.
SYNOPSIS OF THE WHOLE WORK I9
In the next articles, 28 & 29, I refute a further pair of arguments advanced by others;
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
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
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-
ciple of induction itself, as far as Art. 44, I 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,
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,
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
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
Imatter. -
In Art. 63, after summing up all that has been said about the Law of Continuity, I
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:
which is to be called “force.” Then, in Art. 74, I prove that this force is a mutual one, &
that it acts in opposite directions; the proof is by induction. From this, in Art. 75, I
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.
*
28
so
32
39
45
73
74
75
2O • SYNOPSIS TOTIUS OPERIS
8I
88
9I
IOO
IOI
IO4
Io6
Eo usque virium legem deduco, ac definio ; tum num. 8I eruo ex ipsa lege consti-
tutionem elementorum materiæ, quæ 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 inquiro in illud, an hæc elementa, ut simplicia esse
debent, ita etiam inextensa esse debeant, ac exposita illa, quam virtualem extensionem
appellant, eandem excludo inductionis principio, & difficultatem evolvo tum eam, quæ peti
possit ab exemplo ejus generis extensionis, quam in anima indivisibili, & simplice per aliquam
corporis partem divisibilem, & extensam passim admittunt : vel omnipræsentiæ Dei : tum
eam, quæ 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, nec
quietem omnimodam in Natura haberi usquam, nec adesse semper omnimodam inter
tempus, & spatium analogiam. Hic autem ingentem colligo ejusmodi determinationis
fructum, ostendens usque ad num. 9I, quantum prosit simplicitas, indivisibilitas, inextensio
elementorum materiæ, ob summotum transitum a vacuo continuo per saltum ad materiam
continuam, ac ob sublatum limitem densitatis, quæ 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 & gravissimæ difficultates plurimæ evanescunt, &
infinitum actu existens habetur nullum, sed in possibilibus tantummodo remanet series
finitorum in infinitum producta.
His definitis, inquiro usque ad num. 99 in illud, an ejusmodi elementa sint censenda
homogenea, an heterogenea : ac primo quidem argumentum pro homogeneitate saltem 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. •
Ea ad probationem Theoriæ pertinent ; qua absoluta, antequam inde fructus colli-
gantur multiplices, gradum hic facio ad evolvendas difficultates, quæ vel objectæ jam sunt,
vel objici posse videntur mihi, primo quidem contra vires in genere, tum contra meam
hanc expositam, comprobatamque virium legem, ac demum contra puncta illa indivisibilia,
& inextensa, quæ ex ipsa ejusmodi virium lege deducuntur.
Primo quidem, ut iis etiam faciam satis, qui inani vocabulorum quorundam sono
perturbantur, a num. IOI ad IO4 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
tractatio de Motibus, qui a datis viribus etiam sine immediato impulsu oriuntur. A num.
IO4 ad Io6 ostendo, nullum committi saltum in transitu a repulsionibus ad attractiones,
& vice versa, cum nimirum per omnes intermedias quantitates is transitus fiat. Inde vero
ad objectiones gradum facio, quæ totam curvæ formam impetunt. Ostendo nimirum usque
ad num. II6, 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, quæ, quo altioris sunt gradus, eo in
pluribus punctis rectam secare possunt, & eo in immensum plures sunt numero ; haberi
potius, ubi curva quæritur, quæ vires exprimat, indicium pro curva ejus naturæ, ut rectam
in plurimis punctis secet, adeoque plurimos secum afferat virium transitus a repulsivis ad
attractivas, quam pro curva, quæ nusquam axem secans attractiones solas, vel solas pro
distantiis omnibus repulsiones exhibeat : sed vires repulsivas, & multiplicitatem transituum
esse positive probatam, & deductam totam curvæ formam, quam itidem Ostendo, non esse
ex arcubus natura diversis temere coalescentem, sed omnino simplicem, atque eam ipsam
SYNOPSIS OF THE WHOLE WORK 2 I
So far I have been occupied in deducing and settling the law of these forces. Next,
in Art. 81, I derive from this law the constitution of the elements of matter. These must be
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-
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
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
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
empty sound of certain terms, I show, in Art. IOI to IO4, 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. IO4 to IO6, I show that no sudden change takes place in passing from repulsions to
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
curve. I show indeed, from here on to Art. II6, 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;
8I
88
9I
IOO
IOI
IO4.
Ioô
22 * SYNOPSIS TOTIUS OPERIS
I 2 I
I24
I 3 I
I4O
I43
I 53
I65
I66
I68
simplicitatem in Supplementis evidentissime demonstro, exhibens methodum, qua deveniri
possit ad æquationem ejusmodi curvæ simplicem, & uniformem ; licet, ut hic ostendo, ipsa
illa lex virium possit mente resolvi in plures, quæ per plures curvas exponantur, a quibus
tamen omnibus illa reapse unica lex, per unicam illam continuam, & in se simplicem curvam
componatur. -
A num. I 2 I refello, quæ objici possunt a lege gravitatis decrescentis in ratione reciproca
duplicata distantiarum, quæ nimirum in minimis distantiis attractionem requirit crescentem
in infinitum. Ostendo autem, ipsam non esse uspiam accurate in ejusmodi ratione, nisi
imaginarias resolutiones exhibeamus ; nec vero ex Astronomia deduci ejusmodi legem
prorsus accurate servatam in ipsis Planetarum, & Cometarum distantiis, sed ad summum ita
proxime, ut differentia ab ea lege sit perquam exigua : ac a num. I 24 expendo argumentum,
quod pro ejusmodi lege desumi possit ex eo, quod cuipiam visa sit omnium optima, &
idcirco electa ab Auctore Naturæ, ubi ipsum Optimismi principium ad trutinam revoco, ac
excludo, & vero illud etiam evinco, non esse, cur omnium optima ejusmodi lex censeatur :
in Supplementis vero Ostendo, ad quæ potius absurda deducet ejusmodi lex, & vero etiam
aliæ plures attractionis, quæ imminutis in infinitum distantiis excrescat in infinitum.
Num. I3I a viribus transeo ad elementa, & primum Ostendo, cur punctorum inexten-
sorum ideam non habeamus, quod nimirum eam haurire non possumus per sensus, quos
solæ massæ, & 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 Leibnitianæ
monades recidant, sed sublata extensione continua difficultatem auferre illam omnem, quæ
jam olim contra Zenonicos objecta, nunquam est satis soluta, qua fit, ut extensio continua
ab inextensis effici omnino non possit.
Num. I40 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,
nec ullum haberi extensum continuum. A num. I43 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
quædam evolvo continuitatis primo aspectu violatæ, in quibusdam proprietatibus luminis,
ac in aliis quibusdam casibus, in quibus quædam crescunt per additionem partium, non (ut
ajunt) per intussumptionem. * - -
A num. I 53 ostendo, quantum hæc 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 inquiro, & quæ inde præjudicia profluant, expono. Postremo
autem loco num. I65 innuo, qui fieri possit, ut puncta inextensa, & a se invicem distantia,
in massam coalescant, quantum libet, cohærentem, & iis proprietatibus præditam, quas in
corporibus experimur, quod tamen ad tertiam partem pertinet, ibi multo uberius pertrac-
tandum ; ac ibi quidem primam hanc partem absolvo.
PARS II
Num. I66 hujus partis argumentum propono ; sequenti vero I67, quæ potissimum in
curva virium consideranda sint, enuncio. Eorum considerationem aggressus, primo quidem
usque ad num. I72 in ipsos arcus inquiro, quorum alii attractivi, alii repulsivi, alii asym-
ptotici, ubi casuum occurrit mira multitudo, & in quibusdam consectaria notatu digna, ut
& illud, cum ejus formæ 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. I21, I start to give a refutation of those objections that may be raised from
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. I24 onwards, I examine the value of the argument that can be drawn
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. f
In Art. 131 I pass from forces to elements. I first of all show the reason why we may
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. I40 I show that the principle of induction yields no argument against these
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. I43 onwards,
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-
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
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. 166 I state the theme of this second part; and in Art. 167 I declare what
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
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
I 2 I
I24
I3 I
I4O
I43
I53
I66
I68
24 SYNOPSIS TOTIUS OPERIS
I 72
I79
I89
204
2O9
2I4
22 I
222
223
228
238
24O
elementi. Ad. num. I79 areas contemplor arcubus clausas, quæ respondentes segmento axis
cuicunque, esse possunt magnitudine utcunque magnæ, vel parvæ, sunt autem mensura
incrementi, vel decrementi quadrati velocitatum. Ad num. I89 inquiro in appulsus curvæ
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.
Num. I89 a consideratione curvæ ad punctorum combinationem gradum facio, ac
primo quidem usque ad num. 204 ago de systemate duorum punctorum, ea pertractans,
quæ 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
cohæsionis varia genera, fermentationes, conflagrationes, emissiones vaporum, proprietates
luminis, elasticitatem, mollitiem. -
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
diversæ pariunt combinationes punctorum, utut tantummodo numero trium, persequor
usque ad num. 209. Hinc usque ad num. 2 I4 quædam evolvo, quæ pertinent ad vires
ortas in singulis ex actione composita reliquorum duorum, & quæ 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 Naturæ
explicationem futurum sit usui, significo. Usque ad num. 22 I 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 :
tum 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 nec attrahitur, nec repellitur ; cujusmodi discrimen cum in massis haberi debeat multo
majus, illud indico, num. 222, quantus inde itidem in Physicam usus proveniat.
Hic 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 quædam exurgunt, quæ multo generaliores redduntur inferius, ubi
in tribus etiam punctis tantummodo adumbrantur, quæ pertinent ad virgas rigidas, flexiles,
elasticas, ac ad vectem, & ad alia plura, quæ itidem inferius, ubi de massis, multo generaliora
fiunt. Demum usque ad num. 238 contemplor tria puncta posita non in directum, sive in
æquilibrio sint, sive in perimetro ellipsium quarundam, vel curvarum aliarum ; in quibus
mira occurrit analogia limitum quorundam cum limitibus, quos habent bina puncta in axe
curvæ primigeniæ ad se invicem, atque ibidem multo major varietas casuum indicatur pro
massis, & specimen applicationis exhibetur ad soliditatem, & liquationem per celerem
intestinum motum punctis impressum. Sequentibus autem binis numeris generalia quædam
expono de systemate punctorum quatuor cum applicatione ad virgas solidas, rigidas, flexiles,
ac ordines particularum varios exhibeo per pyramides, quarum infimæ ex punctis quatuor,
superiores ex quatuor pyramidibus singulæ coalescant.
A num. 24o ad massas gradu facto usque a num. 264 considero, quæ ad centrum gravi-
tatis pertinent, ac demonstro generaliter, in quavis massa esse aliquod, & esse unicum :
ostendo, quo pacto determinari generaliter possit, & quid in methodo, quæ 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
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
far as Art. 189, I 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.
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
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
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-
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
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.
At this point then, in Art. 223, I pass from the forces derived from two points to the
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
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
172
179
2O4.
209
2I4.
22 I
222
223
228
238
240
26 SYNOPSIS TOTIUS OPERIS
277,
264
265
266
276
278
279
289
297
3o7
3 I3
32 I
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 inquiro : tum vero ejusdem
ope demonstro admodum expedite, & accuratissime celebre illud Newtoni theorema de
statu centri gravitatis per mutuas internas vires numquam turbato.
Ejus tractionis fructus colligo plures : conservationem ejusdem quantitatis motuum in
Mundo in eandem plagam num. 264, æqualitatem actionis, & reactionis in massis num. 265,
collisionem corporum, & communicationem motus in congressibus directis cum eorum
legibus, inde num. 276 congressus obliquos, quorum Theoriam a resolutione motuum reduco
ad compositionem num. 277, quod sequenti numero 278 transfero ad incursum etiam in
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, quæ prima fronte virium resolutionem requirere videntur.
A num. 289 ad 297 leges expono compositionis virium, & resolutionis, ubi & illud
notissimum, quo pacto in compositione decrescat vis, in resolutione crescat, sed in illa priore
conspirantium summa semper maneat, contrariis elisis ; in hac posteriore concipiantur
tantummodo binæ vires contrariæ adjectæ, quæ consideratio nihil turbet phænomena ;
unde fiat, ut nihil inde pro virium vivarum Theoria deduci possit, cum sine iis explicentur
omnia, ubi plura itidem explico ex iis phænomenis, quæ pro ipsis viribus vivis afferri solent.
A num. 297 Occasione inde arrepta aggredior quædam, quæ 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 incidentiæ, & 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, quæ ad luminis reflexionem, & refractionem explican-
dam, definiendamque ex Mechanica requiritur, relationem itidem vis absolutæ ad relativam
in obliquo gravium descensu, & nonnulla, quæ ad Oscillationum accuratiorem Theoriam
necessaria sunt, prorsus elementaria, diligenter expono. .
A num. 307 inquiro in trium massarum systema, ubi usque ad num. 3I3 theoremata
evolvo plura, quæ pertinent ad directionem virium in singulis compositarum e binis
reliquarum actionibus, ut illud, eas directiones vel esse inter se parallelas, vel, si utrinque
indefinite producantur, per quoddam commune punctum transire omnes : tum usque ad
32 I theoremata alia plura, quæ 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, distantiæ ipsarum a massa tertia, sinus
anguli, quem singularum directio continet cum sua ejusmodi distantia, & massæ ipsius eam
habentis compositam vim, ad distantiam, sinum, massam alteram ; vires autem motrices
habere tantummodo priores rationes duas elisa tertia. -
Eorum theorematum fructum colligo deducens inde usque ad num. 328, quæ ad
æquilibrium pertinent divergentium utcumque virium, & ipsius æquilibrii centrum, ac
nisum centri in fulcrum, & quæ ad præponderantiam, Theoriam extendens ad casum etiam,
quo massae non in se invicem agant mutuo immediate, sed per intermedias alias, quæ 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 deduco pro machinis omnibus : tum omnium vectium genera evolvo,
ut & illud, facta suspensione per centrum gravitatis haberi æquilibrium, sed in ipso centro
debere sentiri vim a fulcro, vel sustinente puncto, æqualem summæ ponderum totius
systematis, unde demum pateat ejus ratio, quod passim sine demonstratione assumitur,
nimirum systemate quiescente, & impedito omni partium motu per æquilibrium, 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-
servation of the same quantity of motion in the Universe in one plane; in Art. 265 the
equality of action and reaction amongst masses; then the collision of solid bodies, and the
communication of motions in direct impacts & the laws that govern them, & from that,
in Art. 276, oblique impacts; in Art. 277 I reduce the theory of these from resolution of
motions to compositions, & in the article that follows, Art. 278, I pass to impact on to a
fixed plane; from Art. 279 to Art. 289 I show that there can be no real resolution of forces
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, I state the laws for the composition & resolution of forces;
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
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
far as Art. 32 I, 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
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. . -
264
265
266
276
277
278
279
289
297
307
3-I 3
32 I
28 SYNOPSIS TOTIUS OPERIS
328
344
3
4
7
348
39I
398
4. I9
A num. 328 ad 347 deduco ex iisdem theorematis, quæ pertinent ad centrum Oscilla-
tionis quotcunque massarum, sive sint in eadem recta, sive in plano perpendiculari ad axem
rotationis ubicunque, quæ Theoria per systema quatuor massarum, excolendum aliquanto
diligentius, uberius promoveri deberet & extendi ad generalem habendum solidorum nexum,
qua re indicata, centrum itidem percussionis inde evolvo, & ejus analogiam cum centro
Oscillationis exhibeo. ' - -
Collecto ejusmodi fructu ex theorematis pertinentibus ad massas tres, innuo num. 347,
quæ mihi communia sint cum ceteris omnibus, & cum Newtonianis potissimum, pertinentia
ad summas virium, quas habet punctum, vel massa attracta, vel repulsa a punctis singulis
alterius massæ ; tum a num. 348 ad finem hujus partis, sive ad num. 358, expono quædam,
quæ 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, præter alias curvas, exponi posse per Logisticam,
& in fluidis gravitate nostra terrestri præditis pressiones haberi debere ut altitudines ;
deinde vero attingo illa etiam, quæ pertinent ad velocitatem fluidi erumpentis e vase, &
expono, quid requiratur, ut ea sit æqualis velocitati, quæ acquiretur cadendo per altitudinem
ipsam, quemadmodum videtur res obtingere in aquæ effluxu : quibus partim expositis,
partim indicatis, hanc secundam partem concludo. -
PARS III
Num. 358 propono argumentum hujus tertiæ partis, in qua omnes e Theoria mea
generales materiæ proprietates deduco, & particulares plerasque : tum usque ad num. 37I
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
transitu per substantias intimas sine vera compenetratione, & mira quædam phænomena
huc pertinentia explico admodum expedite. Inde ad mum. 375 de extensione ago, quæ
mihi quidem in materia, & corporibus non est continua, sed adhuc eadem præbet phæno-
menæ sensibus, ac in communi sententia ; ubi etiam de Geometria ago, quæ vim suam in
mea Theoria retinet omnem : tum ad num. 383 figurabilitatem persequor, ac molem,
massam, densitatem singillatim, in quibus omnibus sunt quædam Theoriæ meæ propria
scitu non indigna. De Mobilitate, & Motuum Continuitate, usque ad num. 388 notatu
digna continentur : tum usque ad num. 39I ago de æqualitate actionis, & reactionis, cujus
consectaria vires ipsas, quibus Theoria mea innititur, mirum in modum confirmant.
Succedit usque ad num. 398 divisibilitas, quam ego ita admitto, ut quævis massa existens
numerum punctorum realium habeat finitum tantummodo, sed qui in data quavis mole
possit esse utcunque magnus ; quamobrem divisibilitati in infinitum vulgo admissæ sub-
stituo componibilitatem in infinitum, ipsi, quod ad Naturæ phænomena explicanda
pertinet, prorsus æquivalentem. His evolutis addo num. 398 immutabilitatem primorum
materiæ elementorum, quæ cum mihi sint simplicia prorsus, & inextensa, sunt utique
immutabilia, & ad exhibendam perennem phænomenorum seriem aptissima.
A num. 399 ad 406 gravitatem deduco ex mea virium Theoria, tanquam ramum
quendam e communi trunco, ubi & illud expono, qui fieri possit, ut fixæ in unicam massam
non coalescant, quod gravitas generalis requirere videretur. Inde ad num. 4I9 ago de
cohæsione, qui est itidem veluti alter quidam ramus, quam ostendo, nec in quiete con-
sistere, nec in motu conspirante, nec in pressione fluidi cujuspiam, nec in attractione
maxima in contactu, sed in limitibus inter repulsionem, & attractionem ; ubi & problema
generale propono quoddam huc pertinens, & illud explico, cur massa fracta non iterum
coalescat, cur fibræ ante fractionem distendantur, vel contrahantur, & innuo, quæ ad
cohæsionem pertinentia mihi cum reliquis Philosophis communia sint.
A cohæsione gradum facio num. 419 ad particulas, quæ ex punctis cohærentibus
efformantur, de quibus ago usque ad num. 426, & varia persequor earum discrimina :
SYNOPSIS OF THE WHOLE WORK 29
3-
From Art. 328 to Art. 347, I deduce from these same theorems, others that relate to
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 cºntre of percussion, & I show the analogy between it & the centre
of oscillation. - -
I obtain all such results from theorems relating to three masses. After that, in Art.
247, 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,
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
of the special, properties of matter from my Theory. Then, as far as Art. 371, I deal some-
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-
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
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.
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;
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. º
From Art. 399 to Art. 406, I derive gravity from my Theory of forces, as if it were a
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
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, I pass on from cohesion to particles which are formed from a number of
cohering points; & I consider these as far as Art. 426, & investigate the various distinctions
328
358
360
371
375
383
388
39 I
398
399
4O6
4I9
3o SYNOPSIS TOTIUS OPERIS
426
446
45O
452
467
47I
472
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 diversæ vires unius particulæ in aliam, ac
itidem diversæ in diversis partibus ejusdem particulæ respectu diversarum partium, vel
etiam respectu ejusdem partis particulæ 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 : tum illud
addo, particulas eo difficilius dissolubiles esse, quo minores sint ; debere autem in gravitate
esse penitus uniformes, quæcunque punctorum dispositio habeatur, & in aliis proprietatibus
plerisque debere esse admodum (uti observamus) diversas, quæ diversitas multo major in
majoribus massis esse debeat.
A num. 426 ad 446 de solidis, & fluidis, quod discrimen itidem pertinet ad varia
cohæsionum 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 æqualibus ; illorum vero ex inæqualitate 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 ipsæ tenacissimæ sint, totum etiam
Atomistarum, & Corpuscularium systema a mea Theoria repeti`ita, ut id nihil sit aliud,
nisi unicus itidem hujus veluti trunci fœcundissimi ramus e diversa cohæsionis ratione
prorumpens. Demum ostendo, cur non quævis massa, utut constans ex homogeneis
punctis, & circa se maxime in gyrum mobilibus, fluida sit ; & fluidorum resistentiam quoque
attingo, in ejus leges inquirens.
A num. 446 ad 450 ago de iis, quæ itidem ad diversa pertinent soliditatis genera, nimirum
de elasticis, & mollibus, illa 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 ; hæc a limitum frequentia, atque ingenti vicinia, qua fiat, ut ex
uno ad alium delata limitem puncta, ibi quiescant itidem respective, ut prius. Tum vero
de ductilibus, & malleabilibus ago, ostendens, in quo a fragilibus discrepent : Ostendo autem,
hæc omnia discrimina a densitate nullo modo pendere, ut nimirum corpus, quod
multo sit altero densius, possit tam multo majorem, quam multo minorem soliditatem, &
cohæsionem habere, & quævis ex proprietatibus expositis æque possit cum quavis vel majore,
vel minore densitate componi.
Num. 45o inquiroin vulgaria quatuor elementa ; tum a num. 45 I ad num. 467 persequor
chemicas operationes ; num. 452 explicans dissolutionem, 453 præcipitationem, 454, & 455
commixtionem plurium substantiarum in unam : tum num. 456, & 457 liquationem binis
methodis, 458 volatilizationem, & effervescentiam, 46I emissionem effluviorum, quæ 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.
A fermentatione num. 467 gradum facio ad ignem, qui mihi est fermentatio quædam
substantiæ lucis cum sulphurea quadam substantia, ac plura inde consectaria deduco usque
ad num. 47I ; tum ab igne ad lumen ibidem transeo, cujus proprietates præcipuas, ex
quibus omnia lucis phænomena oriuntur, propono num. 472, ac singulas a Theoria mea
deduco, & fuse explico usque ad num. 5o3, 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
æquales inde ad 484, refractionem ad 487, tenuitatem num. 487, calorem, & ingentes
intestinos motus allapsu tenuissimæ lucis genitos, num. 488, actionem majorem corporum
oleosorum, & sulphurosorum in lumen num. 489 : tum num. 490 ostendo, nullam resist-
SYNOPSIS OF THE WHOLE WORK - 3I
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, I consider solids & fluids, the difference between which is
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
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
Art. 451 to Art. 467, I treat of chemical operations; I explain solution in Art. 452, preci-
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
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
which, from which all the phenomena of light arise, I set forth in Art. 472 ; & I deduce
& 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
426
446
450
452
467
47I
472
32 SYNOPSIS TOTIUS OPERIS
503
entiam veram pati, ac num. 49I 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 deduco duas diversas dispositiones ad æqualia 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 incidentiæ sit major, ac num.
499 & 5oo expono, unde discrimen in intervallis vicium, ex quo uno omnis naturalium
colorum pendet Newtoniana Theoria. Demum num. 5oI miram attingo crystalli
Islandicæ proprietatem, & ejusdem causam, ac num. 5o2 diffractionem expono, quæ est
quædam inchoata refractio, sive reflexio. -
Post lucem ex igne derivatam, quæ ad oculos pertinet, ago brevissime num. 5o3 de
sapore, & odore, ac sequentibus tribus numeris de sono : tum aliis quator de tactu, ubi
etiam de frigore, & calore : deinde vero usque ad num. 5I4 de electricitate, ubi totam
Franklinianam Theoriam ex meis principiis explico, eandem ad bina tantummodo reducens
principia, quæ ex mea generali virium Theoria eodem fere pacto deducuntur, quo præcipi-
tationes, atque dissolutiones. Demum num. 5I4, ac 515 magnetismum persequor, tam
directionem explicans, quam attractionem magneticam.
Hisce expositis, quæ ad particulares etiam proprietates pertinent, iterum a num. 516
ad finem usque generalem corporum complector naturam, & quid materia sit, quid forma,
quæ censeri debeant essentialia, quæ accidentialia attributa, adeoque quid transformatio
sit, quid alteratio, singillatim persequor, & partem hanc tertiam Theoriæ meæ absolvo.
De Appendice ad Metaphysicam pertinente innuam hic illud tantummodo, me ibi
exponere de anima illud inprimis, quantum spiritus a materia differat, quem nexum anima
habeat cum corpore, & quomodo in ipsum agat : tum de DEO, ipsius & existentiam me
pluribus evincere, quæ nexum habeant cum ipsa Theoria mea, & Sapientiam inprimis, ac
Providentiam, ex qua gradum ad revelationem faciendum innuo tantummodo. Sed hæc
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
next four articles, I consider touch, & in connection with it, cold & heat also. After that,
as far as Art. 514, I deal with electricity; here I explain the whole of the Franklin theory
by means of my principles; I reduce this theory to two principles only, & these are
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
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,
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.
503
504.
507
5II
5I4
516
Cujusmodi systema» I.
Theoria exhibeat.
In quo conveniat
cum systemate
Newtoniano, &
Leibnitiano.
In quo differat a
Leibnitiano & ipsi
præstet.
In quo differat a
Newtoniano & ipsi
præstet.
"' PHILOSOPHIÆ NATURALIS THEORIA
PARS I
Theoriae expositio, ama/ytica deductio, & vimdicatio.
ÉTQIRIUM mutuarum Theoria, in quam incidi jam ab Anno 1745, dum e
|| notissimis principiis alia ex aliis consectaria eruerem, & ex qua ipsam
simplicium materiæ 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 præcipuis 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, quæ pro aliis punctorum distantiis a
se invicem aliæ sint ; & quidem ex ipso itidem Newtono non ejusmodi vires tantummodo,
quæ ipsa puncta determinent ad accessum, quas vulgo attractiones nominant ; sed etiam
ejusmodi, quæ 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 Opticæ Quæstione proposuit, ac exemplo transitus a positivis
ad negativa, qui habetur in algebraicis formulis, illustravit. Illud autem utrique systemati
commune est cum hoc meo, quod quævis particula materiæ 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
oppositæ, qui casus est infinities improbabilis, motus in iis omnibus aliquis inde ortus
habeatur.
3. Distat autem a Leibnitiana Theoria longissime, tum quia nullam extensionem
continuam admittit, quæ ex contiguis, & se contingentibus inextensis oriatur : in quo
quidem difficultas jam olim contra Zenonem proposita, & nunquam sane aut soluta satis,
aut solvenda, de compenetratione omnimoda inextensorum contiguorum, eandem vim
adhuc habet contra Leibnitianum systema : tum 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 Naturæ analogia
ducit, ac chemicæ 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 sufficientis usque adeo a Leibnitianis deprædicata, meo quidem judicio,
nihil omnino possunt.
4. Distat itidem a Newtoniano systemate quamplurimum, tum in eo, quod ea, quæ
Newtonus in ipsa postremo Quæstione Opticæ conatus est explicare per tria principia,
gravitatis, cohæsionis, fermentationis, immo & reliqua quamplurima, quæ 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 : tum 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
STHE following Theory of mutual forces, which Ilit upon as far back as the year
& Cº. 1745, whilst I was studying various propositions arising from other very
sºlº ſº well-known principles, & from which I have derived the very constitu-
jº sº 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
which is founded the theory of Leibniz ; & also to the mutual forces, which vary as the
distances of the points from one another vary, the characteristic of the theory of Newton ;
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 formulae. 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,
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,
because it explains by means of a single law of forces all those things that Newton himself,
in the last of his ‘Questions on Optics’, 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
The kind of sys-
tem the Theory
presents.
What there is in
it common to the
systems of New-
ton & Tleibniz.
How it differs from,
& surpasses, the
theory of Leibniz.
How it differs from,
& surpasses, the
theory of Newton.

36 . PHILOSOPHIÆ NATURALIS THEORIA
Ubi de ipsa ctum
ante ; & quid pro-
missum. -
Qua occasione hoc
de ipsa conscrip-
tum opus.
Erima, elementa in-
divisibilia inexten-
Sa, nec contigua.
infinitum, quo distantiæ in infinitum decrescant. Unde illud necessario consequitur, ut nec
cohæsio a contactu immediato oriatur, quam ego quidem longe aliunde desumo ; nec ullus
immediatus, &, ut illum appellare soleo, mathematicus materiæ contactus habeatur, quod
simplicitatem, & inextensionem inducit elementorum, quæ ipse variarum figurarum voluit,
& partibus a se invicem distinctis composita, quamvis ita cohærentia, ut nulla Naturæ vi
dissolvi possit compages, & adhæsio labefactari, quæ adhæsio ipsi, respectu virium nobis
cognitarum, est absolute infinita.
5. Quæ ad ejusmodi Theoriam pertinentia hucusque sunt edita, continentur disserta-
tionibus meis, De viribus vivis, edita Anno I745, De Lumine A. I748, De Lege Continuitatis
A. I754, De Lege virium in matura existentium A. I755, De divisibilitate materiæ, & principiis
corporum A. I757, ac in meis Supplementis Stayanæ Philosophiæ versibus traditæ, cujus primus
Tomus prodiit A. I755 : eadem autem satis dilucide proposuit, & amplissimum ipsius per
omnem Physicam demonstravit usum vir e nostra Societate doctissimus Carolus Benvenutus
in sua Physicae Generalis Synopsi edita Anno I754. In ea Synopsi proposuit idem & meam
deductionem æquilibrii binarum massarum, viribus parallelis animatarum, quæ ex ipsa mea
Theoria per notissimam legem compositionis virium, & æqualitatis inter actionem, & reac-
tionem, fere sponte consequitur, cujus quidem in supplementis illis § 4. ad lib. 3. mentionem
feci, ubi & quæ in dissertatione De centro Gravitatis edideram, paucis proposui ; & de centro
Oscillationis agens, protuli aliorum methodos præcipuas quasque, quæ ipsius determinationem
a subsidiariis tantummodo principiis quibusdam repetunt. Ibidem autem de æquilibrii
centro agens illud affirmavi : In Natura nullae sunt rigidae virgae, inflexiles, & omni gravitate,
ac inertia carentes, adeoque nec revera ullae leges pro iis conditae ; & si ad genuina, & simpli-
cissima maturae principia, res exigatur, invenietur, omnia pendere a compositione virium, quibus in
se invicem agunt particulae materiae ; a quibus mimirum viribus omnia Naturæ phænomena
proficiscuntur. Ibidem autem exhibitis aliorum methodis ad centrum oscillationis perti-
nentibus, promisi, me in quarto ejusdem Philosophiæ tomo ex genuinis principiis investiga-
turum, ut æquilibrii, sic itidem Oscillationis centrum.
6. Porro cum nuper occasio se mihi præbuisset inquirendi in ipsum oscillationis centrum
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 & æquilibrium, ac omne
vectium genus, & momentorum mensura pro machinis, & oscillationis centrum etiam pro
casu, quo Oscillatio fit in latus in plano 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 Theoriæ meæ specimen quoddam exponere ; sed paullatim
excrevit opusculum, ut demum & Theoriam omnem exposuerim ordine suo, & vindicarim,
& ad Mechanicam prius, tum ad Physicam fere universam applicaverim, ubi & quæ maxima
notatu digna erant, in memoratis dissertationibus ordine suo digessi omnia, & alia adjeci
quamplurima, quæ vel olim animo conceperam, vel modo sese obtulerunt scribenti, & omnem
hanc rerum farraginem animo pervolventi.
7. Prima elementa materiæ mihi sunt puncta prorsus indivisibilia, & inextensa, quæ in
immenso vacuo ita dispersa sunt, ut bina quævis 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 materiæ punctorum sit nulla,
idem prorsus spatii vulgo concepti 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
dissertations, De Viribus vivis, issued in 1745, De Lumine, 1748, De Lege Continuitatis,
1754, De Lege virium in natura existentium, 1755, De divisibilitate materia, & principiis
corporum, 1757, & in my Supplements to the philosophy of Benedictus Stay, issued in verse,
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 Physica
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
gentro 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 ;
& so, neither are there really any laws founded on them. If the matter is worked back to the
genuine & 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; & 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
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
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 obtruded
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-
extended points; they are so scattered in an immense vacuum that every two of them are
separated from one another by a definite interval; this interval can be indefinitely
increased or diminished, but can never vanish altogether without compenetration of the
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
When & where I
have already dealt
with this theory ;
& a promise that I
made.
The occasion that
led to my writing
this work on the
matter.
The primary ele-
ments are indivi-
sible, non-extended
& they are not
contiguous.
38 PHILOSOPHIÆ NATURALIS THEORIA
Eorum inertiæ vis
cujusmodi.
Eorundem v ir e s
mutuæ in aliis
distantiis a, tt r a c-
tivæ, in aliis re-
pulsivæ : v i ri u m
ejusmodi exempla.
Virium earundem
lex.
haberi veram, ac omnimodam conpenetrationem. Quamobrem non vacuum ego quidem
admitto disseminatum in materia, sed materiam in vacuo disseminatam, atque innatantem.
8. In hisce punctis admitto determinationem perseverandi in eodem statu quietis, vel
motus uniformis in directum (a) in quo semel sint posita, si seorsum singula in Natura
existant ; vel si alia alibi extant puncta, componendi per notam, & communem metho-
dum compositionis virium, & motuum, parallelogrammorum ope, præcedentem motum
cum mo-[5]-tu quem determinant vires mutuæ, quas inter bina quævis puncta agnosco
a distantiis pendentes, & iis mutatis mutatas, juxta generalem quandam omnibus com-
munem legem. In ea determinatione stat illa, quam dicimus, inertiæ vis, quæ, an a
libera pendeat Supremi Conditoris lege, an ab ipsa punctorum natura, an ab aliquo iis
adjecto, quodcunque, istud sit, ego quidem non quæro ; nec vero, si velim quærere, in-
veniendi spem habeo ; quod idem sane censeo de ea virium lege, ad quam gradumjam facio.
9. Censeo igitur bina quæcunque materiæ puncta determinari æque in aliis distantiis
ad mutuum accessum, in aliis ad recessum mutuum, quam ipsam determinationem appello
vim, in priore casu attractivam, in posteriore repulsivam, eo nomine non agendi modum, sed
ipsam determinationem exprimens, undecunque proveniat, cujus vero magnitudo mutatis
distantiis mutetur & ipsa secundum certam legem quandam, quæ per geometricam lineam
curvam, vel algebraicam formulam exponi possit, & oculis ipsis, uti moris est apud Mechanicos
repræsentari. Vis mutuæ a distantia pendentis, & ea variata itidem variatæ, atque ad omnes
in immensum & magnas, & parvas distantias pertinentis, habemus exemplum in ipsa
Newtoniana generali gravitate mutata in ratione reciproca duplicata distantiarum, quæ
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 mutuæ variatæ secundum
distantias, & a determinatione ad recessum migrantis in determinationem ad accessum, &
vice versa. Ibi enim si duæ 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, donec in quadam distantia
evanescat, & fiat prorsus nulla ; tum distantia adhuc aucta, incipit determinatio ad accessum,
quæ 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 intermediæ
laminæ plicatæ ; sed hic physicam rei causam non moror, & solum persequor exemplum
determinationis ad accessum, & recessum, quæ determinatio in aliis distantiis alium habeat
nisum, & migret etiam ab altera in alteram.
�a
Io. Lex autem virium est ejusmodi, ut in minimis distantiis sint repulsivæ, atque eo
majores in infinitum, quo distantiæ ipsæ minuuntur in infinitum, ita, ut pares sint extinguen-
[6]-dæ cuivis velocitati utcunque magnæ, 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 : tum adhuc, aucta distantia, mutentur in
attractivas, primo quidem crescentes, tum 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, donec, ubi ad aliquanto
majores distantias ventum sit, incipiant esse perpetuo attractivæ, & ad sensum reciproce
(a) Id quidem respectu ejus spatii, in quo continemur mos, & omnia quæ nostris observari sensibus possunt, corpora ;
quod quiddam spatium si quiescat, nihil ego in ea re a reliquis differo ; si forte moveatur motu quopiam, quem motum
ax hujusmodi determinatione sequi debeant ipsa materiae puncta ; tum haec mea erit quaedam non absoluta, sed respectiva
inertiae vis, quam ego quidem exposui & in dissertatione De Maris æstu & in Supplementis Stayanis Lib. I. § 13 ;
ubi etiam illud occurrit, quam ob causam ejusmodi respectivam inertiam excogitarim, &5 quibus rationihus evinci putem,
absolutam omnimo demonstrari non posse ; sed ea huc 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
same state of rest, or of uniform motion in a straight line, (a) in which they are initially
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
to approach one another at some distances, & in an equal degree recede from one another at
other distances. This determination I call “force’; in the first case “attractive’, in the
second case ‘repulsive ’; this term does not denote the mode of action, but the propen-
sity itself, whatever its origin, of which the magnitude changes as the distances change;
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. -
Io. Now the law of forces is of this kind; the forces are repulsive at very small dis-
tances, & become indefinitely greater & greater, as the distances are diminished indefinitely,
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 changed 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, which 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 P In that case this force of inertia that I postulate is not absolute, but relative 5 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 absolute. But these things do
not concern us at present.
The nature of the
force of inertia that
they possess.
The mutual forces
between them are
attractive at Some
distances & repul-
sive at others ; ex-
amples of forces of
this kind.
The law of forces
for the points.
40 PHILOSOPHIÆ NATURALIS THEORIA
Legis simplicitas
exprimibilis p e r
continuam curvam.
Forma, curvæ ips-
ius.
Abscissæ exprimen-
tes distantias, Or-
dinatæ exprimen-
tes vires.
Mutationes ordina-
tarum, & virium iis
expressarum.
proportionales quadratis distantiarum, atque id vel utcunque augeantur distantiæ etiam in
infinitum, vel saltem donec ad distantias deveniatur omnibus Planetarum, & Cometarum
distantiis longe majores. •
II. Hujusmodi lex primo aspectu videtur admodum complicata, & ex diversis legibus
temere inter se coagmentatis coalescens ; at simplicissima, & prorsus incomposita esse potest,
expressa videlicet per unicam continuam curvam, vel simplicem Algebraicam formulám, uti
innui superius. Hujusmodi curva linea est admodum apta ad sistendam oculis ipsis ejusmodi
legem, nec requirit Geometram, ut id præstare possit: satis est, ut quis eam intueatur
tantummodo, & in ipsa ut in imagine quadam solemus intueri depictas res qualescunque,
virium illarum indolem contempletur. In ejusmodi curva eæ, quas Geometræ abscissas
dicunt, & sunt segmenta axis, ad quem ipsa refertur curva, exprimunt distantias binorum
punctorum a se invicem : illæ vero, quæ dicuntur ordinatæ, ac sunt perpendiculares lineæ
ab axe ad curvam ductæ, referunt vires : quæ quidem, ubi ad alteram jacent axis partem,
exhibent vires attractivas ; ubi jacent ad alteram, repulsivas, & prout curva accedit ad axem,
vel recedit, minuuntur ipsæ 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 curvæ aliquis ad rectam quampiam axi perpendicularem
in infinitum productam semper magis accedit ita ultra quoscumque limites, ut nunquam in
eam recidat, quem arcum asymptoticum appellant Geometræ, ibi vires ipsæ in infinitum
eXCreSCunt.
I2. Ejusmodi curvam exhibui, & exposui in dissertationibus De viribus vivis a Num. 5 I,
De Lumine Num. 5, De Lege virium in Naturam existentium a Num. 68, & in sua Synopsi
Physicae Generalis P. Benvenutus eandem protulit a Num. Io8. En brevem quandem ejus
ideam. In Fig. I, Axis C'AC habet in puncto A asymptotum curvæ rectilineam AB
indefinitam, circa quam habentur bini curvæ rami hinc, & inde æquales, 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, donec ad ipsum deveniat alicubi in E ; tum 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, quem 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 TpjV, jacens ex parte opposita axis respectu primi cruris, quod alterum crus ipsum
habet axem pro asymptoto, & ad ipsum accedit ad sensum ita, ut distantiæ ab ipso sint in
ratione reciproca duplicata distantiarum a recta BA.
I 3. Si ex quovis axis puncto a, b, d, erigatur usque ad curvam recta ipsi perpendicularis
ag, br, dh, segmentum axis Aa, Ab, Ad, dicitur abscissa, & refert distantiam duorum materiæ
punctorum quorumcunque a se invicem ; perpendicularis ag, br, dh, dicitur Ordinata, &
exhibet vim repulsivam, vel attractivam, prout jacet respectu axis ad partes D, vel oppositas.
I4. Patet autem, in ea curvæ forma Ordinatam ag augeri ultra quoscunque limites, si
abscissa Aa, minuatur pariter ultra quoscunque limites ; quæ si augeatur, ut abeat in Ab,
ordinata minuetur, & abibit in br, perpetuo imminutam in accessu b ad E, ubi evanescet :
tum aucta abscissa in Ad, mutabit Ordinata directionem in dh, ac ex parte opposita augebitur
prius usque ad F, tum decrescet per il usque ad G, ubi evanescet, & iterum mutabit
directionem regressa in mm ad illam priorem, donec post evanescentiam, & directionis
mutationem factam in omnibus sectionibus I, L, N, P, R, fiant ordinatæ 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
41

42
PHILOSOPHIAE NATURALIS THEORIA
H
É

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.
II. A law of this kind will seem at first sight to be very complicated, & to be the result
of combining together several different laws in a haphazard sort of way; but it can be of
the simplest kind & not complicated in the slightest degree; it can be represented for
instance by a single continuous curve, or by an algebraical formula, as I intimated above.
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 Viribus vivis
(Art. 51), De Lumine (Art. 5), De lege virium in Natura existentium (Art. 68); and Father
Benvenutus published the same thing in his Synopsis Physica Generalis (Art. IOS). This
will give you some idea of its nature in a few words.
The simplicity of
the law can be re-
presented by means
of a continuous
Cllr Vē.
The form of the
CUITVe.
In Fig. I the axis CAC 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-
pendicular to it to meet the curve, such as ag, br, or dh then the segment of the axis, Aa,
Ab, or Ad, is called the abscissa, & represents the distance of any two points of matter from
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
beyond all bounds, if the abscissa Aa is in the same way diminished beyond all bounds; &
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 dh, & 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 A0, Av. Hence it is perfectly evident that, by a curve of this kind, we can
The abscissae re-
present distances,
& the Ordinates
forces.
Change in the or-
dinates & the forces
that they represent.
44 PHILOSOPHIÆ NATURALIS THEORIA
Discrimen hu us
legis v irium a
gravitate N.e w-
tOniana : ejus usus
in Physica : ordo
pertractandorum.
viribus, vel motibus componatur vis, vel
Occasio inveniendæ
Theoriæ ex consid-
eratione impulsus.
Origo ejusdem ex
oppositione impul-
sus immediati cum
lege Continuitatis.
Læsio legis Continu-
itatis necessaria, si
corpus velocius im-
mediate incurrat in
minus velox.
repulsivas, & imminutis in infinitum distantiis auctas in infinitum, auctis imminutas, tum
evanescentes, abeuntes, mutata directione, in attractivas, ac iterum evenescentes, mutatasque
per vices : donec demum in satis magna distantia evadant attractivæ ad sensum in ratióne
reciproca duplicata distantiarum.
I 5. Hæc virium lex a Newtoniana gravitate differt in ductu, & progressu curvæ eam
exprimentis quæ nimirum, ut in fig. 2, apud Newtonum est hyperbola DV gradus tertii,
jacens tota citra axem, quem nuspiam
secat, jacentibus omni-[8]-bus ordinatis Β
vj, op, bt, ag ex parte attractiva, ut
c. • i e � � s A.
idcirco nulla habeatur mutatio e positivo c A] a b O w . C
ad negativum, ex attractione in repulsi- Í -
onem, vel vice versa ; cæterum utraque
per ductum exponitur curvæ continuæ
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
motus quidam ope parallelogrammorum,
quorum latera exprimant vires, vel mo- FIG 2.
tus componentes, & quod vires ejusmodi •
in punctis singulis, tempusculis singulis æqualibus, inducant velocitates, vel motus proportion-
ales sibi, omnes mihi profluunt generales, & præcipuæ quæque particulares proprietates cor-
porum,uti etiam superius innui, nec ad singulares proprietates derivandas in genere affirmo, eas
haberi per diversam combinationem, sed combinationes ipsas evolvo, & geometrice demon-
stro, quæ e quibus combinationibus phænomena, & corporum species oriri debeant. Verum
antequam ea evolvo in parte secunda, & tertia, Ostendam in hac prima, qua via, & quibus
positivis rationibus ad eam virium legem devenerim, & qua ratione illam elementorum
materiæ simplicitatem eruerim, tum quæ difficultatem aliquam videantur habere posse,
dissolvam. - -
I6. Cum anno I745 De Viribus vivis dissertationem conscriberem, & omnia, quæ 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, quæ juxta communem omnium Mechanicorum sententiam
velocitates vel generant, vel utcunque inducunt proportionales sibi, & tempusculis, quibus
agunt, uti est gravitas, elasticitas, atque aliæ vires ejusmodi ; coepi aliquanto diligentius
inquirere in eam productionem velocitatis, quæ per impulsum censetur fieri, ubi tota
velocitas momento temporis produci creditur ab iis, qui idcirco percussionis vim infinities
majorem esse censent viribus omnibus, quæ pressionem solam momentis singulis exercent.
Statim illud mihi sese obtulit, alias pro percussionibus ejusmodi, quæ nimirum momento
temporis finitam velocitatem inducant, actionum leges haberi debere.
I7. Verum re altius considerata, mihi illud incidit, si recta utamur ratiocinandi methodo,
eum agendi modum submovendum esse a Natura, quæ nimirum eandem ubique virium
legem, ac eandem agendi rationem adhibeat : impulsum nimirum immediatum alterius
corporis in alterum, & immediatam percussionem haberi non posse sine illa productione
finitæ velocitatis facta momento temporis indivisibili, & hanc sine saltu quodam, & læsione
illius, quam legem Continuitatis appellant, quam quidem legem in Natura existere, & quidem
satis [9j valida ratione evinci posse existimabam. En autem ratiocinationem ipsam, qua
tum quidem primo sum usus, ac deinde novis aliis, atque aliis meditationibus illustravi, ac
confirmavi.
I8. Concipiantur duo corpora æqualia, quæ moveantur in directum versus eandem
plagam, & id, quod præcedit, habeat gradus velocitatis 6, id vero, quod ipsum persequitur
gradus I 2. Si hoc posterius cum sua illa velocitate illæsa 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 I2 ad 9, illo a 6 ad 9, sine ullo transitu per intermedios gradus II, & 7 ; Io, &
8 ; 93, & 83, &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. -
15. This law of forces differs from the law of gravitation enunciated by Newton in
the construction & development of the curve that represents it ; thus, the curve given in
Fig. 2, which is that according to Newton, is DV, a hyperbola of the third degree, lying
altogether on one side of the axis, which it does not cut at any point; all the ordinates,
such as vs, op, bt, ag lie on the side of the axis representing attractive forces, & there-
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
roduced 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. -
I6. In the year 1745, I was putting together my dissertation De Viribus vivis, & had
derived everything that they who adhere to the idea of Leibniz, & the greater number of
those who measure ‘living forces’ by means of velocity only, derive from these ‘living
forces’; as, I say I had derived everything directly & solely from the velocity generated by
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
we employ a straightforward method of argument, such a mode of action must be with-
drawn from Nature, which in every case adheres to one & the same law of forces, & the
same mode of action. I came to the conclusion that really immediate impulsive action of
one body on another, & immediate percussion, could not be obtained, without the pro-
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. -
18. Suppose there are two equal bodies, moving in the same straight line & in the
same direction; & let the one that is in front have a degree of velocity represented by
6, & the one behind a degree represented by 12. If the latter, i.e., the body that was be-
hind, should ever reach with its velocity undiminished, & come into absolute contact with,
the former body which was in front, then in every case it would be necessary that, at the
very instant of time at which this contact happened, the hindermost body should diminish
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, II & 7, Io & 8, 93 & 84, & so on, For it cannot possibly happen
Difference between
this law of forces
& Newton’s law of
gravitation ; it s
use in Physics ;
the Order in which
the subjects are to
be taken.
The occasion that
led to the discovery
of my Theory from
the consideration
of impulsive action.
The Call Se Of
the investigation
was the opposition
raised to the Law
of Continuity by
the idea of direct
impulse.
Violation of the
Law of Continuity,
if a body moving
more Swiftly comes
into actual con-
tact with another
body moving more
slowly.
46 PHILOSOPHIÆ NATURALIS THEORIA
Objectio petita a
negatione durorum
* corporum.
Ea uti non posse,
qui admittunt ele-
menta solida, &
dura.
Extensionem con-
tinuam requirere
primos poros, &
parietes solidos, ac
duros.
Læsio legis Con-
tinuitatis saltem
in primis superfi-
ciebus, vel punctis.
temporis particulam ejusmodi mutatio fiat per intermedios gradus, durante contactu. Si
enim aliquando alterum corpus jam habuit 7 gradus velocitatis, & alterum adhuc retinet
II ; toto illo tempusculo, quod effluxit ab initio contactus, quando velocitates erant I 2, & 6,
ad id tempus, quo sunt II, & 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
præcedens contactum, & subsequens, est indivisibilis limes, ut punctum apud Geometras
est limes indivisibilis inter duo continuæ lineæ segmenta, mutatio velocitatum facta fuerit
per saltum sine transitu per intermedias, læsa penitus illa continuitatis lege, quæ itum ab
una magnitudine ad aliam sine transitu per intermedias omnino vetat. Quod autem in
corporibus æqualibus diximus de transitu immediato utriusque ad 9 gradus velocitatis,
recurrit utique in iisdem, vel in utcunque inæqualibus 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.
I9. Sunt, qui difficultatem omnem submoveri posse censeant, dicendo, id quidem ita se
habere debere, si corpora dura habeantur, quæ nimirum nullam compressionem sentiant,
nullam mutationem figuræ ; & quoniam hæc a multis excluduntur penitus a Natura ; dum
se duo globi contingunt, introcessione, [Io] & compressione partium fieri posse, ut in ipsis
corporibus velocitas immutetur per omnes intermedios gradus transitu facto, & omnis
argumenti vis eludatur.
2o. At inprimis ea responsione uti non possunt, quicunque cum Newtono, & vero etiam
cum plerisque veterum Philosophorum prima elementa materiæ omnino dura admittunt, &
solida, cum adhæsione infinita, & impossibilitate absoluta mutationis figuræ. Nam in primis
elementis illis solidis, & duris, quæ in anteriore adsunt sequentis corporis parte, & in præce-
dentis posteriore, quæ nimirum se mutuo immediate contingunt, redit omnis argumenti vis
prorsus illæsa.
2 I. Deinde vero illud omnino intelligi sane non potest, quo pacto corpora omnia partes
aliquas postremas circa superficiem non habeant penitus solidas, quæ idcirco comprimi
omnino 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
inextricabiles ; 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 determinatæ crassitudinis paries usque ad
primum porum, poris utique carens ; vel quomodo, quod eodem recidit, nullus sit extimus,
& superficiei externæ omnium proximus porus, qui nimirum si sit aliquis, parietem habeat
utique poris expertem, & compressionis incapacem, in quo omnis argumenti superioris vis
redit prorsus illæsa.
22. At ea etiam, utcunque penitus inintelligibili, sententia admissa, redit omnis eadem
argumenti vis in ipsa prima, & ultima corporum se immediate contingentium superficie, vel
si nullæ continuæ superficies congruant, in lineis, vel punctis. Quidquid enim sit id, in quo
contactus fiat, debet utique esse aliquid, quod nimirum impenetrabilitati occasionem
præstet, & cogat motum in sequente corpore minui, in præcedente augeri ; id, quidquid est,
in quo exeritur impenetratibilitatis vis, quo fit immediatus contactus, id sane velocitatem
mutare debet per $altum, 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 II degrees; thus, during
the whole time that has passed since the beginning of contact, when the velocities were
respectively 12 & 6, until the time at which they are II & 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
saying that this is just as it should be, if hard bodies, such as indeed experience no com-
pression or alteration of shape, are dealt with ; whereas by many philosophers 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-
ton, & indeed many of the ancient philosophers as well, admit the primary elements of
matter to be absolutely hard & solid, possessing infinite adhesion & a definite shape that it
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
not have some of their last parts just near to the surface perfectly solid, & on that account
altogether incapable of being compressed. If matter is continuous, it may & must be sub-
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,
then the whole force of the same argument applies to the first or last surface of the bodies
that are in immediate contact with one another ; or, if there are no continuous surfaces
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
An objection de-
rived from denying
the existence of
hard bodies.
This reply cannot
be made by those
who admit solid &
hard elements.
Continuous exten-
Sion requires pri-
mary pores & walls
bounding them,
Solid & hard.
Violation of the
Law of Continuity
takes place, at any
rate, in prime sur-
faces or points.
48 PHILOSOPHIÆ NATURALIS THEORIA
Objectio petita a
v U c e m a S S a, &
?motus, quæ super-
ficiebus, & punctis
non conveniant.
Responsionis ini-
tium : superficiem,
lineam, punctum,
posita, extensione
continua, esse ali-
quid.
Quo pacto nomen
in a s s ae p O s S i t,
7motus debeat cOn-
venire superficie-
bus, lineis, punctis.
Motu m p a s sim
tribui p u n ct i s ;
fore, ut in eo læda-
t u r Continuitatis
lex. -
Fore, ut ea lædatur
saltem in velocitate
punctorum.
Objectio ex impene-
trabilitate admissa.
in minimis parti-
culis, & ejus confu-
tatio.
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 materiæ. Est nimirum realis affectio quædam corporis, videlicet
ejus limes ultimus realis, superficies, realis superficiei limes linea, realis lineæ limes punctum,
quæ affectiones utcunque in iis sententiis sint prorsus inseparabiles [II] ab ipso corpore,
sunt tamen non utique intellectu confictæ, sed reales, quæ 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 quædam, vel modi.
23. Est, qui dicat, nullum in iis committi saltum idcirco, quod censendum sit, nullum
habere motum, superficiem, lineam, punctum, quæ massam habeant nullam. Motus, inquit,
a Mechanicis habet pro mensura massam in velocitatem ductam ; massa autem est super-
ficies baseos ducta in crassitudinem, sive altitudinem, ex. gr. in prismatis. Quo minor est
ejusmodi crassitudo, eo minor est massa, & motus, ac ipsa crassitudine evanescente, evanescat
oportet & massa, & motus.
24. Verum qui sic ratiocinatur, inprimis ludit in ipsis vocibus. Massam vulgo appellant
quantitatem materiæ, & motum corporum metiuntur per massam ejusmodi, ac velocitatem.
At quemadmodum in ipsa geometrica quantitate tria genera sunt quantitatum, corpus, vel
solidum, quod trinam dimensionem habet, superficies quæ binas, linæ, quæ unicam, quibus
accedit lineæ limes punctum, omni dimensione, & extensione carens ; sic etiam in Physica
habetur in communi corpus tribus extensionis speciebus præditum ; superficies realis extimus
corporis limes, prædita binis ; linea, limes realis superficiei, habens unicam ; & ejusdem
lineæ indivisibilis limes punctum. Utrobique alterum alterius est limes, non pars, & quatuor
diversa genera constituunt. Superficies est nihil corporeum, sed non & nihil superficiale,
quin immo partes habet, & augeri potest, & minui ; & eodem pacto linea in ratione quidem
superficiei est nihil, sed aliquid in ratione lineæ ; ac ipsum demum punctum est aliquid in
suo genere, licet in ratione lineæ sit nihil. .
25. Hinc autem in iis ipsis massa quædam considerari potest duarum dimensionum, vel
unius, vel etiam nullius continuæ dimensionis, sed numeri punctorum tantummodo, uti
quantitas ejus genere designetur ; quod si pro iis etiam usurpetur nomen massæ generaliter,
motus quantitas definiri poterit per productum ex velocitate, & massa ; si vero massæ nomen
tribuendum sit soli corpori, tum motus quidem corporis mensura erit massa in velocitatem
ducta ; superficiei, lineæ, punctorum quotcunque motus pro mensura habebit quantitatem
superficiei, vel lineæ, vel numerum punctorum in velocitatem ducta ; sed motus utique iis
omnibus speciebus tribuendus erit, eruntque quatuor motuum genera, ut quatuor sunt
quantitatum, solidi, superficiei, lineæ, 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.
[12] 26. Et quidem ipsi Mechanici vulgo motum tribuunt & superficiebus & lineis, &
punctis, ac centri gravitatis motum ubique nominant Physici, quod centrum utique punctum
est aliquod, non corpus trina præditum dimensione, quam iste ad motus rationem, &
appellationem requirit, ludendo, ut ajebam, in verbis. Porro in ejusmodi motibus exti-
márum saltem 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, & massæ, si factum ex
velocitate, & massa, evanescente una e tribus dimensionibus, evanescit ; remanet utique
velocitas reliquarum dimensionum, quæ remanet, si eæ reapse remanent, uti quidem omnino
remanent in superficie, & ejus velocitatis mutatio haberi deberet per saltum, ac in ea violari
continuitatis lex jam toties memorata.
28. Hæc quidem ita evidentia sunt, ut omnino dubitari non possit, quin continuitatis
lex infringi debeat, & saltus in Naturam induci, ubi cum velocitatis discrimine ad se invicem
accedant corpora, & ad immediatum contactum deveniant, si modo impenetrabilitas
corporibus tribuenda sit, uti revera est. Eam 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 argumenti
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-
sidered that a surface, a line or a point, having no mass, cannot have any motion. He may
say that motion has, according to Mechanicians, as its measure, the mass multiplied by the
velocity; also mass is the surface of the base multiplied by the thickness or the altitude,
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.
Mass is commonly called quantity of matter, & the motion of bodies is measured by mass
of this kind & the velocity. But, just as in a geometrical quantity there are three kinds of
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,
& of no extension. So also in Physics, a body is considered to be endowed with three
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
of one, or even of no continuous dimension, but only numbers of points, just as quantity of
this kind is indicated. Now, if for these also, the term mass is employed in a generalized
sense, we shall be able to define the quantity of motion by the product of the velocity &
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 &
points, & Physicists universally speak of the motion of the centre of gravity; this centre is
undoubtedly some point, & not a body endowed with three dimensions, which the objector
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
the velocity & the mass vanishes when one of the three dimensions vanish, there will still
remain the velocity of the remaining dimensions; & this will persist so long as the dimen-
sions persist, as they do persist undoubtedly in the case of a surface. Hence the change
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
of Continuity is infringed, & that a sudden change is introduced into Nature, when bodies
approach one another with a difference of velocity & come into immediate contact, if only
we are to ascribe impenetrability to bodies, as we really should. And this property too,
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
E
Objection derived
from the terms mass
and motion, which
do not accord with
surfaces & points.
Commencement of
the answer to this ;
a Surface, or a line,
Or a point, is some-
thing real, if con-
tinuous extension
is supposed to ex-
ist,
The manner in
which the term
%ass may, and the
term motus is bound
to, apply to surfaces,
lines, & points.
Motion is ascribed
to points indiscri-
minately ; the Law
of Continuity is vio-
lated by doing so.
It is at least a fact
that this law is vio-
lated by the idea
of the velocity of
points.
Objection derived
from the admission
of impenetrability
in very Small par-
ticles, & its refuta-
tion.
5o PHILOSOPHIÆ NATURALIS THEORIA
Objectio a v o c e
motus assum p ta
pro mutatione ;
c o n futatio ex
realitate m o t u s
localis.
Q u i Continuitatis,
legem summover-
int.
Theoriæ e x ortu s,
ea lege, uti fieri
debet, retenta.
Lex Continuitatis
quid sit : discri-
men inter status,
& incrementa.
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, & analogiæ, quam unam habemus
in Physica investigandis generalibus naturæ legibus idoneam, cujus inductionis vis quæ sit,
& quibus in locis usum habeat, quorum locorum unus est hic ipse impenetrabilitatis ad
minimas quasque particulas extendendæ, inferius exponam.
29. Fuit itidem e Leibnitianorum familia, qui post evulgatam Theoriam meam cen-
suerit, difficultatem ejusmodi amoveri posse dicendo, duas monades sibi etiam invicem
occurrentes cum velocitatibus quibuscunque oppositis æqualibus, post ipsum contactum
pergere moveri sine locali progressione. Eam progressionem, ajebat, revera omnino nihil
esse, si a spatio percurso æstimetur, cum spatium sit nihil ; motum utique perseverare, &
extingui per gradus, quia per gradus extinguatur energia illa, qua in se mutuo agunt, sese
premendo invicem. Is itidem ludit in voce motus, quam adhibet pro mutatione quacunque,
& actione, vel actionis modo. Motus localis, & velocitas motus ipsius, sunt ea, quæ ego
quidem adhibeo, & guæ ibi abrumpuntur per saltum. Ea, ut evidentissime constat, erant
aliqua ante contactti & post contactum mo-[I3]-mento temporis in eo casu abrumpuntur ;
nec vero sunt nihil*ffcet 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, vellentius ; est aliquid non imaginarie tantummodo, sed
realiter diversum ; in eo vero per immediatum contactum saltus utique induceretur in eô
casu, quo ego superius sum usus.
3o. Et sane summus nostri ævi Geometra, & Philosophus Mac-Laurinus, cum etiam ipse
collisionem 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 fostris 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, & immediato corporum
contactu ille ne dubitari quidem posse arbitrabatur, (nec 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.
3 I. Ast ego cum ipsam continuitatis legem aliquanto diligentius considerarim, &
fundamenta, 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, quæ ex ipsa continuitate
servata sponte profluebant, directa ratiocinatione delatus sum ad eam, quam superius
exposui, virium mutuarum legem, quæ consectaria suo quæque ordine proferam, ubi ipsa,
quæ ad continuitatis legem retinendam argumenta me movent, attigero.
32. Continuitatis lex, de qua hic agimus, in eo sita est, uti superius innui, ut quævis
quantitas, dum ab una magnitudine ad aliam migrat, debeat transire per omnes intermedias
ejusdem generis magnitudines. Solet etiam idem exprimi nominandi transitum per gradus
intermedios, quos quidem gradus Maupertuisius ita accepit, quasi vero quædam exiguæ
accessiones fierent momento temporis, in quo quidem is censuit violari jam necessario legem
ipsam, quæ utcunque exiguo saltu utique violatur nihilo minus, quam maximo ; cum
nimi-[14]-rum magnum, & parvum sint tantummodo respectiva ; & jure quidem id censuit ;
si nomine graduum incrementa magnitudinis cujuscunque momentanea intelligerentur.



A THEORY OF NATURAL PHILOSOPHY 5 I
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
Theory, expressed his opinion that this kind of difficulty could be removed by saying that
two monads colliding with one another with any velocities that were equal & opposite
would, after they came into contact, go on moving without any local progression. He
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
had considered the collision of solid bodies & observed that there is nothing which could
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, &
pondering over the fundamental ideas on which it depends, came to the conclusion that
it certainly could not be withdrawn altogether out of Nature. Hence, since it had to be
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
intimated above, any quantity, in passing from One magnitude to another, must pass through
all intermediate magnitudes of the same class. The same notion is also commonly expressed
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
Objection to the
term motus being
used for a change ;
refutation from the
reality of local mo-
tion.
There are some who
would deny the
Law of Continuity.
The origin of my
Theory, retaining
this Law, as should
be done.
The nature of the
Law of Continuity;
distinction between
States & incre-
ments.
52 PHILOSOPHIAE NATURALIS THEORIA
Geometriae usus ad
eam exponendam :
momenta punctis,
tempora continua
lineis expressa.
Fluxus ordinata
transeuntis per
m a gn itu di nes
omnes intermedias.
Idem in quantitate
variabili expressa :
aequiv o ca ti o in
voce gradus.
Status singulos
momentis, in cre-
menta vero utoun-
que p a rv a te m-
puSculis continuis
respondere.
Transitus sine sal-
tu, etiam a positivis
ad negativa per ni-
hilum, quod tamen
non est vere nihil-
mu, sed quidam
realis status,
mentis, quae sunt indivisibiles earum partium
Verum idita 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 quadam alia linea CDE. Exprimat prior ex iis tempus, uti solet
utique in ipsis horologiis circularis peripheria
ab indicis cuspide denotata tempus definire. - L N l A'
Quemadmodum in Geometria in lineis G º pre- SL
puncta sunt indivisibiles limites continuarum -
linee partium, non vero partes linea ipsius;
ita in tempore distinguenda erunt partes
continui temporis respondentes ipsis linea
partibus, continua itidem & ipsae, a mo-
O
N
A
e - º
limites, & punctis respondent; nec inpos-
terum alio sensu agens de tempore momenti
nomen adhibebo, quam eo indivisibilis
limitis; particulam vero temporis utcunque
exiguam, & habitam etiam pro infinitesima, E
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 eius quantitatis magnitudo
FG, HI; momentis autem intermediis aliis K, M, aliaemagnitudines, KL, MN, respondebunt;
ac si a puncto G ad I continua, & finita abeat pars linea CDE, facile patet & accurate de-
monstrari potest, utcunque eadem contorqueatur, nullum fore punctum K intermedium,
cui aliqua ordinata KL non respondeat; & e converso nullam fore ordinatam magnitu-
dinis intermedia inter FG, HI, quae alicui puncto inter F, H intermedio non respondeat.
FIG. 3.
35. Quantitas illa variabilis per hano 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 interpuncta L, N arcus ipse LN non mutet recessum a recta AB
in accessum ; ducta LO ipsi parallela, habebitur quantitas NO, quae in schemate exhibito
est incrementum magnitudinis ejus quantitatis continuo variata. Quo minor est ibi
temporis particula KM, eo minus est id incrementum NO, &illa 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-II51-telligi illud
incrementum NO, quanquam aliquando etiam ille status, illa 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.
36. Sed omissis aequivocationibus ipsis, illud, quod ad rem facit, est accessio incremen-
torum facta non momento temporis, sed tempusculo continuo, quod est particula continui
temporis. Utcunque exiguum sit incrementum ON, ipsi semper respondet tempusculum
quoddam KM continuum. Nullum est in linea punctum M ita proximum puncto K, ut sit
primum post ipsum ; sed vel congruunt, vel intercipiuntlineolam 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 interjacet 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 linea ipsam exprimentis, &
quacunque utcunque exigua accessione, inveniri potest tempusculum continuum, quo ea
accessio advenerit.
37. Atque sic quidem intelligitur, quo pacto fieri possit transitus per intermedias
magnitudines omnes, per intermedios status, per gradus intermedios, quin ullus habeatur
saltus utcunque exiguus momento temporis factus. Notari illud potest tantummodo,
mutationem fieri alicubi per incrementa, ut ubi KL abit, in MN per NO; alicubi per
decrementa, ut ubi K'L' abeat in N'M' per O'N'; quin immo si linea CDE, quae 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.
Let AB be any straight line (Fig. 3), to which as axis let any other line CDE be referred.
Let the first of them represent the time, in the same manner as it is customary to specify
the time in the case of circular clocks by marking off the periphery with the end of a pointer.
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
to it ordinates FG, HI, to meet the line CD ; any of these ordinates can be taken to repre-
sent a quantity that is continuously varying. To any instant of time F, or H, there will
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
accordance with the Law of Continuity; for, from the magnitude FG, which it has at
the instant of time F, to the magnitude HI, which corresponds to the instant H, it passes
through all intermediate magnitudes KL, MN, which correspond to the intermediate
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
increments is accomplished, not in an instant of time, but in a small interval of con-
tinuous time, which is a part of continuous time. However small the increment ON may
be, there always corresponds to it some continuous interval KM. There is no point M
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
through all intermediate magnitudes, through intermediate states, or through intermediate
stages, without any sudden change being made, no matter how small, in an instant of time.
It can merely be remarked that change in some places takes place by increments (as when
KL becomes MN by the addition of NO), in other places by decrements (as when K'L'
Explanation by the
use of geometry;
instants represen-
ted by points, con-
tinuous intervals of
time by lines.
The flux of the Or-
dinate as it passes
through all inter-
mediate values.
The same holds
good for the vari-
able quantity so
represented ; equi-
vocation in the use
of the term Step.
Single states cor-
respond to instants,
but
however small to
increments
intervals of con-
tinuous time.
Passages without
Sudden change,
from positive to
negative through
zero ; zero how-
ever is not really
nothing, but a cer-
tain real state.
54. PHILOSOPHIÆ NATURALIS THEORIA
Proponitu r p r o-
banda existentia,
legis Continuitatis.
Ejus probatio ab
inductione sat i s
ampla.
variationis exhibit, alicubi secet rectam, temporis AB, potest ibidem evanescere magnitudo,
ut ordinata M'N', puncto M' allapso ad D evanesceret, & deinde mutari in negativam PQ,
RS, habentem videlicet directionem contrariam, quæ, quo magis ex oppositæ parte crescit,
eo minor censetur in ratione priore, quemadmodüm in ratione possessionis, vél divitiarum,
pergit perpetuo se habere pejus, qui iis omnibus, quæ habebat, absumptis, æs alienum
contrahit perpetuo majus. Et in Geometria quidem habetur a positivo ad negativa
transitus, uti etiam in Algebraicis formulis, tam transeundo per nihilum, quam per infinitum,
quos ego transitus persecutus sum partim in dissertatione adjecta meis Sectionibus Comicis,
partim in Algebra § 14, & utrumque simul in dissertatione De Lege Continuitatis ; sed in
Physica, ubi nulla quantitas in infinitum excrescit, is casus locum non habet, & non, nisi
transeundo per nihilum, transitus fit a positi-[16]-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 consideratione quadam tantummodo, in qua negativa etiam, qui sunt
veri status, in se positivi, ut ut ad priorem seriem pertinentes negativo quodam modo,
negativa appellentur.
38. Exposita hoc pacto, & vindicata continuitatis lege, eam in Natura existere plerique
Philosophi arbitrantur, contradicentibus nonnullis, uti supra innui. Ego, cum in eam
primo inquirerem, censui, eandem omitti omnino non posse ; si eam, quam habemus unicam,
Naturæ analogiam, & inductionis vim consulamus, ope cujus inductionis eam demonstrare
conatus sum in pluribus e memoratis dissertationibus, ac eandem probationem adhibet
Benvenutus in sua Synopsi Num. II9 ; in quibus etiam locis, prout diversis occasionibus
conscripta sunt, repetuntur non nulla. - - -
39. LOngum hic esset singula inde excerpere in ordinem redacta : satis erit exscribere
dissertationis De lege Continuitatis numerum I38. Post inductionem petitam præcedente
numero a Geometria, quæ nullum uspiam habet saltum, atque a motu locali, in quo nunquam
ab uno loco ad alium devenitur, nisi ductu continuo aliquo, unde consequitur illud, dis-
tantiam a dato loco nunquam mutari in aliam, neque densitatem, quæ utique a distantiis
' pendet particularum in aliam, nisi transeundo per intermedias ; fit gradus in eo numero ad
motuum velocitates, & ductus, quæ magis hic ad rem faciunt, nimirum ubi de velocitate
agimus non mutanda per saltum in corporum collisionibus. Sic autem habetur : “ Quin
immo in motibus ipsis continuitas seryatur etiam in eo, quod motus omnes in lineis cQntinuis
fiunt nusquam abruptis. Plurimos ejusmodi motus videmus. Planetæ, & cometæ 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
gontinuo mqtu 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 obveniens, qui quidem motus in hypothesi Telluris motæ a
parabolicis plurimum distant, & curvam continuam exhibent etiam pro casu projectionis
accurate verticalis, quo, quiescente penitus Tellure, & nulla ventorum yi. deflectente motum,
haberetur [I 7] ascensus rectilineus, vel descensus. Immo omnes alii motus a gravitate
pendentes, omnes ab elasticitate, a vi magnetica, continuitatem itidem servant; cum eam
servent vires illæ ipsæ, quibus gignuntur. Nam gravitas, cum decrescat in ratione reciproca
duplicata distantiarum, & distantiæ 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, tam in lineis quæ describuntur, quam in velocitatibus, quæ 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 Comic 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-
phers to exist in Nature, though there are some who deny it, as I mentioned above. I,
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. I 19). 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
these essays ; it will be sufficient if I give Art. I 38 of the dissertation De Lege Continuitatis.
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
I propose to prove
the existence of the
Law of Continuity.
Proof by induction
Sufficient for the
purpose.
56 PHILOSOPHIÆ NATURALIS THEORIA
Duplex inductionis
genus, ubi & cur
vim habeat inductio
incompleta.
& in aliis mille ejusmodi, in quibus mutationes velocitatis fiunt gradatim, nec retro cursus
reflectitur, nisi imminuta velocitate per omnes gradus. Ea diligentissime continuitatem
servat omnia. Hinc nec ulli in naturalibus motibus habentur anguli, sed semper mutatio
'directionis fit paullatim, nec vero anguli exacti habentur in corporibus ipsis, in quibus
utcunque videatur tenuis acies, vel cuspis, microscopii saltem ope videri solet curvatura,
quam etiam habent alvei fluviorum semper, habent arborum folia, & frondes, ac rami, habent
lapides quicunque, nisi forte alicubi cuspides continuæ 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
continuitas non servetur, qui omnino exhiberi non poterit.”
40. Inductio amplissima tum ex hisce motibus, ac velocitatibus, tum ex aliis pluribus
exemplis, quæ habemus in Natura, in quibus ea ubique, quantum observando licet depre-
hendere, continuitatem vel observat accurate, vel affectat, 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 hic inserere partem numeri I 34, & totum I 35, disserta-
tionis De Lege Continuitatis. Sic autem habent ibidem : “ Inprimis ubi generales Naturæ
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-[18]-ræ legibus stabiliendis locum habere non potest. Habet locum laxior
quædam inductio, quæ, 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 quæ prima
fronte contraria videantur, re accuratius perspecta, cum illa lege possint omnia conciliari ;
licet, an eo potissimum pacto concilientur, immediate innotescere, nequaquam possit. Si
eæ conditiones habeantur ; inductio ad legem stabiliendam censeri debet idonea. Sic quia
videmus corpora tam multa, quæ habemus præ manibus, aliis corporibus resistere, ne in
eorum locum adveniant, & loco cedere, si resistendo sint imparia, potius, quam eodem
perstare simul ; impenetrabilitatem corporum admittimus ; nec obest, quod quædam
corpora videamus intra alia, licet durissima, insinuari, ut oleum in marmora, lumen in
crystalla, & gemmas. Videmus enim hoc phænomenum facile conciliari cum ipsa impene-
trabilitate, dicendo, per vacuos corporum poros ea corpora permeare. (Num. I 35).
Præterea, quæcunque proprietates absolutæ, nimirum quæ 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, quæ pendeant a ratione totius, seu multitudinis, contradistincta a ratione partis.
Primum evincitur ex eo, quod magna, & parva sunt respectiva, ac insensibilia dicuntur ea,
quæ respectu nostræ molis, & nostrorum sensuum sunt exigua. Quare ubi agitur de
proprietatibus absolutis non respectivis, quæcunque communia videmus in iis, quæ 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 analogiæ
læsio, poterat illa 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, quæ 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,
as well as from a large number of other examples, such as we have in Nature, where Nature
in every case, as far as can be gathered from direct observation, maintains continuity or
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. I 34 & the whole of Art. I35 from my dissertation De Lege
Continuitatis. The passage runs thus: “Especially when we investigate the general laws
Induction of a two-
fold kind ; when
& why incomplete
induction has vali-
dity.
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.
I35). 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
58 PHILOSOPHIÆ NATURALIS THEORIA
Et impenetrabili-
tatem, & contin-
uitatem evinci per
inductionem : ad
ipsam quid requira-
tur.
Eju8 applicatio ad
impenetrabilitatem.
Similis ad continu-
itatem : duo cas-
uum genera, in
quibus ea, videatur
lædi.
quasdam prudentes regulas fiat, successum habere solet. Cum id indicium fallere possit ;
fieri potest, ut committatur error, sed contra ipsum errorem habebitur præsumptio, ut
etiam in jure appellant, donec 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 [I 9] eo, quod habeatur in majoribus massis, non debemus inferre,
eam 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 materiæ, 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 ; nec ea, ob rationem nimirum eandem, a toto, vel aggregato debet
transferri ad partes. Est de ratione totius, ut partes habeat, nec totum sine partibus haberi
potest. Est de ratione figurabilis, & extensi, ut habeat aliquid, quod ab alio distet, adeoque,
ut habeat partes ; hinc eæ proprietates, licet in quovis aggregato particularum materiæ,
sive in quavis sensibili massa, inveniantur, non debent inductionis vi transferri ad particulas
quascunque.”
4I. Ex his patet, & impenetrabilitatem, & continuitatis legem per ejusmodi inductionis
genus abunde probari, atque evinci, & illam quidem ad quascunque utcunque exiguas
particulas corporum, hanc ad gradus utcunque exiguos momento temporis adjectos debere
extendi. Requiritur autem ad hujusmodi inductionem primo, ut illa proprietas, ad quam
probandam ea adhibetur, in plurimis casibus observetur, aliter enim probabilitas esset exigua ;
& ut nullus sit casus observatus, in quo evinci possit, eam violari. Non est necessarium illud,
ut in iis casibus, in quibus primo aspectu timeri possit defectus proprietatis ipsius, positive
demonstretur, eam 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.
42. Id ipsum fit, ubi per inductionem impenetrabilitas corporum accipitur pro generali
lege Naturæ. Nam impenetrabilitatem ipsam magnorum corporum observamus in exemplis
sane innumeris tot corporum, quæ pertractamus. Habentur quidem & casus, in quibus eam
violari quis crederit, ut ubi oleum per ligna, & marmora penetrat, atque insinuatur, & ubi
lux per vitra, & gemmas traducitur. At præsto est conciliatio phænomeni cum impenetra-
bilitate, petita ab eo, quod illa corpora, in quæ se ejusmodi substantiæ insinuant, poros
habeant, quos eæ permeent. Et quidem hæc conciliatio exemplum habet manifestissimum
in spongia, quæ 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 reliquæ inductionis vi, ut dicere debeamus, eo
potissimum pacto se rem habere, & ne ibi quidem violari generalem utique impenetrabilitatis
legem. *
[2o] 43. Eodem igitur pacto in lege ipsa continuitatis agendum est. Illa tam ampla
inductio, quam habemus, debet nos movere ad illam generaliter admittendam etiam pro iis
casibus, in quibus determinare immediate per observationes non possumus, an eadem
habeatur, uti est collisio corporum ; ac si sunt casus nonnulli, in quibus eadem prima fronte
violari videatur ; ineunda est ratio aliqua, qua ipsum phænomenum 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 eam ut exemplum quoddam ad confirmationem quandam inductionis
majoris adhibui. Posterior, ut sæpe & illa 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
of Continuity can be proved by a kind of induction of this type; & the former must be
extended to all particles of bodies, no matter how small, & the latter to all additional steps,
however small, made in an instant of time. Now, in the first place, to use this kind of
induction, it is required that the property, for the proof of which it is to be used, must be
observed in a very large number of cases; for otherwise the probability would be very
small.
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
as a law of Nature through inductive reasoning. For we observe this impenetrability of
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
induction that we possess should lead us to admit in general this law even in those cases in
which it is impossible for us to determine directly by observation whether the same law
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.
Also it is required that no case should be observed, in which it can be proved that
Both impenetra-
bility & continuity
can be de m on-
strated by induc-
tion ; what is re-
quired for this pur-
pose.
Application of in-
duction to impene-
trability.
Similar application
to continuity; two
classes of cases in
which there seems
to be violation.
6o PHILOSOPHIAE NATURALIS THEORIA
Exemplum geome-
tricum primi gene-
ris, ubi nos inter-
medias magnitu-
dines omittimus.
Quando id accidat
exempla physica,
dierum, & Oscilla-
tionum consequen-
tium.
Exempla secundi
generis, ubi mutatio
sit celerrime, sed
nOn momento tem-
poris.
Applicatio ipsorum
ad alia, nominatim
ad effluxum aquæ
e Va,S€.
44. In axe curvæ cujusdam in fig. 4. sumantur segmenta AC, CE, EG æqualia, &
erigantur ordinatæ AB, CD, EF, GH. Areæ BACD, DCEF, FEGH videntur continuæ
cujusdam seriei termini ita, ut ab illa BACD ad DCEF, & inde ad FEGH immediate
transeatur, & tamen secunda a prima, ut Æ^
& tertia a secunda, differunt per quanti- H
tates finitas : si enim capiantur CI, EK
æquales BA, DC, & arcus BD transferatur
in IK ; area DIKF erit incrementum se-
cundæ supra primam, quod videtur imme-
diate advenire totum absque eo, quod
unquam habitum sit ejus dimidium, vel
quævis alia pars incrementi ipsius ; ut idcirco
a prima ad secundam magnitudinem areæ
itum sit sine transitu per intermedias. At
ibi omittuntur a nobis termini intermedii,
qui continuitatem servant ; si enim ac æqualis
AC motu continuo feratur ita, ut incipiendo
ab AC desinat in CE ; magnitudo areæ BACD per omnes intermedias bacd abit in magnitu-
dinem DCEF sine ullo saltu, & sine ulla violatione continuitatis. -
45. Id sane ubique accidit, ubi initium secundæ magnitudinis aliquo intervallo distat
ab initio primæ ; sive statim veniat post ejus finem, sive quavis alia lege ab ea disjungatur.
Sic in physicis, si diem concipiamus intervallum temporis ab occasu ad occasum, vel etiam
ab Ortu ad occasum, dies præcedens a sequenti 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, quæ eandem latitudinem geographicam
habent ; ea singula loca suam habent durationem diei, & omnium ejusmodi dierum initia,
ac fines continenter fluunt ; donec ad eundem redeatur locum, cujus præ-[21]-cedens dies
est in continua illa 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 aere ; sequens Oscillatio per finitam magnitudinem distat a præcedente ; sed &
initium & finis ejus finito intervallo temporis distat a præcedentis initio, & fine, ac intermedii
termini continua serie fluente a prima oscillatione ad secundam essent ii, qui haberentur, si
primæ, & secundæ Oscillationis arcu in æqualem partium numerum diviso, assumeretur via
confecta, vel tempus in ea impensum, interjacens 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.

F
i
!
|
Í
C C E. G
FIG. 4.
46. Secunda classis casuum est ea, in qua videtur aliquid momento temporis peragi,
& tamen peragitur tempore successivo, sed perbrevi. Sunt, qui objiciant pro violatione
continuitatis casum, quo quisquam manu lapidem tenens, ipsi statim det velocitatem
quandam finitam : alius objicit aquæ e vase effluentis, foramine constituto aliquanto infra
superficiem ipsius aquæ, 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 globo, qui ab
elastro sibi relicto propellatur : quo elasticitas est major, eo citius, sed nunquam momento
temporis velocitas in globum inducitur. -
47. Hæc exempla illud præstant, quod aqua per poros spongiæ ingressa respectu
impenetrabilitatis, ut ea responsione uti possimus in aliis casibus omnibus, in quibus accessio
aliqua magnitudinis videtur fieri tota momento temporis ; ut nimirum dicamus fieri tempore
A THEORY OF NATURAL PHILOSOPHY 6I
44. In the axis of any curve (Fig. 4) let there be taken the segments AC, CE, EG equal
to one another ; & let the ordinates AB, CD, EF, GH be erected. The areas BACD, DCEF,
FEGH seem to be terms of some continuous series such that we can pass directly from BACD
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
by a definite interval from the beginning of the first ; whether it comes immediately after
the end of the first or is disconnected from it by some other law. Thus in physics, if we
look upon the day as the interval of time between sunset & sunset, or even between sunrise
& 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
instant of time, but still it is really done in a continuous, but very short, interval of time.
There are some who bring forward, as an objection in favour of a breach of continuity, the
case in which a man, holding a stone in his hand, gives to it a definite velocity all at once;
another raises an objection that favours a breach of continuity, in the case of water flowing
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,
which we employed in the matter of impenetrability; so that we can make use of this reply
in all other cases in which some addition to a magnitude seems to have taken place entirely in
an instant of time. Thus, without doubt we may say that it takes place in an exceedingly
Geometric al ex-
ample of the first
kind, where we
omit intermediate
magnitudes.
When this will
happen : physical
examples in the
case of consecutive
days, or consecutive
oscillations.
Examples of the
Second class, in
which the change
is very rapid, but
does not take place
in an instant of
time.
Application of
the se to other
cases; particularly
to the flow of water
from a vessel,
62 PHILOSOPHIÆ NATURALIS THEORIA
Transitus ad meta-
physicam probati-
onem : limes in
continuis unicus,
ut in Geometria.
Idem in tempore
& in quavis serie
continua : eviden-
tius in quibusdam.
Inde cur motus lo-
calis non fiat, nisi
per lineam contin-
ll 2lIm.
brevissimo, utique per omnes intermedias magnitudines, ac illæsa penitus lege continuitatis.
Hinc & in aquæ effluentis exemplo res eodem redit, ut non unico momento, sed successivo
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 quærimus, 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 illa velocitas produceretur ; illa enim
non a percussione aliqua, sed a pressione superincumbentis aquæ 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.
48. Illæsa igitur esse debet continuitatis lex, nec ad eam evertendam contra inductionem,
tam uberem quidquam poterunt casus allati hucusque, vel iis similes. At ejusdem con-
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 præcedentia 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 continuæ partes latitudine caret ; punctum continuæ
lineæ segmenta discriminans, dimensione omni : nec 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.
49. Eodem autem pacto idem debet accidere etiam in tempore, ut nimirum inter tempus
continuum præcedens, & continuo subsequens unicum habeatur momentum, quod sit
indivisibilis terminus utriusque ; nec 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,
quæ continuo tempore duret, haberi debet series quædam magnitudinum ejusmodi, ut
momento temporis cuivis respondeat sua, quæ præcedentem cum consequente conjungat,
& ab illa per aliquam determinatam magnitudinem differat. Quin immo in illo quantitatum
genere, in quo [23] binæ 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 duæ magnitudines, postrema seriei præcedentis, & 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.
5o. Id quidem satis patebit in ipso locali motu, in quo habetur phænomenum omnibus
sane notissimum, sed cujus ratio non ita facile aliunde 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, quæ numero
infinities infinitæ 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 lineæ, 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 hic ea accipio, contigua esse non possint, uti superius exposui. 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 cannot 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
hitherto brought forward, nor others like them, have any power at all to controvert this
law in opposition to induction so copious. Moreover I discovered another argument, a
metaphysical one, in favour of this continuity, & published it in my dissertation De Lege
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
a preceding continuous time & the next following there should be a single instant, which
is the indivisible boundary of either. There cannot be two instants, as we intimated above,
contiguous to one another; but between one instant & another there must always intervene
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
the phenomenon is perfectly well known to all; the reason for it, however, is not so easily
derived from any other source, whilst it follows most clearly from this idea. A body can
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
Passing to a meta-
physical proof, we
have a single limit
in the case of con-
tinuous things, as in
geometry.
Similarly for time
& any continuous
Series ; more evi-
dent in some than
in others.
Hence the reason
why local motion
only occurs in a
continuous line.
64 PHILOSOPHIÆ NATURALIS THEORIA
primo casu in omnibus iis infinitis intermediis momentis nullibi esset id corpus, in secundo
casu idem esset eodem illo momento in binis locis, adeoque replicaretur ; interio haberetur
replicatio non tantum respectu eorum binorum moméntorum, sed omnium etiam inter-
mediorum, in quibus nimirum omnibus id corpus esset in binis locis. Cum igitur corpus
Illustratio ejus
argumenti ex Geo-
metria : ratiocina-
tione metaphysica,
pluribus exemplis.
Objectio ab esse, &
f^on esse conjun-
gendis in creatione,
& annihilatione, ac
ejus solutio.
existens nec nullibi esse possit, nec simul in locis pluribus ; illa viæ mutatio, & ille saftus
haberi omnino non possunt.
5I. Idem ope Geometriæ magis adhuc oculis ipsis subjicitur. Exponantur per rectam
AB tempora, ac per ordinatas adlineas CD, EF, abrìptas alicubi, diversistatus rei cujuspiam.
Ductis ordinatis DG, EH, vel punctum H jaceret post G, ut in Fig. 5 ; vel cùm ipso
congrueret, ut in 6; yel ipsum præcederet, ut in 7. In primo casu nulla respondéret
ordinata omnibus punctis rectæ GH; in secundo binæ responderent GD, & HE eidem puncto
G ; in tertio vero binæ HI, & HE puncto H, binæ GD, GK puncto G, & binæ LM, LN
È
D L•^
Κ F
Ð
!
Σ
C. - C.
ATaT-E R —äg—5 A—g-t-5
FIG. 5. FIG. 6. FIG. 7.
puncto cuivis intermedio L ; nam ordinata est relatio quædam distantiæ, quam habet
punctum curvæ cum puncto axis sibi respondente, adeoque ubi jacent in recta eadem
perpendiculari axi bina curvarum puncta, habentur binæ ordinatæ respondentes eidem
puncto axis. Quamobrem si nec o-[24]-mni statu carere res possit, nec haberi possint
status simul bini ; necessario consequitur, saltum illum committi non posse. Saltus ipse, si
deberet accidere, uti vulgo fieri concipitur, accideret binis momentis G, & H, quæ 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 hic linea finita sine
puncto primo, & postremo, quod sit ejus limes, & superficies sine linea esse non potest ; unde
fit, ut in casu figuræ 6 binæ ordinatæ 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, nec densitas, quia duæ simul haberentur distantiæ, vel duæ densitates,
quod utique sine replicatione haberi non potest ; caloris itidem, & frigoris mutatio in
thermometris, ponderis atmosphæræ mutatio in barometris, non fit per saltum, quia binæ
simul altitudines mercurii in instrumento haberi deberent eodem momento temporis, quod
fieri utique non potest ; cum quovis momento determinato unica altitudo haberi debeat,
ac unicus determinatus caloris gradus, vel frigoris ; quæ quidem theoria innumeris casibus
pariter aptari potest.
52. Contra hoc argumentum videtur primo aspectu adesse aliquid, quod ipsum prorsus
evertat, & tamen ipsi illustrando idoneum est maxime. Videtur nimirum inde erui,
impossibilem esse & creationem rei cujuspiam, & interitum. Si enim conjungendus est
postremus terminus præcedentis seriei cum primo sequentis ; in ipso transitu a non e$$e ad
esse, vel vice versa, debebit utrumque conjungi, ac idem simul erit, & non erit, quod est
absurdum. Responsio in promptu est. Seriei finitæ realis, & existentis, reales itidem, &
existentés termini esse debent ; non vero nihili, quod nullas proprietates habet, quas exigat,
Hinc si realium statuum seriei altera series realium itidem statuum succedat, quæ 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,
nec cum hoc esse suum non esse conjunget simul ; at si densitas certa per horam duret, tum
momento temporis in aliam mutetur duplam, duraturam itidem per alteram sequentem
horam ; momento temporis, [25] quod horas dirimit, binæ debebunt esse densitates simul,
nimirum & simplex, & dupla, quæ sunt reales binarum realium serierum termini.

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.
Let times be represented by the straight line AB, & diverse states of any thing by
ordinates drawn to meet the lines CD, EF, which are discontinuous at some point. If the
ordinates 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
exemplify it. It seems that from this argument it follows that both the creation of any
thing, & its destruction, are impossible. For, if the last term of a series that precedes is to
be connected with the first term of the series that follows, then in the passage from a state
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 peal & 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.
F
Illustration of this
argument from geo-
metry; the line of
reasoning being
metaphysical, with
Several examples.
A difficulty raised
over the connecting
together of existence
& non-existence at
the time of creation
or annihilation ; &
its Solution.
66 PHILOSOPHIÆ NATURALIS THEORIA
ɤ'* 53. Id ipsum in dissertatione De lege virium in Natura existentium satis, ni fallor,
luculenter exposui, ac geometricis figuris illustravi, adjectis nonnullis, quæ eodem recidunt,
& quæ in applicatione ad rem, de qua agimus, & in cujus gratiam hæc omnia ad legem con-
tinuitatis pertinentia allata sunt, proderunt infra ; libet autem novem ejus dissertationis
numeros huc transferre integros, incipiendo ab Octavo, sed numeros ipsos, ut & schematum
numeros mutabo hic, ut cum superioribus consentiant.
È. gette eX 54. “ Sit in fig. 8 circulus GMM'm, qui referatur ad datam rectam AB per ordinatas
É °"* HM ipsi rectæ perpendiculares ; uti itidem perpendiculares sint binæ tangentes EGF,
E'G'F/. Concipiantur igitur recta quædam indefinita ipsi rectæ AB perpendicularis, motu
quodam continuo delata ab A ad B. Ubi ea habuerit, positionem quamcumque CD, quæ
præcedat tangentem EF, vel C'D', quæ consequatur tangentem E'F' ; ordinata ad circulum
nulla erit, sive erit impossibilis, & ut Geometræ
loquuntur, imaginaria. Ubicunque autem ea sit / . / / 4/

inter binas tangentes EGF, E'G'F', in HI, H'I', DI F| | I IF [D
occurret circulo in binis punctis M, m, vel M', m', 4%T>
& habebitur valor ordinate HM, Hm, vel H'M', / 111 111
H'm'. Ordinata quidem ipsa respondet soli inter-
vallo EE': & si ipsa linea AB referat tempus ; G
momentum E est limes inter tempus præcedens g
continuum AE, quo ordinata non est, & tempus M • IV
continuum EE' subsequens, quo ordinata est ; punc- P LL^ SJ >^
/
tum E' est limes inter tempus præcedens EE', quo
ordinata est, & subsequens E'B, quo non est. Vita
igitur quædam ordinatæ est tempus EE' ; ortus À
habetur in E, interitus in E'. Quid autem in A. •,
ipso ortu, & interitu ? Habetur-ne quoddam ejje C, E H H E” C^
ordinatæ, an non esse ? Habetur utique esse, nimi- FIG. 8.
rum EG, vel E'G', non autem mom esse. Oritur -
tota finitæ magnitudinis ordinata EG, interit tota finitæ magnitudinis E'G', nec tamen
ibi conjungit esse, & mom esse, nec ullum absurdum secum trahit. Habetur momento E
primus terminus seriei sequentis sine ultimo seriei præcedentis, & habetur momento E'
ultimus terminus seriei præcedentis sine primo termino seriei sequentis.”
solutio ex meta- 55. “ Quare autem id ipsum accidat, si metaphysica consideratione rem perpendimus,
Ę* °* statifî patebit. Nimirum veri nihili nullæ sunt'véræ proprietates : entis realis veræ, &
reales proprietates sunt. Quævis realis series initium reale debet, & finem, sive primum, &
ultimum terminum. Id, quod non est, nullam habet veram proprietatem, nec proinde sui
generis ultimum terminum, aut primum exigit. Series præcedens ordinatæ nullius, ultimum
terminum non [26] habet, series consequens non habet primum : series realis contenta
intervallo EE', & primum habere debet, & ultimum. Hujus reales termini terminum illum
nihili per se se excludunt, cum ipsum esse per se excludat non esse.”
B
Illustratio ulterior 56. ** Atque id quidem manifestum fit magis : si consideremus seriem aliquam
geometrica. præcedentem realem, quam exprimant ordinatæ ad lineam continuam PLg, quæ respondeat
toti tempori AE ita, ut cuivis momento C ejus temporis respondeat ordinata CL. Tum
vero si momento E debeat fieri saltus ab ordinata Eg ad ordinatam EG : necessario ipsi
momento E debent respondere binæ ordinatæ EG, Eg. Nam in tota linea PLg non potest
deesse solum ultimum punctum g ; cum ipso sublato debeat adhuc illa 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 ; quæ si non interjaceat ; jam illa puncta in unicum coalescunt. Quare non potest
deesse nisi lineola aliqua gL ita, ut terminus seriei præcedentis sit in aliquo momento C
præcedente momentum E, & disjuncto ab eo per tempus quoddam continuum, in cujus
tempóris momentis omnibus ordinata sit nulla.”
£i; tig*!£t . 57. “ Patet igitur discrimen inter transitum a vero nihilo, nimirum a quantitate
iätiöïém. imaginaria, ad e$$e, & 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 magnitudo realis ab una quantitate ad
A THEORY OF NATURAL PHILOSOPHY 67
53. I explained this very point clearly enough, if I mistake not, in my dissertation
De lege virium in Natura existentium, & I illustrated it by geometrical figures; also I made
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
means of ordinates HM drawn perpendicular to that straight line; also let the two tan-
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 H'I', 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 : 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
matter metaphysically. Thus, to absolute nothing there belong no real properties; but
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 themselvcs
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
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 gll, 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.”
57. “Evidently, then, there is a distinction between passing from absolute nothing,
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
The source from
which the solution
is to be borrowed.
Solution derived
from a geometrical
example.
Solution from a
metaphysical con-
sideration.
Further illustration
by geometry.
Application to crea-
tion & annihilation.
68 PHILOSOPHIÆ NATURALIS THEORIA
Aliquando videri
nihilum id, quod
est aliquid.
Ordinatam nullam,
ut & distantiam
nullam existentium
esse compenetra-
tionem.
Ad idem pertinere
seriei realis genus
eam distantia, m
nullam, & aliquam.
Alia, quæ videntur
nihil, & sunt ali-
quid : discrimen
inter radicem ima-
ginariam, & zero.
G congreditur, ordinata CL videtur abire in
aliam transire debet per saltum ; momento temporis, quo saltus committitur, uterque
terminus haberi deberet. Manet igitur illæsum argumentum nostrum metaphysicum pro
exclusione saltus a creatione & annihilatione, sive ortu, & interitu.”
58. “At hic 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 præcedentibus, licet alio
nomine insignitus.”
[27] 59. “ Sit in Fig. 9. Linea AB, ut prius, ad quam linea quædam 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,
quæ evanescet, ubi puncto C abeunte in E, ipsa CD abibit in EF, tum in positione ulteriori
rectæ perpendicularis HI, vel abibit in nega-
tivam HM, vel retro positiva regredietur D F i
in HM'. Ubi linea altera cum altera coit, •.
& punctum E alterius cum alterius puncto T5~
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 quædam
punctorum C, L ; H, M : in situ EF
habetur eorundem punctorum compene-
tratio. Distantia est relatio quædam
binorum modorum, quibus bina puncta
existunt ; compenetratio itidem est relatio binorum modorum, quibus ea existunt,
quæ compenetratio est aliquid reale ejusdem prorsus generis, cujus est distantia, constituta
nimirum per binos reales existendi modos.”
6o. “ Totum discrimen est in vocabulis, quæ 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 ortæ a binis localibus existendi modis. Diversa
vero habent nomina ad arbitrarium instituta, cum aliæ ex ejusmodi relationibus, ut CL,
dicantur distantiæ positivæ, relatio EG dicatur compenetratio, relationes HM dicantur
distantiæ negativæ. Sed quoniam, ut a decem palmis distantiæ demptis 5, relinquuntur 5,
ita demptis aliis 5, habetur nihil (non quidem verum nihil, sed nihil in ratione distantiæ a
nobis ita appellatæ, cum remaneat compenetratio) ; ablatis autem aliis quinque, remanent
quinque palmi distantiæ negativæ ; ista omnia realia sunt, & ad idem genus pertinent ; cum
eodem prorsus modo inter se differant distantia palmorum IO a distantia palmorum 5, hæc
a distantia nulla, sed reali, quæ compenetrationem importat, & hæc a distantia negativa
palmorum 5. Nam ex prima illa quantitate eodem modo devenitur ad hasce posteriores per
continuam ablationem palmorum 5. Eodem autem pacto infinitas ellipses, ab infinitis
hyperbolis unica interjecta parabola discriminat, quæ 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.”
PIG. 9.
[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 præcedentem velocitatem, & ita porro ;
plurimum hæc discrepant a vero non esse. Casus ordinatæ respondentis lineæ EF in fig. 9,
differt plurimum a casu Ordinatæ circuli respondentis lineæ CD figuræ 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 impossibile ; usque
adeo in hoc secundo casu habetur verum nihilum, omni reali proprietate carens ; in illo
primo habetur aliquid realibus proprietatibus præditum, quod ipsis etiam solutionibus
problematum, & constructionibus veras sufficit, & reales determinationes ; cum realis, non
imaginaria sit radix equationis cujuspiam, quæ sit = o, sive nihilo æqualis.”

*--- 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
ideas for the consideration of production & destruction, it seems also that sometimes
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
point G belongs to the line PL, & E to the line AB, both being produced to meet one
another at this point); & suppose that PL either goes on beyond the point as GM, or
recoils along GM'. Then the straight line CD will contain the ordinate CL, which will
vanish when, as the point C gets to E, CD attains the position EF; & after that, in the
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.
Two local modes of existence can constitute an infinite number of relations, some of one
sort & some of another. All of these differ from one another, & yet agree with one
another in a high degree; for 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,
is some real state ; so is ‘no’ velocity a real state of an existent point, namely, a propensity
to remain in the same place; so also is ‘no’ force a real state of an existent point, namely,
a propensity to retain the velocity that it has already ; & so on. All these differ from
a state of non-existence in the highest degree. The case of the ordinate corresponding
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.
equal to nothing, is something that is real, & is not an imaginary thing.”
For the root of any equation that = 0, or is
Sometimes what is
really something ap-
pears to be nothing.
When the ordinate
is nothing, just as
when the distance
between two exis-
tent things is no-
thing, there is com-
penetration. -
This ‘no ' distance
belongs to the same
kind of series of
real quantities as
“some distance.
Other things that
seem to be nothing,
and yet are really
Something ;
tinction between
an imaginary root
& zero.
d is-
7o PHILOSOPHIÆ NATURALIS THEORIA
Conclusio pro solu-
tione ejus objec-
tionis.
Applicatio legis
continuitatis a, d
collisionem corpo-
flIIIl.
62. “ Firmum igitur manebit semper, & stabile, seriem realem quamcunque, quæ
continuo tempore finito duret, debere habere & primum principium, & ultimum finem
realem, sine ullo absurdo, & sine conjunctione sui esse cum non esse, si forte duret eo solo
tempore : dum si præcedenti etiam exstitit tempore, habere debet & ultimum terminum
seriei præcedentis, & primum sequentis, qui debent esse unicus indivisibilis communis limes,
ut momentum est unicus indivisibilis limes inter tempus continuum præcedens, & subsequens.
Sed hæc de ortu, & interitu jam satis.” •
63. Ut igitur contrahamus jam vela, continuitatis lex & inductione, & metaphysico
argumento abunde nititur, quæ idcirco etiam in velocitatis communicatione retineri omnino
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, quæ
' videri possent mutatæ per saltum. Quod autem pertinet ad metaphysicum argumentum, si
D u o velocitatum
genera, potentialis,
& actualis.
Einas velocitates
tum actuales, tum
potentiales simul
haberi non posse,
ne detur, vel exiga-
tur compenetratio.
toto tempore ante contactum subsequentis corporis superficies antecedens habuit I2 gradus
velocitatis, & sequenti 9, saltu facto momentaneo ipso initio contactus ; in ipso momento ea
tempora dirimente debuisset habere & I2, & 9 simul, quod est absurdum. Duas enim
velocitates simul habere corpus non potest, quod ipsum aliquanto diligentius demonstrabo.
64. Velocitatis nomen, uti passim usurpatur a Mechanicis, æquivocum est ; potest
enim significare velocitatem actualem, quæ nimirum est relatio quædam in motu æquabili
spatii percursi divisi per tempus, quo percurritur ; & potest significare [29] quandam, quam
apto Scholiasticorum vocabulo potentialem appello, quæ nimirum est determinatio, ad
actualem, sive determinatio, quam habet mobile, si nulla vis mutationem inducat, percur-
rendi motu æquabili determinatum quoddam spatium quovis determinato tempore, quæ
quidem duo & in dissertatione De Viribus Vivis, & in Stayanis Supplementis distinxi,
distinctione utique necessaria ad æquivocationes evitandas. Prima haberi non potest
momento temporis, sed requirit tempus continuum, quo motus fiat, & quidem etiam motum
æquabilem 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, quæ sit
tantummodo determinatio ad actualem, quam quidem ipsam intelligo, ubi in collisione
corporum eam nego mutari posse per saltum ex hoc posteriore argumento.
65. Jam vero velocitates actuales non posse simul esse duas in eodem mobili, satis patet ;
quia oporteret, id mobile, quod initio dati cujusdam temporis fuerit in dato spatii puncto,
in omnibus sequentibus occupare duo puncta ejusdem spatii, ut nimirum spatium percursum
sit duplex, alterum pro altera velocitate determinanda, adeoque requireretur actualis
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 nec 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 tum 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 eæ componuntur in unicam
ita, ut singulæ constituant statum mobilis, qui status respectu dispositionum, quas eo
momento, in quo tum est, habet ipsum mobile, complectentium omnes circumstantias
præteritas, & præsentes, est tantummodo conditionatus, non absolutus ; nimirum ut con-
tineant determi-[3o]-nationem, quam ex omnibus præteritis, & præsentibus 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,
which lasts for some finite continuous time, is bound to have a first beginning & a final
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
both on induction & on metaphysical reasoning; & On that account it should be retained
in every case of communication of velocity. So that indeed there can never be any passing
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
may mean actual velocity, that is to say, a certain relation in uniform motion given by
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 Viribus Vivis & 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
time in the same moving body. For, then it would be necessary that the moving body,
which at the beginning of a certain time occupied a certain given point of space, should at
all times afterwards occupy two points of that space; so that the space traversed would be
twofold, the one space being determined by the one velocity & the other by the other.
Thus an actual replication would be required; & this we can clearly prove in a perfectly
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
C on clusi on in
favour of a solution
of this difficulty.
Application of the
Law of Continuity
to the collision of
solid bodies.
Two kinds of velo-
city, potential &
actual.
It is impossible
for a body to
have two velocities,
either actual or
potential, unless it
is given, or we are
forced to admit,
that there is com-
penetration.
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 PHILOSOPHIAE NATURALIS THEORIA
Quovis momento
punctum existens
d e b e re ha b e r e
statum realem ex
genere velocitatis
potentialis.
Non posse mom-
ento temporis tran-
siri ab una veloci-
t at e ad ali a m,
demonstra tu r, &
vindicatur.
temporis ; nisi aliunde eiusmodi determinatio per conjunctionem alterius causae, quae tum
agat, veliam egerit, mutaretur, & loco ipsius alia, quae composita dicitur, succederet. Sed
status absolutus resultans ex omnibus eo momento praesentibus, & praeteritis 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, & praesentibus est absolutus, licet sit itidem conditionatus pro
futuris : si nimirum eadem, vel aliae causae agentes sequentibus momentis non mutent
determinationem, 8 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 praeteritis, & praesentibus circumstantiis absolutos 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 praeteritis, &
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 eius momenti, 8 omnium praeteritarum ; nam toto praecedenti tempore habita
fuisset realis series statuum cum illa priore, & toto sequenti deberet haberi cum illa
posteriore, adeoque eo momento, simul utraque, cum neutra series realis sine reali suo
termino stare possit. r
66. Praeterea corporis, vel puncti existentis potest utique nulla esse velocitas actualis,
saltem accurate talis; si nimirum difformem habeat motum, quod ipsum etiam semper in
Natura accidit, ut demonstrari posse arbitror, sed huc non pertinet; at semper utique
haberi debet aliqua velocitas potentialis, vel saltem 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 hic addam in fine inter nonnulla, quae hic
etiam supplementa appellabo, & occurrent primo, ac secundo loco. Sed id ipsum itidem
nequaquam huc pertinet. Iis etiam penitus praetermissis, eruitur e reliquis, quae diximus,
admisso etiam ut existente, vel possibili in Natura motu uniformi, 8 quiete, utramque
velocitatem habere conditiones necessarias ad [31] hoc, ut secundum argumentum pro
continuitatis lege superius allatum vim habeat suam, nec ab una velocitate ad alteram abiri
possit sine transitu per intermedias.
67. Patet autem, hinc illud evinci, nec interire momento temporis posse, nec oriri
velocitatem totam corporis, vel puncti non simul intereuntis, vel orientis, nec huc transferri
posse, quod de creatione, 8 morte diximus ; cum nimirum ipsa velocitas nulla corporis, vel
puncti existentis, sit non purum nihil, ut monui, sed realis quidam status, qui simul cum
alio reali statu determinata illius intereuntis, vel orientis velocitatis deberet conjungi; unde
etiam fit, ut nullum effugium haberi possit contra superiora argumenta, dicendo, quando a
I2 gradibus velocitatis transitur ad 9, durare utique priores 9, & interire reliquos tres, in
quo nullum absurdum sit, cum nec in illorum duratione habeatur saltus, nec 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 I2 gradus velocitatis non sunt quid
compositum e duodecim rebus inter se distinctis, atque disiunctis, quarum 9 manere possint,
3 interire, sed sunt unica determinatio ad existendum in punctis spatii distantibus certo
intervallo, ut palmorum 12, 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 ordinata HE communis ipsi usque ad D, sed
sunt duae ordinata, quarum prima constitit in relatione distantiae, puncti curva D a puncto
axis G, secunda in relatione puncti curvae E a puncto axis H, quod estibi idem, acpunctum 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
point, may be nothing; that is to say, if the motion is non-uniform. Now, this always
is the case in Nature; as I think can be proved, but it does not concern us at present. But,
at any rate, it is bound to have some potential velocity, or at least some state, which,
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
velocity of a body cannot perish or arise in an instant of time, nor for a point that does
not perish or arise along with it ; nor can our arguments with regard to production &
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
At any instant an
existing point must
have a real state
arising from a kind
of potential velo-
city.
Rigorous proof that
it is impossible to
pass from one velo-
city to another in
an instant of time.
74. PHILOSOPHIÆ NATURALIS THEORIA.
Cur adhibita col-
lisio pergentium in
eandem plagam pro
Theoria deducenda.
Quo pacto mutata
velocitate poten-
tiali per saltum,
non mutetur per
saltum actualis.
Relationem distantiæ punctorum D, & G constituunt duo reales modi existendi ipsorum,
relationem distantiæ punctorum D. & E duo reales modi existendi ipsorum, & relationem
distantiæ punctorum H, & E duo reales modi existendi ipsorum. T Hæc 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
præcisiva, & indefinita idea stat mihi idea spatii continui ; & intermedii modi possibiles inter
G, & D sunt pars intermediorum inter E, & H. Præterea omissis etiam hisce omnibus ipse
ille saltus a velocitate finita ad nullam, vel a nulla ad finitam, haberi non potest. •
68. Atque hinc ego quidem potuissem etiam adhibere duos globos æquales, qui sibi
invicem occurrant cum velocitatibus æqualibus, quæ nimirum in ipso contactu deberent
momento temporis interire ; 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 I2 velocitatis gradibus
assequatur alterum præcedentem cum 6; ut nimirum abeundo ad velocitatem aliam
quamcunque haberetur saltus ab una velocitate ad aliam, in quo evidentius esset absurdum.
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 eam
metiamur spatio, quod conficitur, diviso per tempus, transitus utique fieret per omnes
intermedias, quod sic facile ostenditur ope Geometriæ. In fig. IO designent AB, BC bina
Irregularitas alia,
1n expressione act-
ualis velocitatis.
B, semper eadem habebitur velocitatis men-
tempora ante & post contactum, & momento quolibet H sit velocitas potentialis illa major
HI, quæ æquetur velocitati primæ AD ; quovis autem momento Q posterioris temporis sit
velocitas potentialis minor QR, quæ æquetur D I . L V N E Y
velocitati cuidam datæ CG. Assumpto quovis ---;
tempore HK determinatæ magnitudinis, area N | -
IHKL divisa per tempus HK, sive recta HI, X P, R T. 'G
exhibebit velocitatem actualem. Moveatur 1-
tempus HK versus B, & donec K adveniat ad F

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 nec erit MN æqualis priori AD, nec BF, - I -
Ê minor pìà ï FE ; sed AH K M B OG S C
facile demonstrari potest (b), capta VE æquali - FIG. Io. -
IL, vel HK, sive MO, & ducta recta VF, quæ secet MN in X, quotum ex illo divisione
prodeuntem fore MX, donec, abeunte toto illo tempore ultra B in QS, jam area QRTS
divisa per tempus QS exhibeat velocitatem constantem QR. - -
76. Patet igitur in ea consideratione a velocitate actuali præcedente 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 tum
etiam, si velocitas aliqua actualis debeat agnosci, & determinari spatio diviso per tempus ;
pro aliis, atque aliis temporibus assumptis pro mensura aliæ, atque aliæ velocitatis actualis
mensuræ ob-[33]-veniant, secus ac accidit in motu semper æquabili, quam ipsam ob causam,
velocitatis actualis in motu difformi nulla est revera mensura accurata, quod supra innui
sed ejus idea præcisa, ac distincta æquabilitatem motus requirit, & idcirco Mechanici in
difformibus motibus ad actualem velocitatem determinandam adhibere solent spatiolum
infinitesimo tempusculo percursum, in quo ipso motum habent pro æquabili.
(b) Si enim producatur OP usque ad NE in 1^, erit E?^ = VN, ob VE = MO =N1^. Est autem
VE : VN :: EF : NX; quare VN × EF = VE × NX, sive posito E1^ pro VN, & MO pro VE, erit
E1^ ×EF =MO × NX. Totum MN1^O est MO × MN, pars FE1^P est = E1^ × EF. Quare residuus
gnomon NMOPFE est MO×(MN— NX), sive est MO × MX, quo diviso per MO habetur 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
another with equal velocities, which in truth at the moment of contact would have to be
destroyed in an instant of time. But, in order to avoid the very considerations just stated
with regard to the passage from a real state to another real state (when we pass from a
definite velocity to none), I have preferred to employ in all my dissertations a ball having
12 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
velocity, if the collision occurs with any given difference of velocities whatever. In the
actual velocity, measured by the space traversed divided by the time, the change will at any
rate be through all intermediate stages; & this can easily be shown to be so by the aid of
Geometry. -
In Fig. Io 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
preceding actual velocity HI to the subsequent velocity QR is made through all intermediate
velocities such as MX, which will be determined by the continuous straight line VF. There
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 H.K. 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 1°, then E1 = WN; for VE = MO = Wr. Moreover
WE: WN=EF: NX ; and therefore WN.EF=/E.NX. Hence, replacing VN by E1’, and VE hy MO, we have
E1’EF=MO.NX. Now, the whole MNTO = MO.MN, and the part FETP= E1’.E.F. Hence the remainder
(the gnomon NMOPFE) = MO.(MN – NX) = MO.MX: and this, on division hy MO, will give MX.
Why the collision -
of bodies moving in
the same direction
is employed for the
purpose of deduc-
ing my Theory.
How, supposing
that there were a,
sudden change in
the potential velo-
city, there might
not be a sudden
change in the act-
ual velocity.
A further irregu-
larity in the repre-
sentation of actual
velocity.
76 PHILOSOPHIÆ NATURALIS THEORIA
Conclu d it u r a d
contactum imme-
diatum non posse
deveniri cum differ-
entia, velocitatum.
Eromovenda, ana-
lysis eo excluso.
Debere ante con-
tactum haberi mu- '
tationem velocita-
tis, adeoque vim,
quæ mutat.
Eam vim debere
eSSe mutuam, &
agere in partes op-
positas.
7I. At velocitas potentialis, quæ singulis momentis temporis respondet sua, mutaretur
utique per saltum ipso momento B, quo deberet haberi & ultima velocitatum præcedentium
BE, & prima sequentium BF, quod cum haberi nequeat, uti demonstratum est, fieri non
potest per secundum ex argumentis, quæ adhibuimus pro lege continuitatis, ut cum illa
velocitatum inæqualitate deveniatur ad immediatum contactum ; atque id ipsum excludit
etiam inductio, quam pro lege continuitatis in ipsis quoque velocitatibus, atque motibus
primo loco proposui. - *
72. Atque hoc demum pacto illud constitit evidenter, non licere continuitatis legem
deserere in collisione corporum, & illud admittere, ut ad contactum immediatum deveniatur
cum illæsis binorum corporum velocitatibus integris. Videndum igitur, quid necessario
consequi debeat, ubi id non admittatur, & hæc analysis ulterius promovenda.
73. Quoniam ad immediatum contactum devenire ea corpora non possunt cum præce-
dentibus velocitatibus ; oportet, ante contactum ipsum immediatum incipiant mutari
velocitates ipsæ, & 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, quæcunque illa sit. Causa vero mutans statum corporis in ordine ad
motum, vel quietem, dicitur vis ; habebitur igitur vis aliqua, quæ effectum gignat, etiam
ubi illa duo corpora nondum ad contactum devenerint.
74. Ad impediendam violationem continuitatis satis esset, si ejusmodi vis ageret in
alterum tantummodo e binis corporibus, reducendo præcedentis velocitatem ad gradus I2,
vel sequentis ad 6. Videndum igitur aliunde, an agere debeat in alterum tantummodo, an
in utrumque simul, & quomodo. Id determinabitur per aliam Naturæ legem, quam nobis
inductio satis ampla Ostendit, qua nimirum evincitur, omnes vires nobis cognitas agere
utrinque & æqualiter, & in partes oppositas, unde provenit principium, quod appellant
actionis, & reactionis æqualium ; est autem fortasse quædam actio duplex semper æqualiter
Hinc dic e n d a m.
e sse repulsivam :
quærendam ejus
legem.
Ea vi debere totum
velocita, t u m dis-
crimen elidi ante
contactum.
agens in partes oppositas. Ferrum, & magnes æque se mutuo trahunt ; elastrum binis
globis æqualibus interjectum æque utrumque urget, & æqualibus velocitatibus propellit ;
gravitatem ipsam generalem mutuam esse osten-[34]-dunt errores Jovis, ac Saturni potissi-
mum, ubi ad se invicem accedunt, uti & curvatura orbitæ lunaris orta ex ejus gravitate in
terram comparata cum æstu maris orto ex inæquali partium globi terraquei gravitate in
Lunam. Ipsæ nostræ vires, quas nervorum ope exerimus, semper in partes oppositas agunt,
nec 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, eam ibi vim in
utrumque corpus agere, quæ actio ad æqualitatem non reducet inæquales illas velocitates,
nisi augeat præcedentis, minuat consequentis corporis velocitatem ; nimirum nisi in iis
producat velocitates quasdam contrarias, quibus, si solæ essent, deberent a se invicem
recedere : sed quia eæ componuntur cum præcedentibus ; hæc utique non recedunt, sed
tantummodo minus ad se invicem accedunt, quam accederent.
75. Invenimus igitur vim ibi debere esse mutuam, quæ ad partes oppositas agat, & quæ
sua natura determinet per sese illa corpora ad recessum mutuum a se invicem. . Hujusmodi
igitur vis ex nominis definitione appellari potest vis repulsiva. - Quærendum jam ulterius,
qua lege progredi debeat, an imminutis in immensum distantiis ad datam quandam mensuram
deveniat, an in infinitum excrescat ?
76. Ut in illo casu evitetur saltus ; satis est in allato exemplo ; si vis repulsiva, ad quam
delati sumus, extinguat velocitatum differentiam illam 6 graduum, antequam ad contactum
immediatum corpora devenirent : quamobrem possent utique devenire ad eum contactum
eodem illo momento, quo ad æqualitatem velocitatum deveniunt. At si in alio quopiam
casu corpus sequens impellatur cum velocitatis gradibus 2o, corpore præcedente cum suis 6 ;
A THEORY OF NATURAL PHILOSOPHY 77
71. The potential velocity, each corresponding to its own separate instant of time,
would certainly be changed suddenly at that instant of time B; & at this point we are
bound to have both the last of the preceding velocities, BE, & the first of the subsequent
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
Law of Continuity in the collision of bodies, & admit the idea that they can come into
immediate contact with the whole velocities of both bodies unaltered. Hence, we must
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
at first, it is necessary that those velocities should commence to change before that immediate
contact; & either that of the body that follows should be diminished, or that of the one
going in front should be increased, or that both these changes should take place together.
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
this kind should act on one of the two bodies only, altering the velocity of the body in
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
directions; one which from its very nature imparts to those bodies a natural propensity
for mutual recession from one another. Hence a force of this kind, from the very meaning
of the term, may be called a repulsive force. We have now to go further & find the law
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 example under consideration, if the repulsive force, to which our arguments have led
us, should destroy that difference of 6 degrees in the velocities before the bodies should
have come into immediate contact. Hence they might possibly at least come into contact
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
The conclusion is
that immediate
contact with a dif-
ference of velocities
cannot be attained.
Immediate contact
being barred, the
analysis is to be car-
ried further.
There must be then,
before contact, a
change in the velo-
city; & therefore
Some force that
causes the change.
The force must be
mutual, & act in
Opposite directions.
Hence the force
must be termed
repulsive ; the law
governing it is now
to be found.
The whole differ-
ence between the
velocities must be
destroyed by the
force before con-
tact. -
78 PHILOSOPHIÆ NATURALIS THEORIA
Eam vim debere
augeri in infinitum,
imminutis, & qui-
dem in infinitum,
distantiis : habente
virium curva, ali-
quam asymptotum
in origine abscissa-
1TUAIIl.
Vim in majoribus
distantiis esse at-
tractivam, curva
secante axem in
aliquo limite.
tum vero ad contactum deveniretur cum differentia velocitatum majore, quam graduum 8.
Nam illud itidem amplissima inductione evincitur, vires omnes nobis cognitas, quæ 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
Mechanicæ, quæ inde motuum leges eruit, quas experimenta in pendulis, in projectis
gravibus, in aliis pluribus comprobant, & Astronomia confirmat in cælestibus motibus.
Quamobrem illa vis repulsiva, quæ 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 illa 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 I4, 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, quæ 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 I4 gradibus discriminis, qui habebantur.
77. Quando autem huc 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, quæ imminutis
distantiis crescat in infinitum, atque ita crescat, ut par sit extinguendæ cuicunque velocitati,
utcunque magnæ. Devenimus igitur ad vires repulsivas imminutis distantiis crescentes
in infinitum, nimirum ad arcum illum asymptoticum ED curvæ 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, quæ habemus præ manibus, quædam data distantia quæcunque
esse ultimus limes virium in infinitum excrescentium, quo casu asymptotus AB non transiret
per initium distantiæ binorum corporum, sed tanto intervallo post ipsum, quantus esset
ille omnium distantiarum, quas remotiores particulæ possint acquirere a se invicem, limes
minimus ; sed aliquem demum esse debere extremum etiam asymptoticum arcum curvæ
habentem pro asymptoto rectam transeuntem per ipsum initium distantiæ, sic evincitur ;
si nullus ejusmodi haberetur arcus ; particulæ materiæ minores, & primo collocatæ in
distantia minore, quam esset ille ultimus limes, sive illa distantia asymptoti ab initio
distantiæ binorum punctorum materiæ, 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-
guendæ cuivis, utcunque magnæ. Ad summum in curva virium haberi possent plures
asymptotici arcus, alii post alios, habentes ad exigua intervalla asymptotos inter se parallelas,
qui casus itidem uberrimum aperit contemplationibus foecundissimis campum, de quo
aliquid inferius ; sed aliquis arcus asympto-[36]-ticus postremus, cujusmodi est is, quem
in figura I proposui, haberi omnino debet. Verum ea perquisitione hic omissa, pergendum
est in consideratione legis virium, & curvæ eam exprimentis, quæ habentur auctis distantiis.
78. In primis gravitas omnium corporum in Terram, quam quotidie experimur, satis
evincit, repulsionem illam, quam pro minimis distantiis invenimus, non extendi ad distantias
quascunque, sed in magnis jam distantiis haberi determinationem ad accessum, quam vim
attractivam nominavimus. Quin immo Keplerianæ leges in Astronomia tam feliciter a
Newtono adhibitæ ad legem gravitatis generalis deducendam, & ad cometas etiam traductæ,
A THEORY OF NATURAL PHILOSOPHY
79
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PHILOSOPHIAE NATURALIS THEORIA
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A THEORY OF NATURAL PHILOSOPHY 8I
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
that, what happens in the second case when compared with the first, will happen also in
a third case, in which the velocity of the body that follows is once more increased, when
compared with the second case; & so on, & so on. Hence, in order to guard against any
sudden change at all in every case whatever, Nature will necessarily have taken measures
for this purpose by means of a force of such a kind that, as the distances are diminished the
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
experience, proves sufficiently that the repulsion that we found for very small distances
does not extend to all distances; but that at distances that are now great there is a
propensity for approach, which we have called an attractive force. Moreover the Keplerian
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,
G
The force must in.
crease indefinitely,
as the distances are
diminished, a 1 so
indefinitely ; the
curve of forces has
an asymptote at the
origin of abscissae.
The force at greater
distances is attract-
ive, the curve cut-
ting the ax is at
Some limit point.
82 PHILOSOPHIÆ NATURALIS THEORIA
Plures esse debere,
immo p 1 u r i m os
transitus, & limites.
Hinc tota curvæ
forma cum binis
asymptotis, & plu-
ribus flexibus, ac
sectionibus.
Hinc elementorum :
primorum materiæ
simplicitas carens
partibus.
Solutio objectionis
petitæ ex eo, quod
vires repulsivas
habere possent non
puncta singula, se
particulæ primi-
geniæ.
satis ostendunt, gravitatem vel in infinitum, vel saltem 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 eam tertii gradus hyperbolam, cujus ordinatæ sunt in ratione reciproca
duplicata distantiarum, qui nimirum est ille arcus STV figuræ 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.
79. Duos alios nobis indicat limites ejusmodi, sive alias duas intersectiones, ut G & I,
phænomenum vaporum, qui oriuntur ex aqua, & aeris, qui a fixis corporibus gignitur ;
cum in iis ante nulla particularum repulsio fuerit, quin immo fuerit attractio, ob
cohærentiam, qua, una parte retracta, altera ipsam consequebatur, & in illa tanta expansione,
& elasticitatis vi satis se manifesto prodat repulsio, ut idcirco a repulsione in minimis distantiis
ad attractionem alicubi sit itum, tum inde iterum ad repulsionem, & iterum inde ad generalis
gravitatis attractiones. Effervescentiæ, & fermentationes adeo diversæ, in quibus cum
adeo diversis velocitatibus eunt, ac redeunt, & jam ad se invicem accedunt, jam recedunt
a se invicem particulæ, indicant utique ejusmodi limites, atque transitus multo plures ;
sed illos prorsus evincunt substantiæ molles, ut cera, in quibus compressiones plurimæ
acquiruntur cum distantiis admodum adversis, in quibus, tamen omnibus limites haberi
debent ; nam, anteriore parte ad se attracta, posteriores eam sequuntur, eadem propulsa,
illæ recedunt, distantiis ad sensum non mutatis, quod ob illas repulsiones in minimis
distantiis, quæ contiguitatem impediunt, fieri alio modo non potest, nisi si limites ibidem
habeantur in iis omnibus distantiis inter attractiones, & repulsiones, quæ nimirum requi-
runtur ad hoc, ut pars altera alteram consequatur retractam, vel præcedat propulsam.
80. Habentur igitur plurimi limites, & plurimi flexus curvæ hinc, & inde ab axe præter
duos arcus, quorum prior ED in infinitum protenditur, & asymptoticus est, alter STV,
[37] si gravitas generalis in infinitum protenditur, est asymptoticus itidem, & ita accedit
ad crus illud hyperbolæ 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 curvæ
virium, qualem initio proposui, directa ratiocinatione a Naturæ phænomenis, & genuinis
principiis deducta. Remanet jam determinanda constitutio primorum elementorum
materiæ ab iis viribus deducta, quo facto omnis illa Theoria, quam initio proposui, patebit,
nec erit arbitraria quædam hypothesis, ac licebit progredi ad amovendas apparentes quasdam
difficultates, & ad uberrimam applicationem ad omnem late Physicam qua exponendam,
qua tantummodo, ne hoc opus plus æquo excrescat, indicandam.
81. Quoniam, imminutis in infinitum distantiis, vis repulsiva augetur in infinitum ;
facile patet, nullam partem materiæ posse esse contiguam alteri parti : vis enim illa repulsiva
protinus alteram ab altera removeret. Quamobrem necessario inde consequitur, prima
materiæ elementa esse omnino simplicia, & a nullis contiguis partibus composita. Id
quidem immediate, & necessario fluit ex illa constitutione virium, quæ in minimis distantiis
sunt repulsivæ, & in infinitum excrescunt.
82. Objicit hic fortasse quispiam illud, fieri posse, ut particulæ primigeniæ materiæ
sint compositæ quidem, sed nulla Naturæ vi divisibiles a se invicem, quarum altera tota
respectu alterius totius habeat vires illas in minimis distantiis repulsivas, vel quarum pars
quævis respectu reliquarum partium ejusdem particulæ non solum nullam habeat repulsivam
vim, sed habeat maximam illam attractivam, quæ ad ejusmodi cohæsionem requiritur :
eo pacto evitari debere quemvis immediatum impulsum, adeoque omnem saltum, & con-
tinuitatis læsionem. At in primis id esset contra homogeneitatem materiæ, de qua agemus
infra : nam eadem materiæ 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. 1. 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.
9. The phenomenon of vapour arising from water, & that of gas produced from
fixed bodies lead us to admit two more of these limit-points, i.e., two other intersections,
say, at G & I. Since in these there would be initially no repulsion, nay rather there
would be an attraction due to cohesion, by which, when one part is retracted, another
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
on the curve, first on one side & then on the other side of the axis, in addition to two
arcs, one of which, ED, is continued to infinity & is asymptotic, & the other, STV, is
asymptotic also, provided that universal gravitation extends to infinity. It approximates
to the form of the hyperbola of the third degree mentioned above so closely that the
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
indefinitely diminished, it is quite easy to see clearly that no part of matter can be contiguous
to any other part; for the repulsive force would at once separate one from the other.
Therefore it necessarily follows that the primary elements of matter are perfectly simple,
& 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
particles of matter are composite, but that they cannot be disintegrated by any force in
Nature; that one whole with regard to another whole may possibly have those forces
that are repulsive at very small distances, whilst any one part with regard to any other part
of the same particle may not only have no repulsive force, but indeed may have a very
great attractive force such as is required for cohesion of this sort; that, in this way, we
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
There are bound to
be many, nay, very
many of these pas-
sages, with corre-
sponding limit-
points.
Hence we get the
whole form of the
curve, with two
asymptotes, many
flexures & many
intersections with
the axis.
The simplicity of
the primary ele-
ments of matter ;
they are altogether
without parts.
Solution of the ob-
jection derived from
the assertion that
single points can-
not have repulsive
forces, but t h a t
primary particles
can have them.
84 PHILOSOPHIÆ NATURALIS THEORIA
An elementa sint
extensa : argumen-
ta pro virtuali eor-
um extensione.
Excluditur virtu-
alis extensio prin-
cipio inductionis
rite applicato.
IResponsio ad exem-
plum animæ & Dei.
Itidem ad analo-
giam cum quiete.
aliarum omnium attractivam in iisdem distantiis, quod analogiæ adversatur. Deinde si a
Deo agente supra vires Naturæ sejungerentur illæ partes a se invicem, tum ipsius Naturæ
vi in se invicem incurrerent ; haberetur in earum collisione saltus naturalis, utut præsup-
ponens aliquid factum vi agente supra Naturam. Demum duo tum cohæsionum 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
cohæsionis ; adeoque multo minus simplex, & minus uniformis evaderet Theoria.
[38] 83. Simplicitate & incompositione elementorum definita, 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 majus etiam occupent
spatium, ac eo loco, quo unum stet, possint, eo remoto, 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, & extensæ, præsentem toti etiamnum arbitrantur.
Deum autem ipsum præsentem ubique credimus per totum utique divisibile spatium,
quod omnia corpora occupant, licet ipse simplicissimus sit, nec ullam prorsus compositionem
admittat. Videtur autem sententia eadem inniti cuidam etiam analogiæ loci, ac temporis.
Ut enim quies est conjunctio ejusdem puncti loci cum serie continua omnium moment-
orum ejus temporis, quo quies durat : sic etiam illa virtualis extensio est conjunctio unius
momenti temporis cum serie continua omnium punctorum spatii, per quod simplex illud
ens virtualiter extenditur ; ut idcirco sicut illa quies haberi creditur in Natura, ita & hæc
virtualis extensio debeat admitti, qua admissa poterunt utique illa primæ materiæ elementa
esse simplicia, & tamen non penitus inextensa.
84. At ego quidem arbitror, hanc itidem sententiam everti penitus eodem inductionis
principio, ex quo alia tam multa hucusque, quibus usi sumus, deduximus. Videmus enim
in his corporibus omnibus, quæ observare possumus, quidquid distinctum occupat locum,
distinctum esse itidem ita, ut etiam satis magnis viribus adhibitis separari possint, quæ
diversas occupant spatii partes, nec ullum casum deprehendimus, in quo magna hæc corpora
partem aliquam habeant, quæ eodem tempore diversas spatii partes occupet, & eadem
sit. Porro hæc proprietas ex natura sua ejus generis est, ut æque cadere possit in
magnitudines, quas per sensum deprehendimus, ac in magnitudines, quæ infra sensuum
nostrorum limites sunt ; res nimirum pendet tantummodo a magnitudine spatii, per quod
haberetur virtualis extensio, quæ 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 materiæ particulas, ut ne illæ quidem
ejusmodi habeant virtualem extensionem.
[39] 85. Exempla, quæ adduntur, petita ab anima rationali, & ab omnipræsentia
Dei, nihil positive evincunt, cum ex alio entium genere petita sint ; præterquam quod nec
illud demonstrari posse censeo, animam rationalem non esse unico tantummodo, simplici,
& inextenso corporis puncto ita præsentem, ut eundem locum obtineat, exerendo inde
vires quasdam in reliqua corporis puncta rite disposita, in quibus viribus partim necessariis,
& partim liberis, stet ipsum animæ commercium cum corpore. Dei autem præsentia
cujusmodi sit, ignoramus omnino ; quem sane extensum per spatium divisibile nequaquam
dicimus, nec 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. - -
86. Quod autem pertinet ad analogiam cum quiete, sunt sane satis valida argumenta,
quibus, ut supra innui, ego censeam, in Natura quietem nullam existere. Ipsam nec 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,
there can be no doubt as to whether they are also non-extended or whether, although
simple, they have an extension of the kind that is termed virtual extension by the
Scholastics. For there were some, especially among the Peripatetics, who admitted elements
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
principle of induction, by which we have hitherto deduced so many results which we have
employed. For we see, in all those bodies that we can bring under observation, that
whatever occupies a distinct position is itself also a distinct thing; so that those that occupy
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
soul & the omnipresence of GoD, prove nothing positively; for they are derived from
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
strong to lead us to believe, as I remarked above, that there is no such thing in Nature
as absolute rest. Indeed, I proved that such a thing could not be, by a direct argument
Whether the ele-
ments are extended;
certain arguments
in favour of virtual
extension.
Virtual extension
is excluded by the
principle of in-
duction correctly
applied.
Reply to the
parallel case of the
Soul & GOD.
Again with regard
to the analogy with
rest.
86 • PHILOSOPHIÆ NATURALIS THEORIA
existere, argumento quodam positivo ex numero combinationum possibilium infinito
contra alium finitum, demonstravi in Stayanis Supplementis, ubi de spatio, & tempore
quæ juxta num. 66 occurrent infra Supplementorum § I, & § 2 ; numquam vero eam
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 quæcunque materiæ particula,
uti luminis particula quæcunque ; reliquæ omnes utcunque remotæ, quæ inde positionem
ab illa mutant, mutant & gravitatem, ac proinde moventur motu aliquo exiguo, sed sane
motu. In ipsa Telluris quiescentis sententia, quiescit quidem Tellus ad sensum, nec tota
In quo deficiat ana-
logia loci, & tem-
poris.
Inextensio u t i 1 i s
a d excludendum
transitum momen-
taneum a densitate
nulla ad summam.
Itidem ad hoc, ut
densitatis a u g e r i
possit, ut p ot e st
minui in infinitum.
ab uno in alium transfertur locum ; at ad quamcunque crispationem maris, rivuli decursum,
muscæ volatum, æquilibrio dempto, trepidatio oritur, perquam exigua illa 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.
87. Sed nec ea ipsa analogia, si adesset, rem satis evinceret ; cum analogiam inter tempus,
& locum videamus in aliis etiam violari : nam in iis itidem paragraphis Supplementorum
demonstravi, nullum materiæ punctum unquam redire ad punctum spatii quodcunque,
in quo semel fuerit aliud materiæ punctum, ut idcirco duo puncta materiæ nunquam
conjungant idem [4o] punctum spatii ne cum binis quidem punctis temporis, dum quam-
plurima binaria punctorum materiæ conjungunt idem punctum temporis cum duobus
punctis loci ; nam utique coexistunt : ac præterea tempus quidem unicam dimensionem
habet diuturnitatis, spatium vero habet triplicem, in longum, latum, atque profundum.
88. Quamobrem illud jam tuto inferri potest, hæc primigenia materiæ elementa, non
solum esse simplicia, ac indivisibilia, sed etiam inextensa. Et quidem hæc ipsa simplicitas,
& inextensio elementorum præstabit commoda sane plurima, quibus eadem adhuc magis
fulcitur, ac comprobatur. Si enim prima elementa materiæ sint quædam partes solidæ,
ex partibus compositæ, vel etiam tantummodo extensæ virtualiter, dum a vacuo spatio
motu continuo pergitur per unam ejusmodi particulam, fit saltus quidam momentaneus
a densitate nulla, quæ habetur in vacuo, ad densitatem summam, quæ habetur, ubi ea
particula spatium occupat totum. Is vero saltus non habetur, si elementa simplicia sint,
& inextensa, ac a se invicem distantia. Tum enim omne continuum est vacuum tantum-
modo, & in motu continuo per punctum simplex fit transitus a vacuo continuo ad vacuum
continuum. Punctum illud materiæ occupat unicum spatii punctum, quod punctum
spatii est indivisibilis limes inter spatium præcedens, & consequens. Per ipsum non
immoratur mobile continuo motu delatum, nec ad ipsum transit ab ullo ipsi immediate
proximo spatii puncto, cum punctum puncto proximum, uti supra diximus, nullum sit ;
sed a vacuo continuo ad vacuum continuum transitur per ipsum spatii punctum a materiæ
puncto occupatum. - •
89. Accedit, quod in sententia solidorum, extensorumque elementorum habetur illud,
densitatem corporis minui posse in infinitum, augeri autem non posse, nisi ad certum limitem
in quo incrementi lex necessario abrumpi debeat. Primum constat ex eo, quod eadem
particula continua dividi possit in particulas minores quotcunque, quæ idcirco per spatium.
utcunque magnum diffundi potest ita, ut nulla earum sit, quæ aliquam aliam non habeat
utcunque libuerit parum a se distantem. Atque eo pacto aucta mole, per quam
eadem illa massa diffusa sit, eaque aucta in ratione quacunque minuetur utique
densitas in ratione itidem utcunque magna. Patet & alterum : ubi enim omnes
particulæ ad contactum devenerint ; densitas ultra augeri non poterit. Quoniam
autem determinata quædam 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. I4 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
satisfactorily; since we see that in other ways the analogy between space & time is impaired.
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
are not only simple & indivisible, but also that they are non-extended. Indeed this
very simplicity & non-extension of the elements will prove useful in a really large number
of cases for still further strengthening & corroborating the results already obtained.
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
that the density of a body can be diminished indefinitely, but cannot be increased except
up to a certain fixed limit, at which the law of increase must be discontinuous. The first
comes from the fact that this same continuous particle can be divided into any number
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,
Where the analogy
of space and time
fails.
Non-extension use-
ful in excluding
an instantaneous
passage from ‘no’
density to a very
great One. -
Also for the idea
that density can be
increased, as it can
be de creased,
indefinitely.
88 PHILOSOPHIÆ NATURALIS THEORIA
Et ad excludendum
COntinuum exten-
sum, & in infinitum
in existentibus.
In ext e n s i o n e m
admitti oportere ;
qu æ re n d u m de
homogeneitate.
Homogenei tat e m
suaderi ab homo-
geneitate primi, &
ultimi asymptotici
cruris pro punctis
omnibus.
Nihil contra deduci
ex principio indis-
cernibilium, & rati-
Onis sufficientis.
in [41] ratione quacunque lineola quæcunque secari sane possit : adeoque uti nullus est
limes raritatis auctæ, ita etiam nullus erit auctæ densitatis.
90. Sed & illud commodum accidet, quod ita omne continuum coexistens eliminabitur
e Natura, in quo explicando usque adeo desudarunt, & fere incassum, Philosophi, nec idcirco
divisio ulla realis entis in infinitum produci poterit, nec hærebitur, ubi quæratur, an numerus
partium actu distinctarum, & separabilium, sit finitus, an infinitus ; nec alia ejusmodi
sane innumera, quæ in continui compositione usque adeo negotium facessunt Philosophis,
jam habebuntur. Si enim prima materiæ elementa sint puncta penitus inextensa, &
indivisibilia, a se invicem aliquo intervallo disjuncta ; jam erit finitus punctorum numerus
in quavis massa : nam distantiæ omnes finitæ erunt ; infinitesimas enim quantitates in se
determinatas nullas esse, satis ego quidem, ut arbitror, luculenter demonstravi & in disser-
tatione De Natura, &5 Usu infinitorum, ac infimite parvorum, & in dissertatione De Lege
Continuitatis, & alibi. Intervallum quodcunque finitum erit, & divisibile utique in
infinitum per interpositionem aliorum, atque aliorum punctorum, quæ 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 quæcunque, quæ 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 præteriri. Hoc autem pacto, sublato
ex existentibus omni actuali infinito, innumeræ sane difficultates auferentur.
9I. Cum igitur & positivo argumento, a lege virium positive demonstrata desumpto,
simplicitas, & inextensio primorum materiæ elementorum deducatur, & tam multis aliis
vel indiciis fulciatur, vel emolumentis inde derivatis confirmetur ; ipsa itidem admitti
jam debet, ac supererit quærendum illud tantummodo, utrum hæc elementa homogenea
censeri debeant, & inter se prorsus similia, ut ea initio assumpsimus, an vero heterogenea,
ac dissimilia.
92. Pro homogeneitate primorum materiæ elementorum illud est quoddam veluti
principium, quod in simplicitate, & inextensione conveniant, ac etiam vires quasdam habeant
utique omnia. Deinde curvam ipsam virium eandem esse omnino in omnibus illud indicat,
vel etiam evincit, quod primum crus repulsivum impenetrabilitatem secum trahens, &
postremum attractivum gravitatem definiens, omnino communia in omnibus sint : nam
corpora omnia æque impenetrabilia sunt, & vero etiam æque gravia pro quantitate materiæ
suæ, uti satis [42] evincit æqualis velocitas auri, & plumæ cadentis in Boyliano recipiente
Si reliquus curvæ arcus intermedius esset difformis in diversis materiæ punctis ; infinities
probabilius esset, difformitatem extendi etiam ad crus primum, & ultimum, cum infinities
plures sint curvæ, quæ, cum in reliquis differant partibus, differant plurimum etiam in
hisce extremis, quam quæ in hisce extremis tantum modo tam arcte consentiant. Et hoc
quidem argumento illud etiam colligitur, curvam virium in quavis directione ab eodem
primo materiæ elemento, nimirum ab eodem materiæ puncto eandem esse, cum & primum
impenetrabilitatis, & postremum gravitatis crus pro omnibus directionibus sit ad sensum
idem. Cum primum in dissertatione De Viribus 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 particulæ directionibus,
habetur utique ex diverso numero, & positione punctorum eam componentium, qua de
re inferius aliquid.
93. Nec vero huic homogeneitati opponitur inductionis principium, quo ipsam
Leibnitiani oppugnare solent, nec principium rationis sufficientis, atque indiscernibilium,
quod superius innui numero 3. Infinitam Divini Conditoris 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 illa 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.
90. The theory of non-extension is also convenient for eliminating from Nature all
idea of a coexistent continuum—to explain which philosophers have up till now laboured
so very hard & generally in vain. Assuming non-extension, no division of a real entity
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, {5 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
directly proved, we have both deduced the simplicity & non-extension of the primary
elements of matter, & also we have strengthened the theory by evidence pointing towards
it, or corroborated it by referring to the advantages to be derived from it; this theory
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
speak some foundation derived from the fact that all of them agree in simplicity & non-
extension, & also that they are all endowed with forces of some sort. Now, that this
curve of forces is exactly the same for all of them is indicated or even proved by the fact
that the first repulsive branch necessitating impenetrability, & the last attractive branch
determining gravitation, are exactly the same in all respects. For all bodies are equally
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 Viribus Vivis, 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
from the principle of induction, by means of which the followers of Leibniz usually raise
an objection to it; nor from the principle of sufficient reason, & of indiscernibles, that
I mentioned above in Art. 3. I am indeed quite convinced, & a great many other
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
Also for excluding
the idea of a con-
tinuum in existing
things, that is
extended & infinite.
No n - extension
must be admitted ;
we have now to
investigate homo-
geneity.
Homogeneity for
all points to be ad-
vocated from a
consideration of
the homogeneity
of the first & last
a S y m p to t i c
branches of the
curve of forces.
Nothing to be
brought a ga in st
this from the doc-
trines of indiscern-
ibles & ‘sufficient
reason.’
9O PHILOSOPHIAE NATURALIS THEORIA
JPosse etiam puncta
convenire in iis,
differre in aliis.
INon valere hic prin-
cipium inductionis
a massis : eas de-
ferre ex diversis
combinationibus.
Physica ratio dis-
Criminis in pluribus
massis ut in foliis.
principium falsum omnino esse censeo, ac ejusmodi, ut omnem veræ 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, quæcunque existunt,
necessario existunt, & quæcunque 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 hæc ipsa fusius pertractavi
tum in aliis locis pluribus, tum in Stayanis Supplementis, ubi etiam illud ostendi, id prin-
cipium nullum habere usum posse in iis ipsis casibus, in quibus adhibetur, & prædicari solet
tantopere, atque id idcirco, quod nobis non innotescant rationes omnes, quas tamen
oporteret utique omnes nosse ad hoc, ut eo principio uti possemus, affirmando, nullam
esse rationem sufficientem pro hoc potius, quam pro illo [43] alio : sane in exemplo illo
ipso, quod adhiberi solet, Archimedis hoc principio æquilibrium determinantis, ibidem
Ostendi, ex ignoratione causarum, sive rationum, quæ postea detectæ sunt, ipsum in suæ
investigationis progressu errasse plurimum, deducendo per abusum ejus principii sphæricam
figuram marium, ac Telluris. -
94. Accedit & illud, quod illa puncta materiæ, licet essent prorsus similia in simplicitate,
& extensione, ac mensura virium, pendentium a distantia, possent alias habere proprietates
metaphysicas diversas inter se, nobis ignotas, quæ ipsa etiam apud ipsos Leibnitianos
discriminarent.
95. Quod autem attinet ad inductionem, quam Leibnitiani desumunt a dissimilitudine,
quam observamus in rebus omnibus, cum nimirum nusquam ex. gr. in amplissima silva reperire
sit duo folia prorsus similia ; ea sane me nihil movet ; cum nimirum illud discrimen sit
proprietas relativa ad rationem aggregati, & nostros sensus, quos singula materiæ 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.
Alphabeti diversimodo combinatis formantur voces omnes, quibus huc usque usa sunt
Omnia idiomata, quæ extiterunt, & quibus omnia illa, quæ possunt existere, uti possunt.
Quid si numerus earum existeret tanto major, quanto major est numerus punctorum.
materiæ 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 præscindendo ab æquilibrio virium, est infinitus, cum ipso æquilibrio est immen-
sus) ; tanto major est improbabilitas duarum massarum omnino similium, quam omnium
aliquantisper saltem inter se dissimilium.
96. Et quidem accedit illud etiam, quod alicujus dissimilitudinis in aggregatis physicam
quoque rationem cernimus in iis etiam casibus, in quibus maxime inter se similia esse
deberent. Cum enim mutuæ vires ad distantias quascunque pertineant ; status uniuscu-
jusque puncti pendebit saltem aliquantisper a statu omnium aliorum punctorum, quæ
sunt in Mundo. Porro utcunque puncta quædam sint parum a se invicem remota, uti
sunt duo folia in eadem silva, & multo magis in eodem ramo ; adhuc tamen non eandem
prorsus relationem distantiæ, & virium habent ad reliqua omnia materiæ puncta, quæ
[44] sunt in Mundo, cum non eundem prorsus locum obtineant ; & inde jam in aggregato
discrimen aliquod oriri debet, quod perfectam similitudinem omnino impediat. Sed illud
eam inducit magis, quod quæ 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 9I
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
similar as regards simplicity & extension, & in having the measure of their forces depen-
dent on their distances, might still have other metaphysical properties different from one
another, & unknown to us; & these distinctions also are made by the followers of
Leibniz. -
95. As regards the induction which the followers of Leibniz make from the lack of
similitude that we see in all things, (for instance such as that there never can be found in
the largest wood two leaves exactly alike), their argument does not impress me in the
slightest degree. For that distinction is a property that is concerned with reasoning for
an aggregate, & also with our senses; & these senses single elements of matter cannot
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
dissimilarity in groups for those cases too, in which they ought to be especially similar to
one another. For since mutual forces pertain to all possible distances, the state of any
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.
It is possible for
points also to agree
in these properties
but to disagree in
others. '
The principle does
not hold good here
of induction from
masses; they differ
On a C count Of
different combina-
tions of their parts.
Physical reason for
the difference in
Several masses, as
in leaves.
92 PHILOSOPHIÆ NATURALIS THEORIA
Similitudine quali-
cunque in aliquibus
magis probarihomo-
geneitatem, quam
dissimilitudine he-
terogeneitatem.
Homogeneitatem
ab analysi Naturæ
insinuari : exem-
plum a, libris, lit-
teris, punctulis.
Applicatio exempli
ad Naturæ analy-
simì,
97. Patet igitur, varietatem illam a numero pendere combinationum possibilium in
numero punctorum necessario ad sensationem, & circumstantiarum, quæ ad formationem
massæ sunt necessariæ, adeoque ejusmodi inductionem extendi ad elementa non posse.
Quin immo illa tanta similitudo, quæ cum exigua dissimilitudine commixta invenitur in
tam multis corporibus, indicat potius similitudinem ingentem in elementis. Nam ob
tantum possibilium combinationum numerum, massæ 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 nullæ massæ, ne tantillum quidem, similes provenire deberent. Cum
elementa multo minus dissimilia esse debeant, quam aggregata elementorum, multo
magis ad elementorum homogeneitatem valere debet illa quæcunque similitudo, quam
in corporibus observamus, potissimum in tam multis, quæ ad eandem pertinent speciem,
quam. ad homogeneitatem eorundem tam exiguum illud discrimen, quod in aliis tam
multis observatur. Rem autem penitus conficit illa tanta similitudo, qua superius usi
sumus, in primo crure exhibente impenetrabilitatem, & in postremo exhibente gravitatem
generalem, quæ crura cum ob hasce proprietates corporibus omnibus adeo generales, adeo
inter se in omnibus similia sint, etiam reliqui arcus curvæ exprimentis vires omnimodam
similitudinem: indicant pro corporibus itidem omnibus.
98. Superest, quod ad hanc rem pertinet, illud unum iterum hic monendum, quod
ipsum etiam initio hujus Operis innui, ipsam Naturam, & ipsum analyseos ordinem nos
ducere ad simplicitatem & homogeneitatem elementorum, cum nimirum, quo analysis
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 litteræ, non ductu atramenti
continuo, sed punctulis rotundis nigricantibus, & ita parum a se invicem remotis, ut inter-
valla non nisi ope microscopii discerni possent, & quidem ipsæ litterarum formæ pro typis
fieri pos-[45]-sent ex ejusmodi rotundis sibi proximis cuspidibus constantes. Concipiatur
ingens quædam bibliotheca, cujus omnes libri constent litteris impressis, ac sit incredibilis
in ea multitudo librorum conscriptorum linguis variis, in quibus omnibus forma charac-
terum sit eadem. Si quis scripturæ ejusmodi, & linguarum ignarus circa ejusmodi libros,
quos omnes a se invicem discrepantes intueretur, observationem institueret cum diligenti
contemplatione ; primo quidem inveniret vocum farraginem quandam, quæ voces in
quibusdam libris occurrerent sæpe, cum eædem in aliis nusquam apparent, & inde lexica
posset quædam componere totidem numero, quot idiomata sunt, in quibus singulis Omnes
ejusdem idiomatis voces reperirentur, quæ quidem numero admodum pauca essent, discri-
mine illo ingenti tot, tam 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 tam yarias voces constare ex 24
tantummodo diversis litteris, discrimen aliquod inter se habentibus in ductu linearum,
quibus formantur, quarum combinatio diversa pareret omnes illas yoces tam varias, ut
earum combinatio libros efformaret usque adeo magis a se invicem discrepantes. Et ille
quidem si aliud quodcunque sine microscopio examen institueret, nullum aliud inyeniret
magis adhuc simile elementorum genus, ex quibus, diversa ratione cgmbinatis orirentur
ipsâ litteræ ; at microscopio arrepto, intueretur utique illam ipsam litterarum cqmposi-
tionem e punctis illis rötundis prorsus homogeneis, quorum sola diversa positio, ac
distributio litteras exhiberet.
99. Hæc mihi quædam imago videtur esse eorum, quæ cernimus in Natura. Tam
multi, tam varii illi libri corpora sunt, & quæ ad diversa pertinent regna, sunt tanquam
diversis conscripta linguis. Horum omnium Chemica analysis principia quædam inyenit
minus inter se difformia, quam sint libri, nimirum voces, Hæ tamen ipsæ 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 ihveniret, & ultima juxta Theoriam meam deveniret ad homogenea punctula,
quæ ut illi circuli nigri litteras, ita ipsa diversas diversgrum corporum particulas per solam
dispositionem diversäm efformarent? usque adeo analogia ex ipsa Naturæ consideratione
A THEORY OF NATURAL PHILOSOPHY
93
97. It is clear then that this variety depends on the number of possible combinations
to be found for the number of points that are necessary to make the mass sensible, &
of the circumstances that arenecessary for the formation of the mass ; & so it is not
possible that the induction should be extended to the elements. Nay rather, the great
similarity that is found accompanied by some very slight dissimilarity in so many bodies
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
this connection that one thing, which I have already remarked at the beginning of this
work, namely, that Nature itself & the method of analysis lead us towards simplicity &
homogeneity of the elements; since in truth the farther the analysis is pushed, the fewer
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
books, so many in number & so different in character are bodies, & those which belong
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
Homogeneity is to
be demonstrated
from any sort of
similitude in some
cases more than
heterogeneity from
dissimilarity.
Homogeneity is
Suggested by an
analysis of Nature;
example taken
from books, letters
and dots.
Application of the
illustration to the
analysis of Nature.
94 PHILOSOPHIÆ NATURALIS THEORIA
Transitus a pro-
batione Theoriæ ad
objectiones.
Legem virium non
inducere actionem
in distans, nec esse
Occultam qualita-
tem.
Quid adhuc lateat :
admittendam om-
nino : quo pacto
evitetur hic actio
in distans.
Sine im pu 1 sione
melius explicatam
esse hucusque Na-
turam, & melius ex-
plicandam. impost-
€IüIO.
derivata non ad difformitatem, sed ad conformitatem elementorum nos ducit.
IOO. Atque hoc demum pacto ex principiis certis & vulgo receptis, per legitimam,
consectariorum seriem devenimus ad omnem illam, quam initio proposui, Theoriam,
nimirum ad legem virium mutuarum, & ad constitutionem primorum materiæ elementorum
ex illa ipsa virium lege derivatorum. [46] Videndum jam superest, quam uberes inde
fructus per universam late Physicam colligantur, explicatis per eam unam præcipuis cor-
porum proprietatibus, & Naturæ phænomenis. Sed antequam id aggredior, præcipuas
quasdam e difficultatibus, quæ contra Theoriam ipsam vel objectæ jam sunt, vel in oculos
etiam sponte incurrunt, dissolvam, uti promisi.
IOI. Contra vires mutuas illud solent objicere, illas esse occultas quasdam qualitates,
vel etiam actionem in distans inducere. His satisfit notione virium exhibita numero 8,
& 9. Illud unum præterea hic addo, admodum manifestas eas esse, quarum idea admodum
facile efformatur, quarum existentia positivo argumento evincitur, quarum effectus multi-
plices continuo oculis observantur. Sunt autem ejusmodi hæ 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, quæ 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 particulæ se invicem repellunt, ubi gravia descendunt, hi
motus, inquam, quotidie incurrant in oculos. Patet itidem saltem in genere forma curvæ
ejusmodi vires exprimentis. Hæc omnia non occultam, sed patentem reddunt ejusmodi
virium legem.
Io2. Sunt quidem adhuc quædam, quæ ad eam pertinent, prorsus incognita, uti est
numerus, & distantia intersectionum curvæ cum axe, forma arcuum intermediorum, atque
alia ejusmodi, quæ quidem longe superant humanum captum, & quæ ille solus habuit
omnia simul præ oculis, qui Mundum condidit ; sed id omnino nil officit. Nec 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 appellarit,
& multo minus ejusdem existentiam negabit idcirco, quod admodum probabile sit, plures
alias latere ipsius proprietates, olim forte detegendas, uti aliæ tam multæ subinde detectæ
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 prævenimus, 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 pün-[47]-cto. Illud sane mihi est evidens, nihilo magis occultam esse,vel explicatu,
& captú difficilém productionem motus per hasce vires pendentes a certis distantiis, quam
sit productio motus vulgo concepta per immediatum impulsum, ubi ad mgtum determinat
impenetrabilitas, quæ itidem vel a corporum natura, vel a libera conditoris lege repeti
debet.
Io3. Et quidem hoc potius pacto, quam per impulsionem, in motuum causas, & leges
inquirendum esse, illud etiam satis indicat, quod ubi huc usque, impulsione omissa, vires
adhibitæ sunt a distantiis pendentes, ibi sane tantummodo accurate definita sunt omnia,
atque determinata, & ad calculum redacta cum phænQmenis congruunt ultra, quam sperare
licéret, accuratissime. Ego quidem ejusmodi in explicando, ac determinando felicitatem
nusquam alibi video in universa Physica, nisi tantummodo in Astronomia mechanica, quæ
abjectis vorticibus, atque omni impulsione submota, per gravitatem generalem absolvit
ominia, ac in Theoria luminis, & colorum, in quibus per vires in aliqua distantia agentes,
& reflexionem, & refractionem, & diffractionem Newtonus exposuit, ac prigrum duarum,
potissimum leges omnes per calculum, & Geometriam determinavit, & ubi illa etiam, quæ
ád 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
legitimate series of deductions, we have arrived at the whole of the Theory that I enunciated
at the start; that is to say, at a law of mutual forces & the constitution of the primary
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.
IoI. The objection is frequently brought forward against mutual forces that they
are some sort of mysterious qualities or that they necessitate action at a distance. This
is answered by the idea of forces outlined in Art. 8, & 9. In addition, I will make just
one remark, namely, that it is quite evident that these forces exist, that an idea of them
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,
Passing from the
proof of the Theory
to the considera-
tion of objections
against it.
The law of forces
does not necessi-
tate action at a
distance, nor is it
some mysterious
quality.
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
altogether ignorant, such as the number & distances of the intersections of the curve
with the axis, the shape of the intervening arcs, & other things of that sort; these indeed
far surpass human understanding, & He alone, Who founded the universe, had the whole
before His eyes. But truly there is no reason on that account, why a thing, whose existence
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.
Iog. Now, that the investigation of the causes & laws of motion are better made
by my method, than through the idea of impulse, is sufficiently indicated by the fact that,
where hitherto we have omitted impulse & employed forces depending on the distances,
only in this way has everything been accurately defined & determined, & when reduced
to calculation everything agrees with the phenomena with far more accuracy than we
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
What is so far un,
known ; the theory
to be admitted in
all detail; the way
in which the idea
of action at a dis-
tance is eliminated.
As far as we have
gone, Nature has
been more clearly
explained without
the idea of impulse;
and what follows
will be so too.
96 PHILOSOPHIAE NATURALIS THEORIA
Non fieri saltum in
transitu a vi at-
tractiva ad repul-
sivam.
Nullum esse post-
remum attractionis,
& primum repulsio-
nis gradum, qui si
essent, adhuc tran-
sire per omnes in-
termedios.
Objectio ab appar-
enti compositione
curvæ, & duobus
virium generibus.
EResponsio : vi m
repulsivam positive
demonstrari præter
attractionem.
fere intactas, ac alia multa admodum feliciter determinantur, explicanturque, quod & ego
præstiti in dissertatione De Lumine, & præstabo hic in tertia parte ; cum in ceteris Physicæ
partibus plerumque explicationes habeantur subsidariis quibusdam principiis innixæ &
vagæ 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. -
Io4. Solent & illud objicere, in hac potissimo Theoria virium committi saltum illum,
ad quem 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 quæcunque vis repulsiva utcunque parva ; tum quæcunque vis attractiva.
Inter eas intercedunt omnes vires repulsivæ minores usque ad zero, in quo habetur deter-
minatio ad conservandum præcedentum statum quietis, vel motus uniformis in directum :
tum omnes vires attractivæ a ze-[48]-ro usque ad eam determinatam vim, & omnino nullus
erit ex hisce omnibus intermediis statibus, quem aliquando non sint habitura puncta, quæ
a repulsione abeunt ad attractionem. Id ipsum facile erit contemplari in fig. I, in qua a
vi repulsiva br ad attractionem dh itur utique continuo 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 ; tum omnes oppositas
majores usque ad posteriorem dh. Qui in ea veluti imagine mentis oculos defigat, is omnem
apparentem difficultatem videbit plane sibi penitus evanescere.
Io5. Quod autem additur de postremo repulsionis gradu, & primo attractionis nihil
sane probaret, quando etiam essent aliqui ii gradus postremi, & primi ; nam ab altero
eorum transiretur ad alterum per intermedium illud zero, & ex eo ipso, quod illi essent
postremus, ac primus, nihil omitteretur intermedium, quæ tamen sola intermedii omissio
continuitatis legem evertit, & saltum inducit. Sed nec habetur ullus gradus postremus,
aut primus, sicut nulla ibi est Ordinata postrema, aut prima, nulla lineola omnium minima.
Data quacunque lineola utcunque exigua, aliæ illa 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 hic innui, & quam ego
idcirco initio meæ dissertationis De Lege Continuitatis abunde exposui.
Io6. Videri potest cuipiam saltem illud, ejusmodi legem virium, & curvam, quam in
fig. I protuli, esse nimium complicatam, compositam, & irregularem, quæ nimirum coalescat
ex ingenti numero arcuum jam attractivorum, jam repulsivorum, qui inter se nullo pacto
cohæreant ; rem eo redire, ubi erat olim, cum apud Peripateticos pro singulis proprietatibus
corporum singulæ qualitates distinctæ, & pro diversis speciebus diversæ formæ substantiales
confingebantur 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.
Io7. Inprimis quod ad hoc postremum pertinet, satis patet, per positivam meæ
Theoriæ probationem immediate evinci repulsionem ita, ut a minore attractione repeti
omnino non possit ; nam duæ materiæ particulæ si etiam solæ in Mundo essent, & ad se
invicem cum aliqua velocitatum inæqualitate accederent, deberent utique ante contactum
ad æqualitatem devenire vi, quæ 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.
IO4. It is very frequently objected that, in this Theory more especially, a sudden change
is made in the forces, whilst the theory is to be accepted for the very purpose of avoiding
such a thing. For it is said that the transition from attractions to repulsions is made
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 brto 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. -
IoS. Further, as regards what is said in addition about the last stage of repulsion &
the first stage of attraction, it would really not matter, even if there were these so called
last & first stages; for, from one of them to the other the passage would be made through
the one intermediate stage, namely zero; since it passes zero, & because they are the first
& last, therefore no intermediate stage is omitted. Nevertheless the omission of this
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.
IO6. It may seem to some that at least a law of forces of this nature, & the curve
expressing it, which I gave in Fig. I, is very complicated, composite & irregular, being
indeed made up of an immense number of arcs that are alternately attractive & repulsive,
& that these are joined together according to no definite plan; & that it reduces to
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.
IO7. First of all, as regards the last objection, it is clear enough from what has been
directly proved in my Theory that the existence of repulsion has been rigorously demonstrated
in such a way that it cannot possibly be derived from the idea of a smaller attraction. For
two particles of matter, if they were also the only particles in the universe, & approached
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
There is no sudden
change in the tran-
sition from an at-
tractive to a repul-
Sive force.
There is no last
stage of attraction,
and no first for re-
pulsion ; and even
if there were, the
passage would be
made through all
intermediate stages.
Objection raised
against the appar-
ent composite cha-
racter of the curve,
and the two kinds
of forces.
In reply ; it is
possible to prove
directly the exist-
ence of a repulsive
force apart from
attraction.
98 PHILOSOPHIÆ NATURALIS THEORIA
Hinc nihil obstare,
si diversi sint gene-
ris ; sed esse ejus-
dem, uti sunt posi-
tiva, & negativa.
Probatio hujus a
progressu, & re-
gressu, in fluvio.
Probatio ex Alge-
bra, & Geometria :
applicatio ad omnes
quantitates varia-
biles.
Io8. Deinde vero quod pertinet ad duas diversas species attractionis, & repulsionis ;
id quidem licet ita se haberet, ni-[49]-hil sane obesset, cum positivo argumento evincatur
& repulsio, & attractio, uti vidimus ; at id ipsum est omnino falsum. Utraque vis ad
eandem pertinet speciem, cum altera respectu alterius negativa sit, & negativa a positivis
specie non differant. Alteram negativam esse respectu alterius, patet inde, quod tantum-
modo differant in directione, quæ in altera est prorsus opposita directioni 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, & minus non differunt specie. Nam a positivo per continuam subtrac-
tionem, nimirum diminutionem, habentur prius minora positiva, tum zero, ac demum
negativa, continuando subtractionem eandem. .
Io9. Id facile patet exemplis solitis. Eat aliquis contra fluvii directionem versus locum
aliquem superiori alveo proximum, & singulis minutis perficiat remis, vel vento Ioo hexapedas,
dum a cursu fluvii retroagitur per hexapedas 4o ; is habet progressum hexapedarum 6o
singulis minutis. Crescat autem continuo impetus fluvii ita, ut retroagatur per 5o, tum per
60, 70, 80, 90, IOO, I Io, I2o, &c. Is progredietur per 5o, 4o, 3o, 2o, Io, nihil ; tum
regredietur per IO, 2o, quæ erunt negativa priorum ; nam erat prius Ioo—5o, IOO—6o,
IOO —7O, IOO—8o, IOO—90, tum IOO — IOO = o, IOO— IIo, = — Io, IOO— I2O = — 2o, et ita
porro. Continua imminutione, sive subtractione itum est a positivis in negativa, a
progressu ad regressum, in quibus idcirco eadem species mansit, non duæ diversæ.
I Io. Idem autem & algebraicis formulis, & geometricis lineis satis manifeste ostenditur.
Sit formula Io— x, & pro x ponantur valores 6, 7, 8, 9, Io, II, 12, &c. ; valor formulæ
exhibebit 4, 3, 2, I, o, — I, —2, &c., quod eodem redit, ubi erat superius in progressu, &
regressu, qui exprimerentur simul per formulam Io— x. Eadem illa formula per continuam
mutationem valoris x migrat e valore positivo in negativum, qui æque ad eandem formulam
pertinent. Eodem pacto in Geometria in fig. -
II, si duæ lineæ MN, OP referantur invicem E
per ordinatas AB, CD, &c. parallelas inter se, O
secent autem se in E ; continuo motu ipsius
ordinatæ a positivo abitur in negativum, mutata
directione AB, CD, quæ hic habentur pro
positivis, in FG, HI, post evanescentiam in E.
Ad eandem lineam continuam OEP æque
pertinet omnis ea ordinatarum series, nec est G I
altera linea, alter locus geometricus OE, ubi
ordinatæ sunt positivæ, ac EP, ubi sunt nega- FIG. II.
tivæ. 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 ; illa etiam quantitas semper positiva manebit. Sed si mutet latus linea,
vel formula valoris signum ; ipsa illa quantitatis debebit itidem ejusmodi mutationem
An habeatur trans-
itus e positivis in
negativa ; investi-
gatio ex sola, curv-
arum natura.
habere. Ut autem a formulæ, vel lineæ exprimentis natura, & positione respectu axis
mutatio pendet ; ita mutatio eadem a natura quantitatis illius pendebit ; & ut nec duæ
formulæ, nec duæ lineæ speciei diversæ sunt, quæ positiva exhibent, & negativa ; ita nec in
ea quantitate duæ erunt naturæ, duæ species, quarum altera exhibeat positiva, altera
negativa, ut altera progressus, altera regressus ; altera accessus, altera recessus ; & hic altera
attractiones, altera repulsiones exhibeat ; sed eadem erit, unica, & ad eandem pertinens
quantitatis speciem tota.
II I. Quin immo hic 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, yel
repulsivarüm tantummodo, an utrumque simul: hac sane ratiocinatione ad eam perquisi-
tiónem uti liceret. Erit utique aliqua linea continua, quæ 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
were a fact that they were so, it would not matter in the slightest degree, since by rigorous
argument the existence of both attraction & repulsion is proved, as we have seen ; but
really the supposition is untrue. Both kinds of force belong to the same species; for one
is negative with regard to the other, & a negative does not differ in species from positives.
That the one is negative with regard to the other is evident from the fact that they only
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.
Io9. The matter is easily made clear by the usual illustrations. Suppose a man
to go against the current of a river to some place on the bank up-stream; & suppose
that he succeeds in doing, either by rowing or sailing, IOO fathoms a minute, whilst he
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, IOO, IIo, 120,
&c. fathoms per minute. His forward motion will be successively 50, 40, 30, 20, IO 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, IOO – 60, IOO – 70, IOO - 80, IOO – 90, then IOO – IOO (which = 0),
then Ioo – IIo (which = – IO), IOO – I2O (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.
IIo. Further, the same thing is shown plainly enough by algebraical formulae, &
by lines in geometry. Consider the formula IO – w, & for x substitute the values, 6,
7, 8, 9, Io, II, 12, &c.; then the value of the formula will give in succession 4, 3, 2,
I, O,
progressive & retrograde motion, which may be expressed by the formula IO – 3, 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. II, 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 formulae, 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. - -
- III. Lastly, this is the proper place for me to bring forward an argument that I used
in the dissertation De Lege Continuitatis ; by it indeed it is proved that a theory of attractive
& repulsive forces for different distances is far more reasonable than one of attractive
forces only, or of repulsive forces only. Let us imagine that we are quite ignorant of the
kind of forces that exist in Nature, whether they are only attractive or only repulsive, or
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
Hence it does not
matter if they are
of different kinds ;
but as a matter of
fact they are of the
same kind, just as
a positive and a
negative are so.
Demonstration by
means of progres-
sive and retrograde
motion on a river.
Proof from algebra
and geometry; ap-
plication to a 1 1
variable quantities.
– I, - 2, &c.; & this comes to the same thing as we had above in the case of the
Whether there can
be a transition
from positive to
negative ; inves-
tigation by means
of the nature of the
curve only.
IOO PHILOSOPHIÆ NATURALIS THEORIA
Transitum deduci
ex eo, quod plures
sint curvæ, quas
recte secent, quam
eæ, quaS non se-
cent. -* •
Ulterior per qui-
sitio : curvarum
genera : quo alti-
Ores, eo in pluribus
punctis secabiles
a recta.
Quo altiores, eo
itidem in immen-
sum plures in eo-
dem genere.
Deductio inde plu-
rimarum intersec-
tionum, axis, &
curvæ exprimentis
V1reS.
Curvam virium pro-
positam posse esse
simplicem : in quo
sita sit curvarum
simplicitas.
secuerit ; vires erunt alibi attractivæ, alibi repulsivæ ; vel ubique attractivæ tantum, aut
repulsivæ tantum. Videndum igitur, an sit rationi consentaneum magis, lineam ejus
naturæ, & positionis censere, ut axem alicubi secet, an ut non secet.
II2. Inter rectas axem rectilineum unica parallela ducta per quodvis datum punctum
non secat, omnes aliæ numero infinitæ secant alicubi. Curvarum nulla est, quam infinitæ
numero rectæ secare non possint ; & licet aliquæ curvæ ejus naturæ sint, ut eas aliquæ rectæ
non secent ; tamen & eas ipsas aliæ infinitæ numero rectæ secant, & infinitæ numero curvæ,
quod Geometriæ sublimioris peritis est notissimum, sunt ejus naturæ, ut nulla prorsus sit
recta linea, a qua possint non secari. Hujusmodi ex. gr. est parabola illa, cujus Ordinatæ
sunt in ratione triplicata abscissarum. Quare infinitæ numero curvæ sunt, & infinitæ
numero rectæ, quæ sectionem necessario habeant, pro quavis recta, quæ non habeat, & nulla
est curva, quæ sectionem cum axe habere non possit. Ergo inter casus possibiles 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, quæ vires exprimat, esse unam ex iis,
quæ axem secant, quam ex iis, quæ 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.
II 3. Sed eodem argumento licet ulterius quoque progredi, & primum etiam difficultatis
caput amovere, quod a sectionum, & idcirco etiam arcuum jam attractivorum, jam repulsi-
vorum multiplicitate desumitur. Curvas lineas Geometræ in quasdam classes dividunt
ope analyseos, quæ earum naturam exprimit per illas, quas Analystæ appellant, æquationes,
& quæ ad varios gradus ascendunt. Aequationes primi gradus exprimunt rectas ; æquati-
ones secundi gradus curvas primi generis ; æquationes tertii gradus curvas secundi generis,
atque ita porro ; & sunt curvæ, quæ omnes gradus transcendunt finitæ Algebræ, & quæ
idcirco dicuntur transcendentes. Porro illud demonstrant Geometræ in Analysi ad
Geometriam applicata, lineas, quæ exprimuntur per æquationem primi gradus, posse
secari a recta in unico puncto ; quæ æquationem 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.
II4. Jam vero curvæ primi generis sunt tantummodo tres conicæ sectiones, ellipis,
parabola, hyperbola, adnumerato ellipsibus etiam circulo, quæ 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 curvæ
genus, eo plures in eo genere sunt curvæ, progressione ita in immensum crescente, ut ubi
aliquanto altius ascenderit genus ipsum, numerus curvarum omnem superet humanæ
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, & quæ fuerunt, aut sunt, & quæ esse possunt.
II 5. Inde jam pronum est argumentationem hujusmodi instituere. Numerus
linearum, quæ axem secare possint in punctis quamplurimis, est in immensum major earum
numero, quæ 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.
I I6. Sed omissa ista conjecturali argumentatione quadam, formam curvæ exprimentis
vires positivo argumento a phænomenis Naturæ deducto nos supra determinavimus cum
plurimis intersectionibus, quæ transitus ejusmodi quamplurimos exhibeant. Nec ejusmodi
curva debet esse e pluribus arcubus temere compaginata, & compacta : diximus enim,
A THEORY OF NATURAL PHILOSOPHY IOI
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.
II2. Amongst straight lines there is only one, drawn parallel to the rectilinear axis,
through any given point that does not cut the axis; all the rest (infinite in number) will
cut it somewhere. There is no curve that an infinite number of straight lines cannot cut;
& although there are some curves of such a nature that some straight lines do not cut them,
yet there are an infinite number of other straight lines that do cut these curves; & there
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, &
first of all remove the chief point of the difficulty, that is derived from the multiplicity
of the intersections, & consequently also of the arcs alternately attractive & repulsive.
Geometricians divide curves into certain classes by the help of analysis, which expresses
their nature by what the analysts call equations; these equations rise to various degrees.
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. -
II.4. Now there are only three curves of the first class, namely the conic sections, the
parabola, the ellipse & the hyperbola; the circle is included under the name of ellipse;
& these three curves were known to the ancient geometricians also. Newton was the
first of all persons to enumerate the curves of the second class, & there are about eighty
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,
Intersection is to:
be inferred from
the fact that there
are more lines that
cut a straight line
than lines that do
not.
Further investiga-
tion ; classes of
curves ; the higher
their order, the
more the points in
which a straight
line can cut them.
As the class gets
higher, the number
of curves of that
class becomes im-
mensely greater.
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. -
II5. From what has been said above we are led to set up the following line of argument.
The number of lines that can cut the axis in very many points is immensely greater than
the number of those that can cut it in a few points only, or in a single point. Hence, when
the line representing the law of forces is in question, it will appear to one, who otherwise
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
determined, by what has been said above, the form of the curve representing forces by a
rigorous argument derived from the phenomena of Nature, & that there are very many
intersections which represent just as many of these transitions. Further, a curve of this
Hence we deduce
that there are very
many intersections
of the axis and the
curve representing
forces.
It may be that the
curve of forces is
simple ; the charac-
teristic of simplicity
in curves.
IO2 PHILOSOPHIÆ NATURALIS THEORIA
Problema continens
naturam curvæ ana-
lytice exprimendam.
Conditiones
problematis.
ej u s
notum esse Geometris, infinita esse curvarum genera, quæ ex ipsa natura sua debeant axem
in plurimis secare punctis, adeoque & circa ipsum sinuari ; sed præter hanc generalem
responsionem desumptam a generali curvarum natura, in dissertatione De Lege Virium im
Natura existentium ego quidem directe demonstravi, curvam illius ipsius formæ, cujusmodi
ea est, quam in fig. I exhibui, simplicem esse posse, non ex arcubus diversarum cürvarum
compositam. Simplicem autem ejusmodi curvam affirmavi esse posse : eam enim simplicem
appello, quæ tota est uniformis naturæ, quæ in Analysi exponi possit per æquationem non
resolubilem in plures, e quarum multiplicatione eadem componatur cujuscunque demum
ea curva sit generis, quotcunque habeat flexus, & contorsiones. Nobis quidem altiorum
generum curvæ videntur minus simplices ; quia nimirum nostræ humanæ 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, quæ lineæ a recta recedunt magis, & discrepant, illas habemus pro compositis, &
magis ab ea simplicitate, quam nobis confinximus, recedentibus. At vero lineæ continuæ,
& uniformis naturæ omnes in se ipsis sunt æque simplices ; & aliud mentium genus, quod
cujuspiam ex ipsis proprietatem aliquam æque evidenter intueretur, ac nos intuemur
congruentiam rectarum, illas maxime simplices esse crederet curvas lineas, ex illa earum
proprietate longe alterius Geometriæ sibi elementa conficeret, & ad illam ceteras referret
lineas, ut nos ad rectam referimus ; quæ quidem mentes si aliquam ex. gr. parabolæ pro-
prietatem intime perspicerent, atque intuerentur, non illud quærerent, quod nostri
Geometræ quærunt, ut parabolam rectificarent, sed, si ita loqui fas est, ut rectam
parabolarent. •
I I7. Et quidem analyseos ipsius profundiorem cognitionem requirit ipsa investigatio
æquationis, qua possit exprimi curva ejus formae, quæ meam exhibet virium legem.
Quamobrem hic tantummodo exponam conditiones, quas ipsa curva habere debet, & quibus
æquatio ibi inventa satis facere [53] debeat. (•) Continetur autem id ipsum num. 75,
illius dissertationis, ubi habetur hujusmodi Problema : Invenire maturam curvae, cujus
abscissis exprimentibus distantias, ordinatae exprimant vires, mutatis distantiis utcumque
mutatas, & in datis quotcumque limitibus transeuntes e repulsivis in attractivas, ac ex attractivis
in repulsivas, in minimis autem distantiis repulsivas, &5 ita crescentes, ut sint pares extinguendae
cuicumque velocitati utcumque magnae. Proposito problemate illud addo : quoniam posuimus
mutatis distantiis utcunque mutatas, complectitur propositio etiam rationem quae ad rationem
reciprocam duplicatam distantiarum accedat, quantum libuerit, in quibusdam satis magnis
distantiis.
I 18. His propositis numero illo 75, sequenti numero propono sequentes sex conditiones,
quæ requirantur, & sufficiant ad habendam curvam, quæ quæritur. Primo : ut sit regularis,
ac simplex, & non composita ex aggregato arcuum diversarum curvarum. Secundo : ut $ecet
axem C'AC figurae I. tantum in punctis quibusdam datis ad bimas distantias AE', AE ; AG',
AG ; & ita porro aequales (d) hinc, & inde. Tertio : ut singulis abscissis respondeant singulæ
ordinatae. (e) Quarto : ut sumptis abscissis aequalibus hinc, &5 inde ab A, respondeant ordinatae
(c) Qui velit ipsam rei determinationem videre, poterit hic in fine, ubi Supplementorum, § 3. exhibebitur $olutio
problematis, quæ in memorata dissertatione continetur a num. 77. ad I Io. Sed &$ numerorum ordo, &5 figurarum
mutabitur, ut cum reliquis hujusce operis cohaereat.
Addetur praeterea eidem §. postremum scholium pertinens ad quaestionem agitatam ante hos aliquot annos Parisiis ;
am vis mutua inter materiæ particulas debeat omnino exprimi per solam aliquam distantiae potentiam, an po$$it per
aliquam ejus functionem ; & constabit, posse utique per functionem, ut hic ego praesto, quae uti superiore numero de curvis
est dictum, est in se aeque simplex etiam, ubi mobis potentias ad ejus expre$sionem adhibentibus videatur admodum
composita. •
(d) Id, ut & quarta conditio, requiritur, ut curva utrinque sit sui similis, quod ipsam magis uniformem reddit 3
quamquam de illo crure, quod est citra a$ymptotum AB, mihil est, quod soliciti simus ; cum ob vim repulsivam imminutis
distantiis ita in infimitum excrescentem, non possit abscissa distantiam exprimens umquam evadere zero, {5 abire in
negativam. » •
(e) Nam singulis distantiis singulæ vires respondent.
A THEORY OF NATURAL PHILOSOPHY IO3
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
Virium 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 Maris 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.
II.7. The investigation of the equation, by which a curve of the form that will represent
my law of forces can be expressed, requires a deeper knowledge of analysis itself. Wherefore
I will here do no more than set out the necessary requirements that the curve must fulfil
& those that the equation thereby discovered must satisfy.() It is the subject of Art. 75
of the dissertation De Lege Virium, where the following problem is proposed. Required
to find the nature of the curve, whoſe abscissae represent distances & whoſe ordinates represent
forces that are changed as the distances are changed in any manner, & pass from attractive
forces to repulsive, {5 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. - - -
II8. In addition to what is proposed in this Art. 75, I set forth in the article that
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, 85 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, 4G", AG, and so on, are equal (d) in pairs on each side of A [see p. 80].
(iii) To each abscissa there shall correspond one ordinate & one only. (e)
(iv) To equal abscissae, 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 so at 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 IIo. But here both the numbering of the articles & 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 ; 85 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, & 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 & then become negative.
(e) For to each distance one force, & and only one, corresponds. -
Problem dealing
with the analytical
expression of the
nature of the curve.
The conditions of
the problem.
IO4 PHILOSOPHIÆ NATURALIS THEORIA
Curvæ virium re-
solutio in attrac-
tionem gravitatis
Newtonianam, &
alia m quandam
V1IY).
Hujus posterioris
vis resolutio in alias
plures.
Non obesse theo-
ria m gravitatis;
cujus lex in minimis
distantiis l o c u m
non habet.
E x plan et a ru m
apheliis erui eam
legem quamprox-
ime, non accurate.
aequales. Quinto : ut habeant rectam AB pro asymptoto, area asymptotica BAED existente (!)
infinita. Sexto : ut arcus binis quibuscumque inter$ectionibus terminati po$sint variari, ut
libuerit, &5 ad quascumque distantias recedere ab axe C'AC, ac accedere ad quoscumque quarum-
cumque curvarum arcus, quantum libuerit, eos $ecando, vel tangendo, vel osculando ubicumque,
& quomodocumque libuerit.
[54] I I9. Verum quod ad multiplicitatem virium pertinet, quas diversis jam Physici
nominibus appellant, illud hic 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, quæ habeat accurate ordinatas in ratione reciproca
duplicata distantiarum, quæ quidem erit quædam velut continuatio cruris VTS, tum singulis
ordinatis ag, dh curvæ virium expressæ in fig. I. adjungere ordinatas hujus novæ hyperbolæ
ad partes AB incipiendo a punctis curvæ g, h, & eo pacto Orietur nova quædam curva, quæ
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 illa. Tum poterit dici, puncta omnia materiæ habere gravitatem decrescentem
accurate in ratione reciproca duplicata distantiarum, & simul habere vim aliam expressam
ab illa nova curva : nam idem erit, concipere simul hasce binas leges virium, ac illam
præcedentem unicam, & iidem effectus orientur.
I2o. Eodem pacto hæc 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 æqualia ordinatis
respondentibus novæ legi assumptæ. Hoc pacto habebuntur plures etiam vires diversæ,
quod aliquando, ut in resolutione virium accidere diximus, inserviet ad faciliorem deter-
minationem effectuum, & ea erit itidem vera virium resolutio quædam ; sed id omne erit
nostræ mentis partus quidam ; nam reipsa unica lex virium habebitur, quam in fig. I.
exposui, & quæ ex omnibus ejusmodi legibus componetur. -
121. Quoniam autem hic mentio injecta est gravitatis decrescentis accurate in ratione
reciproca duplicata distantiarum ; cavendum, ne cui difficultatem aliquam pariat illud,
quod apud Physicos, & potissimum apud Astronomiæ mechanicæ 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
particulæ, omnin6 plurimum discrepat a gravitate, quæ sit in ratione reciproca duplicata
áistantiarum. Nam & vapores, qui tantam exercent vim ad se expandendos, repulsionem
habent utique in illis minimis distantiis a se invicem, non attractionem ; & ipsa attractio,
quæ in cohæsione se prodit, est illa quidem in immensum major, quam, quæ ex generali
gravitate consequitur; cum ex ipsis Newtoni compertis attractio grayitati respondens [55]
in globos 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 particulæ diametro totius Terræ, adeoque penitus insen-
sibilis. Et idcirco Newtonus aliam admisit vim pro cohæsione, quæ decrescat in ratione
majore, quam sit reciproca duplicata distantiarum ; & multi ex Newtonianis admiserunt
ę ç 4, b • . . • « * « • • •
V1IY). respondentem huic formulæ 7,3 -+ a, cujus prior pars respectu posterioris s1t 1n
immensum minor, ubi x sit in immensum major unitate assumpta ; sit vero major, ubi *
sit in immensum minor, ut idcirco in satis magnis distantiis evanescente ad sensum prima
parte, vis remaneat quam proxime in ratione reciproca duplicáta distantiarum x, in minimis
vero distantiis sit quam proxime in ratione reciproca triplicata: usque adeo ne apud
Newtonianos quidem servâtur omnino accurate ratio duplicata distantiarum.
122. Demonstravit quidem Newtonus, in ellipsibus planetariis, eam, quam Astronomi
lineam apsidum nominant, & est axis ellipseos, habituram ingentem motum, si ratio yirium
a recipröca duplicata distantiarum aliquanto magis aberret, cumque ad sensum quiescant
(f) Id requiritur, quia in Mechanica demonstratur, aream curvæ, cujus abscissæ exprimant distantias, & ordinatæ
vires, exprimere incrementum, vel decrementum quadrati velocitatis : quare ut illae vires sint pares extinguendæ veloci-
tati cuivis utcunque magnæ, debet illa area e$$e omni finita major.
A THEORY OF NATURAL PHILOSOPHY
IO5
(v) The straight line AB shall be an asymptote, and the asymptotic area BAED shall be
infinite.() -
(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
by different names, it can also here be observed that, if anyone wants to consider one of these
séparately, the curve though it is of itself quite one-fold can yet be divided into several
parts by a sort of mental & fictitious resolution of the forces. Thus, for instance, if
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 WTS. 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 oc, & 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. -
120. In the same manner this new curve can be divided into two others, or several
others, by considering some other force, in some way or other accurately obeying certain
fixed laws, & at the same time altering the curve just obtained by translating the points of it
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,
Resolution of the
curve of forces into
t he Newtonian
attraction of
gravitation and
some other force.
The resolution of
this latter force
into several other
forces.
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. 1, & 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
inverse ratio of the squares of the distances, it is to be remarked that no one should make
any difficulty over the fact that, amongst physicists & more especially those who deal with
celestial mechanics, it is considered as an established fact that gravitation decreases accurately
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 lay 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 ſw" + b x*; in this the first term is immensely less than the second, when &
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 & ; 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
Astronomers call the apsidal line, i.e., the axis of the ellipse, would have a very great motion,
if the ratio of the forces varied to any great extent from the inverse ratio of the squares
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 abscissae represent distances
& 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.
The the or y of
gravitation is not
in opposition ; this
law does not hold
good at very small
distânces.
The law follows
very nearly, but
not a c curately,
from the aphelia of
the planets.
Io6 PHILOSOPHIÆ NATURALIS THEORIA
Idem ex reliqua.
Astronomia, : posse
autem hanc legem
accedere ad illam
quantum libuerit.
in earum orbitis apsidum lineæ, intulit, eam 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 nec omnino
quiescunt illæ apsidum lineæ, 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 ortæ ex mutua
planetarum actione in se invicem ; at illud utique huc usque nondum demonstratum est,
illum 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 quæratur motus trium corporum in se mutuo
agentium in ratione reciproca duplicata distantiarum, & utcunque projectorum, ac illæ
ipsæ adhuc admodum imperfectæ solutiones, quæ prolatæ huc 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,
nec scitur, quam multi sint, nec 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
ipsis motibus lineæ apsidum Lunæ colligi sensibilem recessum a ratione reciproca duplicata
distantiæ, & Eulerus in dissertatione De Aberrationibus jovis, & Saturni, quæ premium
retulit ab Academia Parisiensi an. I748, censuit, in ipso Jove, & Saturno haberi recessum
admodum sensibilem ab illa ratione ; sed id quidem ex calculi defectu non satis producti
sibi accidisse Clairautius ipse agnovit, ac edidit ; & Eulero aliquid simile fortasse accidit :
nec ullum habetur positivum argumentum pro ingenti recessu gravitatis generalis a ratione
duplicata distantiarum in distantia Lunæ, & multo magis in distantia planetarum. Vero
nec ullum habetur argumentum positivum pro ratione ita penitus accurata, ut discrimen
Difficultas a Mau-
pertuisiana perfec-
tione maxima New-
tonianæ legis.
Prima responsio:
nec cognosci fines
omnes, & perfect-
iones, ac seligi et-
iam minus perfecta
in gratiam perfec-
tionum.
sensum omnem prorsus effugiat. At & si id haberetur ; nihil tamen pati posset inde
Theoria mea ; cum arcus ille meæ curvæ postremus VT possit accedere, quantum libuerit,
ad arcum illius hyperbolæ, quæ exhibet legem gravitatis reciprocam quadratorum dis-
tantiæ, 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.
124. Nec vero quidquam ipsi meæ virium Theoriæ obsunt meditationes Maupertuisii,
ingeniosæ illæ quidem, sed meo judicio nequaquam satis conformes Naturæ legibus circa
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 singulæ particulæ. Demonstravit enim Newtonus, globos, quorum singuli
paribus a centro distantiis homogenei sint, & quorum particulæ minimæ se attrahant in
ratione reciproca duplicata distantiarum, se itidem attrahere in eadem ratione distantiarum
reciproca duplicata. Ob hasce perfectiones hujus Theoriæ virium ipse censuit hanc legem
reciprocam duplicatam distantiarum ab Auctore Naturæ selectam fuisse, quam in Natura
esse vellet. -
125. At mihi quidem inprimis nec unquam placuit, nec placebit sane unquam in
investigatione Naturæ causarum finalium usus, quas tantummodo ad meditationem quandam,
contemplationemque, usui esse posse abitror, ubi leges Naturæ aliunde innotuerint. Nam
nec perfectiones omnes innotescere nobis possunt, qui intimas rerum naturas nequaquam
inspicimus, sed externas tantummodo proprietates quasdam agnoscimus, & fines omnes,
quös Naturæ 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
he had obtained from the motions of the line of apses for the Moon a sensible discrepancy
from the inverse square of the distance. Also Euler, in his dissertation De Aberrationibus
jovis, & Saturni, which carried off the prize given by the Paris Academy, considered that
in the case of Jupiter & Saturn there was quite a sensible discrepancy from that ratio.
But Clairaut found out, & proclaimed the fact, that his result was indeed due to a defect
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
meditations of Maupertuis ; these are certainly most ingenious, but in my opinion in no
way sufficiently in agreement with the laws of Nature. Those meditations of his, I mean,
with regard to the law of forces decreasing in the inverse ratio of the squares of the distances;
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.
I25. Now, in the first place I was never satisfied, nor really, shall I ever be satisfied,
with the use of final causes in the investigation of Nature; these I think can only be employed
for a kind of study & contemplation, in such cases as those in which the laws of Nature
have already been ascertained from other methods. For we cannot possibly be acquainted
with all perfections; for in no wise do we observe the inmost nature of things, but all we
know are certain external properties. Nor is it at all possible for us to see & know all
the intentions which the Author of Nature could and did set before Himself when He founded
The same thing is
to be deduced from
the rest of astro-
nomy ; moreover
this law of mine
can approximate
to the other as
nearly as is desired.
Objection arising
from the greatest
perfection, accord-
ing to Maupertuis,
of the Newtonian
law.
First reply to this ;
all the aims and
perfections are not
known ; and even
a less perfect might
be selected for the
sake of greater per-
fection.
IO8 PHILOSOPHIÆ NATURALIS THEORIA
Eandem legem nec
perfectam esse, nec
in corporibus, non
utique accura te
sphæricis habere
locum.
Objectio ex præ-
judicio pro impul-
sione, & ex testi-
monio sensuum :
responsio ad hanc
posteriorem.
Felicius explicari
omnia sine impul-
sione : eam nus-
quam positive pro-
bari.
videre, & nosse Qmnino non possumus. Quin immo cum juxta ipsos Leibnitianos inprimis,
aliqsque omnes defensores acerrimos principii rationis sufficientis, & Mundi perfectissimi,
qui inde consequitur, multa quidem in ipso Mundo sint mala, sed Mundus ipse idcircò
sit optimus, quod ratio boni ad malum in hoc, qui electus est, omnium est maxima ; fieri
utique pQterit, ut in ea ipsius Mundi parte, quam hic, & nunc contemplamur, id, quod
electum fuit, debuerit esse non illud bonum, in cujus gratiam tolerantür 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, eam pro Natura fuisse electam, & constitutam.
126. At nec revera perfectissima est, quin immo meo quidem judicio est omnino
imperfecta, & tam ipsa, quam aliæ plurimæ leges, quæ requirunt attractionem imminutis
distantiis crescentem in ratione reciproca duplicata distantiarum, ad absurda deducunt
plurima, vel saltem ad inextricabiles difficultates, quod ego quidem tum alibi etiam, tum
inprimis demonstravi in dissertatione De Lege Virium in Natura existentium a num. 59: (g)
Accedit autem illud, quod illa, quæ videtur ipsi esse perfectio maxima, quod nimirum
eandem sequantur legem globi integri, quam particulæ minimæ, nulli fere usui est in
Natura ; si res accurate adfexactitudinem absolutam exigatur ; cum nulli in Natura sint
accurate perfecti globi paribus a centro distantiis homogenei, nam præter non exiguam
inæqualitatem 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 saltem analogia, præter
scabritiem superficiei, quæ utique est aliqua, satis patet, ipsa rotatione circa próprium
axem induci in omnibus compressionem aliquam, quæ ut ut exigua, exactam globositatem
impedit, adeoque illam assumptam perfectionem maximam corrumpit. Accedit autem
& illud, quod Newtoniana determinatio rationis reciprocæ duplicatæ 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 tantümmodo
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 reciprocæ duplicatæ distantiarum, adeoque nec
illius perfectionis creditæ maxime perfectam, absolutamque applicationem.
•
[58] 127. Demum & illud nonnullis difficultatem parit summam in hac Theoria
Virium, quod censeant, phænomena omnia per impulsionem explicari debere, & immedi-
atum contactum, quem ipsum credant evidenti sensuum testimonio evinci ; hinc hujusmodi
nostras vires immechanicas appellant, & eas, ut & Newtonianorum generalem gravitatem,
vel idcirco rejiciunt, quod mechanicæ 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, quæ eo
in genere habeamus præjudicia, & unde : cum nimirum ipsis sensibus tribuamus id,
quod nostræ ratiocinationis, atque illationis vitio est tribuendum. Satis erit hic monere
illud, ubi corpus ad nostra organa satis accedat, vim repulsivam, saltem 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, quæ in
communi sententia excitaretur ; quam ob rem ab iis sensationibus, quæ in hac ipsa
Theoria Virium haberentur, nullum utique argumentum desumi potest contra ipsam,
quod ullam vim habeant utcunque tenuem.
I 28. Quod pertinet ad explicationem phænomenorum per impulsionem immediatam,
monui sane superius, quanto felicius, ea prorsus omissa, Newtonus explicarit Astronomiam,
& Opticam ; & patebit inferius, quanto felicius phænomena quæque præcipua sine ulla
immediata impulsione explicentur. Cum iis exemplis, tum aliis, commendatur abunde
ea ratio explicandi phænomena, quæ adhibet vires agentes in aliqua distantia. Ostendant
(g) Quæ huc pertinent, & continentur novem numeris ejus Dissertationis incipiendo a 59, habentur in fine Supplem.
§ 4.
A THEORY OF NATURAL PHILOSOPHY
I09
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
my opinion, it is altogether imperfect. Both it, & several other laws, that require
attraction at very small distances increasing in the inverse ratio of the squares of the distances
lead to very many absurdities; or at least, to insuperable difficulties, as I showed in the
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 Expeditione
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
they consider that all the phenomena must be explained by impulse and immediate contact;
this they believe to be proved by the clear testimony of the senses. So they call forces
like those I propose non-mechanical, and reject them, just as they also reject the universal
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.
I28. As regards the explanation of phenomena by means of immediate contact I,
indeed, mentioned above how much more happily Newton had explained Astronomy and
Optics by omitting it altogether; and it will be evident, in what follows, how much more
happily every one of the important phenomena is explained without any idea of immediate
contact. Both by these instances, and by many others, this method of explaining phenomena,
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 be found at the end of this work as Supplement IV. - -
This law is neither
perfect, nor does it
hold good for bod-
ies that are not
exactly spherical.
Objection founded
on a prejudice for
impulse, and on the
testimony of the
Senses ; reply to
this latter.
Everything is more
happily explained
without the idea of
impulse ; and the
latter is nowhere
rigorously proved
to exist.
IIO PHILOSOPHIÆ NATURALIS THEORIA
Vires hujus Theo-
riæ pertinere ad ve-
rum, nec occultum
mechanismum.
Discrimen in ter
contactum mathe-
maticum, & physi-
cum : hunc dici
proprie contactum.
Transitus ab ob-
jectionibus contra
Theoriam virium
ad objectiones con-
tra puncta.
Objectio ab
puncti
idea.
inextensi,
qua caremus : re-
sponsio : unde idea.
extensionis sit orta.
isti vel unicum exemplum, in quo positive probare possint, per immediatam impulsionem
communicari motum in Natura. Id sane ii præstabunt nunquam ; cum oculorum testi-
monium ad excludendas distantias illas minimas, ad quas primum crus repulsivum pertinet,
& contorsiones curvæ circa axem, quæ oculos necessario fugiunt, adhibere non possint ; cum
e contrario ego positivo argumento superius excluserim immediatum contactum omnem,
& positive probaverim, ipsum, quem ii ubique volunt, haberi nusquam.
I 29. De nominibus quidem non esset, cur solicitudinem haberem ullam ; sed ut &
in iisdem aliquid præjudicio cuidam, quod ex communi loquendi usu provenit, illud
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. Quæ Archimedes de
æquilibrio tradidit, quæ Galilæus de li-[59]-bero gravium descensu, ac de projectis, quæ
de centralibus in circulo viribus, & oscillationis centro Hugenius, quæ 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 saltem mente seclusa.
Ubicunque vires agant, quæ motum materiæ 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 quem hæc virium genera pertinent multo magis,
quæ idcirco appellari jure possunt vires Mechanicæ, & quidquid per illas fit, jure affirmari
potest fieri per Mechanismum, nec vero incognitum, & occultum, sed uti supra demonstra-
vimus, admodum patentem, a manifestum.
I3o. Eodem etiam pacto in omnino propria significatione usurpare licebit vocem com-
tactus ; licet intervallum semper remaneat aliquod ; quanquam ego ad æquivocationes evi-
tandas soleo distinguere inter contactum Mathematicum, in quo distantia sit prorsus nulla,
& contactum Phy$icum, in quo distantia sensus effugit omnes, & vis repulsiva satis magna
ulteriorem accessum per nostras vires inducendum impedit. Voces ab hominibus institutæ
sunt ad significandas res corporeas, & corporum proprietates, prout nostris sensibus subsunt,
iis, quæ continentur infra ipsos, nihil omnino curatis. Sic planum, sic læve 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 læve. Eodem pacto & nomen
contactus ab hominibus institutum est, ad exprimendum physicum illum contactum tantum-
modo, sine ulla cura contactus mathematici, de quo nostri sensus sententiam ferre non
possunt. Atque hoc quidem pacto si adhibeantur voces in propria significatione illa, quæ
ipsarum institutioni respondeat ; ne a vocibus quidem ipsis huic Theoriæ virium invidiam
creare poterunt ii, quibus ipsa non placet.
I 31. Atque hæc de iis, quæ contra ipsam virium legem a me propositam vel objecta
sunt hactenus, vel objici possent, sint satis, ne res in infinitum excrescat. Nunc ad illa
transibimus, quæ contra constitutionem elementorum materiæ inde deductam se menti
offerunt, in quibus itidem, quæ maxime notatu digna sunt, persequar.
132. Inprimis quod pertinet ad hanc constitutionem elementorum materiæ, sunt
sane multi, qui nulló pacto in animum sibi possint inducere, ut admittant puncta prorsus
indi-[6o]-visibilia, & inextensa, quod nullam se dicant habere posse e9rum ideam. At id
hominum genus præjudiciis quibusdam tribuit multo plus æquo. Ideas omnes, saltem
eas, quæ ad materiam pertinent, per sensus hausimus. Porro sensus nostri nunquam
potuerunt percipere singüla elementa, quæ nimirum vires exerunt nimis tenues ad movendas
fibras, & propagandum motum ad cerebrum : massis indiguerunt, sive . elementorum
aggregatis, quæ ipsas impellerent collata vi. Hæc omnia aggregata constabant partibus,
qüarum partium extremæ sumptæ hinc, & inde, debebant a se invicem distare per aliquod
intervallüm, nec ita exiguum. Hinc factum est, ut nullam unquam per sensus acquirere
otuerimus ideam pertinentem ad materiam, quæ simul & extensionem, & partes, ac
âivisibilitatem 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
III
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
in that too there is something that, from the customary manner of speaking, gives rise to
a kind of prejudice, I think it should be observed that Mechanics was certainly never
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
sense; the interval may always remain something definite. Although, in order to avoid
ambiguity, I usually distinguish between mathematical contact, in which the distance is
absolutely nothing, & physical contact, in which the distance is too small to affect our
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.
I31. I have now said sufficient about those objections that either up till now have
been raised, or might be raised, against the law of forces that I have proposed ; otherwise
the matter would grow beyond all bounds. Now we will pass on to objections against
the constitution of the elements of matter derived from it, which present themselves to the
mind; & in these also I will investigate those that more especially seem worthy of remark.
I32. First of all, as regards the constitution of the elements of matter, there are indeed
many persons who cannot in any way bring themselves into that frame of mind to admit
the existence of points that are perfectly indivisible and non-extended; for they say that
they cannot form any idea of such points. But that type of men pays more heed than
is right to certain prejudices. We derive all our ideas, at any rate those that relate to
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, -
The forces in this
Theory refer to a
real and not to an
Occult mechanism.
Distinction be-
tween mathema-
tical and physical
COntact ; the latter
to be more properly
called contact.
Passing on from
objections against
my Theory of
forces to objections
against points.
Objection to the
idea of non-ex-
tended points,
which we postu-
late ; reply; the
origin of the idea
of extension.
II2 PHILOSOPHIÆ NATURALIS THEORIA
Id ea m puncti
debere acquiri per
reflexionem : quo-
modo ejus idea
negativa acquira-
tur.
Quomodo ejus idea
positiva acquiri
possit per limites,
& limitum inter-
sectionem.
Natura inextensi,
quod non potest
esse inextenso con-
tiguum in lineis.
Eadem in punctis :
idea puncti geo-
metrici translata
ad physicum, &
materiale.
Punctorum exist-
entiam a l iu n d e
demonstrari : per
ideam acquisitam
ea tantum concipi.
I33. Quamobrem ad concipiendum punctum indivisibile, & inextensum ; non debemus
consulere ideas, quas immediate per sensus hausimus ; sed eam nobis debemus efformare
per reflexionem. - Reflexione adhibita non ita difficulter efformabimus nobis ideam ejusmodi.
Nam inprimis 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 materiæ, quæ deest in loco foraminis.
I 34. Verum & positivam quandam indivisibilis, & inextensi puncti ideam poterimus
comparare nobis ope Geometriæ, & ope illius ipsius ideæ extensi continui, quam per sensus
hausimus, & quam inferius Ostendemus, fallacem esse, ac fontem ipsum fallaciæ ejusmodi
aperiemus, quæ 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 illa
erit utique limes inter partem dexteram & sinistram, longus quidem pedes duos, quanta
erat plani longitudo, at latitudinis omnino expers : nam ab altera parte immediate motu
continuo transitur ad alteram, quæ, si illa sectio crassitudinem haberet aliquam, non esset
priori contigua. Illa 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, translato plano, movebitur, & motu suo lineam describet, longam quidem,
sed latitudinis expertem.
I 35. Quo autem melius ipsius indivisibilis natura concipi possit ; quærat a nobis
quispiam, ut aliam faciamus ejus planæ massæ sectionem, quæ priori ita sit proxima, ut
nihil prorsus inter utramque intersit. Respondebimus sane, id fieri non posse : vel enim
inter novam sectionem, & veteram intercedet aliquid ejus materiæ, ex qua planum con-
tinuum constare concipimus, vel nova sectio congruet penitus cum præcedente. En
quomodo ideam acquiremus etiam ejus naturæ 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 illa sectio promoveatur tantummodo
per spatium latitudinis sibi æqualis. Utcunque exiguus fuerit motus, jam ille novus
sectionis locus distabit a præcedente per aliquod intervallum, cum sectio sectioni contigua
esse non possit.
I 36. Hæc si ad concursum sectionum transferamus, habebimus utique non solum ideam
puncti indivisibilis, & inextensi, sed ejusmodi naturæ puncti ipsius, ut aliud punctum sibi
contiguum habere non possit, sed vel congruant, vel aliquo a se invicem intervallo distent.
Et hoc pacto sibi & Geometræ 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 materiæ, quod hoc secundum habebit
proprietates reales vis inertiæ, & virium illarum activarum, quæ cogent duo puncta ad se
invicem accedere, vel a se invicem recedere, unde fiet, 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. -
I 37. En igitur per reffexionem acquisitam ideam punctorum realium, materialium,
indivisibilium, inextensorum, quam inter ideas ab infantia acquisitas per sensus incassum
quærimus. Idea ejusmodi non evincit eorum existentiam. Ipsam quam nobis exhibent
positiva argumenta superius facta, quod nimirum, ne admittatur in collisione corporum
saltus, quem & inductio, & impossibilitas binarum velocitatum diversarum habendarum
omnino ipso momento, quo saltus fieret, excludunt, oportet admittere in materia vires,
quæ repulsivæ sint in minimis distantiis, & iis in infinitum imminutis augeantur in infinitum ;
A THEORY OF NATURAL PHILOSOPHY II3
133. Hence for the purpose of forming an idea of a point that is indivisible & non-
extended, we cannot consider the ideas that we derive directly from the senses; but we
must form our own idea of it by reflection. If we reflect upon it, we shall form an idea
of this sort for ourselves without much difficulty. For, in the first place, when we have con-
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
the aid of geometry, and by the help of that idea of an extended continuum that we derive
from the senses; this we will show below to be a fallacy, & also we will open up the very
source of this kind of fallacy, which nevertheless will lead us to a perfectly clear idea of
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.
Suppose someone should ask us to make another section of the plane mass, which shall lie
so near to the former section that there is absolutely no distance between them. We
should indeed reply that it could not be done. For either between the new section &
the old there would intervene some part of the matter of which the continuous plane was
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
have not only the idea of an indivisible & non-extended point, but also an idea of the
nature of a point of this sort; which is such that it cannot have another point contiguous
to it, but the two either coincide or else they are separated from one another by some interval.
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
points can be acquired; whilst we seek for it in vain amongst those ideas that we have
acquired since infancy by means of the senses. But an idea of this sort about things does
not prove that these things exist. That is just what the rigorous arguments given above
point out to us; that is to say, because, in order that in the collision of solids a sudden
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
The idea of a point
must be obtained
by reflection ; how
a negative idea of
it may be acquired.
How a positive idea
can be acquired by
means of boun-
daries, and inter-
Sections of boun-
daries.
The nature of a
no n - extended
thing, which
cannot lie next to
an extended thing
as far as lines are
concerned.
The same thing for
points; the idea of
a geometrical point
transferre d to a
physical and mate-
rial point.
The existence of
points must be
otherwise demon-
strated ; they can
merely be thought
of through acquir-
ing an idea of them.
II4 PHILOSOPHIÆ NATURALIS THEORIA
simplicia,
& inextensa a b
aliis quoque ad-
missa : se d iis
præsta re h a. n c
eorum theoriam.
Huncta,
Impugnatur con-
ciliatio extensionis
formatæ ab inex-
tensis petita ab
impenetrabilitate.
I n d u ct io n e m a.
sensibilibus c o m-
positis, & extensis
haud valere contra
puncta simplicia, &
inextensa.
Per ipsam etiam
exclusionem inex-
tensi vi inductionis
habitam ipsum ex-
tensum excludi.
unde fit, ut duæ particulæ materiæ sibi [62] invicem contiguæ esse non possint : nam illico
vi illa repulsiva resilient a se invicem, ac particula iis constans statim disrumpetur, adeoque
prima materiæ elementa non constant contiguis partibus, sed indivisibilia sunt prorsus,
atque simplicia, & vero etiam ob inductionem separabilitatis, ac distinctionis eorum, quæ
occupant spatii divisibilis partes diversas, etiam penitus inextensa. Illa idéa acquisita per
reflexionem illud præstat 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.
I 38. Ceterum simplicium, & inextensorum notionem non ego primus in Physicam
induco. Eorum ideam habuerunt veteres post Zenonem, & Leibnitiani monades suas &
simplices utique volunt, & inextensas ; ego cum ipsorum punctorum contiguitatem auferam,
& distantias velim inter duo quælibet materiæ puncta, maximum evito scopulum, in quem
utrique incurrunt, dum ex ejusmodi indivisibilibus, & inextensis continuum extensum
componunt. Atque ibi quidem in eo videntur mihi 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 lævam, quin
ea extrema se contingant ; in quo, præter contiguitatem indivisibilium, & inextensorum
impossibilem, uti supra demonstravimus, quam tamen coguntur admittere, si rem altius
perpenderint ; videbunt sane, se ibi illam ipsam globuli inter duos globulos interjacentis
ideam admiscere.
I 39. Nec ad indivisibilitatem, & inextensionem elementorum conjungendas cum
continua extensione massarum ab iis compositarum prosunt, ea, quæ nonnulli ex Leibniti-
anorum familia proferunt, de quibus egi in una adnotatiuncula adjecta num. I 3. dissertationis
De Materiae Divisibilitate, & Principiis Corporum, ex qua, quæ eo pertinent, huc libet
transferre. Sic autem habet : Qui dicunt, monades mom compenetrari, quia matura sua
impenetrabiles sunt, ii difficultatem nequaquam amovent ; nam si & matura sua impenetrabiles
sunt, & continuum debent componere, adeoque contigua esse ; compenetrabuntur simul, & mom
compenetrabuntur, quod ad absurdum deducit, & ejusmodi entium impossibilitatem evincit.
Ex omnimodae inextensionis, &5 contiguitatis motione evincitur, compenetrari debere argumento
contra Zemomistas instituto per tot saecula, &5 cui numquam satis responsum est. Ex matura,
quae im [63] iis supponitur, ipsa compenetratio excluditur, adeoque habetur contradictio, &
absurdum.
I4o. Sunt alii, quibus videri poterit, contra hæc ipsa puncta indivisibilia, & inextensa
adhiberi posse inductionis principium, a quo continuitatis legem, & alias proprietates
derivavimus supra, quæ nos ad hæc indivisibilia, & inextensa puncta deduxerunt. Videmus
enim in materia omni, quæ se uspiam nostris objiciat sensibus, extensionem, divisibilitatem,
partes ; quamobrem hanc ipsam proprietatem debemus transferre ad elementa etiam per
inductionis principium. Ita ii : at hanc difficultatem jam superius præoccupavimus, 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 nec ad sensibilitatem extenditur.
I4I. Sed etiam si extenderetur, esset adhuc nostræ Theoriæ causa multo melior in eo,
quod circa extensionem, & compositionem partium negativa sit. Nam eo ipso, quod
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 II5
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
into physics. The ancients from the time of Zeno had an idea of them, & the followers
of Leibniz indeed suppose that their monads are simple & non-extended. I, since I do
not admit the contiguity of the points themselves, but suppose that any two points of
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.
139. Those arguments that some of the Leibnitian circle put forward are of no use
for the purpose of connecting indivisibility & non-extension of the elements with continuous
extension of the masses formed from them. I discussed the arguments in question in
a short note appended to Art. I 3 of the dissertation De Materia Divisibilitate and
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 compenetrated, because
they are by nature impenetrable, by no means remove the difficulty. For, if they are both by
nature impenetrable, 85 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 ;
{5 this leads to an absurdity & 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 & an absurdity.
140. There are others, who will think that it is possible to employ, for the purpose
of opposing the idea of these indivisible & non-extended points, the principle of induction,
by which we derived the Law of Continuity & other properties, which have led us to
these indivisible & non-extended points. For we perceive (so they say) in all matter,
that falls under our notice in any way, extension, divisibility & parts. Hence we must
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.
I4I. But even if this point is reached, there would only be all the more reason for our
Theory from the fact that it denies extension and composition by parts. For, from the very
fact that, if continuity be admitted, continuity of the elements is excluded by legitimate
argument, it follows that continuity ought to be absolutely excluded in all cases. For in
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
Simple and non-
extended points
are admitted by
others as well ; but
my Theory about
them is the best.
The deduction from
impenetrability of
a conciliation of
extension with its
formation from non-
extended things.
Induction derived
from things that
are sensible, Com-
pound, and , ex-
tended are of no
avail for the pur-
pose of opposing
simple and non-
extended things.
Extension itself is
excluded by the
exclusion of non-
extension, obtained
by the force of in-
duction.
II6 PHILOSOPHIÆ NATURALIS THEORIA
Cujusmodi con-
tinuum in hac
Theoria admittatur:
quid sit spatium,
& tempus.
lUbi habeat con-
tinuitatem Natura
ubi affectet tan-
tummodo.
assumpta veritate contradictoriæ propositionis ; cum enim ambæ simul veræ 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 materiæ
elementis, & in extensione, defectum hunc haberi vel inde eruitur : nec oberit
quidquam principium inductionis physicæ, quod utique non est demonstrativum, nec vim
habet, nisi ubi aliunde non demonstretur, casum illum, quem inde colligere possumus,
improbabilem esse tantummodo, adhuc tamen haberi, uti aliquando sunt & falsa veris
probabiliora.
I42. Atque hic quidem, ubi de continuitate seipsam excludente mentio injecta est,
notandum & illud, continuitatis legem a me admitti, & probari pro quantitatibus, quæ
magnitudinem mutent, quas nimirum ab una magnitudine ad aliam censeo abire non posse,
nisi transeant per intermedias, quod elementorum materiæ, quæ magnitudinem nec mutant,
nec ullam habent variabilem, continuitatem non inducit, sed argumento superius facto
penitus summovet. Quin etiam ego quidem continuum nullum agnosco coexistens, uti &
supra monui ; nam nec spatium reale mihi est ullum continuum, sed [64] imaginarium
tantummodo, de quo, uti & de tempore, quæ in hac mea Theoria sentiam, satis luculenter
exposui in Supplementis ad librum I. Stayanæ Philosophiæ (b). Censeo nimirum quodvis
materiæ 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 curo. 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 distantiæ cujusdam determinatæ, & in spatio determinatæ 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 infinitos modos possibiles pertinentes ad quodvis aliud, mihi est unus, qui cum eo
inducat in tempore relationem coexistentiæ ita, ut existentiam habere uterque non possit,
quin simul habeant, & coexistant ; in spatio vero, si existunt simul, inducant relationem
compenetrationis, reliquis omnibus inducentibus relationem distantiæ temporariæ, vel
localis, ut & positionis cujusdam localis determinatæ. Quoniam autem puncta materiæ
existentia habent semper aliquam a se invicem distantiam, & numero finita sunt ; finitus est
semper etiam localium modorum coexistentium numerus, nec ullum reale continuum
efformat. Spatium vero imaginarium est mihi possibilitas omnium modorum localium
confuse cognita, quos simul per cognitionem præcisivam 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 tam minimæ, quam maximæ
distantiæ 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 materiæ puncta, quæ si existerent,
compenetrationis relationem inducerent, ut eodem pacto idem nomino momentum tem-
poris temporarios modos omnes, qui relationem inducunt coexistentiæ. Sed de utroque
plura in illis dissertatiunculis, in quibus & analogiam persequor spatii, ac temporis
multiplicem.
[65] 143. Continuitatem igitur agnosco in motu tantummodo, quod est successivum
çjuid, nön coexistens, & in eó itidem solo, vel ex eo solo in corporeis saltem entibus legem
Continuitatis admitto. Atque hinc patebit clarius illud etiam, quod superius innui,
Naturam ubique continuitatis legem vel accurate observare, vel affectare saltem. Servat in
motibus, & distantiis, affectat in aliis casibus multis, quibus continuitas, uti etiam supra
definivimus, nequaquam convenit, & in aliis quibusdam, in quibus haberi omnino non potest
continuitas, quæ primo aspectu sese nobis objicit res non aliquanto intimius inspectantibus,
ac perpendentibu§ : ex. gr. quando Sol oritur supra horizontem, si concipiamus Solis discum
(h) Binæ dissertatiunculæ, quæ huc pertinent, inde excerptæ habentur hic Supplementorum § I, $ 2, quarum mentio
facta est etiam superius num. 66, §§ 86. -
A THEORY OF NATURAL PHILOSOPHY II7
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 beinferred, 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
of continuity, it should be observed that the Law of Continuity is admitted by me, &
proved for those quantities that change their magnitude, but which indeed I consider
cannot pass from one magnitude to another without going through intermediate stages;
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.(b) 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.
- I43. Hence I acknowledge continuity in motion only, which is something successive
and not co-existent; & also in it alone, or because of it alone, in corporeal entities at any
rate, lies my reason for admitting the Law of Continuity. From this it will be all the
more clear that, as I remarked above, Nature accurately observes the Law of Continuity,
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 I & II : these have
been already referred to in Arts. 66 & 86 above.
The sort of Con-
tinuum that is
admitted in this
Theory; the nature
of space and time.
Where there is con-
tinuity in Nature ;
where Nature does
no more than at-
tempt to maintain
it.
II8 PHILOSOPHIÆ NATURALIS THEORIA
Exempla, continu-
it at is apparentis
tantum : unde ea
Ortum ducat.
Motuum omnium
continuit a s in
1 in eis continuis
nusquam in ter-
ruptis, aut mutatis.
Apparens saltus in
diffusione reflexi,
ac refracti luminis.
ut continuum, & horizontem ut planum quoddam ; ascensus Solis fit per omnes magnitudines
ita, ut a primo ad postremum punctum & segmenta solaris disci, & chordæ 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 illa continuitas huic casui non convenit, & habetur tantummodo
in distantiis punctorum singulorum componentium eam massam ab illo imaginario plano.
Natura tamen etiam hic continuitatem quandam affectat, cum nimirum illa punctula ita
sibi sint invicem proxima, & ita ubique dispersa, ac disposita, ut apparens quædam ibi etiam
continuitas habeatur, ac in ipsa distributione, a qua densitas pendet, ingentes repentini
saltus non fiant. *
I44. Innumera ejus rei exempla liceret proferre, in quibus eodem pacto res pergit.
Sic in fluviorum alveis, in frondium flexibus, in ipsis salium, & crystallorum, ac aliorum
corporum angulis, in ipsis cuspidibus unguium, quæ acutissimæ in quibusdam animalibus
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, quæ continuam superficiem non efformant,
& in quibus, si quævis 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.
I45. Atque hinc fiet manifestum, quid respondendum ad casus quosdam, qui eo
pertinent, & in quibus violari quis crederet [66] continuitatis legem. Quando plano aliquo
speculo lux excipitur, pars refringitur, pars reflectitur : in reflexione, & refractione, uti eam
olim creditum est fieri, & etiamnum a nonnullis creditur, per impulsionem nimirum, &
incursum immediatum, fieret violatio quædam continui motus mutata linea recta in aliam ;
sed jam hoc Newtonus advertit, & ejusmodi saltum abstulit, explicando ea phænomena per
vires in aliqua distantia agentes, quibus fit, ut quævis particula luminis motum incurvet
paullatim in accessu ad superficiem reflectentem, 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 materiæ
punctum a Mundi initio unicam quandam continuam descripsit orbitam, pendentem a
continua illa virium lege, quam exprimit figura I, quæ ad distantias quascunque protenditur ;
quam quidem lineæ continuitatem nec liberæ 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, quæ non permittant ejusmodi linearum nobis ita simplicium accuratas
orbitas in motibus.
I46. Et quidem ut in iis omnibus, & aliis ejusmodi Natura semper in mea Theoria
accuratissimam continuitatem observat, ita & hic in reflexionibus, ac refractionibus luminis.
At est aliud ea in re, in quo continuitatis violatio quædam haberi videatur, quam, qui rem
altius perpendat, credet primo quidem, servari itidem accurate a Natura, tum 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, tum ipsum prisma sensim convertatur ; ubi ad certum devenitur in
conversione angulum, lux, quæ 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 æqualem angulo
A THEORY OF NATURAL PHILOSOPHY II9
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
which the matter comes about in the same manner. Thus, in the channels of rivers, the
bends in foliage, the angles in salts, crystals and other bodies, in the tips of the claws that
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
matter, in which it might be considered that the Law of Continuity was violated. When
light falls upon a plane mirror, part is refracted & part is reflected. In reflection &
refraction, according to the idea held in olden times, & even now credited by some people,
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
in my Theory observes the most accurate continuity, so also is this done here in the case
of the reflection and refraction of light. But there is another thing in this connection,
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
Examples of con-
tinuity that is
merely apparent;
its origin.
The continuity of
motions in con-
tinuous lines is
nowhere inter-
rupted or altered.
Apparent discon-
tinuity in diffusion
of reflected and re-
fracted light.
12O PHILOSOPHIÆ NATURALIS THEORIA
Apparens concili-
atio cum lege con-
tinuitatis per radios
irregulariter disper-
SOS. -
Cur ea apparens
tantum : vera, COn-
ciliatio per contin-
uitatem viæ cujus-
vis punctilucis.
incidentiæ, & jacentem citra, sine ulla reflexione in angulis intermediis minoribus ab ipsa
superficie ad ejusmodi finitum angulum. •
I47. Huic cuidam velut læsioni continuitatis videtur responderi posse per illam lucem
quæ reflectitur, vel refrin-[67]-gitur irregulariter in quibusvis angulis. Jam olim enim
observatum est illud, ubi lucis radius reflectitur, non reflecti totum ita, ut angulus
reflexionis æquetur angulo incidentiæ, 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 quem incurrit radius, quod utique non fieret, nisi e solaribus illis directis radiis
etiam ad oculum ipsius radii devenirent, egressi in omnibus iis directionibus, quæ ad omnes
oculi positiones tendunt ; licet ibi quidem satis intensum lumen non appareat, nisi in
directione faciente angulum reflexionis æqualem incidentiæ, in qua resilit maxima luminis
pars. Et quidem hisce radiis redeuntibus in angulis hisce inæqualibus egregie utitur
Newtonus in fine Opticæ 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 illum 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 illæsa perseverare continuitas.
I48. Verum si adhuc altius perpendatur res ; patebit in illa intermedia serie non haberi
accuratam continuitatem, sed apparentem quandam, quam Natura affectat, non accurate
servat illæsam. Nam lumen in mea Theoria non est corpus quoddam continuum, quod
diffundatur Continuo per illud omne spatium, sed est aggregatum punctorum a se invicem
disjunctorum, atque distantium, quorum quodlibet suam percurrit viam disjunctam a
Quo pacto servetur
çontinuitas in qui-
busdam casibus, in
quibus videtur lædi.
Generalis responsio
ad casus similes in-
de eruta.
proximi via per aliquod intervallum. Continuitas servatur accuratissime in singulorum
punctorum viis, non in diffusione substantiæ non continuæ, & quo pacto ea in omnibus iis
motibus servetur, & mutetur, mutata inclinatione incidentiæ, via a singulis punctis descripta
sine saltu, satis luculenter exposui in secunda parte meæ dissertationis De Lumine a num. 98.
Sed hæc ad applicationem jam pertinent Theoriæ ad Physicam. -
I49. Haud multum absimiles sunt alii quidam casus, in quibus singula continuitatem
observant, non aggregatum utique non continuum, sed partibus disjunctis constans.
Hujusmodi est ex. gr. altitudo cujusdam domus, quæ ædificatur de novo, cui cum series
nova adjungitur lapidum determinatæ cujusdam altitudinis, per illam additionem repente
videtur crescere altitudo domus, sine transitu per altitudines intermedias : & si dicatur id
non esse Naturæ opus, sed artis ; potest difficultas transferri facile ad Naturæ 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 finitæ videntur
acquiri simul totæ sine [68] transitu per intermedias magnitudines. In iis casibus
continuitas servatur in motu singularum partium, quæ accedunt. Illæ per lineam quandam
continuam, & continua velocitatis mutatione accedunt ad locum sibi deditum : quin immo
etiam posteaquam eo advenerunt, pergunt adhuc moveri, & nunquam habent quietem nec
absolutam, nec respectivam respectu aliarum partium, licet jam in respectiva positione
sensibilem mutationem non subeant : parent nimirum adhuc viribus omnibus, quæ
respondent omnibus materiæ punctis utcunque distantibus, & actio proximarum partium,
quæ novam adhæsionem 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 æstimatione quadam pendet nostrorum sensuum satis crassa ; ut licet perpetuo
accedant illæ partes, & pergant perpetuo mutare positionem respectu ipsius massæ ; tum
incipiant censeri ut pertinentes ad illam domum, vel massam : cum desinit respectiva
mutatio esse sensibilis, quæ sensibilitatis cessatio fit ipsa etiam quodammodo per gradus
omnes, & continuo aliquo tempore, non vero per saltum.
I 5o. Hinc distinctius ibi licebit difficultatem omnem amovere dicendo, non servari
mutationem continuam in magnitudinibus earum rerum, quæ continuæ non sunt, &
magnitudinem non habent continuam, sed sunt aggregata rerum disjunctarum ; vel in iis
rebus, quæ a nobis ita censentur aliquod totum constituere, ut magnitudinem aggregati non
A THEORY OF NATURAL PHILOSOPHY
I2 I
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.
I47. It seems that an explanation of this apparent breach of continuity can be given
by means of light that is reflected or refracted irregularly at all sorts of angles. For long ago
it was observed that, when a ray of light is reflected, it is not reflected entirely in such a
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
that intermediate series there is no accurate continuity, but only an apparent continuity;
& this Nature tries to maintain, but does not accurately observe it unimpaired. For,
in my Theory, light is not some continuous body, which is continuously diffused through
all the space it occupies ; but it is an aggregate of points unconnected with & separated
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.
I49. There are certain cases, not greatly unlike those already given, in which each
part preserves continuity, but not so the whole, which is not continuous but composed of
separate parts. For an instance of this kind, take the height of a new house which is being
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. r -.
150. From this consideration we may here in a clearer manner remove all difficulty
by saying that a continuous change is not maintained in the magnitudes of those things,
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
Apparent reconcili-
ation with the Law
of Continuity effec-
ted by means of
irregular dispersion.
Why this is only an
apparent reconcili-
ation ; the true
reconciliation is
through the con-
tinuity of path for
any point of light.
The manner in
which continuity
is maintained in
certain cases in
which it is appar-
ently impaired.
General refutation
for similar cases
derived from this.
I22 PHILOSOPHIÆ NATURALIS THEORIA
Alii casus in quibus
lae d i tu r, alii, in
quibus habetur
solum proxima, non
accurata, continui-
tas.
Conclusio pertinens
ad ea, quæ veram,
& ea, quæ affecta-
tam habent contin-
uitatem.
Difficultates petitæ
a discrimine debito
inter materiam, &
spiritum.
Differre hæc puncta,
a spiriti b u s per
impenetrabilitatem,
sensibilitatem, in-
capacitatem cogit-
ationis.
determinent distantiæ inter eadem extrema, sed a nobis extrema ipsa assumantur jam alia,
jam alia, quæ 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 objeëtis
casibus accessiones partium novæ fiunt, atque ibi nos in usu vocabuli saltüm facimus; ars,
& Natura saltum utique habet nullum.
I5I. Non idem contingit etiam, ubi plantæ, vel animantia crescunt, succo se insinuante
pertubulo$ fibrarum, & prQcurrente, ubi & magnitudo computata per distantias punctorum
maxime distantium transit per omnes intermedias ; cum nimirum ipse procursus fiat
per omnes intermedias distantias. At quoniam & ibi mutantur termini illi, qui distantias
determinant, & nomen suscipiunt altitudinis ipsius plantæ ; vera & accurata cóntinuitas 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 illa alia continuitas quædam apparens, & affectata
tantummodo a Natura, quam intuemur etiam in progressu sübstantiarüm, ut incipiendo ab
inanima-[69]-tis corporibus progressu facto per vegetabilia, tum per quædam fere
semianimalia torpentia, ac demum animalia perfectiora magis, & perfectióra usque ad simios
homini tam similes. Quoniam & harum specierum, ac existentiüm individuorüm 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 discretæ quædam quantitates,
non continuæ ; ut & in Arithmetica series ex. gr. naturalium numerorum non est continua,
sed discreta ; & ut ibi series ad continuam reducitur tantummodo, si generaliter omnes
intermediæ fractiones concipiantur ; sic & in superiore exemplo quædam velut continua
series habebitur tantummodo ; si concipiantur omnes intermediæ species possibiles.
I 52. Hoc pacto excurrendo per plurimos justmodi casus, in quibus accipiuntur
aggregata rerum a se invicem certis intervallis distantium, & unum aliquid continuum non
constituentium, nusquam accurata occurret continuitatis lex, sed per quandam dispersionem
quodammodo affectata, & vera continuitas habebitur tantummodo in motibus, & in iis, quæ
a motibus pendent, uti sunt distantiæ, & vires determinatæ a distantiis, & velocitates a
viribus ortæ ; quam ipsam ob causam ubi supra num. 39 inductionem pro lege continuitatis
assumpsimus, exempla accepimus a motu potissimum, & ab iis, quæ cum ipsis motibus
connectuntur, ac ab iis pendent.
I 53. Sed jam ad aliam difficultatem gradum faciam, quæ non nullis negotium ingens
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, præstituros eadem phænomena, tolli nimirum corpus, & omnem
corporeæ substantiæ notionem sublata extensione continua, quæ sit præcipua materiæ
proprietas ita pertinens ad naturam ipsius ; ut vel nihil aliud materia sit, nisi substantia
prædita extensione continua ; vel saltem idea corporis, & materiæ haberi non possit ; nisi
in ea includatur idea extensionis continuæ. Multa hic quidem congeruntur simul, quæ
nexum aliquem inter se habent, quæ hic seorsum evolvam singula.
I 54. Inprimis falsum omnino est, nullum esse horum punctorum discrimen a spiritibus.
Discrimen potissimum materiæ a spiritu situm est in hisce duobus, quod materia est
sensibilis, & incapax cogitationis, ac voluntatis, spiritus nostros sensus non afficit, & cogitare
potest, ac velle. Sensibilitas autem non ab extensione continua oritur, sed ab impene-
trabilitate, qua fit, ut nostrorum organorum fibræ tendantur a corporibus, quæ ipsis
sistuntur, & motus ad cerebrum pro-[7o]-pagetur. Nam si extensa quidem essent corpora,
sed impenetrabilitate carerent ; manu contrectata fibras non sisterent, nec motum ullum
in iis progignerent, ac eadem radios non reflecterent, sed liberum intra se aditum luci
præberent. Porro hoc discrimen utrumque manere potest integrum, & manet inter mea
indivisibilia hæc 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, nec Oculis inspectantur, nec manibus palpari
possunt. Deinde in meis hisce punctis ego nihil admitto aliud, nisi illam virium legem cum
inertiæ vi conjunctam, adeoque illa 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
a c curate ly, 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
m a de between
matter & spirit.
These points differ
from spirits on
account of their
impenetra bility,
their being sen-
sible, & their inca-
pacity for thought.
I24 PHILOSOPHIÆ NATURALIS THEORIA
Si possibilis sub-
stantia pr æ dita
hisce viribus, &
capax cogitationis;
e a m n e c fore
m a, te ria m, ne c
spiritum.
IN ih i 1 a. m itt i,
amisso argumento
eorum, qui a com-
positione partium
deducunt incapaci-
tatem cogitationis.
Quamobrem discrimen essentiæ illud utrumque, quod inter corpus, & spiritum
agnoscunt omnes, id & ego agnosco, nec vero id ab extensione, & compositione continua
desumitur, sed ab iis, quæ cum simplicitate, & inextensione æque conjungi possunt, &
cohærere cum ipsis.
I 55. At si substantiæ capaces cogitationis & voluntatis haberent ejusmodi virium legem,
an non eosdem præstarent effectus respectu nostrorum sensuum, quos ejusmodi puncta ?
Respondebo sane, me hic non quærere, utrum impenetrabilitas, & sensibilitas, quæ ab iis
viribus pendent, conjungi possint cum facultate cogitandi, & volendi, quæ quidem quæstio
eodem redit, ac in communi sententia de impenetrabilitate extensorum, ac compositorum
relata ad vim cogitandi, & volendi. Illud ajo, notionem, quam habemus partim ex
observationibus tam sensuum respectu corporum, quam intimæ conscientiæ 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 afficiendi 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 massæ constituant
corpora a spiritibus longissime discrepantia. Si possibile sit illud substantiæ genus, quod
& hujusmodi vires activas habeat cum inertia conjunctas, & simul cogitare possit, ac velle ;
id quidem nec corpus erit, nec spiritus, sed tertium quid, a corpore discrepans per capacitatem
cogitationis, & voluntatis, discrepans autem a spiritu per inertiam, & vires hasce nostras,
quæ impenetrabilitatem inducunt. Sed, ut ajebam, ea quæstio huc non pertinet, & aliunde
resolvi debet ; ut aliunde utique debet resolvi quæstio, qua quæratur, an substantia extensa,
& impenetrabilis [71] hasce proprietates conjungere possit cum facultate cogitandi,
volendique. -
I 56. Nec vero illud reponi potest, argumentum potissimum ad evincendum, materiam
cogitare non posse, deduci ab extensione, & partium compositione, quibus sublatis, omne id
fundamentum prorsus corruere, & ad materialismum sterni viam. Nam ego sane non video,
quid argumenti peti possit ab extensione, & partium compositione pro incapacitate cogitandi,
& volendi. Sensibilitas, præcipua corporum, & materiæ proprietas, quæ ipsam adeo a
spiritibus discriminat, non ab extensione continua, & compositione partium pendet, uti
vidimus, sed ab impenetrabilitate, quæ 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 ; singulæ ejus partes
deberent singulas cogitationis partes habere, adeoque nulla pars objectum perciperet ; cum
nulla haberet eam perceptionis partém, 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 materiæ, quæ certa
partium dispositione sit prædita, uti anima rationalis per tam multos Philosophos, ut ut
indivisibilis, in omni corpore, vel saltem in parte corporis aliqua divisibili existit, & ad
ejusmodi præsentiam præstandam certa indiget dispositione partium ipsius corporis, qua
semel læsa 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 inhærentem
divisibili materiæ natura, & determinatio cogitantis haberetur. Unde aperte constat eo
Etiam si quidpiam
amittatur ; theor-
iam positive prob-
ari, & in ea, manere
summum discrimen
inter materiam, &
spiritum. -
argumento amisso, nihil omnino amitti, quod jure dolendum sit.
I 57. Sed quidquid de eo argumento censeri debeat, nihil refert, nec ad infirmandam
Theoriam positivis, & validis argumentis comprobatam, ac e solidissimis principiis directa
ratiocinatione deductam, quidquam potest unum, vel alterum argumentum amissum, quod
ad probandam aliquam veritatem aliunde notam, & a revelatis principiis aut directe, aut
indirecte confirmatam, ab aliquibus adhibeatur, quando etiam vim habeat aliquam, quam,
uti ostendi, superius allatum argumentum omnino non habet. Satis est, si illa Theoria cum
ejusmodi veritate conjungi possit, uti hæc nostra cum immaterialitate spirituum con-
jungitur optime, cum retineat pro materia inertiam, impenetrabilitatem, sensibilitatem,
incapacitatem cogitandi, & pro spiritibus retineat incapacitatem afficiendi sensus nostros
per impenetrabilitatem, & facultatem cogitandi, ac volendi. [72] Ego quidem in ipsius
A THEORY OF NATURAL PHILOSOPHY I 25
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
forces of this kind, is it possible that they would produce the same effects with respect to
our senses, as points of this sort : Truly, I will answer that I do not seek to know in this
connection, whether impenetrability & sensibility, which depend on these forces, can
be conjoined with the faculty of thinking & willing; indeed this question comes to the
same thing as the general idea of the relations of impenetrability of extended & composite
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. -
I56. Now it cannot be ignored that an argument of great importance in proving that
matter is incapable of thought is deduced from extension & composition by parts; &
if these are denied, the whole foundation breaks down, & the way is laid open to materialism.
But really I do not see what in the way of argument can be derived from extension &
composition by parts, to support incapacity for thinking and willing. Sensibility, the chief
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. - -
I57. But whatever opinion we are to form about this argument, it makes no difference,
nor can it weaken a Theory that has been corroborated by direct & valid arguments, &
deduced from the soundest principles by a straightforward chain of reasoning, if we leave
out one or other of the arguments, which have been used by some for the purpose of
testing some truth that is otherwise known & confirmed by revealed principles either
directly or indirectly; even when the argument has some validity, which, as I have shown,
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
If it were possible
that there was a
substance that was
both endowed with
these forces & was
capable of thought,
it would be neither
matter nor spirit.
Nothing is lost
even if we dismiss
the argument of
those who deduce
in capa city for
thought from com-
position by parts.
Even if something
is thus neglected,
the Theory can be
proved in a direct
manner, & there
will still remain in
it the greatest dif-
ference between
matter & spirit.
I26 PHILOSOPHIÆ NATURALIS THEORIA
Sensus omnino falli
in illa tanta con-
tinuitate in exten-
sionis, quam nobis
ingerunt.
F o n s præjudici-
Orum : haberi pro
nullis in se, quæ
sunt nulla in nostris
sensibus : eorum
exempla.
EOrum correctio
ubi deprehendatur,
rem a 1 i o et ia m
modo c u m sen-
suum apparentia
conciliari posse.
materiæ, & corporeæ substantiæ definitione ipsa assumo incapacitatem cogitandi, & volendi,
& dico corpus massam compositam e punctis habentibus Tvim inertiæ conjunctam cum
viribus activis expressis in fig. I, & cum incapacitate cogitandi, ac volendi, qua definitione
admissa, evidens est, materiam cogitare non posse ; quæ erit metaphysica quædam conclusio,
ea definitione admissa, certissima : tum ubi solæ rationes physicæ adhibeantur, dicam, hæc
corpora, quæ meos afficiunt sensus, esse materiam, quod & sensus afficiant per illas utique
vires, & non cogitent. Id autem deducam inde, quod nullum cogitationis indicium
præstent ; quæ erit conclusio tantum physica, circa existentiam illius materiæ ita definitæ,
æque physice certa, ac est conclusio, quæ dicat lapides non habere levitatem, quod nunquam
eam prodiderint ascendendo sponte, sed semper e contrario sibi relicti descenderint.
I 58. Quod autem pertinet ad ipsam corporum, & materiæ ideam, quæ videtur exten-
sionem continuam, & contactum partium involvere, in eo videntur mihi quidem Cartesiani
inprimis, qui tantopere contra præjudicia pugnare sunt visi, præjudiciis 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, tum ex
diversa gravitate specifica, quæ a diversa multitudine vacuitatum oritur utique, tum ex
eo, quod per illa insinuentur substantiæ 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.
I 59. Quamobrem jam ejusmodi nostrorum sensuum testimonium, vel potius noster
eorum ratiociniorum usus, in hoc ipso genere suspecta esse debent, in quo constat nos
decipi. Suspicari igitur licet, exactam continuitatem sine ullis spatiolis, ut in majoribus
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 præjudiciorum
fons, & origo præcipua, ut quidquid in nostris sensibus est nihil, habeamus pro nihilo
absoluto. Sic utique per tot sæcula 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 quæstionem longe aliunde resolvendam esse,
quam per sensus, in quibus debent eædem 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. Tam mutatio, quam
permanentia fieri possunt duplici modo : mutatio, primo si nobis immotis objectum movea-
tur ; & permanentia, si id ipsum stet : secundo, illa, si objecto stante moveamur nos ; hæc, 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
tum quidem motum ipsum pro nullo, nisi aliunde admoneamur de eodem motu per causas,
quæ nobis sint cognitæ, ut ubi provehimur portu, quo casu vector, qui jam diu assuevit ideæ
littoris stantis, & navis promotæ per remos, vel vela, corrigit apparentiam illius, terraeque
urbesque recedunt, & sibi, non illis, motum adjudicat.
16o. Hinc Philosophus, ne fallatur, non debet primis hisce ideis acquirere, quas e
sensationibus haurimus, & ex illis deducere consectaria sine diligenti perquisitione, ac in ea
quæ ab infantia deduxit, debet diligenter inquirere. Si inveniat, easdem illas sensuum
perceptiones duplici modo æque fieri posse ; peccabit utique contra Logicæ etiam naturalis
ieges, si alterum modum præ 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
extension, it seems to me indeed that in this matter the Cartesians in particular, who have
appeared to impugn prejudgments with so much vigour, have given themselves up to these
prejudgments more than anyone else. We obtain the idea of bodies through the senses :
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.
I59. Hence such evidence of our senses, or rather our employment of such arguments,
must now lie open to suspicion in that class, in which it is known that we have been deceived.
We may then suspect that accurate continuity without the presence of any little empty
spaces—such as is certainly absent from bodies of considerable size, although our senses
seem to remark its presence—is also nowhere existent in any of their smallest particles ;
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 85 buildings recede,”
he attributes the motion to himself and not to them. º
I60. Hence, the philosopher, to avoid being led astray, must not seek to obtain from
these primary ideas that we derive from the senses, or deduce from them, consequential
theorems, without careful investigation; & he must carefully study those things that
he has deduced from infancy. If he find that these very perceptions by the senses can
come about in two ways, one of which is as probable as the other ; then he will certainly
commit an offence against the laws of natural logic, if he should proceed to choose one
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
The senses are
altogether at fault
in the greatness of
the continuity of
extension that they
force us to believe.
The origin of pre-
judgments ; things
considered as no-
thing, which are
nothing So far as
the Senses are con-
cerned ; examples
of these.
Correction of those
things, where it is
known th a t t he
matter cannot be
brought in to
agreement with
what is apparent
to the senses in
Some other way.
I28 PHILOSOPHIÆ NATURALIS THEORIA
Ordo idearum, quas
hausimus circa cor-
pus: primas habitas
esse per tactum.
Quæ fuerint tum
consideranda : in-
fantia ad eas re-
flexiones, inepta : in
quo ea sita sit.
Præjudicia in d'e
orta extensionis
continuæ ut essen-
tialis, odorum, &c.,
ut accidentalium.
sensationes habebuntur eædem, sive materia constet punctis prorsus inextensis, & distantibus
inter se per intervalla minima, quæ sensum fugiant, ac vires ad illa 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 præcipua discrimina,
cum punctis hisce indivisibilibus conveniant, & quidem multo sane melius, quam in communi
sententia de continua extensione materiæ. Quamobrem errabit contra rectæ ratiocinationis
usum, qui ex præjudicio ab hujusce conciliationis, & alterius hujusce sensationum nostrarum
causæ ignoratione inducto, continuam extensionem ut proprietatem necessariam corporum
omnino credat, & multo magis, qui censeat, materialis substantiæ ideam in ea ipsa continua
extensione debere consistere.
I6I. Verum quo magis evidenter constet horum præjudiciorum origo, afferam hic
dissertationis De Materiae Divisibilita-[74]-te, & Primcipiis Corporum, numeros tres inci-
piendo a I4, 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, materiæ, rei corporeæ, rei materialis, nos hausisse ex sensibus. Porro ideæ primæ
omnium, quas circa corpora acquisivimus per sensus, fuerunt omnino eæ, 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 cœpimus, multo minor sub initium frequentia fuit. Ideæ autem, quas per
tactum habuimus, ortæ sunt ex phænomenis hujusmodi. Experiebamur palpando, vel
temere impingendo resistentiam vel a nostris, vel a maternis membris ortam, quæ cum
nullam interruptionem per aliquod sensibile intervallum sensui objiceret, obtulit nobis ideam
impenetrabilitatis, & extensionis continuæ : cumque deinde cessaret in eadem directione,
alicubi resistentia, & secundum aliam directionem exerceretur ; terminos ejusdem quanti-
tatis concepimus, & figuræ ideam hausimus.”
162. “ Porro oriebantur hæc phænomena a corporibus e materia jam efformatis, non a
singulis materiæ particulis, e quibus' ipsa corpora componebantur. Considerandum
diligenter erat, num extensio ejusmodi esset ipsius corporis, non spatii cujusdam, per quod
particulæ corpus efformantes diffunderentur : num ea particulæ ipsæ iisdem proprietatibus
essent præditæ : num resistentia exerceretur in ipso contactu, an in minimis distantiis sub
sensus non cadentibus vis aliqua impedimento esset, quæ id ageret, & resistentia ante ipsum
etiam contactum sentiretur : num ejusmodi proprietates essent intrinsecæ ipsi materiæ, ex
qua corpora componuntur, & necessariæ : an casu tantum aliquo haberentur, & ab extrinseco
aliquo determinante. Hæc, & alia sane multa considerare diligentius oportuisset : sed erat
id quidem tempus maxime caliginosum, & obscurum, ac reflexionibus minus obviis minime
aptum. Præter organorum debilitatem, occupabat animum rerum novitas, phænomenorum
paucitas, & nullus, aut certe satis tenuis usus in phænomenis 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 phænomenorum inopia, in hac efformandi systematis difficultate, in hoc exiguo
reflexionum usu, magis etiam, quam in organorum imbecillitate, arbitror, sitam esse
infantiam.” -
[75] 163. “ In hac tanta rerum caligine ea prima sese obtulerunt animo, quæ minus
ältaTindagine, minus intentis reflexionibus indigebant, eaque ipsa ideis toties repetitis altius
impressa sunt, & tenacius adhæserunt, & quendam veluti campum nacta prorsus vaguum,
& àdhuc immunem, suo quodammodo jure quandam veluti possessionem inierunt. Inter-
valla, quæ sub sensum nequaquam cadebant, pro nullis habita : ea, quorum ideæ semper
simul conjunctæ excitabantur, habita sunt pro iisdem, vel arctissimo, & necessario nexu
inter se conjunctis. Hinc illud effectum est, ut ideam extensionis continuæ, ideam
A THEORY OF NATURAL PHILOSOPHY I29
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.
I61. Now in order that the source of these prejudices may be the more clearly known,
I will here quote, from the dissertation De Materia Divisibilitate & Principii Corporum,
three articles, commencing with Art. I4, where we have:—“Even if we allow (a thing quite
opposed to my way of thinking) that some ideas are innate & are not acquired through
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
matter, not from the single particles of matter of which the bodies themselves were composed.
It would have to be considered carefully whether such extension was a property of the
body itself, & not of some space through which the particles forming the body were
diffused; whether the particles themselves were endowed with the same properties;
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
were those which required a less deep study & less intent investigation; & these, since
the ideas were the more often renewed, made the greater impression & became fixed
the more firmly in the mind, & as it were took possession of, so to speak, a land that they
found quite empty & hitherto immune, by a sort of right of discovery. Intervals, which
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
K
Order of the ideas
which we obtain
about bodies ; the
first ideas come
through the sense
of touch.
Such things de-
mand reflection at
the time; inepti-
tude of infancy for
Such reflection; on
what they may be
founded.
Thence prejudg-
ments are derived
that continuity of
extension is an
essential, but that
continuity of odours
&c., is accidental.
I3o PHILOSOPHIÆ NATURALIS THEORIA
Einæ propositiones
dissertationis totam
Theoriam continen-
tis.
Qu9 pacto con-
geries punctorum
coalescant in massas
tenaces : transitus
ad partem secun-
dam.
impenetrabilitatis prohibentis ulteriorem motum in ipso tantum contactu corporibus
affinxerimus, & ad omnia, quæ ad corpus pertinent, ac ad materiam, ex qua ipsum constat,
temere transtulerimus : quæ ipsa cum primum insedissent animo, cum frequentissimis, immo
perpetuis phænomenis, & experimentis confirmarentur ; ita tenaciter sibi invicem
adhæserunt, ita firmiter ideæ corporum immixta sunt, & cum ea copulata ; ut ea ipsa pro
primis corporibus, & omnium corporearum rerum, nimirum etiam materiæ corpora compo-
nentis, ejusque partium proprietatibus maxime intrinsecis, & ad naturam, atque essentiam
earundem pertinentibus, & tum habuerimus, & nunc etiam habeamus, nisi nos præjudiciis
ejusmodi liberemus. Extensionem nimirum continuam, impenetrabilitatem ex contactu,
compositionem ex partibus, & figuram, non solum naturæ corporum, sed etiam corporeæ
materiæ, & singulis ejusdem partibus, tribuimus tanquam proprietates essentiales : cætera,
quæ serius, & post aliquem reflectendi usum deprehendimus, colorem, saporem, odorem
sonum, tanquam accidentales quasdam, & adventitias proprietates consideravimus.”
I64. Ita ego ibi, ubi Theoriam virium deinde refero, quam supra hic exposui, ac ad
præcipuas corporum proprietates applico, quas ex illa deduco, quod hic præstabo in parte
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, & non potius constare e
punctis prorsus indivisibilibus a se per aliquod intervallum distantibus ; nec ulla ratio seclusis
praejudiciis suadet extensionem ipsam continuam potius, quam compositionem e punctis prorsus
indivisibilibus, inextensis, &5 mullum continuum extensum constituentibus. 2. Sunt argumenta,
&$ $atis valida illa quidem, quae hanc compositionem e punctis indivisibilibus evincant extensioni
ipsi continuae praeferri oportere, -
I65. At quodnam extensionis genus erit istud, quod e punctis inextensis, & spatio
imaginario, sive puro nihilo [76] constat ? Quo pacto Geometria locum habere poterit,
ubi nihil habetur reale continuo extensum ? An non punctorum ejusmodi in vacuo
innatantium congeries erit, ut quædam nebula unico oris flatu dissolubilis prorsus sine ulla
consistenti figura, solidate, resistentia ? Hæc quidem pertinent ad illud extensionis ,&
cohæsionis genus, de quo agam in tertia parte, in qua Theoriam applicabo ad Physicam, ubi
istis ipsis difficultatibus faciam satis. Interea hic illud tantummodo innuo in antecessum, me
cohæsionem 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, quæ immutatis distantiis oriuntur, sint
satis magnæ, 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 cohæreant inter se, tuebuntur utique positionem suam, & massam
constituent formæ tenacissimam, ac eadem prorsus phænomena exhibentem, quæ exhiberent
solidæ massulæ 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 I31
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
expounded in the previous articles of this work, and I applied the theory to the principal
properties of bodies, deducing them from it; & this I will set forth in the third part
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,
& 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, 85 forming
no extended continuum of any ſort. 2. There are arguments, 85 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
points & imaginary space, i.e., out of pure nothing How can Geometry be upheld
if no thing is considered to be actually continuously extended ? Will not groups of points,
floating in an empty space of this sort be like a cloud, dissolving at a single breath, &
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. -
A pair of proposi-
tions of the disser-
tation containing
the whole of my
Theory.
The manner in
which groups of
points coalesce into
tenacious maSSes :
& then we pass on
to the second part,
Ante applicationem
ad Me c h a nicam
consideratio curvæ.
Quid in ea con-
siderandum.
Diversa a r c u u m
g e n e r a : a r c u s
asymptotici etiam
numero infiniti.
Arcu s intermedii.
Arcus prostremus
gravitatis fortasse
non asymptoticus.
[77] PARS II
T%eoriae .App/icato ad Mechamicam
I66. Considerabo in hac secunda parte potissimum generales quasdam leges æquilibrii
& motus tam punctorum, quam massarum, quæ ad Mechanicam utique pertinent, & ad
plurima ex iis, quæ in elementis Mechanicæ passim traduntur, ex unico principio, & adhibito
constanti ubique agendi modo, demonstranda viam sternunt pronissimam. Sed prius
præmittam nonnulla quæ pertinent ad ipsam virium curvam, a qua utique motuum,
phænomena pendent omnia. • . -
I67. In ea curva consideranda sunt potissimum tria, arcus curvæ, area comprehensa
inter axem, & arcum, quam generat ordinata continuo fluxu, ac puncta illa, in quibus
CurVa SeCat aX€m. -
I68. Quod ad arcus pertinet, alii dici possunt repulsivi, & alii attractivi, prout nimirum
jacent ad partes cruris asymptotici ED, vel ad contrarias, ac terminant ordinatas exhibentes
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 naturæ, quam exposuimus, posset habere
alia itidem asymptotica crura, quot libuerit, ut si ordinata mm in H abeat in infinitum.
Sunt nimirum curvæ continuæ, & uniformis naturæ, quæ asymptotos habent plurimas,
& habere possunt etiam numero infinitas. ()
[78] I69. 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 PefqR. -
I7o. Si gravitas generalis legem vis proportionalis inverse quadrato distantiæ, quam
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.(£) Tum vero inter
(i) Sit ex. gr. in fig. 12. cyclois continua CDEFGH &c., quam gemeret punctum peripheriæ circuli continuo revoluti
supra rectam AB, quae natura sua protenditur utrinque in infinitum, adeoque im infinitis punctis C, E, G, I, &c. occurrit
basi AB. Si ubicumque ducatur quaevis ordinata PQ, productaturque in R ita, ut sit PR tertia post PQ, & datam quampiam
rectam ; punctum R erit ad curvam continuam constantem totidem ramis MNO, VXY, &c., quot erunt arcus Cycloidales
CDE, EFG, &c., quorum ramorum singuli habebunt bina crura asymptotica, cum ordinata PQ im accessu ad omnia puncta,
C, E, G, &c. decrescat ultra quoscumque limites, adeoque ordinata PR crescat ultra limites quoscumque. Erunt hic quidem
omnes asymptoti CK, EL, GS &c. parallelæ inter $e, & perpendiculares basi AB, quod in aliis curvis non est necessarium,
cum etiam divergentes utcumque possint esse. Erunt autem & totidem numero, quot puncta illa C, E, G &$c., nimirum
infinitae. Eodem autem pacto curvarum quarumlibet singuli occursus cum axe in curvis per eas hac eadem lege gemitis
bima crura a$ymptotica generant, cruribus ipsis jacentibus, vel, ut hic, ad eamdem axis partem, ubi curva genetrix ab eo
regreditur retro post appulsum, vel etiam ad partes oppositas, ubi curva genetrix ip$um $ecet, ac transiliat : cumque po$sit
eadem curva altiorum generum secari in punctis plurimis a recta, vel contingi ; poterunt utique haberi & rami a$ymptotici
in curva eadem continua, quo libuerit dato numero.
(k)Nam ex ipsa Geometrica continuitate, quam persecutus sum in dissertatione De Lege Continuitatis, & in dissertatione
De Transformatione Locorum Geometricorum adjecta Sectionibus Comicis, exhibui necessitatem gemeralem secundi
illius cruris asymptotici redeuntis ex infinito. Quotiescunque enim curva aliqua saltem algebraica habet asymptoticum
crus aliquod, debet necessario habere & alterum ipsi respondens, & habens pro asymptoto eamdem rectam : sed id habere
I32
A THEORY OF NATURAL PHILOSOPHY
I33.
FIG. 12.


I34 PHILOSOPHIE NATURALIS THEORIA
ſº
>
ſº
; :)
G\
:
=


PART II -
JApplication of the Theory to Mechanics
166. I will consider in this second part more especially certain general laws of
equilibrium, & motions both of points & masses; these certainly belong to the science of
Mechanics, & they smooth the path that is most favourable for proving very many of those
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;
namely, the arcs of the curve, the area included between the axis & the curve swept out
by the ordinate by its continuous motion, & those points in which the curve cuts the axis.
168. As regards the arcs, some may be called repulsive, & others attractive, according
indeed as they lie on the same side of the axis as the asymptotic branch ED or on the opposite
side, & terminate ordinates that represent repulsive or attractive forces. The first arc
ED must certainly be asymptotic on the repulsive side of the axis, & continued indefinitely.
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 mm 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
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 PejqR. -
170. If universal gravity obeys the law of a force inversely proportional to the square of
the distance (which, as I remarked in the first part, it only obeys as nearly as possible, but
not exactly), sensibly unchanged only throughout the planetary & cometary system, it will
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, 85 thus
meets the base AB in an infinite number of points such as C, E, G, I, &c. If 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 & some given straight line ; then the point
R will trace out a continuous curve consisting of as many branches, MNO, WXY, &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, 85 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 & 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, I 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 & has the same straight line
I35
Consideration of
the curve before
proceeding with
the application to
Mechanics.
The points we have
to consider with
regard to it.
The different kinds
of arcs ; a symp-
totic arcs may even
be infinite in num-
€ſ.
Intermediate arcs.
The ultimate arc
representing grav-
ity possibly not
asymptotic.
I 36 PHILOSOPHIÆ NATURALIS THEORIA
Series CufVa Tu τη
similium, cum serie
Mundorum mag-
nitudine propor-
tionalium.
Omissis sublimiori-
bus, progressus ad
a.Tea S.
Cuicu n q u e axis
se g m e n t o posse
aream respondere
utcunque magnam
vel parvam : partis
Secundæ demon-
stratio.
alios casus innumeros, qui haberi possent, unum censeo speciminis gratia hic non omitten-
dum ; incredibile enim est, quam ferax casuum, quorum singuli sunt notatu dignissimi,
unica etiam hujusmodi curva esse possit.
I7I. Si in fig. I4 in axe C'C sint segmenta AA', A'A'^ numero quocunque, quorum
posteriora sint in immensum majora respectu præcedentium, & per singula transeant,
asympto-[79]-ti AB, A'B', A'B'' perpendiculares axi ; possent inter binas quasque asymptotos
esse curvæ ejus formæ, quam in fig. I habuimus, & quæ exhibetur hic in DEFI &c., D'E'F'I',
&c., in quibus primum crus ED esset asymptoticum repulsivum, postremum SV attractivum,
in singulis vero intervallum EN, quo arcus curvæ contorquetur, sit perquam exiguum
respectu intervalli circa S, ubi arcus diutissime perstet proximus hyperbolæ habenti
ordinatas in ratione reciproca duplicata distantiarum, tum 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 collocato quocunque punctorum numero inter binas quascunque
asymptotos, 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 repulsivo, & inde attractivo ; & 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 materiæ extra ipsum posito accedentes, quantum libuerit, ad æquales, & parallelas
quæ idcirco nihil turbarent respectivum ipsius statum internum.
172. Sed ea jam pertinent ad applicationem ad Physicam, quæ quidem hic innui
tantummodo, ut pateret, quam multa notatu dignissima considerari ibi possent, & quanta
sit hujusce campi foecunditas, in quo combinationes possibiles, & possibiles formæ sunt
sane infinities infinitæ, quarum, quæ ab humana mente perspici utcunque possunt, ita
sunt paucæ respectu totius, ut haberi possint pro mero nihilo, quas tamen omnes unico
intuitu præsentes vidit, qui Mundum condidit, DEUS. Nos in iis, quæ consequentur,
simpliciora tantummodo quædam plerumque consectabimur, quæ nos ducant ad phæno-
mena iis conformia, quæ in Natura nobis pervia intuemur, & interea progrediemur ad
areas arcubus respondentes.
I73. Aream curvæ propositæ cuicunque, utcunque exiguo, axis segmento respondentem
posse esse utcunque magnam, & aream respondentem cuicunque, utcunque magno, [80]
posse esse utcunque parvam, facile patet. Sit in fig. I 5, MQ segmentum axis utcunque
parvum, vel magnum ; ac detur area utcunque magna, vel parva. Ea applicata ad MQ
exhibebit quandam altitudinem MN ita, ut, ducta NR parallela MQ, sit MNRQ æqualis
areæ datæ, adeoque assumpta QS dupla QR, area trianguli MSQ erit itidem æqualis areæ
datæ. Jam vero pro secundo casu satis patet, posse curvam transire infra rectam NR,
uti transit XZ, cüjus area idcirco esset minor, quam area MNRQ; nam esset ejus pars.
potest vel ex eadem parte, vel ex opposita ; & crus ipsum jacere potest vel ad easdem plagas partis utriuslibet cum priore
crure, vel ad oppositas, adeoque cruris redeuntis ex infinito positiones quatuor esse po$$unt. Si in fig. I3 crus ED abeat
in infinitum, existente asymptoto ACA', potest regredi ex parte A vel ut HI, quod crus jacet ad eamdem plagam, vel ut
ΚL, quod jacet ad oppositam ; & 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 a$ymptotica rectilineam habentia asymptotum esse non possint, nisi numero pari, ut & radices imaginariae in
aequationibus algebraicis.
Verum hic in curva virium, in qua arcus semper debet progredi, ut singulis distantiis, sive abscissis, singulae vires,
sive ordinatæ 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 137

138
PHILOSOPHIAE NATURALIS THEORIA
i

A THEORY OF NATURAL PHILOSOPHY I39
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. I4, there are any number of segments AA', A'A", of which each that
follows is immensely great with regard to the one that precedes it ; & if through each
point there passes an asymptote, such as AB, A'B', A"B", perpendicular to the axis; then
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,
A series of similar
curves, with a series
of universes propor-
tional in magnitude.
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; &
indeed I only mentioned them here to show how many things there may be well worth
considering in that section, & how great is the fertility of this field of investigation, in
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
Leaving out more
abstruse matters,
We pass On to areas.
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. .
I73. It is easily shown that the area corresponding to any segment of the axis, however
small, can be anything, no matter how great; & the area corresponding to any segment,
however great, can be anything, no matter how small. In Fig. 15, let MQ be a segment of the
axis, no matter how small, or great ; & let an area be given, no matter how great, or
small. If this area is applied to MQ a certain altitude MN will be given, such that, if NR
is drawn parallel to MQ, then MNRQ will be equal to the given area; & thus, if QS is
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, where the arm lies on the same side of the asymptote or as KL
which lies on the opposite side of its 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 ; & the
method of generation given above will yield examples of the second & 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 & 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 jourth & second cases are
left ; 85 these we will make use of later.
To any segment of
the axis there may
Correspond any
a rea, how ever
great or however
Small ; proof of the
Second part of this
assertion.
I4O PHILOSOPHIÆ NATURALIS THEORIA
Quin immQ 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
quQ$gunque determinatQs, 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. 7 Pos-
set autem ea area esse minor quacunque data ;
etiamsi QV esset asymptotus, qua de Tre 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 illum ferat, quemad-
modum figura exhibet, extra triangulum MSQ, quo
casu patet, aream curvæ respondentem intervallo MQ
fore majorem, quam sit area trianguli MSQ, adeoque
quam sit area data ; erit enim ejus trianguli area
pars areæ pertinentis ad curvam. Quod si curva
etiam secaret alicubi axem, ut in H inter M, & Q,
tum 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.
I75. Area asymptQtica clausa inter asymptotum, & ordinatâm qüamvis, ut in fig. I
BAag, potest esse vel infinita, vel finita magnitudinis cujusvis ingeritis, vel exiguæ.` Id
quidem etiam geometrice demonstrari potest, sed multò facilius demonstratur calculo
integrali admodum elementari ; & in Geometriæ sublimioris elementis habentur theoremata,
ex quibus id admodum facile deducitur (). Generaliter nimi-[81]-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.
I76. Hoc, quod de areis dictum est, necessarium füit ad applicationem ad Mechanicam,
ut nimirum habeatur scala quædam velocitatum, quæ in acéessu puncti cujusvis ad aliud
punctum, vel recessu generantur, vel eliduntur ; prout ejus motus conspiret directione vis,
vel sit ipsi contrarius. Nam, quod innuimus & süpra in adnot. (í) ad mium. I 18., ubi vires
exprimuntur per Qrdinatas, & spatia per abscissas, area, quam texit ordinata, exprimit
incrementum, vel decrementum quadrati velocitatis, quod itidem ope Geometriæ démon-
stratur facile, & demonstravi tam in dissertatione De Viribus Vivis, quam in Stayanis
Supplementis ; sed multo facilius res conficitur ope calculi integralis. (n)
zy + A. Quoniam incipit area in A, in
D e m o n s t ratio
primæ.
Aream asympto-
ticam posse esse
infinitam, vel fini-
tam magnitudinis
cujuscunque.
Areas exprimere
incrementa, v e 1
decrementa, quad-
ati velocitatis.
G
FIG. 15.
(1) Sit Aa in Fig. I =x, ag=y ; ac sit x"y* = I ; erit y = x*/*, y dx elementum areæ = x*/*dx, cujus integrale
f, ę ę � * _ 4
-*- * * * + A, addita constanti A, sive ob x-"'* = y, habebitür
71—711
origine abscissarum ; si m — m fuerit numerus positivus, adeoque m major, quam m ; area erit finita, ac valor A = o ;
area vero erit ad rectangulum Aa ×ag, ut in ad m — m, quod rectangulum, cum ag po$$it esse magna, &5 parva, ut libuerit,
potest esse magnitudinis cujusvis. Is valor fit infinitus, si facto m = m, divisor evadat = o ; adeoque multo magis fit
infinitus valor areæ, si m sit major, quam m. Unde constat, aream fore infinitam, quotiescumque ordinatae crescent in
ratione reciproca simplici, & majore ; secus fore finitam. -
(m) Sit u vis, c celeritas, t tempus, s spatium : erit u dt = dc, cum celeritatis incrementum sit proportionale vi, &
� ę � � G. ę C.
tempusculo ; ac erit c dt = ds, cum $patiolum confectum respondeat velocitati, &5 tempusculo. Hinc eruitur dt = f,
t/,
& pariter dt =#, adeque # =*, & c dc = u ds.
in hypothesi, quod ordinata sit u, & spatium s sit abscissa, est areola respondens spatiolo ds confecto. Igitur incrementum
quadrati velocitatis conspirante vi, adeoque decrementum vi contraria, respondet areæ respondenti spatiolo percurso quovis
infimitesimo tempusculo ; & proinde tempore etiam quovis finito incrementum, vel decrementum quadrati velocitatis
respondet areæ pertinenti ad partem axis referentem spatium percursum. • -
Hinc autem illud sponte consequitur : si per aliquod spatium vires in singulis punctis eaedem permaneant, mobile autem
adveniat cum velocitate quavis ad ejus imitium ; differentiam quadrati velocitatis finalis a quadrato velocitatis initialis
fore semper eamdem, quæ idcirco erit tota velocitas finalis in casu, in quo mobile initio illius spatii haberet velocitatem
mullam. Quare, quod nobis erit inferius usui, quadratum velocitatis finalis, conspirante vi cum directione motus, aequabitur
binis quadratis binarum velocitatum, ejus, quam habuit initio, & ejus, quam acquisivisset in fine, si imitio ingressum fuisset
sine ulla velocitate.
Porro 2c dc est incrementum quadrati velocitatis cc, & u ds

A THEORY OF NATURAL PHILOSOPHY I4I
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 MaV 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 Qayſ,
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. () In
general, it is true, an area of this kind is infinite ; namely when the ordinate increases in
the simple inverse ratio of the abscissae, 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 th
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 (f) to Art. I 18, 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 Viribus Vivis
& in the Supplements to Stay's Philosophy; but the matter is much more easily made
out by the aid of the integral calculus.(m) -
(1) In Fig. I let Aa = x, ag = y; & let w”y” = I. Then will y = x *, & the element of area y dw = w m/*
dx : the integral of this is #" "T"--A, where a constant A is added; or, since wº"—y, we shall have *- wy + A.
-* 71–777 -
Now, since the area is initially A, at the origin of the abscissae, if n—m happened to be a positive number, &
thus n greater than m, then the area will be finite, & the value of A will be = 0. Also the area will be to
the rectangle Aa.ag as n is to n-m; tº 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, 85 the distance. Then will u dt = dc, since the increment
of the velocity is proportional to the force, {5 to the small interval of time. Also e dt = dx, since the distance traverted
corresponds with the velocity & the small interval of time. Hence it follows that dt = dc/u, & similarly di = ds/c,
& therefore de/u = d/º, & 6 de = u dº. Further, 2c dº is the increment of the square of the velocity cº, & u d.
on the hypothesis that the ordinate represents u, & 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, &
the decrement when opposite in direction to the force, is represented by the area corresponding to dº, 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 represent; 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 tº the square of the initial velocity will always be the same : & 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 he velocity which
it had at the beginning tº 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. - -
Proof of the first
part. -
A n a symptotic
area may be either
infinite or equal to
anyº finite area
whatever.
The areas represent
the increments or
decrements of the
Square of the velo-
city.
I42 PHILOSOPHIÆ NATURALIS. THEORIA
Atque id ipsu m,
licet segmenta axis
, sint dimidia spatio-
rum percursorum a
singulis punctis.
Si areæ sint partim
attractivæ, partim
repulsivæ, assumen-
dam esse differen-
tiam earundem.
Appulsus ad axem
curvæ secantis, vel
tangentis : sectio-
mum seu limitum
duo genera.
In quo conveniant
inter se, in quo
differant : limites
cohæsionis, & non
cohæsionis.
Duo genera con-
tactuum.
I77. Duo tamen hic tantummodo notanda sunt ; primo quidem illud : si duo
puncta ad se invicem accedant, vel a se invicem recedant in ea recta, quæ ipsa conjungit,
segmenta illius [82] axis, qui exprimit distantias, non expriment spatium confectum ; nam
moveri debebit punctum utrumque : adhuc tamen illa segmenta erunt proportionalia ipsi
spatio confecto, eorum nimirum dimidio ; quod quidem satis est ad hoc, ut illæ areæ adhuc
sint proportionales incrementis, vel decrementis quadrati velocitatum, adeoque ipsa
exprimant.
I78. Secundo loco notandum illud, ubi areæ respondentes dato cuipiam spatio sint
partim attractivæ, partim repulsivæ, earum differentiam, quæ 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 areæ
omnes attractivæ simul, omnes repulsivas simul superent, an deficiant, & quantum ; inde
enim, & a velocitate, quæ habebatur initio, erui poterit quod quæritur.
I79. Hæc quidem de arcubus, & areis ; nunc aliquanto diligentius considerabimus
illa axis puncta, ad quæ curva appellit. Ea puncta vel sunt ejusmodi, ut in iis curva axem
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, & hæc 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 sequenti sectione
debet necessario ex parte attractiva transire ad repulsivam, ac vice versa ; patet, limites
fore alternatim prioris illius, & hujus posterioris generis. -
18o. Porro limites prioris generis, a limitibus posterioris ingens habent inter se dis-
crimen. Habent illi quidem hoc commune, ut duo puncta collocata in distantia unius
limitis cujuscunque nullam habeant mutuam vim, adeoque si respective quiescebant, pergant
itidem respective quiescere. At si, ab illa respectiva quiete dimoveantur ; tum 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 distantiâ minuatur ; habebunt in limite prioris generis vim repulsivam, quæ obstabit
uteriori accessui, & urgebit puncta ad mutuum recessum, quem 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 illa priore distantia, quæ erat
major, adhuc magis sponte fugient. Pariter si distantia augeatur, in primo limitum genere
a vi attractiva, quæ 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 $e magis adhuc
fugient, adeoque a minore illa priore distantia sponte magis recedent. . Hinc illos prioris
generis limites, qui mutuæ positionis tenaces sunt, ego quidem appellavi limites cohæ$ionis,
& secundi generis limites appellavi limites mom cohæ$ionis.
181. Illa puncta, in quibus curva axem tangit, sunt quidem terminus quidam yirium,
quæ 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 repulsivo ; repulsiones evanescunt, sed post contactum remanent itidem
repulsiones ; ac si ab arcu attractivo, attractionibus evanescentibus attractiones iterum
imimediate succedunt. Duo puncta collocata in ejusmodi distantia respective quiescunt;
A THEORY OF NATURAL PHILOSOPHY I43
177. However, there are here two things that want noting only. The first of them
is this, that if two points approach one another or recede from one another in the straight
line joining them, the segments of the axis, which expresses distances, do not represent
the distances traversed; for both points will have to move. Nevertheless the segments
will still be proportional to the distance traversed, namely, the half of it; & this indeed is
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
given interval are partly attractive & partly repulsive, their difference, obtained by
subtracting the sum of all those that are repulsive from the sum of those that are attractive,
or vice versa, will represent the increment, or the decrement, of the square of the velocity,
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
manner those points of the axis to which the curve approaches. These points are either
such that the curve cuts the axis in them, for instance, the points E, G, I, &c. in Fig. I :
or such that the curve only touches the axis at the points. Points of the first kind are
those in which there is a transition from repulsions to attractions, or vice-versa; & these
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.
180. Further, there is a distinction between limit-points of the first & those of the
second kind. The former kind have this property in common; namely that, if two points
are situated at a distance from one another equal to the distance of any one of these limit-
points from the origin, they will have no mutual force; & thus, if they are relatively
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
of forces, which decrease on both sides, as approach to these points takes place, beyond
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
The same result
holds good even
when the segments
of the axis are the
halves of the dis-
tances traversed by
single points,
If the areas are
partly attractive &
partly repulsive,
their difference
must be taken.
Approach of the
Curve to the axis
when it cuts or
touches it ; t w O
kinds of intersec-
tion's or limit-
points.
In what they agree
& in what they
differ ; the limit-
points of cohesion
& Of non-cohesion.
Two kinds of con-
tact.
I 44 PHILOSOPHIAE NATURALIS THEORIA
Limites cohaesionis
validi, vel languidi
pro forma curva
prope sectionem.
Posse limites esse
quotcunque nume-
ro, utcunque proxi-
mos, vel remotos
invicem, & respectu
originis abscissa-
rum, positos ordine
quoCunque.
Quae positio rectae
tangentis curvam
in limite rarissima,
quae frequentissima.
Arcus exigui hinc
& inde aequales, &
similes.
Transitus per infi-
nitum cru ribus
asymptoticis.
sed in primo casu resistunt soli compressioni, non etiam distractioni, & in secundo resistunt
huic soli, non illi. -
182. Limites cohaesionis possunt esse validissimi, & languidissimi. Si curva ibi quasi
ad perpendiculum 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 cohaesionis exhibet in fig. 1 arcus tNy, secundum cNx. In illo
assumptis in axe Nz, Nu utcunque exiguis, possunt vires zt, uy, & area Nzt, Nuy esse
utcumque magnae, adeoque, mutatis utcunque parum distantiis, possunt haberi vires ab
ordinatis expressa utcunque magna, qua vi comprimenti, vel distrahenti, quantum libuerit,
valide resistant, vel area utcunque magnae, quae velocitates quantumlibet magnas
respectivas elidant, adeoque sensibilis mutatio positionis mutua impediri potest contra
utcunque magnam velvim prementem, vel celeritatem ab aliorum punctorum actionibus
impressam. In hoc secundo genere limitum cohaesionis, assumptis etiam majoribus
segmentis Nz, Nu, possunt & vires ze,ux, & areae Nzc, Nux, esse quantum libuerit exiguae,
& idcirco exigua itidem, quantum libuerit, resistentia, quae mutationem vetet.
I83. Possunt autem hi limites esse quocunque, utcunque magno numero; cum
demonstratum sit, posse curvam in quotcunque, & quibuscunque punctis axem secare.
Possuntidcirco etiam esse utcunque inter se proximi, vel remoti, ut [84] alicubi intervallum
inter duos proximos limites sit etiam in quacunque ratione majus, quam sit distantia
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, tum 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.
184. In utroque limitum genere fieri potest, ut curva in ipso occursu cum axe pro
tangente habeat axem ipsum, ut habeat ordinatam, ut aliam rectam aliquam inclinatam.
In primo casu maxime ad axem accedit, & initio saltem languidissimus est limes; in secundo
maxime recedit, & initio saltem 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, velpotius nullos, constat exeo, quod flexus contrarii puncta
in quovis finito arcu data curva cujusvis numero finito esse debent, utin Theoria curvarum
demonstrari potest, & alia puncta sunt infinita numero, adeoque illa cadere in intersectiones
est infinities improbabilius. Possunt tamen saepe cadere prope limites: nam in singulis
contorsionibus curvae saltem 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, 8 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.
185. Hi quidem sunt limites per intersectionem curvae cum axe, viribus evanescentibus
in ipso 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 I45
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
at the point almost at right angles, & goes off to a considerable distance from it, they
are very strong. But if it cuts the axis at a very small angle & recedes from it but little,
then they will be very weak. The arc tRNy in Fig. I represents the first kind of limit-
points of cohesion, and the arc cMa: the second kind. At the point N, if Nz, Nu are
taken along the axis, no matter how small, the forces zt, uy, & the areas Nzt, Nuy 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,
Nu are taken of considerable size even, then it is possible for both the forces ze, uy, &
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;
for it has been proved that the curve can cut the axis in any number of points, & anywhere.
Therefore it is possible for them to be either close to or remote from one another, without
any restriction whatever, so that the interval between any two consecutive limit-points
at any place shall even bear to the distance of the first of the two from A, the origin of
abscissae, a ratio that is greater than unity. ... In other words, in any interval, either very
small or very large, there may be an exceedingly large number of them so close to one
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
meets the axis, may have the axis itself as its tangent, or the ordinate, or any other straight
line inclined to the axis. In the first case it approximates very closely to the axis, &
close to the point at any rate it is a very weak limit-point; in the second case, it departs
from the axis very sharply, & close to the point at any rate it is a very strong limit-point.
But these two cases must be of very rare occurrence, if indeed they ever occur. For, since
at the point the curve is bound to cut the axis & go on, & thus be cut in the same point
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
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
T.
The limit-points of
cohesion are strong
or weak according
to the form of the
curve n e a r the
point of inter-
section.
The limit-points
are indefinite a s
regards number,
their proximity to
or remoteness from
one another, & the
order of their occur-
rence with respect
to the origin of ab-
Scissae.
What position of
the straight line
touching the curve
at a limit-point is
most unusual, &
what most fre-
quent ; Small arcs
On each side of the
limit-point are
equal & similar.
Passage through
infinity for asymp-
totic branches.
I46 PHILOSOPHIÆ NATURALIS THEORIA
Quatuor e o r u m
genera ; bini re-
spondentes contac-
tibus, bini limiti-
hus, alter cohæs-
ionis, alter non
cohæsionis.
INullum in Natura,
admittendum præ-
ter postremum, nec
vero eum 1psum
utcunque.
curvæ arcus. Diximus supra num. 168. adnot. (i), quando crus asymptQticum abit in
infinitum, debere ex infinito regredi crus aliud habens pro asymptoto eandem rectam, &
posse regredi cum quatuor diversis positionibus pendentibus a binis partibus ipsius rectæ,
& binis plagis pro singulis rectæ partibus ; sed cum nostra curva debeat semper pr9gredi,
diximus, relinqui pro ea binas ex ejusmodi quatuor positionibus pro quovis crure abeunte
in infinitum, in qüibus nimirum regressus fiat ex plaga opposita. Quoniam vero, progre-
diente curva, abire potest in infinitum tam crus repulsivum, quam crus attractivum , jam
iterum fiunt casus quatuor possibiles, quos exprimunt figuræ 16, 17, 18, & 19, in quibus
omnibus est axis ACB, asymptotus DCD', cfus recedens in infinitum EKF, regrediens
ex infinito GMH.


M.
Dy G
FIG. I6.
/
D
FIG. 17.
186. In fig. 16. cruri repulsivo EKF succedit itidem repulsivum GMH ; in fig. I7
repulsivo attractivum ; in 18 attractivo attractivum ; in I9 attractivo repulsivum. Primus
& tertius casus respondent contactibus. Ut enim in illis evanescebat vis ; sed directionem
non mutabat ; ita & hic abit quidem in infinitum, sed directionem non mutat. Repulsioni
IK in fig. I6 succedit repulsio LM ; & attractioni in fig. I8 attractio. Quare ii casus non
habent limites quosdam. Secundus, & quartus habent utique limites ; nam in fig. I7
repulsioni IK succedit attractio LM ; & in fig. I9 attractioni repulsio ; atque idcirco
secundus continet limitem cohaesionis, quartus limitem non cohaesionis.
187. Ex istis casibus a nostra curva censeo removendos esse omnes præter solum
quartum ; & in hoc ipso removenda omnia crura, in quibus ordinata crescit in ratione
minus, quam simplici reciproca distantiarum a limite. ' Ratio excludendi est, ne haberi
aliquando vis infinita possit, quam & per se se absurdam censeo, & idcirco præterea, 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 collocato 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 eæ crescant in ratione reciproca minus,
quam simplici distantiarum CI,CL ; area FKICD, vel GMLCD erit finita, adeoque
punctum impulsum versus C velocitate majore, quam quæ respondeat illi areæ, debet
transire per omnes virium magnitudines usque ad vim absolute infinitam in C, quæ ibi
[86] præterea & attractiva esse deberet, & repulsiva, limes videlicet omnium & attracti-
varum, & repulsivarum ; idcirco ne hic 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 I47
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.
I68, 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.
D D|\G
E.
H
E
L
B
Fº Fl|Dſ
FIG. I.8. FIG 19.
186. In Fig. 16, 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. 16, 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,
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-
SIOIR,
None of these ex-
Cept the last admis-
sible in Nature ; &
not even that in
general.
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.

I48 PHILOSOPHIÆ NATURALIS THEORIA
Transitus per eum
limitem impossi-
bilis : in quibus
distantiis constet,
eum non haberi.
Transitus ad puncta
materiæ, & massas.
Quies in limitibus;
motus puncti positi
extra ipsos.
Motus post proxi-
mum limitem super-
atum, & oscillatio.
Casus oscillationis
majoris trans plures
limites.
188. Quando habeatur hic quartus casus in nostra curva cum ea conditione ; tum
quidem nullum punctum collocatum ex altera parte puncti C poterit ad alteram transilire,
quacunque velocitate ad accessum impellatur versus alterum punctum, vel ad recessum
ab ipso, impediente transitum area repulsiva infinita, vel infinita attractiva. Inde vero
facile colligitur, eum casum non haberi saltem in ea distantia, quæ 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 nostræ sensibi-
litatis sphæram, vel ultra omnes telescopicas fixas, vel citra microscopicas moleculas.
I89. Expositas hisce, quæ ad curva virium pertinebant, aggrediar simpliciora quædam,
quæ maxime notatu digna sunt, ac pertinent ad combinationem punctorum primo quidem
duorum, tum trium, ac deinde plurium in massa etiam coalescentium, ubi & vires mutuas,
& motus quosdam, & vires, quas in alia exercent puncta, considerabimus.
I90. Duo puncta posita in distantia æquali distantiæ limitis cujuscunque ab origine
abscissarum, ut in fig. I. AE, AG, AI, &c, (immo etiam si curva alicubi axem tangat, æquali
distantiæ 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 æqualia, prout fuerint süb
arcu attractivo, vel repulsivo. Quoniam autem vis manebit semper usque ad proximum
limitem directionis ejusdem ; pergent progredi in ea recta, quæ ipsa urgebat prius, usque
ad distantiam limitis proximi, motu semper accelerato, juxta legem expositam num. I76,
ut nimirum quadrata velocitatum integrarum, quæ acquisitæ jam sunt usque ad quodvis
momentum (nam velocitas initio ponitur nulla) respondeant areis clausis inter ordinatam
respondentem puncto axis terminanti abscissam, quæ exprimebat distantiam initio motus,
& ordinatam respondentem puncto axis terminanti abscissam, quæ exprimit distantiam
pro eo sequenti 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, quæ habebatur prius, adeoque
perget acceleratio prioris motus.
I9I. Proximus limes erit ejus generis, cujus generis diximus limites cohæsionis, in quo
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 præcedentibus viribus, incipiet,
velocitas motus minui vi contraria priori, sed motus in eadem directione perget ; donec
sub sequenti arcu obtineatur area curvæ æqualis illi, quam habebat prior arcus ab initio
motus usque ad limitem ipsum. Si ejusmodi æqualitas obtineatur alicubi sub arcu
sequente ; ibi, extincta omni præcedenti 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, quem fuerant prætergressa ; tum amittendo, quas acquisiverant usque ad dis-
tantiam, quam habuerant initio ; viribus nimirum iisdem occurrentibus in ingressu, &
areolis curvæ iisdem per singula tempuscula exhibentibus quadratorum velocitatis incre-
menta, vel decrementa eadem, quæ fuerant antea decrementa, vel incrementa. Ibi autem
iterum retro cursum reflectent, & oscillabunt circa illum cohæsionis limitem, quem fuerant
prætergressa, quod facient hinc, & inde perpetuo, nisi aliorum externorum punctorum
viribus perturbentur, habentia velocitatem maximam in plagam utramlibet in distantia
ipsius illius limitis cohæsionis. -
192. Quod si ubi primum transgressa sunt proximum limitem cohæsionis, offendant
arcum ita minus validum præcedente, qui arcus nimirum ita minorem concludat aream,
quam præcedens, ut tota ejus area sit æqualis, vel etiam minor, quam illa præcedentis
arcus area, quæ habetur ab ordinata respondente distantiæ habitæ initio motus, usque ad
A THEORY OF NATURAL PHILOSOPHY I49
188. When, if ever, this fourth case occurs in our curve, then indeed no point situated
on either side of the point C will be able to pass through it to the other side, no matter
what the velocity with which it is impelled to approach towards, or recede from, the other
point; for the infinite repulsive area, or the infinite attractive area, will prevent such
passage. Now, it can easily be derived from this, that this case cannot happen at any rate
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.
189. Having thus set forth these matters relating to the curve of forces, I will now
discuss some of the simpler things that are more especially worth mentioning with regard
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
from the origin of abscissae, like AE, AG, AI, &c. in Fig. I (or indeed also where the
curve touches the axis anywhere, equal to the distance of the point of contact from the
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. I76; 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. wº
191. The next limit-point will be one of the kind we have called limit-points of cohesion,
namely, one in which, if the distance is increased by repulsion, then attraction follows;
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
happened to come to an arc representing forces so much weaker than those of the preceding
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
Passage through a
limit-point of this
kind is impossible ;
distances at which
it is certain that
there are no such
limit-points.
We now pass on to
points of matter, &
Iſla,SSCS.
Rest at limit-
points; motion of
a point situated
without them.
Motion after the
next limit-point is
passed ; oscillation.
The case of a larger
Oscillation through
Several limit-points.
I5o PHILOSOPHIÆ NATURALIS THEORIA
Velocitatis muta-
tiones alternæ: ubi
ea habeat maxi-
mum, & minimum
ubi extingui possit
Circa quos limites
oscillatio m ajo r
esse debeat, & unde
pendeat ejus mag-
nitudo.
Accessum debere
sisti saltem a primo
arcu re p ul si v o,
r e c e s s u m posse
haberi in infinitum :
casus n o t a b i 1 is
exiguæ differentiæ
velocitatis ingentis.
limitem ipsum ; tum vero devenient ad distantiam alterius limitis proximi priori, qui
idcirco erit limes non cohæsionis. Atque ibi quidem in casu æqualitatis illarum arearum
consistent, velocitatibus prioribus elisis, & nulla vi gignente novas. At in casu, quo tota
illa area sequentis arcus fuerit minor, quam illa pars areæ præcedentis, appellent ad dis-
tantiam ejus limitis motu quidem retardato, sed cum aliqua velocitate residua, quam
distantiam idcirco prætergressa, & nacta vires directionis mutatæ jam conspirantes cum
directione sui motus, non, ut ante, oppositas, accelerabunt motum usque ad distantiam
limitis proxime sequentis, quam prætergressa procedent, sed motu retardato, ut in priore ;
& si area sequentis arcus non sit par extinguendæ ante suum finem toti [88] velocitati,
quæ fuerat residua in appulsu ad distantiam limitis præcedentis non cohæsionis, & quæ
acquisita est in arcu sequenti usque ad limitem cohæsionis proximum ; tum puncta
appellent ad distantiam limitis non cohæsionis sequentis, ac vel ibi sistent, vel progredientur
itidem, eritque semper reciprocatio quædam motus perpetuo accelerati, tum retardati ;
donec 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.
I93. Porro in hujusmodi motu patet illud, dum itur a distantia limitis cohæsionis
ad distantiam limitis non cohæsionis, velocitatem semper debere augeri ; tum post
transitum per ipsam debere minui, usque ad appulsum ad distantiam limitis non cohæsionis,
adeoque habebitur semper in ipsa velocitate aliquod maximum in appulsu ad distantiam
limitis cohæsionis, & minimum in appulsu ad distantiam limitis non cohæsionis. Quamo-
brem poterit quidem sisti motus in distantia limitis hujus secundi generis ; si sola existant
illa duo puncta, nec ullum externum punctum turbet illorum motum : sed non poterit
sisti in distantia limitis illius primi generis ; cum ad ejusmodi distantias deveniatur semper
motu accelerato. Præterea patet & illud, si ex quocunque loco impellantur velocitatibus
æqualibus vel alterum' versus alterum, vel ad partes oppositas, debere haberi reciprocationes
easdem auctis semper æque velocitatibus utriusque, dum itur versus distantiam limitis
primi generis, & imminutis, dum itur versus distantiam limitis secundi generis.
I94. Patet & illud, si a distantia limitis primi generis dimoveantur vi aliqua, vel non
ita ingenti velocitate impressa, Oscillationem fore perquam exiguam, saltem si quidam
validus fuerit limes ; nam velocitas incipiet statim minui, & ei vi statim vis contraria
invenietur, ac puncta parum dimota a loco suo, tum 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 quem motus primo retardari incipiet. Quin immo si arcus proximus
hinc, & inde ab ejusmodi limite secundi generis concluserit aream ingentem, ac majorem
pluribus sequentibus contrariæ directionis, vel majorem excessu eorundem supra areas
interjacentes directionis suæ ; tum 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, quæ post oc-[89]-currunt, & quadratum
velocitatis vel augent, vel minuunt quantitate sibi proportionali. -
I 95. Utcunque magna sit velocitas, qua dimoveantur a distantia limitum illa duo
puncta, utcunque validos inveniant arcus conspirantes cum velocitatis directione, si ad
se invicem accedunt, debebunt utique alicubi motum retro reflectere, vel saltem sistere,
quia saltem advenient ad distantias illas minimas, quæ respondent arcui asymptotico,
cujus area est capax extinguendæ cujuscunque velocitatis utcunque magnæ. At si
recedant a se invicem, 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 I5I
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
of a limit-point of cohesion to a limit-point of non-cohesion the velocity is bound to be
always increasing; then after passing through the latter it must decrease up to its arrival at the
distance of a limit-point of non-cohesion. Thus, there will always be in the velocity a
maximum on arrival at a distance equal to that of a limit-point of cohesion, & a minimum
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
a limit-point of the first kind by some force (especially when the velocity thus impressed
is not extremely great), then the Oscillation will be exceedingly small, at least so long as the
limit-point is a fairly strong one. For the velocity will commence to be diminished
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
Alternate changes
of velocity; where
it has a maximum
value, & a mini-
mum value ; where
it may be destroyed.
The limit-points
about which the
Oscillation must be
larger; & the thing
on which its magni-
tude depends.
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
a distance equal to that of any limit-point, no matter how strong are the arcs they come
upon, which are in the same direction as that of the velocity; yet, if they approach one
another, they are bound somewhere to have their motion reversed, or at least to come
to rest; for, at all events, they must finally attain to those very small distances that correspond
to an asymptotic arm, the area of which is capable of destroying any velocity whatever,
no matter how great. But, if they recede from one another, it may happen that they come
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
Approach is bound
to cease at any rate
owing to the first
repulsive arc, but
Separation can go
on indefinitely ; a
noteworthy case
of the very small
difference for a very
great velocity.
I 52 PHILOSOPHIÆ NATURALIS THEORIA
Demonstratio ad-
modum simplex.
sivas, usque ad languidissimum illum arcum postremi cruris gravitatem exhibentis. Tum
vero motus acquisitus ab illo arcu nunquam poterit a sequentibus sisti, & puncta illa recedent
à se invicem in immensum : quin immo si ille arcus répulsivus cum sequentibus repulsivis
ingentem habeat areæ excessum supra arcus sequentes áttractivos ; cum ingenti velocitate
pergent puncta in immensum recedere a se invicem ; & licet ad initium ejus tam validi
arcus repulsivi deveniant puncta cum velocitatibus non parum diversis ; tamen velocitates
recessuum post noyum ingens illud augmentum erunt parum admodum discrepantes a
se invicem : nam si ingentis radicis quadrato addatur qüadratum radicis multo minoris,
quamvis non exiguæ; radix extracta ex summa parum admodum differet a radice priore.
196. Id quidem ex Euclidea etiam Geometria manifestum fit. Sit in fig. 2o AB
linea longior, cui addatur ad perpendiculum BC, multo minor, quam fit ipsa ; tum centro A,
intervallo AC, fiat semicirculus occurrens AB hinc, & inde in E, D. Quadrato AB
addendo quadratum BC habetur quadratum AC, sive AD ; & tamen hæc excedit præce-
dentem radicem AB per solam BD, quæ semper est minor, quam BC, & est ad ipsám, ut
Quid accidat binis
punctis, cum sunt
Sola, quid possit
accidere actionibus
aliorum externis.
Si limites sint a se
invicem re m oti,
m ut a, ta, multum
distantia re diri
retro : secus, si
sint proximi.
Superiorum usus in
IPhysica.
est ipsâ ad totam BE. Exprimat AB velocitatem, quam in punctis quiescentibus gigneret
arcus ille repulsivus per suam aream, una cum differentia `omnium sequentium arcuum
repulsiv9rum supra omnes sequentes attractivos : exprimat autem BC velocitatem, cum
qua advenitur ad distantiam respondentem
initio ejus arcus : exprimet AC velocitatem,
quæ 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, quæ
permanebit ad sensum eadem post recessum in
1ÌYÌIÌ1€I)SlJIY). -
I97. Hæc accident binis punctis sibi relictis, vel impulsis [90] in recta, qua junguntur,
cum oppositis velocitatibus æqualibus, quo casu etiam facile demonstratur, punctum,
quod illorum distantiam bifariam secat, debere quiescere ; nunquam in hisce casibus
poterit motus extinguiin adventu ad distantiam limitis cohæsionis, & 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 illa, poterit res multo
aliter se habere. Ubi ex. gr. a se recedunt, & velocitates recessus augeri deberent in accessu
ad distantiam limitis cohæsionis ; potest externa compressio illam velocitatem minuere,
& extinguere in ipso appulsu ad ejusmodi distantiam. Potest externa compressio cogere
illa puncta manere immota etiam in ea distantia, in qua se validissime repellunt, uti duæ
cuspides elastri manu compressæ detinentur in ea distantia, a qua sibi relictæ statim
recederent : & simile quid accidere potest vi attractivæ per vires externas distrahentes.
I98. Tum vero diligenter notandum discrimen inter casus varios, quos inducit varia
arcuum curvæ natura. Si puncta sint in distantia alicujus limitis cohæsionis, circa quem
sint arcus amplissimi, ita, ut proximi limites plurimum inde distent, & multo magis etiam,
quam sit tota distantia proximi citerioris limitis ab origine abscissarum ; tum 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 repulsivo permaneat, vel
attractivo. At si ibi frequentissimi limites, curva sæpissime secante axem ; tum quidem
post compressionem, vel distractionem ab externa vi factam, poterunt sisti in multo minore,
vel majore distantia, & adhuc esse in distantia alterius limitis cohæsionis sine ullo conatu
ad recuperandum priorem locum.
A Éb
EIG. 2o.
I99. Hæc omnia aliquanto fusius considerare libuit, quia in applicatione ad Physicam
magno usui erunt infra hæc ipsa, & multo magis hisce similia, quæ massis respondent
habentibus utique multo uberiores casus, quam bina tantummodo habeant puncta. Illa
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 I 53
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
AB be a fairly long line, to which is added, perpendicular to it, BC, which is much less
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
along the straight line joining them with velocities that are equal & opposite; in such
a case it can be easily proved that the middle point of the distance between them is bound
to remain at rest. The motion in the cases we have discussed can never be destroyed
altogether on arrival at a distance equal to that of a limit-point of cohesion, & much less
will the two points be able to stop at a distance apart that is not equal to that of some
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,
which arise from the various natures of the arcs of the curve. If our points are at a distance
of any limit-point of cohesion, on each side of which the arcs are very wide, so that the
nearest limit-points are very far distant from it, & also much more so than the nearest
limit-point to the left is distant from the origin of abscissae ; they may, under the action
of an external force causing either compression or tension, be reduced after many alternations
to a distance, either less, or greater, than the original distance, in such a way that they
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
at length; for they will be of great service later in the application of the Theory to physics,
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
The demonstration
is perfectly simple.
What may happen
to two points when
they are by them-
selves; what may
happen to them
when under the
actions of other
points external to
them.
If the limit-points
are far apart, there
is a tendency to
return if the dis-
tance suffers a ny
considerable change;
but this is not the
case when the limit-
points are very close
together.
The use of the
above facts in phy-
sics.
I 54 PHILOSOPHIÆ NATURALIS THEORIA
M o t u s binorum
punctorum oblique
projectorum.
Casus, in quo duo
puncta, debeant de-
scribere spirales
circa medium im-
motum.
Theorema de statu
puncti medii, & gen-
eraliter in massis
c e n t r i gravitatis
perseverante.
Accessum alterius
e binis ad planum
quodvis alterius
æquari recessui ex
vi mutua.
Transitus ad syste-
ma, p u n c t o r u m
trium : bina gener-
alia problemata.
repulsivo cum velocitatibus ingentibus, quæ ubi jam ad ingentes deventum est distantias,
parum admodum a se invicem differant, nec ad sensum mutentur quidquam per immensa
intervalla, luminis emissionem, & propagationem uniformem, ac ferme eandem celeritatem
in quovis ejusdem speciei radio fixarum, Solis, flammæ, cum exiguo discrimine inter
diversos coloratos radios ; illa vis permanens post compressionem ingentem, vel diffractionem
elasticitati explicandæ in-[9I]-serviet ; quies ob frequentiam limitum, sine conatu ad
priorem recuperandam figuram, mollium corporum ideam suggeret ; quæ quidem hic
innuo in antecessum, ut magis hæreant animo, prospicienti jam hinc insignes eorum usus.
2oo. Quod si illa duo puncta projiciantur oblique motibus contrariis, & æqualibus
per directiones, quæ cum recta jungente ipsa illa duo puncta angulos æquales efficiant ;
tum vero punctum, in quo recta illa conjungens secatur bifariam, manebit immotum ;
ipsa autem duo puncta circa id punctum gyrabunt in curvis lineis æqualibus, & contrariis,
quæ data lege virium per distantias ab ipso puncto illo immoto (uti daretur, data nostra
curva virium figuræ I, cujus nimirum abscissæ exprimunt distantias punctorum a se invicem,
adeoque eorum dimidiæ 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. § I 9.
2OI. Hic illud notabo tantummodo, inter infinita curvarum genera, quæ describi
possunt, cum nulla sit curva, quæ assumpto quovis puncto pro centro virium describi
non possit cum quadam virium lege, quæ definitur per Problema directum virium
centralium, esse innumeras, quæ 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, quæ illa jungit (qui quidem casus accurati occursus in ea recta est infinities
improbabilior casu deflexionis cujuspiam, cum sit unicus possibilis contra infinitos), non
recedant retro, sed circa punctum spatii medium immotum gyrent perpetuo sibi deinceps
semper proxima, intervallo etiam sub sensus non cadente ; qui quidem casus itidem
diligenter notandi sunt, cum sint futuri usui, ubi de cohæsione, & mollibus corporibus
agendum erit. -
2o2. Si utcunque alio modo projiciantur bina puncta velocitatibus quibuscunque ;
potest facile ostendi illud : punctum, quod est medium in recta jungente ipsa, debere
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 dissertatione De Centro Gravitatis proposui, quam ipsam demonstrationem
hic etiam inferius exhibebo. - -
[92] 2o3. Interea hic illud postremo loco adnotabo, quod pertinet ad duorum
punctorum motum ibi usui futurum : si duo puncta moveantur viribus mutuis tantummodo,
& ultra ipsa assumatur planum quodcunque ; accessus alterius ad illud planum secundum
directionem quamcunque, æquabitur recessui alterius. Id sponte consequitur ex eo,
quod eorum absoluti motus sint æquales, & contrarii ; cum inde fiat, ut ad directionem
aliam quamcunque redacti æquales itidem maneant, & contrarii, ut erant ante. Sed de
æquilibrio, & motibus duorum punctorum jam satis.
204. Deveniendo ad systema trium punctorum, uti etiam pro punctis quotcunque,
res, si generaliter pertractari deberet, reduceretur ad hæc duo problemata, quorum alterum
pertinet ad vires, & alterum ad motus : I. Data positione, & distantia mutua eorum punc-
torum, invenire magnitudinem, & directionem vis, qua urgetur quodvis ex ipsis, compositae a
viribus, quibus urgetur a reliquis, quarum singularum virium lex communis datur per curvam
figuræ primae. 2. Data illa lege virium figuræ primae invenire motus eorum, punctorum,
qüorum' singula cum datis velocitatibus projiciantur ex datis locis cum datis directionibus.
Primum facile solvi potest, & potest etiam ope curvæ figuræ I determinari lex virium
A THEORY OF NATURAL PHILOSOPHY I55
& 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
opposite to one another, in directions making equal angles with the straight line joining
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
curves that can be described, there are an innumerable number which will either re-enter
their paths, or wind in spirals; for there is no curve that, having taken any point whatever
for the centre of forces, cannot be described with some law of forces, which is determined
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,
it can readily be proved that the middle point of the line joining them must remain at
rest or move uniformly in a straight line; and that about this point, whether it is at rest
or is moving uniformly, the oscillations or descriptions of the curved paths, referred to
above, must take place. But this, more generally, is a property relating to masses, of any
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 Centro Gravitatis; this demonstration I will also give here in the articles that
follow.
2O3. Lastly, I will here mention in passing something that refers to the motion of
two points, which will be of use later, in connection with that subject. If two points
move subject to their mutual forces only, & any plane is taken beyond them both, then
the approach of one of them to that plane, measured in any direction, will be equal to the
recession of the other. This follows immediately from the fact that their absolute motions
are equal & opposite; for, on that account, it comes about that the resolved parts in any
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.
2O4. When we come to consider systems of three points, as also systems of any number
of points, the whole matter in general will reduce to these two problems, of which the
one refers to forces and the other to motions. I. Being given the position and the mutual
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
The motion of two
points projected
obliquely. -
The case in which
the two points are
bound to describe
spirals about the
motionless middle
point.
Theorem on the
steady state of the
Central point &,
m or e generally,
of the centre of
gravity in the case
of masses.
The approach of
One of the two
points towards any
plane is equal to
the recession of the
Other from it, on
aCCOunt of the
mutual force.
Extension to a
system of three
points; two general
problems.
156 PHILOSOPHIÆ NATURALIS THEORIA
generaliter pro omnibus distantiis assumptis in quavis recta positionis datæ, a que id tam
geometrice determinando per puncta curvas, quæ ejusmodi legem exhibeant, ac déterminent
sive magnitudinem vis absolutæ, sive magnitudines binarum virium, in quas ea concipiatur
resoluta, & quarum altera sit perpendicularis datæ illi rectæ, altera secúndum illam agat ;
quam exhibendo tres formulas analyticas, quæ id præstent. 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 massæ punctorum, est illud ipsum
celeberrimum problema quod appellant trium corporum, usque adeo quæsitum per hæc
nostra tempora, & non nisi pro peculiaribus quibusdam casibus, & cum ingentibus limita-
tionibus, nec adhuc satis promoto ad accurationem calculo, solutum a paucissimis nostri
ævi Geometris primi ordinis, uti diximus num. 122.
$,£,£? 205. Pro hoc secundo casu illud est notissimum, si tria puncta sint in fig. 2I A, C, B,
É"äÄ"jî & distantia AB duorum divisa semper bifariam 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 generali theore-
mate de centro gravitatis, cujus & superius
injecta est mentio, & de quo age-[93]-mus
infra pro massis quibuscunque. Hinc si sibi re-
linquantur, accedet C ad E, & rectæ 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 adverso sint, in quo motu tam directio D
rectæ AB, quam directio rectæ CD, & ejus incli-
natio ad AB, plerumque mutabitur. EIG. 2I.
#.i; 206. Quod pertinet ad inveniendam vim pro quacunque positione puncti C respectu
e'binis viriÉÉ` punctorum A, & B, ea facile sic invenietur. In fig. I assumendæ essent abscissæ in axe
æquales rectis AC, BC figuræ 2 I, & erigendæ ordinatæ ipsis respondentes, quæ vel ambæ
essent ex parte attractiva, vel ambæ ex parte repulsiva ; vel prima attractiva, & secunda
repulsiva ; vel prima repulsiva & secunda attractiva. In primo casu sumendæ essent CL,
CK ipsis æquales (figura 2 I exhibet minores, ne nimis 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 completo parallelogrammo LCKF,
vel MCNH, vel LCMI, vel KCNG, diameter CF, vel CH, vel CI, vel CG exprimeret
directionem, & magnitudinem vis compositæ, qua urgetur C a reliquis binis punctis.
Methodus constru- 207. Hinc si assumantur ad arbitrium duo loca quæcunque punctorum A, & B, ad
Ę",3; quæ referendum sit tertium C; ducta quavis recta DEC indefinita, ex quovis ejus puncto
mat vim ejusmödi, posset erigi recta ipsi perpendicularis, & æqualis 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, quæ exprimeret
vim redactam ad directionem DC per perpendiculum FO, ut CO ; alteram, quæ exprimeret
vim perpendicularem OF : nam eo pacto haberentur etiam directiones vis absolutæ ab
iis compositæ per ejusmodi binas ordinatas. Oporteret autem ipsam ordinatam curvæ
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
rectæ DC, vel ad oppositam, pro secunda.
Expressio magis 2o8. Hoc pacto datis locis A, B pro singulis rectis egressis e puncto medio D duæ
É* P**"* haberentur divérsæ curvæ, quæ diversas admodum exhiberent virium leges; ac si quære-
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. 1, 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. I22. sº
205. As for this second case, it is very well known that, if in Fig. 21, A,C,E, are three
points, & the distance between two of them, A & B, is always bisected at D, & CD is joined,
& DE is taken equal to one third of DC, then, however these points move under the influence
of the forces compounded from the forces of any projection whatever & the mutual forces,
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
regard to the points A & B, that is easily effected in the following manner. Take, in Fig. 1,
abscissae measured along the axis equal to the straight lines AC & BC of Fig. 21 ; draw
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,
& to these the third point C is referred ; & if any straight line DEC is drawn of indefinite
length; then from any point of it a straight line can be erected perpendicular to it, &
equal to the diagonal of the parallelogram, for instance CF in the first case. From these
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
the point D, we should obtain distinct curves; & these would represent altogether different
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,
Theorem with re-
gard to the motion
of a point under
the action of two
other points.
Determination of
this force that is
compounded from
two forces.
The method of
constructing a
Curve which will in
general express a
force of this sort.
A more general
expression by means
of a surface.
or express in general all the forces pertaining to all points such as C, wherever they might
I 58 PHILOSOPHIAE NATURALIS THEORIA
Methodus determi-
nandi vim composi-
tam ex viribus re-
spicientibus puncta
quotcunque.
puncta C, ubicumque collocata ; oporteret erigere in omnibus punctis C rectas normales
plano 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 illa fuisset erecta supra, vel infra planum; & altera pariter vires perpen-
diculares. Ejusmodi locus geometricus, si algebraice tractari deberet, esset ex iis, quoS
Geometrae tractant tribus indeterminatis per unicam aequationem inter se connexis; ac
data aequatione ad illam primam curvam figurae I, posset utique inveniri tam aequatio ad
utramlibet curvam respondentem singulis rectis DC, constans binis tantum indeterminatis,
" aequatio determinans utramlibet superficiem simul indefinite per tres indetermin-
atas. (m)
[95] 209. Si pro duobus punctis tantummodo agentibus in tertium daretur numerus
quicunque punctorum positorum in datis locis, ac agentium in idem punctum, posset utique
constructione simili inveniri vis, qua singula agunt in ipsum collocatum in quovis assumpto
loci puncto, ac vis ex ejusmodi viribus composita definiretur tam directione, quam
magnitudine, per notam virium compositionem. Posset etiam analysis adhiberi ad expri-
mendas curvaS per aequationes duarum indeterminatarum pro rectis quibuscunque, & (9)
si omnia puncta jaceant in eodem plano, superficies per aequationem 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 diversa 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, qua dabuntur
data inclinatione DC, 85 punctis B, A, ac pariter dabuntur 5 DP, DQ. Dicatur praterea DC = x, 85 dabuntur analytice
CQ, CP. Quare ob angulos rectos P, Q, dabuntur etiam analytice CB, CA. Denominentur CK=u, CL =z, CF = y.
Quoniam datur AB, 5 dantur analytice AC, CB, dabitur analytice ex applicatione Algebra ad Trigonometriam
sinus anguli ACB per x, 5 datas quantitates, qui est idem, ac sinus anguli CKF complementi ad duos rectos. Datur
autem idem ex datis analytice valoribus CK = u, KF = CL = z, CF = y; quare habetur ibi una dequatio per x, y,
z, u, 5 constantes. Si praterea valor CB ponatur pro valore abscissae in aequatione curva figura. I ; acquiritur altera
aequatio per valores CK, CB, sive per x,u, 5 constantes. Eodem pacto invenietur ope dequationis curva figura. I tertia
aequatio per AC, & CL, adeoque per x, z, 85 constantes. Quare jam habebuntur dequationes tres per x,u,z,y, 65 con-
stantes, quae, eliminatis u, 63 z, reducentur ad unicam per x,y, 65 constantes, ac ea primam illam curvam definiet.
Quod si quaratur aquatio ad secundam curvam, cujus ordinata est CO, vel tertiam, cujus ordinata OF,
BP o o -
inveniri itidem poterit Nam datur analytice sinus anguli DCB = ci 85 in triangulo FCK datur analytice
sinus FCK =" X sin CKF. Quare datur analytice etiam sinus C -
differentiae OCF, adeoque (5 eius cosinus, t5 inde, ac ex CF datur
analytice OF, vel CO. Si igitur altera ex illis dicatur p, acquiri-
tur nova aequatio, cujus ope una cum superioribus eliminari
poterit praeterea una alia indeterminata ; adeoque eliminata
CF = y, habebitur unica dequatio per x,p, & constantes, quae
exhibebit utramlibet e reliquis curvis determinantibus legem
virium CO, vel OF.
Pro aequatione cum binis indeterminatis, quae exhibebit locum
ad superficiem, ducatur CR perpendicularis ad AB, 3 dicatur
DR = x, RC = q, denominatis, ut prius, CK = u, CL = z,
CF = v; 5 quoniam dantur AD, DB; dabuntur analytice per x, A D - R E
85 constantes AR, RB, adeoque per x, q, t5 constantes AC, CB, 65 - -
factis omnibus reliquis, ut prius, habebuntur quatuor dequationes
per x,q,u,z,y,p, 3 constantes, quae eliminati valoribus u,z,y, G)
reducentur ad unicam datam per constantes, 85 tres indeterminatas
ac,p,q, sive DR, RC, 5 CO, vel OF, qua exhibebit quaesitum
locum ad superficiem. -
Calculus quidem esset immensus, sed patet methodus, qua deveniri possit ad aquationem quaesitam. Mirum autem,
quanta curvarum, 5 superficierum, adeoque 5 legum virium varietas obvenerit, mutata tantummodo distantia AB binorum
punctorum agentium in tertium, qua mutata, mutatur tota lex, S aquatio. - - -
(o) Haec conditio punctorum jacentium in eodem plano necessaria fuit pro loco ad superficiem, 85 pro aequatione, qua
legem virium exhibeat per aquationem indeterminatarum tantummodo trium : at si puncta sint plura, S in eodem plano
non jaceant, quod punctis tantummodo tribus accidere omnino non potest; tum vero locus ad superficiem, 85 aequatio trium
indeterminatarum non sufficit, sed ad eam generaliter exprimendam legem Geometria omnis est incapax, S analysis indiget
aequatione indeterminatarum quatuor. Primum patet ex eo, quod si manentibus punctis A, B, exeat punctum C ex dato
quodam plano, pro quo constructus sit locus ad superficiem ; liceret convertere circa rectam AB planum illud cum superficie
curva legem virium determinante, donec ad punctum C deveniret planum ipsum : tum 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,
FIG. 22.

A THEORY OF NATURAL PHILOSOPHY I59
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.() e
209. If instead of only two points acting upon a third we are given any number of
points situated in given positions, & acting on the same point, it would be possible, by a
similar construction in each case, to find the force, with which each acts on the point
situated in any chosen position; & the force compounded from forces of this kind would
be determined, both in position & magnitude, by the well-known method for composition
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,E,F,L,O be in the same positions as in Fig. 21, 85 let BP, AQ be drawn
perpendicular to CD ; then these will be known, if the inclination of CD & the positions of A & B are known: &
so also will DP & DQ be known. Further, suppose DC = x, then CQ & CP will be given analytically. Hence on
account of the right angles at P & Q, CB & CA will also be given analytically. Suppose CK = u, CL = z, CF = y.
Since AB is known, & 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 w & known quantities ; & 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
CK = u, KF = CL = 2, CF = y. Hence there is obtained in this case an equation involving w,y,z,w, & constants.
The method of
determining the
force compounded
from the forces due
to any number of
points. The great
number & variety
of laws.
If, in addition, the value CB is substituted for the value of the abscissa in the equation of the curve in Fig. 1, another
equation will be obtained in terms of the values of CK, CB, i.e. in terms of x, u, & 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 & CL, i.e., in terms of
w,z, & constants. Now, ºnce there will be thus obtained three equations in terms of x,y,z,w, & constants, these, on
eliminating u,z, will reduce to a single equation involving w.y, & constants ; & this will be the equation defining
the first curve. - -
Again, if 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 fine of the angle DCB is analytically
given, being equal to BP/CB; 3 from the triangle FCK, the fine of the angle FCK is given, being equal to
sin CKF.(FK/CF). Therefore the sine of the difference OCF is also given analytically, 85 therefore also its cosine; &
from this & 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 & 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 w.p, &
constants, which will be that of one or other of the remaining curves determining the law of forces for CO or ÓF.
For an equation in three unknowns, which will repreſent the surface, draw CR perpendicular to AB, & let DR=w
tº RC = q = {3, as before, let CK = u, CL = z, CF = y. Then, since AD, DB are given, AR & RB are also given
analytically in terms of x tº constants : & therefore AC & CB are given in terms of w.q, & constants : & # all
the rest of the work is done as before, four equations will be obtained in terms of 3,4,11,2,3,p, $5 constants. These, on
eliminating the values u,2,3, will reduce to a single equation in terms of constants & the three unknowns w;p,q, or DR,
RC, tº 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 & surfaces, & therefore of laws of force, would be
met with, if merely the distance between A & B, the two points which act upon the third, is changed; for if this
alone is changed, the whole law is altered & 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,
& for the equation, which will express the law of the forces by an equation involving only three unknowns. If the points
4re numerous, tº they do not all lie in the same plane (which is quite impossible in the case of only three point), then
indeed a surface locus, & an equation in three unknowns, will not be sufficient; indeed, to expreſs the law generally,
the whole of geometry is powerless, & analysis requires an equation in four unknowns. The first point is clear from
the fact that if, whilst the points A & 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 them, 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,
I6o - PHILOSOPHIÆ NATURALIS THEORIA
Vis in latus in
exiguis distantiis,
ac ejus usus pro
solidis : in magnis
nulla : in iis summa
virium simplicium.
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
Theoriæ fœcunditatem ostenderet.
2IO. Ego hic simpliciora quædam, ac faciliora, & usum habitura in sequentibus, ac in
applicatione ad Physicam inprimis attingam tantummodo ; sed interea quod ad generalem
pertinet determinationem expositam, duo adnotanda proponam. Primo quidem in ipsa
trium punctorum combinatione occurrit jam hic nobis præter vim determinantem ad
accessum, & recessum, vis urgens in latus, ut in fig. 2 I, præter vim CF, vel CH, vis CI,
vel CG. Id erit infra magno usui ad explicanda solidorum phænomena, in quibus,
inclinato fundo virgæ solidæ, tota virga, & ejus vertex moventur in latus, ut certam ad
basim positionem acquirant. Deinde vero illud : hæc omnia curvarum, & legum discrimina
tam quæ [97] pertinent ad diversas directiones rectarum DC, data distantia punctorum
, B, quam quæ 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 figuræ 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
figuræ 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,
& quæcunque fuerit inclinatio CD ad AB, erit semper ad sensum eadem, directa ad sensum
ad punctum D, ad sensum proportionalis reciproce quadrato distantiæ DC ab ipso puncto
D, & ad sensum dupla ejus, quam in curva figuræ I requireret distantia DC.
• .
At secundum sit manifestum ex eo, quod si puncta agentia sint etiam omnia in eodem plano, & punctum, cujus vis
composita quaeritur, in quavis recta posita extra ipsum planum, relationes omnes distantiarum a reliquis punctis, ac
directionum, a quibus pendent vires singulorum, & compositio ipsarum virium, longe aliæ e$$ent, ac in quavis recta in eodem
plano posita, uti facile videre est. Hinc pro quovis puncto loci`ubicumque assumpto sua responderet vis composita, & quarta
aliqua plaga, seu dimensio, præter longum, latum, & profundum, requireretur ad ducendas ex omnibus punctis spatii rectas
iis viribus proportionales, quarum rectarum vertices locum continuum aliquem exhiberent determinantem virium legem.
Sed quod Geometria non assequitur, assequeretur quarta alia dimensio mente concepta, ut si conciperetur $patium totum
plenum materia continua, quod in mea sententia cogitatione tantummodo effingi potest, §9 ea eget in omnibus spatii punctis
Jemsitatis diversae, vel diversi pretii ; tum illa diversa densitas, vel illud pretium, vel quidpiam ejusmodi, exhibere posset
legem virium ipsi respondentium, quae nimirum ipsi essent proportionales. Sed ibi iterum ad determinandam directionem
vis compositae non esset satis resolutio in duas vires, alteram secundum rectam transeuntem per datum punctum ; alteram
ipsi perpendicularem ; ed requirerentur tres, nimirum vel omnes secundum tres datas directiones, vel tendentes per rectas,
quae per data tria puncta transeant, vel quavis alia certa lege definitas : adeoque tria loca ejusmodi ad spatium, quarta
âliqua dimensione, vel qualitate affectum requirerentur, quæ tribus ejusmodi plusquam Geometricis legibus vis compositae
legem definirent, tum quod pertinet ad ejus magnitudinem, tum quod ad directionem.
Verum quod mom assequitur Geometria, assequeretur Analysis ope æquationis quatuor indeterminatarum : si enim
conciperetur planum, quod libuerit, ut ACB, & in eo quævis recta AB, ac in ipsa recta quodvis punctum D ; tum quovis
hujus segmento DR appellato x, quavis recta RC ipsi perpendiculari y, quavis tertia perpendiculari ad totum planum z,
per hasce tres indeterminatas involveretur positio puncti spatii cujuscumque, in quo collocatum esset punctum materiae,
cujus vis quaeritur. • • • •
* Pumciorum agentium utcumque collocatorum ubicumque vel intra id planum, vel extra, possent definiri positiones per
ejusmodi tres rectas, datas utique pro singulis, si eorum positiones dentur. Per ea$, & per illas x,y,z, po$$et tutique haberi
distantia cujuscumque ex iis punctis agentibus, & positione datis, a puncto indefinite accepto ; adeoque ope æquationis
figuræ 1 posset haberi analytice per æquationes quasdam, ut supra, vis ad singula agentia puncta pertinens, & per easdem
Âcta, ejüs etiam directio resoluta in tres parallelas illis x,y,z. Hinc haberetur analytice omnium summa pro singulis
ejusmodi directionibus per aliam aequationem derivatam ab ejus summæ denominatione, ea nimirum fa£ta = u, ac expunctis
6mnibus subsidiariis vàloribus, methodo non absimili ei, quam adhibuimus superius pro loco ad superficiem, deveniretur ad
unam aequationem constitutam illis quatuor indeterminatis x,y,z, u, £ constantibus ; ac tres ejusmodi aequationes pro tribus
directionibus vim omnem compositam definirent. Sed hæc innuisse sit $atis, quæ nimirum & altiora sunt, & ob ingentem
complicationem casuum, ac nostræ humanæ mentis imbecillitatem nulli nobis inferius futura sunt usui.
A THEORY OF NATURAL PHILOSOPHY I61
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
cases, such as will be of use in what follows, & later when considering the application to
Physics. But meanwhile, I will enunciate two theorems, applying to the general deter-
mination set forth above, which should be noted. Firstly, in the case of the combination
of three points, we have here already met with, in addition to a force inducing approach
& recession, i.e., in Fig. 21, in addition to a force CF or CH, a force CI or CG, urging
the point C to one side. This will be of great service to us in explaining certain phenomena
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
The lateral force at
very small dis-
tances, & its use in
the consideration
of solids ; the
a bsence of this
for ce at great
distances, the sum
of the simple forces
in the latter case.
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, & the point for
which the resultant force is required, lies in any straight line situated without that plane, even them 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 ; 85 a fourth region, or dimension, in addition to length, breadth, 85 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 continuous matter, which in my opinion can only be a mental
fiction) ; 85 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, 85 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 ; “5 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, 85 in it any straight line AB, 85 in this
straight line any point D ; then, calling any segment of it w, any straight line perpendicular to it y, 85 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 85 wherever they may be situated, either within the plane or without
it, could be defined by three straight lines of this sort ; 85 these would in all cases be known for each point, if the positions
of the points are given. By means of these, {5 the former straight lines denoted by x,y,z, there could be 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, 85 2. 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) ; 85, 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, & 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, {5, on account of the enormous
complexity of cases, & the disability of the human intelligence, will not be of any use to us later.
M
I62 PHILOSOPHIÆ NATURALIS THEORIA
Demonstratio post-
remi theorematis.
Discrimen in gens
virium, quas mas-
s u 1 a exercet in
massulam proxi-
mam, conformitas
Summa in remo-
tarum viribus, quæ
S u n t directe u t
massæ, & reciproce,
ut quadrata dis-
tantiarum.
Unde necessaria
Omnium corporum
uniformita s in
gravitate, diffor-
mitas in aliis in-
numeris proprieta-
tibus.
2 I I. Id quidem facile demonstratur. Si enim AB respectu DC sit perquam exigua,
angulus ACB erit perquam exiguus, & a recta CD ad sensum bifariam sectus : distantiæ AC,
CB erunt ad se invicem ad sensum in ratione æqualitatis, adeoque & vires CL, CK ambæ
attractivæ debebunt ad sensum æquales esse inter se, & proinde LCKF ad sensum rhombus,
diametro CF ad sensum secante angulum LCK bifariam, quæ rhombi proprietas est, &
ipsa CF congruente cum CO, ac (ob angulum FCK insensibilem, & CKF ad sensum
æqualem duobus rectis) æquali ad sensum binis CK, KF, sive CK, CL, simul sumptis ;
quæ singulæ cum sint quam proxime in ratione reciproca duplicata distantiarum CB,
BA ; erunt & eadem, & earum summa ad sensum in ratione reciproca duplicata distantiæ
CD.
2I2. Porro id quidem commune est etiam massulis constantibus quocunque punctorum
numero. Mutata illarum combinatione, vis composita a viribus singulorum agens in
punctum distans a massula ipsa per intervallum perquam exiguum, nimirum ejusmodi,
in quo curva figuræ I circa axem contorquetur, debet mutare plurimum tam intensitatem
suam, quam directionem, & fieri utique potest, quod infra etiam in aliquo simpliciore
casu trium punctorum videbimus, ut in alia combinatione punctorum massulæ pro eadem
distantia a medio repulsiones prævaleant, in alia attractiones, in alia oriatur vis in latus ad
perpendiculum, ac in eadem constitutione massulæ pro diversis directionibus admodum
diversæ sint vires pro eadem etiam distantia a medio. At in magnis illis distantiis, in
quibus singulorum punctorum vires jam attractivæ sunt omnes, & directiones, ob molem
massulæ tam exiguam respectu ingentis distantiæ, 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 rectæ tendentis ad mediam massulam, &
æquabitur vis ipsa ad sensum summæ virium omnium punctorum constituentium ipsam
massulam, adeoque erit attractiva semper, & ad sensum proportionalis in diversis etiam
massulis numero punctorum directe, & quadrato distantiæ a medio massulæ 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 illa solicitetur, sed massula alia, cujus vis componatur e singulis viribus
singulorum suorum punctorum, quod tamen in massula etiam respectu massulæ admodum
remotæ evanescet, singulis ejus punctis vires habentibus ad sensum æquales, & agentes
in eadem ad sensum directione ; unde fiet, ut vis motrix ejus massulæ solicitatæ, orta ab
actionibus illius alterius remotæ massulæ, sit ad sensum proportionalis numero punctorum,
quæ habet ipsa, numero eorum, quæ habet altera, & quadrato distantiæ, quæcunque sit
diversa dispositio punctorum in utralibet, quicunque numerus.
2 I 3. Mirum sane, quantum in applicatione ad Physicam hæc animadversio habitura
sit usum ; nam inde constabit, cur omnia corporum genera gravitatem acceleratricem
habeant proportionalem massæ, in quam tendunt, & quadrato distantiæ, adeoque in
superficie Terræ aurum, & pluma cum æquali celeritate descendant seclusa resistentia, vim
autem totam, quam etiam pondus appellamus, proportionalem præterea massæ suæ, adeoque
in ordine ad gravitatem nullum sit discrimen, quæcunque differentia habeatur inter corpora,
quæ gravitant, & in quæ gravitant, sed ad solam demum massam, & distantiam res omnis
deveniat ; at in iis proprietatibus, quæ 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 aliæ sensibiles ejusmodi qualitates pro tactu, tot cohæsionum tam diversa genera,
secretiones, nutritionesque, fermentationes, conflagrationes, displosiones, dissolutiones,
præcipitationes, ac alii effectus Chemici omnes, & mille alia ejusmodi, quæ diversa corpora
a se invicem discernunt, in iis, inquam, tantum sit discrimen, & vires tam variæ, ac tam
A THEORY OF NATURAL PHILOSOPHY I63
211. The latter theorem can be easily demonstrated. For, if AB is very small compared
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,
whatever their number may be. The force compounded from the several forces acting
on a point, whose distance from the mass is very small, i.e., such a distance as that for which,
in Fig. 1, the curve is twisted about the axis, must be altered very greatly if the combination
of the points is altered; & this is so, both as regards its intensity, & as regards its direction.
It may even happen, as will be seen later in the more simple case of three points, that in
one combination of the points forming the little mass, & for one & the same distance from
the mean point, repulsions will preponderate, in another case attractions, & in another
case there will arise a perpendicular lateral force. Also for the same constitution of the
mass, for the same distance from the mean point, there may be altogether different forces
for different directions.
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
Iſla SS.
213. It is indeed wonderful what great use can be made of this consideration in the
application of my Theory to Physics; for, from it it will be clear why all classes of bodies
have an accelerating gravity, proportional to the mass on which they act, & to the square
of the distance [inversely]; & hence that, on the surface of the Earth, a piece of gold & a
feather will descend with equal velocity, when the resistance of the air is eliminated. It
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,
But, for considerable distances, where the forces due to the several
Proof of the latter
theorem.
There is a huge
difference in the
forces which a small
maSS exerts On a
Small mass that
is very near to it;
but the greatest
possible uniformity
in the forces due
to remote masses;
these vary directly
as the masses, &
inversely as the
Square of the dis-
tances.
Hence we h a ve
necessarily, for all
bodies, uniformity
in the case of
gravity, & non-
uniformity in the
case of numerous
other properties.
I64 PHILOSOPHIAE NATURALIS THEORIA
Vis in duo puncta
puncti p o siti in
re ct a j u n gente
ipsa, vel in recta
secante hanc bi-
fariam, 8 ad angu-
lo s rectos directa
secundum eandem
recta.m.
Constructio curvae
exhibentis legem
casus posterioris,
Constructionis de-
monstratio.
Plures ejus curvae
proprietates,
varii motus, qui tam varia phaenomena, & omnes specificas tot corporum differentias
inducunt, consensu Theoriae hujus cum omni Natura sane admirabili. Sed hac, quae
huc usque dicta sunt ad massas pertinent, & ad amplicationem ad Physicam : interea
peculiaria quaedam persequar ex innumerisiis, quae per-I99]-tinent ad diversas leges binorum
punctorum agentium in tértium. - -
214. Si libeat considerare illas leges, quae oriuntur in recta perpendiculari ad AB
ducta per D, velin ipsa AB hinc, & inde producta, in primis facile est videre illud, direc-
tionem vis compositae utrobique fore eandem cum ipsa recta sine ulla vi in latus, & sine ulla
declinatione a recta, quae tendit ad ipsum D, vel ab ipso. Pro recta AB res constat per
sese; nam vires illae, quae ad bina ea puncta pertinent, vel habebunt directionem eandem,
vel oppositas, jacente ipso tertio puncto in directum cum utroque e prioribus: unde
fit, ut vis composita aequetur summae, vel differentiae 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, aequales erunt, & proinde
vel ambae attractivae, ut CL, CK, vel ambae repulsiva, 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 aequalitatem triangulorum DCA, DCB,
patet, ipsas CF, CH debere cum eadem congruere. Quamobrem in hisce casibus evane-
scitº vis illa, perpendicularis FO, quae in praecedentibus binis figuris habebatur, ac in iis
per unicam aequationem res omnis absolvitur (?), quarum ea, qua ad posteriorem casum
pertinet, admodum facile invenitur.
215. Legem pro recta perpendiculari rectae jungenti duo puncta, & aeque distanti ab
utroque exhibet fig. 24, quae vitandae confusionis causa exhibetur, ubi sub numero 24
habetur littera B, sed quod ad eius 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, qua vires compositas ibi exhibet per ordinatas, constructa est ex
fig. I, quod fieri potest, inveniendo vires singulas singulorum punctorum, tum vim com-
positam ex iis more consueto juxta [Iool generalem constructionem numeri 205; sed
etiam sic facilius idem praestatur; centro Y intervallo cuiusvis abscissae Ad figurae I in-
veniatur in figura 24 sub littera A in recta CC punctum d. sumaturque de versus Y
aequalis ordinata dh figurae 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 illa 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.
216. Demonstratio facilis est: si enim ducatur d'X, & 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
illa dh figurae I fuerit itidem attractiva, vel repulsiva.
217. Porro ex ipsa constructione patet, si centro Y, intervallis AE, AG, AI figurae I
inveniantur in recta CAC hujus figurae positae sub littera B puncta E, G, I, &c, ea fore
limites respectu novae curvae; & eodem pacto reperiri posse limites E', G", I', &c. ex parte
opposita A; in iis enim punctis evanescente de figura eiusdem 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, 85 ad angulos rectos ex rhombi natura.
Dicatur CD = x, CF = y, DB = a, t3 erit CB = Vaa-F xx, 8 CD = x.CB = Vaa + xx:: CI = 33.CK =i
v/aa F xx, quo valore posito in aequatione curva figura 1 pro valore ordinata, 9 Vaa F xx pro valore abscissa, habebitur
immediate aquatio nova per x, y, 3 constantes, quae eiusmodi curvam determinabit,
A THEORY OF NATURAL PHILOSOPHY
I65
FIG. 24.B.
[Not drawn to Scale.]

I66
PHILOSOPHIAE NATURALIS THEORIA
['9'[eos Oļ ūAAæıp qoN]
ºgłº z ºbIſI

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. -
2I4. 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 fººt .." it.
with the line itself without any lateral force or any declination from the straight line which ſº ºn
is drawn towards or away from D. In the case of AB itself the matter is self-evident; *::::. tº:
for the forces which pertain to the two points either have the same direction as one another, right 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 ... ." ...sº
confusion the curve itself is given in Fig. 24B, 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 do 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 8°ing construction.
line dA bisecting the angle Xd 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 ºf curve of this
Fig. 24B 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. 24B, there is a change from the attractive
(p) For, if in Fig. 23, LK is drawn, it will cut FC somewhere, in I say : {5 it will be at right angles to it
on account of the nature of a rhombus. Suppose CD = x, CF = y, DB = a then CB = A/(a”-- w”), & we have
CD (or x.): CB (or V(a” + x*) = CI (or #y) : CK, ... CK = y. V(a” + x*)/2x ;
85 if this value is substituted in the equation of the curve in Fig. I instead of the ordinate, 85 A/(a” + x*) for the
abscissa, we shall get straightaway a new equation in x, y, & constants ; 85 this will determine a curve of the kind
under consideration.
I68 PHILOSOPHIAE NATURALIS THEORIA
Constructio curvæ
exhibentis legem
casus prioris.
Ejus curvæ pro-
prietates : discri-
mina p r o mutata
distantia, puncto-
rum : collatio cum
curva casus alterius.
Tria genera hujus
casus notatu dig-
nissima.
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 oppositæ vires se destruent, adeoque
ibi debere curvam axem secare ; ac licet distantiæ AX, AY fuerint perquam exiguæ, ut
idcirco repulsiones singulorum punctorum evadant maximæ ; tamen prope A vires erunt
perquam exiguæ ob inclinationes duarum virium ad XY ingentes, & contrarias ; & si ipsæ
AY, AX fuerint non majores, quam sit AE figuræ 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 abscissæ
paullo majores, quam sit hæc YA. Postrema crura TpV,T'p'V', patet, fore attractiva ;
& si in figura I fuerint asymptotica, fore asymptotica etiam hic ; sed in A nullum erit
asymptoticum crus.
218. At curva quæ exhibet in fig. 25 legem virium pro recta CC' transeunte per duo
puncta X, Y, est admodum diversa a priore. Ea facile construitur : satis est pro quovis
ejus puncto d assumere in fig. I duas abscissæ æquales, alteram Yd hujus figuræ, alteram
Xd ejusdem, & sumere hic dh æqualem [IoI] summæ, vel differentiæ binarum ordinatarum
pertinentium ad eas abscissas, prout fuerint ejusdem directionis, vel contrariæ, & eam
ducere ex parte attractiva, vel repulsiva, prout ambæ ordinatæ figuræ 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 mutatæ respectu CC', ut in fig. superiore diximus, ad
partes oppositas A debet esse aliud g'd', habens asymptotum c'b' transeuntem per X;
ac utrumque crus debet continuari usque ad A, ubi curva secabit axem. Hoc postremum
patet ex eo, quod vires oppositæ 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, quæ
provenit ab X, manente finita ; adeoque tam summa, quam differentia debet esse vis
repulsiva respectu Y, & proinde attractiva respectu A, quæ 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'.
2I9. Poterit autem etiam arcus curvæ interceptus asymptotis bYc, b'Xc' sive cruribus
dg, d'g' secare alicubi axem, ut exhibet figura 26 ; quin immo & in locis pluribus, si nimirum
AY sit satis major, quam AE figuræ I, ut ab Y habeatur alicubi citra A attractio, & ab X
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 æqualis AE figuræ I, in fig. 26 æqualis AI, ejusdem quæ variatio
usque adeo mutavit figuræ genitæ ductum ; & assumptis aliis, atque aliis distantiis punc-
torum X, Y, aliæ, atque aliæ curvæ novæ provenirent, quæ inter se collatæ, & cum illis,
quæ habentur in recta CAC" perpendiculari ad XAY, uti est in fig. 24 ; ac multo magis
cum iis, quæ 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
attractivo TpV, ubique conjuncta cum tanto discrimine in arcu se circa axem contorquente.
220. Verum ex tanto discriminum numero unum seligam maxime notatu dignum,
& maximo nobis usui futurum inferius. Sit in fig. 27C—'AC axis idem, ac in fig. I, & quin-
que arcus consequenter accepti alicubi GHI, IKL, LMN, NOP, PQR sint æquales
prorsus inter se, ac similes. Ponantur autem bina puncta B', B hinc, & inde ab A in fig. 28
[Io2] ad intervallum æquale dimidiæ amplitudini unius e quinque iis arcubus, uti uni
GI, vel IL ; in fig. 29 ad intervallum æquale integræ ipsi amplitudini ; in fig. 3o ad
intervallum æquale duplæ ; sint autem puncta L, N in omnibus hisce figuris eadem, &
quæratur, quæ futura sit vis in quovis puncto g in intervallo LN in hisce tribus posi-
tionibus punctorum B', B.
A THEORY OF NATURAL PHILOSOPHY 169
C. Qix - St.
Ul, & –º S.
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S. 5 - -“S. Sº S.
Tri .

170
THEORIA
PHILOSOPHIAE NATURALIS
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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 clearly 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
CC’, when it passes through the points X,Y, is quite different from the one just considered.
It is easily constructed; it is sufficient, for any point d upon it, to take, in Fig. I, two
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 abscissae, according
as they are in the same direction or in opposite directions; &, according as each ordinate,
Construction for
the curve express-
ing the law for the
first case.
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 asymptote 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 the axis. 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 by c, increases indefinitely :
& it thus follows that the arc g4 will be asymptotic with respect to Ye ; & 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 by c,
b'Xc', i.e., between the branches dg, d'g', to cut the axis somewhere, as is shown in Fig. 26;
nay rather, it may cut it in more places than one, for instance, if AY is sufficiently greater
than AE in Fig. I ; so that, at some place on the near side of A, there is obtained an attraction
from the point Y & a repulsion from the point X, or a repulsion from X greater than the
repulsion from Y. Besides, by a mere inspection of the last two figures, it will be evident
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
notice in a high degree, which also will be of the greatest service to us later. In Fig. 27,
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
a distance equal to the whole of this width ; &, in Fig. 30, at a distance equal to double
the width ; also let the points L,IN 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'.
The properties of
this curve ; differ-
ences corresponding
to changed dis-
tances between the
points; compari-
Son with the curve
obtained in the
other case.
Three classes of
this case that are
well worth remark.
PHILOSOPHIAE NATURALIS THEORIA
IDeterminatio v is
Compositæ in iis-
dem.
In alia dispositione
vim 1n tractu con-
tinuo nullam, in
alia attractionem,
in alia repulsionem,
manente distantia ;
usus in Physica
SüIn ImUlS.
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-
á.IllII1.
22 I. Si in Fig. 27 capiantur hinc, & inde ab ipso g intervalla æqualia intervallis AB',
AB reliquarum trium figurarum ita, ut ge, gi respondeant figuræ 28 ; gc, gm figuræ 29;
ga, go figuræ 3o ; patet, intervallum ei fore æquale amplitudini LN, adeoque Le, Ni
æquales fore dempto communi Li, sed puncta e, i debere cadere sub arcus proximos
directionum contrariarum ; ob arcuum vero æqualitatem fore æqualem vim ef vi contrariæ
il, adeoque in fig. 28 vim ab utraque compositam, respondentem puncto g, fore nullam.
At quoniam gc, gm integræ amplitudini æquantur ; cadent puncta c, m sub arcus IKL,
NOP, conformes etiam directione inter se, sed directionis contrariæ respectu arcus LMN,
eruntque æquales mN, cI ipsi gL, adeoque attractiones mm, cd, & repulsioni gh æquales,
& inter se ; ac idcirco in figura 29 habebitur vis attractiva gh composita ex iis binis dupla
repulsivæ figuræ 27. Demum cum ga, go sint æquales duplæ amplitudini, cadent puncta
a, o sub arcus GHI, PQR conformis directionis inter se, & cum arcu LMN, eruntque
pariter binæ repulsiones ab, op æquales repulsioni gh, & inter se. Quare vis ex iis com-
positae pro fig. 3o erit repulsio gh dupla repulsionis gh figuræ 27, & æqualis attractioni
figuræ 29.
222. Inde igitur jam patet, loci geometrici exprimentis vim compositam, qua bina
puncta B', B agunt in tertium, partem, quæ respondet intervallo eidem LN, fore in prima
e tribus eorum positionibus propositis ipsum axem LN, in secunda arcum attractivum
LMN, in tertia repulsivum, utroque recedente ab axe ubique duplo plus, quam in fig.
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 æqualem 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 centro particulæ constantis iis duobus punctis sit omnino eadem. Possunt autem
in omnibus hisce casibus puncta B', B esse simul in arctissimis limitibus cohæsionis 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 curvæ derivatæ utique continuæ, deductæ nimirum certa lege a curva
continua, possit congruere accurate cum recta ; at poterunt ea omnia ad æqualitatem
accedere, quantum [Io3] libuerit ; poterunt hæc ipsa discrimina haberi ad sensum per
tractus continuos aliis modis multo adhuc pluribus, immo etiam pluribus in immensum,
ubi non duo tantummodo puncta, sed immensus eorum numerus constituat massulas,
quæ 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 quædam substantiæ
inter se commixtæ ingentem acquirant intestinarum partium motum per effervescentiam,
& fermentationem, quæ 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 præterfluente distantiam alia aliis partibus
solidis adhæreant, & per alias valvulas transmittantur, aliis libere progredientibus. Sed
adhuc multa supersunt notatu dignissima, quæ pertinent ad ipsum etiam adeo simplex
trium punctorum systema.
223. Jaceant in figura 3I tria puncta A,D, B, in directum : ea poterunt respective
quiescere, si omnibus mutuis viribus careant, quod fieret, si tres distantiæ AD, DB, AB
omnes essent distantiæ 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 I73
C’ & | \| 6 g i \|In O R C
K O
FIG. 27.
C’ I - | -*— |- + |N C
B^ A £ g
C’
TE7TATE
FIG. 29.
C’ }
B’ A
FIG. 30.

I74 PHILOSOPHIAE NATURALIS THEORIA
|-|- N
g
C'. | ! g
B’ A B L N
h
M\
FIG .29.
FIG. 30.





A THEORY OF NATURAL PHILOSOPHY I75
221. If, in Fig. 27, we take, on either side of this point g, intervals that are equal to
the intervals AB", AB of the other three figures; so that ge, gi correspond to Fig. 28;
ge, 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 Li, we have Le & 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 go, 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, mM, cI will be equal to gL ; &
thus the attractions mn, cd will be equal to the repulsion gh, & 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 & 0 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 gh, & 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
geometrical locus representing the resultant force, with which two points B", B act
upon a third, corresponding to the same interval LN, will be the axis LN itself in the first
of the three stated positions of the points; in the second position it will be an attractive
arc LMN, & in the third a repulsive arc; each of these will recede from the axis at all
points along it to twice the corresponding distance shown in Fig. 27. So, for any position
of the point g in the whole interval LN, the force will be nothing at all in the first of the
three cases, an attraction in the second, & a repulsion in the third. This latter will be
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
Determination of
the resultant force
in these three cases.
In One arrange-
ment, there is, in a
continuous region,
no force at all, in
another there is an
attraction & in a
third a repulsion
the distance re-
maining constant;
this result is of the
highest utility in
Physics.
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
with regard to one another if they lack all mutual forces; & this would be the case, if the
three distances AD, DB, AB were all distances corresponding to limit-points. In addition,
relative rest could be obtained owing to elimination of equal & opposite forces. Further,
there will be three different cases with regard to the mutual forces. For, either the middle
point D is attracted by each of the outside points A & B, or it is repelled by each of them,
or it is attracted by one of them & repelled by the other. In the last case, it is evident
that relative rest could not obtain ; for the middle point must then be moved towards the
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
Another instance
of no force in the
case of three points
situated in a
Straight line at the
distances
Sponding to limit-
p O in t s. Three
others, in two of
which the absence
of a resultant force
arises from an eli-
mination of equal
& opposite forces.
C O Iſ I e— ,
I76 - PHILOSOPHIÆ NATURALIS THEORIA
In eorum altero
nisus ad recuper-
andam positionem,
in altero ad magis
ab ea recedendum,
si incipiant inde
removeri.
cum repulsione BS component vires AQ, BT,
Theoria generalior
indicata : t r iu m
punctorum jacen-
tium in directum :
vis maxima ad
cOnservandam dis-
tantiam.
Quid ubi vis exter-
na urgeat in latus:
idea virgæ rigidæ,
& virgæ flexilis.
res haberi: nam vires attractivæ, vel repulsivæ, quas habet medium punctum, possunt
esse æquales ; tum autem extrema puncta debebunt itidem attrahi a medio in primo casu,
repelli in secundo ; quæ si se invicém e contrario æque repellant in casu primo, attrahant
in secundo ; poterunt mutuæ vires elidi omnes.
224. Adhuc autem ingens est discrimen inter hosce binos casus. Si nimirum puncta
illa a directione rectæ lineæ 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 binæ repulsiones CM, CN adhuc
habebuntur in secundo casu in ipso primo
recessu a D (licet eæ mutatis jam satis distan-
tiis BD, AD in BC, AC, evadere possint at-
tractiones) & vim com-[Io4}-ponent direc-
tam per CH contrariam directioni tendenti ,
ad rectam AB. At in primo casu habebuntur
attractiones CL, CK, quæ component vim Q.
CF directam versus AB, quo casu attractio
AP cum repulsione AR, et attractio BV,
quibus puncta A, B ibunt obviam puncto C |
redeunti ad rectam transituram per illud R A
punctum E, quod est in triente rectæ DC, -
& de quo supra mentionem fecimus num. 205.
225. Hæc Theoria generaliter etiam non rectilineæ tantum, sed & cuivis positioni
trium massarum applicari potest, ac applicabitur infra, ubi etiam generale simplicissimum,
ac fœcundissimum theorema eruetur pro comparatione virium inter se : sed hic interea
evolvemus nonnulla, quæ 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, quæ illa dimovere tentet,
vel utcunque magna velocitas impressa fuerit ad ea e suo respectivo statu deturbanda.
Nam vires ejusmodi esse possunt, ut tam in eadem directione ipsius rectæ, quam in
directione ad eam perpendiculari, adeoque in quavis obliqua etiam, quæ in eas duas resolvi
cogitatione potest, validissimus exurgat conatus ad redeundum ad priorem locum, ubi inde
discesserint puncta. Contra vim impressam in directione ejusdem rectæ satis est, si pro
puncto medio attractio plurimum crescat, aucta distantia ab utrolibet extremo, & plurimum
decrescat eadem imminuta ; ac pro utrovis puncto extremo satis est, si repulsio decrescat
plurimum aucta distantia ab extremo, & attractio plurimum crescat, aucta distantia a
medio, quod secundum utique fiet, cum, ut dictum est, debeat attractio medii in ipsum
crescere, aucta distantia. Si hæc ita se habuerint, ac vice versa ; differentia virium vi
FIG. 3I.
• ¢ � te … ¢ *® • • . $.3;
extrinsecæ 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.
226. Quod si vis urgeat perpendiculariter, ut ex.gr. punctum medium D moveatur
per rectam DC perpendicularem ad AB ; tum vires CK, CL possunt utique esse ita validæ,
ut vis composita CF sit post recessum, quantum libuerit, exiguum satis magna ad ejusmodi
vim elidendam, vel ad extinguendam velocitatem impressam. In casu vis, quæ constanter
urgeat, & punctum D versus C, & puncta A, B ad partes oppositas, habebitur inflexio ; ac
in casu vis, quæ agat in eadem directione rectæ jungentis puncta, habebitur contractio,
seu distractio ; sed vires resistentes ipsis poterunt esse ita validæ, ut & inflexio, & contractio,
vel distractio, sint prorsus insensibiles ; [Io5] ac si actione externa velocitas imprimatur
punctis ejusmodi, quæ flexionem, vel contractionem, aut distractionem inducat, tum
ipsa puncta permittantur sibi libera ; habebitur Oscillatio quædam, angulo jam in alteram
plagam obverso, jam in alteram oppositam, ac longitudine ejus veluti virgæ constantis iis
tribus punctis jam aucta, jam imminuta, fieri poterit ; ut oscillatio ipsa sensum omnem
effugiat, quod quidem exhibebit nobis ideam virgæ, 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
instance, if the points are moved a small distance out of the direct straight line, so that
the middle point D, say, is now slightly off the straight line AB, being transferred to C,
then, if left to itself, it will recede still further from it in the first case, & will approach
it once more in the second case. Or, if it is acted on by some external force, it will endeavour
to recover its position & will resist the force acting on it. For two repulsions, CM, CN,
will at first be obtained in the second case, at the first instant of motion from the 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
of the three points in a straight line but also any position whatever. This application
will be made in what follows, where also a general theorem, of a most simple & fertile nature
will be deduced for comparison of forces with one another. But for the present we will
consider certain points that have to do with this more simple case of three points. First
of all, it may come about that three points of this kind may maintain a position practically
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
along the line DC perpendicular to AB, then the forces CK, CL, can in any case be so
strong that the resultant force CF may become, after a recession of any desired degree
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
In one of these
cases there is an
endeavour towards
a recovery of posi-
tion, & in the other
towards a further
recession from it,
if they are initially
moved out of that
position.
Enunciation of a
more general theory
for three points
lying in a straight
line ; possibility of
a very great force
tending to conser-
vation of distance.
What happens if
the external force
does not act along
the straight line ;
idea of a rigid, &
of a non-rigid rod.
178 PHILOSOPHIÆ NATURALIS THEORIA
habet accurata tales, sed tantummodo ad sensum. Quod si vires sint aliquanto debiliores,
tum vero & inflexio ex vi externa mediocri, & oscillatio, ac tremor erunt majores, & jam
hinc ex simplicissimo trium punctorum systemate habebitur species quædam satis idonea
ad sistendum animo discrimen, quod in Natura observatur quotidie oculis, inter virgas
rigidas, ac eas, quæ sunt flexiles, & ex elasticitate trementes. w-
Systemate inflexo
per vires parallelas
vis puncti medii
contraria extremis,
& æqualis eorum
Summae,
Postremum theo-
rema generale, ubi
etiam tria puncta
non jaceant in di-
rectum.
Æquilibrium trium
punctorum non in
directum jacentium
impossibile sine vi
externa, nisi sint
in distantiis limi-
tum : cum iis qui
nisus ad retinen-
dam formam syste-
matis.
Elegans t h e o r i a.
puncti siti in peri-
metro ellipsis binis
aliis occupantibus
foco : vis nulla in
verticibus axium.
227. Ibidem si binæ vires, ut AQ, BT fuerint perpendiculares ad AB, vel etiam
utcunque parallelæ inter se, tertia quoque erit parallela illis, & æqualis earum summæ,
sed directionis contrariæ. Ducta enim CD parallela iis, tum ad illam KI parallela BA,
erit ob CK, VB æquales, triangulum CIK æquale simili BTV, sive TBS, adeoque CI æqualis
BT, IK æqualis BS, sive AR, vel QP. Quare si sumpta IF æquali AQ ducatur KF ; erit
triangulum FIK æquale AQP, ac proinde FK æqualis, & parallela AP, sive LC, & CLFK
parallelogrammum, ac CF, diameter ipsius, exprimet vim puncti C utique parallelam
viribus AQ, BT, & æqualem earum summæ, sed directionis contrariæ. Quoniam vero
est SB ad BT, ut BD ad DC ; ac AQ ad AR, ut DC ad DA ; erit ex æqualitate 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.
228. Ea, quæ hoc postremo numero demonstravimus, æque pertinent ad actiones
mutuas trium punctorum habentium positionem mutuam quamcunque, etiam si a rectilinea
recedat quantumlibet ; nam demonstratio generalis est : sed ad massas utcunque inæquales,
& in se agentes viribus etiam divergentibus, multo generalius traduci possunt, ac traducentur
inferius, & ad æquilibrii leges, & vectem, & centra Oscillationis ac percussionis nos deducent.
Sed interea pergemus alia nonnulla persequi pertinentia itidem ad puncta tria, quæ in
directum non jaceant.
229. Si tria puncta non jaceant in directum, tum vero sine externis viribus non poterunt
esse in æquilibrio ; nisi omnes tres distantiæ, quæ latera trianguli constituunt, sint dis-
tantiæ limitum figuræ I. Cum enim vires illæ mutuæ non habeant [Io6] directiones
oppositas ; sive unica vis ab altero e reliquis binis punctis agat in tertium punctum, sive
ambæ ; haberi debebit in illo tertio puncto motus, vel in recta, quæ jungit ipsum cum
puncto agente, vel in diagonali parallelogrammi, cujus latera binas illas exprimant vires.
Quamobrem si assumantur in figura I tres distantiæ limitum ejusmodi, ut nulla ex iis sit
major reliquis binis simul sumptis, & ex ipsis constituatur triangulum, ac in singulis angu-
lorum cuspidibus singula materiæ puncta collocentur ; habebitur systema trium punctorum
quiescens, cujus punctis singulis si imprimantur velocitates æquales, & parallelæ ; 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 cohæsionis, 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 cohæsionis, erunt ejusmodi, ut
mutata positione : ab ipsa etiam sponte magis discedat systema punctorum eorundem. Sed
consideremus jam casus quosdam peculiares, & elegantes, & utiles, qui huc pertinent.
23o. Sint in fig. 32 tria puncta A,E,B ita collocata, ut tres distantiæ AB, AE, BE sint
distantiæ limitum cohæsionis, & postremæ duæ
sint æquales. Focis A, B concipiatur ellipsis
transiens per E, cujus axis transversus sit FO,
conjugatus EH, centrum D : sit in fig. I AN
æqualis semiaxi transverso hujus DO, sive BE,
vel AE, ac sit DB hic minor, quam in fig. I
amplitudo proximorum arcuum LN, NP, & sint
in eadem fig. I arcus ipsi NM, NO similes, &
æquales ita, ut ordinatæ uy, zt, æque distantes
ab N, sint inter se æquales. Inprimis si punctum
materiæ sit hic in E ; nullum ibi habebit vim,
cum AE, BE sint æquales distantiæ AN limitis
N figuræ I ; ac eadem est ratio pro puncto
collocato in H. Quod si fuerit in O, itidem
erit in æquilibrio. Si enim assumantur in fig. I
Az, Au æquales hisce BO, AO ; erunt Nz, Nu
illius æquales DB, DA hujus, adeoque & inter se. Quare & vires illius zt, uy erunt æquales
inter se, quæ cum pariter oppositæ directionis sint, se mutuo elident ; ac eadem ratio est
pro collocatione in F. Attrahetur hic utique A, & repelletur B ab O ; sed si limes, qui
FIG. 32.
„respondet distantiæ AB, sit satis validus ; ipsa puncta nihil ad sensum discedent a focis

A THEORY OF NATURAL PHILOSOPHY I79
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
to AB, or parallel to one another in any manner, then the third force would also be parallel
to them, equal to their sum, but in the opposition direction. For, if CD is drawn parallel
to the forces, & KI parallel to BA to meet CD in I, then, since CK & VB are equal to
one another, the triangle CIK will be equal to the similar triangle BTV, or to the triangle
TBS ; & therefore CI will be equal to BT, IK to BS or AR or QP. Hence if IF is taken
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 æquali 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
of three points having any relative positions whatever, even if it departs from a rectilinear
position to any extent you may please. For the demonstration is general; &, further, the
results can be deduced much more generally for masses that are in every manner unequal,
& 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
of an external force they cannot be in equilibrium ; unless all three distances, which form
the sides of the triangle, are those corresponding to the limit-points in Fig. 1. For, since
the mutual forces do not have opposite directions, either a single force from one of the
remaining two points acts on the third, or two such forces. Hence there must be for that
third point some motion, either in the direction of the straight line joining it to the acting
point, or along the diagonal of the parallelogram whose sides represent those two forces.
Therefore, if in Fig. I we take three limit-distances of such a kind, that no one of them is
greater than the other two taken together, & if from them a triangle is formed & at each
vertical angle a material point is situated, then we shall have a system of three points at rest.
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,B,B be so placed that the three distances AB,
AE, BE correspond to limit-points of cohesion, & let the two last be equal to one another.
Suppose that an ellipse, whose foci are A & B, passes through E ; let the transverse axis of
this be FO, & the conjugate axis EH, & the centre D. In Fig. I, let AN be equal to
the transverse semiaxis DO of Fig. 32, that is equal to BE or AE; also in the latter figure
let DB be less than the width of the successive arcs LN, NP of Fig. I ; also, in Fig. 1, let
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, Nu 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
In a system dis-
torted by parallel
forces the force on
the middle point is
in the opposite
direction to that of
the outside forces,
and equal to their
Sll Iſl.
The last theorem in
general, even when
the three points do
not lie in a straight
line.
Equilibrium of
three points that
do not lie in a
straight line is im-
possible without
the presence of an
external force,
unless the points
are at distances
corresponding to
limit-points; the
endeavour, in this
case, to conserve
the form of the
system.
An elegant theory
for a point situated
in the perimeter of
an ellipse, each of
the other two
being placed in a
focus ; no force at
the ends of the
a XCS.
I8O PHILOSOPHIAE NATURALIS THEORIA
In reliquis puncti
perimetri vis direc-
ta per ipsam peri-
metrum versus ver-
tices axis conju-
gati.
Analogia verticum
binorum a X iu m
cum limitibus cur-
væ virium.
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 illa duo.
23 I. Manet igitur immotum, ac sine vi,
punctum collocatum tam in verticibus axis con-
jugati ejus ellipseos, quam in verticibus axis
transversi ; & si ponatur in quovis puncto C
[Io7] perimetri ejus ellipseos, tum ob AC, CB
simul æquales in ellipsi axi transverso, sive duplo
semiaxi DO ; erit AC tanto longior, quam ipsa
DO, quanto BC brevior ; adeoque si jam in fig.
I sint Au, Az, æquales hisce AC, BC ; habe-
buntur ibi utique uy, zt itidem æquales inter se.
Quare hic attractio CL æquabitur repulsioni
CM, & LIMC erit rhombus, in quo inclinatio IC
secabit bifariam angulum LCM ; ac proinde si
ea utrinque producatur in P, & Q ; angulus ACP,
qui est idem, ac LCI, erit æqualis angulo BCQ,
qui est ad verticem oppositus angulo ICM. Quæ FIG. 32.
cum in ellipsi sit notissima proprietas tangentis
relatæ ad focos ; erit ipsa PQ tangens. Quamobrem dirigetur vis puncti Cin 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 hic jam licebit contemplari in hac curva perimetro vicissitudinem
limitum prorsus analogorum limitibus cohæsionis, & non cohæsionis, qui habentur in axe
rectilineo curvæ primigeniæ figuræ 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 cohæsionis ; at in
Quando limites
çontrario m o d O
positi : casus ele-
gantissimi alterna-
tionis pluriu m
limitum in peri-
metro ellipseos.
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 cohæsionis. -
233. Contrarium accideret, si DO æquaretur distantiæ limitis non cohæsionis : tum
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 cohæsionis, illi e contrario limites non cohæsionis. Sed
adhuc major analogia in perimetro harum ellipsium habebitur cum axe curvæ primigeniæ
figuræ I ; si fuerit DO æqualis distantiæ limitis cohæsionis AN illius, & DB in hac major,
quam in fig. I amplitudo NL, NP ; multo vero magis, si ipsa hujus DB superet plures
ejusmodi amplitudines, ac arcuum æqualitas maneat hinc, & inde per totum ejusmodi
spatium. Ubi enim AC hujus figuræ fiet æqualis abscissæ AP illius, etiam BC hujus fiet
pariter æqualis AL illius. Quare in ejusmodi loco habebitur limes, & ante ejusmodi locum
versus A distantia [Io8] longior AC habebit repulsionem, & BC brevior attractionem,
ac rhombus erit KGNC, & vis dirigetur versus O. Quod si alicubi ante in loco adhuc
propriore O distantiæ AC, BC æquarentur abscissis AR, AI figuræ 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 æquatur distantiæ cujuspiam limitis cohæsionis, & distantia
punctorum a centro ellipseos, sive ejus eccentricitas est major, quam intervallum dicti
limitis a pluribus sibi proximis hinc, & inde, ac maneat æqualitas arcuum, habebuntur in
singulis quadrantibus perimetri ellipeos tot limites, quot limites transibit eccentricitas
hinc translata in axem figuræ I, a limite illo nominato, qui terminet in fig. I semiaxem
transversum hujus ellipseos ; ac prætererea habebuntur limites in verticibus amborum
ellipseos axium ; eritque incipiendo ab utrovis vertice axis conjugati in gyrum per ipsam
perimetrum is limes primus cohæsionis, tum illi proximus esset non cohæsionis, deinde

A THEORY OF NATURAL PHILOSOPHY I8I
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.
231. A point, then, which is situated at one of the vertices of the conjugate axis of
the ellipse or at one of the vertices of the transverse axis remains motionless & under the
action of no force. If it is placed at any point C in the perimeter of the ellipse, then, since
AC, CB taken together are in the ellipse equal to the transverse axis, or double the semi-
axis DO, AC will be as much longer than DO as BC is shorter. Hence, if in Fig. I Au,
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 IC 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
as being perfectly analogous to those of cohesion & non-cohesion, which were obtained in
the rectilinear axis of the primary curve of Fig. 1. There will be certain limit-points at
E, F, H, O, in which there is no force, whilst in all intermediate points such as C there
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
to a limit-point of non-cohesion. For then the smaller distance BC would have an
attraction CK, & the greater distance AC a repulsion CN; the resultant force along the
diagonal CG of the rhombus CNGK would in the same way have its direction along the
tangent to the ellipse, & at the vertices of either axis there would be certain limit-points;
but a point situated in the perimeter would tend towards the vertices of the transverse
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.
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
At remaining points
of the perimeter
the force directed
along the perimeter
is towards the ver-
tices of the conju-
gate axis.
Analogy between
the vertices of the
two axes & the
limit-points of the
curve of forces.
When the limit
points are disposed
in the opposite
way ; most elegant
instances of alter-
nation of several
limit-points in the
perimeter of the
ellipse.
For where AC in the present figure becomes equal to the abscissa AP
I82 PHILOSOPHIAE NATURALIS THEORIA
Perimetri plurium
ellipsium aequiva-
lentes limitibus.
Demonstratio.
Alias curvas ellip-
sibus substituen-
das : ampla , pro-
blematum se ges,
sed minus utilis :
immensa combina-
tionum varietas.
Conversio totius
Systematis illaesi :
impulsu per peri-
metrum ellips e o s
oscillatio : idea
liquationis, S con-
glaciationis.
alter cohaesionis, & ita porro, donec redeatur ad primum, ex quo incoeptus fuerit gyrus,
vi in transitu per quemvis ex ejusmodi limitibus mutante directionem in oppositam. Quod
si semiaxis hujus ellipseos aequetur distantiae limitis non cohaesionis figurae I ; res eodem
ordine pergit cum hoc solo discrimine, quod primus limes, qui habetur in vertice semiaxis
conjugati sit limes non cohaesionis, tum eundo in gyrum ipsi proximus sit cohaesionis limes,
deinde iterum non cohaesionis, & ita porro.
234. Verum est adhuc alia quaedam analogia cum iis limitibus; si considerentur
plures ellipses iisdem illis focis, quarum semiaxes ordine suo aequentur distantiis, in altera
cujuspiam e limitibus cohaesionis figurae I, in altera limitis non cohaesionis ipsi proximi,
& ita porro alternatim, communis autem illa eccentricitas sit adhuc etiam minor quavis
amplitudine arcuum interceptorum limitibus illis figura I, ut nimirum singulae ellipsium
perimetri habeant quaternos tantummodo limites in quatuor verticibus axium. Ipsae
ejusmodi perimetri tota- erunt quidam veluti limites relate ad accessum, & recessum a
centro. Punctum collocatum in quavis perimetro habebit determinationem ad motum
secundum directionem perimetri eiusdem; 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.
235. Sint enim in fig. 33. ellipsium FEOH, F"E'O'H', F"E"O"Hº 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 essentin perimetro, adeoque in fig. I factis Au, Az majoribus, quam
essent prius, decrescet repulsio zt, crescet attractio uy; ac proinde hic in parallelogrammo
LCMI eritattractio 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'I accedetmagis 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 tertiae perimetri: atque inde patet, quod fuerat propositum.
-
236. Quoniam arcus hinc, &inde a quovis limite non sunt prorsus aequales; quanquam,
ut supra observavimus num. 184, exigui arcus ordinatas ad sensum aequales hinc, & inde
habere debeant; curva, per cuius tangentem perpetuo dirigatur vis, licet in exigua eccen-
tricitate debeat esse ad sensum ellipsis, tamen nec in iis erit ellipsis accurate, nec in
eccentricitatibus majoribus ad ellipses multum accedet. Erunt tamen semper aliquae
curvae, quae determinent continuam directionem virium, & curvae etiam, quae trajectoriam
describendam definiant, habita quoque ratione viscentifugae: atque hic quidem uberrima
seges succrescit problematum Geometria, & Analysi exercendae aptissimorum ; sed omnem
ego quidem eiusmodi perquisitionem omittam, cujus nimirum ad Theoriae applicationem
usus mihi idoneus occurrit nullus ; & quae huc 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.
a
237. At praeterea est & alius insignis, ac magis determinatus fructus, quem exejusmodi
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 [I Iol vis, qua ipsum
retinetur in eo vertice, esse admodum ingens pro utcunque exigua dimotione ab eo loco,
A THEORY OF NATURAL PHILOSOPHY

184
PHILOSOPHIAE NATURALIS THEORIA

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
number of ellipses having the same foci, of which the semiaxes are in order equal to the
distances corresponding to limit-points in Fig. I, namely to one of cohesion for one, to
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. 1,
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',
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
perimeter F'E'O'H'; then AC, BC will be greater than if they were drawn to the perimeter.
Hence, in Fig. 1, since Au, 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 C'I' 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,
& yet, as has been mentioned above in Art. 184, very small arcs on either side are bound
to have approximately equal ordinates; the curve, along the tangent to which the force
is continually directed, although for small eccentricity it must be practically an ellipse,
yet will neither be an ellipse accurately in this case, nor approach very much to the form
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
to be derived from considerations of this kind, & one that will be of service in the application
of the Theory to Physics. For instance, if the two points A & B are at a distance corresponding
to a limit-point of cohesion that is sufficiently strong, & the third point situated at the
vertex E of the conjugate axis is at a distance from the other two which corresponds to
a limit-point of cohesion that is also sufficiently strong, then the force retaining the point-
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
The perimeters of
several elli pses
equivalent to limit-
points.
Demonstration.
point C be situated somewhere outside the middle
Other curves to
be substituted for
ellipses; an ample
crop of theorems,
but not of much use;
great variety of
combinations.
Rotation of the
whole system
intact ; oscillation
along the perimeter
of the ellipse due to
an impulse ; the
idea of liquefaction
& congelation.
I86 PHILOSOPHIÆ NATURALIS THEORIA
ut sine ingenti externa vi inde magis dimoveri non possit. Tum 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 quædam soliditatis cujus & supra injecta est mentio. At
si stantibus in fig. 32 punctis A, B per quaspiam vires externas,
quæ eorum motum impediant, vis aliqua exerceatur in E ad
ipsum a sua positione deturbandum ; donec ea fuerit medio-
cris, dimovebit illud non nihil ; tum, illa cessante, ipsum se resti-
tuet, & oscillabit hinc, & inde ab illo vertice per perimetrum
curvæ cujusdam proximæ arcui elliptico. Quo major fuerit vis
externa dimovens, eo major oscillatio fiet ; 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 sivis externa coegerit
percurrere totum quadrantem, & transilire ultra verticem axis
transversi ; tum vero gyrabit punctum circumquaque per totam
perimetrum motu continuo, quem a vertice axis conjugati ad
FIG. 34.
verticem transversi retardabit, tum ab hoc ad verticem conjugati accelerabit, & ita porro,
Systema punctorum
quatuor, in eodem
plano cum distan-
tiis limitum, suæ
formæ tenax.
Alia ratio system-
a t i s punctorum
quatuor in eodem
plano cum idea
virgæ rigidæ, &
flexilis : systema
eorundem formæ
pyramidalis: ordi-
nes varii particula-
rum pyramidalium.
nec sistetur periodicus conversionis motus, nisi exteriorum punctorum impedimentis
occurrentibus, quæ sensim celeritatem imminuant, & post ipsos éjusmodi 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, tum iterum refrigescentes, agitatione
sensim cessante per vires, quibus igneæ particulæ emittuntur, & evolant, positionem prio-
rem recuperant, ac tenacissime iterum servant, & tuentur ? Sed hæc de trium punctorum
systemate hucusque dicta sint satis. «*
238. Quatuor, & multo magis plurium, punctorum systemata multo plures nobis
variationes objicerent ; si rite ad examen vocarentur ; sed de iis id unum innuam. Ea
quidem in plano eodem possunt positionem mutuam tueri tenacissime ; si singulorum
distantiæ a reliquis æquentur distantiis limitum satis validorum figuræ I : neque enim
in eodem plano positionem respectivam mutare possunt, aut aliquod ex iis exire e plano
ducto per reliqua tria, nisi mutet distantiam ab aliquo e reliquis, cum datis trium punctorum
distantiis mutuis detur triangulum, quod constituere debent, tum datis distantiis quarti
a duobus detur itidem ejus positio respectu eorum in eodem plano, & detur distantia ab
eorum tertio, quæ, si id punctum exeat e [III] priore plano, sed retineat ab iis duobus
distantiam priorem, mutari utique debet, ut facili negotio demonstrari potest.
239. Quin immo in ipsa ellipsi considerari possunt puncta quatuor, duo in focis, &
alia duo hinc, & inde a vertice axis conjugati in ea distantia a se invicem, ut vi mutua
repulsiva sibi invicem elidant vim, qua juxta præcedentem Theoriam urgentur in ipsum
verticem ; quo quidem pacto rectangulum quoddam terminabunt, ut exhibet fig. 35, in
punctis A, B, C, D. Atque inde si supra angulos quadratæ basis assurgant series ejusmodi
punctorum exhibentium series continuas rectangulorum, habebitur quædam adhuc magis
præcisa idea virgæ solidæ, in qua si basis ima inclinetur ; statim ominia superiora puncta
movebuntur in latus, ut rectangulorum illorum positionem retineant, & celeritas conversionis
erit major, vel minor, prout major fuerit, vel minor vis illa in latus, quæ ubi fuerit aliquanto
languidior, multo serius progredietur vertex, quam fundum,
& inflectetur virga, quæ inflexio in omni virgarum genere
apparet adhuc multo magis manifesta, si celeritas conversionis
fuerit ingens. Sed extra idem planum possunt quatuor puncta
collocata ita, ut positionem suam validissime tueantur, etiam
ope unicæ distantiæ limitis unici satis validi. Potest enim fieri
pyramis regularis, cujus latera singula triangularia habeant
ejusmodi distantiam. Tum ea pyramis constituet particulam • ©
quandam suæ figuræ tenacissimam, quæ in puncta, vel pyra- A E
mides ejusmodi aliquanto remotiores ita poterit agere, ut ejus
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 eam componen-
C. D.
PIG. 35.

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-
tions, if they were examined carefully one after the other ; but I will only mention one thing
about such systems. It is possible that such systems, in one plane, may conserve their rela-
tive positions very tenaciously, if the distances of each from the rest are equal to the dis-
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
foci, & the other two on either side of a vertex of the conjugate axis at such a distance from
one another, that the mutual repulsive force between them will cancel the force with which
they are urged towards that vertex, according to the preceding theorem. Thus, they are
at the vertices of a rectangle, as is shown in Fig. 35, where they occupy the points A,B,C,D.
Hence, if we have a series of points of this kind to stand above the four angles of the quadratic
base, so as to represent continuous series of rectangles, we shall obtain from this supposition
a more precise idea than hitherto has been possible of a solid rod, in which, if the lowest
set of 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;
A system of four
points, in one plane
at distances corre-
sponding to limit-
points, which con-
serves its form.
A further consider-
ation of a system
of four points in
connection with
the idea of rigid &
flexible rods ; a
system of four
points in the form
of a pyramid ;
different arrange-
ments of particular
pyramids.
I 88 PHILOSOPHIÆ NATURALIS THEORIA
T ra n s i t u s
massas : quid
a d
Cen-
trum gravitatis :
theoremata hic de
eo demonstrando.
Definitio c e n t r i
g ra v it a t i s
pendens a b
I1O11
idea,
gravitatis : ejus
Congruentia,
idea communi.
C UIIÍl
tium, qua fit, ut vires in easdem ab externis punctis impressæ multo magis inæquales inter se
sint, quam fuerint in punctis constituentibus particulas ordinis primi ; ac eodem pacto ex
his secundi ordinis particulis fieri possunt particulæ ordinis tertii adhuc minus tenacés figuræ
suæ, atque ita porro, donec ad eas deventum sit multo majores, sed adhuc multo magis
mobiles, atque variabiles, ex quibus pendent chemicæ operationes, & ex quibus hæc ipsa
crassigra corpora componuntur, ubi idipsum accideret, qu6d Newtonus in postrema Opticæ
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 grádum.
240. In massis primum nobis se offerunt considerandæ elegantissimæ sane, ac & fœcund-
issimæ, & utilissimæ proprietates centri gravitatis, quæ quidem e nostra Theoria sponte
propemodum fluunt, aut saltem ejus ope evidentissime demonstrantur. Porro centrum
gravitatis a gravium æquilibrio nomen accepit suum, a quo etiam ejus consideratio ortum
duxit; . sed id quidem a gravi-[I I2]-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 ; tum demonstrabo accuratissime,
in quavis massa haberi aliquod gravitatis centrum, idque unicum, quod quidem passim
omittere solent, & perperam ; deinde ad ejus proprietatem præcipuam exponendam
gradum faciam, demonstrando celeberrimum theorema a Newtono propositum, centrum
gravitatis commune massarum, sive mihi punctorum quotcunque, & utcunque disposi-
torum, quorum singula moveantur sola inertiæ vi motibus quibuscunque, qui in singulis
punctis uniformes sint, in diversis utcunque diversi, vel quiescere, vel moveri uniformiter
in directum : tum vero mutuas actiones quascunque inter puncta quælibet, vel omnia
simul, nihil omnino turbare centri communis gravitatis statum quiescendi vel movendi
uniformiter in directum, unde nobis & actionis, ac reactionis æqualitas in massis quibusque,
& principia collisiones corporum definientia, & alia plurima sponte provenient. Sed
aggrediamur ad rem ipsam. w
24I. Centrum igitur commune gravitatis punctorum quotcunque, & utcunque
dispositorum, appellabo id punctum, per quod si ducatur planum quodcunque ; summa
distantiarum perpendicularium ab eo plano punctorum omnium jacentium ex altera
ejusdem parte, æquatur summa distantiarum ex altera. Id quidem extenditur ad quas-
cunque, & quotcunque massas ; nam eorum singulæ punctis utique constant, & omnes
simul sunt quædam punctorum diversorum congeries. Nomen traxit ab æquilibrio
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, & distantiæ perpen-
diculari a plano verticali ducto per axem ipsum ; unde fit, ut ubi ejusmodi vires, vel, ut
ea vocant, momenta virium hinc, & inde æqualia fuerint, habeatur æquilibrium. Porro
ipsa pondera in nostris gravibus, in quibus gravitatem concipimus, ac etiam ad sensum
experimur, proportionalem in singulis quantitati materiæ, & agentem directionibus inter
se parallelis, proportionalia sunt massis, adeoque punctorum eas constituentium numero ;
quam ob rem idem est, ea pondera in distantias ducere, ac assumere summam omnium
distantiarum omnium punctorum ab eodem plano. Quod si igitur respectu aggregati
cujuscunque punctorum materiæ quotcunque, & quomodocunque dispositorum sit aliquod
punctum spatii ejusmodi, ut, ducto per ipsum quovis plano, summa distantiarum ab illo
punctorum jacentium ex parte altera æquetur summæ distantiarum jacentium ex altera ;
concipiantur autem singula ea puncta animata viribus æqualibus, & parallelis, cujusmodi sunt
vires, quas in nostris gravibus concipimus ; illud utique consequitur, [II 3] suspenso 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 æquilibrio ; atque illud ipsum requiri, ut in æquilibrio sit. Si enim vel
unicum planum ductum per id punctum sit ejusmodi, ut summæ illæ distantiarum non
sint æquales hinc, & inde ; converso systemate omni ita, ut illud planum evadat verticale,
jam non essent æquales inter se summæ momentorum hinc, & inde, & altera pars alteri
præponderaret. Verum hæc quidem, uti supra monui, fuit occasio quædam 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-
tion are certain really very elegant, as well as most fertile & useful properties of the centre of
gravity. These indeed come forth almost spontaneously from my Theory, or at least are
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,
situated in any positions whatever, that point which is such that, if through it any plane
is drawn, the sum of the perpendicular distances from the plane of all the points lying on
one side of it is equal to the sum of the distances of all the points on the other side of it.
The definition applies also to masses, of any sort or number whatever ; for each of the
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, & the whole 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
Passing on to
masses; what is a
centre of gravity ?
Theorems to be
proved concerning
it at this point.
Definition of the
centre of gravity
independent of any
idea of gravitation ;
the agreement of
this definition with
the usual idea.
I9O PHILOSOPHIÆ NATURALIS THEORIA
Corollarium g e n-
erale pertinens ad
summas distanti-
a r u m o m n i u m
punctorum massæ
a plano transeunte
per centrum gravi-
tatis æquales utrin-
que.
Ei n a theoremata
per t in e nt ia a d
planum parallelum
plano distantiarum
æquali u m cum
eorum demonstra-
tionibus.
C o m p 1 e me n tum
demonstrationis ut
e x t e n d a t u r a d
omnes casus.
ad solas massas animatas viribus æqualibus, & parallelis, cujusmodi concipiuntur a nobis
in nostris gravibus, licet ne in ipsis quidem accurate sint tales. Quamobrem assumpta
superiore definitione, quæ a gravitatis, & æquilibrii natura non pendet, progrediar ad
deducenda inde corollaria quaædam, quæ nos ad ejus proprietates demonstrandas deducant.
242. Primo quidem si aliquod fuerit ejusmodi planum, ut binæ summæ distantiarum
perpendicularium punctorum omnium hinc & inde acceptorum æquenter inter se :
æquabuntur & summæ distantiarum acceptarum secundum quancunque aliam directionem
datam, & communem pro omnibus. Erit enim quævis distantia perpendicularis ad quanvis
in dato angulo inclinatam semper in eadem ratione, ut patet. Quare & summæ illarum
ad harum summas erunt in eadem ratione, ac æqualitas summarum alterius binarii utriuslibet
secum trahet æqualitatem 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 plano habenti æquales
hinc, & inde distantiarum summas ; summa distantiarum omnium punctorum jacentium
ex parte altera superabit summam jacentium ex altera, excessu æquali distantiæ planorum
acceptæ secundum directionem eandem ductæ in numerum punctorum : & vice versa si
duo plana parallela sint, ac is excessus alterius summæ supra summam alterius in altero
ex iis æquetur eorum distantiæ ductæ in numerum punctorum ; planum alterum habebit
oppositarum distantiarum summas æquales. Id quidem facile concipitur ; si concipiatur,
planum distantiarum æqualium moveri versus illud alterum planum motu parallelo
secundum eam directionem, secundum quam sumuntur distantiæ. In eo motu distantiæ
singulæ ex altera parte crescunt, ex altera decrescunt continuo tantum, quantum promo-
vetur planum, & si aliqua distantia evanescit interea ; jam eadem deinde incipit tantundem
ex parte contraria crescere. Quare patet excessum omnium citeriorum [I I4] distantiarum
supra omnes ulteriores æquari progressui plani toties sumpto, quot puncta habentur, &
in regressu destruitur e contrario, quidquid in ejusmodi progressu est factum, atque idcirco
ad æqualitatem reditur. Verum ut demonstratio
quam accuratissima evadat, exprimat in fig. 36 recta
AB planum distantiarum æqualium, & 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.
Rectæ autem per ipsa ductæ in directione data
quacunque, occurrant rectæ AB in M, H, K, &
rectæ CD in N, I, L ; ac sit quædam reacta direc-
tionis ejusdem ipsis AB, CD occurrens in O, P.
Patet, ipsam OP fore æqualem ipsis MN, HI, KL.
Dicatur jam summa omnium punctorum E primæ
classis E, & distantiarum omnium EM summa e ;
punctorum F secundæ classis F, & distantiarum f ;
punctorum G tertiæ classis summa G, & distantiarum [D
earundem g; distantia vero OP dicatur O. Patet, sum-
mam omnium MN fore E × O; summam omnium
HI fore F × O ; summam omnium KL fore G × O ; erit autem quævis EN = EM -|-MN ;
quævis FI = HI — FH ; quævis GL = KG — KL. Quare summa omnium EN erit
3-|- E ×O ; summa omnium FI = F × O — f, & summa omnium GL = g — G × O ;
adeoque summa omnium distantiarum punctorum jacentium citra planum CD, primæ nimi-
rum, ac secundæ classis, erit e + E × O + F × O — f, & summa omnium jacentium ultra,
nimirum classis tertiæ, erit g — G × O. Quare excessus prioris summæ supra secundam
erit e + E × O + F × O — f — g -|- G × O ; adeoque si prius fuerit e = f -+- g ;
deleto e — f — g, totus excessus erit E × O + F × O -- G × O, sive (E + F -- G) X O,
summa omnium punctorum ducta in distantiam planorum ; & vice versa si is excessus
respectu secundi plani CD fuerit æqualis huic summæ ductæ in distantiam O, Qportebit
e — f — g æquetur nihilo, adeoque sit e = f + g, nimirum respectu primi plani AB summas
distantiarum hinc, & inde æquales esse.
244. Si aliqua puncta sint in altero ex iis planis, ea superioribus formulis contineri
possunt, concepta zero singulorum distantia a plano, in quo jacent ; sed & ii casus involvi
facile possent,`concipiendo alias binas punctorum classes ; quorum priora sint in priore
plano AB, posteriora in posteriore CD, quæ quidem nihil rem turbant : nam prioris classis
OęA
BI .
FIG. 36.

A THEORY OF NATURAL PHILOSOPHY I9I
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
perpendicular distances of all the points on either side of it taken together are equal to
one another, then the sums of the distances taken together in any other given direction,
that is the same for all of them, will also be equal to one another. For, any perpendicular
distance will evidently be in the same ratio to the corresponding distance inclined at a
given angle. Hence the sums of the former distances will bear the same ratio to the sums
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
distances on either side are equal, then the sum of the distances of all the points lying
on the one side of it will exceed the sum for those lying on the other side by an amount
equal to the distance between the two planes measured in the like direction multiplied
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
General corollary
relating to the sums
of the distances of
all the points of a
mass, from a plane
passing through
the centre of grav-
ity, being equal on
either side of it.
Two the or em S
relating to a plane
parallel to the
plane of equal
distances ; & their
demonstrations.
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 & 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 EXO;
the sum of all the HI’s will be FX O ; the sum of all the KL’s will be GX O ; also
in every case, EN = EM + MN, FI = HI — FH, & GL = KG — KL. Hence the sum
of the EN’s will be e -- EXO, the sum of the FI’s will be FX O-f, & 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 e -- EXO + F X O-f; & 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 e -- EXO + F X O – f – g + G × O. Therefore, if at
first we had e = f + g, then, on omitting e – f – g, we have the total excess equal to
EXO + FX O + G XO, or (E + F + G) × O, 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
be included in the foregoing formulae, if we suppose that the distance for each of them
is zero distance from the plane in which they lie. Then these cases may also be included
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
Completion of the
proof, so as to in-
clude all possible
Ca,SeS.
I92 PHILOSOPHIÆ NATURALIS THEORIA
Theoremata pro
plano posito ultra
O m n i a p u n c t a :
eOrum extensio ad
quævis plana.
distantiæ a priore plano erunt omnes simul zero, & a posteriore æquabuntur distantiæ O
ductæ in eorum numerum, quæ summa accedit priori summæ punctorum jacentium citra ;
posterioris autem classis distantiæ a priore erant prius simul æquales summæ ipsorum
ductæ itidem in O, & deinde fiunt nihil ; adeoque [II 5] summæ distantiarum punctorum
jacentium ultra, demitur horum posteriorum punctorum summa itidem ducta in O, &
proinde excessui summæ citeriorum supra summam ulteriorum accedit summa omnium
punctorum harum duarum classium ducta in eandem O.
245. Quod si planum parallelum plano distantiarum æqualium jaceat ultra omnia
puncta ; jam habebitur hoc theorema : Summa omnium distantiarum punctorum omnium ab
eo plano aequabitur distantiae planorum ductae in omnium punctorum summam, & si fuerint duo
plana parallela ejusmodi, ut alterum jaceat ultra omnia puncta, & summa omnium di$tanti-
arum ab ipso aequetur distantiae planorum ductae in omnium punctorum numerum ; alterum illud
planum erit planum distantiarum æqualium. Id sane patet ex eo, quod jam secunda sum-
ma pertinens ad puncta ulteriora, quæ nulla sunt, evanescat, & excessus totus sit sola prior
summa. Quin immo idem theorema habebitlocum pro quovis plano habente etiam ulteriora
puncta, si citeriorum distantiæ habeantur pro positivis, & ulteriorum pro negativis ; cum
Cuivis plano
veniri posse paral-
lelum planum dis-
tantiarum æqua-
lium.
Theorema præci-
puum si tria plana
distantiarum æqua-
lium habeant uni-
C u m p u n ct u m
commune ; reliqua.
omnil për id tran-
seuntia erunt ejus-
modi.
Demonstratio ejus-
dem.
in-
nia sint plana distantiarum æqualium,
• occurrant in recta PM, postrema duo
nimirum summa constans positivis, & negativis sit ipse excessus positivorum supra negativa ;
quo quidem pacto licebit considerare planum distantiarum æqualium, ut planum, in quo
summa omnium distantiarum sit nulla, negativis nimirum distantiis elidentibus positivas.
246. Hinc autem facile jam patet, dato cuivis plano haberi aliquod planum parallelum,
quod sit planum distantiarum æqualium ; quin immo data positione punctorum, &5 plano illo
ipso, facile id alterum definitur. Satis est ducere a singulis punctis datis rectas in data
directione ad planum datum, quæ dabuntur ; tum a summa omnium, quæ jacent ex parte
altera, demere summam omnium, si quæ sunt, jacentium ex opposita, ac residuum dividere
per numerum punctorum. Ad eam distantiam ducto plano priori parallelo, id erit planum
quæsitum distantiarum æqualium. Patet autem admodum facile & illud ex eadem
demonstratione, & ex solutione superioris problematis, dato cuivis plano non nisi unicum
esse posse planum distantiarum æqualium, 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
A C N B
G}/ /||VI
coalescentium. Id fiet ope sequentis
/â
/

theorematis; si per quoddam punctum tran-
seant tria plana distantiarum aequalium se
mom in eadem communi aliqua recta secam- i
tia ; omnia alia plana transeuntia per illud
idem punctum erunt itidem distantiarum
aequalium plana. Sit enim in fig. 37, ejus-
modi punctum C, per quod transeant tria
plana GABH, XABY, ECDF, quæ om-
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 æqualium, si illa priora
ejusmodi sint. Concipiatur quodcunque
punctum P ; & per ipsum P concipiatur
tria plana parallela planis DCEF, ABYX,
GABH, quorum sibi priora duo mutuo
Κ
-—
F
FIG. 37.
in recta PV, primum cum tertio in
recta PO : ac primum occurrat plano
GABH in MN, secundum vero eidem -
in MS, plano DCEF in QR, ac plano 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 plano KICL secundum datam direc-
tionem BA erit summa omnium PV, quæ resolvitur in tres summas, omnium PR, omnium
RT, omnium TV, sive eæ, ut figura exhibet in unam colligendæ sint, sive, quod in aliis
plani novi inclinationibus posset accidere, una ex iis demenda a reliquis binis, ut habeatur
omnium PV summa. Porro quævis PR est distantia a plano DCEF secundum eandem
eam directionem ; quævis RT est æqualis QS sibi respondenti, quæ ob datas directiones
laterum trianguli SCQ est ad CQ, æqualem MN, sive PO, distantiæ a plano XABY secundum
A THEORY OF NATURAL PHILOSOPHY I93
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
side of all the points then the following theorem is obtained. The sum of all the distances
of all the points from this plane will be equal to the distance between the planes multiplied by
the sum of all the points ; & if there were two parallel planes, such that one of them lies beyond
all the points, 85 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
Theorems for a
plane lying beyond
a 11 the points;
extension of these
theorems to a ny
plane whatever.
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
another plane parallel to it, which is a plane of equal distances ; further, if we are given the
position of the points, 85 also the plane is given, then the parallel plane is easily determined.
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 &
explanation, I will proceed to prove that there is a centre of gravity for any set of points,
no matter how they are dispersed or what the number of masses may be into which they
coalesce, or where these masses may be situated. The proof follows from the theorem —
If through any point there pass three planes of equal distances that do not all cut one another in
some common line then all other planes passing through this same point will also be planes of equal
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
BA, will be equal to the sum of all the PV’s; & this can be resolved into the three sums,
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; & each RT 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
O
Given any plane,
there can be found
a plane of equal
distances, parallel
to it,
The important
theorem, that, if
three plane s of
e qual distances
have a common
point, then any
other plane through
the point will be of
the same nature.
Proof of the theo-
rem,
I94 PHILOSOPHIÆ NATURALIS THEORIA
[Haberi se m p er
aliquod gravitatis
centrum, atque id
esse unicum.]
INecessitas demon-
strandi haberi sem-
per centrum gravi-
atis.
C e ntru m enim
magnitudinis non
semper haberi.
Ubi h æ c primo
demonstrata.
datam directionem DC, in ratione data ; & quævis VT est itidem in ratione data ad TS
æqualem PM, distantiæ a plano 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 positivæ directionum PR, PO, PM a suis negativis destruuntur in illis tribus
æqualium distantiarum planis, etiam omnes positivæ PR, RT, TV a suis negativis destru-
entur, adeoque & omnes PV positivæ a suis negativis. Quamobrem planum LCIK erit
planum distantiarum æqualium. Q.E.D. -
249. Demonstrato hoc theoremate jam sponte illud consequitur, in quavis punctorum
congerie, adeoque ma$$arum utcumque dispersarum summa, haberi $emper aliquod gravitatis
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 quæcunque, ut OPM, RPM, RPO. Tum singulis poterunt per
num. 246 inveniri plana parallela, [I I7] quæ sint plana distantiarum æqualium, quorum
priora duo si sint DCEF, XABY, se secabunt in aliqua recta CE parallela illorum inter-
sectioni 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
plano, adeoque nec illa illi erit, sed in ipsum alicubi incurret. Transibunt igitur per
punctum C tria plana distantiarum æqualium, adeoque per num. 247 & aliud quodvis
planum transiens per punctum idem C erit planum æqualium distantiarum pro quavis
directione, & idcirco etiam pro distantiis perpendicularibus ; ac ipsum punctum C juxta
definitionem num. 24I, 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 æqualium, quod est
contra id, quod num. 246 demonstravimus.
25o. Demonstrandum necessario fuit, haberi aliquod gravitatis centrum, atque id
esse unicum ; & perperam id quidem a Mechanicis passim omittitur ; si enim id non
ubique adesset, & non esset unicum, in paralogismum incurrerent quamplurimæ Mechanic-
orum ipsorum demonstrationes, qui ubi in plano duas invenerunt rectas, & in solidis tria
plana determinantia æquilibrium, in ipsa intersectione constituunt gravitatis centrum, &
supponunt omnes alias rectas, vel omnia alia plana, quæ per id punctum ducantur, eandem
æquilibrii proprietatem habere, quod utique fuerat non supponendum, sed demonstrandum.
Et quidem facile est similis paralogismi exemplum præbere in alio quodam, quod magni-
tudinis centrum appellare liceret, per quod nimirum figura sectione quavis secaretur in
duas partes æquales inter se, sicut per centrum gravitatis secta, secatur in binas partes
æquilibratas in hypothesi gravitatis constantis, & certam directionem habentis plano
secanti parallelam. .
25 I. Erraret sane, qui ita definiret centrum magnitudinis, tum determinaret id ipsum
in datis figuris eadem illa methodo, quæ pro centri
gravitatis adhibetur. Is ex. gr. pro triangulo ABG
in fig. 38 sic ratiocinationem institueret. Secetur
AG bifariam in D, ducaturque BD, quæ utique
ipsum triangulum secabit in duas partes æquales.
Deinde, secta AB itidem bifariam in E, ducatur GE,
quam itidem constat, debere secare triangulum in
partes æquales duas. In earum igitur concursu C
habebitur centrum magnitudinis. Hoc invento si
progrederetur ulterius, & haberet pro æqualibus
partes, quæ alia sectione quacunque facta per C *
obtinentur ; erraret pessime. Nam ducta ED, jam A [D - G
constat, fore ipsam ED parallelam BG, & ejus dimi- -
diam ; adeoque similia fore triangula [I I8] ECD, EIG. 38.
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 æqualitatis.
252. Nimirum quæcunque punctorum, & massarum congeries, adeoque & figura
quævis, in qua concipiatur punctorum numerus auctus in infinitum, donec figura ipsa
evadat continua, habet suum gravitatis centrum ; centrum magnitudinis infinitæ earum
non habent ; & illud primum, quod hic accuratissime demonstravi, demonstraveram jam

A THEORY OF NATURAL PHILOSOPHY IQ5
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
from it that, for any group of points, & therefore also for a ſet of masses scattered in any manner,
there exists a centre of gravity, 85 there is but one ; 85 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,
& that there is only one in every case; & this proof is everywhere omitted by Mechanicians,
quite unjustifiably. For, if there were not one in every case, or if it were not unique,
very many of the proofs given by these Mechanicians would result in fallacious argument.
There is always
one centre of grav-
ity; and only one.
The need for
proving that there
is a centre of grav-
ity in every case.
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
& then proceed to determine it in given figures by the same method as that used for the
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 ECD, 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
of points is supposed to be indefinitely increased until the figure becomes continuous,
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
For there is not
always a centre of
magnitude.
e
Where the first
proof of this was
given.
I 96 PHILOSOPHIÆ NATURALIS THEORIA
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.
Difficultas demon-
strationis in eom-
muni methodo.
olim methodo aliquanto contractiore in dissertatione De Centro Gravitatis ; hujus vero
secundi exemplum hic patet, ac in dissertatione De Centro Magnitudinis, priori illi addita
in secunda ejusdem impressione, determinavi generaliter, in quibus figuris centrum
magnitudinis habeatur, in quis desit ; sed ea ad rem præsentem non pertinent.
253. Ex hac generali 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 binæ massæ, quarum
centra gravitatis sint in fig. 39 in A, & B. Si per rectam AB ducatur planum quodvis, id
debet esse planum distantiarum æqualium re-
spectu utriuslibet. Quare etiam respectu
summæ omnium punctorum ad utrumque
simul pertinentium distantiæ omnes hinc, &
inde acceptæ æquantur inter se ; ac proinde id
etiam respectu summæ debet esse planum dis-
tantiarum æqualium, & centrum commune
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 plano secundum directionem AB punctorum pertinentium ad massam
A, si a positivis demantur negativæ, æqualis per num. 243 numero punctorum massæ A
ducto in AC, & summa pertinentium ad B numero punctorum in B ducto in BC ; quæ
producta æquari debent inter se, cum omnium distantiarum summæ positivæ 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. Tum harum commune centrum sic inventum conjungitur cum
centro tertiae, & dividitur, distantia in ratione reciproca summae massarum priorum ad massam
tertiam, & ita porro. Quin immo possunt seorsum inveniri centra gravitatis binarum
quarumvis, ternarum, demarum quocumque [II 9] ordine, tum bimaria conjungi cum ternariis,
demariis, aliisque, ordine itidem quocumque, &$ $emper eadem methodo devenitur ad centrum
commune gravitatis massae totius. Id patet, quia quotcunque massæ considerari possunt
pro massa unica, cum agatur de numero punctorum massæ tantummodo, & de summa
distantiarum punctorum omnium ; summæ massarum constituunt massam, & summæ
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 summae
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, quæ omnes a recta BD
secantur bifariam, ut FH, centra gravitatis sunt in eadem recta, adeoque & areæ ab iis
contextæ centrum gravitatis est tam 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 meæ Theoriæ communia jam cum
vulgaribus elementis, hic 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 hic & illud fluit, areas FAGH, FBH licet inæquales, habere
tamen æquales summas distantiarum omnium suorum punctorum ab eadem recta FH.
EIG. 39.
256. In communi methodo alio modo se res habet. Posteaquam inventum est in
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 Centro Gravitatis; & a case of the second is here clearly shown ; & in the dissertation
De Centro 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 bearing on the matter now
in question. - ſº
253. From this general determination of the centre of gravity it is readily deduced that
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
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
of finding the common centre of gravity of ſeveral masses. First of all the centres of two of
them are joined, {5 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 ;
{5 ſo on. Indeed, we may find the centres of gravities of any groups of two, three, or ten, in
any order, {5 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:-If the
centres of gravity of ſeveral masses all lie in one & the ſame straight line, then the centre of
gravity of the whole set will also lie in the same straight line. This indicates a method for
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
common centre of gravity of the masses A & B has been found, for the third mass, whose
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,
Hence to determine
the position of the
common centre of
two maSSeS.
Hence, the usual
method for any
number of masses.
Hence, a theorem,
by the help of
which the centre
of gravity for con-
tinuous figures may
be investigated.
The difficulty of
proof in the ordi-
nary method.
198 PHILOSOPHIÆ NATURALIS THEORIA
Similis difficultas
in summa, & mul-
tiplicatione plurium
numerorum, & in
vi composita, ex
pluribus: methodus
componendi simul
OÎÎ1I1€S.
Consensus ej u s
methodi cum com-
muni per parallelo-
gramma.
D e m o n str a, t i o
generalis methodi.
sectionis punctum pro centro gravitatis. Nisi
generaliter demonstratum fuisset, haberi sem-
per aliquod, & esse unicum gravitatis cen-
trum ; oporteret hic iterum demonstrare id
novum sectionis punctum fore idem, ac illud
prius : sed per singulos casus ire, res infi-
nita esset, cum diversæ 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. II4.
[I2o] 257. Atque hic illud quidem accidit,
quod in numerorum summa, & multiplica-
tione experimur, ut nimirum quocunque ordine 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
multiplicatione potest res itidem demonstrari etiam generaliter, sed ea huc non pertinent.
Pertinet autem huc magis aliud ejusmodi exemplum petitum a compositione virium, in
qua itidem si multæ vires componantur communi methodo componendo inter se duas per
diagonalem parallelogrammi, cujus latera eas exprimant, tum 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
composito metior vim, ut e spatiolo, quod dato tempusculo vi aliqua percurreretur, solet
ipsa vis simplex quælibet æstimari. 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, & magnitudo
toto tempusculo perseveret eadem, fore in fine ejus tempusculi in eo loci puncto, in quo esset,
si singulæ eadem intensitate, & directione egissent aliæ post alias totidem tempusculis, quot
sunt vires, cessante omninova solicitatione, & omni velocitate jam producta a vi qualibet post
suum tempusculum : tum rectam, quæ conjungit primum illud punctum cum hoc postremo,
assumo pro mensura vis ex omnibus compositæ, quæ 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 ; assumo
itidem, fore in eo puncto loci, in quo esset, si altero tempusculo percurreret spatiolum, ad
quod determinatur ab illa velocitate, altero spatiolum, ad quod determinatur a vi, sive aliis
totidem tempusculis percurreret spatiola, ad quorum singula determinatur a viribus singulis.
258. Huc recidere methodum compon-
endi per parallelogramma facile constat ; si
enim in fig. 4I componendi sint plures motus,
vel vires expressæ a rectis PA, PB, PC, &c, &
incipiendo a binis quibusque PA, PB, eæ com-
ponantur per parallelogrammum PAMB, tum
vis composita PM cum tertia PC per parallelo-
grammum PMNC, & ita porro ; [I2I] patet,
ad idem loci punctum N per hæc parallelo-
gramma definitum debere devenire punctum
mobile, quod prius percurrat PA, tum AM par-
allelam, & æqualem PB ; tum MN parallelam,
& æqualem PC, atque ita porro additis quot-
cunque aliis motibus, vel viribus, quæ per
nova parallela, & æqualia 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, tum vi
PO cum vi PB per novum parallelogrammum, quod itidem haberet cuspidem in N ; sed
eo deveniretur alia via PAON. Hoc autem ipsum, quod tam multis viis, quam multas diversæ
plurium'compositiones motuum, ac virium exhibere possunt, eodem semper deveniri debeat, sic
generaliter demonstro. Si assumantur ultra omnia puncta, ad quæ 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
PIG. 4o.


A THEORY OF NATURAL PHILOSOPHY I99
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. II.4, 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
we find, for instance, no matter what the order is in which the numbers are taken, whether
they are taken singly, & added to the number already obtained, or multiplied, or whether
the addition or multiplication is made with a group of several of them ; the same number
is arrived at finally after all those that have been given have been used each once. Now
in addition it is easily seen that the result obtained is the same ; & for multiplication also
the matter can be easily demonstrated; but we are not concerned with these proofs here.
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
comes to the same thing. For, if, in Fig. 41, the several motions or forces to be compounded
are represented by PA, PB, PC, &c.; &, beginning with any two of them, PA & PB, these
are compounded by means of the parallelogram PAMB, then the resultant force PM is
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
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
A similar difficulty
exists in the case of
a Sum or a product
of Several num-
bers ; & also in a
force compounded
from Several forces;
the method of
Compounding them
all at one time.
Agreement of this
method with the
usual One by means
of the parallelo-
gram.
General proof of
the method.
2O0 PHILOSOPHLE NATURALIS THEORIA
Theorema de statu
c e n t r i gravitatis
manente etiam ubi
agant utcunque
Vires mutuæ, ac
eju s demonstra-
tionis initium.
Progressus ejusdem
demonstrationis
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, & æqualis. Quare summa ejusmodi accessuum, ac
Summa recessuum erit eadem in fine omnium tempusculorum, quocunque ordine dispon-
antur lineolæ hæ parallelæ, & æquales 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 plano, erit eadem. Inde autem
sponte jam fluit id, quod demonstrandum erat, nimirum punctum illud esse idem semper.
Si enim ad duo puncta duabus diversis viis deveniretur, assumpto plano perpendiculari ad
rectam, quæ illa duo puncta jungeret, distantia perpendicularis ab ipso non esset utique
eadem pro utroque, cum altera distantia deberet alterius esse pars.
260. Porro similis admodum est etiam methodus, qua utor ad demonstrandum
præclarissimum Newtoni theorema, in quod coalescunt simul duo, quæ superius innui, &
huc reducuntur. Si quotcumque materiae puncta utcumque disposita, & in quotcumque utcumque
disjunctas massas coale$centia habeant velocitates quascumque cum directionibus quibuscumque,
& praeterea urgeantur viribus mutuis quibuscumque, quae in binis quibusque punctis æqualiter
agant in plagas oppositas ; centrum commune gravitatis omnium vel quiescet, vel movebitur
uniformiter in directum eodem motu, quem haberet, si nulla adesset mutua punctorum actio in
se invicem. Hoc autem theorema sic generaliter, & admodum facile, ac luculenter demon-
stratur. [I22] Concipiamus vires singulas per quodvis determinatum tempusculum
servare directiones suas, & magnitudines : in fine ejus tempusculi punctum materiæ
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, & præterea unum, ac primo
tempusculo habeant omnia puncta motus debitos velocitatibus illis suis, quas habent initio
ipsius, singula singulos ; tum assignato quovis e sequentibus tempusculis cuivis binario,
habeat binarium quodvis tempusculo sibi respondente motum debitum vi mutuæ, quæ
agit inter bina ejus puncta, ceteris omnibus quiescentibus. In fine postremi tempusculi
omnia puncta materiæ 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. -
261. Concipiatur jam ultra omnia ejusmodi puncta planum quodcunque. Primo
ex illis tot assumptis tempusculis alia puncta accedent, alia recedent ab eo plano, & summa
omnium accessuum punctorum omnium demptis omnibus recessibus, si qua superest, vel
vice versa summa recessuum demptis accessibus, divisa per numerum omnium punctorum,
æquabitur accessui perpendiculari ad idem planum, vel recessui centri gravitatis communis ;
cum summa distantiarum perpendicularium tam 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 plano nunquam mutata ; quia ob æquales & contrarios punctorum
motus, alterius accessus ab alterius recessu æquali eliditur. Quamobrem in fine omnium
tempusculorum ejus distantia erit eadem, & accessus ad planum erit idem, qui esset, si
solæ adfuissent ejusmodi velocitates, quæ habebantur initio ; adeoque etiam cum omnes
vires simul agunt, in fine illius unici tempusculi habebitur distantia, quæ haberetur, si
vires illæ mutuæ non egissent, & accessus æquabitur summæ accessuum, qui haberentur ex
solis velocitatibus, demptis recessibus. Si jam consideretur secundum tempusculum in
quo simul agant vires mutuæ, & velocitates ; debebunt considerari tria genera motuum :
primum eorum, qui proveniunt a velocitatibus, quæ 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,
quæ ob mutatas jam positiones concipiantur aliis, directionibus agere per totum secundum
tempusculum. Porro quoniam hi posteriorum duorum generum motus [I23] sunt in
singulis punctorum binariis contrarii, & æquales ; illi itidem distantiam centri gravitatis
ab eodem plano, & 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
of Newton, is exactly similar ; in it the two noted above are combined, & come to the
same thing. If any number of points of matter, disposed in any manner, {5 coalescing
to form any number of ſeparate masses in any manner, have any velocities in any direction ;
{5 if, in addition, the points are under the influence of any mutual forces whatever, these forces
acting on each pair of points equally in opposite directions ; then the common centre of gravity
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
place, for these little intervals of time of which we have assumed the number stated, some
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
Theorem relating
to the permanent
State of the centre
of gravity even
when there are
mutu al for c e S
acting ; the first
steps of the proof.
Continuation of
the demonstration.
2O 2 PHILOSOPHIAE NATURALIS THEORIA
Progressus ulterior.
Demonstra t i O n is
finis.
Coralla, r i u m d e
quantitate motus
in eandem plagam
c o n s e r v a, t a, in
Mundo.
Æqualitas actionis
& reactionis in
massis inde orta.
sed ea habentur, sicuti haberentur, si semper durarent solæ illæ velocitates, quæ 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 præcedentia tempuscula, ex
quibus omnibus, & ex nova actione vis mutuæ, componitur quovis tempusculo motus
puncti cujusvis : sed omnia ista inducunt motus contrarios, & æquales, adeoque summa
accessuum, vel recessuum ortam ab illis solis initialibus velocitatibus non mutant.
262. Quod si jam tempusculorum magnitudo minuatur in infinitum, aucto itidem
in infinitum intra quodvis finitum tempus eorundem numero, donec 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 plano distantiam perpendicularem, quam haberet ex solis
velocitatibus habitis initio ejus temporis, si nullæ deinde egissent mutuæ vires ; & accessus
ad illud planum, vel recessus ab eo, æquabitur summæ 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 æquabili, sunt in ratione eadem earundem temporis
partium ; ac proinde & eorum summæ in eadem ratione sunt.
263. Inde vero prona jam est theorematis demonstratio. Ponamus enim, centrum
gravitatis quiescere quodam tempore, tum moveri per aliquod aliud tempus. Debebit
utique aliquo momento temporis esse in alio loci puncto, diverso 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, tum accepto plano aliquo perpendiculari ipsi productæ
ultra omnia puncta, centrum gravitatis ad id planum accederet secunda continui ejus
temporis parte per intervallum æquale illi rectæ, & 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 quæ transit, non jacent in
directum, sumantur tria in dire-[I24]-ctum non jacentia, & ducatur recta per prima duo,
quæ per tertium non transibit, adeoque per ipsam duci poterit planum, quod non transeat
per tertium, tum 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 æquale distantiæ
a priore plano, adeoque accessus iterum proportionales temporibus non fuissent. Demum
motus erit æquabilis. Si enim ultra omnia puncta concipiatur planum perpendiculare
rectæ, per quam movetur ipsum centrum commune gravitatis, jacens ad eam 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.
264. Ex eodem fonte, ex quo profluxit hoc generale theorema, sponte fluit hoc aliud
ut consectarium : quantitas motus in Mundo conservatur semper eadem, si ea computetur
secundum directionem quacumque ita, ut motus secundum directionem oppositam consideretur
ut megativus, ejusmodi motuum contrariorum summa subtracta a summa directorum. Si enim
consideretur eidem directioni perpendiculare planum ultra omnia materiæ puncta,
quantitas motus in ea directione est summa omnium accessuum, demptis omnibus recessibus,
quæ summa tempusculis æqualibus manet eadem, cum mutuæ vires inducant accessus,
& recessus se mutuo destruentes ; nec ejusmodi conservationi obsunt liberi motus ab anima
nostra producti, cum nec ipsa vires ullas possit exerere, nisi quæ agant in partes oppositas
æqualiter juxta num. 74.
265. Porro ex illo Newtoniano theoremate statim jam profluit lex actionis, & reactionis
æqualium pro massis omnibus. Nimirum si duæ massæ quæcunque in se invicem agant
viribus quibuscunque mutuis, & inter singula punctorum binaria æqualibus ; binæ illæ
A THEORY OF NATURAL PHILOSOPHY 2O3
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
of intervals in any given finite time being thus indefinitely increased, until we acquire
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.
For, let us suppose that the centre of gravity is at rest for a certain time, & then moves
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,
there is obtained immediately the following also, as a corollary. The quantity of motion
in the Universe is maintained always the same, so long as it is computed in some given direction
in such a way that motion in the oppoſite direction is considered negative, 85 the sum of the
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
action & reaction for all masses. Thus, if any two masses act upon one another with any
mutual forces, which are also equal for each pair of points, the two masses will acquire,
Further steps in
the demonstration.
Conclusion of the
demonstration.
Corollary with
regard to the
Conservation of the
quantity of motion
in the Universe in
a given direction.
Equality of action
re a cition for
masses the result
of this theorem.
2o4 PHILOSOPHIÆ NATURALIS THEORIA
, Inde leges collisi-
onum : discrimen,
virium in corpori-
b u s e la sticis, &
mollibus.
Præparatio pro col-
lisionibus globorum,
planorum, circu-
lorum.
massæ acquirent ab actionibus mutuis summas motuum æquales in partes contrarias, &
celeritates acquisitæ ab earum centris gravitatis in partes oppositas, componendæ 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 inertiæ 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 præter
priores motus ex vi inertiæ uniformes, ob actionem mutuam adhuc magis ad hoc commune
centrum accedet alterum ex iis, vel ab eo recedet ; accedet & alterum, [I25] vel recedet,
accessibus, vel recessibus reciproce proportionalibus ipsis massis. Nam accessus ipsi, vel
recessus, sunt differentiæ distantiarum habitarum cum actione mutuarum virium a distantiis
habendis sine iis, adeoque erunt & ipsi in ratione reciproca massarum, in qua sunt totæ
distantiæ. Quod si per centrum commune gravitatis concipiatur planum quodcumque,
cui quæpiam data directio non sit parallela ; summa accessuum, vel recessuum punctorum
omnium massæ utriuslibet ad ipsum secundum eam directionem demptis oppositis, quæ
est summa motuum secundum directionem eandem, æquabitur accessui, vel recessui centri
gravitatis ejus massæ 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, quæ jungit
centra massarum, sunt in ratione reciproca ipsarum massarum. Quare productum accessus,
vel recessus centri primæ massæ per numerum punctorum, quæ habentur in ipsa, æquatur
producto accessus, vel recessus secundæ per numerum punctorum, quæ in ipsa continentur ;
nimirum ipsæ motuum summæ in illa directione computatorum æquales sunt inter se,
in quo ipsa actionis, & reactionis æqualitas est sita.
266. Ex hac actionum, & reactionum æqualitate sponte profluunt leges collisionis
corporum, quas ex hoc ipso principio Wrennus olim, Hugenius, & Wallisius invenerunt
simul, ut in hac ipsa lege Naturæ exponenda Newtonus etiam memorat Principiorum
lib. I. Ostendam autem, quo pacto generales formulæ inde deducantur tam pro directis
collisionibus corporum mollium, quam pro perfecte, vel pro imperfecte elasticorum.
Corpora mollia dicuntur ea, quæ resistunt mutationi figuræ, seu compressioni, sed compressa
nullam exercent vim ad figuram recuperandam, ut est cera, vel sebum : corpora elastica,
quæ figuram amissam recuperare nituntur ; & si vis ad recuperandam sit æqualis vi ad
non amittendam ; dicuntur perfecte elastica, quæ quidem, ut & perfecte mollia, nulla,
ut arbitror, sunt in Natura ; si autem imperfecte elastica sunt, vis, quæ in amittenda,
ad vim, quæ in recuperanda figura exercetur, datam aliquam rationem habet. Addi solet
& tertium corporum genus, quæ dura dicunt, quæ 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, quæ
minus, ac minus comprimantur, donec compressio evadat nulla ; & ita quæ de mollibus
dicentur, aptari poterunt duris multo meliore jure, quam alii elasticorum leges ad ipsa
transferant, considerando elasticitatem infinitam ita, ut figura nec mutetur, nec se restituat ;
[126] nam si figura non mutetur, adhuc concipi poterit, impenetrabilitatis vi amissus
motus, ut amitteretur in compressione ; sed ad supplendam vim, quæ exeritur ab elasticis
in recuperanda figura, non est, quod concipi possit, ubi figura recuperari non debet. Porro
unde corpora mollia sint, vel elastica hic non quæro ; id pertinet ad tertiam partem,
quanquam id ipsum innui superius num. I99 ; sed leges quæ in eorum collisionibus observari
debent, & ex superiore theoremate fluunt, expono. Ut autem simplicior evadat res,
considerabo globos, atque hos ipsos circumquaque circa centrum, in eadem saltem ab
ipso centro distantia, homogeneos, qui primo quidem concurrant directe ; nam deinde
ad obliquas etiam collisiones faciemus gradum.
267. Porro ubi globus in globum agit, & ambo paribus a centro distantiis homogenei
sunt, facile constat, vim mutuam, quæ est summa omnium virium, qua singula alterius
puncta agunt in singula puncta alterius, habituram semper directionem, quæ jungit centra ;
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
collision of bodies, which some time ago Wren, Huygens & Wallis derived from this very
principle at about the same time, as Newton also mentioned in the first book of the Principia,
when expounding this law of Nature. Now I will show how general formulae may be
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
at equal distances from their centres, it is readily shown that the mutual force, which is
the sum of all the forces with which each of the points of the one acts on each of the points
of the other, must always be in the direction of the line joining the two centres. For,
Hence the laws for
Collision ; the dis-
tinction be tween
the forces for elas-
tic bodies & soft
bodies.
Preparation for the
consideration of
collisions of spheres,
planes & circles.
2o6 PHILOSOPHIÆ NATURALIS THEORIA
Formulæ pro cor-
pore molli incur-
re n t e in m O 1 1 e
lentius progrediens
in eandem plagam.
Ejus extensio ad
Omnes casus : cele-
rit a s amissa, vel
acquisita.
Transitus ad elas-
ticorum collisiones.
nam in ea recta jacent centra ipsorum globorum, quæ 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,
quæ 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 quævis teres, sive circa axem quendam
rotunda, & in quovis plano perpendiculari axi homogenea, vel etiam circulus simplex,
agat, vel concipiatur agens in globum, vel punctum in ipso axe constitutum ; res eodem
redit.
268. Præcurrat jam globus mollis cum velocitate minore, quem alius itidem mollis
consequatur cum majore ita, ut centra ferantur in eadem recta, quæ illa conjungit, & hic
demum incurrat in illum, quæ dicitur collisio directa. Is incursus mihi quidem non fiet
per immediatum contactum, sed antequam ad contactum deveniant, vi mutua repulsiva
comprimentur partes posteriores præcedentis, & anteriores sequentis, quæ compressio fiet
semper major, donec ad æquales celeritates devenerint ; tum enim accessus ulterior desinet,
adeoque & ulterior compressio ; & quoniam corpora sunt mollia, nullam aliam exercent
vim mutuam post ejusmodi compressionem, sed cum æquali illa velocitate pergunt moveri
porro. Hæc æqualitas velocitatis, ad quam reducuntur ii duo glo-[127]-bi, una cum
æqualiate actionis, & reactionis æqualium, rem totam perficient. Sit enim massa, sive
quantitas materiæ, globi præcurrentis = 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 æquales, hæc quantitas erit eadem etiam post collisionem, post
quam motus totus utriusque massæ, erit CQ + cq. Quoniam autem progrediuntur cum
æquali celeritate ; celeritas illa habebitur ; si quantitas motus dividatur per totam
� • e * , e ... CQ + c ■ • ę
quantitatem materiæ ; quæ idcirco erit $i;'. Nimirum ad habendam velocitatem
communem post collisionem, oportebit ducere singulas massas in suas celeritates, &
productorum summam dividere per summam massarum.
269. Si alter globus q quiescat ; satis erit illius celeritatem c considerare = o : & si
moveatur motu contrario motui prioris globi ; satis erit illi valorem negativum tribuere ;
ut adeo & hic, & in sequentibus formula inventa pro illo primo casu globorum in eandem
progredientium plagam, 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 + 4, & CQ + 4 ,
- - Q + q Q + q
ad eundem denominatorem, ac habebitur -
Cq — cq, s. CQ—cQ
Q + q QTq?
ex quibus deducitur hujusmodi theorema : ut summa massarum ad massam alteram, ita
differentia celeritatum ad celeritatem ab altera acquisitam, quæ in eo casu accelerabit motum
præcurrentis & retardabit motum consequentis.
27o. Ex hisce, quæ pertinent ad corpora mollia, facile est progredi ad perfecte elastica.
In iis post compressionem maximam, & mutationem figuræ inductam ab ipsa, quæ habetur,
ubi ad æquales velocitates est ventum, agent adhuc in se invicem bini globi, donec deveniant
ad figuram priorem, & hæc actio duplicabit effectum priorem. Ubi ad sphæricam figuram
deventum fuerit, quod fit recessu mutuo oppositarum superficierum, quæ in compressione
ad se invicem accesserant, pergent utique a se invicem recedere aliquanto magis eædem
superficies, & figura producetur, sed opposita jam vi mutua inter partes ejusdem globi
incipient retrahi, & productio perget fieri, sed usque lentius, donec ad maximam quandam
productionem de-[I28]-ventum fuerit, quæ deinde incipiet minui, & globus ad sphæricam
accedet iterum, ac iterum comprimetur quodam oscillatorio, ac partium trepidatione
hinc, & inde a figura sphærica, 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
body which is following it with a greater velocity, in such a manner that their centres are
travelling in the same straight line, namely that which joins them ; & finally let the latter
impinge upon the former ; this is termed direct impact. This impact, in my opinion
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 + cq)/(Q + 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
equal to zero; also, if it is moving in a direction opposite to that of the first sphere, we
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 + cq)/(Q -- g) & (CQ + cq)/(Q + q) – c to a common denominator, when we
shall obtain the formulae (C4— 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
are perfectly elastic. For such bodies, after the maximum compression has taken place,
& 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
Formulae for a soft
body impinging
upon another Soft
body proceeding
more slowly in the
Same direction.
Extension to all
Cases; velocity lost
Or gained.
Transition to im-
pact between elas-
tic bodies.
2o8 PHILOSOPHIÆ NATURALIS THEORMA
Formulæ pro per-
fecte elasticis.
Formulæ pro im-
perfecte elasticis.
C a su s, in qu o
g 1 o b u s perfecte
elasticus incurrit in
alium.
cohæsionis 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 sphæricam, post quem superficies
alterius postica & alterius antica in centra jam retractæ 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 mutuæ actionis effectus in
recuperanda, quantus fuit in amittenda figura.
27 I. Duplicato igitur effectu, globus ammittet celeritatem aC — 2a, & globus q
Q + q
eg:o, Quare illius celeritas post collisionem erit C —
CQ — Cq + 2cq.
ę e � 2CQ—2cQ, _ cq — cQ + 2CQ
sive — s;;— ; hujus vero erit c -+- IQIA • — Q + q
eandem plagam, vel globus alter quiescet, vel fient in plagas oppositas ; prout determinatis
valoribus Q,q, C, c, formulæ valor evaserit positivus, nullus, vel negativus.
272. Quod si elasticitas fuerit imperfecta, & vis in amittenda ad vim in recuperanda
figura fuerit in aliqua ratione data, erit & effectus prioris ad effectum posterioris itidem in
ratione data, nimirum in ratione subduplicata prioris. Nam ubi per idem spatium agunt
vires, & velocitas oritur, vel extinguitur tota, ut hic respectiva velocitas extinguitur in
compressione, oritur in restitutione figuræ, quadrata velocitatum sunt ut areæ, quas
describunt ordinatæ viribus proportionales juxta num. I76, & hinc areæ erunt in ratione
virium, si, viribus constantibus, sint constantes & ordinatæ, cum inde fiat, ut scalæ celeri-
tatum ab iis descriptæ sint rectangula. Sit igitur rationis constantis illarum virium ratio
subduplicata m ad m, & erit effectus in amittenda figura ad summam effectuum in tota
^m -+- m
2Cq — 2cq
Q + q
, & motus fient in
acquiret celeritatem
satis erit,
collisione, ut m ad m -+- m, quæ ratio si ponantur esse I ad r, ut sit r =
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 acquisitæ in partes contrarias, & componendæ cum velocitatibus [I29] prioribus.
rCq — rcq
TQTqT°
, adeoque velocitas illius post congressum erit C —
Erit nimirum illa quæ pertinet ad globum Q =
rCQ — rcQ
361, 6
ę - qr — 7" 6
hujus c -+- TQTqT -
ac tum ex hisce formulis, tum e superioribus quam plurima elegantissima theoremata
deducuntur, quæ quidem passim inveniuntur in elementaribus libris, & ego ipse aliquanto
uberius persecutus sum in Supplementis Stayanis ad lib. 2, § 2 ; sed hic satis est, fundamenta
ipsa, & primarias formulas derivasse ex eadem Theoria, & ex proprietatibus centri gravitatis,
ac motuum oppositorum æqualium, deductis ex Theoria eadem ; nec nisi binos, vel ternos
evolvam casus usui futuros infra, antequam ad obliquam collisionem, ac reflexionem motuum
gradum faciam.
& quæ pertinet ad globum
rCq — rcq
Q + q ? &
; quæ formulæ itidem reducuntur ad eosdem denominatores ;
q, erit =
273. Si globus perfecte elasticus incurrat in globum itidem quiescentem, erit, c = o,
«• τ* • e» • e» C -_- •
adeoque velocitas contraria priori pertinens ad incurrentem, quæ erat *£*, er1t
2Cq (• � � • 2CQ — 2cQ «• 2CQ
• –—–. er1t – :
Q + q * velocitas acquisita a quiescente, quæ erat Q + q ? Q + q ? unde
habebitur hoc theorema : ut summa massarum ad duplam ma$$am quiescentis, vel incurrentis,
ita celeritas incurrentis, ad celeritatem amissam a secundo, vel acquisitam a primo ; & si
Casus triplex globi
incurrentis in pla-
num immobile.
massæ æquales fuerint, fit ea ratio æqualitatis ; ac proinde globus incurrens totam suam
velocitatem amittit, acquirendo nimirum æqualem contrariam, a qua ea elidatur, & globus
quiescens acquirit velocitatem, quam ante habuerat globus incurrens.
274. Si globus imperfecte elasticus incurrat in globum quiescentem immensum, &
qui hábeatur pro absolute infinito, cujus idcirco superficies habetur pro plana, in formula
rCQ — rcQ
Q +
infiniti, & proinde s*; evadat = o, tota formula evanescit, adeoque ipse haberi potest
pro plano immobili. In formula vero velocitatis, quam in partem oppositam acquiret
- rCq — rcq
Q + q *
Hinc habetur rj, sive rC, nimirum ob r =
velocitatis acquisitæ a globo quiescente , cum evanescat Q respectu q absolute
globus incurrens, evadit c = o, [130] & Q evanescit itidem respectu q.
qm —H 11 fit (* -{- *)« C, cujus prima pars £; X C,
7;
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
(2C4 – 264)/(Q + q), & the sphere q will gain a velocity equal to (2CQ — 2C)/(Q + q).
Hence, the velocity of the former after impact will be C – (2C1 – 2cq)/(Q+ q) or
(CQ – C4 + 2.c4)/(Q-H 4), & the velocity of the latter will be c + (2CQ – 2cQ)/(Q -- q)
or (cq – c.9 + 2CO)/(Q + q). The motions will be in the original direction, or one of the
spheres may come to rest, or the motions may be 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
some given ratio to the force during recovery of shape, then the effect corresponding to
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 1 : r, so that r = (m+n)/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 (rC4 – req)/(Q + q), & that for the sphere q will be (rCQ — re())/(Q + q).
Hence, the velocity of the former after impact will be C – (rCº. — req)/(Q + q) & the
velocity of the latter will be c + (rCQ — reQ)/(Q -- q); & these formulae also can be
reduced to common denominators. From these formulae, 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
c = 0, & the velocity, in the direction opposite to the original velocity, for the striking
body, which was (2C4 – 2cq)/(Q + q), will in this case be 2C4/(Q -- 4); whilst the velocity
gained by the body that was at rest, which was shown to be (2CQ – 26Q)/(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
may be considered as absolutely infinite, & therefore its surface may be taken to be a plane;
then, in the formula for the velocity gained by a sphere at rest, (rCQ — reC)/(Q -- 4),
since Q vanishes in comparison with q which is absolutely infinite, & thus Q/(Q + q) = 0,
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 — req)/(Q + q), we have c = 0,
& Q also vanishes in comparison with q. Hence we obtain rCq/q, or rC; that is to say,
since r = (m -- n)/m, we have C X (m -- n)/m, of which the first part, C X m/m, or C,
P
Formula for per-
fectly elastic bodies.
Formulae for imper-
fectly elastic bodies.
Case of a perfectly
elastic sphere strik-
ing another.
Threefold case of
a Sphere impinging
On an immovable
plane.
2 IO PHILOSOPHIÆ NATURALIS THEORIA
Summa q u a d ra-
torum velocitatis
ductorum in massas
manens in perfecte
elasticis,
Collisionis obliquæ
communis metho-
du s p er virium
resolutionem.
Compositio virium
resolu t i o n i s u b-
stituta.
sive C, est illa, quæ amittitur, sive acquiritur in partem oppositam in comprimenda figura,
& ; × C est illa, quæ acquiritur in recuperanda, ubi si fit n = o, quod accidit nimirum
in perfecte mollibus; habetur sola pars prima; si m = m, quod accidit in perfecte elasticis,
eSt £; × C = C, secunda pars æqualis primæ; & in reliquis casibus est, ut m ad m, ita
illa pars prima C, sive præcedens velocitas, quæ per primam partem acquisitam eliditur,
ad partem secundam, quæ remanet in plagam oppositam. Quamobrem habetur ejusmodi
theorema. Si incurrat ad perpendiculum in plamum immobile globus perfecte mollis, acquirit
velocitatem contrariam aequalem suae priori, & quiescat ; si perfecte elasticus, acquirit duplam
suae, nimirum æqualem in compre$sione, qua motus omnis sistitur, & æqualem in recuperanda
figura, cum qua resilit ; si fuerit imperfecte elasticus in ratione m ad n, in illa eadem ratione
erit velocitas priori suae contraria acquisita, dum figura mutatur, quae priorem ipsam velocitatem
extinguit, ad velocitatem, quam acquirit, dum figura restituitur, & 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, quæ fuerat ante ipsum. Nam velocitates
_ 24 - 2Q .__ /* \ •
post congressum sunt C σΊ, * (C — c), & e + dii, x (C— c) ; quadrata ducta
in massas continent singula ternos terminos : primi erunt QCC-+- qcc ; secundi erunt
(— CC + Cc) × ę, & (cC — cc) × ę, quorum summa evadit g eo -+- 2Cc—
4Qq . se 4Qqq 44 a-A
cc) X ā,; postrem erunt ÖÄ ^ (CC — 2Cc -+- cc), & (ÖTj ^ (CC
2Cc -+- cc), sive simul *;£;o, [I3I] X (CC — 2Cc+ cc), vel ę, (CC—2Cc-+- cc),
quod destruit summam secundi terminorum binarii, remanente sola illa QCC -+- qcc,
summa quadratorum velocitatum præcedentium ducta in massas. Sed hæc æqualitas nec
habetur in mollibus, nec in imperfecte elasticis.
276. Veniendo jam ad congressus obliquos, deveniant dato tempore bini globi A, C
in fig. 42 per rectas quascunque AB, CD, quæ illorum velocitates metiantur, ia B, & D ad
physicum contactum, in quo jam sensibi-
lem effectum edunt vires mutuæ. Com- M l-___ ___ O
muni methodo collisionis effectus sic de- z T\R Q P ;•
finitur. Junctis eorum centris per rectam A. W /
BD, ducantur, ad eam productam, qua A. \,
opus est, perpendicula AF, CH, & com- /
pletis rectangulis AFBE, CHDG resolvan- „^ \ \
tur singuli motus AB, CD in binos ; ille , ^ k~% K \
quidem in AF, AE, sive EB, FB, hic vero 4 - -
in CH, CG, sive GD, HD. Primus utro- IN IF I X £/\ D H S
bique manet illæsus ; secundus FB, & HD *-. - B |
collisionem facit directam. Inveniantur
per legem collisionis directæ velocitates BI,
DK, quæ juxta ejusmodi leges superius
expositas haberentur post collisionem di- A E. G C.
versæ pro diversis corporum speciebus, -
& componantur cum velocitatibus expositis
per rectas BL, DQ jacentes in directum cum EB, GD, & illis æquales. His peractis
expriment BM, DP celeritates, ac directiones motuum post collisionem.
277. Hoc pacto consideratur resolutio motuum, ut vera quædam resolutio in duos,
quorum alter illæsus perseveret, alter mutationem patiatur, ac in casu, quem figura exprimit,
extinguatur penitus, tum iterum alius producatur. At sine ulla vera resolutione res
vere accidit hoc pacto. Mutua vis, quæ agit in globos B, D, dat illis toto collisionis tempore
velocitates contrarias BN, DS æquales in casu, quem figura exprimit, binis illis, quarum
altera vulgo concipitur ut elisa, altera ut renascens. Eæ compositæ cum BO, DR jacentibus
in directum cum AB, CD, & æqualibus iis ipsis, adeoque exprimentibus effectus integros
præcedentium velocitatum, exhibent illas ipsas velocitates BM, DP. Facile enim patet,
fore LO æqualem AE, sive FB, adeoque MO æqualem 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 inertiæ, & ea composita cum nova velocitate, quam generant vires, quæ
agunt in collisione,
/
FIG. 42.
A THEORY OF NATURAL PHILOSOPHY - 2 II
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 = 0, which is the case for perfectly soft bodies, there is only the first part ;
if m = m, 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. If 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
& elegant theorem, discovered by Huygens for perfectly elastic solids. Namely, that the
sum of the squares of the velocities, each multiplied by the corresponding mass, remains
the same after the impact as it was before it. Now, the velocities after impact are
C — o: q × (C – ), & c + 20 × (C – 6); the squares of these, multiplied by the
masses contain three terms each; the first are QCC & qcc : the second are
(— CC + C) × 4Q4 & (cC — ce) × 4Qq & the sum of these reduces to
Q —H q Q -- q’
(— CC+ 2Ce — cc) × dº; the last are º, (CC – 2CC + cc), & *; X
(CC — 2C. + cc), or added together 4(Q wº ; Qq × (CC – 2Cc + cc), or
d; × (CC – 2Co+ ce), which will cancel the sum of the second terms; hence all
that remains is QCC + qoc, 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 &
Cat 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
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
a certain real resolution into two parts, of which the one part persisted unchanged, & the
other part suffered alteration; & in 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.
The sum of the
squares of the
velocities, e a ch
multiplied by the
corresponding mass,
remains unaltered
in the case of
perfectly elastic
bodies.
The usual method
for oblique impact
by means of reso-
lution of forces.
Composition of
forces substituted
for resolution.
2I2 PHILOSOPHIÆ NATURALIS THEORIA
$sit, ii); 278. Idem etiam mihi accidit, ubi oblique globus incurrit in planum, sive consideretur
£tiam, ubi globus motus, qui haberi debet deinde, sive percussionis obliquæ energia respectu perpendicularis.
#i' P'*""" Deveniat in fig. 43 globus A cum directione obliqua AB ad planum [i32] CD consideratum
- utimmobile, quod contingat physice in N, & concipiatur planùm GI päräïléîum priori ductum
per centrum B, ad quod appellet ipsum centrum, & à 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 illæsam, secundam destrui a
resistentia plani : tum perseverare illam
solam per BI æqualem ipsi FB; si corpus
incurrens sit perfecte molle, vel componi
cum alia in perfecte elasticis BE æquali
priori EB, in imperfecte elasticis Be,
quæ ad priorem EB habeat rationem
datam, & percurrere in primo casu BI,
in secundo BM, in tertio Bm. At in
mea Theoria globus a viribus in illa
minima distantia agentibus, quæ ibi sunt
repulsivæ, acquirit secundum direc-
tionem NE perpendicularem plano re- L.
pellenti CD in primo casu velocitatem
BE, æqualem illi, quam acquireret, si � ce ζ »
cum velocitate EB perpendiculariter advenisset per EB, in secundo BL ejus duplam, in
tertio BP, quæ ad ipsam habeat illam rationem datam r ad I, sive m -+- m ad m, & habet
qeinde velocitatem compositam ex velocitate priore manente, ac expressa per BO æqualem
AB, & positam ipsi in directum, ac ex altera BE, BL, BP, ex quibus constat, componi illas
ipsas BI, BM, Bm, quas prius ; cum ob IO æqualem AF, sive EB, & IM, Im æquales BE,
Be, sive EL, EP, totæ etiam BE, BP, BL totis OI, OM, Om sint æquales, & parallelæ. «•
Ubique in hac 279. Res mihi per compositionem virium ubique eodem redit, quo in communi
É „£t methödo per earumi resolutionem. Resolutionem $olent vulgo admittere in motibus,
Š, e$$e quos vQcant impeditos, ubi vel planum subjectum, vel ripa ad latus procursum impediens,
É'*" **" iit in fluviorumi alveis, vel filuih, aut virgâ sustentans, üt in pendülorum oscillationibus,
impedit motum secundum eam directionem, qua agunt, velocitates jam conceptæ, vel
vires ; ut & virium resolutionem agnoscunt, ubi binæ, 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 plano 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 illa unica velocitas, vel vis æquivaleat, 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,
nec a virga vere rigida, & inflexili sustentante, sed semper a viribus mutuis, repulsivis in
primo casu, attractivis in secundo ; semper habebitur nova velocitas, vel vis æqualis, &
cQntraria illi, quam communis methodus elisam dicit, quæ cum [133] tota yelocitate, yel
vi, obliqua composita eundem motum, vel idem æquilibrium restituet, ac idem omnino
erit, in effectuum computatione considerare partes illas binas, & alteram, vel utramque
impeditam, ac considerare priorem totam, aut velocitatem, aut vim, cQmpositam cum iis
novis contrariis, & æqualibus illi parti, vel illis partibus, quæ dicebantur elidi. In id autem,
quod yel inferne, vel superne motum massæ cujuspiam impedit, yel vim, non aget pars
illa prioris velocitatis, vel illius vis, quæ concipitur resoluta, sed velocitas orta a vi mutua,
& contraria velocitati illi novæ genitæ in eadem massa, a vi mutua, vel ipsa vis mutua, quæ
semper debet agere in partes contrarias, & cui occasionem præbet illa determinata distantia
major, vel minor, quam sit, quæ limites, & æquilibrium constitueret.
FIG. 43.
ɤ, ";.; 28o. Id quidem abunde apparet in ipso superiore exemplo. Ibi in fig. 43 globus
figur£te,i5 pla- (quem concipamus mollem) advenit oblique per AB, & oblique impeditur a plano ejus
num immobile. progressus. Non est velocitas perpendicularis AF, vel EB, quæ extinguitur, durante AE,
vel FB, uti diximus ; nec illa ursit planum CD. Velocitas AB occasionem dedit globo
accedendi ad planum CD usque ad eam exiguam distantiam, in qua vires variæ 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 Be, 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 Bm. 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 + 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 Bm,
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, beingas 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
Composition sub-
Stituted also for
resolution in 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.
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 ;
A case in point
where a soft sphere
impinges on a n
immovable plane.
2 I4 PHILOSOPHIAE NATURALIS THEORIA
Aliud globi descen-
dentis per planum
inclinatum.
Aliud in pendulo.
Alia, ratio com-
ponendi vires in
eodem casu.
Aliud exemplum in
globo sustentato a.
binis planis. Diffi-
cultas communis
methodi in eodem.
larem tangenti. At in mea Theoria res hoc pacto
donec ex omnibus actionibus conjunctis impediretur accessus ad ipsum planum, sive
perpendicularis distantiæ ulterior diminutio. Illæ vires agent simul in directione perpen-
diculari ad ipsum planum juxta num. 266: debebunt autem, ut impediant ejusmodi
ulteriorem accessum, producere in ipso globo velocitatem, quæ composita cum tota BO
perseverante in eadem directione AB, exhibeat velocitatem per BI parallelam CD. Quoniam
vero triangula rectangula AFB, BIO æqualia erunt necessario ob AB, BO æquales ; erit
BEIO parallelogrammum, adeoque velocitas perpendicularis, quae cum priore velocitate
BO debeat componere velocitatem per rectam parallelam plano, debebit necessario esse
contraria, & æqualis illi ipsi EB perpendiculari eidem plano, in quam resolvunt vulgo
velocitatem AB. Interea vero vis, quæ semper agit in partes contrarias æqualiter, urserit
planum tantundem, & omnes in eo produxerit effectus illos, qui vulgo tribuuntur globo
advenienti cum velocitate ejusmodi, ut perpendicularis ejus pars sit EB.
28 I. 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 - E
plano CD, qua urgetur ipsum planum, quod eum N /
sustinet ; alteram BI, parallelam eidem plano, N •=-/ '
quæ obliquum descensum accelerat. In mea *\ N 4%
Theoria gravitas cogit globum semper magis ac-
cedere ad planum CD ; donec distantia ab eodem
evadat ejusmodi ; ut vires mutuæ [I34] repul- / J*\ \ . '
sivæ agant, & illa quidem, quæ agit in B, sit / N » I
ejusmodi ut composita cum BO exhibeat BI R« -
parallelam plano ipsi, adeoque non inducentem *\
ulteriorem accessum, sit autem perpendicularis \ j A
plano ipsi. Porro ejusmodi est BE, jacens in \ 'O
directum cum RB, & ipsi æqualis, cum nimirum
debeat esse parallela, & æqualis OI. Vis autem
æqualis 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 ; tum 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 præterea etiam tensionem fili agnoscunt ortum -
a vi centrifuga, quæ exprimitur per DA perpendicu- >7C
FIG. 44.
procedit. Globus ex B abit ad D per vires tres com-
positas simul cum velocitate præcedente ; prima e
viribus est vis gravitatis BO ; secunda attractio versus
C orta a tensione fili, vel virgæ, expressa per BE paral-
lelam, & æqualem OI, adeoque RB, quæ solæ compo-
nerent vim BI; tertia est attractio in C expressa per
BH æqualem AD orta itidem a tensione fili respond-
ente vicentrifugæ, & incurvante motum. Adest præ-
terea velocitas præcedens, quam exprimit BK æqualis
IA, ut sit BI æqualis 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 aliæ exillis causis agerent post alias : gravitate agente
veniret per BO, vi BE abiret per OI, velocitate BK abiret per IA ipsi æqualem, vi BH
abiret per AD. Quamobrem res tota itidem peragitur sola compositione virium, & motuum.
283. Porro si sumatur EG æqualis BH ; tum tota attractio orta a tensione fili erit
BG, quæ 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), tum 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
meæ Theoriæ cum omnium usitatis Mechanicæ elementis communia sunt, posteaquam
compositionis hujus cum illa resolutione æquivalentia est demonstrata.
284. Quæ de motu diximus facto vi oblique, sed non penitus impedita, eadem in
æquilibrio habent locum, ubi omnis impeditur motus. Innitatur globus gravis B in fig.
46 binis planis AC, CD, quæ accurate, vel in mea Theoria [I35] physice solum, contingat
FIG. 45.



A THEORY OF NATURAL PHILOSOPHY 2 I 5
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 straightline 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
art is EB. -
p 281. The same thing happens also in the rest of the cases mentioned above. Let a
heavy sphere descend along the inclined plane CD, in Fig. 44; the descent takes place,
according to the customary idea, in the following manner. Gravity, represented by BO,
is resolved into two parts, the one, BR, perpendicular to the plane CD, acts upon the 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
descend obliquely along the circular arc BD. Now, in the usual method, gravity, represented
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
the tension of the string will be BG, which previously was considered only as being com-
pounded of two parts acting in the same straight line ; & it comes to the same thing
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,
as those we have already spoken of with respect to motion derived from a force acting
obliquely, but not altogether impeded. In Fig. 46, a heavy sphere is supported by two
planes AC, CD, which actually, or in my Theory physically only, it touches at H & F ;
Another case in
point, that of a
sphere descending
along an inclined
plane.
The pendulum is
another case in
point.
Another manner of
compounding the
forces in the case
just considered.
Another example ;
a sphere supported
by two planes;
difficulty with the
usual method in
this case.
2 I6 PHILOSOPHIÆ NATURALIS THEORIA
in H, & F, & gravitatem referat recta verticalis BO, ac ex puncto O ad rectas BH, BF
ducantur rectæ OR, OI parallelæ ipsis BF, BH, & producta sursum BKtantundem, ducantur
ex K ipsis BF, BH parallelæ KE, KL usque ad easdem BH, BF; ac patet, fore rectas BE,
BL æquales, & cQntrarias BR, BI. In communi methodo resolutiohis 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 æqualis esse debet, cum uterque sit complementum HBF ad duos rectos, altér ob
parallelogrammum, alter ob angulos BHC, BFC rectos ; fieri potest, ut singula latera
Solutio in ipsa
methodo communi :
in hac Theoria
nullum ipsi difficul-
tati esse locum.
Aliud in globo sus-
penso filis obliquis.
Conclusio gene-
ralis pro hac
theoria, quæ omnia
præstat per solam
compositionem.
IResolutio tantum
mente c o n c e pta,
sæpe utilis ad con-
tra h e n d a s solu-
tiones.
BR, RO, sive BI, sint, quantum libuerit, longiora quam BO ; vires singulæ, quæ urgent
illa plana, possunt esse, quantum libuerit, majores, quam sola gravitas : mirafitur mülti,
fieri posse, ut gravitas per solam ejusmodi applicatiónem tantùm quodammodo supra se
assurgat, & effectum tanto majorem edat. -
285. Difficultas ejusmodi in communi etiam sententia evitari facile potest exemplo
vectis, de quo agemus infra, in quo sola applicatio vis in multo majore distantia collocátæ
multo majorem effectum edit. 'Verum in mea Theoria ne ullus quidem difficultati est
locus. Non resolvitur revera gravitas in duas vires BR, BI, quarum $ingulæ plana urgeant,
$ed gravitas inducit ejusmodi accessum ad ea plana, in quo vires repulsivæ pérpendiculares
ipsis planis agentes in globum componant vim BK æqualem, & contrariam gravitati BO,
quam sustineat, & ulteriorem accessum impediat. Ad id præstandum requiruntur illæ
vires BE, BL æquales, & contrariæ hisce BR, BI, quæ rem cónficiunt. Sed quoniam vires
sunt mutuæ, habebuntur repulsiones agentes in ipsa plana contrariæ, & æquales illis ipsis
BE, BL, adeoque agent vires expressæ per illas ipsas BR, BI, in quas communis methodus
gravitatem resolvit. -
286. Quod si globus gravis P in fig. 47 e filo BP pendeat, ac sustineatur ab obliquis
filis AB, DB, exprimat autem BH gravitatem, & sit BK ipsi contraria, & æqualis, ac sint
HI, KL parallelæ DB, & HR, KE parallelæ filo AB; communis methodus resolvit gravi-
tatem BH in duas BR, BI, quæ a filis sustineantur, & illa tendant ; sed ego compono vim
BK gravitati contrariam, & æqualem e viribus BE, BL, quas exerunt attractivas puncta
fili, quæ ob pondus P delatum deorsum sua gravitate ita distrahuntur a se invicem, donec
habeantur vires attractivæ componentes ejusmodi vim contrariam, & æqualem gravitati.
287. Quamobrem per omnia casuum diversorum genera pervagati jam vidimus, nullam
esse uspiam in mea Theoria veram aut virium, aut motuum resolutionem, sed omnia
prorsus phænomena pendere a sola compositione virium, & motuum, adeo-[I36]-que
naturam eodem ubique modo simplicissimo agere, componendo tantummodo vires, &
motus plures, sive edendo simul eum effectum, quem ederent illæ omnes causæ 3. si aliæ
post alias effectus ederent suos æquales, & eandem habentes directionem cum iis, quos
singulæ, si solæ essent, producerent. Et quidem id generale esse Theoriæ meæ, 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, quæ ab aliis omnibus pendent
materiæ punctis. i e • • ę
288. Quanquam autem habeatur revera sola compositio virium ; licebit adhuc vires
imaginatione nostra resolvere in plures, quod sæpe demonstrationes theorematum, &
solutionem problematum contrahet mirum in modum, ac expeditiores reddet, & elegant-
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 illæ vires compQnentes ; manifestum est, substitutione harum pro illa nihil
turbari conclusiones, quæ inde deducuntur : & si post resolutionem ejusmodi inveniatur
vis contraria, & æqualis alicui e viribus, in quas vis illa data resolvitur ; illa haberi potest
pro nulla consideratis solis reliquis, si in plures resoluta fuit, vel sola altera reliqua, si
resoluta fuit in duas. Nam componendo vim, quæ resolvitur, cum illa contraria uni ex iis,
in quas resolvitur, eadem vis provenire debet Qmnino, quæ oritur componendo , simul
reliquas, quæ fuerant in resolutione sociæ illius elisæ, 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

2I8
PHILOSOPHIAE NATURALIS THEORIA
[02
FIG. 47.


A THEORY OF NATURAL PHILOSOPHY 2 I9
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
by comparing the case of the lever, with which we will deal later; in it the mere application
of a force situated at a much greater distance gives a far greater effect. But with my
Theory there is no occasion for any difficulty of the sort. For there is no actual resolution
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
up by inclined strings AB, DB, & gravity is represented by BH; let BK be equal & opposite
to it, & let HI, KL be parallel to the string DB, & HR, KE parallel to the string AB. The
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
in my Theory any real resolution either of forces or of motions; but that all phenomena
depend on composition of forces & motions alone. Thus, nature in all cases acts in the
same most simple manner, by compounding many forces & motions only; that is to say,
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
we may mentally resolve our forces into several; & this will often shorten the proofs of
theorems & the solution of problems in a wonderful manner, & render them more elegant
& less cumbrous; for we may assume instead of a single force the forces from which it is
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.
A n swer to the
difficulty by the
ordinary method ;
in my Theory
there is no occasion
for any difficulty.
Explanation in the
case of a sphere
suspended by in-
clined strings.
General summing
up in favour of
this Theory, which
gives everything
by composition
alone.
Resolution, al-
though on ly a
mental fiction, is
yet often useful in
shortening, so lu-
tions.
22O PHILOSOPHIÆ NATURALIS THEORIA
Methodus generalis
resolvendi vim in
alias quotcunque.
Evolutio resolu-
t i o n is in d u a s.
tantum.
Cur vis composita,
sit minor com-
ponentibus simul
sumptis.
Cur ea crescere
videatur in reso-
lutione : nihil inde
posse deduci pro
viribus vivis.
Satis patere ex hac
T h e o r i a, V i ^ e s
Vivas in Naturæ
mullas eSSe.
289. Porro quod pertinet ad resolutionem in plures vires, vel motus, facile est ex
iis, quæ dicta sunt num. 257 definire legem, quæ ipsam resolutionem rite dirigat, ut
habeantur vires, quæ datam aliquam componant. Sit in fig. 48, vis quæcunque, vel motus
AP, & incipiendo ab A ducantur quotcunque, & cujuscunque longitudinis rectæ AB, BC,
CD, DE, EF, FG, GP, continuo inter se connexæ ita, ut incipiant ex A, ac desinant in P ;
& si ipsis BC, CD, &c. ducantur parallelæ, & æquales Ac, Ad, &c. ; vires omnes AB, Ac,
Ad, Ae, Af, Ag, Ap component vim AP ; unde patet illud : ad componendam vim
quamcunque posse assumi vires quotcunque, & quascunque, quibus assumptis determinari
poterit una alia præterea, quæ compositionem perficiat ; nam poterunt duci rectæ AB,
BC, CD, &c. parallelæ, & æquales datis quibuscunque, & ubi postremo deventum fuerit
ad aliquod punctum G, satis erit addere vim expressam per GP.
[I37] 290. Eo autem generali casu continetur particularis casus resolutionis in vires
tantummodo duas, quæ potest fieri per duo quævis latera trianguli 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, & æqualis BP, ac binæ vires AB, AC componunt vim AP: atque
idem dicendum de motibus.
29I. Ejusmodi resolutio illud etiam palam faciet ; cur vis composita a viribus non
in directum jacentibus, sit minor ipsis componentibus, quæ nimirum sunt ex parte sibi
invicem contrariæ, & elisis mutuo contrariis, & æqualibus, remanet in vi composita summa
virium conspirantium, vel differentia oppositarum pertinentium ad componentes. Si
enim in fig. 5o, 5 I, 52 vis AP compohatur ex viribus AB, AC, quæ sint latera parallelogrammi
ABPC, & ducantur in AP perpendicula BE, CF, cadentibus E, & F inter A, & P in fig. 5o,
in A, & P in fig. 5 I, extra in fig. 52 ; satis patet, fore in prima, & postrema æqualia triangula
AEB, PFC, adeoque vires EB, FC contrarias, & æquales elidi ; vim vero AP in primo
casu esse summam binarum virium conspirantium AE, AF, æquari unicæ AF in secundo,
& fore differentiam in tertio oppositarum AE, AF.
292. In resolutione quidem vis crescit quodammodo ; quia mente adjungimus alias
oppositas, & æquales, quæ adjunctæ cum se invicem elidant, rem non turbant. Sic in
fig. 52 resolvendo AP in binas AB, AC, adjicimus ipsi AP binas AE, PF contrarias, &
præterea in directione perpendiculari binas EB, FC itidem contrarias, & æquales. Cum
resolutio non sit realis, sed imaginaria tantummodo ad faciliorem problematum solutionem ;
nihil inde difficultatis afferri potest contra communem methodum concipiendi vires, quas
huc usque consideravimus, & quæ momento temporis exercent solum nisum, sive pressionem ;
unde etiam fit, ut dicantur vires mortuæ, & idcirco solum continuo durantes tempore
sine contraria aliqua vi, quæ illas elidat, velocitatem inducunt, ut causæ velocitatis ipsius
inductæ : nec inde argumentum ullum desumi poterit pro admittendis illis, quas Leibnitius
invexit primus, & vires vivas appellavit, quas hinc potissimum necessario saltem concipiendas
esse arbitrantur nonnulli, ne nimirum in resolutione virium habeatur effectus non æqualis
suæ causæ. Effectus quidem non æqualis, sed proportionalis esse debet, non causæ, sed
actioni causæ, ubi ejusmodi actio contraria aliqua actione non impeditur vel tota, vel ex
parte, quod accidit, uti vidimus, in obliqua compositione : ac utcunque & aliæ responsiones
sint in communi etiam sententia pro casu resolutionis ; [I38] in mea Theoria, cum ipsa
resolutio realis nulla sit, nulla itidem est, uti monui, difficultas.
293. Et quidem tam ex iis, quæ huc usque demonstrata sunt, quam ex iis,
quæ consequentur, satis apparebit, nullum usquam esse ejusmodi virium vivarum indicium,
nullam necessitatem ; cum omnia Naturæ phænomena pendeant a motibus, & æquilibrio,
adeoque a viribus mortuis, & velocitatibus inductis per earum actiones, quam ipsam ob
causam in illa dissertatione De Viribus Vivis, quæ hujus ipsius Theoriæ occasionem mihi
præbuit ante annos I 3, affirmavi, Vires Vivas in Natura nullas e$$e, & multa, quæ ad eas
probandas proferri solebant, satis luculenter exposui per solas velocitates a viribus non
vivis inductas.
A THEORY OF NATURAL PHILOSOPHY - 22T
FIG. 48. • * FIG. 49.
FIG. 50. FIG. 51.
FIG. 52.



222 PHILOSOPHIAE NATURALIS THEORIA
FIG. 48. FIG. 49.
FIG. 5O. FIG. 5.I.
FIG. 52.



A THEORY OF NATURAL PHILOSOPHY 223
289. Further, as regards resolution into several forces or motions, it is easy, from
what has been said in Art. 257, to determine a law which will rightly govern such resolution,
so that the forces which compose any given force may be obtained. In Fig. 48, let AP
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, Af, 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
the general case. This can be accomplished by means of any two sides of any triangle.
In Fig. 49, if AP is the given force, & any triangle ABP is constructed, then the force AP
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 brings out clearly the reason why a force compounded
from forces that do not lie in the same straight line is less than the sum of these components.
These are indeed partly opposite to one another ; &, when the equals & opposites have
cancelled one another, there remains in the force compounded of them the sum of the
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
this is that mentally we add on other equal & opposite forces, which taken together cancel
one another, & thus do not have any disturbing effect. Thus, in Fig. 52, by resolving
the force AP into the two forces AB, AC, we really add to AP the two equal & opposite
forces AE, PF, &, in addition, in a direction at right angles to AP, the two forces EB, FC,
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,
as well as from what will follow, that there is nowhere any sign of such living forces, nor
is there any necessity. For all the phenomena of Nature depend upon motions & equilibrium,
& thus from dead forces & the velocities induced by the action of such forces. For this
reason, in the dissertation De Viribus 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,
A general method
for resolving a force
into any number of
other forces.
Derivation of the
principle of resolu-
tion in two direc-
tions only.
Why the resultant
force is less than
the two component
for c e s taken to-
gether.
The reason why the
for c e see m s to
increase in resolu-
tion : no argument
for living forces to
be derived from
this.
It is sufficient to
prove from my
Theory that there
do not exist in
Nature any living
forces.
224 - PHILOSOPHIÆ NATURALIS THEORIA
Impactus obliquus
globi el a stici in
quatuor g l o b o s,
qui pro iis afferri
solet.
Ejus explicatio in
hac Theoria sine
viribus vivis per
Solam compositi-
0I1€m.
Q u id notandum
idcirco, quod nulli
sunt globi continui,
aut plana continua,
aut mathematicus
contactus.
L e x • r efl e xionis
perfecte, & imper-
fecte elasticorum.
294. Unum hic proferam, quod pertinet ad collisionem globorum elasticorum obliquam,
quæ compositionem resolutioni substitutam illustrat. Sint in fig. 53 triangula ADB.
BHG, GML rectangula in D, H, M ita, ut latera BD, GH, LM, sint æqualia singula
dimidiæ basi AB, ac sint BG, GL, LQ parallelæ AD, BH, GM. Globus A cum velocitate
AB = 2 incurrat in B in globum C sibi æqualem jacentem in DB producta : ex collisione
obliqua dabit illi velocitatem CE = I, æqualem suæ BD, quam amittet, & progredietur
per BG cum velocitate = AD = V 3. Ibi eodem pacto si inveniat globum I, dabit ipsi
velocitatem IK = I, amissa sua GH, & progredietur per GL cum V 2 ; tum ibi dabit,
globo O velocitatem OP = I, amissa sua LM, & abibit cum LQ = I, quam globo R,
directe in eum incurrens, communicabit. Quare, ajunt, illa vi, quam habebat cum veloci-
tate = 2, communicabit quatuor globis sibi æqualibus vires, quæ junguntur cum
velocitatibus singulis = I ; ubi si vires fuerint itidem singulæ = I, erit summa virium =
4, quæ cum fuerit simul cum velocitate = 2, vires sunt, non ut simplices velocitates in
massis æqualibus, sed ut quadrata velocitatum.
295. At in mea Theoria id argumentum nullam sane vim habet. Globus A non
transfert in globum C partem DB suæ velocitatis AB resolutæ in duas DB, TB, & cum
ea partem suæ vis. Agit in globos vis nova mutua in partes oppositas, quæ alteri imprimit
velocitatem CE, alteri BD. Velocitas prior globi A expressa per BF positam in directum
cum AB, & ipsi æqualem, 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, componunt
velocitates GH, & GN ; LM, & LS ; LQ, & QL, sine ulla aut vera resolutione, aut
translatione vis vivæ, Natura in omni omnino casu, & in omni corporum genere agente
prorsus eodem pacto. -
296. Sed quod attinet ad collisiones corporum, & motus [I39] reflexos, unde digressi
eramus ; inprimis illud monendum duco ; cum nulli mihi sint continui globi, nulla plana
continua ; pleraque ex illis, quæ dicta sunt, habebunt locum tantummodo ad sensum,
& proxime tantummodo, non accurate ; nam intervalla, quæ habentur inter puncta,
inducent inæqualitates sane multas. Sic etiam in fig. 43. ubi globus delatus ad B incurrit
in CD, mutatio viæ 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 constanti lege se extendunt, sed in majoribus distantiis sunt insensibiles ;
nisi massa, in quam tenditur, sit enormis, ut est totius Terræ massa in quam sensibili vi
tendunt gravia. At ubi globus advenerit satis prope planum CD ; incipiet incurvari
etiam via centri, quæ quidem, jam attracto, jam repulso globo, serpet etiam, donec alicubi
repulsio satis prævaleat 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
æquipollentiam veræ compositionis virium cum imaginaria resolutione), & retro etiam
motum reflectat. -
297. Et quidem si vires in accessu ad planum, ac in recessu a plano fuerint prorsus
æquales inter se ; dimidia curva ab initio sensibilis curvaturæ usque ad minimam distantiam
a plano erit prorsus æqualis, & similis reliquæ dimidiæ curvæ, quæ habebitur inde usque
ad finem curvaturæ sensibilis, ac angulus incidentiæ erit æqualis 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
æqualia circa angulos rectos secum trahant æqualitatem angulorum ABF, MBI, quorum
alter dicitur angulus incidentiæ, & alter reflexionis, ubi in imperfecte elasticis non habetur
ejusmodi æqualitas, sed tantummodo constans ratio inter tangentem anguli incidentiæ,
& tangentem anguli reflexionis, quæ nimirum ad radios æquales BF, BI sunt FA, & Im,
& sunt juxta denominationem, quam supra adhibuimus num. 272, & retinuimus huc usque,
ut m ad m, -
A THEORY OF NATURAL PHILOSOPHY 225
S P
N
R Q L -
K
|I-2'
M
G F
HS
D
Byrº E

FIG. 53. -
226 PHILOSOPHIAE NATURALIS THEORIA
S P
.N
O
R Q l-
- - K
|
M *
w G F
H\
D
B C E.

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, LO 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 = 1,
which is equal to its own velocity BD, which it loses; & it itself will go on along BG with
a velocity equal to AD = V3. It will then come to the sphere I, will give to it a velocity
IK = 1, losing its own velocity GH, & will go on along GL with a velocity equal to V2.
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, LO & 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
sphere S, an ex-
a mple 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 PHILOSOPHIÆ NATURALIS THEORIA
Eadem facta vi
agente in aliqua
distantia, consider-
a t a c u r v a, t u r a.
semitæ.
Quid, si planum sit
asperum : a p p 1 i-
catio ad reflexionem
lucis.
Quid in impactu
obliquo globi mol-
lis in planum :
velocitas amissa,
quæ manet illæsa
in curvatura con-
tinua.
Theorem at a pro
vi accelerante de-
scensum, vel retar-
dante ascensum in
planis inclinatis, &
pendulis.
298. Curvaturam in reflexione exhibet figura 54, ubi via puncti mobilis repulsi a plano
CO est ABQDM, quæ circa B, ubi vires incipiunt esse sensibiles, incipit ad sensum incurvari,
& 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 plano distantiis
vires æquales sunt ; satis patet, & accu-
ratissime demonstrari [I4o] etiam pos-
set, ubi semel deventum sit alicubi, ut
in Q, ad directionem parallelam plano,
debere deinceps describi arcum QD
prorsus æqualem, & similem arcui QB,
& ita similiter positum respectu plani
CO, ut ejus inclinationes ad ipsum
planum in distantiis æqualibus ab eo,
& a Q hinc, & inde sint prorsus æquales ;
quam ob causam tangentes BN, DP,
quæ sunt quasi continuationes rectarum AB, MD, angulos faciunt ANC, MPO æquales,
qui deinde habentur pro angulis incidentiæ, & reflexionis.
299. Si planum sit asperum, ut Figura exhibet, & ut semper contingit in Natura ; .
æqualitas illa virium utique non habetur. At si scabrities sit satis exigua respectu ejus
distantiæ, ad quam vires sensibiles protenduntur ; inæqualitas ejusmodi erit perquam
exigua, & anguli incidentiæ, & reflexionis æquales erunt ad sensum. Si enim eo intervallo
concipiatur sphæra 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 massæ, satis exiguam inæqualitatem poterunt
inducere ; & proinde sensibilem æqualitatem angulorum incidentiæ, & 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 sphæræ. Atque
hæc quidem, qui diligentius perpenderit, videbit sane, & lucem a vitro satis lævigato resilire
debere cum angulo reflexionis æquali ad sensum angulo incidentiæ ; licet & ibi pulvisculus
quo poliuntur vitra, relinquat sulcos, & monticulos, sed perquam exiguos etiam respectu
distantiæ, ad quam extenditur sensibilis actio vitri in lucem ; sed respectu superficierum,
quæ ad sensum scabræ sunt, debere ipsam lucem irregulariter dispergi quaqua versus.
3oo. Pariter ubi globus non elasticus deveniat per AB in eadem illa fig. 43, & deinde
debeat sine reflexione excurrere per BQ, non describet utique rectam lineam accurate,
sed serpet, & saltitabit non nihil : erit tamen recta ad sensum : velocitas vero mutabitur
ita ; ut sit velocitas prior AB ad posteriorem BI, ut radius ad cosinum inclinationis OBI
rectæ 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 [I4I] continua, uti fit in curvis
continuis, ea summa evanescit, & nulla fit velocitas amissio ex inflexione continua orta, sed
vis perpetua, quæ tantummodo ad habendam curvaturam requiritur perpendicularis ipsi
curvæ, nihil turbat velocitatem, quam parit vis tangentialis, si qua est, quæ motum perpetuo
acceleret, vel retardet ; ac in curvilineis motibus quibuscunque, qui habeantur per quas-
cunque directiones virium, semper resolvi potest vis illa, quæ agit, in duas, alteram
perpendicularem curvæ, alteram secundum directionem tangentis, & motus in curva per
hanc tangentialem vim augebitur, vel retardabitur eodem modo, quo si eædem vires agerent,
& motus haberetur in eadem recta linea constanter. Sed hæc jam meæ Theoriæ communia
sunt cum Theoria vulgari.
3o I. Communis est itidem in fig. 44, & 45 ratio gravitatis absolutæ BO ad vim BI, quæ
obliquum descensum accelerat, vel ascensum retardat, quæ est, ut radius ad sinum anguli
BOI, vel OBR, sive cosinum OBI. Angulum OBI est in fig. 44, quem continet directio
BI, quæ est eadem, ac directio plani CD, cum linea verticali BO, adeoque angulus OBR
est æqualis 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 hæc theoremata : Vis accelerans descensum, vel retardans ascensum in planis
FIG. 54.

A THEORY OF NATURAL PHILOSOPHY 229
298. Fig. 54 illustrates the curvature in reflection; here we have the path of a moving
point repelled by a plane CO represented by ABQDM ; this, near B, where the forces
begin to be sensible, begins to be appreciably curved, & leaves off at the same distance
from the plane, near the point D. The path, indeed, if there is always repulsion, will be
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,
& from Q 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,
there will in no case be this equality of forces. But if the roughness is sufficiently slight
in comparison with that distance, over which sensible forces are extended, such inequality
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
reflection has to continue along BQ, it will not describe a perfectly accurate straight line,
but will wind irregularly to some extent ; yet the line will be to all intents a straight
line. Moreover, the velocity will be changed in such a way that the previous velocity
AB will be to the new velocity BI, as the radius is to the cosine of OBI the inclination
of the straight line BO to the plane CD ; & the previous velocity is to the difference
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
BI, which accelerates the descent or retards the ascent; & this ratio is equal to that of the
radius to the sine of the angle BOI, or OBR, or the cosine of OBI. The angle OBI is, in
Fig. 44, that which is contained by the direction BI, which is the same as the direction of
the plane CD, with the vertical line BO; & thus the angle OBR is equal to the inclination
of the plane to the horizon; & the same angle OBR, in Fig. 45, is that which is contained
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,
The case of a force
acting at a con-
siderable distance ;
consideration of
the curvature of
the path.
What if the plane
is rough ; applica-
tion to the reflec-
tion of light.
What happens in
the case of oblique
impact of a soft
sphere ; the velo-
city lost, which re-
mains unimpaired
in continuous cur-
vature.
The or em S with
regard to the force
acceler a ting de-
scent or retarding
ascent in the cases
of the inclined
plane & of the
pendulum.
23O PHILOSOPHIÆ NATURALIS THEORIA
Applicatio Theoriæ
ad refractionem :
tres casus veloci-
tatis normalis ex-
tinctæ, imminutæ,
auctæ.
Primo reflexionem
induci. -
Secundo re f ra, c-
t i o n e m cum ac-
céssu ad superficiem
refringentem, tertio
itidem refractio-
nem, s e d c u m
I€C€SSll.
Ra t io constans
sinus anguli inci-
dentiæ, ad sinum
anguli refracti.
inclinatis, vel ubi oscillatio fit in arcu circulari, est ad gravitatem ab$olutam, ibi quidem ut
simus inclinationis ipsius plani, hic vero ut simus anguli, quem cum verticali linea continet recta
jungens punctum oscillans cum puncto suspensionis, ad radium. E quorum theorematum
priore fluunt omnia, quæ Galilæus tradidit de descensu per plana inclinata ; ac e posteriore
omnia, quæ pertinent ad oscillationes in circulo ; quia immo etiam ad Oscillationes factas
in curvis quibuscunque pondere per filum suspenso, & curvis evolutis applicato ; ac eodem
utemur infra in definiendo centro Oscillationis.
302. Hisce perspectis, applicanda est etiam Theoria ad motuum refractionem, ubi
continentur elementa mechanica pro refractione luminis, & occurrit elegantissimum
theorema a Newtono inventum huc pertinens. Sint in fig. 55 binæ superficies AB, CD
parallelæ inter se, & punctum mobile quodpiam extra illa plana nullam sentiat vim, inter
ipsa vero urgeatur viribus quibuscunque, quæ tamen & semper habeant directionem
perpendicularem ad ipsa plana, & in æqualibus distantiis ab altero ex iis æquales sint ubique ;
ac mobile deferatur ad alterum ex iis, ut AB, directione quacunque GE. Ante appulsum
feretur motu rectilineo, & æquabili, cum nulla urgeatur vi : ejus velocitatem exprimat EH,
quæ erecta ER, perpendiculari ad AB, resolvi poterit in duas, alteram perpendicularem
ES, alteram parallelam HS. Post ingressum inter alia duo [I42] 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 prævalente viribus eadem
directione agentibus ; vel perstat potius aucta.
3o3. In primo casu, ubi primum velocitas perpendicularis extincta fuerit alicubi in
X, punctum mobile reflectet cursum retro per XI, & iisdem viribus agentibus in regressu,
quæ egerant in progressu, acquiret velocitatem perpendicularem IL æqualem amissæ ES,
quæ composita cum parallela LM, æquali priori HS, exhibebit obliquam IM in recta nova
IK, quam describet post egressum, & erunt æquales anguli HIL, MES, adeoque & anguli
IKIB, GEA; quod congruit cum iis, quæ in fig. 54. sunt exhibita, & pertinent ad
reflexionem. -
3o4. In secundo casu prodibit ultra superficiem ulteriorem CD, sed ob velocitatem
perpendicularem OP minorem priore ES, parallelam vero PN æqualem priori HS, erit
angulus ONP minor, quam EHS, adeoque inclinatio VOD ad superficiem in egressu minor
inclinatione GEA in ingressu. Contra vero in tertio casu ob op majorem ES, angulus
uoD erit major. In utroque autem hoc casu differentia quadratorum velocitatis ES, &
OP vel op, erit constans, per num. I77 in adn. m, quæcunque fuerit inclinatio GE in
ingressu, a qua inclinatione pendet velocitas perpendicularis SE.
3o5. Inde autem facile demonstratur, fore sinum anguli incidentiæ HES, ad sinum
anguli refracti PON (& quidquid dicitur de iis, quæ designantur litteris PON, erunt com-
münia iis, quæ exprimuntur litteris pom) in ratione constanti, quæcunque fuerit inclinatio
, rectæ incidentis GE. Sumatur enim HE constans, quæ exprimat velocitatem ante
incidentiam : exprimet HS velocitatem parallelam, quæ erit æqualis rectæ PN exprimenti
velocitatem parallelam post refractionem; ac ES, OP expriment velocitates perpendiculares
ante, & post, quarum quadrata habebunt differentiam constantem. Sed ob HS, PN semper
æquales, différentia quadratorum HE, ON æquatur differentiæ 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 ; quæ quidem ratio est eadem,
ac sinu§ 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 æqualia, ut hic HS, PN, erunt reciproce ut bases.
A THEORY OF NATURAL PHILOSOPHY 23 I
G K
R
H S L M
|
A E
X B
C D
O O N
P \,
A) II

FIG. 55.
232
PHILOSOPHLE 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 fine of the inclination of the plane to the radius, 85 in
the second case as the fine of the angle between the vertical & 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
of motions, in which are contained the mechanical principles of the refraction of light;
here also we find a most elegant theorem discovered by Newton, referring to the subject.
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
suppose it is subject to any forces, so long as these always have a direction perpendicular
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
X, the moving point will follow a return path along XI ; & as the same forces act in the
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,
since the perpendicular velocity OP is now less than the previous one ES, whilst the parallel
velocity is the same as the previous one HS, the angle ONP will be less than the angle EHS,
& therefore the inclination to the surface, VOD, on egress, will be less than the inclination,
GEA, on ingress. On the other hand, in the third case, since op is greater than ES, 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
is to the sine of the angle of refraction HON (& whatever is said with regard to these angles,
denoted by the letters PON, will hold good for the angles denoted by the letters pon), in
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. -
Application of the
Theory to refrac-
tion ; the three
cases in which the
normal velocity is
extinguished, or
di minished, or
increased.
In the first case
reflection is in-
duced.
In the second case
we have refraction
& nearer approach
to the refracting
surface ; in the
third, refraction &
recession from the
surface.
The constant ratio
of the sine of the
angle of incidence
to the sine of the
angle of refraction.
234. PHILOSOPHIAE NATURALIS THEORIA
Eatio sinuum con-
stans, & ratio velo-
citatum reciproca
rationis sinuum.
Transitus ad Theo-
rema, quod huic
Operi occasionem
dedit.
Consideratio direc-
tionis virium, qui-
bus tres massæ in
se mutuo agunt.
306. Quamobrem in refractionibus, quæ hoc modo fiant motu libero per intervallum
inter duo plana parallela, in quo vires paribus distantiis ab altero eorum pares sint, ratio
sinus anguli incidentiæ, sive anguli, quem facit via ante incursum cum rectâ perpendiculari
plano, ad sinum anguli refracti, quem facit via post egressum itidem cum vérticali, est
constans, quæcunque fuerit inclinatio in ingressu. Præterea vero habetur & illud, fore
celeritates absolutas ante, & post in ratione reciproca eorum sinuum. Sunt enim
ejusmodi velocitates ut HE, ON, quæ sunt reciproce ut illi sinus.
307. Hæc quidem ad luminis refractiones explicandas viam sternunt, ac in Tertia
Parte videbimus, quo pacto hypothesis hujusce theorematis applicetur particulis luminis.
Sed interea considerabo vires mutuas, quibus in se invicem agant tres massæ, ubi habebuntur
generalius ea, quæ pertinent etiam ad actiones trium punctorum, & quæ a num. 225, &
228 huc reservavimus. Porro si integræ vires alterius in alteram diriguntur ad ipsa centra
gravitatis, referam hic ad se invicem vires ex integris compositas ; sed etiam ubi vires aliam
directionem habeant quancunque ; si singulæ resolvantur in duas, alteram, quæ se dirigat
a centro ad centrum ; alteram, quæ sit ipsi perpendicularis, vel in quocunque dato angulo
obliqua ; omnia in prioribus habebunt itidem locum.
308. Agant in se invicem in fig. 56 tres massæ, quarum centra gravitatis sint A, B, C,
viribus 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 saltem ad partes oppositas producta ingreditur triangulum, & secat
illa angulum internum ACB, hæc 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. Primæ 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 tam priores binæ,
quam posteriores, jacent ad eandem partem lateris AB, & vel ambæ ingrediuntur triangulum
tendentes versus ipsum, vel ambæ extra ipsum etiam productæ abeunt, & tendunt ad
partes oppositas directionis Ce respectu AB. Secundæ CZ debent respondere repulsiones
BT, AL, quæ cum repulsionibus BR, AI, constituunt BS, AK, cum attractionibus BN,
AE constituunt BM, AD, ac tam priores binæ, quam posteriores jacent ad eandem plagam
respectu AB, & ambarum [I44] directiones vel productæ ex parte posteriore ingrediuntur
triangulum, sed tendunt ad partes ipsi contrarias, ut CZ, vel extra triangulum utrinque
abeunt ad partes oppositas directioni CZ respectu AB. Quod si habeatur CX, quam
exponunt CV, CY, tum illi respondent BP, & AL, ac si prima conjungitur cum BN, jam
habetur BO ingrediens triangulum ; si BR, tum habetur quidem BQ, cadens etiam ipsa
extra triangulum, ut cadit ipsa CX; sed secunda AL jungetur cum AI, & habebitur AK,
quæ producta ad partes A ingredietur triangulum. Eodem autem argumento cum vi Cb
Theorema pertinens
ad directiones vir-
1lìIm.
Theorema elegan-
tius ad eas perti-
nens cum ejus de-
monstratione.
vel cónjungitur AF ingrediens triangulum, vel BS, quæ producta ad B triangulum itidem
ingreditur. Quamobrem semper aliqua ingreditur, & tum de reliquis binis redeunt, quæ
dicta sunt in casu virium Ce, CZ.
3o9. Habetur igitur hoc theorema. Quando tres massæ in se invicem agunt viribus
directis ad centra gravitatis, vis composita saltem unius habet directionem, quae saltem producta
ad partes oppositas secat angulum internum trianguli, & ipsum ingreditur : reliquae autem
duae vel simul ingrediuntur, vel simul evitant, & semper diriguntur ad eamdem plagam respectu
lateris jungentis earum duarum massarum centra ; ac in primo casu vel omnes tres tendunt ad
interiora irianguli 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
jüngentis eas binas massas tendunt in plagas oppositas ei, in quam tendit vis illa prioris ma$$æ.
3 Io. Sed est adhuc elegantius theorema, quod ad directionem pertinet, nimirum :
Omnium trium compositarum virium directiones utrimque productæ transeunt per idem punctum :
& si id jaceat intra triangulum ; vel omnes simul tendunt ad ipsum, vel omnes simul ad partes
ipsi conirarias : si vero jaceat extra triangulum ; binae, quarum directiones non ingrediuntur
A THEORY OF NATURAL PHILOSOPHY
235
FIG. 56.
M T
S

236
PHILOSOPHIAE NATURALIS THEORIA
FIG. 56.

A THEORY OF NATURAL PHILOSOPHY 237
306. Hence, in refractions, which arise in this way from a free motion between two
parallel planes, where the forces at equal distances from one or the other of them are equal,
the ratio of the sine of the angle of incidence, or the angle made by the path before refraction,
with a straight line perpendicular to the plane, to the sine of the angle of refraction, or
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
Part we shall see the manner in which the hypothesis of the above theorem may be applied
to particles of light. In the meanwhile, I will consider the mutual forces, with which
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
one another with mutual forces directed to their centres ; & first of all let the directions
of the forces be considered. The force on the point C, from the two attractive forces
CV, Cd will be Ce; that from CY, Ca, both repulsive, will be CZ; & the direction of
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 aGY, 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
with forces directed towards their centres of gravity, the resultant force, in at least one case,
will have a direction which, produced backwards if necessary, will divide an internal angle
of the triangle into two parts, 85 fall within the triangle. Also the remaining two forces will
either both fall within, or both without, the triangle & 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.
3Io. But there is a still more elegant theorem with regard to the directions of the
forces, namely:-The directions of all three resultant forces, when produced each way, pass
through the same point. If this point lies within the triangle, all three forces tend towards
it, or all three away from it; but, if it lies without the triangle, those two forces, which do not
The ratio of the
sines Constant :
the ratio of the
velocities the
reciprocal ratio of
the sines.
Passing on to the
theorem which
gave rise to this
work.
Investigation of
the directions of
the for ce S with
which three maSSes
act upon One an-
Other.
Theorem relating
to the directions of
the forces.
A still more ele-
gant the ore m
with regard to the
directions of the
forces; & its de-
monstration.
Corollariu m pro
ca s u directionum
parallelaru
Aliud generale ter-
tiæ directionis datæ
datis binis
IIl.
Theorema præcip.
uum de magnitu-
dine, quo
Operi occasionem
dedit. Eju
s t ratio e
sima.
d t O t i
s demon-
xpeditis.
238 - PHILOSOPHIÆ NATURALIS THEORIA
triangulum, tendunt ad ipsum, ac tertia, cujus directio triangulum ingreditur, tendit ad partes
ipsi contrariias ; vel illae binae ad partes ipsi contrarias, & tertia ad ipsum.
Prima pars, quod omnes transeant per idem punctum, sic demonstratur. In figura
quavis. a 57 ad 62, quæ omnes casus exhibent, vis pertinens ad C sit ea, quæ triangulum
ingreditur, ac reliquæ binæ HA, QB concurrant in D : oportet demonstrare, vim étiam,
quæ 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 æqualem Cd;
res erit perfecta : ducta enim dT remanebit CVTd parallelogrammum, per cujus diagonalem
CT dirigetur vis composita ex CV, Cd. Ejusmodi autem æqualitas 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
ma$$as, quarum ea puncta sunt centra gravitatum, est ex actione, & reactione æqualibus
ratio massæ B ad C ; secunda sim PQB, sive ABD, ad sim PBQ, sive CBD ; tertia A ad
B: quarta sim HAG, sive CAD, ad sim GHA, sive BAD: quinta C ad A. Tres rationes,
in quibus habentur massæ, componunt rationem B × A × C ad C × B × A, quæ est I ad
I, & remanet ratio $im ABD × sim CAD ad sim CBD × sim BAD. Pro sim ABD, & yjm
BAD, ponantur AD, & BD ipsis proportionales ; ac pro sinu CAD, & sim CBD ponantur
sim ACD × CD , yim BCD × CD
AD 9 BD
sim ACD × CD ad sim BCD × CD sive sim ACD, vel CTV, qui ipsi æquatur ob VT,
CA parallelas, ad sim BCD, sive VCT, nimirum ratio ejusdem illius CV ad VT. Quare
VT æquatur Cd, CVTd est parallelogrammum, & vis pertinens ad C, habet directionem
itidem transeuntem per D. -
Secunda pars patet ex iis, quæ 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, & 6o intra triangulum,
in 6I, & 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 Comicarum Elementis, Elementorum tomo 3.
, ipsis æquales ex Trigonometria, & habebitur ratio
3I I. Quoniam evadentibus binis HA, QB parallelis, punctum D abit in infinitum
& tertia CT evadit parallela reliquis binis etiam ipsa juxta easdem leges ; patet illud : Si
binæ ex ejusmodi directionibus fuerint parallelæ inter se ; erit iisdem parallela & tertia :
ac illa, quae jacet inter directiones virium transeuntes per reliquas binas, quae idcirco in eo
casu appellari potest media, habebit directionem oppositam directionibus reliquarum comformibus
111ter $e.
3I2. Patet autem, datis binis directionibus virium, dari semper & tertiam. Si enim
illæ sint parallelæ ; erit illis parallela & tertia : si autem concurrant in aliquo puncto ;
tertiam determinabit recta ad idem punctum ducta : sed oportet, habeant illam conditionem,
ut tam binæ, quæ triangulum non ingrediantur, quam quæ ingrediantur, vel simul tendant
ad illud punctum, vel simul ad partes ipsi contrarias.
313. Hæc quidem pertinent ad directiones : nunc ipsas earum virium magnitudines
inter se comparabimus, ubi statim occurret elegantissimum illud theorema, de quo
mentionem feci num. 225 : Vires acceleratrices binarum quarumvis e tribus massis in se
mutuo agentibus sunt in ratione composita ex tribus, [I46] nimirum ex directa sinuum angulorum
quos continet recta jungens ipsarum centra gravitatis cum rectis ductis ab iisdem centris ad centrum
tertiae massae ; reciproca sinuum angulorum, quos directiones ipsarum virium continent cum
iisdem rectis illas jumgemtibus 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 massæ A ad massam B : tertia sinus IHA, sive HAG, vel CAD, ad sinum
HIA, sive CAB : eæ rationes, permutato solo ordine antecedentium, & consequentium,
sunt rationes sinus CBA ad sinum CAB, quæ est illa prima e rationibus propositis directa ;
sinus CAD ad sinum CBD, quæ est secunda reciproca : & massæ A ad massam B, quæ
est tertia itidem reciproca. Eadem autem est prörsus 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


240 PHILOSOPHIAE NATURALIS THEORIA

A THEORY OF NATURAL PHILOSOPHY 24I
fall within the triangle, tend towards it, & the third, whose direction does not fall within the
triangle, tends away from it, or the former two tend away from the point & 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, CVTā 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 B. × A × C to C X B X A,
which is that of I to I ; & there remains the ratio of sin/ABD X sinCAD to sinCBD X
sinEAD. For sin/ABD & sinBAD substitute AD & BD, which are proportional to them ;
& for sincAD & sincBD substitute sin/ACD X CD/AD & sinBCD X CD/BD, which are
equal to them by trigonometry. There will be obtained the ratio of sin/ACD X CD to
sinBCD X CD, or sin/ACD to sinBCD ; &, since VT & CA are parallel, this ratio is equal to
that of sincIV to sinWCT, that is, to the ratio of CV to VT. Therefore VT is equal to
Cd, CVTā 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 Comicarum Elementa, the third volume of my
Elementa Matheseos.
3II. Now, since the point D will go off to infinity, when two of the forces, HA &
QB, happen to be parallel, & the third also, according to the same laws, becomes parallel
to the other two, we have this theorem. If two of these forces are parallel to one another,
the third also is parallel to them ; 85 that force, which lies between the directions of the other
two, ö 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
direction also of the third force is given in all cases. For if the former are parallel, the
third will be parallel to them ; & if the former meet at a point, the straight line joining
the third mass to this point will determine the third direction. But this condition holds ;
namely, that the two which do not fall within the triangle, or the pair which do fall within
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
another the magnitudes of these forces. We immediately come to that most elegant
theorem, which has already been mentioned in Art. 225. The accelerating effects of any
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 ; 85 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 sincBA is to sincAB, which is the first direct ratio
of those enunciated ; as sinCAD to sinCBD, 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,
r . . * R
Corollary for the
case of parallel
directions.
Another general
theorem ; the direc-
tion of the third
force is given when
the directions of
the other two are
given.
Fundamental theo-
rem concerning
magnitude which
gave rise to the
whole of this work.
242 PHILOSOPHLE NATURALIS THEORIA
Corollarium sim-
plex pro v ir i b u s
ipsis.
Ratio virium mo-
tricium.
Patio virium acce-
leratricium, ubi eæ
diriguntur ad ali-
q u o d commune
punctum.
Alia expressio tam
virium motricium,
quam acceleratri-
cium in eodem casu.
Hic debere haberi
ea, quæ habentur
in compositione, &
resolutione virium.
A lia ex p r e s s i o
rationum earundem
virium.
agitur, angulis quibusvis substitui possunt, uti sæpe est factum, & fiet imposterum, eorum
complementa ad duos rectos, quæ eosdem habent sinus.
314. Inde consequitur, e$$e ejusmodi vires reciproce, ut massas ductas in suas distantias
a tertia ma$$a, $5 reciproce, ut simus, quos earum directiones continent cum iisdem rectis ;
adeoque ubi eae ad ejusmodi rectas inclinentur in angulis æqualibus, esse tantummodo reciproce,
ut producta ma$$arum 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
illa, habentur tres rationes reciprocæ, quas exprimit ipsum theorema hic propositüm.
Porro ubi anguli æquales sunt, sinus itidem sunt æquales, adeoque eorum sinuüm ratio
fit I ad I.
315. /ires autem motrices sunt in ratione composita ex binis tantummodo, nimirum
directa sinuum angulorum, quos continent distantiae a tertia massa cum distantia a se invicem ;
§5 reciproca sinuum angulorum, quos continent cum iisdem distantiis directiones virium ; vel
in ratione compo$ita ex reciproca illarum distantiarum, & reciproca horum posteriorum sinuum :
ac $i inclimationes ad distantias sint æquales, in sola ratione reciproca distantiarum. Nam
vires motrices sunt summæ omnium virium determinantium celeritatem in punctis
omnibus secundum eam directionem, secundum quam movetur centrum gravitatis commune,
quæ idcirco sunt præterea directe, ut massæ, sive ut numeri punctorum ; adeoque ratio
directa, & reciproca massarum mutuo eliduntur.
3I6. Præterea vires acceleratrices, si alicubi earum directiones concurrunt, sunt ad se
invicem im ratione composita ex reciproca massarum, & reciproca sinuum angulorum, quibus
inclimamtur ad directionem tertiae ; & vires motrices in hac poste-[I47]-riore tantum. Nam
ob latera proportionalia sinubus angulorum oppositorum, erit AC × sim CAD = CD ×
sin CDA ; & pariter CB × sim CBA = CD × sim CDB. Quare ob CD communem,
sola ratio sinuum ADC, BDC, quibus directiones AD, BD inclinantur ad CD, æquatur
compositæ ex rationibus sinuum CAD, CBD, & distantiarum CA, CB, quæ ingrediebantur
rationem virium B, & A ; ac eodem pacto AC × sin ACD= AD × sim ADC, & AB ×
sim ABD = AD × sim ADB, adeoque AC × sim ACD ad AB × sim ABD, ut sinus ADC
ad sinum ADB, quibus directiones CD, BD inclinantur ad AD ; & eadem est demonstratio
pro sinubus ADB, EDB assumpto communi latere BD.
317. Si ducatur MO parallela DA, occurrens BD, CD in M, O, & compleatur parallelo-
grammum DMON ; erunt vires motrices in C, B, A ad se invicem, ut rectae DO, DM, DN,
& vires acceleratrices praeterea in ratione massarum reciproca. Est enim ex præcedenti vis
motrix in C ad vim in B, ut sim BDA ad sim CDA, vel ob AD, OM parallelas, ut sim DMO
ad sim DOM, nimirum ut DO ad DM, & simili argumento vis in C ad vim in A, ut DO
ad DN. Vires autem motrices divisæ per massas evadunt acceleratrices. Quamobrem si,
tres vires agerent in idem punctum cum directionibus, quas habent eae vires motrices, & essent
iis proportionales ; binae componerent vim oppo$itam, & æqualem tertiae, ac essent in aequilibrio.
Id autem etiam directe patet : nam vires BQ, AH componuntur ex quatuor viribus BR,
BP, AI, AG, quæ si ducantur in massas suas, ut fiant motrices ; evadit prima æqualis, &
contraria tertiæ, quam idcirco elidit, ubi deinde AH, BQ componantur simul, & in ejusmodi
compositione remanent BP, AG, ex quarum oppositis, & æqualibus CV, Cd componitur
tertia CT.
318. Hinc in hisce viribus motricibus habebuntur omnia, quæ habentur in compositione
virium ; dummodo capiatur [resolutio1 compositæ contraria. Si nimirum resolvantur
singulæ componentes in duas, alteram secundum directionem tertiæ, alteram ipsi perpen-
dicularem, hæ posteriores elidentur, illæ priores conficient summam æqualem tertiæ, ubi
ambæ eandem directionem habent, uti sunt binæ, quæ simul ingrediantur, vel simul evitent
triangulum ; nam in iis, quarum altera ingreditur, altera evitat, tertia æquaretur differen-
tiæ ; & facile tam hic, quam in ratione composita, res traducitur ad resolutionem in
aliam quamcunque directionem datam, præter directionem tertiæ, binis semper elisis, &
reliquarum accepta summa ; si rite habeatur ratio positivorum, & negativorum.
, AB × ED AE
* ÄDIX BD' AD*
& acceleratrices praeterea [I48] in ratione reciproca massarum. Nam ex Trigonometria
319. Est & illud utile : tres vires motrices in C, B, A sunt inter se, u
BE
BD?
eSt AB _ sim ADB , AE _ sin ADE
` BD T jjm BAD' ` ED T jjm EAD'
communis : erit sim ADB ad sin ADE, ut
Quare cum divisor sim BAD, & sin EAD sit
AB AE
Äî *' iij
, vel, ducendo utrunque terminum

244 PHILOSOPHIE NATURALIS THEORIA

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 i.º.º.º.
sines of the angles between their directions & these distances ; & thus, if they are inclined g
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 CBA, 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 "**
to one another ; 85 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 & 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 tºº ~
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 * *
angles, we have AC. sincAD = CD. sincDA, & similarly, CB. sincBA = CD. sincDB.
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. sin/ACD = AD. sin/ADC, & AB. sin/ABD = AD. sin/ADB, & therefore AC. sin/ACD
is to AB. sin ABD 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. If 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 tº. *...* ‘. .
as the straight lines DO, DM, DN, 85 for the accelerations, we haze 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 * *
force for B as sin BDA is to sincDA; that is to say, since AD, OM are parallel, as sinDMO
is to sinDOM, 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 {5 opposite
to the third, & 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 force.
composition of forces, so long as resolution is considered to be the inverse of composition. **
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 pºº
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, 85 A are in Another expression
the ratio of AB.ED/AD&BD, AE/AD, BE/BD, & the accelerations have, in addition, ...”. “”
the inverse ratio of the masses. For, by trigonometry, we have AB/BD = sinADB/sinEAD,
& AE/ED = sin/ADE/sinEAD. Hence, since the divisors sinBAD, & sinEAD are equal,
it follows that sin/ADB is to sin/ADE as AB/BD is to AE/ED ; or, multiplying each term
246 1PHILOSOPHIÆ NATURALIS THEORIA
Expressio simpli-
Cor pro casu paral-
lelismi.
Applicatio ratio-
num superiorum ad
centrum æquilibrii.
Determinatio vis,
quam fulcrum sus-
tinet.
Consideratio mas-
sarum etiam inter-
me d ia r u m, quæ
COnnectant massas
v i r i b u s externis
præditas, & positas
in æquilibrio.
ED AB × ED AE
in AD° ut AD x BD ad AD' Simili autem argumento est itidem sim BDA. sim BDE
. . AB × ED BE. €X QuO patent ©
' ' AD × BD ' BD ? quo patent omnia.
320. Si punctum D abeat in infinitum, directionibus virium evadentibus parallelis ;
ratio rectarum ED, AD, BD, ad se invicem evadit ratio æqualitatis. Quare in eo casu
illæ tres vires sunt ut AB, AE, EB, in quibus prima æquatur summæ reliquarum. Conci-
piantur rectæ parallelæ directioni virium ductæ per omnium trium massarum centra
gravitatis, quarum massarum eam, quæ jacuerit inter reliquarum binarum parallelas diximus
mediam : ac si ducantur in quavis alia directione data rectæ ab iis massis ad illas parallelas ;
erunt ejusmodi distantiæ ab iis parallelis, ut ipsæ AB, EB, ad quas erunt singulæ in ratione
data, ob datas directiones. Quare pro viribus parallelis habetur hujusmodi theorema :
Vires parallelæ motrices binarum quarumvis ex tribus massis sunt inter se reciproce ut distantiae
a directione communi transeunte per tertiam : vires autem acceleratrices praeterea in ratione
reciproca massarum, &5 media est directionis contrariae respectu reliquarum, ac vis media
motrix aequatur reliquarum summae, utralibet vero extrema differentiae.
32 I. Hoc theorema primo quidem exhibet centrum æquilibrii, viribus utcunque
divergentibus, vel convergentibus. Si nimirum sint tres massæ A, B, C (& nomine massarum
etiam intelligi possunt singula puncta), quarum binæ, ut A, & B, solicitentur viribus
motricibus externis ; poterunt mutuis viribus illas elidere, ac esse in æquilibrio, & eas
elident omnino, mutatis, quantum libuerit, parum mutuis distantiis ; si fuerint ante
applicationem earum virium externarum in satis validis limitibus cohæsionis, ac vis massæ
C elidatur fulcro opposito in directione DC, vel suspensione contraria : dummodo binæ
illæ vires ductæ in massas habeant conditiones requisitas in superioribus, ut nimirum ambæ
tendant ad idem punctum, vel ab eodem, aut si fuerint parallelæ, ambæ eandem directionem
habeant, ubi simul ambæ ingrediantur, vel simul ambæ evitent triangulum ABC : ubi
vero altera ingrediatur triangulum, altera evitet, tendat altera ad punctum concursus,
altera ad partes illi oppositas : vel si fuerint parallelæ, habeant directiones [149] oppositas :
& si parallelæ fuerint ; sint inter se, ut distantiæ 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.
322. Determinabitur autem admodum facile per ipsa theoremata etiam vis, quam
sustinebit fulcrum C, quæ in casu parallelismi æquabitur summæ, vel differentiæ reliquarum,
prout ibi fuerit media, vel extrema : & in casibus reliquis omnibus æquabitur summæ
pariter, vel differentiæ reliquarum ad suam directionem reductarum, reliquis binis in
resolutione priorum sociis se per contrariam directionem, & æqualitatem elidentibus.
323. Habebitur igitur, quidquid pertinet ad æquilibrium virium agentium in eodem
plano, & connexarum non per virgas inflexiles carentes omni vi præter cohæsionem, uti
eas vulgo concipiunt, sed hisce viribus mutuis. Et Theoria quidem habebit locum tum
hic, tum in sequentibus ; licet massæ A, B, C non agant in se invicem immediate, sed sint
aliæ massæ intermediæ, quæ ipsas jungant. Nam si inter massam B, & C sint aliæ massæ
nullis externis viribus agitatæ, & positæ in æquilibrio cum hisce massis, & inter se, ac prima,
quæ venit post B, agat in ipsam vi motrice æquali BP, aget & B in ipsam vi æquali : quare
debebit illa ad servandum æquilibrium urgeri a secunda, quæ est post ipsam, vi æquali
in partes contrarias. Hinc æquali contraria aget tertia in secundam, ut secunda in
æquilibrio sit, & ita porro, donec deveniatur ad C, ubi habebitur vis motrix æqualis motrici,
quæ erat in B, & erunt vires BP, CV acceleratrices in ratione reciproca massarum B, & C,
cum vires illæ motrices æquales sint producta ex acceleratricibus ductis in massas. At si
circumquaque sint massæ quotcunque cum vacuis quibuscunque, ac ubicunque interjectis,
quæ 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 illæ
externæ QB, HA concipiantur resolutæ singulæ 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 sinBDA is to sinBDE 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
parallel to one another, the ratios of the straight lines ED, AD, BD finally become ratios
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 ;
& 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
any forces, whether diverging or converging. For instance, if A, B, C are three masses
(& in the term masses, single points can also be understood to be included), of which two,
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
on the fulcrum placed at C; this, in the case of parallelism, will be equal to the sum or
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
in one plane, & connected, not by inflexible rods lacking all force except cohesion, but by
these mutual forces. The Theory holds good indeed, not only here, but also in what
follows; that is to say, although the bodies A, B, C may not act upon one another directly,
yet there are other intermediate masses which connect them. For, if between A & B
there were other masses not influenced by any external forces, & placed in equilibrium
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
A more simple
expression for the
case of parallelism.
Application of the
above ratios to the
centre of equili-
brium.
Determination of
the force on the
fulcrum.
Investigation of the
case in which there
are also interme-
diate masses con-
necting the masses
upon which the ex-
ternal forces act, &
placed in equili-
brium.
248 PHILOSOPHIAE NATURALIS THEORIA
Qui motus, ubi non
habeatur æ quili-
brium. .
Extensio ad æqui-
librium quotcunque
massarum, & inde
principium generale
pro machinis, &
ratio momentorum.
Applicatio ad om-
nia vectium genera.
Consectaria, d o c-
trinæ de vectibus,
& principium pro
statera, ; cur totum.
pondus cOnsidere-
tur, ut collectum
in centre gravitatis.
quæ illa tria puncta conjungunt ; poterit elisis mutuo reliquis omnibus æquilibrium
constituentibus deveniri ad vires in punctis binis, ut A, & C, acceleratrices contrarias viribus
BP, BR, & reciproce proportionales massis ipsarum respectu massæ B; licet ipsæ proveniant
a massis quibusvis etiam non in eadem directione sitis, & agentibus in latus : nam per
ejusmodi resolutionem, & ejusmodi virium considerationem adhuc habetur æquilibrium
totius systematis affecti in illis tribus punctis per illas tres vires, cum assumantur in iis
tantummodo vires motrices contrariæ, & æquales : unde fit, ut etiam illæ, quæ præterea
ad has in illis considerandas assumuntur, & per quas connectuntur cum reliquis massis,
se mutuo elidant.
[I5o] 324. Quod si vires ejusmodi non fuerint in ea ratione inter se ; non poterunt
puncta B, & A esse in æquilibrio, sed consequetur motus secundum directionem ejus, quæ
prevalet : ac si omnis motus puncti C fuerit impeditus ; habebitur conversio circa ipsum C.
325. Quod si non in tribus tantummodo massis habeantur vires externæ, sed in pluribus ;
licebit considerare quanvis aliam massam carentem omni externa vi, & eam concipere
connexam cum singulis reliquarum massis, & massa C per vires mutuas, ac habebitur itidem
Theoria pro æquilibrio omnium, cum positione omnium constanter servata etiam sine
ulla figuræ mutatione, quæ sensu percipi possit. Quin immo si singulæ vires illæ externæ
resolvantur in duas, quarum altera urgeat in directione rectæ transeuntis per C, ac elidatur
vi proveniente a solo puncto C, & altera agat perpendiculariter ad ipsam, ut habeatur
æquilibrium in singulis ternariis ; oportebit esse singulas vires novæ massæ assumptæ 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, quæ in massam
assumptam agunt directionibus contrariis, se mutuo elidere ad habendum æquilibrium.
Quare debebit summa omnium productorum earum virium, quæ urgent conversione in
unam plagam, per ipsarum distantias a centro conversionis, æquari summæ productorum
earum, quæ urgent in plagam oppositam, per distantias ipsarum, ut habeatur æquilibrium ;
cumque 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 æquari summæ 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.
326. Casus trium tantummodo massarum exhibet vectem, cujus brachia sint utcunque
inflexa. Quod si tres massæ 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. Tum vero vires externæ debebunt esse unius
directionis, & contrariæ directioni vis mediæ, & binæ quævis ex iis erunt ad se invicem
reciproce, ut distantiæ a tertia. Inde autem oriuntur tria genera vectium : si -fulcrum,
vel hypomochlium, sit in medio in E, vis in altero extremo A, [I5I] resistentia in altero
B ; vis ad resistentiam est, ut BE, distantia resistentiæ 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, quæ 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 distantiæ EB ad distantiam majorem AB, cujus idcirco
momentum, seu energia, augetur in ratione suæ distantiæ AB ad EB, ut nimirum possit
tanto majori resistentiæ æquivalere. Si demum fuerit quidem fulcrum in altero extremo
B, & resistentia in A, vis prior in E ; tum 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.
327. Quod si perticæ utcunque inclinatæ applicetur pondus in aliquo puncto E, & bini
humeros supponant in A, & B, sentient ponderis partes inæquales in ratione reciproca
distantiarum ab ipso ; & si e contrario bina pondera suspendantur in A, & B utcunque
inæqualia, assumpto autem puncto E, cujus distantiæ a punctis A, & B sint in ratione
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
be in equilibrium; but motion would follow in the direction of that which preponderated;
also if all motion of the point C were prevented, then there would be rotation about C.
325. Now if we have external forces acting, not on three masses only, but on several,
we can consider any one mass to be without an external force, & suppose that this mass
is connected to each of the others, & to the mass C, by mutual forces ; & the Theory will
hold good for the equilibrium of them all, with the position of them all constantly maintained
without any change of figure so far as can be observed. Further, if all the external forces
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
in any manner. But if the three masses lie in one straight line, they will form a rectilinear
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.
327. Now, if to a long pole, inclined at any angle to the horizontal, a weight is applied
at any point E ; & if two men place their shoulders under the pole at A & B ; then they
will support unequal parts of the weight, in the inverse ratio of their distances from it.
Conversely, if two unequal weights of any sort are suspended from A & B, & a point E is
taken whose distances from the points A & B are in the inverse ratio of the weights, & so
The nature of the
motion when equili-
brium does not
obtain.
|Extension to the
equilibrium of any
number of maSSes ;
& thence a general
principle for
machines & the
ratio of moments.
Application to all
kinds of levers.
Consequences of
this doctrine of
levers, & the prin-
ciple of the steel-
yard. The reason
why the whole may
be considered as if
collected at the
centre of gravity.
25O PHILOSOPHLE NATURALIS THEORIA
Thcoriam exhibere
egregie itidem cen-
trum Oscillationis.
Quid ipsum sit.
Preparatio ad solu.
tionem problematis
quærentis ipsum
centrum.
ex Q in A proportionalem ON, erit ex theoremate numeri
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 æquilibrium, & in E habebitur vis æqualis summæ 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
æquilibrium, & ibi sentietur vis æqualis summæ ponderum, cum ob naturam centri
gravitatis debeant esse singula pondera, seu massæ ductæ in suas perpendiculares distantias
a linea verticali, quam etiam vocant lineam directionis, hinc, & inde æqualia. Nam vires
ponderum sunt parallelæ, & in iis juxta num. 32o satis est ad æquilibrium, si vires motrices
sint reciproce proportionales distantiis a directione virium transeunte per tertium punctum :
sentietur autem in suspensione vis æqualis summæ ponderum. Atque inde fluit, quidquid
vulgo traditur de æquilibrio 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 æquilibrium impedito ejus
descensu tantummodo. Gravitas omnium punctorum non applicatur ad centrum gravi-
tatis, nec ibi ipsa agit per sese ; sed ejusmodi esse debent distantiæ punctorum totius
systematis, ut inter fulcrum, & punctum ipsi imminens habeatur vis quædam æqualis
summæ virium omnium parallelarum, & directa ad partes oppositas directionibus illarum.
[I52] 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 tam cito, quam requirunt propiora, neque tam lente, quam
remotiora, sed actio mutua debebit accelerare hæc, retardare illa. Erit autem aliquod
punctum, quod nec accelerabitur, nec retardabitur, sed oscillabit, tanquam si esset solum.
Illud dicitur centrum oscillationis. Determinatio illius ab Hugenio primum est facta, sed
precario, & non demonstrato principio : tum alii alias itidem obliquas inierunt vias, ac
præcipuas quasque methodos huc usque notas persecutus sum in Supplementis Stayanis
§ 4 lib. 3. En autem ejus determinationem simplicissimam ope ejusdem theorematis
numer1 3 I 3.
329. Sint plures massæ, quarum una A in fig. 63, mutuis viribus singulæ connexæ cum
P, cujus motus sit impeditus suspensione, vel fulcro, & cum massa Q jacente in quavis
recta PQ, cujus massæ Q motus a mutuo nexu nihil turbetur, quæ nimirum sit in centro
oscillationis. Porro hic cum massas pono in punctis spatii A, P, Q, intelligo vel puncta
singula, vel quævis aggregata punctorum, quæ concipiantur, ut compenetrata in iis punctis.
Velocitati jam acquisitæ in descensu nihil obstabit is nexus, - g É>
cum ea sit proportionalis distantiæ a puncto suspensionis P, 273
nisi quatenus per eum nexum retrahentur omnes massæ 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, quæ jungit massam cum A
P, altera sit ipsi perpendicularis, idem punctum P sustinebit X.
etiam priorem illam, posterior autem determinabit massas
ad motus AN, QM, perpendiculares ipsis AP, QP, ac pro- N /
portionales per num. 3oI 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 obliquæ, sed ex parte impeditæ a
nexu, revera percurrat massa A ; quoniam Q non turbatur, G} G}
adeoque percurrit totum suum spatium QM ; erit QM ad >J
AO, ut QP ad AP. Demum actio ex A in Q ad actionem Mi R
3 I4 ut est Q × QP ad A × AP, & omnes ejusmodi actiones FIG. 63.
ab omnibus massis in Q debebunt evanescere, positivis & negativis valoribus se mutuo
elidentibus. Ex illis tribus proportionibus, & hac æqualitate res omnis sic facillime
expeditur. -
33o. Dicatur QM = V, sinus APR = a, sinus QPR = q. Erit ex prima proportione
q : a : : QM — vs AN-;x V. [153] Ex secunda QP. AP : : QM = V. Ao=;$xv.
Quare ON =(£-; ) × V. Sed ex tertia - -
s q QP t AP A x AP A x AP. V.
. . ON . . ( 4 . **£ a X A X Af' _ A X AF*\ J _V_
ox or. A x AP : oN-(; §)x V. ( q. QP )*o*op.

A THEORY OF NATURAL PHILOSOPHY 25 I
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 º: º:
oscillate more quickly, & longer ones more slowly. Hence when several weights are centre of cºilº
connected together, one nearer to the axis of oscillation, & another more remote from it, *W*
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, ...?.
or by a fulcrum ; also let A be connected with a mass Q lying in a straight line PQ, & let Fºº mains
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 ... "
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 PHILOSOPHIAE NATURALIS THEORIA
Evolutio casus pon-
derum jacentium in
eadem recta cum
puncto suspension-
1S.
Et casus jacentium
extra.
Initium applica-
tionis ad oscillati-
ones in latus pon-
derum jacentium in
eodem plano.
Finis ejusdem cum
formula generali.
Corollarium
pro
positione centri
o scillationis, S&
gravitatis ex eadem
parte a
suspenSIOn1s.
puncto
que erit actio in Q ex nexu cum A. Ateodem 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, & deberente 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 APº + B x BP2 &c.
º sºm- QP 5
q
A X AP2 + B x BP2 &c.
- a X A X AP + b X B X BP &c.
33I. Sint jam primo omnes massa in eadem recta linea cum puncto suspensionis
P, & cum centro oscillationis Q ; & angulus QPR aequabitur cuivis ex angulis APR, ac ejus
A x APº + B x BPº &c.
A X AP + B x BP &c. ”
quae est ipsa formula Hugeniana pro ponderibus jacentibus recta transeunte per centrum
Suspensionis.
& inde QP = q x
sinus q singulis sinubus a, b&c. Quare pro eo casu formula evadit
332. Quod si iaceant extra eiusmodi rectam in plano POR perpendiculari ad azem
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
d P = A X AP2 + B X BP2 &c
se evada OP = 1 A vºi i
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
7. A X APº -- B X BP2 &c.
8 M X GP
Sed si summa massarum dicatur M,
[154] 333. Is valor erit variabilis pro varia inclinatione ob valores sinuum q, & g
variatos, nisi QP transeat per G, quo casu sit q = g; & quidem ubi G accedit in infinitum
ad PR, decrescente gin 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 º evanescit. Id vero accidit, quia in appulsu G ad verticalem totum systema
vim acceleratricem in infinitum imminuit, & lentissime acceleratur 3. adeoque ut radius
PQ adhuc obliquus sit ipsi in ea particula oscillationis infinitesima isochronus, nimirum
aeque 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. o º e - a e º
334. Quare ut habeatur pendulum simplex constantis longitudinis, & in quacunque
inclinatione isochronum composito, debet radius PQ ita assumi, ut transeat per centrum
gravitatis G, quo unico casu fit constanter q = g, & formula evadit constans
QP = A x AP2 - B x BPº se, quae est formula generalis pro oscillationibus in latus
M X GP i si a
massarum quotcumque, & quomodocunque collocatarum in eodem plano perpendiculari
ad axem rotationis, qui casus generaliter continet casum massarum jacentium in eadem
recta transeunte per punctum suspensionis, quem prius eruimus.
335. Inde autem pro hujusmodi casibus plura corollaria deducuntur. In primis
patet: gravitas centrum debere jacere in recta, que a centro suspensioni ducitur per centrum
oscillationis, uti demonstratum est num. 334. Sed & debet jacere ad eandem partem cum
ipso centro oscillationis. Nam utcunque mutetur situs massarum per illud planum,
manentibus puncto suspensionis P & centro gravitatis G, signum valoris quadrati cuiusvis
AP, BP manebit semper idem. Quare formula valoris sui signum mutare non poterit;
A THEORY OF NATURAL PHILOSOPHY 253
will be ( × A × AP A ãº) X gºor In the same manner, if there were another
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
(a × A × AP + b x B. × BP + &c.)/q = (A X AP* + B x BPs -- &c.)/QP;
. A X AP* + B X BP2 + &c.
and hence QP = q. a × 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
of suspension P, & so with the point of oscillation Q; then the angle QPR will be equal
to any one of the angles like APR, & its sine q will be equal to any one of the sines a, b
&c. Hence for this case the formula reduces to
A × AP* + B BP -- &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.
5
332. But if the masses lie outside of any such line, in the plane POR, perpendicular
to the axis of rotation passing through P, suppose that G is the common centre of gravity
of all the masses, & let perpendiculars AA’, GG’, QQ' be drawn to PR. Then, since the
radius (= 1): a = AP : AA', therefore AA' = a X AP: & in a similar manner, QQ' =
q × QP, & GG’ = g × 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
QP = A X AP2 -- B X BP2 + &c.
- T *ATXAATEX BBTE &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' + &c. = M × GG’ = M × g × GP;
& therefore we obtain the value of the radius QP, in a form that is independent of the
inclination, namely,
A × AP2 + BxBP2 + &c.
M × GP
333. The value obtained will vary with various inclinations, owing to the varying
values of the sines q &g, unless QP passes through G.; in which case q = g. Indeed, when
G approaches indefinitely near to PR, & g thus decrease indefinitely, if PQ does not pass
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
at any inclination with the composite pendulum, the radius PQ must be so taken that it
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 AP* + B × BP2 + &c.
$. * M × GP e
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
proved above. First of all, it is clear that —The centre of gravity must lie in the straight
line joining the centres of oscillation & suspension; this has been proved in Art. 335. But
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
7 ×
Derivation of the
case of weights
hanging in the
Same Straight line
with the point of
Suspension. -
The case of when
the masses are not
on this line.
Commencement of
the application to
Oscillations to one
side of bodies lying
in the same plane.
Conclusion of the
Same, with a gene-
ral formula.
Corollary with re-
Spect to the
Centres of oscillation
& gravity on the
Same side of the
point of Suspension.
254 PHILOSOPHIAE NATURALIS THEORIA
Centrum gravitatis
debere esse inter
bina reliqua ex iis
punctis.
adeoque si in uno aliquo casu jaceat Q respectu P ad eandem plagam, ad quam jacet G;
debebit iacere semper. Jacet autem ad eandem plagam in casu, in quo concipiatur, omnes
massas abire in ipsum centrum gravitatis, quo casu pendulum evadit simplex, 8 centrum
oscillationis cadit in ipsum centrum gravitatis, in quo sunt massa. Jacebit igitur semper
ad eandem partem cum G.
[I55] 336. Deinde debet centrum gravitatis jacere inter punctum
suspensionis, 85 centrum oscillationis. Sint enim in fig. 64
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
plano, vel in earum quadrata, designetur praefixa litera I soli
termino pertinente ad massam A, ut contractiores evadant
g - ſ.A X AP2
Erit ex formula inventa PO = si S GP
Porro est AG° = APº + GP2 – 2 GP x Pa, adeoque
- APº = AGº – GP2 + 2 GP x Pa,
& ſ.Ax GP2 est M x GP2,
ob GP constantem ; ac .A x Pa = M X GP, cum Pa sit
aequalis distantiae massa a plano perpendiculari rectae QP
transeunte per P, & eorum productorum summa aequetur distan-
tiae centri gravitatis ductae in summam massarum ; adeoque
P
demonstrationes.
FIG. 64.
ſ.A x 2 GP x Pa erit = 2 M x GP2.
Valor constans pro-
ducti ex binis dis-
tantiis centri gravi-
tatis ab iisdem.
Manente p un ct o
suspension is &
centro gravitatis,
InaIlere Centrum
oscillationis.
Centrum oscilla-
tionis, 8 punctum
suspensionis recip-
ITO Ca,T1.
Altera ex iis dis-
tanti i s e vanes-
cente, abire alteram
in infinitum.
Suspensione facta
per centrum gravi-
tatis, nullum haberi
motum.
Quae distantia cen-
tri oscillationis
omnium minima
pro data positione
mutua II la SS3,Ill1 Il
datarum : maxi-
mam haberi nullam.
Quare J.A X AP” A X AG” – M X GPº + 2 M x GPº I.A X AG - GP.
M X GP M X GP M X G
if io i o ſ.A X AGº
Eritigitur PQ major, quam PG, excessu GQ= M GP
337. Exillo excessu facile constat, mutato utcunque puncto suspensionis, rectangulum
sub binis distantiis centri gravitatis ab ipso, & a centro oscillationis fore constans. Cum
o
enim sit QG – º ", ei GO x GP – º Aº
M X GP º - M
& habetur hujusmodi elegans theorema : singula massa ducantur in quadrata suarum
distantiarum a centro gravitatis communi, 85 dividatur omnium eiusmodi productorum summa
per summam massarum, ac habebitur productum sub bimis distantiis centri gravitatis a centro
suspensionis 5 a centro oscillationis.
338. Inde autem primo eruitur illud; manente puncto suspensionis, 85 centro gravitatis,
debere etiam centrum oscillationis manere nihil mutatum ; utcunque totum systema, servata
espectiva omnium massarum distantia, 5 positione ad se invicem convertatur intra idem
planum circa ipsum gravitatis centrum ; nam illa GP inventa eo pacto pendet tantummodo
a distantiis, quas singulae massa habent a centro gravitatis.
339. Sed & illud sponte consequitur : Centrum oscillationis, 85 centrum suspensionis
reciprocari ita, ut, si fiat suspensio per id punctum, quod fuerat centrum oscillationis, evadat
oscillationis [156] centrum illud, quod fierat punctum suspensionis; 5 alterius distantia a
centro gravitatis mutata, mutetur 5 alterius distantia in eadem ratione reciproca. Cum enim
earum distantiarum rectangulum debeat esse constans; si prosecunda ponatur valor, quem
habuerat prima ; debet pro prima obvenire valor, quem habuerat secunda, & altera debet
aequari quantitati constanti divisa per alteram. -
34o. Consequitur etiam illud: Altera ex iis bini, distantiis evanescente, abibit altera
in infinitum, nisi omnes massa in unico puncto sint simul compenetrata. Nam sine ejusmodi
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 finito, crescit in infinitum ;
si divisor in infinitum decrescat. - -
341. Hinc vero iterum deducitur: Suspensione facta per ipsum centrum gravitatis
nullum motum consequi. Evanescit enim in eo casu distantia centri gravitatis a puncto
suspensionis, adeoque distantia centri oscillationis crescit in infinitum, & celeritas
oscillationis evadit nulla.
342. Quoniam utraque distantia simul evanescere non potest, potest autem centrum
oscillationis abire in infinitum ; nulla erit maxima e longitudinibus penduli simplicis
isochroni pendulo facto per suspensionem dati systematis; sed aliqua debet esse minima,
suspensione quadam inducente omnium celerrimam dati systematis oscillationem. Ea
vero minima debet esse, ubi illae binae distantiae aequantur inter se : ibi enim evadit minima
earum summa, ubi altera crescente, 8 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
, quod productum est constans,

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
{5 the centre of oscillation. For, in Fig. 64, let the points A, P, G, O be the same points as
in Fig. 63; & let AG, AQ, & Aa be drawn perpendicular to PQ. Then, the sum of all
the masses, each multiplied into its distance from some chosen straight line or plane, or
into their squares, may be designated by the letter ſ 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 = ſ.A X AP* /M × GP. Now AG* = AP* + GP2 — 2GP × Pa, &
therefore AP2 = AG2 – GP2 + 2GP × Pa; & J.A X GP2 = M × GP8, since GP is
constant; also J.A X Pa = M × 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 × 2GP × Pa = 2M × GP*. Therefore
2 /i _ J.(A X AG” – M × GP2 + 2M × GP2) J.A. × AG”
ſ.A X AP2/M × GP = M X GP = ºrº- + GP.
Hence PQ will be greater than PG; & the excess GQ will be equal to J.AXAG”/MXGP.
337. From the value of this excess, it is readily seen that, however the point of
suspension may be changed, the rectangle contained by the two distances of the centre
of gravity from it & from the centre of oscillation, will be constant. For, since
QG = ſ.A X AG2/M × GP, it follows that GQ X GP =ſ.A X AG”/M ; & this product
is constant. Hence we have the following elegant theorem —If each of the masses is multi-
plied by the square of its distance from the common centre of gravity, 85 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 & oscillation.
338. Now, from this theorem, we can derive first of all the following theorem. If
the centre of gravity & the centre of suspension remain unchanged, then also the centre of
oscillation must remain quite unchanged ; no matter how the whole system is rotated about the
centre of gravity, in the ſame plane, so long as the mutual distances of all the masses & their
position with regard to one another are preserved. For, the value of GP found in the manner
above depends solely on the distances of the several masses from their centre of gravity.
339. But there is another theorem that also follows immediately. The centre of
oscillation & the centre of suspension are mutually related to one another in such a fashion
that, if the suspension is made from the point which formerly was the centre of oscillation, then
the new centre of oscillation will prove to be that point which was formerly the centre of suspension;
85 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
b
#. 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,
unless all the masses are condensed at a single point. For, unless there is condensation of
this kind, the sum of all the products formed from the masses & the squares of their distances
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
gravity, no motion will ensue. For, in this case, the distance of the centre of gravity
from the centre of suspension vanishes and so the distance of the centre of oscillation
increases indefinitely, & therefore the speed of the oscillation becomes zero.
342. Since both distances cannot vanish together, but the centre of oscillation can
go off to infinity, there cannot be a maximum among the lengths of a simple pendulum
isochronous with the pendulum made by the suspension of the given system ; but there
must be a minimum, since there must be one suspension of the given system which will give
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
their distances remains constant, if for the second there is substituted that which the
Of the three points,
the centre of gra-
vity must lie be.
tween the other
two.
The value of the
product of the
distances of the
centre of gravity
from the other two
Centres is constant.
If the centre of
Suspension & the
centre of gravity
remain unchanged,
So also will the
centre of oscilla-
tion.
The centre of oscil-
lation & the centre
of Suspension are
reversible.
If one of the dis.
tances vanishes, the
Other will become
infinite.
If the suspension
is made from the
centre of gravity,
there is no motion.
To find the least
dist a n ce of the
centre of oscillation
for a given position
of the masses with
regard to on e
another ; there is
no maximum dis. “
tance,
256 PHILOSOPHIÆ NATURALIS THEORIA
Superiora habere
locum t an t u m-
modo, ubi omnes
massae sint in eo-
dem plano perpen-
diculari a d axem
rotationis : transi-
tus a d centrum
percussionis.
Eræparatio ad in-
v e n ie n d u m cen-
trum percussionis
massarum jacen-
tium in e a, de m
1recta. .
Calculus cum ejus
determinatione.
igitur A « AP × PQ + B × BP × PQ &c.
Determinatio v i s
percussionis in ipso
CentrO.
infinitesimum erit ad alterius decrementum in ratione ipsarum, nec ea æquari poterunt
inter se, nisi ubi ipsæ distantiæ inter se æquales fiant. Tum vero illarum productum
evadit utriuslibet quadratum, & longitudo penduli simplicis isochroni æquatur eorum
summae ; ac proinde habetur hujusmodi theorema : Singulae massae ducantur im quadrata suarum
distantiarum a centro gravitatis, ac productorum summa dividatur per summam massarum :
& dupla radix quadrata quoti exhibebit minimam penduli simplicis isochroni longitudinem.
Vel Geometrice sic : Pro quavis massa capiatur recta, quae ad distantiam cujusvis massae a
centro gravitatis sit in ratione subduplicata ejusdem massae ad massarum summam : inveniatur
recta, cujus quadratum aequetur quadratis omnium ejusmodi rectarum simul : & ipsius duplum
dabit quaesitum longitudinem mediam, quae brevissimam praestet oscillationem.
343. Hæc quidem omnia locum habent, ubi omnes massæ sint in unico plano perpen-
diculari ad axem rotationis, ut ni-[I57]-mirum singulæ massæ possint connecti cum centro
suspensionis, & centro oscillationis. At ubi in diversis sunt planis, vel in plano 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 (4) ; & relatio virium, quæ in latus agant extra planum, in quo tres e massis
jaceant, quæ perquisitio est operosior, sed multo foecundior, & ad problemata plurima
rite solvenda magni usus ; sed quæ 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 massæ A, B, C, D connexæ inter se in recta P.
quadam, quæ concipiatur revoluta circa punctum P in ea situm, & I
quæratur 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, singulæ massæ habebunt velocitates Aa, Bb
&c. proportionales distantiis a puncto P, adeoque singularum motus,
qui per mutuas vires motrices extingui debent, poterunt exprimi per
A × AP, B × BP &c. Quare vires motrices in iis debebunt esse pro-
portionales iis motibus. Concipiantur singulæ connexæ cum punctis
P, & Q, & quoniam velocitas puncti P erat nulla ; ibi omnium ac-
tionum summa debebit esse = o : summa autem earum, quæ habentur
in Q, elidetur a vi externa percussionem sustinente.
345. Quoniam actiones debent esse perpendiculares eidem rectæ
jungenti massas, erit per theorema numeri 3 I 4, ut PQ ad AQ, ita
A
C
£)
actio in A = A × AP, ad actionem in P = A. X £;^ AQ, sive ob -
AQ = PQ — AP, erit ea actio [I58] A × AP *,£-^ <^P*. Eodem FIG. 65.
- B × BP × BQ — B × PB2
pacto actio in P ex nexu cum B erit , & ita porro. Iis omnibus
PQ
positis=o, divisor communis PQ abit, & omnia positiva æquantur negativis. Erit
= A × AP* -+- B × BP* &c. ; quare
- 2 2 e •
PQ A × AP -|- B × BP &c, quæ formula est eadem, ac formula centri oscilla-
A × AP -- B × BP &c. * • • ©
tionis, ac habetur hujusmodi theorema : Distantia centri percussionis a puncto conver-
sionis aequatur distantiae cemtri oscillationis a puncto suspensionis ; adeoque hic locum habent
in hoc casu, quæcunque de centro oscillationis superius dicta sunt.
346. Quod si quis quærat vim percussionis in Q, hic habebit
• • A × AP2
QP. AP : : AxAP. -;;;*-
venientur vires ex reliquis : adeoque summa virium erit
, quæ erit vis in Q ex nexu cum A. Eodem pacto in-
A × AP2 -|- B × BP2
PQ
(q) Systema binarum massarum cum binis punctis commexarum, & inter se, sed adhuc in eodem plano jacentium,
persecutus fueram ante aliquot annos ; quod sibi a me communicatum exhibuit in sua Synopsi Physicæ Generalis
P. Benvemutus, ut ibidem ipse innuit. Id inde excerptum habetur hic im Supplementis § 5. •
Habetur autem post idem supplementum & Epistola, quam delatus Florentiam scripsi ad P. Scherfferum, dum hoc
ipsum opus relictum Viennæ ante tres menses jam ibidem imprimeretur, quae quidem adjecta est in ipsa prima editione
in fine operis. Ibi & theoriam trium massarum extendi ad casum massarum quatuor ita ; ut inde generaliter deduci
possit & æquilibrium, & centrum oscillationis, & centrum percussionis, pro massis quotcunque, & utcumque dispositis.
&c.,



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 —If each mass is multiplied by the square of
the distance from the centre of gravity, {5 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
to the axis of rotation, so that each of the masses can be connected with the point of
suspension & the centre of oscillation. But, when they are in different planes, or all in
a plane that is not perpendicular to the axis of rotation, it is necessary to connect each
of the masses with a pair of points on the axis & with the centre of oscillation : & we thus
have the case of a system of four masses acting upon one another (4), & the relation between
the forces which act to one side, out of the plane in which three of the masses lie. This
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,
which is supposed to be rotated about a point P situated in it; it is required to find in
this straight line a point Q such that, if its motion is prevented, then the whole motion of
the masses is also prevented through the mutual actions. This point is called the centre
of percussion. Now, since the whole system rotates round P, each of the masses will have
velocities, such as Aa, Bb, &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. × 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,
we shall have, by Art. 314, PQ to AQ as the action on A, which is equal to A X AP, is
to the action on P; hence the latter is equal to AX AP × AQ/PQ, or, since AQ = PQ – AP,
this action will be equal to (A X AP × PQ–AX AP*)/PQ. In the same way, the
action on P due to the connection with B is equal to (B X BP × PQ – B X BP*)/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 × AP × PQ —- B X BP × PQ –– &c. = A X AP2 -- B X BP2 + &c. Hence
PO - A X A. H × BP2 + &c. -
T A X AP + B X BP + &c.
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
is to the force on Q due to the connection with A; hence this latter is equal to AXAP*/PQ.
In the same way we can find the forces due to the rest; and thus the sum of all
the forces will be (A X AP* + B × BP* + &c.)/PQ. Now, since PQ is equal to
(q) I investigated the system of two masses connected with two points & with one another, yet all lying in the
same plane, several years ago : &S, when I had communicated the matter to Father Benvenuto, he expounded it in his
Synopsis Physicae 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 I wrote to Father Scherſfer when I
reached Florence, whilst this work, which I had left in his hands at Vienna three months before, was in the press there;
{5 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 four masses, in such a manner that from it it is possible to deduce, in a perfectly general way, the equilibrium,
the centre of oscillation, & the centre of percussion for any number of masses disposed in any manner whatever.
S
The theorems given
above only hold
good when all the
masses are in the
same plane perpen-
dicular to the axis
of rotation ; let us
pass on to the centre
of percussion.
Preparation for
finding the centre
of percussion for
masses lying in the
same straight line.
The calculation
giving the deter-
mination of this
Centre.
, which is the same formula as the formula for the
Determination of
the force of per-
cussion a t the
centre of percus-
Sion.
258 PHILOSOPHIÆ NATURALIS THEORIA
Omitti hic multa
quæ adhanc Theo-
riam pertinerent,
ad quam pertinet
universa Mechanica,
Pressio fluidorum
si puncta sint in
recta verticali.
Eadem p u n ct is
utcunque dispersis,
& cum omnibus
directionibus agens.
ob PQ = A × AP2 -|- B × BP2 &c.
A × AP + B × BP &c. *
nimirum ejusmodi vis erit æqualis summæ virium, quæ requiruntur ad sistendos omnes
motus massarum A, B, &c., cum illis diversis velocitatibus progredientium, videlicet ejusmodi,
quæ in massa percussionem excipiente possit producere quantitatem motus æqualem
toti motui, qui sistitur in massis omnibus, quod congruit cum lege actionis, & reactionis
æqualium, & cum conservatione ejusdem quantitatis motus in eandem plagam, de quibus
sive
summa illa erit A × AP + B × BP &c. ;
egimus num. 265, & 264.
347. Haberent hic locum alia sane multa, quæ pertinent ad summas virium, quibus
agunt massæ, compositarum e viribus, quibus agunt puncta, vel a Newtono, vel ab aliis
demonstrata, & magni usus in Mechanica, & Physica : hujusmodi sunt ea omnia, quæ
Newtonus habet sectione I 2, & I 3 libri I Princip. de attractionibus corporum sphæricorum,
& non sphæricorum, quæ componantur ex attractionibus particularum ; ubi habentur
præclarissima theoremata tam 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 [I59] orbem sphæricum, vel ellipticum vacuum nullas vires
sentit, elisis contrariis ; intra globos plenos punctum habet vim directe proportionalem
distantiæ 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, quæ
Mac-Laurinus tradit de sphæroide elliptico potissimum, quæ Clairautius de attractionibus
pro tubulis capillaribus, quæ D'Alembertus, Eulerus, aliique pluribus in locis
persecuti sunt ; quin omnis Mechanica, quæ agit vel de æquilibrio, vel de moti-
bus, seclusa omni impulsione, huc pertinet, & ad diversos arcus reduci potest curvæ
nostræ, qui possunt esse quantumlibet multi, habere quascunque amplitudines, sive
distantias limitum, & areas quæ sint inter se in ratione quacunque, ac ad curvas
quascunque ibi accedere, quantum libuerit ; sed res in immensum abiret, & satis
est, ea Omn1a 1nnu1sse.
348. Addam nonnulla tantummodo, quæ 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 æquilibrio. Inter
primum punctum A, & secundum ipsi proximum debebit esse vis repulsiva, quæ
æquetur vi externæ puncti A. Quare urgebitur punctum secundum hac vi repul-
siva, & præterea vi externa sua. Hinc vis repulsiva inter secundum, & tertium
punctum debebit æquari vi huic utrique, adeoque erit æqualis summæ virium ex-
ternarum puncti primi, & secundi. Adjecta igitur sua vi externa tendet deorsum
cum vi æquali summæ virium externarum omnium trium ; & ita porro progred- E
iendo usque ad B, quodvis punctum urgebitur deorsum vi æquali summæ 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 externæ exprimat recta FH in fig. 67, & FEGH
faciem ipsi parallelam ; adhuc facile demonstrari potest
componendo, vel resolvendo vires ; sed & per se patet,
vires repulsivas, quas debebit ipsa basis exercere in par-
ticulas sibi propinquas, & ad quas vis ejus mutua perti-
nebit, fore æquales summæ omnium superiorum virium
externarum ; atque id erit commune tam solidis, quam
fluidis. At quoniam in fluidis particulæ possunt ferri .:;:::::::;;:::$;
directione quacunque, quod unde proveniat, videbimus ÉÉ
in tertia parte ; quævis particula, ut ibidem videbimus, . j3;
in omnem plagam urgebitur viribus æqualibus, & urgebit
sibi proximas, quæ pressionem in alias propagabunt ita,
A.
$
i
j
s ve •
es I * • s * • *
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•• N s E e, •X • • • § 3 «• û
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• XI° • •„* s
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•, • • • • •
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Κ
ut, quæ sint in eodem plano LI, parallelo FH, in cujus ÉÉÉÉÉÉÉÉÉ
directione [160] nulla vis externa agit, vires ubique ÉÉÉÉÉÉÉÉÉÉÉ
eædem sint. Quamobrem quævis particula sita ubicun- É:::::§3;
que in ea recta in N, habebit eandem vim tam versus ÉÉÉ
s. 5: •¥. 3• :£; :::::::: ę •• J •aXXX• • •• •<*• •
planum EF, quam versus planum EG, & versus FH, quam ÉÉÉÉÉÉÉÉÉÉÉ
habet particula collocata in eadem linea in MK etiam, E
ubi addantur parietes AM, CK paralleli FE, cum planis R.
ç» se � ■* IG. 67.
LM, KI, parallelis FH, nimirum vi, quæ respondet
altitudini MA : ac particula sita in O prope basim FH urgebitur, ut quaquaversum, ita
Xs5$*J£. w •
& versus ipsam, iisdem viribus, quibus particula sita in BD sub AC. -Ipsam urgebunt







A THEORY OF NATURAL PHILOSOPHY 259
(A X AP* +B × BP* + &c.)/(A × AP + B X BP + &c.), this sum will be equal to
A X AP +B X BP + &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
of forces, with which masses act, these being compounded from the forces with which
points act; such as have been proved by Newton & others; & things that are of great use
in Mechanics & Physics. Of this kind are all those which Newton has in the 12th & 13th
sections of The First Book of the Principia concerning the attractions of spherical bodies,
& 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 of 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
fluids. Suppose we have a set of points, in Fig. 66, lying in a straight line, extended in any
direction AB, under the action of some force external to the system of points; & suppose
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
throughout a parallelepiped, & if, in Fig. 67, FH denotes the base of the parallelepiped,
which is perpendicular to the direction of the external force, & FEGH is a face parallel
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 straightline, 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
Many things per-
taining to the
Theory must here
be omitted ; for the
whole of Mecha-
nics pertains to this
Theory.
Pressure of fluids
when the points are
all in a vertical
line.
The same for points
disposed in any
manner, & acting
in all directions.
26o PHILOSOPHIÆ NATURALIS THEORIA
Inde, cur exiguo
fluidi pondere fieri
possit ingens pres-
sio.
Pressio fluidorum
cum compressione
s e n s i b i l i u n d e
proveniat in hac
Theoria.
Compressio aeris
a qua vi proveniat :
aquæ compressio
cur ad sensum
nulla : unde muta-
tio in vapores tam
elasticos.
particulæ in eodem plano horizontali jacentes, & accedet ad omnes fluidi, & baseos particu-
las, dome$ vi contraria elidatur vis ejüs tota ab ejusmodi pressione derivata. Quamobrem
basis FH a fluido tanto minore FLMACKIH $entiet pfessionem, quam sentiret a toto
fluid9 FEGH : superficies autem LM sentiet a particulis N vim æqualém vi massae LEAM,
accedentibus ad ipsam particulis, donec vis mutua repulsiva ei vi æquétur.
350. Hinc autem patet, cur in fluidis nostris gravitate præditis basis FH sentiat
pressionem tant9. majqrem massæ fluidæ incumbentis pondere, & cur pondere perquam
exiguo fluidi AMKC elevetur pondus collocatum supra LM etiam immaiie, ubi repagulum
LM sit ejusmodi, ut pressioni fluidi parere possit, quemadmodum sunt coriaêeî. At
totum yas FLMACKIH bilanci impositum habebit póndus æquale ponderi suo, & fluidi
contenti tantummodo : nam superficies vasis LM, KI horizóntali§ vi repulsiva mutua
urgebit $ur$um, quantum urget deorsum puncta omnia N versus O, & illa pressio tantundem
imminuit vim, quam in bilancem exercefvas, ac tota vis ipsius habebitur dempta pressione
sursum superficiei LM, KI a pressione fundi FH facta deorsum : & pariter se mutuo elident
yires exergitæ in parietes oppositos, Atque hæc Theoria poterit applicari facile aliis etiam
figuris quibuscunque. Respondebit semper pressio superficiei, & fóti ponderi fluidi, quod
habeat basim illi superficiei æqualem, & altitudinem ejusmodi, quæ üsque ad supremam
superficiem pertinet inde accepta in directione illius externæ vis.
35 I. Quod si vires particularum repulsivæ sint ejusmodi, ut ad eas multum augendas
requiratur mutatio distantiæ, quæ ad distantiam totam habeat rationem sensibilem ; tum
vero compressio massæ erit sensibilis, & densitas in diversis
altitudinibus admodum diversa: sed iniisdem horizontalibus M.
planis eadem. Si vero mutatio sufficiat, quæ rationem habet
prorsus insensibilem ad totam distantiam ; tam 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, quæ 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 rectæ perpendiculares DE, BF itidem in
quacunque ratione minoris inæqualitatis utcunque magna :
poterit utique arcus MN curvæ exprimentis mutuas par- -
ticularum vires transire per illa puncta F, F, & exhibere quodcunque pressionis incre-
mentum cum quacunque pressione utcunque magna, vel utcunque insensibili.
352. Compressionem ingentem experimur in aere, quæ in eo est proportionalis vi
comprimenti. Pro eo casu demonstravit Newtonus Princ. Lib. 3. prop. 23, vim particularum
repulsivam mutuam debere esse in ratione reciproca simplici distantiarum. Quare in iis
distantiis, quas habere possunt particulæ aeris perseverantis cum ejusmodi proprietate,
& 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
hyperbolæ conicæ Apollonianæ. 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 eædem particulæ debent esse prope limites, nam & distractioni resistit aqua.
Infinita sunt curvarum genera, quæ 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
distantiæ AD. Demonstratur passim, subtangentem logisticæ ad intervallum ordinatarum
exhibens rationem duplam esse proxime ut I4 ad IO ; & eadem subtangens ad intervallum,
quod exhibeat ordinatas in quacunque magna ratione inæqualitatis, habet in omnibus
logisticis rationem eandem. Si igitur minuatur subtangens logisticæ, quantum libuerit ;
minuetur utique in eadem ratione intervallum BD respondens cuicunque rationi ordina-
tarum BF, DE, & accedet ad æqualitatem, 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

E
`~ N
D C
FIG. 68.
A
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,
in our fluids possessed of gravity, to a pressure so much greater than the weight of the
superincumbent fluid ; & why by a very small weight of fluid, like AMKC, the weight
collected above LM can be upheld, even though this is immensely great, when the restraint
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
increase them to any sensible extent, a change of distance is required, which bears a sensible
ratio to the whole distance; then the compression of the mass will also be sensible, & the
density at different heights will be quite different; nevertheless, they will still be the
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; take 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 compressing force. For this case, Newton proved, in prop. 3, of the Third Book of
his Principia, that the mutual repulsive force between the particles must be inversely
proportional to the first power of the distance. Hence, for these distances, which the
particles of air can have as it persists with a property of this kind, & does not induce another
form (for Newton remarked that an air could from being volatile become fixed, & Hales
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 near 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 Io; & 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
Hence the reason
why in a very small
amount of fluid
there can exist a
very great pressure.
The source of
pressure for fluids
with sensible com-
pression according
to this Theory.
The force that
gives rise to the
compression of air ;
the reason for the
incompressibility of
water ; the source
of the change in
highly elastic va-
pours.
262 PHILOSOPHIÆ NATURALIS THEORIA
Ubi pressio propor-
tionalis altitudini,
& unde.
Quomodo fiat aC-
celeratio in effluxu.
Unde v e 1 o citas
aquæ effluentis sub-
duplicata, altitudi-
nis. -
longitudinis, & vires in singulis punctis res-
Q u id requiritur,
ut velocitas sit
æqualis illi, quæ
habetur ca, d e n d O
per altitudinem.
in quacunque ingenti ratione auctæ cum compressione utcunque exigua, & ratione densi-
tatum utcunque accedente ad æqualitatem. Verum ubi ordinata ED jam satis exigua
fuerit, debet curva recedere plurimum ab arcu logisticæ, ad quem accesserat, & qui in
infinitum protenditur ex parte eadem, ac debet accedere ad axem AC, & ipsum secare,
ut habeantur deinde vires attractivæ, quæ ingentes etiam esse possunt ; tum post exiguum
intervallum debet haberi alius arcus [I62] repulsivus, recedens plurimum ab axe, qui
exhibeat vires illas repulsivas ingentes, quas habent particulæ aqueæ, ubi in vapores abierunt
per fermentationem, vel calorem. -
353. In casu densitatis non immutatæ ad sensum, & virium illarum parallelarum
æqualium, uti eas in gravitate nostra concipimus, pressiones erunt ut bases, & altitudines ;
nam numerus particularum paribus altitudinibus respondens erit æqualis, adeoque in
diversis altitudinibus erit in earum ratione ; virium autem æqualium summæ erunt ut
particularum numeri. Atque id experimur in omnibus homogeneis fluidis, ut in Mercurio,
& aqua.
354. Ubi facto foramine liber exitus relinquitur ejusmodi massæ particulis, erumpent
ipsæ velocitatibus, quas acquirent, & quæ respondebunt viribus, quibus urgentur, & spatio,
quo indigent, ut recedant a particulis se insequentibus ; donec vis mutua repulsiva jam
nulla sit. Prima particula relicta libera statim incipit moveri vi illa repulsiva, qua
premebatur a particulis proximis : utcunque parum illa recesserit, jam secunda illi proxima
magis distat ab ea, quam a tertia, adeoque movetur in eandem plagam, differentia virium
accelerante motum ; & eodem pacto aliæ 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, donec ea evadat nulla ;
quin immo etiam aliquanto plus æquo a se invicem deinde recedunt particulæ, & jam
attractivis viribus retrahuntur, accedentes iterum, non quod retro redeant, sed quod
anteriores moveantur jam aliquanto minus velociter, quam posteriores ; tum iterum aucta
vi repulsiva incipiunt accelerari magis, & recedere, ubi & oscillationes habentur quædam
hinc, & inde. -
355. Velocitates, quæ remanent post exiguum quoddam determinatum spatium, in
quo vires mutuæ, vel nullæ jam sunt, vel æque augentur, & minuuntur, pendent ab area
durvæ, cujus axis partes exprimant non distantias, a proxima particula, sed tota spatia ab
initio motus percursa, & ordinatæ in singulis punctis axis exprimant vires, quas in iis habebat
particula. Velocitates in effluxu aquæ experimur in ratione subduplicata altitudinum,
adeoque subduplicata virium comprimentium.
Id haberi debet, si id spatium sit ejusdem
pondentibus ejus spatii sint in ratione primæ
illius vis. Tum enim areæ totæ erunt ut ipsæ
vires initiales, & proinde velocitatum quadrata,
ut ipsæ vires. Infinita sunt curvarum genera,
quæ rem exhibere possunt ; verum id ipsum
ad sensum exhibere potest etiam arcus al-
terius logisticæ cujuspiam amplioris illa, quæ
exhibuit distantias singularum particularum. sy :*
Sit ea in fig. 69 MFIN. Tota ejus area FIG. 69:
infinita ad partes CN asymptotica a quavis . I »
ordinata [163] æquatur producto sub ipsa ordinata, & subtangente constanti. Quare
ubi ordinataTÌÉD jam est perquam exigua respectu ordinatarum BF, HI , tota area
CDEN respectu CBFN insénsibilis erit, & areæ CBFN, CHIN. integræ accipi poterunt
pro areis FBDE, IHDE, quæ idcirco erunt, ut vires initiales BF, HI. • • ¢
356. Inde quidem habebuntur quadrata celeritatum proportignalia pressionibus, sive
altitúdinibus. Ut autem velocitas absóluta sit æqualis illi, quam particula acquireret cadendg
a superficie suprema, quod in aqua experimur ad sensum ; debet præterea tota ejusmodi
area æquari rectangulo facto sub recta exprimente yim gravitatis, unius particulæ, sive vis
repulsivæ, quam in se mutuo exercent binæ particulæ, quæ se primo repellunt, sustinente
inferiore gråvitatem superioris, & sub tota altitudine. Deberet eo casu esse totum pondus
BF ad iliam vim, ut ést altitudo tota fluidi ad subtangentem logisticæ, si FE est ipsius
logisticæ arcus. Est autem pondus BF ad gravitatem primæ particulæ, ut numerus
pärticularum in ea altitudiné ad unitatem, âdeoque ut.£adem illa tota altitudo ad
distantiam primarum particularum. Quare subtangens illius logisticæ deberet æquari

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
forces, such as we suppose our gravity to be, the pressures will be proportional to the bases
& the altitudes. For, the number of particles corresponding to equal altitudes will be
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.
54. When, on making an opening, a free exit is left for the particles of a mass, they
burst forth with the velocities which they acquire & which correspond to the forces urging
Where the pressure
is proportional to
the altitude, & the
reason for this.
How acceleration
in 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
mutual forces are either nothing or are equally increased & diminished, depend on the
area of the curve, of which parts of the axis represent not the distances from the next
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
the pressures, or the altitudes. Now, in order that the absolute velocity may be equal
to that which the particle would acquire in falling from the upper surface, as is found
to be approximately the case for water, we must have, in addition, that the whole of such
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
Why the velocity
of effluent water is
the sub-duplicate
of the height.
What is required
so that the velocity
shall be equal to
that acquired in
falling from the
given altitude.
264 PHILOSOPHIÆ NATURALIS THEORIA
illi distantiæ primarum particularum, quæ quidem subtangens erit itidem idcirco perquam
eXigua.
Tentandum , an in 357. An in omnibus fluidis habeatur ejusmodi absoluta velocitas & an quadrata
Š* $,$ velocitatum in effluxu respondeant altitudinibus ; per experimenta videndum est, ut
äïíåíem íériáí. constet, an curvæ 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. Hæc interea speciminis loco sint satis ad immensam quandam
hujusce campi fœcunditatem indicandam utcunque.
A THEORY OF NATURAL PHILOSOPHY 265
the primary particles; & thus the subtangent must also be itself very small on this
aCCOllnt.
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 sº *A.
experimentally; in 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 it.” 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
App/icatio Theoriæ ad Physicam
Ę? 358. In secunda hujusce Operis parte, dum Theoriam meam applicarem ad
$£"££ Mechanicam, multa identidem immiscui, quæ applicationi ad Physicam sternerent viam,
iim, tum de discrim- & vero etiam ad eandem pertinerent ; at hic, quæ pertinent ad ipsam Physicam, ordinatius
ine i n t e r VariaS persequar ; & primo quidem de generalibus agam proprietatibus corporum, quas omnes
species. o . . • «• • ç © � «• © �
omnino exhibet illa lex virium, quam initio primæ partis exposui ; tum ex eadem præcipua
discrimina deducam, quæ inter diversas observamus corporum species, & mutationes
ι• (… ę 9 q. • • p p » »
quæ ipsis accidunt, alterationes, atque transformationes evolvam.
Enumeratio earum, 359. Primum igitur agam de Impenetrabilitate, de Extensione, de Figurabilitate,
£.£'*"* *"* de Mole, Massa, & Densitate, de Ineftia, de Mobilitate, de Continuitate motuum, de
Æqualitate Actionis & Reactionis, de Divisibilitate, & Componibilitate, quam ego divisi-
bilitati in infinitum substituo, de Immutabilitate primorum materiæ elementorum, de
Gravitate, de Cohæsione, quæ quidem generalia sunt. Tum agam de Varietate Naturæ,
& particularibus proprietatibus corporum, nimirum de varietate particularum, & massarum
multiplici, de Solidis, & Fluidis, de Elasticis, & Mollibus, de Principiis Chemicarum
Operationum, ubi de Dissolutione, Præcipitatione, Adhæsione, & Coalescéntia, de Fermen-
tatione, & emissione Vaporum, de Igne, & emissione Luminis ; ac ipsis præcipuis Luminis
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, quæ ad
rei perquisitionem aptissima mihi videatur.
Impenetra bilitas 36o. Impenetrabilitas corporum a mea Theoria omnino sponte fluit ; si enim in
Ę, '" "** minimis distantiis agunt vires répulsivæ, quæ iis in infinitum immifiutis crescant in infinitum
ita, ut pares sint extinguendæ cuilibet velocitati utcunque magnæ, utique non potest
ulla finita vis, aut velocitas efficere, ut distantia duorum punctorum evanescat, quod
requiritur ad compenetrationem ; sed ad id præstandum infinita Divina virtus, quæ
infinitam vim exerceat, vel infinitam producat velocitatem, sola sufficit.
Ę. [165] 361. Præter hoc impenetrabilitatis genus, quod a viribus repulsivis oritur, est
Ę & áliud, quod provenit ab inextensione punctorum, & quod evolvi in dissertationibus
De Spatio, & Tempore, quas ex Stayanis Supplementis huc transtuli, & habetur hic in fine
Supplementorum § I, & 2. Ibi enim ex eo, quod in spatio continuo numerus punctorum
loci sit infinities infinitus, & numerus punctorum materiæ finitus, erui illud : nullum
punctum materiæ occupare unquam punctum loci, non solum illud, quod tunc occupat
aliud materiæ punctum, sed nec illud, quod vel ipsum, vel ullum aliud materiæ 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, quæ aliquod punctum massæ projectæ dirigant per rectam transeuntem per
aliquod punctum massæ, contra quam projicitur, est utique finitus ; cum numerus
punctorum in utraque massa finitus sit ; at numerus projectionum, quæ dirigant puncta
omnia per rectas nulli secundæ massæ puncto occurrentes, est infinities infinitus, ob puncta
spatii in quovis plano infinities infinita. Quamobrem, habita etiam ratione infinitorum
continui temporis momentorum, est infinities improbabilior primus casus secundo ; & in
quacunque projectione massæ contra massam nullus habebitur immediatus occursus puncti
materiæ cum altero puncto materiæ, 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
in also at the same time many things which opened the road for an application to Physics,
& really even belonged to the latter. In this part I will investigate in a more ordered
manner those things that belong to Physics. First of all, I will deal with general properties
of bodies; & these will be given by that same law of forces that I enunciated at the beginning
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,
Mass, Density, Inertia, Mobility, Continuity of Motions, the Equality of Action &
Reaction, Divisibility, & Componibility (for which I substitute infinite divisibility), the
Immutability of the primary elements of matter, gravity, & Cohesion; all these are general
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
forces act at very small distances, & these forces increase indefinitely as the distances
decrease, so that they are capable of destroying any velocity however large ; then there
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
is also another kind, which comes from the inextension of the points; this I discussed in
the dissertations De Spatio, ö Tempore, which I have abstracted from the Supplement
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 sif 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
We will first of all
here deal with the
general properties
of bodies, & then
with the difference
between the several
Species.
Enumera tion of
the matters to be
dealt with, & the
Order in which they
will be taken.
The origin of Im-
penetrability, ac-
C ording to this
Theory.
Another kind of
Impenetra bility,
pe culiar to this
Theory.
268 PHILOSOPHIÆ NATURALIS THEORIA
Sine viribus repul.
sivis debere haberi
compenetratione m
apparentem. Quid
eæ præsten
t in par.
ticulis, & velo quo.
dam, potissimum si
habeantur
toti.
aSymp-
Solidum indissolu-
bile, & impermea-
bile.
Alia ratio acqui-
rendi impenetra.
bilitatem, & nexum
p er asymptotos
remotas ab origine
abscissarum.
In iis & aliis casi-
bus resistentia, con-
tinua, sine continuo
faciente
absoluta.
meabilitas.
vim, &
1 m p er-
362. Si vires repulsivæ non adessent ; omnis massa libere transiret per aliam quanvis
massam, ut lux per vitra, & gemmas transit, ut oleum per marmora insinuatur ; atque id
semper fieret sine ulla vera compenetratione. Vires, quæ ad aliquod intervallum extend-
untur satis magnæ, impediunt ejusmodi liberum commeatum. Porro hic duo casus
distinguendi sunt ; alter, in quo curva virium non habeat ullum arcum asymptoticum
cum asymptoto perpendiculari ad axem, præter illum 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, quæ habeat arcum citra se attractivum, ultra repulsivum cum area infinita,
ut juxta num. I88 puncta posita in minore distantia non possint acquirere distantiam
majorem, nec, quæ in majore sunt, minorem ; tum vero particula composita ex punctis
in minore distantia positis, esset prorsus impenetrabilis a particula posita in majore distantia
ab ipsa, nec ulla finita velocitate posset cum illa commisceri, & in ejus locum irrumpere ;
& si duæ habeantur [I66] asymptoti ejusmodi satis proximæ, quarum citerior habeat ulterius
crus repulsivum, ulterior citerius attractivum cum areis infinitis, tum duo puncta collocata
in distantia a se invicem intermedia inter distantias earum asymptotorum, nec possent
ulla finita vi, aut velocitate acquirere distantiam minorem, quam sit distantia asymptoti
citerioris, nec majorem, quam sit ulterioris ; & cum eæ duæ asymptoti possint esse utcunque
sibi invicem proximæ ; illa puncta possent esse necessitata ad non mutandam distantiam
intervallo utcunque parvo. Si jam in uno plano sit series continua triangulorum æqui-
laterorum habentium eas distantias pro lateribus, & in singulis angulis poneretur quicunque
numerus punctorum ad distantiam inter se satis minorem ea, qua distentillæ duæ asymptoti,
vel etiam puncta singula ; fieret utique velum quoddam indissolubile, quod tamen esset
plicatile in quavis e rectis continentibus triangulorum latera, & posset etiam plicari in
gyrum more veterum voluminum.
363. Si autem sit solidum compositum ex ejusmodi velis, quorum alia ita essent aliis
imposita, ut punctum quodlibet superioris veli terminaret pyramidem regularem habentem
pro basi unum e triangulis veli inferioris, & in singulis angulis collocarentur puncta, vel
massæ punctorum ; id esset solidissimum, & ne plicatile quidem ; etiamsi crassitudo
unicam pyramidum seriem admitteret. Possent autem esse dispersa inter latera illius
veli, vel hujus muri, puncta quotcunque, nec eorum ullum posset inde egredi ad distantiam
a punctis positis in angulis veli, vel muri, majorem illa distantia ulterioris asymptoti. Quod
si præterea ultra asymptotum ulteriorem haberetur area repulsiva infinita ; nulla externa
puncta possent perrumpere nec murum, nec velum ipsum, vel per vacua spatiola transire,
utcunque magna cum velocitate advenirent ; cum nullum in triangulo æquilatero sit
punctum, quod ab aliquo ex angulis non distet minus, quam per latus ipsius trianguli.
364. Quod si ejusmodi binæ asymptoti inter se proximæ sint in ingenti distantia a
principio abscissarum, & in distantia media inter earum binas distantias ab ipso initio
ponantur in cuspidibus trianguli æquilateri tria puncta materiæ, tum in cuspide pyramidis
regularis habentis id triangulum æquilaterum pro basi ponantur quotcunque puncta, quæ
inter se minus distent, quam pro distantia illarum asymptotorum ; massula constans hisce
punctis erit indissolubilis ; cum nec ullum ex iis punctis possit acquirere distantiam a
reliquis, nec reliqua inter se distantiam minorem distantia asymptoti citerioris, & majorem
distantia ulterioris, & ipsa hæc particula impenetrabilis a quovis puncto externo materiæ,
cum nullum ad reliqua illa tria puncta possit ita accedere, si distat magis, vel recedere, si
minus, ut acquirat distantiam, quam habent puncta ejus massæ. Ejusmodi massis ita
cohibitis per terna puncta ad maximas distantias sita posset integer constare Mundus,
qui ha-[I67]-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.
365. In omnibus hisce casibus habetur in massa non continua vis ita continua, ut
nulla ne apparens quidem compenetratio, & permixtio haberi possit æque, ac in communi
sententia de continua impenetrabilis materiæ extensione. Quod autem in illo velo, vel
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. I88, 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
ca, S eS, we have
continuous resist-
ance without im-
agining a continu-
ous force, & also
absolute impene-
trability.
27o PHILOSOPHIÆ NATURALIS THEORIA
Sine asymptoto om-
nes substantias per-
meabiles fore ab
aliis si iis satis
magnas velocitas
i m p r i m a t u r.
Exemplum globuli
ferrei inter mag-
metes transeuntis.
Diversi effectus re-
late ad magnetes
pro diversa veloci.
tate ejus globuli.
Inde facilis explica-
tio phænomeni, quo
globus sclopeto ex-
• p l o s u s perforat
plana mobilia, nec
movet : cur lumini
data tanta veloci-
tas.
Cur in cinere re-
m a ne a t i 1 l æ s a.
forma plantæ avo-
lante parte volatili
per ignem.
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.
366. At in primo illo casu, in quo nulla habetur ejusmodi asymptotus præter primam,
res longe alio modo se haberet. Patet in eo casu illud, si velocitas imprimi possit massæ
cuipiam satis magna ; fore, ut ea transeat per massam quancunque sine ulla perturbatione -
suarum partium, & sine ulla partium alterius ; nam vires, ut agant, & motum aliquem
finitum sensibilem gignant, indigent continuo tempore, quo imminuto in immensum,
uti imminuitur, si velocitas in immensum augeatur, imminuitur itidem in immensum
earum effectus. Rei ideam exhibebit globulus ferreus, qui debeat transire per planum,
in quo dispersæ sint hac, illac plurimæ massæ magneticæ vim habentes validam satis. Si
is globus cum velocitate non ita ingenti projiciatur per directionem etiam, quæ 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, prætervolabit utique, nullo sensibili damno ejus
velocitati illato. .
367. Quin immo ibi considerandum & illud ; si velocitas ejus fuerit exigua, ipsum
globum facile sisti, exiguo motu a vi mutua æquali, seu reactione, impresso magnetibus,
quo per solam plani fractionem, & mutuas eorum vires impedito, exigua in eorum position-
ibus mutatio fiat. Si velocitas impressa aliquantulum creverit ; tum mutatio in positione
magnetum major fiet, & adhuc sistetur globuli motus ; sed si velocitas fuerit multo major,
globulus autem transeat satis prope aliquas e massis magneticis ; 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, avulsæ, a cæteris, quæ ob
actiones in majore distantia minores, & brevitatem temporis, remaneant ad sensum immotæ,
& nihil turbatæ. Sed si velo-[I68]-citas ipsa adhuc augeretur, quantum est opus, eo
deveniri posset ; ut massa utcunque proxima in globuli transitu nullum sensibilem motum
auferret illi, & ipsa sibi acquireret.
368. Porro ejusmodi exemplum intueri licet, ubi globus aliquis contra obstaculum
aliquod projicitur, quod, si satis magnam velocitatem habet, concuti totum, & diffringit
ac eo majorem effectum edit, quo major est velocitas, ut in muris arcium accidit, qui
tormentariis globis impetuntur. At ubi velocitas ad ingentem quandam magnitudinem
devenerit ; nisi satis solida sit compages obstaculi, sive vires cohæsionis satis validæ ; jam
non major effectus fit, sed potius minor, foramine tantum excavato, quod æquetur ipsi
globo. Id experimur ; si globus ferreus explodatur sclopeto contra portam ligneam,
quæ licet semiaperta sit, & summam habeat super suis cardinibus mobilitatem ; tamen
nihil prorsus commovetur ; sed excavatur tantummodo foramen æquale ad sensum diametro
globi, quod in mea Theoria multo facilius utique intelligitur, quam si continuo nexu partes
perfecte solidæ inter se complicarentur, & conjungerentur. Nimirum, ut in superiore
magnetum casu, particulæ globi secum abripiunt particulas ligni, ad quas accesserunt
magis, quam ipsæ ad sibi proximas accederent, & brevitas temporis non permisit viribus
illis, a quibus distantium ligni punctorum nexus præstabatur, ut in iis motus sensibilis
haberetür, qui nexum cum aliis sibi proximis a vi mutua ortum dissolveret, aut illis, &
toti portæ $atis sensibilem motum cómmunicaret. Quod si velocitas satis adhuc augeri
posset ; ne iis quidem avulsis massa per massam transvolaret, nulla sensibili mutatiqne
facta, & sine vera compenetratione haberetur illa apparens compenetratio, quam habet
lumen, dum per homogeneum spatium liberrimo rectilineo motu progreditur ; quam
ipsam fortasse ob causam Divinus Naturæ Opifex tam immanem luci velocitatem voluit
imprimi, quantam in ea nobis ostendunt eclipsés Jovis satellitum, & annua fixarum aberratio,
ex quibus Roemerus, & Bradleyus deprehenderunt, lumen semjquadrante horæ Percurrere
distántiam æqualem distantiæ Solis à Terra, sive plura milliariorum millia singulis arteriæ
pulsibus.
369. Ac eodem pacto, ubi herbarum forma in cinere cum tenuissimis filamentis remanet
intacta, avolantibus oleosis partibus omnibus sine ulla læsione structuræ illarum, id quidem
admodum facile intelligitur, qui fiat : ibi nova vis excitata ingentem velocitatem parit
brevi tempore, quæ 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
would be a far different result. In this case, it is evident that, if a sufficiently great velocity
can be given to any mass, it would pass through any other mass without any perturbation
of its own parts, or of the parts of the other. For, the forces have no continuous time
in which to act & produce any finite sensible motion; since if this time is diminished
immensely (as it will be diminished, if the velocity is immensely increased), the effect of
the forces is also diminished immensely. We can illustrate the idea by the example of an
iron ball, which is required to pass across a plane, in which lie scattered in all positions
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
very small, the ball might easily be brought to rest, a slight motion due to an equal mutual
force or reaction being communicated to the magnets; but this latter being prevented
merely by the friction of the plane, the change in their positions would be very small.
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
projected against an obstacle; if the velocity is sufficiently great, it agitates the whole &
breaks it to pieces; & the effect produced is the greater, the greater the velocity, as is the
case for the walls of forts bombarded with cannon-balls. But when the velocity reaches a
certain very great magnitude, unless the fabric of the obstacle is sufficiently solid or the
forces of cohesion sufficiently great, there will now be no greater effect, rather a less, a
hole only being made, equal to the size of the ball. Let us consider this; suppose an iron
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.
369. In the same way, when the form of stalks remain intact in the ash with their
finest fibres, after that the oleose parts have all been driven off without any breaking down
of their structure, what happens can be quite easily understood. Here, a new force being
excited produces in a brief space of time a mighty velocity, which prevents all that other
effect arising from the mutual forces between the oily & the ashy parts; the oily particles
If there were no
a sympt O tes, all
Substances would
be permeable by
One a not her, if
Sufficiently great
velocity is given
to them. Example,
an iron globe pass-
ing between mag-
netS.
Relatively diverse
effects with regard
to the magnets, due
to diverse velocities
of the ball.
He n c e a n e a sy
explanation of the
p he no men on in
which a ball fired
from a cannon will
perforate a mov-
able plane without
moving it ; & why
Such a gre a t
velocity is given to
light.
The reason why
in the ash there
remains unimpaired
the form of the
plant after that the
volatile part has
been driven off by
the action of fire.
272 PHILOSOPHIÆ NATURALIS THEORIA
Compenetratio ap-
parens, quæ habe-
retur, si possemus
nobis imprimere
velocitatem satis
magnam.
Extensio necessario
profluens a viribus
repulsivis.
Extensum ejusmodi
esse physice, non
mathematice con-
tinuum : re a l e m
esse : in quo id
consistat.
cineres, oleaginosis particulis inter terreas cum hac apparenti compenetratione liberrime
avolantibus sine ullo immediato impactu, & incursu.
370. Quod si ita res habet; liceret utique nobis per occlusas ingredi portas, & per
durissima transvolare murorum se-[169]-pta $ine ullo obstaculo, & sine ulla vera compene-
tratione, nimirum, satis magnam velocitatem nobis ipsis possemus imprimere, quod si
Natura nobis permisisset, & velocitates corporum, quæ habemius præ manibus, ac fiostrorum
digitorum celeritates solerent esse satis magfiae ; apparentibus ejusmodi continuis
compenetrationibus assueti, nullam impenetrabilitatis haberemus ideam, quam mediocritati
mo$trarum virium, & velocitatum, ac experimentis hujus generis a sinu materno, & prima
infantia usque adeo frequentibus, & perpetuo repetiti§ debemus omnem.
373. Ex impenetrabilitate oritur extensio. Ea sita est in eo, quod aliæ partes sint
extra alias : id autem necessario haberi debet ; si plura puncta idem $patii punétum simul
occupare non possint. Et quidem si nihil aliunde sciremus de distributióne punctorum
materiæ : ex regulis probabilitatis constaret nobis, dispersa esse per spatium extensum
in longum, latum, & profundum, atque ita constaret, ut de eo dubitâre omnino non liceret,
adeoque haberemus extensionem in longum, latum, & profundum ex eadem etiam solâ
Theoria deductam. Nam in quovis plano pro quavis récta linea infinita sunt curvarum
genera, quæ eadem directione egressæ e dato puncto extenduntur in longum, & latum
respectu ejusdem rectæ, & pro quavis ex ejusmodi curvis infinitæ sunt curvæ, quæ ex illo
puncto egressæ 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 cäsu 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 plano plura, quam tria.
372. Hæc quidem extensio non est mathematice, sed physice tantum continua: at
de præjudicio, ex quo ideam omnino continuæ extensionis ab infantia nobis efformavimus,
satis dictum est in prima Parte a num. I 58; ubi etiam vidimus, contra meam Theoriam
non posse afferri argumenta, quæ 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. Nec vero illud yim ullam contra me habet, quqd nonnulli adhibent,
dicentes, hujusmodi extensionem nullam esse, cum constet punctis penitus inexten-[17o]-sis,
& yacuo spatio, quod est purum nihil. Constat per me, nqn solis punctis, sed punctis
habentibus relationes distantiarum a se invicem : eæ relationes in mea Theoria non
constituuntur a spati9 vacuo, intermedio, quod spatium nihil est actu existens, sed, est
aliquid solum possibile a ngbis 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 hic 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 cqn-
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 continuo extenso,
quod potius etiam independenter a meis argumentis difficultates habet quamplurimas.
Id extensum, quod admitto, est ejusmodi, ut puncta materiæ alia sint, extra alia, ac
distantias habeant aliquas inter se, nec omnia jaceant in eadem recta, nec in eodem plano
omnia, sint vero multa ita proxima, ut eorum interyalla omnem sensum effugiant. In eo
sita est extensio, quam admitto, quæ erit reale quidpiam, non imaginarium, & erit physice
COntlnua.
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
hardest walled enclosures without any resistance, & without any real compenetration;
that is to say, if we could impress upon ourselves a sufficiently great velocity. Now if
Nature allowed us this, & the velocities of bodies which are around us, & the speed of our
fingers were usually sufficiently great, we, being accustomed to such continuous apparent
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
some parts are outside other parts; & this of necessity must be the case, if several points
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
matter of the prejudgment, from which we have formed for ourselves the idea of absolutely
continuous extension from infancy, enough has been said in the First Part, starting with
Art. 158. There, too, we saw that there could not be brought forward against my Theory
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.
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. I42, & 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.
T
For these disputants assume that their non-extended points
Apparent compene-
tration, such as
would be obtained
if we were able to
give ourselves a
velocity great
enough.
Extension neces-
Sarily arising from
repulsive forces.
Such extension is
physical ly, not
mathematically,
Continuous; it is
real ; of what
consists.
it.
274 PHILOSOPHIÆ NATURALIS THEORIA
Quomodo existat
Geometria sublato
continuo actu exist-
ente.
Quid in ea imagi-
narium, quid reale :
elegans analogi a
loci cum tempore
in ordine ad æqual-
itatis mensuras.
Figurabilitas orta.
ab ext e n s io me :
quid sit figura, &
quam V a g a, &
incerta sit ejus idea
etiam in communi
sententia.
373. At erit fortasse, qui dicet, sublata extensione absolute mathematica tolli omnem
Geometriam. Respondeo, Geometriam non tolli, quæ considerat relationes inter distantias,
& inter intervalla distantiis intercepta, quæ 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, quæ existunt ;
an autem inter ea, quæ possunt existere, habeantur, omnino ignoro. Sed aliquid ejusmodi
in communi etiam sententia accidit. Nulla existit revera in Natura recta linea, nullus
circulus, nulla ellipsis, nec in ejusmodi lineis accurate talibus fit motus ullus, cum omnium
Planetarum, & Terræ in communi sententia motus habeantur in curvis admodum compli-
catis, atque altissimis, &, ut est admodum probabile, transcendentibus. Nec vero in
magnis corporibus ullam habemus superficiem accurate planam, & continuam, aut sphæricam,
aut cujusvis e curvis, quas Geometræ contemplantur, & plerique ex iis ipsis, qui solida
volunt elementa, simplices ejusmodi figuras ne in ipsis quidem elementis admittent.
374. Quamobrem Geometria tota imaginaria est, & idealis, sed propositiones hypo-
theticæ, quæ inde deducuntur, [I7I] sunt veræ, & si existant conditiones ab illa assumptæ,
existent utique & conditionata inde eruta, ac relationes inter distantias punctorum
imaginarias ope Geometriæ ex certis conditionibus deductæ, semper erunt reales, & tales,
quales eas invenit Geometria, ubi illæ ipsæ conditiones in realibus punctorum distantiis
existant. Ceterum ubi de realibus distantiis agitur, nec illud in sensu physico est verum,
ubi punctum interiacet aliis binis in eadem recta positis, a quibus æque distet, binas illas
distantias fore partes distantiæ punctorum extremorum juxta ea quæ 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, quæ ablato intermedio reali puncto
manet prorsus eadem. Illæ duæ sunt partes illius tertiæ tantummodo in imaginario, &
geometrico statu, qui concipit indefinite omnes possibiles intermedios existendi modos
locales, & per eam cognitionem abstractam concipit continua intervalla, ac eorum partes
assignat, & ope ejusmodi conceptuum ratiocinationes instituit ab assumptis conditionibus.
petitas, quæ, 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 æqualis distantiæ aliorum duorum, situm est in
ipsa natura illorum modorum, quibus existunt, non in eo, quod illi modi, qui eam individuam
distantiam constituunt, transferri possint, ut congruant. Eodem pacto relatio duplæ,
vel triplæ distantiæ habetur immediate in ipsa essentia, & natura illorum modorum. Vel
si potius velimus illam referre ad distantiam æqualem ; dici poterit, eam esse duplam
alterius, quæ talis sit, ut si alteri ex alterius punctis ponatur tertium novum ad æqualem
distantiam ex parte altera ; distantia nova hujus tertii a primo sit æqualis illi, quæ duplæ
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
nec in tempore potest unquam ; adeoque nec æqualitas per congruentiam in statu reali
haberi potest, nec ratio dupla per partium æqualitatem. Ubi decempeda transfertur
ex uno loco in alium, succedunt alii, atque alii punctorum extremorum existendi modi,
qui relationes inducunt distantiarum ad sensum æqualium : ea æqualitas a nobis supponitur
ex causis, nimirum ex mutuo nexu per vires mutuas, uti hora hodierna ope egregii horologii
comparatur cum hesterna, itidem æqualitate supposita ex causis, sed loco suo divelli, &
ex uno die in alterum hora eadem traduci nequaquam potest. Verum hæc omnia ad
Metaphysicam potius pertinent, & ea fusius cum omnibus [I72] loci, ac temporis relationibus
persecutus sum in memoratis dissertationibus, quas hic in fine subjicio.
375. Ex extensione oritur figurabilitas, cum qua connectitur moles, & densitas
supposita massa. Quoniam puncta disperguntur per spatium extensum in longum, latum,
& profundum ; spatium, per quod extenduntur, habet suos terminos, a quibus figura
pendet. Porro figuram determinatam ab ipsa natura, & existentem in re, possunt agnoscere
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,
then the whole of Geometry is demolished. I reply, that Geometry is not demolished,
since it deals with relations between distances, & between intervals intercepted in these
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
that are deduced from it are true, if the conditions assumed by it exist, & also the conditional
things deduced from them, in every case; & the relations between the imaginary distances
of points, derived by the help of geometry from certain conditions, will always be real,
& such as they are found to be by geometry, when those conditions exist for real distances
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;
How Geometry can
Stand, if an actually
existing continuum
is excluded. .
The imaginary &
the real parts of
Geometry; an ele-
gant analogy be-
tween place & time
aS regards measures
of equality.
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 &,
when we have conceived the idea of mass, density. Since points are scattered through
extended space in length, breadth & depth, the space through which they are extended
has its boundaries; & upon these boundaries depends shape. Further, it is in the elements
alone that a shape, determinate by its very nature, & existing of itself, can be acknowledged
by those who suppose the elements to be solid, compact & continuous; & by those who
Figurability arises
from extension ;
the nature of shape,
& how vague the
idea of it is, even in
the opinion usually
held.
276 PHILOSOPHIÆ NATURALIS THEORIA
& qui ab inextensis extensum continuum componi posse arbitrantur, ubi nimirum tota
Quanto magis in
hac Theoria.
M o 1 e s a figura
pendens : incerta
eju s id e a & in
sententia communi,
& multo magis in
hac Theoria.
Massa : quid in ejus
idea incertum ob
materiam exteram
immixtam. Omnia
corp o r a constare
partibus diversæ
naturæ.
illa materia superficie continua quadam terminetur.' Ceterum in corporibus hisce, quæ
nobis sub sensum cadunt, idea figuræ, quæ 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 ; Imprimis hoc
ip$um momem figurae terrestris, quod certam quamdam, ac determinatam significationem videtur
habere, habet illam quidem admodum incertam, & vagam. Superficies illa, quae maria, &
lacus, & fluvios, ac montes, & campos, vallesque terminat, est illa quidem admodum, nobis
$altem, irregularis, &5 vero etiam instabilis : mutatur enim quovis utcumque minimo undarum,
&5 glebarum motu, nec de hac Telluris figura agunt, qui in figuram Telluris inquirunt : aliam
ip$i $ub$tituunt, quae regularis quodammodo sit, sit autem illi priori proxima, quae nimirum
abrasis haberetur montibus, collibusque, vallibus vero oppletis. At haec iterum terrestris figurae
motio vaga admodum e$t, & incerta. Uti enim infinita sunt curvarum regularium genera,
quae per datum datorum punctorum numerum transire po$sint, ita infinita sunt genera curvarum
superficierum, quae Tellurem ita ambire po$sint, atque concludere, ut vel omnes, vel datos
contingant in datis punctis montes, collesque, vel si per medios transire colles, ac montes debeat
$uperficies quaedam ita, ut regularis sit, & tantumdem materiae concludat extra, quantum
vacui aeris infra sese concludat usque ad veram hanc nobis irregularem Telluris superficiem,
quam intuemur : infinitae itidem, & a se invicem diversae admodum superficies haberi possunt,
quae problemati $atisfaciamt, atque eae ejusmodi etiam, ut nullam, quae sensu percipi possit,
præ $e ferant gibbositatem, quae ipsa vox non ita determinatam continet ideam.
376. Hæc ego ibi de Telluris figura, quæ omnino pertinent ad figuram corporis
cujuscunque in communi etiam sententia de continua extensione materiæ : nam omnium
fere corporum superficies hic apud nos utique multo magis scabræ sunt pro ratione suæ
magnitudinis, quam Terra pro ratione magnitudinis suæ, & vacuitates internas habent
quamplurimas. Ve-[173]-rum in mea Theoria, res adhuc magis indefinita, & incerta est.
Nam infinitæ sunt etiam superficies curvæ continuæ, in quibus tamen omnia jacent puncta
massæ cujusvis : quin immo infinitæ numero curvæ sunt lineæ, quæ per omnia ejusmodi
puncta transeant. Quamobrem mente tantummodo confingenda est quædam superficies,
quæ omnia puncta includat, vel quæ pauciora, & a reliquorum coacervatione remotiora
excludat, quod æstimatione quadam morali fiet, non accurata geometrica determinatione.
Ea superficies figuram exhibebit corporis ; atque hic jam, quæ ad diversa figurarum genera
pertinent ; id omne mihi commune est cum communi Theoria de continua extensione
mater 12e.
377. A figura pendet moles, quæ nihil est aliud, nisi totum spatium extensum in
longum, latum, & profundum externa superficie conclusum. Porro nisi concipiamus
superficiem illam, quam innui, quæ figuram determinet ; nulla certa habebitur molis
idea : quin immo si superficiem concipiamus tortuosam illam, in qua jaceant puncta omnia ;
jam moles triplici dimensione prædita erit nulla ; si lineam curvam concipimus per omnia
transeuntem : nec duarum dimensionum habebitur ulla moles. Sed in eo itidem incerta
æstimatione indiget sententia communis ob interstitia illa vacua, quæ habentur in omnibus
corporibus, & scabritiem, juxta ea, quæ diximus, de indeterminatione figuræ. Hic autem
itidem, concepta superficie extima terminante figuram ipsam, quæ 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 sphæricæ figuræ superficiem respectu molis : in figuris autem
similibus molem esse in ratione triplicata laterum homologorum, & superficiem in
duplicata, ex quibus pendent phænomena sane multa, atque ea inprimis, quæ pertinent
ad resistentiam tam fluidorum, quam solidorum.
$.
378. Massa corporis est tota quantitas materiæ pertinentis ad id corpus, quæ quidem
mihi erit ipse numerus punctorum pertinentium ad illud corpus. At hic jam oritur
indeterminatio quædam, vel saltem summa difficultas determinandi massæ ideam, nec id
tantum in mea, verum etiam in communi sententia, ob illud additum punctorum pertimentium
ad illud corpus, quod heterogeneas substantias excludit. Ea de re sic ego quidem in Stayanis
Supplementis § IO Lib. I : Nam admodum difficile est determinare, quae sint illae substan-
tiae heterogeneæ, quae non pertinent ad corporis comstitutionem. Si materiam $pectemus ; ea
& mihi, & aliis plurimis homogenea est, & solis ejus diversis combinationibus diversae 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 & 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 ; & moreover it is
also unstable ; for it changes with the slightest motion of the waves & 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 & the hills levelled off, whilst the valleys
are filled up. Now once more the idea of this figure of the Earth is vague & 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 & circumscribe it in such a manner that they touch all the mountains & hills, or at
least certain given ones ; or, if you like, some surface is bound to pass through the middle of the
hills & mountains in such a way that it cuts off as much matter outside itself, as it encloses
empty air-spaces within it & our true surface of the Earth, to our eyes, so irregular. Also,
there can be an infinite number of surfaces, & these too quite different from one another, which
satisfy the problem ; 85 all of them, too, of such a kind that they have no manifest humps, as
far as can be detected ; & this term even contains no true definiteness.
376. These are my words in that dissertation with regard to the figure of the Earth;
& they apply in general to the figure of any body also, if considered according to the usual
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,
extended in length, breadth & depth, which is included by the external surface. Further,
unless we picture that surface which I mentioned as determining the shape, there can be
no definite idea of volume. Nay indeed, if we think of the tortuous surface in which all
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; &
in my Theory this is precisely the same thing as the number of points that go to form the
body. Here now we have a certain indefiniteness, or at least the greatest difficulty, in
forming a definite idea of mass ; & that, not only in my theory, but in the usual theory as
well, on account of the addition of the words points that go to form the body; this excludes
heterogeneous substances. On this point indeed, I made the following remarks in the
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, {5 in that of very many others, homogeneous ; & the different species of
The vagueness is
still greater in this
Theory.
Volume depends on
shape ; the idea of
this too is vague in
the usual theory, &
much more so in
mine. '
Mass ; what there
is in the idea of it
that is indefinite
owing to outside
m atter mingling
with it. All bodies
are composed of
parts of different
natureS.
278 PHILOSOPHIAE NATURALIS THEORIA
corporum $pecies. Quare ab ipsa materia non potest desumi discrimem illud inter substantias
pertinentes, &5 non pertinentes. Si autem & diversam [174] illam combinationem spectemus,
corpora omnia, quae observamus, mixta sunt ex substantiis admodum dissimilibus, quae tamem
omnes ad ejus corporis constitutionem pertinent. Id in animalium corporibus, in plantis, in
marmoribus plerisque, oculis etiam patet, in omnibus autem corporibus Chemia docet, quae
mixtionem illam dissolvit.
Plures £t 379. Ex alia parte tenuissima aetherea materia, quæ omnino est aliqua nostro aere varior,
£ £'.'! ad constitutionem massae nequaquam pertinere censetur, ut nec pro corporibus plerisque aer,
poris. qui meatibus intermis interjacet. Sic aer inclusus spongiae meatibus, ad ipsius constitutionem
nequaquam cen$etur pertinere. Idem autem ad multorum corporum constitutionem pertinet :
$altem ad fixam maturam redactus, ut Halesius demonstravit, plures & animalis regni, &
vegetabilis substantias magna sui parte constare aere fixitatem adepto. Rursus substantiae
volatiles, aere ipso tenuiores multo, quae in corporum dissolutione chemica in halitus, & fumos
abeunt, & plures fortasse, quas nos nullo sensu percipimus, ad ipsa corpora pertinebant.
Nec excludi omnia 380. Nec illud a$$umi potest, quidquid solidum, & fixum est, id tantummodo pertinere
Ęgri; ad corporis massam ; quis enim a corporis humani massa sanguinem omnem, & tot lymphas
translató , corpore excludat, a lignis re$ectis succos nondum concretos ? Praeterquam quod massae idea non ad
Š„° “ solida solum corpora pertinet, sed etiam ad fluida, in quibus ipsis'alia tênuiora aliorum densiorum
meatibus interjacent. Nec vero dici potest, pertinere ad corporis constitutionem, quidquid
materiae 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, &5 qui aliquamdiu manet, mutatur deinde.
Hinc indistinctam 38 I. Haec, & alia mihi diligentius perpendenti, illud videtur demum, ideam massae non
ā*'ï'*'* esse accurate determinatam, & distinctam,'sed admodum vagam, arbitrariam, & confusam.
raritas ; utranque Erit massa materia omnis ad corporis constitutionem pertimens ; sed a cra$$a quadam, &
augeri, & minui ę «• «• ® » • e» • • w w*
È';„„.,;;;;; arbitraria æstimatione pendebit illud, quod est pertinere ad ipsam ejus constitutionem. Hæc
in, qua cu n que ego ibi: tum ad molem transeo, de cujus indeterminatione jam hic superius egimus, ac
ratione. deinde ad densitatem, quæ est relatio massæ, 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 ; & quæcunque vulgo proferuntur de comparationibus inter massam, molem, &
densitatem, hæc 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 [I75]
ego nullum habeo limitem densitatis & raritatis, nec maximum, nec minimum ; dum illi
minimam debent aliquam raritatem agnoscere, & maximam densitatem possibilem, utut
finitam, quæ illis idcirco per saltum quendam necessario abrumpitur ; licet nullam agnoscant
raritatem maximam, & minimam densitatem. Mihi enim materiæ puncta possunt &
augere distantias a se invicem, & imminuere in quacunque ratione ; cum data linea quavis,
possit ex ipsis Euclideis elementis inveniri semper alia, quæ 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 quævis massa dividi in
quenvis numerum particularum, quæ dispersæ 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 ; tum vero densitas est maxima, &
raritas minima omnium, quæ haberi possint, & tamen finita est, cum mensura prioris
habeatur, massa finita per finitam molem divisa, & mensura posterioris, divisa mole per
1Y13SSaIm.
Inertia, massarum 382. Inertia corporum oritur ab inertia punctorum, & a viribus mutuis ; nam illud
orta ex in e rtia, υ © � ę © •
punctorum : i p si demonstravimus num. 26o, si puncta quæcunque vel quiescant, vel moveantur directionibus,
respondens conser, & celeritatibus quibuscunque, sed singula æquabili motu ; centrum commune, gravitas
;t tu,* vel quiescere, vel moveri uniformiter in directum, ac vires mutuas quascunque inter eadem
inassæ ünitæ in puncta nihil turbare statum centri communis gravitatis sive quiescendi, sive movendi
1pso. uniformiter in directum. Porro vis inertiæ in eo ipso est sita : nam vis inertiæ 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 & 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 ; 85 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
much less dense than our air, can in no sense be considered to be a constituent part of a body :
nor indeed, in the case of most bodies can the air which is contained in its internal parts. Thus
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,
A large number of
substances do not
pertain to the sub-
stance of a body.
volatile substances, more tenuous than air itself, which go off in vapours & fumes from bodies
chemically decomposed, 85 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 {5 fixed can pertain to the mass of a
body. For who would exclude from the mass of the human body the whole of the blood, & the
large number of watery fluids, or from chips of wood the juices that are not yet congealed 2
Especially as the idea of maſs pertains not only to solids alone but also to fluids ; & in these
some of the more tenuous parts lie in the interstices of the more dense. On the other hand, it
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, & remains for some length of time,
{5 then is moved. +
381. After carefully considering these & other matters, I have come to the conclusion
that the idea of mass is not strictly definite & distinct, but that it is quite vague, arbitrary
& confused. Mass will be the whole of the matter pertaining to the constitution of a body;
but what part of it actually does pertain to its constitution, will depend upon a non-scientific
{5 arbitrary assessment. These are my words; & after that I pass on to volume, the
indefiniteness of which I have already dealt with above, & after that to density, which is
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 breakin 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.
382. The inertia of bodies arises from the inertia of their points & their mutual forces.
For, in Art. 260, it was proved that, if any points are either at rest, or moving in any
directions with any velocities, so long as each of the motions is uniform, then the centre
of gravity of the set will either be at rest or move uniformly in a straight line; & that,
whatever mutual forces there may be between the points, these will in no way affect the
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
Nor can all fluids be
excluded ; nor can
all those things be
included, which
when the body is
moved are carried
with the body.
Hence also the idea
of mass is indefi-
nite. The nature
of density, &rarity;
either of them in
this Theory can be
increased or dimin-
ished to any extent.
The inertia of a
mass arises from
the inertia of its
points; the corre-
sponding conserva-
tion of the state of
the centre of grav-
ity ; the idea of
a mass being con-
densed at its centre
of gravity.
28o PHILOSOPHIAE NATURALIS THEORIA
Mobilitas : qui e s-
cibilitatem I] O 11
haberi, ex cl u sa
prorsus quiete a,
Natura.
Quies"exclusa etiam
a continui tate
omnium motuum :
problema generale
eo pertinens.
Quid curvæ de-
scriptæ a punctis
non habeant. Pro-
ble ma inversum
datis particulis de-
scriptis tempusculo
utcunque parvo.
determinatio perseverandi in eodem statu quiescendi, vel movendi uniformiter in directum :
nisi externa vis cogat statum suum mutare : & cum ex mea Theoria demonstretur, eam
proprietatem debere habere centrum gravitatis massæ cujuscunque compositæ punctis
quotcunque, & utcunque dispositis ; patet, eam deduci pro corporibus omnibus : & hic
illud etiam intelligitur, cur concipiantur corpora tanquam collecta, & compenetrata in
1pso gravitatis centro.
383. Mobilitas recenseri solet inter generales corporum proprietates, quæ quidem
sponte consequitur vel ex ipsa curva virium : cum enim ipsa exprimat suarum ordinatarum
ope determinationes 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 saltem
in Natura, uti constituta est, haberi non posse arbitror, uti exposui num. 86. Eam excludi
Qportere censeo etiam infinitæ improbabilitatis argumento, quo sum usus in ea dissertatione
De Spatio, & Tempore, quam toties jam nominavi, & in Supplementis hic proferam § I,
ubi [I76] evinco, casum, quo punctum aliquod materiæ occupet quovis momento temporis
punctum spatii, quod alio quopiam quocunque occuparit vel ipsum, vel aliud punctum
qu9dcunque, esse infinities improbabilém, considerato nimirum número punctorum materiæ
finito, numero momentorum possibilium infinito ejus generis, cujus sunt infinita puncta
in una recta, qui numerus momentorum bis sumitur, semel cum consideratur puncti dati
materiæ cujuscunque momentum quodvis, & iterum cum consideratur momentum quodvis,
quo aliud qu9dpiam materiæ 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
gQmmune gravitatis, yel quiescens, vel uniformiter progrediens in recta linea, quies actualis
itidem a Natura excluditur.
384. Verum ipsam quietem excludit alia mihi proprietas, quam omnibus itidem
materiæ punctis, & omnium corporum centris gravitatis communem censeo, nimirum
continuitas motuum, de qua egi num. 883, & alibi. Quodvis materiæ punctum seclusis
motibus liberis, qui oriuntur ab imperio liberorum spirituum, debet describere curvam
quandam lineam continuam, cujus determinatio reducitur ad hujusmodi problema generale :
Dato numero punctorum materiæ, ac pro singulis dato puncto loci, quod occupent dato
quopiam momento temporis, ac data directione, & yelocitate motus initialis, si tum primo
projiciuntur, vel tangentialis, si jam ante fuerunt in motu, ac data lege virium expressa
per curvam aliquam continuam, cujusmodi est curva figuræ I, quæ meam hanc Theoriam
continet, inyenire singulorum punctorum trajectorias, lineas nimirum, per quas ea moventur
singula. Id problema mechanicum quam sublime sit, quam omnem humanæ mentis
excedat vim, ille satis intelliget, qui in Mechanica versatus non nihil noverit, trium etiam
corporum motus, admodum simplici etiam vi præditorum, nondum esse generaliter definitos,
uti, monui num. 2o4, & consideret immensum punctorum numerum, ac altissimam curvæ
virium tantis flexibus circa axem circumvolutæ elevationem.
385. Sed licet ejusmodi problema vires omnes humanæ mentis excedat; adhuc tamen
unusquisque Geometra vidébit facile, problema esse prorsus determinatum, & curvas
ejusmodi fore omnes continuas sine ullo saltu, si in lege virium nullus sit saltus. Quin
immo & illud arbitror, in ejusmodi curvis nec ullas usquam cuspides occurrere ; nam nodos
nullos esse consequitur ex eo, quod nullum materiæ punctum redeat ad idem punctum
spatii, in quo ipsum aliquando fuerit, adeoque nullus habeatur regressus, qui tamen ad
niodum est necéssarius. Hujusmodi curvæ necessariæ essent omnes, & mens, [I77] quæ
tantum haberet vim, quanta requiritur ad ejusmodi problemata rite trangtanda, & intimius
perspiciendas solutionés (quæ qüidem mens posset etiam finita esse, si finitus sit punctorum
humierus, & per finitam expressionem sit data notio curvæ exprimentis legem virium) pQsset
ex arcu continuo descripto tempore etiam utcunque exiguo a punctis materiæ Qmnibus
derivare ipsam virium légem, cùm quidam finiti tantummodo positionum numeri finitos
determináre possint numeros punctórum curvæ virium, & arcus continuus legem ipsam
continuam : & fortasse solæ etiam positiones omnium punctorum cum dato arcu continuo
percurso ab unico etiam puncto miotu continuo, exigüo etiam aliquo tempusculQ ad rem
præstandam satis essent. Cognita autem lege virium & positione, ac velocitate, & directione
punctorum omnium dato tempore, posset ejusmodi mens prævidere Qmnes futuros neces-
$arios motus, ac status, & omfiia Nãturæ phænomena necessaria, ab iis utique pendentia,
atque prædicere : & ex unico arcu descripto a quovis puncto, tempore continuo utcunque
A THEORY OF NATURAL PHILOSOPHY 28I
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; &
indeed it follows immediately from the curve of forces. For, since this curve, by means of
its ordinates, represents the propensity to approach or recede, it necessarily requires mobility,
or the possibility of motion, without which approach or recession can certainly not be
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, 85 Tempore, which I have mentioned so many
times already, & which I quote in this work as Supplement, $ 1 ; 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
which I consider is common also to all points of matter & to the centres of gravity of all
bodies; namely, continuity of motion, with which I dealt in Art. 88 & elsewhere. Any
point of matter, setting aside free motions that arise from the action of arbitrary will,
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
intellect, yet any geometer can easily see thus far, that the problem is determinate, & that
such curves will all be continuous without any break in them, so long as there is no discon-
tinuity in the law of forces. Indeed, I think that, in such curves, there never occur
any cusps; for, it follows that there are no nodes, from the fact that no point of matter
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
Mobility; quiesci-
bility is impossible,
since there is no
such thing in
Nature as absolute
rest.
Absolute rest is
excluded also by
the continuity of
all motions; gene-
ral problem with
respect to it.
What do e s not
exist in curves
described by the
points. The inverse
problem, given the
CUITVeS described
in any interval of
time how ever
Small.
282 PHILOSOPHIÆ NATURALIS THEORIA
Cur a b humana
mente solvi non
possit. Quid offi-
ciat ei determina-
tioni libertas : Har-
moniæ præstabilitæ
impugnatio.
Motus liberos om-
n in o a b a nim a
progigni, sed non
imprimi, nisi æqua-
1 iter in partes
oppositas, & sine
saltu. -
Conclusiones d e-
ductæ ; potissimum
exclusio quietis.
Aequalitas action-
is, & reactionis, &
ejus consectaria.
parV9, quem aliqua mens satis comprehenderet, eadem determinare posset reliquum omnem
ejusdem continuæ curvæ tractum utraque e parte in infinitum próductum.
. 386. Nos eo aspirare non possumus, tum ob nostræ mentis imbecillatatem, tum quia
ignoramus numerum, & positionem, ac motum punctorum singulorum (nam nec mótus
absolutos intuemur, sed respectivos tantummodo respectu Telluris, vèl ad summum
respectu systematis planetarii, vel systematis fixarum omnium) tum etiam, quia curvas
illas turbant liberi motus, quos producunt spirituales substantiæ. Harmonia præstabilita
Leibnitianorum ejusmodi perturbationem töllit omnem, saltem respectu animæ nostræ,
cum omne immediatum commercium demat inter corpus, & animam ; & id, quod tantopere
improbatum est in Theoria Cartesiana, quæ bruta redegerat ad automatâ, ad homines
etiam ipsos transfert, quorum motus a machina provenire omnes, & necessarios esse in ea
Theoria, facile constat: & quidem idcirco etiam mihi Theoria displicet plurimum, quam
præterea si admitterem, nullam sane viderem, ne tenuissimam quidem rationem, `quæ
mihi suadere posset, præter animam meam, cujus ideæ per se, & sine ullo immediato 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 quæ 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.
387. Censeo igitur, & id intima vi, qua anima suarum [I78] idearum naturam, &
proprietates quasdam, atque originem novit, constare arbitror, motus liberos corporis ab
anima provenire : ac quemadmodum virium lex necessaria, in ipsa fortasse materiæ natura
sita, ejusmodi est ; ut juxta eam bina materiæ puncta debeant ad se invicem accedere,
vel a se invicem recedere, determinata & quantitate motus, & directione per distantias ;
ita esse alias leges virium liberas animæ, secundum quas debeant quædam puncta materiæ
habentia ejusmodi dispositionem, quæ ad vivum, & sanum corpus organicum requiritur,
ad ipsius animæ nutum moveri ; sed hujusmodi leges itidem censeo requirere illud, ut
nulli materiæ puncto imprimatur motus aliquis, nisi alicui alteri imprimatur alius contrarius,
& æqualis, quod constat ex ipso nisu, quem semper exercemus in partes contrarias, juxta ea,
uæ diximus num. 74 : ac itidem arbitror, & id ipsum diligenti 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 curvæ
descriptæ ; sed motuum continuitas nequaquam turbabitur.
388. Porro inde constat, cur in motibus nullum uspiam deprehendamus saltum, cur
nullum materiæ 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, quæ huc pertinent. Verum simul patebit
& illud, in cujus gratiam hæc congessimus, nullam fore absolutam quietam, in qua nimirum
continuatus ille curvæ descriptæ ductus abrumpatur ea continuitate læsa nihilo minus,
quam læderetur, si curva continua desineret alicubi in rectam.
389. Jam vero ad actionis, & reactionis æqualitatem gradu facto, eam abunde deduximus
a num. 265. pro binis quibusque corporibus ex actione, & reactione æqualibus in punctis
quibuscunque. Cum nimirum mutuæ vires nihil turbent statum centri gravitatis com-
munis, & centra gravitatis binarum massarum debeant cum ipso communi centro 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
distantiæ 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
æqualem summæ motuum computatorum secundum oppositam, quam massa altera acquiret
simul, in quo ipso sita est actionis & reactionis æqualitas, ex qua corporum [I79] collisiones
deduximus in secunda parte, & ex qua multa phænomena 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
task, but also because we do not know the number, or the position & motion of each of these
points (for we do not observe absolute motions, but merely relative motions with respect
to the Earth, or at most those with respect to the planetary system or the system of all
the fixed stars); & there is yet another reason, namely that the free motions produced
by spiritual substances affect these curves. The “pre-established harmony” of the followers
of Leibniz abrogates all such disturbing effect, at least as far as regards our will, since it does
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
this is due to an inner force, by which the mind knows the nature, certain properties & the
origin of its ideas, I think can be easily established. Just as we must have a law of forces,
perhaps involved in the very nature of matter, of such a kind that according to it two points
of matter must approach towards, or recede from, one another with a motion determined
in magnitude & direction by the distance between the points; so there must be other
Why the problem
cannot be solved
by the human
in tellect ; what
ob St a cle to its
determination is
due to freedom ;
argument against
“pre-establish ed
harmony.”
Free motions are
certainly produced
by the mind, but
are not impressed
except equally in
Opposite directions,
& without breach
of continuity
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,
why no point of matter can ever pass from one position to another without passing through
all intermediate positions, why density can in no case be suddenly changed, why reflected
& 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,
fully proved its truth for any two bodies from the equality of the action & reaction between
any two points. For 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. 4
Conclusions de-
duced ; especially
the exclusion of
absolute rest.
Equality of action
& reaction ; its
Consequences.
284 PHILOSOPHIAE NATURALIS THEORIA
Inde a n m o t u s
massæ proveniat
a viribus internis,
an ab externis.
Divisibilitas in im- .
finitum spatii con-
tinui ; & materiæ
itidem si sit con-
tinua, & sine vir-
tuali extensione.
Virtualem exten-
sionem non haberi.
Puncta esse indi-
visibilia ; massas
divisibiles usque ad
certum limit e m
singulas.
Componibilitas i n
infinitum.
Ejus æquivalentia
c u m divisibilitate
in infinitum.
390. Illud unum hic adnotandum censeo, per hanc ipsam legem comprobari plurimum
ipsas vires mutuas inter materiæ particulas, & deveniri ad originem motüum plurimorum,
quæ inde pendet; si nimirum particulæ massæ cujuslibet ingentem habeant motum
reciprocum hac, illac, & interea centrum commune gravitatis iisdem iis motibus careat ;
id sane indicio est, eos motus provenire ab internis viribus mutuis inter puncta ejusdem
massæ. Id vero accidit inprimis in fermentationibus, quæ habentur post quarundam
substantiarum permixtionem, quarum particulæ 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, & Terræ, respectu cujus quiescit vas.
39I. Quod ad divisibilitatem pertinet, eam quidem in infinitum progredientem sine
ullo limite in spatio continuo ille solus non agnoscet, qui Geometriæ etiam elementaris
vim non sentiat, a qua pro ejusmodi divisibilitate in infinitum tam multa, & simplicia, &
perspicua sane argumenta desumuntur. Ubi ad materiam sit transitus ; si, ubi de ea agitur,
quæ distinctas occupant loci partes, distincta etiam sunt ; ab illa spatii continui divisibilitate
in infinitum, materiæ quoque divisibilitas in infinitum consequitur evidentissime, &
utcunque prima materiæ elementa atomos, sive Naturæ vi insectilia censeant multi, ut
& Newtonus ; adhuc tamen absolutam eorum divisibilitatem agnoscunt passim illi ipsi.
392. Materiæ elementa extensa per spatium divisibile, sed omnino simplicia, & carentia
partibus, admiserunt nonnulli e Peripateticis, & est etiam nunc, qui recentiorem Philoso-
phiam professus admittat ; at eam sententiam non ex præjudicio 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 æque, ac illud ipsum est spatium.
393. At in mea Theoria, in qua prima elementa materiæ mihi sunt simplicia, ac inex-
tensa, nullam, eorum divisibilitatem haberi constat. Massæ autem, quæcunque actu
existant, sunt mihi congeries punctorum ejusmodi numero finitæ. Hinc eæ 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. Nec
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 & usu
infinitorum, & infimite parvorum ; tum & 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 : tum
enim nullus est numerus finitus ita magnus, ut alius adhuc finitus ipso major haberi in eo
spatio non possit. Nam inter duo puncta quæcunque 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 novum, &
priora illa 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 Naturæ Conditor determinare voluit, nullos habuit limites,
quos non potuerit prætergredi, nullum ultimum habente terminum serie illa possibilium
finitorum in infinitum crescentium.
395. Hæc componibilitas in infinitum æquivalet divisibilitati in ordine ad explicanda
Naturæ phænomena. Posita divisibilitate materiæ in infinitum, solvitur facile illud
A THEORY OF NATURAL PHILOSOPHY
28;
390. I consider that in this connection it should be remarked that by means of this
law especially the existence of these mutual forces between particles of matter is established,
& that in it we attain to the source of most of the motions, which arises from it. For
instance, considering that the particles of a mass may have an immense reciprocal motion,
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
any limit in continuous space will be denied only by one who does not feel the force of
the most elementary principles of geometry; for, from it may be derived so many simple
& perfectly clear arguments in favour of such infinite divisibility. When we come to
consider matter, if in dealing with it, we take it that what occupies a distinct part of space
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
space, but quite simple & without parts; & at the present day there is one professing a
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-
extended, it is easily seen that there can be no divisibility of the elements. Also masses,
in so far as they actually exist, are to me merely sets of such points finite in number.
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
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, {5 usu infinitorum, & 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 phenomena of Nature. If we postulate infinite divisibility for matter, we have an easy
Hence, the point as
to whether the
motion of a mass
arises from internal
or external forces.
Infinite divisibility
of continuous
Space ; the same of
m a t t er, if it is
continuous, & with-
Out virtual exten-
Sion.
Virtual extension is
non-existent.
Points are indivi-
sible, whilst every
mass is divisible up
to a certain limit.
Infinite
bility.
componi-
The equivalence of
componibility to
infinite divisibility.
286 PHILOSOPHIÆ NATURALIS THEORIA
Demonstratur
ipsa.
€3,
Divis ib i lita s in
Natura immanis;
Immutabilitas pri-
IY) OTUlIÌl elemen-
torum materiæ: or-
dines diversi parti-
cularum minus, ac
problema : Datam massam utcumque parvam, ita distribuere per datum spatium utcumque
magnum, ut in eo mullum $it $patiolum majus dato quocunque utcumque parvo penitus vacuum,
§5 sine ulla ejus materiae 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, & singulæ particulæ destinantur singulis
spatiolis ; quæ 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 suæ particulæ,
vel possunt ejus particulæ segmenta iterum per illud spatiolum utcunque dispergi. In
[181] mea Theoria substituitur hujusmodi aliud problema : Intra datum spatiolum collocare
eum punctorum numerum, qui deinde distribui possit per spatium utcunque magnum ita, ut
in eo nullum $it $patiolum cubicum majus dato quocumque utcumque parvo penitus vacuum, &
quod in se non habeat numerum punctorum utcumque magnum. -
396. Quod in ordine ad explicanda phænomena hoc secundum problema æquivaleat
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 expositæ 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, quem velimus collocari in hoc ipso quovis posteriore spatiolo post
dispersionem, & auctor Naturæ potuit utique intra illud spatiolum primum hunc punctorum
numerum collocare.
397. Jam quod pertinet ad divisibilitatem immanem, quam nobis ostendunt Naturæ
phænomena in coloratis quibusdam corporibus, immanem molem aquæ 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 æque bene explicabitur per componibilitatem
ac in communi Theoria ex illo alio per divisibilitatem materiæ in infinitum.
398. Prima materiæ elementa volunt plerunque immutabilia, & ejusmodi, ut atteri,
atque confringi omnino non possint, ne nimirum phænomenorum ordo, & tota Naturæ
facies commutetur. At elementa mea sunt sane ejusmodi, ut nec immutari ipsa, nec
legem suam virium, ac agendi modum in compositionibus commutare ullo modo possint ;
cum nimirum simplicia sint, indivisibilia, & inextensa. Ex iis autem juxta ea, quæ diximus
num. 239 ad distantias perquam exiguas collocatis in limitibus virium admodum validis
oriri possunt primæ particulæ minus jam tenaces suæ formæ, quam simplicia elementa,
sed ob ingentem illam viciniam adhuc tenacissimæ idcirco, quod alia particula quævis
m i n u s immuta-
bilium.
Gravitas exhibita
a postremo arcu
curvæ accedens ad
INewtonionam quam
p r o xi me : p o s s e
nostro concipiendi
modo fieri absolute
talem.
ejusdem ordinis in omnia simul ejus puncta fere æqualiter agat, & vires mutuæ majores
sint, quam sit discrimen virium, quibus diversa ejus puncta solicitantur ab illa particula.
Ex hisce primi ordinis particulis possunt constare particulæ ordinis secundi ; adhuc minus
tenaces, & ita porro ; quo enim plures compositiones sunt, & majores distantiæ, eo facilius
fieri potest, ut inæqualitas [182] virium, quæ sola mutuam positionem turbat, incipiat
esse major, quam sint vires mutuæ, quæ tendunt ad conservandam mutuam positionem,
& formam particularum ; & tunc jam alterationes, & transformationes habebuntur, quas
videmus in corporibus hisce nostris, & quæ habentur etiam in pluribus particulis postremorum
ordinum, hæc ipsa nova corpora componentibus. Sed prima materiæ elementa erunt
omnino immutabilia, & primorum etiam ordinum particulæ formas suas contra externas
vires validissime tuebuntur.
399. Gravitas etiam inter generales proprietates a Newtonianis inprimis numeratur,
quibus assentior ; dummodo ea reipsa non habeat rationem reciprocam duplicatam
distantiarum extensam ad omnes distantias, sed tantum ad distantias ejusmodi, cujusmodi
sunt eæ, quæ 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, 85 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, 85 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,
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
to us in certain coloured bodies, when they stain an immense volume of water with the same
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
Demonstration.
The immense divi-
sibility in Nature.
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,
& of such a kind that it is quite impossible for them to be subject to attrition or fracture,
unless indeed the order of phenomena & the whole face of Nature were changed. Now,
my elements are really such that neither themselves, nor the law of forces can be changed;
& 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
quite immutable, & particles of the first orders will preserve their forms in opposition to
even very strong forces from without. -
399. Gravity also is counted as a general property, especially by followers of Newton;
& I am of the same opinion, so long as it is not supposed to be in the inverse ratio of
the squares of the distances for all distances, but merely for distances such as those that lie
between the distance of our bodies from the far greatest part of the mass of the Earth,
Immutability of
the primary ele-
ments of matter :
different kinds of
particles, less &
less immutable.
Gravity, as repre-
sented by the last
arc of the curve, ap-
proximates to that
given by the New-
tonian law ; possi-
bility of its being
exactly the same,
according to my
hypothesis,
288 PHILOSOPHIÆ NATURALIS THEORIA
Gravitatem gener-
alem haberi in toto
Solari systemate,
nec posse tribui
pressioni fluidi.
Ea m ex ipsa
Theoria respondere
massæ directe, &
quadratO distantiæ
reciproce.
C o m m e n d a tio
Theoriæ ex con-
formitate omnium
corporum in ea, &
discrimine in totis
aliis.
Omnia fere a gravi-
tate pendentia sunt
comm u n i a. h u i c
Theoriæ cum com-
muni : nonnullo-
rum in ea facilior
deductio.
massæ 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, quæ diximus num. I2I. Ejusmodi
autem gravitas exhibetur ab arcu illo postremo meæ curvæ figuræ I, qui, si gravitas exten-
ditur cum eadem illa lege ad sensum, vel cum aliqua simili, in infinitum, erit asymptoticus.
Posset quidem, ut monui num. II 9, concipi gravitas etiam accurate talis, quæ extendatur
ad quascunque distantias cum eadem lege, & præterea alia quædam vis exposita per aliam
curvam, in quam vim, & in gravitatem accurate reciprocam quadratis distantiæ resolvatur
lex virium figuræ I ; quæ 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 conceptæ legis itidem
repulsiva tanto major, quam vis legis primitivæ figuræ I, quanta esset gravitas ibi conceptâ,
quæ nimirum ab illo additamento vis repulsivae elidi deberet. Sed hæc jam a nostro
concipiendi modo penderent, ac in ipsa mea lege primitiva, & reali, gravitas utique est
generalis materiæ, ac legem sequitur rationis reciprocæ duplicatæ distantiarum, quanquam
non accurate, sed quamproxime, nec ad omnes extenditur distantias ; sed illas, quas exposui.
400. Ceterum gravitatem generalem haberi in toto planetario systemate, ego
quidem arbitror omnino evinci iisdem argumentis ex Astronomia petitis, quibus utuntur
Newtoniani, quæ hic non repeto, cum ubique prostent, & quæ tum alibi ego quidem
congessi pluribus in locis, tum in Adnotationibus ad poema P. Noceti De Aurora Boreali.
Illud autem arbitror evidentissimum, illum accessum ad Solum cometarum, & planetarum
primariorum, ac secundariorum ad primarios, quem videmus in descensu a recta tangente
ad arcum curvæ, & multo magis alios motus a mutua gravitate pendentes haberi omnino
[I83] non posse per ullius fluidi pressionem ; nam ut alia prætermittam sane multa, id
fluidum, quod sola sua pressione tantum possit in ejusmodi globos, multo plus utique posset
occursu suo contra illorum tangentialem velocitatem, quæ omnino deberet imminui per
ejusmodi resistentiam, cum ingenti perturbatione arearum, & totius Astronomiæ Mechanicæ
perversione ; adeoque id fluidum vel resistentiam ingentem deberet parere planetæ, aut
cometæ progredienti, vel ne pressione quidem ullum ipsi sensibilem imprimit motum.
4o I. Ejus autem præcipuæ leges sunt, ut directe respondeat massæ, & reciproce
quadratis distantiarum a singulis punctis massæ ipsius, quod in mea Theoria est admodum
manifestum ita esse debere ; nam ubi ventum est ad arcum illum meæ curvæ, qui gravitatem
refert, vires omnes jam sunt attractivæ, & eandem illam ad sensum sequuntur legem,
adeoque aliæ alias non elidunt contrariis directionibus, sed summa earum respondet ad
sensum summæ punctorum ; nisi quatenus ob inæqualem punctorum distantiam, &
positionem, ad habendam accurate ipsam summam, ubi moles sunt aliquanto majores,
opus erit illa reductione, qua Mechanici utuntur passim, & cujus ope inveniuntur leges,
sécundum quas punctum in data distantia, & positione situm respectu massæ habentis
datam figuram, ab ipsa attrahitur ; ubi, quemadmodum indicavimus num. 347, globus
in globum ita gravitat, ut gravitaret, si totæ eorum massæ essent compenetratæ in eorum
centris : at in aliis figuris longe aliæ leges obveniunt. -
4o2. Verum hic illud maxime Theoriam commendat meam, quod num. 212 nqtandum
dixi, quod videamus tantam hanc conformitatem in vi gravitatis in omnibus massis; licet
eædem in ordine ad alia phænomena, quæ a minoribus distantiis pendent, tantum discrimen
habeant, quantum habent diversa corpora in duritie, colore, sapore, odore, sgno. Nam
diversa combinatio punctorum materiæ inducit summas virium admodum diversas pro
iis distantiis, in quibus adhuc curva virium contorquetur circa axem ; proinde exigua
mutatio distantiæ 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.
4o3. Distinctio gravitatis (quæ est ut massa, in quam tenditur, directe, & quadratum
distafitiæ reciproce) à pondere (quod est præteræ ut massa, quæ gravitat) est mihi eadem,
ac Newtoniamiis, & omnibus Mechanicis; & illa vim acceleratricem exhibet, hoc vim
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. I21. 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. I 19, 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
system, I think is thoroughly demonstrated by those arguments derived from Astronomy
which are used by the disciples of Newton ; these I do not repeat here, since they are set
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 &
inversely as the square of the distances from each of the points of that mass ; & in my
Theory it is quite clear that this must be the case. For, as soon as we reach the arc of
my curve that represents gravitation, all the forces are attractive, & to all intents obey
the same law; & so some of them do not cancel others in opposite directions, but their
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
said ought to be noticed; namely, in the fact that we see so much uniformity in all masses
with regard to the force of gravity; in spite of the fact that these same masses, for the
purpose of other phenomena depending on the smaller distances apart, have differences
so great as those possessed by different bodies as regards hardness, colour, taste, smell &
sound. For, a different combination of the points of matter induces totally different sums
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
produces a huge difference in the former case, & a very great uniformity in the latter,
403. The distinction between gravitation (which is proportional to the mass on which
it acts, directly, & as the square of the distance, inversely) & weight (which is, in addition,
proportional to the mass causing the gravitation) is just the same in my Theory as in that
of Newton & all mechanicians. The former gives the accelerating force, the latter the motive
U
Gravity exists
through out the
whole Solar system,
& it cannot possibly
be explained by
fluid pressure.
Gravit a ti o n, ac-
COrding to my
Theory, varies as
the mass directly
& as the Square
of the dist a n ce
inversely.
Support to be
derive d for my
Theory from the
conformity of all
bodies in having
gravitation, whilst
there are SO many
differences in other
properties.
Nearly everything
depending on grav-
ity in my Theory is
in agreement with
the usual theory :
but the deduction
of some of them is
easier in mine.
29O PHILOSOPHIAE NATURALIS THEORIA
Immobilitas fixa-
ru m quomodo a
Newtonianis expli-
Cetur.
Difficultas residua
sublata a b h a C
Theoria.
Cohaesio: explicatio
p e r quietem, ve l
motus conspirantes.
motricem, cum illa determinet vim puncti gravitantis cujusvis, a qua pendet celeritas
massa ; [184] hoc summam virium ad omnia eiusmodi puncta pertinentium. Pariter
communia mihi sunt, quaecunque pertinet ad gravium motus a Galilaeo, & Hugenio definitos,
nisi quod gravitatis resolutionem in descensu per plana inclinata, 8 in gravibus sustentatis
per bina obliqua plana, vel obliqua fila, reducam ad compositionem juxta num. 284, 8 286,
& centrum oscillationis, una cum centro aequilibrii, & vecte, & libra, & machinarum principiis
deducam e consideratione systematis trium massarum in se mutuo agentium, ac potissimum
a simplici theoremate ad id pertinente, qua fuse persecutus sum a num. 307. Communia
pariter mihi sunt, quacunque habentur in calesti 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 hac postrema problemata rite solvenda, multo tutior, & expeditior mihi panditur via,
quae me eo deducet post considerationem systematis massarum quatuor jacentium etiam
non in eodem plano communi, & connexarum invicem per vires mutuas, uti ad centrum
oscillationis etiam in latus in eodem plano, & ad centrum percussionis in eadem recta tam
facile me deduxit consideratio systematis massarum trium.
4O4. Illud mihi praeterea non est commune, quod pertinet ad immobilitatem stellarum
fixarum, quam contra generalem Newtoni gravitatem vulgo Solent objicere, qua nimirum
debeant ea attractione mutua ad se invicem accedere, & in unicam demum coire massam.
Respondent alii, Mundum in infinitum protendi, & proinde quamvis fixam aeque in omnes
partes trahi. Sed in actu existentibus infinitum absolutum, ego quidem censeo, haberi
omnino non posse. Recurrent alii ad immensam distantiam, quae non sinat motum in
fixis oriundum a vi gravitatis, ne post immanem quidem saeculorum seriem sensu percipi.
Ii in eo verum omnino affirmant; si enim concipiamus fixas Soli nostro aequales & similes,
vel saltem rationem luminum, quae emittunt, non multum discedere a ratione massarum ;
quoniam & vis ipsis massis proportionalis est, ac praeterea tam 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 totalux, quae provenit a fixis omnibus,
ad lucem, quae provenita 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.
4o5. Adhuc tamen, quoniam nostra vita, 8 memoria respectu immensi fortasse subse-
cuturi aevi est itidem fere nihil, [I85] si gravitas generalis in infinitum protendatur cum
eadem illa 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, 8 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 haudita se habere, demonstrari omnino
non potest; adhuc tamen Divinae Providentiae videtur melius consulere Theoria, quae
ejus etiam ruinae universalis evitandae viam aperiat, ut aperit sane mea. Fieri enim potest,
uti notavimus n. 17o, utpostremusille curva 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 eius generis ordines particularum eiusmodi 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. I7I.
406. Hoc pacto difficultas quae a necessario fixarum accessu repetebatur contra
Newtonianam Theoriam, in mea penitus evanescit ac simula gravitate jam gradum fecimus
ad cohaesionem, quam ex generalibus materiae proprietatibus posueram postremo loco.
A THEORY OF NATURAL PHILOSOPHY 29I
force; since the former gives the force of any gravitating point, upon which depends
the yelocity 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.
4O4. In addition to these, there is one thing in which I do not agree, namely, in that
which relates to the immobility of the fixed stars; it is usually objected to the universal
gravitation of Newton, that in accordance with it the fixed stars should by their mutual
attraction approach one another, & in time all cohere into one mass. Others reply to this,
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
number of ages perchance to follow, is almost as nothing, if universal gravitation
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 repeatedly brought against the Newtonian
theory on account of this necessary mutual approach of the fixed stars, disappears altogether
in my Theory. At the same time, we have now passed on from gravity to cohesion, which
The manner in
which the immo-
bility of the fixed
stars was explained
by Newton.
The remaining diffi-
culty taken away in
this Theory.
Cohesion ; explana-
tion by means of
rest,or of motions in
the same direction,
292 PHILOSOPHIÆ NATURALIS THEORIA
Illas ex p o n e r e
effectum, causam
non exhibere.
Explicatio petita
a pressione fluidi;
cur adhiberi non
possit.
Explicatio N e w-
toni ab attractione
in minimis distan-
tiis : cur admitti
non possit.
Cohæsionem explicuerunt aliqui per puram quietam ut Cartesiani alii per motus conspir-
antes, ut Joanfies Bernoulliùs, àc Leibnitiùs, quam explicationem illustrarunt exemplo
illius veli äquæ, quod in fontibus quibusdam cernimus, quqd yelum sit tantummodo ex
conspirante motu guttularum tenuissimarum, & tamen si quis digito velit perrumpere, eo
majórem resistentiam sentit, quo velocitas aquæ effluentis est major, ut idcirco multo
adfiuc major conspirantis mofus velocitas videatur nostrorum cohæsionem corporum
exhibere, quæ non fiisiimmani vi confringimus, ac in partes dividimus. Utraque explicandi
ratio eodem redit, si quietis nomine intélligatur non utique absoluta quies, quæ translata
Tellure a Cartesianis nequaquam admittebatur, sed respectiva : nam etiam conspirantes
motus nihil sunt aliud, nisi quies respectiva illarum partium, quæ conspirant in motibus.
4o7. At neutra eam explicat, quam cohæsionem reipsa dicimus, sed cohæsionis quendam
velut effectum. Ea, quæ cohærent, utique respective quiescunt, sive motus conspirantes
habent, & id quidem ipsum in hac mea Theoria accidit [186] itidem, in qua cum singula
puncta materiæ suam pergant semper eandem continuam curvam describere, ea, quæ
Gohaerent inter se, toto eo tempore, quo cohærent, arcus habent curvarum suarum inter
se proximos, & in arcubus ipsis conspirantes motus. Sed in iis, quæ cohærent, id ipsum,
quòd motus ibi sint conspirantes, non est sine causa pendente a mutuis egrum viribus, quæ
&ausa impediat separationem alterius ab altero, ac in ea ipsa causa stat discrimen cohæren-
tium a Contiguis. Si duo lapides in plano horizontali jaceant, utique habent motum
conspirantem, quem circa Solem habet Tellus ; sed si tertius lapis in alterutrum incurrit,
vel égo ipsum submoveo manu, statim sine ulla vi mutua, quæ separationem impediat,
dividüntur, & motus desinit esse conspirans. Hanc ipsam quærimus causam, dum in
cohæsionem inquirimus. Nec velocitas motus, & exemplum veli aquæ rem conficit.
Motus conspirans duorum lapidum contiguorum cum tota Tellure est utique multo yelocior,
quam motús particularum aquæ proveniens a gravitate in illo velo, & tamen sine ull9,
difficultate separantur. In aqua experimur difficultatem perrumpendi velum, quia illa
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 növis particulis, quæ cum tota sua ingenti respectiva velocitate incurrunt in digitum.
At in corporibus, in quibus partes cohærentes cernimus, eæ partes nullam habent veloci-
tatem respectivam respectu nostri, nec aliæ succedunt aliis fugientibus. Quamobrem
longe aliter in iis se res habet, & oportet invenire causam longe aliam, præter ipsum solum
conspirantem motum, ut explicetur difficultas, quam experimur in iis separandis, & in
inducendo motu non conspirante. .
4o8. Sunt, qui adducant pressionem fluidi cujuspiam tenuissimi, uti pressio atmo-
sphæræ extracto^ aere ex hemisphæriis etiam vacuis ipsorum separationem impedit vi
respondente ponderi ipsius atmosphæræ, quæ vis, cum in yulgaribus cohæsionibus, & vero
etiàm in hemisphæriis bene ad se invicem adductis, sit multis vicibus major, quam pondus
atmosphæræ ipsius, quod se prodit in suspensione mercurii in barometris ; aliud auxilio
advocänt tenuiius fluidum. At inprimis ejus fluidi hypothesis precaria est ; deinde huc
illud redit, quod supra etiam monui, ubi de gravitatis causa egimus, quod nimirum meo
quidem judiGio explicari nullo modo possit, cur illud fluidum, quod sola pressione tantum
potest, iiihil omnino ad sensum possit incursu suo contra celerrimos planetarum, &
çometarum motus. Accedit etiam, quod distractio & compressio fibrarum, quæ habetur
ante fractionem solidorum corporum, ubi franguntur appenso inferne, vel superne imposito
[187] pondere ingenti, non ita bene cum ea sententia conciliari posse videatur.
409. Newtonus adhibuit ad eam rem attractionem diversam ab attractione gravitatis,
quanquam is quidem videtur eam repetere itidem a tenuissimo aliquo fluido comprimente ;
repetit certe sub finem Opticæ 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
only an effect of cohesion. Things which cohere are certainly relatively at rest; or they
have equal motions in the same direction. This is exactly what happens in my Theory also ;
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
But these methods
only explain the
effect & not the
Cause of cohesion.
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
as an explanation. Just as the pressure of the atmosphere, when the air has been abstracted
from a pair of hollow hemispheres, prevents them from being separated with a force
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
kind to gravitation ; although he indeed seems to seek to obtain this attraction from some
compressing fluid of very small density. In fact, he seeks to obtain it, at the end of his
Optics, from a ‘spirit' permeating the inmost substances of bodies; but I never was able
to grasp clearly what he intended by the term ‘spirit”; & even he confessed that the
Explanation sought
from fluid pressure ;
why it is impossible
that this should be
the case.
The reason why it
is impossible to
admit the explana-
tion from attraction
at very small dis-
tances, as given by
Newton.
294 PHILOSOPHIÆ NATURALIS THEORIA
Cohæsionem repe-
tendam a limitibus
V1I1l1m.
Cohæsio d u o r u m
punctorum : limi-
t e s c o hæ sio nis
posse esse quot-
cunque, utcumque
fortes, quocunque
Ordine positos.
In massis numerus
1 i m itu m multo
major : problema
pro iis inveniendis
quomodo s o 1 v e n-
dum.
quidem agendi modum & sibi incognitum esse profitetur. Is posuit ejusmodi attractionem
imminutis distantiis crescentem ita, ut in contactu sit admodum ingens, & ubi primigeniæ
particulæ se in planis continuis, adeoque in punctis numero infinitis contingant, sit infinities
majqr, quam ubi particulæ primigeniæ particulas primigenias in certis punctis numero
finitis contingant, ac eo minor sit, quo pauciores contactus sunt respectu numeri particu-
larum primigeniarum, quibus constant particulæ majores, quæ se contingunt, quorum
contactuum numerus cum eo sit minor, quo altius ascenditur in ordine particularum a
minoribus particulis compositarum, donec deveniatur ad hæc nostra corpora ; inde ipse
deducit, particulas ordinum altiorum minus itidem tenaces esse, & minime omnium hæc
ipsa corpora, quæ malleis, & cuneis dividimus. At mihi positiva argumenta sunt contra
vires attractivas crescentes in infinitum, ubi in infinitum decrescant distantiæ, de quibus
mentionem feci num. I26 ; & ipsa meæ Theoriæ probatio evincit, in minimis distantiis
vires repulsivas esse, non attractivas, ac omnem immediatum contactum excludit : quam-
obrem alibi ego quidem cohæsionis rationem invenio, quam mea mihi Theoria sponte
propemodum subministrat.
4IO. Cohæsio mihi est igitur juxta num. I65 in iis virium limitibus, in quibus transitur
a vi repulsiva in minoribus distantiis, ad attractivam in majoribus ; & hæc quidem est
cohæsio inter duo puncta, qua fit, ut repulsio diminutionem distantiæ impediat, attractio
incrementum, & puncta ipsa distantiam, quam habent, tueantur. At pro punctis pluribus
cohæsio haberi potest, tum ubi singula binaria punctorum sunt inter se in distantiis limitum
cohæsionum, tum ubi vires oppositæ eliduntur, cujusmodi exemplum dedi num. 223.
4I I. Porro quod ad ejusmodi cohæsionem pertinet, multa ibi sunt notatu digna.
Inprimis ubi agitur de binis punctis, tot diversæ haberi possunt distantiæ cum cohæsione,
quot exprimit numerus intersectionum curvæ virium cum axe unitate auctus, si forte sit
impar, ac divisus per duo. Nam primus quidem limes, in quo curva ab arcu asymptotico
illo primo, sive a repulsionibus impenetrabilitatem exhibentibus transit ad primum
attractivum arcum, est limes cohæsionis, & deinde alterni intersectionum limites sunt non
cohæsionis, & cohæ-[I88]sionis, juxta num. I79 ; unde fit, ut si intersectionum se conse-
quentium assumatur numerus par ; dimidium sit limitum cohæsionis. Hinc quoniam
in solutione problematis expositi num. I I7 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 cohæsione. Poterunt autem ejusmodi cohæsiones ipsæ esse diversissimæ
a se invicem soliditatis, ac nexus, limitibus vel validissimis, vel languidissimis utcunque,
rout nimirum ibi curva secuerit axem fere ad perpendiculum, & longissime abierit, vel
potius ad illum inclinetur plurimum, & parum admodum discedat ; nam in priore eorum
casuum vires repulsivæ imminutis, & attractivæ auctis utcunque parum distantiis, ingentes
erunt ; in posteriore plurimum immutatis, perquam exiguæ. Poterunt autem etiam e
remotioribus limitibus aliqui esse multo languidiores, & alii multo validiores aliquibus e
propioribus ; ut idcirco cohæsionis vis nihil omnino pendeat a densitate, sed cohæsio
possit in densioribus corporibus esse vel multo magis, vel multo minus valida, quam in
rarioribus, & id in ratione quacunque.
412. Quæ de binis punctis sunt dicta, multo magis de massis continentibus plurima,
puncta, dicenda sunt. In iis numerus limitum est adhuc major in immensum, & discrimen
utique majus. Inventio omnium positionum pro dato punctorum numero, in quibus tota
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 æquatione curvæ
primæ pro abscissa, ac valoribus itidem assumptis pro viribus singulorum punctorum pro
ordinatis, eruere totidem æquationes, tum reducere vires singulas singulorum punctorum
ad tres datas directiones, & summam omnium eandem directionem habentium in
quovis puncto ponere = o : orirentur æquationes, quæ paullatim eliminatis valoribus
incognitis assumptis, demum ad æquationes perducerent definientes punctorum distantias
necessarias ad æquilibrium, & respectivam quietem, quæ altissimæ 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. - • .
4Io. Cohesion, then, in my opinion is, as I have said in Art. 165, to be ascribed to
the limit-points on the curve of forces, where there is a passage from a repulsive force at
a smaller distance to an attractive force at a greater distance; that is to say, this is the
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.
4II. Further, with regard to such cohesion, there are many points that are worthy
of remark. First of all, in connection with two points, we can have as many different
distances corresponding with cohesion as is represented by the number of intersections
of the curve of forces with the axis (increased by one if perchance the number is odd) divided
by two. For the first limit-point, at which the curve passes from the first asymptotic arc,
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:
Cohesion is to be
ascribed to the
limit-points on the
curve of forces.
Cohesion of two
points; lim it-
points of cohesion
can be anything
whatever as regards
number, strength &
Order of occurrence.
non-cohesion & cohesion, as was shown in Art. I79. 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
to masses made up of a large number of points. In masses, the number of limit-points
is immensely greater still, & the difference between them is greater in every case. The
finding of all the positions for a given number of points, at which the whole mass has a
limit-point of forces, would be a troublesome undertaking; & the calculation necessary for its
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
In masses the
number of limit-
points is much
greater ; how the
problem of finding
the m is to be
Solved.
296 PHILOSOPHIÆ NATURALIS THEORIA
Cur partes solidi
fracti ad se invicem
appressæ non acqui-
rant cohæsionem
priorem, ratio in
Theoria, Newton-
iana.
Ejusdem ratio in
mea Theoria.
Discrimen massæ
primigeniæ, a, binis
frustis etiam lævi-
gatis ad se invicem
appressis.
Distractio, & com-
pressio fi b r a r u m
ante fractionem hic
commode ex ea. .
haberent radices ; nam æquationes, quo altiores sunt, eo plures radices habere possunt, ac
singulis radicibus singuli limites exhiberentur, vel singulæ positiones exhibentes vim nullam.
Inter ejusmodi positiones illæ, in quibus repulsioni in minoribus distantiis habitæ, succe-
derent attractiones in majoribus, exhiberent limites cohæsionis, qui adhuc essent quam
plurimi, & inter se magis diversi, quam limites ad duo tantummodo pun-[189]cta pertin-
entes ; cum in compositione plurium semper utique crescat multitudo, & diversitas casuum.
Sed hæc innuisse sit satis.
4I3. Ubi confringitur massa aliqua, & dividitur in duas partes, quæ prius tenacissime
inter se cohærebant, si iterum illæ partes adducantur ad se invicem ; cohæsio prior non
redit, utcunque apprimantur. Ejus rei ratio apud Newtonianos est, quod in illa divisione
non æque divellantur simul omnes particulæ, ut textus remaneat idem, qui prius : sed
prominentibus jam multis, harum in restitutione contactus impediat, ne ad contactum
deveniant tam multæ particulæ, quam multæ prius se mutuo contingebant, & quam multis
opus esset ad hoc, ut cohæsio fieret iterum satis firma : at ubi satis lævigatæ binæ superficies
ad se invicem apprimantur, sentiri primo resistentiam ingentem dicunt, donec apprimuntur ;
sed ubi semel satis appressæ sint, cohærere 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, quæ in contactu evadat maxima, & in lævigato
marmore satis multi contactus obtineantur simul ; idcirco deinde satis validam cohæsionem
consequi. -
414. Quidquid ipsi de contactibus dicunt, id in mea Theoria dicitur æque de satis
validis cohæsionis limitibus. In scabra superficie satis multæ prominentes particulæ
progressæ ultra limites, in quibus ante sibi cohærebant, repulsionem habent ejusmodi,
quæ impediat accessum reliquarum ad limites illos ipsos, in quibus fuerant ante divulsionem.
Inde fit, ut ibi nimis paucæ simul reduci possint ad cohæsionem particulæ, dum in
lævigatis corporibus adducuntur simul satis multæ. Ubi autem duo marmora, vel duo
quæcunque satis solida corpora, bene complanata, & lævigata sola appressione cohæserunt
invicem, illa 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 particulæ eo motu parallelo delatæ, quæ adhuc
sunt procul a marginibus partium congruentium, vires sentiunt hinc, & inde a particulis
lateralibus, a quibus fere æquidistant, fere æquales, adeoque sentitur resistentia earum
attractionum tantummodo, quas in se invicem exercent marginales particulæ, dum augent
distantias limitum : nam mihi citra limitem quenvis cohæsionis .est repulsio, ultra vero
attractio ; licet ipsi deinde adhuc aliæ & attractiones, & repulsiones possint succedere.
Ubi autem perpendiculariter distrahuntur, debet omnium simul limitum resistentia
V1I1C1.
41 5. Nec vero idem accidit, ubi marmora integra, & nunquam adhuc divisa, inter se
cohærent; tum enim fibræ possunt esse multæ, quarum particulæ adhuc in minori-[I9o]bus
distantiis, & multo validioribus limitibus inter se cohæreant, ad quos sensim devenerint
aliæ post alias iis viribus, quibus marmor induruit, ad quos nunc iterum reduci nequeant
omnes simul, dum marmora apprimuntur, quæ ulteriorum limitum minus adhuc validorum,
sed validorum satis repulsivas vires simul sentiunt, ob quas non possunt denticuli, qui
adhuc supersunt perquam exigui post quamvis lævigationem, in foveolas se immittere, &
ad ulteriores limites validiores devenire ; præterquam quod attritione, & lævigatione illa
plurimarum particularum ordinis proximi massis nobis sensibilibus inducitur discrimen
satis amplum inter massam solidam primigeniam, & binas massas complanatas, lævigatasque
ad se invicem appressas. -
416. Inde autem in mea Theoria satis commode explicatur & distractio, & compressio
fibrarum ante fractionem ; cum nimirum nihil apud me pendeat ab immediato contactu,
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
tenaciously, if the parts are again brought into contact with one another, the previous
cohesion does not return, however much they are pressed together. The reason of this,
according to the followers of Newton, is that in the division all the particles are not equally
torn apart simultaneously, leaving the texture the same as before ; but as many of them
now jut out beyond the rest, the contact between these in restitution prevents as many
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,
The reason given
in the Newtonian
theory to account
for the fact that the
parts of a broken
solid, when brought
closely together, do
not attain their
former cohesion.
in polished marble, a sufficiently great number of contacts was obtained simultaneously;
& in consequence a comparatively great cohesion was obtained.
4I4. All that the Newtonians say with regard to contacts applies in my Theory equally
well with regard to sufficiently strong limit-points of cohesion. In a rough surface, a
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
have not as yet been broken at any time, when they cohere. For, in that case, there may
be many filaments, the particles of which hitherto have been cohering at less distances &
in much stronger limit-points; these limit-points they would gradually reach one after
the other with the forces that have given the marble its hardness; but they cannot be reduced
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.
416. Hence also, in my Theory, we can give a fairly satisfactory explanation of the
distension & compression of fibres that precedes fracture; for, with me, everything depends
not on immediate contact, but on the limit-points, the distance of which is changed by
The reason for the
Same thing accord-
ing to my Theory.
Distinction be-
tween the cases of
a primitive mass &
two pieces that
have been broken
off, polished &
pressed together.
Distension & com-
pression of fibres
before fracture have
a good explanation
from my Theory.
298 PHILOSOPHIÆ NATURALIS THEORIA
Hinc cur solida
corpora, nimio pon-
dere pressa confrin-
ga1ntur.
C o m m u n i a. esse
huic theoriæ tum
communi m u lt a,
quæ pertinent ad
explorandam co-
hæsionis vim, &
resist e nt ia m ad
fractionem in di-
versis positionibus.
vincendam satis magno accessu omnem repulsionem, vel recessu attractionem, requiritur
satis magna vis : quæ quidem repulsio, & attractio in aliis limitibus longe mihi alia est, tam
si vis ipsa consideretur quam si consideretur spatii, per quod ea agit, magnitudo, quæ omnia
pendent a forma, & amplitudine arcuum, quibus hinc, & inde circa axem conítorquetur
mea virium curva. Hinc in aliis corporibus ante fractionem compressiones, & distraótiones
esse possunt longe majores, vel minores, & longe major, vel minor vis requiri potest ad
fractionem ipsam, quæ vis, ubi distantiis immutatis, superaverit maximam àrcus ulterioris
repulsivam vim in recessu, superatis multo magis reliquis omnibus posterioribus viribus
repulsiyis ope celeritatis quoque jam acquisitæ per ipsam vincentem vim, & per attractivas
intermixtas vires, quæ ipsam juvant, defert particulas massam constituentes adillas distantias,
in quibus jam nulla vis habetur sensibilis, sed ad tenuissimum gravitatis arcum acceditur.
417. Hinc autem etiam illud in mea Theoria commodius accidit, quam in communi,
quodin mea statim apparet, cur pila quæcunque utcunque solidi corporis post certa imposita
pondera confringatur, & confringatur etiam solidus globus utrinque compressus ; cum
multo magis appareat, quo pacto textus, & dispositio particularum necessaria ad summam
virium satis validam mutari possit, ubi omnia puncta a se invicem distant in vacuo libero,
quam ubi continuæ compactæ partes se contingant, nec ulla mihi est possibilis solida pila,
quæ Mundum totum, si vi gravitatis in certam plagam feratur totus, sustineat, ut in senténtia
de„gju- extensione materiæ pila perfecte solida utcunque tenuis ad eam rem abunde
Suii1Ceret.
418. Hisce omnibus jam accurate expositis, communia mihi sunt ea omnia, quæ
pertinent ad methodos explorandi per [I9i] experimenta diversam diversorum corporum
cohæsionis vim, quod argumentum diligenter, ut solet, excoluit Musschenbroekius, &
comparandi resistentiam ad fractionem, ubi divisio fieri debeat divulsione perpendiculari
ad superficies divellendas, ut ubi trabi verticali ingens pondus appenditur inferne, cum
resistentia, quæ habetur, ubi circa latus suum aliquod gyrare debeat superficies, quæ
divellitur, quod accidit, ubi extremæ parti trabis horizontalis pondus appenditur ; quam
perquisitionem a Galileo inchoatam, sed sine ulla consideratione flexionis & compressionis
fibrarum, quæ habetur inima parte, alii plures excoluerunt post ipsum ; & in quibus omnibus
discrimina inveniuntur quamplurima. Illud unum hic addam : posse cohæsionem
ingentem acquiri ab iis, quæ per se nullum haberent, nova materia interposita, ut ubi
cineres, qui oleis actione ignis avolantibus inter se inertes remanserunt, oleis novis in massam
cohærentem rediguntur iterum, ac in aliis ejusmodi casibus ; sed id jam pendet a discrimine
inter diversas particulas, & massas, ac pertinet ad soliditatem explicandam inprimis, non
Discrimen i n t er
particulas diversas,
a numero puncto-
ru m , a, m o 1 e, a.
densitate, a figura,
quæ potest esse
quævis, cum quavis
vi ad eam retinen-
dam.
generaliter ad cohæsionem, de quibus jam agam gradu facto a generalibus corporum
proprietatibus ad multiplicem varietatem Naturæ, & proprietates corporum particulares.
4I9. Et primo quidem se hic mihi offert ingens illud plurium generum discrimen,
quod haberi potest inter diversas punctorum congeries, quæ constituunt diversa genera
particularum corpora constituentium. Primum discrimen, quod se objicit, repeti potest
ab ipso numero punctorum constituentium particulam, qui potest esse sub eadem etiam
mole admodum diversus. Deinde moles ipsa diversa itidem esse potest, ac diversa densitas,
ut nimirum duæ particulæ nec massam habeant, nec molem, nec densitatem æqualem.
Deinde data etiam & massa, & mole, adeoque data densitate media particulæ ; potest
haberi ingens discrimen in ipsa figura, sive in superficie omnia includente puncta & eorum
sequente ductum. Possunt enim in una particula disponi puncta in sphæram, in alia in
pyramidem, vel quadratum, vel triangulare prisma. Sumatur figura quæcunque, & in
eam disponantur puncta utcunque : tot erunt ibi distantiæ, quot erunt punctorum binaria,
qui numerus utique finitus erit. Curva virium potest habere limites cohæsionis quot-
cunque, & ubicunque. Fieri igitur potest, ut limites iis ipsis distantiis respondeant, &
tum eam ipsam formam habebit particula, & ejus formæ poterit esse admodum tenax.
Quin immo per unicam etiam distantiam cum repagulo infinitæ resistentiæ, 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
it follows immediately, we have an explanation as to the reason why any pillar whatever,
made of a solid body, is broken when certain weights are imposed upon it; & also why
a solid sphere is crushed when compressed on both sides. For, it is much clearer how the
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. l
418. These matters having now been accurately explained, I proceed in the ordinary
manner in all things that relate to methods of experimental investigation of the different
force of cohesion in different bodies, a mode of demonstration that Mussenbroeck assiduously
practised with his usual care; & methods of comparing the resistance to fracture in the
case when division must take place by a fracture perpendicular to the surfaces to be broken,
such as occur when a great weight is hung beneath a vertical beam, with the resistance
that is obtained in the case when the surface has to rotate about one of its sides, which is
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
there can be amongst different groups of points such as form the different kinds of particles
of which bodies are formed. The first difference that calls our attention can be derived
from the number of points that form the particle; this number can be quite different
within the same volume. Then the volume itself may be different, as also may the
density; for, of course, two particles need not have either equal masses, equal volumes,
or equal densities. Then, even if the mass & the volume be given, that is to say,
the mean density of the particle is given, there may be a huge difference in shape,
that is to say, in the surface enclosing all the points, & conforming with them. For,
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
Hence the reason
why solid bodies
will be broke n
under the pressure
of to o great a
weight.
There are many
points of agreement
between my Theory
& the usual one,
relating to the
investigation of the
forces of cohesion
& resistance to frac-
ture in different
positions.
Distinction be-
tween different
kinds of particles
arising from the
number of points in
them, their volume,
their density, their
sh a pe; for the
latter anything is
possible, & a ny
corresponding force
can be had for the
purpose of main-
taining this shape.
3oO PHILOSOPHIÆ NATURALIS THEORIA
Discrimen in punc-
t o r u m distribu-
tione per figuram
eandem.
Discrimen in vi, qua
figuram conentur
retinere : posse esse
tale m, ut nulla.
finita vi dissolvi
possit.
IParticulæ aliæ se
attrahentes, aliæ
repellentes, a, 1 i æ
inertes inter se.
potest haberi in quavis massa cujuscunque figuræ soliditas etiam infinita, sive vis, quæ
impediret dispositionis mutationem non minorem data quacunque. Nam intra illam
figuram [I92] 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, nec ullum adesset intra illam figuram, vel in ejus
superficie spatii punctum, a quo ad distantiam minorem illa distantia data non haberetur
punctum materiæ aliquod. Possent autem intra massam haberi hiatus ubicunque, &
quotcunque prorsus vacui, inscriptis in solo residuo spatio pyramidibus illis, & in angulis
quibusvis posset haberi quivis numerus punctorum distantium a se invicem minus, quam
distent illæ binæ 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 ullæ sint ejusmodi asymptoti præter illam primam, innumera sunt sane figurarum
genera, in quibus pro dato punctorum numero haberi potest æquilibrium, & cohæsionis
limes per elisionem contrariarum virium, ex solutione problematis indicati num. 4I2.
Hoc discrimen est "maxime notatu dignum.
420. Data etiam figura potest adhuc in diversis particulis haberi discrimen maximum
ob diversam distributionem punctorum ipsorum. Sic in eadem sphæra possunt puncta
esse admodum inæqualiter distributa ita, ut etiam paribus distantiis ex altera parte sint
plurima, ex altera paucissima, vel in diversis locis superficiei ejusdem concentricæ esse
congeries plurimæ punctorum conglobatorum, in aliis eorum raritas ingens, & hæc ipsa
loca possunt in diversis a centro distantiis jacere ad plagas admodum diversas in eadem
etiam particula, & in eadem a centro distantia esse in diversis particulis admodum diversis
modis distributa. Verum etiam si particulæ habeant eandem figuram, ut sphæricam, &
in singulis circumquaque in eadem a centro distantia puncta æqualiter 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, tam quod pertinet ad loca densi-
tatum earundem, quam quod pertinet ad rationem inter diversas densitates, possunt in
infinitum variari. - .
42 I. Hæc omnia discrimina pertinent ad numerum, & distributionem punctorum in
diversis particulis : sed ex iis oriuntur alia discrimina præcipua, quæ maximam corporum,
& phænomenorum varietatem inducunt, quæ nimirum pertinent ad vires, quibus puncta
particulam constituentia agunt inter se, vel quibus tota una particula agit in totam alteram.
Possunt inprimis, & in tanta dispositionum varietate debent, [I93] puncta constituentia
eandem párticulam habere vires cohæsionis admodum inter se diversas, ut aliæ multo
facilius, aliæ multo difficilius dispositionem mutent mutatione, quæ aliquam non ita parvam
rationem habeat ad totum. Est autem casus, in quo possint puncta particulæ cohærere
inter se ita, ut nulla finita vi nexus dissolvi possit, ut ubi adsint asymptotici arcus in curva
primitiva, juxta ea, quæ persecutus sum num. 362.
422. Discrimina autem virium, quas una particula exercet in aliam, debent esse adhuc
plura. Inprimis ex num. 222 patet, fieri posse, ut una particula constans etiam duobus
punctis tertium punctum in iisdem distantiis collocatum ab earum medio attrahat per
totum quoddam intervallum, vel repellat per idem intervallum totum, vel nec usquam in
eo repellat, nec attrahat, conspirantibus in primo casu binis attractionibus, in secund9 binis
repul$ionibus itidem conspirántibus, & in tertio attractione, & repulsione æqualibus §e
mútuo elidentibus. Multo autem magis summa virium totius cujusdam particulæ 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, præva-
leaht, in aliâ repulsiones, in alia vires oppositæ se. mutuo elidant. Inde habebuntur,
particulæ in se invicem agentes viribus admodum diversis, pro diversa sua constitutione
& particulæ ad sensum inertes inter se, quæ quidem persecutus sum ipso num. 222.
A THEORY OF NATURAL PHILOSOPHY 3OI
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 of 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
different particles on account of the different disposition of the points that form it. Thus,
in the same sphere, the points may be quite unequally distributed, in such a way that,
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
particles. From them arise the principal differences that are left for consideration; these
lead to the greatest variety in bodies & in phenomena. Such as those that relate to the forces
with which the points forming a particle act upon one another, or the forces with which
the whole of one particle acts upon the whole of another particle. First of all, the points
forming the same particle may, & in such a great variety of distribution must, have forces of
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
particle exerts upon another particle. First of all, it is evident from Art. 222, that it may
happen that a particle consisting of even two points may attract a third point situated at
the same distances from the middle point of the distance between the two points throughout
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,
Difference in the
distribution of
points within the
same figure.
Difference in the
force with which
particles try to
c on serve the ir
figures ; this may
be such that the
p a rticle can be
broken up by no
finite force.
Some particles
attract, others repel
One a not her, &
Some have no ac-
tion on one an-
other.
302 PHILOSOPHIÆ NATURALIS THEORIA
Particulæ quæ in
certis punctis se
repellant, in aliis
attrahant : qu æ
se urgeant in latus,
quæ circumquaque
eandem vim exer-
ceant.
Quo minores par-
ticulæ, eo difficilius
dissolubiles.
IDiscrimina, inter
particulas oriri ex
punctorum vicinia ;
quanto magis de-
beant differre cor-
pora, quæ ex iis
COnstant.
Quæ natura solido-
rum, & fiuidorum ;
qu id in solidis
rigida, quid virgæ
elasticæ : in fluidis
quid viscosa, quid
humida.
423. Aliud discrimen admodum notabile inter ejusmodi particularum vires est illud,
quod eadem particula ex altera parte poterit datam aliam particulam attrahere, ex altera
repellere ; quin immo possunt esse loca quotcunque in superficie particulæ etiam sphæricæ,
quæ alteram particulam in eadem a centro distantia sitam attrahant, quæ repellant, quæ
nihil agant ; cum nimirum in iis locis possint vel plura, vel pauciora esse puncta, quam in
aliis, & ea ad diversas a centro, & a se invicem distantias collocata. Inde autem & illud
fieri poterit, ut, quemadmodum in iis, quæ vidimus a num. 23I, 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 eam tuendam determinetur,
nec consistere possit, nisi in ea unica positione respectu ipsius, vel in quibusdam determinatis
positionibus, ad quas trudatur ab aliis rejecta. Quod si particula sphærica sit, & in omnibus
concentricis superficiebus puncta æqualiter distributa sint, ad distantias a se invicem perquam
exiguas ; tum ejus, & alterius ejus similis particulæ vires mutuæ 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 Terræ
superficies horizontalis ubique sit, & egregie lævigata ; globus ponderis cujuscunque
posset quayis minima vi rotari per superficiem ipsam, elevari non posset sine vi, quæ totum
ipsius pondus excedat. -
424. In hac actione unius particulæ in aliam generaliter, quo particulæ ipsæ minorem
habuerint molem, eo minus ceteris 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, quæ sola positionem turbare nititur, viribus
æqualibus & parallelis nullam mutuæ positionis mutationem inducentibus, eo erit minor,
quo distantiarum, & directionum discrimen minus erit : atque idcirco, quemadmodum
jam exposui num. 239, inferiorum ordinum particulæ difficilius dissolvi possunt, quam
particulæ ordinum superiorum. »
425. Hæc quidem præcipue notatu digna mihi sunt visa inter particularum ex homo-
geneis etiam punctis compositarum discrimina, quæ tamen, quod ad vires pertinet, intra
admodum exiguos distantiarum limites sistunt : nam pro majoribus distantiis particularum
omnium vires sunt prorsus uniformes, uti ostensum jam est num. 2 I 2, nimirum attractivæ
in ratione reciproca duplicata distantiarum ad sensum. Porro hinc , illud admodum
evidenter consequitur, massas majores ex adeo diversis particulis compositas, nimirum
hæc ipsa nostra majora corpora, quæ sub sensum cadunt, debere esse adhuc multo magis
diversa inter se in iis, quæ ad eorum nexum pertinent, & ad phænomena exhibita a viribus
se extendentibus ad distantias illas exiguas, licet omnia in lege gravitatis generalis, quæ
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.
426. Prima se mihi offerunt solida, & fluida, quorum discrimina quæ sint, & quomodo
a mea Theoria ortum ducant, est exponendum. Solida ita inter se connexa sunt, ut quem-
libet aliquot particularum motum sequantur reliquæ : promotæ, si illæ promoventur :
retractæ, si illæ retrahuntur : conversæ in latus, si linea, in qua ipsæ jacent, directionem
mutet : & in eo soliditas est sita : porro ea dicuntur rigida ; si ingenti etiam adhibita vi
positio, quam habet recta ducta per duas quasvis particulas massæ, respectu rectæ, quæ
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, quæ flexilia sunt, ut elasticæ virgæ, pars una directionem positionis mutat, & [I95]
inclinatur, altera priorem positionem servante : & priora illa franguntur, alia majore,
alia minore vi adhibita ; hæc 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 11OI1
exiguam, ut aqua, vel etiam admodum ingentem, ut mercurius, quorum liquorum particulæ
se in globum etiam conformant mutua particularum suarum attractione, & tamen separantur
admodum facile a se invicem majores eorum massæ, 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
of this sort, namely, that the same particle in one part may exert attraction on another
particle, & repulsion from another part ; indeed, there may be any number of places in
the surface of even a spherical particle, which attract another particle placed at the same
distance from the centre, whilst others repel, & others have no action at all. For, at these
places there may be a greater or less number of points than in other places, & these may
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
particles have, the less, other things being equal, is their relative position affected by another
particle situated at any given distance from it. For the differences in the directions &
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
amongst the differences between particles formed from even homogeneous points, which
yet remained, as far as forces are concerned, within certain very narrow limits. For, as
regards greater distances, the forces of all the particles are quite uniform ; that is to say,
they are attractive forces varying approximately as the inverse square of the distances.
Further, from them it follows perfectly clearly that greater masses, formed from these
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
that exist between solids & fluids & how these arise according to my Theory. Solids are
so connected together that the motion of any number of the particles is followed by the
remaining particles; if the former move forward, so do the latter; if they are retracted,
so are the rest; if a line in which they lie changes its direction, they are moved to one
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
Particles which at
certain points repel
& at others attract ;
Some which urge
one another to one
side, & which exert
the same force to
produce rotation.
The smaller the
particle, the greater
difficulty there is in
breaking it up.
Differences a rise
from the nearness
of points to one
another; how much
more should bodies
formed from them
differ from one
another.
The nature of solids
& fluids; what in
Solids are rigid, &
what elastic rods ;
what in fluids are
viscous, & what are
watery.
3O4 PHILOSOPHIÆ NATURALIS THEORIA .
ut fluida cedant vi cuicunque impressæ, ac cedendo facile moveantur, solida vero nonnisi
tota simul moveri possint, & viribus impressis idcirco resistant magis : quæ autem resistunt
lUnde fluiditas : tria,
fluidorum genera.
lUnde facilis motus
in fluidis primi
generis.
Eadem ratio, & in
reliquis binis : dis-
crimen inter ipsa.
quidem multum, sed non ita multum, ut solida, dicuntur viscosa. Ipsa vero fluida
dicuntur humida, si solido admoto adhærescant, & sicca, si non adhæreant.
427. Hæc omnia phænomena præstari possunt per illa sola discrimina, quæ in diverso
particularum textu consideravimus. Ut enim a fluiditate incipiamus, inprimis in ipsis
fluidis omnes, particulæ in æquilibrio esse debent, dum quiescunt, & si nulla externa vi
comprimantur, vel in certam dirigantur plagam ; id æquilibrium debebit haberi a solis
mutuis actionibus : sed ejusmodi casum non habemus hic in nostris fluidis, quæ incumbentis
massæ 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, & arenulæ, ut illæ, ex quibus etiam horologia clepsydris
veterum similia construuntur, & ad fluidorum naturam accedunt maxime, si satis lævigatam
habeant superficiem, quod in quibusdam granulis cernimus, ut in milio : nam plerumque
scabritiem habent aliquam & inæqualitates, quæ motum difficiliorem 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 diverso modo dispositas, ita diversas habere virium summas in
iisdem etiam centrorum distantiis, ut aliæ se attrahant, aliæ se repellant, aliæ 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.
[196] 428. Porro in primo casu statim apparet, unde facilis ille habeatur motus.
Quoniam, aucta distantia, nulla sensibili vi se attrahunt particulæ ; altera non sequetur
motum alterius ; nisi ubi illa versus hanc promota ita accesserit, ut vi repulsiva mutua,
quemadmodum in corporum collisionibus accidit, cogatur illi loco cedere, quæ cessio, si
satis lævigatæ superficies fuerint, ut prominentes monticuli in exiguos hiatus ingressi motum
non impediant, & sit locus aliquis, versus quem possint vel in gyrum actæ particulæ, vel
elevatæ, vel per apertum foramen erumpentes, loco cedere ; facile fiet, nec alia requiretur
vis ad eum motum, nisi quæ ad inertiæ vim vincendam requiritur, vel si graves particulæ
sint versus externam massam, ut hic versus Tellurem, & fluidum motu impresso debeat
ascendere, vis, quæ requiritur ad vincendam gravitatem ipsam : verum ad vincendam
solam vim inertiæ, satis est quæcunque activa vis utcunque exigua, & ad vincendam gravi-
tatem, in hoc fluidorum genere, si perfecta sit lævigatio ; satis est vis utcunque paullo major
pondere massæ fluidæ ascendentis: quanquam nisi excessus fuerit major ; lentissimus
erit motus ; ipsum autem pondus coget particulas ad se invicem accedere nonnihil, donec
obtineatur vis repulsiva ipsum elidens, uti supra ostendimus num. 348; adeoque in statu
æquilibrii se particulæ, in hoc etiam casu, repellent, sed erunt citra, & propre ejusmodi
limites, ultra quos vis attractiva sit ad sensum nulla. Quod si figura particularum præterea
fuerit sphærica, multo facilior habebitur motus in omnem plagam ob ipsam circumquaque
uniformem figuram. -
429. In secundo, ac tertio genere motus itidem habebitur facilis, si particulæ sphæricæ
sint, & paribus a centro distantiis homogeneæ, ut nimirum vires dirigantur ad centra. In
ejusmodi enim particulis motus quidem unius particulæ circa aliam omni difficultate carebit,
& vires mutuæ solum accessum vel recessum impedient. Hinc impresso motu particulis
aliquot, poterunt ipsæ moveri in gyrum aliæ circa alias, & alia succedere poterit loco ab
alia relicto, quin partes remotiores motum ejusmodi sentiant : quanquam fere semper
fortuita quædam 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
I have already considered in the different texture of particles. For, to begin with fluidity,
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 particles. For, since when the distance is increased the particles do not attract one
another with any sensible force, the one does not follow the motion of the other; except
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 movement, if the particles
are spherical, & homogeneous at equal distances from their centres, that is to say, so that
the forces are directed towards their centres. For, in the case of such particles, the motion
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
X
The origin of fluid-
ity ; three kinds of
fluids.
The source of the
easy movement of .
particles of fluids of
the first kind.
The same argument
also holds good for
the O the r two
kinds ; differences
between them.
3o6
PHILOSOPHIÆ NATURALIS THEORIA
In elasticis fluidis
particulas esse ex-
t ra limit e s s u b
arcubus repulsivis
latis.
In fluidis humidis
lumitem valid u m
c O h æ sio nis fore
proximum, & si
abeat in vapores
debere haberi prope
validissimum arcum
repulsivum.
Motus non obstante
vi mutua facilis,
quod a d m o t u m
aliquot p a rt i c u-
larum non debeant
moveri re m ot æ
simul ut in solidis.
Exemplum in qua-
dam hypothesi glo-
borum gravium.
Applicatio exempli
ad fluida, & solida :
successiva particu-
larum separatio in
fluidis.
5.
motus ad remotiores etiam particulas deveniat, sed eo minor, quo major fuerit earum
distantia. Verum hic notandum erit discrimen ingenis inter duo§ casus,'in quibus partes
fluidi se repellunt, & casus, in quibus se attrahunt.
430. In primo casu particulæ proximæ debebunt se omnino repellere, & vis ex parte
altera elidet vim ex altera ; sed si repente relinquatur libertas ex parte quavis, siné ulla
externa vi, sed sola illa particularum actione mutua, recedent reipsa particulæ a se invicem,
& fluidum dilatabitur ; quin [I97] immo externa vi opus est, ad continendam in eo statu
massam ejusmodi, uti aerem gravitas superioris atmosphæræ continet, vel in vase occluso
yasis, ipsius parietes ; & aucta illa externa vi comprimente augeri poterit compressio,
imminuta imminui. Particulæ illæ inter se non erunt in limitibus quibusdam cohæsionis,
sed erunt sub repulsivo arcu curvæ exprimentis vires compositas particularum ipsarum.
43 I. At in tertio genere particulæ quidem proximæ se mutuo repellent, repulsione
æquali illi vi, quæ necessaria est ad elidendam vim externam, & ad elidendam pressionem,
quæ oritur a remotiorum attractionibus : verum si fluidum est parum admodum compres-
sibile, vel etiam nihil ad sensum, ut aqua ; debent esse citra, & admodum prope limitem,
ultra quem vel immediate, vel potius, si id fluidum neque distrahitur (ut nimirum durante
sua 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 illa prodeat, quæ se in ejusmodi fluidorum massulis prodit ;
licet si iterum id fluidum majore vi abire possit in elasticos vapores, ut ipsa aqua post eum
attractivum arcum ; arcus repulsivus debeat succedere satis amplus, juxta ea, quæ diximus
num. I95.
432. In hoc fluidi genere illud mirum videri potest, quod illa attractiva vis, quæ in
majoribus succedit distantiis, & ille validus cohæsionis limes, qui & compressionem &
rarefactionem impedit, non impediat divisionem massæ, & separationem unius partis massæ
ab alia. At quomodo id facile fieri ibi possit, & non possit in solidis, patebit hoc exemplo.
Concipiatur Terræ superficies sphærica accurate, & bene lævigata, ac gravitas sit ejusmodi,
ut in distantia perquam exigua fiat jam insensibilis, ut vis magnetica in exigua distantia
sensum jam effugit. Sint autem globi multi itidem læves mutua attractiva vi præditi,
quæ vim in totam Terram superet. Si quis unum ejusmodi globum apprehendat, & attollat ;
secundus ipsi adhærebit relicta Terra, & post ipsum ascendet, reliquis per superficiem
Terræ progredientibus, donec 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, quem separat, cum nulla sit difficultas in progressu
reliquorum per superficiem Terræ, 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 præstandam. At si
illi globi deberent elevari simul, ut si simul omnes colligati essent per virgas rigidas ; deberent
utique omnes illæ vires omnium in Terram simul superari, & requireretur vis major omnibus
simul. Res eodem redit, ac ubi fasciculus virgarum [I 98] debeat totus frangi simul, vel
potius debeant aliæ post alias frangi virgæ.
433. Id ipsum est discrimen inter fluida hujus generis, & solida. In his motus parti-
cularum circa particulas liber ob earum uniformitatem permittit, ut separentur aliæ post
alias ; dum in solidis vis in latus, de qua egimus jam in pluribus locis, & anguli prominentes,
ac figurarum irregularitas, impediunt ejusmodi liberum motum, qui fiat sine mutatione
distantiarum, & cogunt divulsionem plurimarum particularum simul : unde oritur
difficultas illa ingens dividendi a se invicem particulas solidas, quæ in divisione fluidorum
est adeo tenuis, ac ad sensum nulla. -
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.
430. In the first case adjacent particles must repel one another, in every instance,
& the force from one part must cancel the force from another part. Moreover, if all at
once freedom of movement is left in any one part, without any external force to prevent it,
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
repulsion being equal to that force that is necessary to cancel the external force, & also
the pressure which arises from the attractions of points further off. But, if the fluid is
only very slightly compressible, or not to any appreciable extent (like water, for example),
then the particles must be on the near side, & quite close to, a limit-point; & on the far
side of this limit-point, either there must follow immediately a comparatively ample attractive
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. I95. •
432. In this kind of fluid it may appear strange that the attractive force which follows
at greater distances, or the strong limit-point of cohesion, which prevents both compression
& rarefaction, does not, either of them, prevent division of the mass or the separation of
one part of it from the other. But the reason why this can take place here, & not in the
case of solids, will become evident on considering the following example. Suppose the
surface of the Earth to be perfectly spherical, & quite smooth; & suppose gravity to be
such, that when the distance becomes very small it becomes insensible, just as magnetic
force practically vanishes at a very small distance. Then, suppose we have a number of
smooth spheres endowed with an attractive force for one another, which exceeds the force
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.
With the former, the free motion of the particles about one another, due to their
uniformity, allows them to be separated one after the other. Whilst, with solids, lateral
force, with which we have already dealt in several places, projecting angles & irregularities
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.
In elastic fluids the
particles are out-
side the limit-
points, & under
wide repulsive arcs.
In watery fluids the
nearest limit-point
must be a very
Strong one, of co-
hesion ; & if the
fluid goes off as a
vapour, there must
be close to it a very
strong repulsive arc.
Mutual force not
hindering, move -
ment is e a sy,
because particles
further away need
not move at the
same time, when
any number of par-
ticles are moved,
as is the case for
Solids. Example in
the hypothesis of
heavy spheres.
Application of the
example to the case
of fluids & solids ;
successive separa-
tion of the particles
in the case of fluids.
3o8 PHILOSOPHLE NATURALIS THEORIA
Exemplum ipsius
in aqua : resisten-
tiam in fluidis ad
separationem fieri
e a n d e m, a c i n
solidis, si velocitas
debeat esse ingens.
Soliditatis causa in
vi, & motu in latus :
exemplum in paral-
lelepipedis.
Idem i n fi g u ris
o m n i b u s : und
d is c r i m e n inte
flexilia, & rigida.
Discrimen i n t e r
flexilia, & fragilia
unde.
434. Successivam hujusmodi separationem particularum aliarum post alias videmus
utique in ipsis aquæ guttis pendentibus, quæ ubi ita excreverunt : ut pondus totius guttæ
superet vim attractivam mutuam partium ipsius ; non divellitur tota simul ingens ejus
aliqua massa, sed a superiore parte, utut brevissimo tempore, attenuatur per gradus ; donec
illud veluti filum jam tenuissimum penitus superetur. Fuerunt prius mille particulæ
in superficie, quæ guttam pendentem connectebant cum superiore parte aquæ, quæ
relinquitur adhærens corpori, ex quo pendebat gutta, fiunt paullo post ibi 900, 8oo, 7oo :
& 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
fasciculo frangantur aliæ post alias. At ubi celerrimo motu in fluidum ejusmodi incurritur
ita ; ut non possint tam brevi tempore aliæ aliis particulæ locum dare, & in gyrum agi ;
tum vero fluida resistunt, ut solida. Id experimur in globis tormentariis, qui ex aqua
resiliunt, in eam satis oblique projecti, ut manente satis magna horizontali velocitate
collisio in perpendiculari fiat more solidorum : ac eandem quoque resistentiam in aqua
Scindenda experiuntur, qui se ex editiore loco in eam demittunt.
435. Hinc autem pronum est videre, unde soliditatis phænomena ortum ducant.
Nimirum ubi particularum figura recedit plurimum a sphærica, vel distributio punctorum
intra particulam inæqualis est, ibi nec habetur libertas illa motus circularis, & omnia, quæ
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 præterea,
ea circa axem quencunque conversa, oportet & illam aliam loco cedere, ac eo abire, ubi
positionem priorem respectivam acquirat ; quod cum & tertia respectu secundæ præstare
debeat, & omnes reliquæ circunquaque circa illam positæ ; patet utique, non posse motum
in eo casu imprimi parti cuipiam systematis ; quin & totius systematis motus consequatur
respectivam po-[I99]-sitionem servantis, quæ est ipsa superius indicata solidorum natura.
Res autem multo adhuc magis manifesta fit, ubi figura multum abludat a sphærica, ut si
sint bina parallelepipeda inter se constituta in quodam cohæsionis 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 quædam continua, quæ 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.
436. Quod de parallelepipedis est dictum, id ipsum ad figuras quascunque transferri
potest inæquales utcunque, quæ ex altero latere possint accedere ad aliam particulam, ex
altero recedere : habebitur semper motus in latus, & habebuntur soliditatis phænomena,
nisi paribus a centro distantiis homogeneæ, & sphæricæ formæ particulæ sint. Verum
ingens in eo motu discrimen erit inter diversa corpora. Si nimirum vires illæ hinc, &
inde a limite, in quo particulæ constitutæ sunt, sint admodum validae ; motus in latus
fiet celerrime, & nulla flexio in virga, aut in massa 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 virgæ, nondum pars summa obtinere
potest positionem jacentem in directum cum ipsa, adeoque habebitur inflexio, quæ quidem
eo erit major, quo major fuerit celeritas conversionis ipsius virgæ, uti omnino per experi-
menta deprehendimus.
437. Nec vero minus facile intelligitur illud, quid intersit inter flexilia solida corpora,
& fragilia. Si nimirum vires hinc, & inde ab illo limite, in quo sunt particulæ, extenduntur
ad satis magnas distantias eædem, 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
after the other, in the case of drops of water hanging suspended; here, as soon as they
have increased up to a point where the weight of the whole drop becomes greater than
the mutual attractive force of its parts, any great part is not torn away as a whole; but
by degrees, though in a time that is exceedingly short, the drop is attenuated at its upper
part, until the neck, which has by now become exceedingly narrow, is finally broken altogether.
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 cleaving the surface. -
435. Further, from what has been said, it can be seen without difficulty whence the
phenomena of solidity derive their origin. For instance, when the shape of the particles
is very far from being spherical, or the distribution of the points within the particle is not
uniform, then there is not that freedom of circular motion; & all things that pertain to
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
figures whatever which are at all irregular, if they can approach another particle at one
side & recede from it on the other side ; there will in every case be motion to one side,
& the phenomena of solidity will be obtained, unless the particles are homogeneous at
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.
37. Nor will it be less easy to understand the reason why there is a difference between
flexible solids & fragile bodies. For instance, if the forces on each side of the limit-point, at
which the particles are, are extended unaltered over sufficiently great distances from it, & the
Example of this in
the case of water ;
the resistance to
separation in fluids
becomes as great as
that in solids, if the
velocity has to be
very great.
The cause of solid-
ity lies in lateral
force & motion ;
example of this in
parallelepipeds.
The same thing for
all shapes; hence
the difference be-
tween flexible &
rigid bodies.
The reason of the
difference between
flexible & fragile
bodies.
3 Io PHILOSOPHIÆ NATURALIS THEORIA
Quid, & unde vis-
cositas.
Organicorum cor-
porum efformatio
per vires in latus
versus certa super-
ficiei puncta.
Atomistarum sys-
tema posse deduci
t o t u m e x h a. c
Theoria, & c u m
illa bene cohærere,
explicata præterea,
cohæsione partium
in atomis.
tum vero, vi externa adhibita utrique extremo, 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 illæ non diu durent hinc, & inde eædem, vel per satis magnum intervallum
sit ingens frequentia limitum ; tum quidem inflexio habebitur sine conatu ad se restitu-
endam, & sine fractione, tam vi adhibita utrique extremo, quam ingenti velocitate impressa
alteri, ut videmus accidere in maxime ductilibus, [2oo] 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, & illa, quæ vulgo appellatur durities, quo vires illæ
hinc & inde statim post limites fuerint majores.
438. Atque hic quidem jam etiam ad discrimen devenimus inter elastica, & mollia ;
Verum antequam ad ea faciamus gradum, adnotabo non nulla, quæ 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 mutationi
figuræ, 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 præter tenacitatem,
quam habent inter se, habent etiam vim, qua adhærent externis corporibus, sed non
omnibus, in quo ad humidos liquores referuntur. Humiditas enim est itidem respectiva.
Aqua, quæ digitis nostris adhæret 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 externi 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.
439. In particulis illis, quæ ad soliditatem requiruntur, invenitur admodum expedita
ratio phænomeni ad solida corpora pertinentis, quod Physicos in summam admirationem
rapit, nimirum dispositio quædam in peculiares quasdam figuras, quæ in salibus inprimis
apparent admodum constantes, in glacie, & in nivium stellulis potissimum adeo sunt
elegantes etiam, & ad certas quasdam leges accedunt, quas itidem cum constanti admodum
figurarum forma in gemmarum succis simplicibus observamus, quæ vero nusquam magis
se produnt, quam in organicis vegetabilium, & animalium corporibus. In hac mea Theoria
in promptu est ratio. Si enim particulæ in certis suæ superficiei partibus quasdam alias
particulas attrahunt, in aliis repellunt ; facile concipitur, cur non nisi certo ordine sibi
adhæreant, 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 præterea eadem particula, eadem sui parte, qua alteram attrahit, alteram
pro ejus varia dispositione repellit ; dum massa plurium particularum temere agitata
prætervolat ; eæ tantummodo sistentur, quæ attrahuntur, & ad ea se applicabunt puncta,
ad quæ maxime attrahuntur, ac in illis hæbebunt, in quibus post accessum maxime tenaces
limites [2oI] nanciscentur ; unde & secretionis, & nutritionis, vegetationis, & certarum
figurarum patet ratio admodum manifesta. Et hæc quidem ad nutritionem, & ad certas
figuras pertinentia jam innueram num. 222, & 423.
440. Quoniam ostensum est, qui fieri possit, ut certam figuram acquirant certa
particularum genera, cujus admodum tenacia sint, si quis omnem veterum corpuscularium
sententiam, quam Gassendus, ac e recentioribus alii secuti sunt, adhibentes variarum
figurarum atomos, ut ad cohæsionem uncinatas, ab hac eadem Theoria velit deducere,
is sane poterit, ut patet, & ejusmodi atomos adhibere ad explicationem eorum omnium
phænomenorum, quæ pendent a sola cohæsione, & inertia, quæ tamen non ita multa sunt :
poterunt autem haberi ejusmodi atomi cum infinita figuræ suæ tenacitate, & cohæsione
mutua suarum partium per solas etiam binas asymptotos illas, de quibus num. 4I9, inter
se satis proximas. Et si curva virium habeat tantummodo in minimis distantiis duas
ejusmodi asymptotos, tum post crus repulsivum ulterioris statim consequatur arcus attrac-
tivus, primo quidem plurimum recedens ab axe cum exiguo recessu ab asymptoto, tum
A THEORY OF NATURAL PHILOSOPHY 3 II
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
we pass on to them, I will mention a few matters that have to do with the nature & properties
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
for a phenomenon pertaining to solid bodies, which is a source of the greatest wonder to
physicists. That is, a disposition in certain special shapes, which in salts especially seem
to be quite constant; in ice, & the star-like flakes of snow more especially, they are wonderfully
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
The n a tu re &
source of viscosity.
The formation of
organic bodies by
means of transverse
forces directed to-
wards certain points
of the surface.
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
acquire certain definite shapes, of which they are quite tenacious; if anyone should wish to
derive from this same theory the whole idea of the ancient corpuscularians, such as Gassendi
& others of the more modern philosophers have followed, employing atoms of various shapes,
hooked together for cohesion; he will certainly be able, as is evident, to use atoms of this sort
to explain all these phenomena that depend upon cohesion alone, & inertia; but the number
of these is not very great. Moreover, atoms of this sort can be had with an infinite tenacity
of shape, & mutual cohesion of their parts, by even the sole assumption of those pairs of
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
The whole of the
system formulated
by the Atomists
can be derived
from this Theory,
with which it
agrees very well ;
in addition, the
cohesion of the
parts of their
atoms is explained
by it.
3 I 2 PHILOSOPHIAE NATURALIS THEORIA
Cu r n o n o m n i a
corpora, sint fluida;
licet omnia, puncta,
sint circumquaque
ejusdem vis.
ad axem regrediens, & accedens statim ad formam gravitati exhibendæ debitam ; haberentur
per ejusmodi curvam atomi habentes impenetrabilitatem, gravitatem, & figuræ suæ
tenacitatem ejusmodi, ut ab ea discedere non possent discessu quantum libuerit parvo ;
cum enim possint illæ duæ asymptoti sibi invicem esse proximæ intervallo utcunque parvo,
posset utique ita contrahi intervallum istud, ut figuræ mutatio æqualis datæ cuicunque
utcunque parvæ mutationi eviteatur. Ubi enim cuicunque figuræ inscripta est series
continua cubulorum, & puncta in singulis angulis posita sunt, mutari non potest figura
externorum punctorum ductum sequens mutatione quadam data, per quam quædam
puncta discedant a locis prioribus per quædam intervalla data, manentibus quibusdam,
ut manente basi, nisi per quædam data intervalla a se invicem recedant, vel ad se invicem
accedant saltem aliqua puncta, cum, data distantia puncti a tribus aliis, detur etiam ejus
positio respectu illorum, quæ mutari non potest, nisi aliqua ex iisdem tribus distantiis
mutetur, unde fit, ut possit data quævis positionis mutatio impediri, impedita mutatione
distantiæ per intervallum ad eam mutationem necessarium. Quod si illæ binæ asymptoti
essent tantillo remotiores a se invicem, tum vero & mutatio distantiæ 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
figuræ 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-[2o2]-dem plurima explicari phænomena,
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, Orlatur 1ngens nexus novus, nimirum SIne regressu a se invicem, unco nimirum
alterius in alterius foramen injecto, & intra illud converso per virium inæqualitatem in
diversas unci partes agentium, ut jam prodire non possit ; nam unci cavitas, & foramen,
seu porus alterius particulæ, posset esse multo amplior, quam pro exigua illa distantia
insuperabili, ut idcirco inseri posset sine impedimento orto a viribus agentibus in minore
distantia. Eædem 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 cohæsionis admodum validos cum tenacitate figuræ non quidem infinita,
sed tamen maxima, ubi, quod illi veteres non explicarunt, cohæsio partium atomorum
inter se, adeoque atomorum soliditas, ut & continuata impenetrabilitatis resistentia, &
gravitas, ex eodem generali derivaretur principio, ex quo & reliqua universa Natura. Illud
unum hic notandum superest, ejusmodi atomos habituras necessario ubique distantiam
a se invicem majorem, quam pro illa insuperabili distantia, ad quam externa puncta devenire
ibi non possunt. -
441. Huc etiam pertinet solutio hujusmodi difficultatis, quæ sponte se objicit: si
omnia materiæ puncta simplicia sunt, & vires in quavis directione circumquaque exercent
easdem ; omnia corpora ex iis utique composita erunt fluida multo potiore jure, quam
fluida esse debeant, quæ globulis constent easdem in omni circum directione vires exercen-
tibus. Huic difficultati hic 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 eórum numero, motus ad sensum idem ; tum utique haberetur idem,
quod habetur in fluidis, & separatis aliis punctis post alia, motus facilis per omnes qmnium
έorporum massas obtineretur. At particulæ primi ordinis ab indivisibilibus pungtis ortæ,
ut & proximorum ordinum particülæ ortæ ab iis, sua ipsa parvitate mglis tueri possunt
juxta num. 424 formam suam, & positionem punctorum : nam differentia, virium exercit-
árum in diver$a earum puncta potest esse perquam exigua, summa virium pr9hibente
tantum accessum unius particulæ ad alteram, quo tamen accessu inæqualitas virium, &
A THEORY OF NATURAL PHILOSOPHY 3 I3
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 beformed 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
all points of matter are simple, & if they exert the same forces in all directions round
themselves; then it is far more natural to expect that all bodies that are composed of such
points would be fluid than that those, which consist of little spheres exerting the same
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,
The reason why all
bodies are not fluid,
although all points
in all directions
round are under
the same force.
3I4 PHILOSOPHIÆ NATURALIS THEORIA
Difficultas d et er-
minandi resisten-
tia m fluidorum :
methodi indirectæ
id præstandi eædem
in hac Theoria ac
in communi.
E i n i resistentiæ
fontes, & utriusque
lex.
Quam legem vide-
antur innuere ex-
perimenta : in vis-
c o sis resistentiam
esse majorem.
Problemata alia ad
resistentiam perti-
n e n tia itid e m
c o m m u n ia huic
Theoriæ.
obliquitas directionum ha-[203]-beatur adhuc satis magna ad vincendas vires mutuas,
mutandam positionem, qua positione manente, manet inæqualitas virium, quas diversa
puncta ejus particulæ exercent in aliam particulam. Ea inæqualitas 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 inæqualitas eo deinde erit
major, quo ad altiores ascenditur particularum ordines, donec deveniatur ad corpora,
quæ a nobis sentiuntur. -
442. Solida externum corpus ad ea delatum intra suam massam non recipiunt, ut
vidimus : at fluida solidum intra se moveri permittunt, sed resistunt motui. Resistentiam
ejusmodi accurate comparare, & ejus leges accurate definire, est res admodum ardua.
Oporteret nosse ipsam virium legem determinate, & numerum, & dispositionem punctorum,
ac habere satis promotam Geometriam, & Analysin ad rem præstandam. 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 æque
cum mea, ac cum communi Theoria, consentire possunt.
443. Ut tamen aliquid innuam etiam de eo argumento, duplex est resistentiæ fons
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 particulæ exercent, dum aliæ in alias incurrunt,
quæ earum motum impediunt, quo casu comprimuntur non nihil particulæ ipsæ etiam
in fluidis non elasticis egressæ e limitibus, & æquilibrio : acquirunt autem motus admodum
diversos, gyrant, & alias impellunt, quæ 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, quæ provenit a motu communicato particulis
fluidi, & quæ 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 particulæ : ut quadratum velocitatis, quia pari densitate eo plures particulæ
dato tempore loco movendæ sunt, quo major est velocitas, nimirum quo plus spatii percur-
ritur, & eo major singulis imprimitur motus, quo itidem velocitas est major. Resistentia
autem, quæ oritur a viribus, quas in se exercent particulæ, si vis ea esset eadem in singulis,
quacunque velocitate [204] moveatur corpus progrediens, esset in ratione temporis, sive
constans : nam plures quidem eodem tempore particulæ eam vim exercent, sed breviore
tempore durat singularum actio, adeoque summa evadit constans. Verum si velocitas
corporis progredientis sit major; particulæ 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.
444. Porro ex experimentis nonnullis videtur erui, resistentiam in nonnullis fluidis
esse partim in ratione duplicate velocitatum, partim in ratione earum simplici, & partim
constantem, sive in ratione momentorum temporis, quanvis ubi velocitas est ingens,
deprehendatur major : & ubi fluiditas est ingens, ut in aqua, ut secundum resistentiæ
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 suæ densitatis, & velocitate
corporis progredientis : nam in ejusmodi fluidis, quæ ad solida accedunt, illud secundum
resistentiæ 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, quæ tamen, si fragilia sunt, & sensibilem compressionem non admittant,
diffringuntur.
445. Jam vero quæcunque a Newtono primum, tum ab aliis demonstrata sunt de
motu corporum, quibus resistitur in variis rationibus velocitatum, ea omnia consentiunt
itidem cum mea Theoria, & hujus sunt loci, ac ad illam pertinent Mechanicæ partem,
quæ agit de motu solidorum per fluida. Sic etiam determinatio figuræ, cui minimum
A THEORY OF NATURAL PHILOSOPHY 3I 5
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
The difficulty of
determining the re-
sistance of fluids;
the in direct
methods for accom-
plishing this are the
same in my Theory
as in the usual one.
of
the
Two sources
resistance, &
laws of each.
& 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 jointly. 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 law that ex-
periments seem to
indicate : the resist-
ance is greater in
viscous fluids.
Other problems
relating to resist-
a n ce th a t are
common also to this
Theory.
3I6 PHILOSOPHIÆ NATURALIS THEORIA
Alia pertinentia
huc pertractata,
in parte secunda :
dis crim e n inter
elastica, & mollia.
Fluid a elastica,
quorum partes non
s u n t in limitibus
cohæsionis ; Omnia
& solida, & fluida
elastica, esse, sed
non dici, quia sensi-
bilem compressio-
nem non patiuntur.
Dura nulla esse :
q u æ dic a nt u r :
u m de fragilitas, &
ductilitas.
resistitur, determinatio vis fluidi solidum impellentis directionibus quibuscunque, mensura
velocitatis inde oriundæ per corporum objectorum resistentiam observatione definitam,
innatatio solidorum in fluidis, ac alia ejusmodi, & mihi communia sunt : sed oportet
rite distinguere, quæ sunt hypothetica tantummodo, ab iis, quæ habentur reapse in
Natura.
446. Ad fluida & solida pertinent itidem, quæcunque in parte secunda demonstrata
sunt de pressione fluidorum, & velocitate in effluxu, quæcumque de æquilibrio solidorum,
de vecte, de centro oscillationis, & percussionis, quæ quidem in Mechanica pertractari
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, quæ ad soliditatem, & fluiditatem pertinent : illud vero, quod pertinet ad
discrimen inter elastica, & mollia, brevi expediam. Elastica sunt, quæ post mutationem
[205] figuræ redeunt ad formam priorem ; mollia, quæ in nova positione perseverant.
Id discrimen Theoria exhibet per distantiam, vel propinquitatem limitum, juxta ea, quæ dicta
num. I99. Si limites proximi illi, in quo particulæ cohærent, hinc, & inde plurimum ab
eo distant, imminuta multum distantia, perstat semper repulsiva vis ; aucta distantia, perstat
vis attractiva. Quare sive comprimatur plus æquo, sive plus æquo distrahatur massa, ad
figuram veterem redit ; ubi rediit, excurrit ulterius, donec contraria vi elidatur velocitas
concepta, ac oritur tremor, & Oscillatio, quæ paullatim minuitur, & extinguitur demum,
partim actione externorum corporum, ut per aeris resistentiam sistitur paullatim motus
penduli, partim actione particularum minus elasticarum, quæ admiscentur, & quæ possunt
tremorem illum paullatim interrumpere frictione, ac contrariis motibus, & sublapsu, quo
suam ipsam dispósitionem nonnihil immutent. . Si autem limites sint satis proximi ; causa
externa, quæ massam comprimit, vel distrahit, posteaquam adduxit particulas ab uno
cohæsionis limite ad alium, ibi eas itidem cogit subsistere, quæ ibidem semel constitutæ
itidem in æquilibrio sunt, & habetur massa mollis. - -
447. Quædam elastica fluida non habent particulas positas inter se in limitibus cohæ-
sionis, sed in distantiis repulsionum, & quidem ingentium, ut aer : sed vel incumbente
pondere, vel parietibus quibusdam impeditur recessus ille, & sunt quodammodo ibidem
in statu violento ; licet semper puncta singula in æquilibrio sint, oppositis repulsionibus
se mutuo elidentibus. Omnia autem & solida, & fluida, quæ videntur nec comprimi, nec
ullas habere vires mutuas inter particulas, sed in limitibus esse, adhuc elastica sunt, sive
vim repulsivam exercent inter particulas proximas, saltem quæ sensibili gravitate sunt
prædita, quæ nimirum vis repulsiva vim gravitatis elidat. Verum ea distantias parum
admodum mutant, mutatione, quæ idcirco sensum omnem effugiat ; quod accidit in
aqua, quæ in fundo putei, & prope superficiem supremam habet eandem ad sensum densi-
tatem, & in metallis, & in marmoribus, & in solidis corporibus passim, quæ pondere majore
imposito nihil ad sensum comprimuntur. Sed ea idcirco appellari non solent elastica, &
ad ejusmodi appellationem non sufficit vis repulsiva etiam ingens inter particulas proximas :
sed etiam requiritur mutatio sensibilis distantiæ respectu distantiæ totalis respondens
sensibili mutationi virium.
448. Dura corpora in eo sensu, in quo a Physicis duritiei nomen accipitur, ut nimirum
figuram nihil prorsus immutent, nulla sunt in mea Theoria, ut & nulla compacta penitus,
ac plane solida, quemadmodum diximus etiam num. 266; sed dura vocat vulgus, quæ
satis magnam exercent vim, ne figuram mutent, sive elastica sint, sive fragilia, sive mollia.
Fragilitas, unde ortum ducat, expositum est paullo su- [2o6]-perius num. 437, & inde etiam
quid ductilitas, ac malleabilitas sit, facile intelligitur. Ductilia nimirum a mollibus non
differunt, nisi in majore, vel minore vi, qua figuram tuentur suam : ut enim mollia pressione
Superiora o m nia
p r ofl u e re € X
Theoria : ejus fœ-
cunditas : illa omnia
a densitate non pen-
dere,
tenui, & ipsis digitis comprimuntur, vel saltem figuram mutant, sed mutatam retinent,
ita ductilia ictu validiore mallei mutant itidem figuram suam veterem, & retinent novam,
quam acquirunt. -
449. Atque hoc demum pacto quæcunque pertinent ad fluidorum, & solidorum diversa
genera, nam & elastica, mollia, ductilia, fragilia eodem referuntur, invenimus omnia in
illo particularum discrimine orto ex sola diversa combinatione punctorum, quam nobis
Theoria 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
were demonstrated with regard to pressure of fluids, & velocity of efflux; & all matters
relating to equilibrium of solids, the lever, the centre of oscillation, & the centre of percussion;
all of which indeed are usually considered in connection with Mechanics. I will but add
that, from the ease of movement of the particles of a fluid about one another, & from their
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
to one another, but are at distances corresponding to repulsions, & these too very great;
for instance, air. But recession is prevented either by superincumbent weight, or by
enclosing walls; these are in some sort of violent condition at these distances, although
each point is always in equilibrium, due to the opposite repulsions cancelling one another.
Moreover, all solids & fluids, which appear neither to suffer compression, nor to have any
mutual forces between their particles, but to be at limit-points, are however elastic; that
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
hardness is accepted by Physicists, namely such as do not suffer the slightest change of shape;
& also there are none that are perfectly compact, or quite solid, as I said in Art. 266. But
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. -
49. Finally, in this way, whatever properties there may be relating to different kinds
of fluids & solids (for elastic, soft, ductile & fragile bodies all come to the same thing), we
have made them all out from the difference between particles that is produced by the
difference in the combination of the points alone; this will be shown by my Theory if
Other matters that
were discussed in
the Second part
really pertain to
this connection ;
distinction between
el a stic & s of t
bodies.
E 1 a stic fluid S
whose particles are
not at limit-points .
of cohesion. All
solids & fluids are
really elastic, but
are not called So,
because they do not
suffer se nsible
compression.
There are no hard
bo di e s ; what
bodies are called
hard ; hence fra-
gility & ductility.
All the above pro-
perties are derived
from my Theory;
all of them do not
depend upon den-
sity.
3I8 PHILOSOPHIÆ NATURALIS THEORIA
Communia quatuor
elementa quid sint.
Chemicarum opera-
tionum genera de-
duci facile ex illo
particularum dis-
crimine : singular-
ium effectuum cau-
sas singulares non
posse cognosc1 a.
mente humana.
& debebit ; si curva primigenia ingentem habeat numerum intersectionum cum axe, &
particulæ primi ordinis, ac reliquæ ordinum superiorum dispositiones, quæ in infinitum
variari possunt, habuerint plurimas, & admodum diversas inter se, ac eas inprimis, quæ ad
hæc ipsa figurarum, & virium discrimina requiruntur. Illud unum hic diligenter notandum
est, quod ipsam Theoriam itidem commendat plurimum, hasce proprietates omnes a densitate
nihil omnino pendere. Fieri enim potest, uti num. I83 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 phænomena eadem figurarum, & virium æque inveniri
possint, ubi multo plura, & ubi multo pauciora puncta massam constituunt. -
45o. Jam vero illa, quæ vulgo elementa appellari solent, Terra, Aqua, Aer, Ignis,
nihil 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, quæ solidi-
tatem aliarum admixtione particularum acquirunt, ut cineres oleorum ope, vel etiam
aliqua mutatione dispositionis internæ, ut in vitrificatione evenit, quæ 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 particulæ, sustinentes vel externæ vis, vel suiipsius ponderis pressionem sine sensibili
distantiarum imminutione. Aer est fluidum elasticum, quem 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, quæ in eo innatant :
sed ejus particulæ satis magna vi se repellunt, [207] & ea repulsiva particularum vis
imminutis distantiis diu perdurat, ac pertinet ad spatium, quod habet ingentem rationem
ad eam tanto minorem distantiam, ad quam compressione reduci potest, & in qua adhuc
ipsa vis crescit, arcu curvæ adhuc recedente ab axe : is vero arcus ad axem ipsum deinde
debet ruere præceps, ut circa proximum limitem adhuc ingentes in eo residuo 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 transformationibus agemus in fine. Ignis etiam est fluidum maxime elasticum,
quod violentissimo intestino 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, quæ ad Chemicas operationes
pertinet, ad quas jam progredior.
45 I. Chemicarum operationum principia ex eodem deducuntur fonte, nimirum ex
illo particularum discrimine, quarum aliæ inter se, & cum quibusdam aliis inertes, alias
ad se attrahunt, alias repellunt constanter per satis magnum intervallum, ubi attractio
ipsa cum aliis est major, cum aliis minor, aucta vero satis distantia, evadit ad sensum nulla ;
quarum itidem aliæ respectu aliarum habent ingentem virium alternationem, quam mutato
nonnihil textu suo, vel conjunctæ, & permixtæ cum aliis mutare possunt, succedente
pro particulis compositis alia virium lege ab ea, quæ in simplicibus observabatur. Hæc
omnia si habeantur ob oculos ; mihi sane persuasum est, facile inveniri posse in hac ipsa
Theoria rationem generalem omnium Chemicarum operationum : nam singulares deter-
minationes effectuum, qui a singulis permixtionibus diversorum corporum, per quas unice
omnia præstantur in Chemia, sive resolvantur corpora, sive componantur, requirerent
intimam cognitionem textus particularum singularum, & dispositionis, quam habent in
massis singulis, ac præterea Geometriæ, & Analyseos vim, quæ humanæ mentis captum
excedit longissime. " Verum illud in genere omnino patet, nullam esse Chemiæ partem,
in qua præter inertiam massæ, & specificam gravitatem, alia virium mutuarum genera
inter particulas non ubique se prodant, & vel invitis incurrant in oculos, quod quidem
vel in sola postrema quæstione Opticæ Newtoni abunde patet, ubi tam multa & attractionum,
A THEORY OF NATURAL PHILOSOPHY 3I9
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
& 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,
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,
from the distinctions between particles; some of these being inert with regard to themselves
& in combination with certain others, some attract others to themselves, some repel others
continuously through a fairly great interval; & the attraction itself with some is greater,
& with others is less, until when the distance is sufficiently increased it becomes practically
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
& intermingled with others; in this case there follows another law of forces for the compound
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
The nature of the
four Elements,
commonly So-
called.
The different kinds
of chemical opera-
tions are readily
derived from the
differences between
particles ; the
special causes of
each of the effects
are beyond the
intelligence of the
human mind.
32O PHILOSOPHIÆ NATURALIS THEORIA
Quid sint : dissolu-
tio & præcipitatio :
prima quomodo fiat,
& quæ sit ejus
CauSa.
Quomodo fiat præ-
cipitatio, & quæ sit
ejus causa.
IPluviam fortasse
esse quoddam præ-
cipitationis genus :
m ira phænomena
c o m mixti o n u m
quomodo explicen-
tur.
& vero etiam repulsionum indicia, atque argumenta proferuntur. Omnia etiam genera
eorum, quæ ad Chemiam pertinent, singillatim persequi, infinitum esset : evolvam
speciminis loco præcipua quædam.
[208] 452. Primo loco se mihi offerunt dissolutio, & ipsi contraria præcipitatio. Immissis in
quædam fluida quibusdam solidis, cernimus, mutuum nexum, qui habebatur inter eorum
particulas, dissolvi ita, ut ipsa jam nusquam appareant, quæ tamen ibidem adhuc
manere in particulas perquam exiguas redacta, & dispersa, ostendit præcipitatio. Nam
immisso alio corpore quodam, decidit ad fundum pulvisculus tenuissimus ejus
substantiæ, & quodammodo depluit. Sic metalla in suis quæque menstruis dissolvuntur,
tum ope aliarum substantiarum præcipitantur : aurum dissolvit aqua regia, quod immisso
etiam communi sale præcipitatur. Rei ideam est admodum facile sibi efformare satis
distinctam. Si particulæ solidi, quod dissolvitur, majorem habent attractionem cum
particulis aquæ, quam inter se ; utique avellentur a massa sua, & singulæ circumquaque
acquirent fluidas particulas, quæ illas ambiant, uti limatura ferri adhæret magnetibus,
ac fient quidam veluti globuli similes illi, quem referret Terra ; si ei tanta aquarum copia
affunderetur, ut posset totam alte submergere, vel illi, quem refert Terra submersa in
aere versus eam gravitante. Si, ut reipsa debet accidere, illa 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, quæ terrulæ ob exiguam molem effugient nostros
sensus, nec vero decident sustentatæ a vi, quæ illas cum circumfuso mari conjungit : sed
globuli illi ipsi constituent quandam veluti continui fluidi massam. Ea est dissolutionis
idea.
453. Quod si jam in ejusmodi fluidum immittatur alia substantia, cujus particulæ
particulas fluidi ad se magis attrahant, & fortasse ad majores etiam distantias, quam
attrahuntur ab illis ; dissolvetur utique hæc secunda substantia, & circa ipsius particulas
affundentur particulæ fluidi, quæ prioris solidi particulis adhærebant, ab illis avulsæ, &
ipsis ereptæ. Illæ igitur nativo pondere intra fluidum specifice levius depluent, & habebitur
præcipitatio. 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, quæ diximus num. 4I 3, non potest iterum
solo accessu, & appressione deveniri ad illos eosdem limites, qui prius habebantur.
454. Hoc pacto dissolutionis, & præcipitationis acquiritur idea admodum distincta ;
& fortasse etiam pluvia est quoddam præcipitationis genus, nec provenit e sola unione par-
[209]-ticularum aquæ, quæ prius tantummodo dispersæ temere fuerint, & ob solam
tenuitatem suam sustentatæ ac suspensæ innataverint. Apparet ibi etiam, qua ratione
binæ substantiæ 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 specificæ 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 phænomeni prima fronte admodum miri in promptu est
causa in mea Theoria : cum particulæ, quæ nimirum se immediate non contingebant,
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 materiæ minueretur moles, victa repulsione mutua
A THEORY OF NATURAL PHILOSOPHY - 32 I
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
certain solids are mixed with certain fluids, we see that the mutual connection which there
used to be between the particles of each is dissolved in such a way that the solids are no
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
of which attract the particles of the fluid to themselves with a stronger force, & perhaps
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
also rain is some sort of precipitation, & does not merely come from the union of particles
of water which previously had been only dispersed at random, & had floated, sustained &
suspended in the air, owing to their extreme tenuity alone. Also, we can now see how
two substances can be mixed together to coalesce into a single mass. This indeed, in the
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
Y
The nature of solu-
tion & precipita-
tion ; how the first
comes about, & its
Call Se.
The manner in
which precipitation
occurs ; & its cause.
Perhaps rain is
Some sort of preci-
pit a tion ; how
certain wonderful
phenomena in con-
nection with mix-
tures are explained.
322 PHILOSOPHIÆ NATURALIS THEORIA
Cu r a d commix-
tionem solidorum
requiratur contu-
sio : quid ad eam
præstetignis : unde
ars separandi me-
talla.
Liqua tio ne m, &
volatili za t i o n e m
fieri posse per agita-
t i o n e m ingentem
p a r t i c u 1 a r u m.
Prima quomodo fiat.
Alia liq uationis
ratio per separa-
tionem partiu m
heterogenearum.
Quomodo fiat vola-
tilizatio : fixatio, &
volatilizatio aeris.
per attractionem magneticam, qua cuspides elastrorum ad se invicem accederent.
455. Ubi solidum cum solido commiscendum est, ut fiat unica massa, ibi quidem
oportet solida ipsa prius contundere, vel etiam dissolvere, ut nimirum exiguæ particulæ
seorsim possint ad exiguas alterius solidi accedere, & cum iis conjungi. Id autem fit
potissimum per ignem, cujus vehementi agitatione, & vero etiam fortasse actione ingenti
mutua inter ejus particulas, & inter quædam peculiaria substantiarum genera, ut olea,
& sulphur, quæ ut gluten quoddam conjungebant inter se vel inertes particulas, vel etiam
mutua repulsione præditas, dissolvit omnium corporum nexus mutuos, & massas omnes
demum, si satis validus sit, cogit liquari, & ad naturam fluidorum accedere. Dissolutarum,
ac liquescentium massarum particulæ commiscentur, & in unam massam coalescunt : ubi
autem sic coaluerunt, possunt iterum sæpe dissimiles separari eadem actione ignis, qui
aliquas prius, alias posterius, cogit minore vi abire per evaporationem, & maxime fixas
majore vi reddit volatiles. Inæqualibus ejusmodi diversarum substantiarum attractionibus,
& inæqualibus adhæsionibus inter earum particulas, omnis fere nititur ars separandi metalla
a terris, cum quibus in fodinis commixta sunt, & alia aliorum ope prius uniendi, tum etiam
a se invicem separandi, quæ omnia singillatim persequi infinitum foret. Generalis omnium
explicatio facile repetitur ab illa, quam exposui, particularum diversa constitutione, quarum
aliæ respectu aliarum inertes sunt, respectu aliarum activitatem habent, sed admodum
diversam, tum [2Io] quod pertinet ad directionem, tum quod ad intensitatem virium.
456 De Liquatione, & volatilizatione dicam illud tantummodo, eas fieri posse etiam
sola ingenti agitatione particularum fluidi cujuspiam tenuissimi, cujus particulæ ad solidi,
& fixi corporis particulas accedant satis, & inter ipsarum etiam intervalla irrumpant ; qui
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 illæ particulæ
vim exercent circunquaque circa illum axem ad sensum eandem, succedentibus sibi invicem
celerrime punctis, & directionibus, in quibus diversæ vires exercentur, qui etiam axes si
celerrime mutentur, irregulari nimirum impulsu, habebitur in iis particulis id, quod
æquivaleat sphæricitati & 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. Hæc fluiditas erit violenta, & desinente
tanta illa agitatione, ac cessante vi, quæ agitationem inducebat, cessabit, ac fluidum etiam
sine admixtione novæ substantiæ poterit evadere solidum. Poterit autem paullatim
cessare motus ille rotationis tam per inæqualitatem exiguam, quæ semper remanet inter
vires in diversis locis particulæ diversas, & obsistit semper nonnihil rotationi, quam per
ipsam expulsionem illius agitatæ substantiæ, ut igneæ, & per resistentiam circumjacentium.
457. Deinde haberi etiam poterit liquatio per subtractionem heterogenearum, &
difformium particularum, quæ magis homogeneas, & ad sphæricitatem accedentes particulas
alligabant quodammodo impedito motu in gyrum. Id sane videtur accidere in pluribus
substantiis, quæ quo magis depurantur, & ad homogeneitatem reducuntur, eo minus
tenaces evadunt, & viscosæ. 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 compositio. -
458. Quod si priore modo liquatio accidat, & in eo motu particulæ a limitibus
cohæsionis, 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 particulæ quæ fixum corpus componebant, erant quidem
inter se in distantiis repulsionum validissimarum, sed per interjacentes particulas alterius
substantiæ cohibebatur illa repulsiva vis superata ab attractione, quam exercebat
in eas nova intrusa particula : si enim hæc agitatione illa excutiatur, vel ab alia, quæ
ipsam attrahat magis, prætervolante ad exiguam distantiam abripia-[2II]-tur ; tum
vero repulsiva vis particularum prioris substantiæ reviviscit quodammodo, & agit,
ac ipsa substantia evadit volatilis, quæ iterum nova earundem particularum intrusione
figitur. Id sane videtur accidere in aere, 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.
55. When a solid has to be mixed with a solid to form a single mass, it is necessary
to first of all crush the solids, or even to dissolve them, so that the exceedingly small particles
of the one can separately approach those of the other solid, & combine with them. Now
this especially takes place in the case of fire; by its vigorous internal movement, & perhaps
too through a very great mutual attraction between its particles & those of certain particular
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.
456. With regard to liquefaction & volatilization, I will only say this : that these
phenomena can take place simply through a violent agitation of some very tenuous fluid,
whose particles approach sufficiently close to the particles of the solid fixed body, & push
into the intervals between them. How this internal motion can happen I will explain,
when I discuss fermentation & effervescence. First of all, owing to the internal agitation,
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,
Why crushing is
necessary for a
mixture of solids ;
the effect of fire in
this respect; whence
the art of separating
metals.
Liquefaction & vol-
atilization can take
place owing to a
very great agitation
of the particles; the
manner in which
the first happens.
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.
457. Secondly, there may be liquefaction through the subtraction of heterogeneous
& non-uniform particles, which bound together the more homogeneous particles which
approximate to sphericity, in such a way as to hinder their rotary motion. This is in
fact seen to happen in 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
particles should go off from the limit-points at which they were to distances a little greater,
& if for these distances there should be a very large repulsive arc, then the particles will fly off
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
A not he r reason
for liquefaction is
through the separa-
tion of heterogene-
ous parts.
How volatilization
takes place ; fixa-
tion & volatilization
of air.
324 PHILOSOPHIÆ NATURALIS THEORIA
Causa agitationis
particul a r u m in
igne, fermenta-
tionibus, efferves-
centiis repetita a
contorsione curvæ
circa axem.
Oscillatio n es in
accessu se m p er
sisti a primo crure
repulsivo, pro re-
cessu bini casus.
In primo cruris
attractivi asympto-
tici semper sisti re-
cessum etiam.
In secundo casu
a r c u s attractivi
ingentis, sed finiti
egressus partis
punctorum excus-
sorum e fine oscil-
lationis sine re-
gressu.
demonstravit per experimenta, partem ingentem lapidum, qui in vesica Oriuntur, &
calculorum in renibus constare puro aere ad fixitatem reducto, qui deinde potest iterum
statum volatilem recuperare : ac halitus inprimis sulphurei, & ipsa respiratio animalium
ingentem aeris copiam transfert a statu volatili ad fixum. Ibi non habetur aeris compressio
sola facta per cellularum parietes ipsum concludentes ; iienim 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 illa diffringeret. Halesius putat, eum inillo
statu amittere elasticitatem suam, quod fieret utique, si particulæ ipsius ad eam 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 illæ particulæ, sed
ab interposita sulphurei halitus particula attrahantur magis, ut paullo ante vidimus in
elastris a globulo magnetico cohibitis, & constrictis. Tum quidem elasticitas in aere ad
fixitatem redacto maneret tota, sed ejus effectus impediretur a prævalente vi. Atque id
quidem animadverti, & monui ante aliquot annos in dissertatione De Turbine, in qua omnia
turbinis ipsius phænomena ab hac aeris fixatione repetii.
459 Porro agitatio illa particularum in igne, ac in fermentationibus, & effervescentiis,
unde oriatur, facile itidem est in mea Theoria exponere. Ut primum crus meæ curvæ
mihi impenetrabilitatem exhibuit, postremum gravitatem, intersectiones autem varia
cohæsionum genera ; ita alternatio arcuum jam repulsivorum, jam attractivorum,
fermentationes exhibet, & evaporationes variorum generum, ac subitas etiam deflagrationes,
& explosiones, illas, quæ occurrunt in Chemia passim, & quam in pulvere pyrio quotidie
intuemur. Quæ autem huc ex Mechanica pertinet, jam vidimus num. I99. Dum ad
se invicem accedunt puncta cum velocitate aliqua, sub omni arcu attractivo velocitatem
augent, sub omni repulsivo minuunt : contra vero dum a se invicem recedunt, sub omni
repulsivo augent, sub omni attractivo minuunt, donec 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.
460. Et in accessu quidem semper devenitur ad arcum repulsivum aliquem parem
extinguendæ velocitati cuilibet utcun-[212]-que magnæ ; devenitur enim saltem ad
primum asymptoticum crus, quod in infinitum protenditur : at pro recessu duo hic casus
occurrunt potissimum considerandi. Vel enim etiam in recessu devenitur ad aliquod
crus asymptoticum attractivum cum area infinita, de cujusmodi casibus egimus jam num.
I95, vel devenitur ad arcum attractivum recedentem longissime, & continentem aream
admodum ingentem, sed finitam. In utroque casu actio punctorum, quæ extra massam
sunt sita, aliorum punctorum massæ intestino illo motu agitatæ Oscillationem augebit
aliorum imminuet, & puncta alia post alia procurrent ulterius versus asymptotum, vel limitem
terminantem attractivas vires : quin etiam actiones mutuæ 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, quæ illa conjungit, deberent, uti
monuimus num. I 92, 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 curvæ in infinitum aucta, itidem nunquam deveniretur ad distantiam illius asymptoti.
Quare in eo primo casu utcunque vehemens esset interna massæ 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.
46I. At in secundo casu, in quo arcus attractivus ille ultimus ejus spatii ingens esset,
sed finitus, posset utique quorundam punctorum in illa agitatione augeri excursus usque
ad limitem, post quem limitem succedente repulsione, jam illud punctum a massa illa
quodammodo velut avulsum avolaret, & motu accelerato recederet. Si post eum limitem
summa arearum repulsivarum esset major, quam summa attractivarum, donec deveniatur
ad arcum illum, qui gravitatem exprimit, in quo vis jam est perquam exigua, & area
asymptotica ulterior in infinitum etiam producta, est finita, & exigua ; tum vero puncti
elapsi recessus ab illa massa nunquam cessaret actione massæ ipsius, sed ipsum punctum
pergeret recedere, donec 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, &
effervescence is also easily explained by my Theory. Just as the first branch of my curve
gives me impenetrability, & the last branch gravitation, & the intersections with the axis
the various kinds of cohesions; SO also the alternation of the arcs, now repulsive, now
attractive, represent fermentations & evaporations of various kinds, as well as sudden
conflagrations & explosions; such things as occur everywhere in chemistry, & what we see
Cause of the agita-
tion of the particles
in fire, fermenta-
tions, & efferves-
cence, derived from
the contortions of
the curve round
the axis.
every day in the case of gunpowder. Those things from Mechanics that belong here
we have already seen in Art. I99. 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
is capable of destroying any velocity however great ; for at least finally the first asymptotic
branch, which goes off to infinity, is reached. But on recession, there are two cases met
with, which have to be considered in this connection. For, on recession, either there is
reached an asymptotic attractive branch having an infinite area, cases of which kind I
dealt with in Art. 194; or else we come to an attractive arc receding very far from the
axis, & containing an exceedingly great but finite area. In either case, the action of points
situated outside the mass will increase the oscillation of some of the points of the mass
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.
.g.
461. Now, in the second case, in which the attractive arc at the end of the space is
very large, but finite, it will indeed be possible for the motion of some points in the
agitation to be increased right up to the limit-point; &, as repulsion follows the limit-point
that point of the mass will now be as it were torn off, & it will fly away & leave the mass
with accelerated motion. If after the limit-point, the sum of the repulsive areas should
be greater than the sum of those that are attractive, that is, until that arc is reached which
represents gravity, where the force then becomes exceedingly small, & the asymptotic area,
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,
Oscillations on ap-
proach are always
stopped by that
first repulsive
branch, but for re-
cession, there are
two cases. In the
first, where there is
an asymptotic at-
tractive branch,
recession also
always stopped.
is
In the Second case,
where there is a
very great but finite
attractive arc, there
will be separation
of Some of the
points at the end
of an Oscillation, &
these will fly off
without returning.
326 PHILOSOPHIAE NATURALIS THEORIA
Inde pro diversa
arCullm, forma
evaporatio lenta.
Vel subita explosio,
č deflagratio ; ac
transformation e S
va riae, avolante
parte mixti.
Concretiones evapo-
rato fluido, 8 certae
figurae residui, ut in
salibus.
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 perseveretad 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.
462. Hic jam plurima considerari possent, & casuum differentium, ac combinationum
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 quaedam evaporatio: puncta quae in fortuita agitatione ad eun finem deveniunt,
erunt pauca respectu totius massa, quae tamen in ingenti massa, 8 eodem fermentationis
statu erunt eodem tempore ad sensum aequali numero, ac, massa imminuta, imminuetur
&is numerus, massa autem diu perseverabit ad sensum nihil mutata. Habebituribi quaedam
velut ebullitio, & vaporum quantitas, acvis in egressu in diversis substantiis variari plurimum
poterit, cum pendeata situ, in quo illa 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.
463. Sed si intervallum, quod massam claudit inter limites accessus, & recessus, sit
perquam exiguum, arcus attractivus postremus non sit ita validus, & succedat arcus repul-
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 massae, agitatio eiusmodi 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 eiicit. 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 particulae ordinum
inferiorum, quae pertinebant ad diversas particulas superiorum, coalescant in particulam
ordinis superioris novi generis: hinc tam multa compositiones, & transformationes in
Natura, & in Chemia in primis: hinc tam multa, tam diversa vaporum genera, 8 in aere
elastico a tam diversis corporibus fixis genito tantum discrimen. Patet ubique immensus
excursui campus: sedeo relicto [214] progredior ad alia nonnulla, quae ad fermentationes,
& evaporationes itidem pertinent.
464. Substantia, quae fuerat dissoluta, non solum per praecipitationem colligituriterum,
ut ubi metalla cadunt suo pondere in tenuem pulvisculum redacta; sed etiam per evapor-
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, & angulosae in omnibus, ac in maxime corrosivis horrendum in
modum cuspidatae, ac serrata, unde & sapores salium acutiores, & aliquorum ex iis, quae
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 & figurae crystallorum & succorum, ex quibus gemmae, & duri lapides
fiunt ubi simplices sunt, & Suam quique figuram affectant, ac aliorum eiusmodi, quae post
evaporationem concrescunt, haberi utique possunt, utibidem 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
of different cases & combinations increases immensely; but we will only mention just a
few of them. When the interval, which encloses the mass between limits of approach
& 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
should be exceedingly small, the last attractive arc may not be so very strong, & a very
strong repulsive arc may follow it. Then indeed, it may happen that, as the mass, which
was in a state of relative rest, coming up to the limit with but a slight motion due to
external points approaching close enough to it to be capable of impressing a non-uniform
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
precipitation, as when metals fall by their own weight reduced to the form of an impalpable
powder, but also by evaporation, as we have said, in the case of salts, which, on the fluid
in which they were dissolved being evaporated, remain behind at the bottom. Nor
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.
Hence from a differ-
ent form of the arcs
comes a slow evapo-
ation.
Or there may be
Sudden explosion
& deflagration; &
various transfor-
mations, as a part
of the mixture flies
off.
Concretions, after
evaporating the
solvent ; definite
shapes in the resi-
dues, as for instance
in Salts.
328 PHILOSOPHIAE NATURALIS THEORIA
Quomodo p o s sit
fermentatio cessar.
Cur quædam sub-
stantiæ fermentent
cum quibusdam, &
non cum aliis ; cur
quædam, ut fer-
mentent, debeant
contundi.
Ignem esse fermen-
tationis g e n u s :
quomodo excitetur
tanta fermentatio
ab exigua scintilla.
Exemplum aviculæ
dimota, arenula, in
summo monte de-
jicientis lapillos,
saxa, r u p e s, &
excitantis in mari
su bje ct o undas
immanes.
465. Fermentatio paullatim minuitur, & demum cessat, cujus imminuti motus causas
attigi pluribus locis, ut num. I97. Eodem autem pertinet illud etiam, quod innui num.
440. Irregularitas particularum, ex quibus corpora constant, & inæqualitas virium,
plurimum confert ad imminuendum, & demum sistendum motum. Ubi nimirum aliquæ
particulæ, vel totæ 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, quæ 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 particulæ, quæ emittuntur in omnes plagas, ut in
evaporatione, vel ubi corpus refrigescit, excussis pluribus igneis particulis, quæ dum evolant
actione paticularum massæ, ipsis massæ particulis procurrentibus motum in partes contrarias
imprimunt, & dum illæ, quæ Oscillationem auxerant, aliæ post alias aufugiunt, illæ, quæ
remanent, sunt, quæ oscillationes ipsas internis, & externis actionibus minuebant.
e*
466. Porro non omnes substantiæ cum omnibus fermentant, sed cum quibusdam
tantummodo : acida cum alcalinis ; & [2I5] quod quibusdam videtur mirum, sunt quædam,
quæ apparent acida respectu unius substantiæ, & alcalina respectu alterius. Ea omnia
in mea Theoria facilem admodum explicationem habent : nam vidimus, particulas quasdam
respectu quarundam inertes esse, cum quibus commixtæ 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, quæ & inflammationem
parit, ac terræmotuum exhibet imaginem quandam, & vulcanorum. Oportuit ferrum in
tenues particulas discerpere, ac ad majorem mixtionem adhuc adhibere aquam.
467. Ignem ego itidem arbitror esse quoddam fermentationis genus, quod acquirat
vel potissimum, vel etiam sola sulphurea substantia, cum qua fermentat materia lucis
vehementissime, si in satis magna copia collecta sit. Ignem autem voco eum, qui non
tantum rarefacit motu suo, sed & calefacit, & lucet, quæ omnia habentur, quando materia
illa sulphurea satis fermentescit. Porro ignis comburit, quia in substantiis combustibilibus
multum adest substantiæ cujusdam, quæ sulphure abundat plurimum, & quæ idcirco
sulphurea appellari potest, quæ vel per lucem in satis magna copia collectam, vel per ipsam
jam fermentescentem sulphuream substantiam satis prægnantem ipsa lucida materia sibi
admotam fermentescit itidem, & dissolvitur, ac avolat. Is ingens motus intestinus
particularum excurrentium fit utique per vires mutuas inter particulas, quæ erant in
æquilibrio : sed mutatis parum admodum distantiis exigui etiam punctorum numeri per
exiguum unius scintillæ, vel tenuissimorum radiorum accessum, jam aliæ vires succedunt,
& per earum reciprocationem perturbatur punctorum motus, qui cito per totam massam
propagatur. - -
468. Imaginem rei admodum vividam habere possumus in sola etiam gravitate.
Emergat e mari satis editus mons, per cujus latera dispositæ sint versus fundum ingentes
lapidum prægrandium moles, tum quo magis ascenditur, eo minores ; donec versus apicem
lapilli sint, & in summo monte arenulæ: sint autem omnia fere in æquilibrio pendentia
ita, ut vi respectu molis exigua devolvi possint. Si avicula in summo monte commoveat
arenulam pede ; hæc decidit, & lapillos 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-[2I6]-cula
æquilibrium arenulæ sustulit vi perquam exigua : reliquos motus gravitas edidit, quæ
occasionem agendi est nacta ex illo exiguo motu aviculæ._ Hæc imago quædam 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, quæ
A THEORY OF NATURAL PHILOSOPHY 329
465. The fermentation diminishes gradually, & at length ceases; I have touched
upon the causes of this diminished motion in several places, for instance, in Art. 197. The
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
them only. Thus, acids ferment only with alkalies; &, what to some seems to be wonderful,
there are some substances that appear to be acid with respect to one substance, & alkaline
with respect to another. Now, all these things have a perfectly easy explanation in my
Theory. For, we have seen that certain particles are inert with regard to certain other
particles, & therefore when these are mixed together there will be no fermentation. With
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
more especially, or even solely by some sulphurous substance, with which the matter
forming light ferments very vigorously, if it is concentrated in sufficiently great amount.
Moreover I apply the term fire to that which not only rarefies through its own motion,
but also produces heat & light; & all these conditions are present when the sulphurous
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
alone. Suppose that from the sea there rises a mountain of considerable height, & that
along the sides of it there lie immense masses of huge stones, & the higher one goes,
the smaller the stones are ; until towards the top the stones are quite small, & at the very
summit they are mere grains of sand. Also suppose that all of these are just in equilibrium,
so that they can be rolled down by a very slight force compared with their whole volume.
If, now, a little bird on the top of the mountain moves with his foot just one grain of the sand,
this will fall, & bring down with it the small stones; these, as they fall, will drag with
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
The manner in
which fermentation
may cease. -
The reason why
S O m e Substances
ferment with cer-
tain substances &
not with others ;
why Some must be
powdered before
they will ferment.
Fire is some sort of
fermentation ; the
manner in which so
great a fermenta-
tion can be excited
by the slightest of
Sparks.
As example, in the
case of a little bird,
by moving a single
grain of sand on
the top of a moun-
tain, hurling down
St O n e S, rocks,
boulders, & exciting
huge waves in the
Sea that lies at the
foot of the moun-
tain.
33O PHILOSOPHIÆ NATURALIS THEORIA
Quæ careant peni-
tus materia sul-
phuræ, a b i g n e
non debere lædi :
hinc fortasse in ipso
Sole posse manere
substantias illæsas.
Exemplum fermen-
tatio nis, quam
Cum aceto habent
aliquæ terræ, aliis
illæsis.
De lumine : senten-
tiam de emissione
luminis præferen-
dam omnimo undis
fluidi elastici.
Eroprietates lumi-
nis, quarum red-
denda est ratio.
quidem perseverat eadem, aucta tantummodo velocitate descensus per novas accelera-
t1OneS.
469. Quod si ignis excitatur tantummodo per sulphureæ substantiæ fermentationem ;
ubi nihil adsit ejus substantiæ, nullus erit metus ab igne. Videmus utique, quo minus
ejusmodi substantiæ corpora habeant, eo minus igni obnoxia esse, ut ex amianto & telæ
fiant, quæ igne moderato purgantur, non comburuntur. Censeo autem idcirco nostras
hasce terrestres substantias ab igne satis intenso dissolvi omnes, & inflammari, quod omnes
ejusmodi substantiæ aliquid admixtum habeant, quod nectat etiam inter se plurimas inertes
particulas. At si corpora haberentur aliqua, quæ nihil ex ejusmodi substantia, haberent
admixtum ; ea in medio igne vehementissimo illæsa perstarent, nec ullum motum acqui-
rerent, quem nimirum nostra hæc 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, quæ vegetent, & vivant sine ulla organici sui textus læsione minima.
Videmus certe maculas superficiei Solis proximas durantes aliquando per menses etiam
plures, ubi nostræ nubes, quibus eæ videntur satis analogæ, brevissimo tempore dissiparentur.
470. Id mirum videbitur homini præjudiciis præoccupato ; nec intelliget, qui fieri
possit, ut vivat aliquid in Sole ipso, in quo tanto major esse debet vis ustoria, dum hic
exiguus radiorum solarium numerus majoribus cavis speculis, vel lentibus collectus dissolvit
omnia. At ut evidenter pateat, cujusmodi præjudicium id sit : fingamus nostra corpora
compacta esse ex illis terris, quas bolos vocant, quæ a diversis aquis mineralibus deponuntur,
quæ cum acidis fermentant, ac omnia corpora, quæ habemus præ manibus, vel ex eadem
esse terra, vel plurimum ex ea habere admixtum. Acetum nobis haberetur loco ignis :
quæcunque 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, quæ acetum non metuant, & in eo
diu perstare possint sine minimo motu, atque sui textus læsione, quo vulgus rem prorsus
absurdam censebit, si audiat, in medio igne, in ipso Sole, posse haberi corpora, quæ [217]
nullam inde læsionem accipiant, sed pacatissime quiescant, & vegetent, ac vivant.
47 I. Hæc quidem de igne ; jam aliquid de luce, quam ignis emittit, & quæ satis
collecta ipsum excitat. Ipsa lux potest esse effluvium quoddam tenuissimum, & quasi
vapor fermentatione ignea vehementi excussus. Et sane validissima, meo quidem judicio,
argumenta sunt, contra omnes alias hypotheses, ut contra undas, per quas olim phænomena
lucis explicare conatus est Hugenius, quam sententiam diu consepultam iterum excitare
conati sunt nuper summi nostri ævi Geometræ, sed meo quidem judicio sine successu (r):
nam explicarunt illi quidem, & satis ægre, paucas admodum luminis proprietates, aliis
intactis prorsus, quas sane per eam hypothesim nullo pacto explicari posse censeo, & quarum
aliquas ipsi arbitror omnino opponi : sed eam sententiam impugnare non est hujus loci,
quód quidem alibi jam præstiti non semel. Mirum sane, quam egregie in effluviorum
emanantium sententia ex mea Theoria profluant omnes tam variæ lucis proprietates, quam
explicationem fuse persecutus sum in secunda parte dissertationis De Lumine : præcipua
capita hic attingam ; interea illud innuam, videri admodum rationi consentaneam
ejúsmodi sententiam materiæ effluentis, vel ex eo, quod in ingenti agitatione, quam habet
ignis, debet utique juxta id, quod vidimus num. I95, evolare copia quædam particularum,
ut in ebullitionibus, effervescentiis, fermentationibus passim evaporationes habentur.
472. Præcipuæ proprietates luminis sunt ejus emissio constans, & ab æquali massa,
ut ab eodem Sole, ab ejusdem candelæ flamma, ad sensum eadem intensitate : immanis
velocitas, nam semidiametrorum terrestrium 2o millia, quanta est circiter Solis a Terra
(r) Cum haec scriberem, nondum prodierant Opera Taurinensis Academiae ; nec vero huc usque, dum hoc Opus
reimprimitur, adhuc videre potui, quae Geometra maximus La Grange hoc in genere protulit.
A THEORY OF NATURAL PHILOSOPHY 33 I
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
none of this matter is present, there will be no danger from fire. We seeindeed, the less
of this substance the bodies have, the less liable they are to be injured by fire; thus, a material
is woven from asbestos, & this is only purified, but not burned, by moderate fire. Further,
I consider that all our earthy substances are broken up by fire, provided it is sufficiently
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
will he be able to understand why it is that anything can live in the Sun, in which
there is bound to be ever so much greater burning force, while on earth an exceedingly
small number of solar rays, collected by fairly large concave mirrors or by lenses, will break
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,
which is given off by fire, & which, when present in sufficient quantity, excites fire. It is
possible that light may be a sort of very tenuous effluvium, or a kind of vapour forced
out by the vigorous igneous fermentation. Indeed, in my judgment, there are very strong
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 (). 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
the intensity is always the same from the same mass, such as from the Sun, or from the
flame of the same candle; its huge velocity, for it traverses a distance equal to twenty
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.
Substances, that
are quite without
sulphurous matter,
are not necessarily
impaired by fire ;
hence, perhaps in
the Sun itself there
may remain Sub-
stances uninjured.
Example, in the
case of fermenta-
tion which some
earths have with
vinegar, while others
are unaffected.
Light ; the theory
of emission of light
to be preferred
altogether before
that of waves in an
elastic fluid.
Those properties of
light for which we
have to find the
Tea,SOIl.
332 PHILOSOPHIÆ NATURALIS THEORIA
Emissio quomodo
fiat : qui fiat, ut
q u æ da m si m u 1
citissime d is so 1-
vantur dum lumen
emittunt, . ut ignis
subitus, quædam,
ut Sol, diutissime
p ersi ste nt sine
Sensibili jactura.
Unde tanta, veloci-
tas ; cum velocita-
tis discrimen exi-
guum, & in radiis
homogeneis multo
minus.
distantia, percurrit semiquadrante horæ ; velocitatum discrimen exiguum 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 æquales in
mutatione medii, parte, quæ reflectitur, eo majore respectu luminis, quo obliquitas
incidentiæ est major ; refractio alterius partis eadem mutatione cum lege constantis rationis
inter sinum incidentiæ, & sinum anguli refracti ; quæ ratio [218] in diversis coloratis
radiis diversa est, in quo stat diversa diversorum coloratorum radiorum refrangibilitas :
dispersio & in reflexione, & in refractione exiguæ 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 quæ nimirum dispositiones maxime
faventes reflexioni, vel refractioni redeunt, æquabilis 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 phænomena laminarum tenuium, & naturalium
colorum tam permanentium, quam variabilium, uti & crassarum laminarum colores, quæ
omnia satis luculenter exposuit in celebri dissertatione De Lumine P. Carolus Benvenuti e
Soc. nostra Scriptor accuratissimus : ac demum illa, quam vocant diffractionem, qua
radii in transitu prope corporum acies inflectuntur, & qui diversum colorem, ac diversam
refrangibilitatem habent, in angulis diversis.
473. Quod pertinet ad emissionem jam est expositum num. I99, & num 46I ; ubi
etiam ostensum est illud, manente eadem massa quæ emittit effluvia, ipsorum multitudinem
dato tempore esse ad sensum eandem. Porro fieri potest, ut massa, quæ lumen emittit,
penitus dissolvatur, ut in ignibus subitis accidit, & fieri potest, ut perseveret diutissime,
Id potissimum pendet a magnitudine intervalli, in quo fit Oscillatio fermentationis, & a
natura arcus attractivi terminantis id intervallum juxta num. I95. Quin immo si Auctor
Naturæ voluit massam vehementissima etiam fermentatione agitatam prorsus indissolubilem
quacunque finita velocitate, potuit facile, id præstare juxta num. 46o per alios asymptoticos
arcus cum areis infinitis, intra quorum limites sit massa fermentescens ; quorum ope ea
colligari potestita, ut dissolvi omnino nequeat, ponendo deinde materiam luminis emittendi,
ultra intervallum earum asymptotorum respectu particularum ejus massæ, & citra arcum
attractivum ingentis areæ, sed non infinitæ, ex quo aliæ lucidæ particulæ evolare possint
post alias. Nec illud, quod vulgo objici solet, tanta luminis effusione debere multum
imminui massam Solis, habet ullam difficultatem, posita illa componibilitate in infinitum
& illa solutione problematis quæ habetur num. 395. Potest enim in spatiolo utcunque
exiguo haberi numerus utcunque ingens punctorum, & omnis massa luminis, quæ diffusa
tam immanem molem occupat, potest in Sole, vel prope Solem occupavisse spatiolum,
quantum libuerit, parvum, ut idcirco Sol post quotcunque sæ-[219]-culorum millia ne
latum quidem unguem decrescat. Id pendet a ratione densitatis luminis ad densitatem
Solis, quæ ratio potest esse utcunque parva ; & quidem pro immensa luminis tenuitate
sunt argumenta admodum valida, quorum aliqua proferam infra.
474. Celeritas utcunque magna haberi potest ab arcubus repulsivis satis validis, qui
occurrant post extremum limitem Oscillationis terminatæ ab arcu ingenti attractivo juxta
num. I94 : nam si inde evadat particula cum velocitate nulla ; quadratum velocitatis
totius definitur ab excessu arearum omnium repulsivarum supra omnes attractivas juxta
num. I78, 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 particulæ luminis ejusdem generis ad finem oscillationis advenient cum
velocitatibus fere nullis : nam eæ, quæ juxta Theoriam expositam num. I95, 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 ;
there also it was shown that, if the mass emitting the effluvia remained the same, then
the amount emitted is practically the same in any given time. Further, it may
happen that the mass emitting the light is completely broken up, as takes place in
sudden flashes of fire; or it may be that this mass persists for a very long time. This
to a very great extent depends on the size of the interval in which the oscillation due to
fermentation takes place, & on the nature of the attractive arc at the end of that interval,
by Art. I95. Nay, if the Author of Nature had wished that a mass, agitated by the most
vigorous fermentation even, should be quite irreducible by any finite force whatever, he
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
repulsive arcs, if these occur after the last limit of oscillation within the confines of a very
great attractive arc, as shown in Art. 194. For if a particle goes off from here with no
velocity, the square of the whole velocity is defined by the excess of all the repulsive areas
over all the attractive, as was shown in Art. 178; &, as this excess can be of any amount
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
How emission takes
place ; how it hap-
pens that some
bodies are very
quickly broken up
at the time they
emit light, like a
Sudden flash of fire,
while others, like
the Sun persist for
a very long time
without any appar-
ent loss.
Whence comes the
great velocity, not-
withstanding the
slight differences in
Velocity, & the still
less differences in
homogeneous rays.
334 PHILOSOPHIÆ NATURALIS THEORIA
lU n de propagatio
rectilinea : incur-
su m immediatum
punctorum 1 u cis,
in puncta, medii
nullum h a, b e ri :
virium in medio
hom o g e n e o exi-
guam inæqualita-
tem eludi a tenui-
tate, & celeritate
luminis.
Si satis magnam
velocitatem habe-
ant ; quævis, solida
et ia m, transitura,
trans alia, s O 1 i d a
sine ulla motuum
perturbatione.
quo si tum, cum avolant, advenirent cum ingenti velocitate, advenissent utique eodem, &
effugissent in oscillatione præcedenti. Demonstravimus autem ibidem, exiguum discrimen
velocitatis in ingressu spatii, in quo datæ vires perpetuo accelerant motum, & generant .
velocitatem ingentem, inducere discrimen velocitatis genitæ perquam exiguum etiam
respectu illius exigui discriminis velocitatis initialis, quod demonstravimus ibi ratione petita a
natura quadrati quantitatis ingentis conjuncti cum quadrato quantitatis multo minoris, quod
quantitatem exhibet a priore illa differentem multo minus, quam sit quantitas illa parva,
cujus quadratum conjungitur. Discrimen aliquod sensibile haberi poterit ; siqua effugiunt,
non sint puncta simplicia, sed particulæ non nihil inter se diversæ : nam curva virium,
qua massa tota agit in ejusmodi particulas, potest esse nonnihil diversa pro illis diversis
particulis, adeoque excessus summæ arearum repulsivarum supra summam attractivarum
potest esse nonnihil diversus & quadratum velocitatis ipsi respondens nonnihil itidem
diversum. Hoc pacto particulæ luminis homogeneæ habebunt velocitatem ad sensum
prorsus æqualem ; particulæ heterogeneæ poterunt habere nonnihil diversam, uti ex
observatione phænomenorum 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 massæ, mutabitur nonnihil : sed
quoniam in fortuita ingenti agitatione massæ totius celerrime succedunt omnes diversæ
positiones punctorum ; summa omnium erit ad sensum eadem, potissimum pro particula
diutius hærente in illo initio suæ fugæ, ad quod advenit, uti diximus, cum velocitate
perquam exigua, ut idcirco homogenearum velocitas, [22o] ubi jam deventum fuerit ad
arcum gravitatis, & vires exiguas, debeat esse ad sensum eadem, & discrimen aliquod haberi
possit tantummodo in heterogeneis particulis a diverso earum textu. Patet igitur, unde
celeritas ingens provenire possit, & si quod est celeritatis discrimen exiguum.
475. Quod pertinet ad propagationem rectilineam per medium homogeneum diaph-
anum, & ad motum liberum sine ullo impedimento a particulis ipsius luminis, vel medii
diaphani, id in mea Theoria admodum facile exponitur, quod in aliis ingentem difficultatem
parit. Et quidem quod pertinet ad impedimenta, si curva virium nullum habeat arcum
asymptoticum perpendicularem axi præter primum ; ostensum est num. 362, sola satis
magna velocitate obtineri posse apparentem compenetrationem duarum substantiarum,
quam tenuitas, & homogeneitas spatii, per quod transitur, plurimum juvat. Quoniam
respectu punctorum materiæ prorsus indivisibilium, & inextensorum infinities infinita
sunt puncta spatii existentia in eodem plano ; infinities infinite est improbabilis pro quovis
momento temporis directio motus puncti materiæ cujusvis accurate versus aliud punctum
materiæ, 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 finito. Verum quidem est, ubicunque oculus collocetur in immensa
propemodum superficie sphæræ circa unam fixam remotissimam descripta, immo intra
ipsam sphæram, videri fixam, & proinde aliquam luminis particulam afficere nostrum
óculum : sed id fit in mea Theoria non quia accurate in omnibus absolute infinitis
directionibus adveniant radii, sed quod pupilla, & fibræ oculorum non unicum punctum
sunt, & vires punctorum particulæ 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 substantiæ
cujusvis, ut in ipsum debeat incurrere. Quamobrem per incursum, & immediatum
impactum nullum punctum luminis aut sistet motum suum, aut deflectet.
476. Id quidem commune est omnibus corporibus, quæ corpora inter se congrediuntur.
Ea nullum habent in mea Theoria punctum immediatum incurrens in aliud punctum ;
quam ob causam & illud ibidem dixi, si nullæ vires mutuæ adessent, debere utique haberi
apparentem quandam compenetrationem omnium massarum ; sed adhuc vel ex hoc solo
capite veram compenetrationem haberi nunquam omnino posse. Vires igitur quæ ad
aliquam distantiam protenduntur, im- [221]-pediunt progressum. Eae yires si circumquaque
essent semper æquales ; nullum impedimentum haberet motus, qui vi inertiæ 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
homogeneous medium, & the free motion, without hindrance, by particles either of the light
or of the transparent medium, is quite easily explained in my Theory, whereas in other
theories it begets a very great difficulty. Also as regards hindrance to this motion, so
long as the curve of forces has no asymptotic arc perpendicular to the axis besides the first,
it has been shown, in Art. 362, that merely with a sufficiently great velocity there can be
obtained an apparent compenetration of two substances; & tenuity & homogeneity of
space traversed will assist this to a very great extent. Now, since, compared with perfectly
indivisible & non-extended points of matter, there are an infinitely infinite number of points
of space existing in the same plane, there is an infinitely infinite improbability that, for
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
one another. In my Theory, they have no point directly colliding with any other point.
For this reason I also stated, in the above-mentioned article, that, if no mutual forces were
present, there is always bound to be an apparent compenetration of all bodies. Yet, from
this article alone, it is utterly impossible that there ever can be real compenetration. Hence,
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
The reas on for
rectilinear propa-
gation ; there can
be no immediate
Collision between
the points of light
& the points of the
medium ; slight
inequality of the
forces in a homo-
geneous medium
are eluded by the
tenuity & the velo-
city of light.
If they possess
great enough velo-
city, a ny bodies,
even solids, will
pass through other
Solids, without any
disturbance of their
motion.
336 PHILOSOPHIÆ NATURALIS THEORIA
Si per asymptoticos
a r c u s particulæ
essent prorsus im-
permeabiles, tu m
recurrendum a d
molem imminutam
quantum oportet.
Asymptoticis iis
cruribus n u 1 1 u m
esse opus : ea po-
tius excludenda :
quam bene omnia
explicentur s i n e
ipsis.
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
finitæ sunt, adeoque & differentia virium secundum diversas directiones ageritium finita
est semper. Igitur utcunque ea sit magna, ipsam finita quædam velocitas elidere potest,
quin permittat ullam retardationem, accelerationem, deviationem, quæ ad datam quampiam
utcunque parvam magnitudinem assurgat : nam vires indigent tempore ad producendam
novam velocitatem, quæ semper proportionalis est tempori, & vi. Hinc si satis magna
velocitas haberetur ; quævis 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, & cohæsionis, quod ibidem illustravi
exemplo ferrei globuli inter magnetes dispersos cum satis magna velocitate libere permeántis,
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.
477. Id quidem ita se habet, si nullæ præter primam asymptoti habeantur, quæ vires
absolute infinitas inducant : nam si per ejusmodi asymptoticos arcus particulæ fiant &
indissolubiles, & prorsus impenetrabiles juxta num. 362 ; tum vero nulla utcunque magna
velocitate posset una particula alteram transvolare, & res eodem recideret, quo in communi
sententia de continua extensione materiæ. Tum nimirum oporteret lucis particulas
minuere, non quidem in infinitum (quod ego absolute impossibile arbitror, quemadmodum
& quantitates, quæ revera infinite parvæ sint in se ipsis tales, ac independenter ab omni
nostro cogitandi modo determinatæ : nec vero earum usquam habetur necessitas in Natura)
sed ita, ut adhuc incursus unius particulæ in aliam pro quovis finito tempore sit, quantum
libuerit, improbabilis, quod per finitas utique magnitudines præstari 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 particulæ 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 latissimæ 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
particulæ rationem, quantum libuerit, majorem illa, quam exprimit ille ingens, sed finitus
accessuum numerus : ac idcirco numerus directionum, per quas possint transire omnia
illa 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. Infinito nusquam
opus erit in Natura, & series finitorum, quæ in infinitum progreditur, semper aliquod
finitum nobis offert ita magnum, vel parvum, ut ad physicos usus quoscunque sufficiat.
478. Quod de particulis inter se collatis est dictum, idem locum habet & in particulis
respectu corporum quoruncunque, potissimum si corpora juxta meam Theoriam constituta
sint particulis distantibus a se invicem, & non continuo nexu colligatis, sive extensionis
vere continuæ illius veli, aut muri continuam infinitam objicientis resistentiam, de quo
egimus num. 362, & 363. Verum ejusmodi asymptoticorum arcuum nulla mihi est
necessitas in mea Theoria, & hic itidem per nexus, ac vires limitum ingentis, quantum
libuerit, quanquam non etiam infiniti valoris, omnia præstari possunt in Natura : & si
principio inductionis inhærere 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 naturæ arcus, ad gravitatem pertinens, indicio sint,
esse & alios alibi in distantiis, quæ 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,
to induce absolutely infinite forces; but if, owing to such asymptotic arcs, the particles
become incapable both of dissolution & penetration, as in Art. 362, then indeed by no
velocity, however great, could one particle pass through another ; & the matter would be
reduced to the same idea, as is held generally about the continuous extension of matter.
Thus, in that case it would be necessary to diminish the size of the particles of light; not
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 willimpinge, 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
If, owing to the
presence of asymp-
totic arcs, particles
|become imperme-
a ble, the n We
should have to fall
back upon diminu-
tion of volume, as
far as was necessary.
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.
78. All that has been said with regard to particles referred to one another, the same
will hold good for particles in reference to any bodies; & especially if the bodies are formed,
in accordance with my Theory, of particles distant from one another, & not bound together
by a continuous connection, or possessing the truly continuous extension of the skin or
wall offering a continuous infinite resistance, with which we dealt in Art. 362, 363. But
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
There is no need
for the asymptotic
branches ; rather,
they should be ex-
cluded ; how well
all things can be
explained without
them.
338 PHILOSOPHIÆ NATURALIS THEORIA
Quomodo rem con-
fi cia, nt velocitas
satis magna, &
æqualitas sensibilis
vir iu m circum-
quaque. Quomodo
hæc in homogeneo
medio habeatur.
Quomodo in gens
velocitas exiguam
inæqualitatem elu-
dat : exemplum in
turbine ligneo non
cadente.
Accedere n ex u m
inter puncta parti-
c u lae : qu id is
præstet.
ERaritatem pluri-
m u m prodesse :
omnes eas quatuor
c a u s a s ha b e re
locum in lumine
non tu r b at o a.
radiis alia, directione
delatis quaquaver-
sum: priores tres
in mediis densiori-
bus pellucidis.
contrahuntur, vel protenduntur. Ceterum si vires omnes finitæ sint, & puncta materiæ
juxta meam Theoriam simplicia penitus, & inextensa : multo sane facilius concipitur,
qui fiat, ut habeatur hæc apparens compenetratio sine ullo incursu, & sine ulla dissolutione
particularum cum transitu aliarum per alias. -
479. Porro duo sunt, quorum singula rem præstare possunt, velocitas satis magna,
quæ nimirum utcunque magnam virium inæqualitatem potest eludere, & virium circum-
quaque positarum æqualitas, quæ differentiam relinquat omnino nullam. Differentia
nunquam sane habebitur omnino nulla, ubi [223] punctum materiæ prætervolet per
quandam punctorum veluti silvam, quorum alia ab aliis distent : necessario enim mutabit
distantiam ab iis, a quibus minimum distat, jam accedens nonnihil, jam recedens. Verum
ubi distributio particularum ad æqualitatem quandam multum accesserit, inæqualitas
virium erit perquam exigua ; si omnium virium habeatur ratio, quas exercent omnia puncta
disposita circa id punctum ad intervallum, ad quod satis sensibiles meæ curvæ vires
protenduntur. Concipiamus enim sphæram quandam, quæ habeat pro semidiametro
illam distantiam, ad quam protenduntur flexus curvæ virium primigeniæ, sive ad quam
vires singulorum punctorum satis sensibiles pertingunt. Si medium satis ad homogenei-
tatem accedat ; secta illa sphæra in duas partes utcunque per centrum, in utraque numerus
punctorum materiæ erit quamproxime idem, & summa virium quam proxime eadem, se
compensantibus omnibus exiguis inæqualitatibus 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 æque in omnem plagam.
480. Quod si accedat ingens velocitas ; multo adhuc minor erit inæqualitatum
effectus, tum quod multo minus habebunt temporis vires ut agant, tum quod in ipso
continuato progressu inæqualitates jam in unam plagam prævalebunt, jam in aliam, quibus
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, inæqualitate ponderis, quæ ad casum determinat, jam ad aliam plagam
jacente, & totam inclinante molem, jam ad aliam, qui, celeritate motus circularis imminuta,
decidit inclinatus, quo exigit præponderantia.
48I. Quod autem homogeneitas medii, & velocitas præstant simul, id adhuc auget
multo magis is nexus, qui est inter materiæ puncta particulam componentia, & æquali ad
sensum velocitate delata, qui mutuis viribus cum accessum ad se invicem punctorum
particulam componentium, & recessum impediat, cogit totam particulam simul trepidare
eo solo motu, quem inducit summa inæqualitatum pertinentium ad puncta omnia, quæ
summa adhuc magis ad æqualitatem accedit : nam in fortuitis, & temere hac, illac dispersis,
vel concurrentibus casu circumstantiis, quo major numerus accipitur, eo inæqualitatum
irregularium summa decrescit magis.
482. Demum raritas medii ad id ipsum confert adhuc magis : quo enim major est
raritas, eo minor occurrit punctorum numerus intra illam sphæram, adeoque eo minor
virium componendarum multitudo, & inæqualitas adhuc multo mi-[224]-nor. Porro
omnes hæ quatuor causæ æqualitatis concurrunt, ubi agitur de radiis collatis cum aliis
radiis : homogeneitas, nam lumen a dato puncto progrediens suam densitatem imminuit
in ratione reciproca duplicata distantiarum a puncto radiante, adeoque in tam exiguo
circunquaque circa quodvis punctum intervallo, quantum est id, ad quod virium actio
sensibilis protenditur, ad homogeneitatem accedit in immensum : celeritas, quæ tanta
est, ut singulis arteriæ pulsibus quævis luminis particula fere bis centum millia Romanorum
milliariorum percurrat: nexus particularum mutuus, nam ipsæ luminis particulæ ad
diversos coloratos radios pertinentes habent perennes proprietates suas, quas, constanter
servant, ut certum refrangibilitatis gradum, & potentiam certo impulsu agitandi oculorum
fibras, per quam certam certi coloris sensationem eliciant : ac demum tenuitas immanis,
qua opus est ad tantam diffusionem, & tam perennem effluxum sine ulla sensibili imminu-
tione solaris massæ, & cujus indicium aliquod proferam paullo inferius. Ubi vero agitur
de lumine comparato cum substantiis pellucidis, per quas pervadit, priora illa tria
tantummodo locum habent respectu particularum luminis, & omnia quatuor respectu
particularum pellucidi corporis, quarum nexus non dissolvitur, nec 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,
a sufficiently great velocity, such as will foil the inequality of the forces, however great
that may be ; & an equality of the forces in all directions, such as will leave the difference
absolutely zero. Now the difference can never really be altogether zero, when a point
of matter passes through, so to speak, a forest of points, which are separated from one
another. For, of necessity, it will change its distance from those points, from which it is
least distant, at one time approaching & at another receding. But when the distribution
of the particles approaches very closely to an equality, the inequality of the forces will be
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
How the matter
C a n be a c com-
plished by a suffi-
ciently large velo-
city & a sensible
equality of the
forces in all direc-
tions. How these
are to be had in
a homogeneous
medium.
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
because the forces will have much less time in which to act, & because in such a continued
progress the inequalities will prevail first on one side & then on the other; & as these follow
one another very quickly, the progress will be still more uniform & nearer to rectilinear
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
together is still further increased by the connection that exists between the points of matter
forming the particle & moving together with practically the same velocity. This connection,
since, through the mutual forces, it prevents the mutual approach or recession of 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
rarity, the smaller the number of points that occur within the sphere imagined above,
& therefore the smaller the number of forces to be compounded, & much smaller still the
inequality. Further, all four of these causes of inequality occur together, when we are
dealing with rays of light in regard to other rays. Homogeneity we have, because light
proceeding from a given point diminishes its density in the inverse ratio of the squares of
the distances from the radiant point; & thus, in the exceedingly small interval round about
any point, whatever the distance may be over which a sensible action of the forces extend,
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
How a very great
velocity foils a
slight inequality;
example, from a
wooden spinning-
top not falling.
In addition there
is a connection
between the points
of a particle ; the
effect of this.
Great effect of
small density ;
all four of these
causes hold for
light undisturbed
by rays proceeding
in any other direc-
tions, & the first
three of them in the
more dense trans-
parent media.
34o PHILOSOPHIÆ NATURALIS THEORIA
, dirimenti
Pelluciditatem oriri
a sola homogenei-
tate : solam hetero-
geneitatem impe-
dire , posse progres-
sum per inæquali-
tatem virium.
indivisibilia puncta prædita insuperabili potentia repulsiva pertingente ad finitam distantiam
esse tam subjecta collisionibus, quam sunt particulæ finitæ magnitudinis, & idcirco nulli
adminiculo esse pro comprehendenda mutua lucis penetratione ; nam sine cruribus illis
asymptoticis posterioribus meæ vires repulsivæ non sunt insuperabiles, nisi ubi puncta
congredi debeant in recta, quæ illa jungit, qui casus in Natura nusquam occurrit.
483. Et vero sola homogeneitas pelluciditatem parit, uti jam olim notavit Newtonus,
nec opacitas oritur ab impactu in partes corporum solidas, & a defectu pororum jacentium
in directum, uti alii ante ipsum plures censuerant, sed ab inæquali textu particularum
heterogenearum, quarum aliæ aliis minus densis, vel etiam penitus vacuis amplioribus
spatiolis intermixtæ satis magnam inducunt inæqualitatem virium, qua lumen in omnes
partes detorquent, ac distrahunt, flexu multiplici, & ambagibus per internos 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 præter ipsam omnem
superiorem Theoriam, quæ rem sola evinceret ; cum nimirum sine inæqualitate virium
I{eflexionem non
oriri ab impactu,
sed ab inæqualitate
virium in mutatione
medii; ubi pro re-
fractionis explica-
tio ne præmissa
pr1nc1p1a.
C o n s i d e ratio
sphæræ, ad quam
e xt e n d i t u r v is
sensibilis agens in
lumen : inde vis
inter bina plana
parallela superficiei
media ;
inter quæ vis agit.
nullum haberi possit libero rectilineo progressui impedimentum. Id sane colligitur ex
eo, quod omnium corporum tenuiores laminæ pellucidæ sunt, uti norunt, qui microscopiis
tractandis assueverunt : id [225] evincunt illæ substantiæ, quæ aliarum poris injectæ
easdem ex opacis pellucidas reddunt, ut charta oleo imbuta fit pellucida, supplente aerem
ipso oleo, cum quo multo minus inæqualiter in lumen agunt particulæ chartæ, quam
agerent soli aeri, vel vacuo spatio intermixtæ. Rem autem oculis subjicit vitrum contusum
in minores particulas, quod sola irregularitate figuræ particularum temere ex contusione
nascentium, & aeris intermixti inæqualitate 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, quæ, dum
concrescunt vacuolis interrupta, illico opaca fiunt.
484. Quamobrem nec reflexio inde ortum ducit, sed habetur etiam in pellucidis
corpóribus ex inæqualitate virium seu repellentium, seu attrahentium, uti in Optica sua
Newtonus tam multis notissimis argumentis demonstravit, quorum unum est illud ipsum
ex asperitate superficiei cujuscunque cujusvis corporis, utcunque nobis, nudo potissimum
inspectantibus oculo, lævis appareat, & perpolita, quod num. 299 exposuimus ; & ex eadem
causa oritur etiam refractio. Si velocitas luminis esset satis magna ; impediret etiam
hujusce inæqualitatis 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 inæqualitatem tanto minorem, quæ habetur in mediis homo-
geneis, non illam tanto majorem, quæ oritur a mediorum discrimine. Quod vero ad
refractionis explicationem ex Mechanica requiritur, exposuimus a num. 392, ubi adhibuimus
principium illud virium inter duo plana parallela agentium æque in distantiis æqualibus
àb eorum utroque, cujus explicationem ad luminis particulas jam expediemus.
485. Concipiatur (£) illa sphærula, cujus semidiameter [226] æquatur distantiæ illi,
ad qüam agunt actione satis sensibili particulæ corporum in lucis particulam, quæ cum
(f) Refert MN in fig. 7o superficiem dirimentem duo media, GE viam radii advenientis, H particulam luminis ;
HE celeritatem, ejus absolutam, HS parallelam, SE perpendicularem, quæ est eo minor, quo radius incidit magis obliquus :
abc est sphæra, intra quam habetur actio sensibilis in particulam H, quae est adhuc tota in priore medio : X, X', X" sumt
loca plura particulae progredientis inter plana AB, CD parallela superficiei MN, sita ad distantiam ab ea æqualem
semidiametro sphæræ Hc. Particula sita inter illa plana ubicumque, ut in X, ea $phæra habebit suum segmentum FRI,
ultra superficiem MN: sit ejus axis RT, & eodem axe $egmentum QTZ priori æquale, ac mn planum per centrum par-
allelum MN. Segmenta mFLn, mQZn ejusdem medii agent æqualiter. Segmenta FRI, QTZ inæqualiter, sed eorum
vires dirigentur per axem TR in alteram e binis plagis oppositis : adeoque & differentia virium dirigetur per eumdem, qui
quidem perpendicularis est utique planis AB, CD. Ea actione via incurva radii sinuatur per XX'X". Prout vis dirigetur
£ertus CD, vel versus AB, curva erit cava versus easdem, &$ im mutatione directionis vis ipsius mutabitur flexus curvæ.
S; autem curva evaserit alicubi parallela plano AB; flectet cursum retro ; nisi id accidat accurate in situ vis = o, qui
A THEORY OF NATURAL PHILOSOPHY 34. I
FIG. 70.

342 PHILOSOPHIAE 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
a lack of pores lying in a straight line, as many others before Newton thought, but from
the unequal structure of heterogeneous particles; of which some are interspersed amongst
others of less density, or even in perfectly empty little spaces, of considerable size,
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
transparent bodies due to the inequality of forces, whether repulsive or attractive. This
was proved by Newton in his Optics by a large number of arguments that are well known ;
one of these is that very reason that was stated in Art. 299, derived from the roughness
of any surface of any body, no matter how smooth & polished it appears to us, especially
when viewed with the naked eye. Refraction also arises from the same cause. If the
velocity of light were great enough, it would prevent even the effect of this inequality
that arises from the different constitution of the media. But, 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.
485. Imagine (f) a sphere, of which the semidiameter is equal to the distance up to
which the particles of a body act upon a particle of light with a fairly sensible action; &
(f) In Fig. 70, MN is the surface of separation between the two media, GE the path of an approaching ray, H a particle
of light, HE its absolute velocity, HS the parallel, SE the perpendicular component, which latter is the less, the more
oblique the incidence of the ray. abc is the small sphere, within which there is sensible action on the particle H, which
is 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 surface MN, and situated at a distance from it equal to the semidiameter of the sphere Ho. If the particle
is situated anywhere between the two planes, as at X, the sphere will have its segment FRL on the far side of the surface
MN. Let the axis of the segment be RT, and let QT2 be a segment having the same axis and equal to the former
segment, and let mn be a plane through the centre parallel to MN. Then the segments mPLn, mOZn, 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
Transparence arises
from homogeneity
alone ; & hetero-
geneity alone is
capable of pre-
venting progressive
motion through in-
equality of forces.
Reflection does not
take place through
impact, but owing
to inequality of
for c e s on the
medium being
changed : where
the principles for
the explanation of
refraction h a ve
been premised.
Consideration of
the sphere whose
radius is the dis-
tance to which the
sensible force of
light extends :
thence the force
between two planes
parallel to the
surface of separa-
tion of the media,
between which the
force acts.
344 PHILOSOPHIÆ NATURALIS THEORIA
Tres casus, qui ex-
hibent reflexionem,
vel refractio n e m
cum recessu a per-
pendiculo, vel ip-
sam refractionem
Cl1Im 3.CC€SSll.
Lumen debere in
corpora, re a, g e r e
æqualiter : h in c
immensa, lucis ten-
uitas : qui effectus
ipsi falso tribuantur
a nonnullis.
lucis particula progrediatur simul. Donec ipsa sphærula est in aliquo homogeneo medio
tota, vires in particulam circunquaque æquales erunt ad sensum, & cum nullus habeatur
immediatus incursus, motus inertiæ vi factus erit ad sensum rectilineus, & uniformis.
Ubi illa sphærula aliquod aliud ingressa fuerit diversæ naturæ medium, cujus eadem moles
exerceat in particulas luminis vim diversam a prioris medii vi ; jam illa pars novi medii,
quæ intra sphærulam immersa erit, non exercebit in ipsam particulam vim æqualem illi,
quam exeret pars sphærulæ ipsi respondens ex altera centri parte, & facile patet, differentiam
virium debere dirigi per axem perpendicularem illis segmentis sphærulæ, per quem singulæ
utriusque segmenti vires diriguntur, nimirum perpendiculariter ad superficiem dirimentem
duo media, quæ illud prius segmentum terminat : & quoniam ubicunque particula sit in
æquali distantia a superficie, illud segmentum erit magnitudinis ejusdem ; vis motum
perturbans in iisdem a superficie illa distantiis eadem erit. Durabit autem ejusmodi vis,
donec ipsa sphærula tota intra novum medium immergatur. Incipiet autem immergi
ipsa sphærula in novum medium, ubi particula advenerit ad distantiam ab ipsius superficie
æqualem radio sphærulæ, & immergetur tota, ubi ipsa particula jam immersa fuerit, ac
ad distantiam eandem processerit. Quare si concipiantur duo plana parallela ipsi superficiei
dirimenti media, quæ superficies in exiguo tractu habetur pro plana, ad distantias citra,
& ultra ipsam æquales radio illius sphærulæ, sive intervallo actionis sensibilis ; particula
constituta inter illa plana habebit vim secundum directionem perpendicularem ipsis planis,
quæ in data distantia ab eorum altero utrovis æqualis erit.
486. Porro id ipsum est id, quod assumpsimus num. 302, & unde derivavimus reflexionis,
ac refractionis legem : nimirum si concipiatur ejusmodi vis resoluta in duas, alteram
parallelam iis planis, alteram perpendicularem : illa vis pot-[227]-est perpendicularem
velocitatem vel extinguere totam ante, quam deveniatur ad planum ulterius, vel imminuere,
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 æquali angulo
incidentiæ : in secundo casu habetur refractio cum recessu a perpendiculo, in tertio
refractio cum accessu ad ipsum, & in utrolibet casu, quæcunque fuerit inclinatio in ingressu,
debet differentia quadratorum velocitatis perpendicularis in ingressu, & egressu esse
constantis cujusdam magnitudinis ex principio mechanico demonstrato num. I76 in adn.
& inde num. 3o5 est erutum illud, sinüm anguli incidentiæ ad sinum anguli refracti debere
esse in constanti ratione, quæ est celeberrima lucis proprietas, cui tota innititur Dioptrica
& præterea illud num. 366 velocitatem in medio præcedente ad velocitatem in medio
sequente esse in ratione reciproca sinuum eorundem.
487. Hoc pacto ex uniformi Theoria deductæ sunt notissimæ, ac vulgares leges
reflexionis, ac refractionis, ex quibus plura consectaria deduci possunt. Imprimis quoniam
debet actio semper esse mutua, dum corpora agunt in lumen ipsum reflectendo, & refrin-
gendo; debet ipsum lumen agere in corpora, ac debet esse velocitas amissa a lumine ad
velocitatem acquisitam a centro gravitatis corporis sistentis lumen, ut est massa corporis
ad massam luminis. Inde deducitur immensa luminis tenuitas : nam massa tenuissima
levissimæ plumulæ suspensæ filo tenui, si impetatur a radio repente immisso, nullum
progressivum acquirit motum, qui sensu percipi possit. Cum tam immanis sit velocitas
amissa a lumine; facile patet, quam immensa sit tenuitas luminis. Newtonus etiam
radiorum impulsioni tribuiit progressum vaporum cometicorum in caudam ; sed eam
ego sententiam satis valido, ut arbitror, argumento rejeci in mea dissertatione De Cometis;
Sünt, qui auroras boreales tribuant halitibus tenuissimis impulsis a radiis solaribus, quod
miror fieri etiam ab aliquo, qui radios putat esse undas tantummodo, nam undæ progressivum
casus est in infinitum improbabilis. Id accidet in aliis radiis citius, in aliis radiis serius, pro diversa absoluta celeritate
radii, pro diversa inclinatione incidentiæ, & pro diversa natura, vel constitutione particulæ, abeuntibus aliis particulis
per QXIK, aliis per QXX'I'K', aliis per QXX'X'I'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 $uperato adhuc summa
actionum usque ad O potest esse ad sensum eadem. Reliqua sunt hic, ut num. 3o6.
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
which we derived the laws of reflection & refraction. Thus, if such a force is supposed
to be resolved into two parts, one parallel & the other perpendicular to the planes, the
latter force may either destroy the whole of the perpendicular velocity before the further
plane is reached, or it may reduce it, or it may increase it. In the first case the particle
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. I76. From which, in Art. 305, I 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
& refraction have been derived; & from these a large number of corollaries can be deduced.
First of all, because the action must always be mutual, so long as bodies act upon light,
reflecting or refracting it, the light must react on the bodies; & the velocity lost by the
light must bear a ratio to the velocity gained by the centre of gravity of the body resisting
the motion of the light, which is equal to the ratio of the mass of the body to the mass of
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"I"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. -
Three cases, ex-
hibiting respective-
ly reflection, refrac-
tion with recession
from the normal,
& refraction with
approach to the
normal.
Light must have
an equal reaction
on the bodies;
hence the extreme
tenuity of light ;
these effects are
falsely attributed
to light itself by
Some people.
346 PHILOSOPHIÆ NATURALIS THEORIA
Tenuissimum mo-
tum im primi a
lumine particulis
c O r p o r u m : cai-
orem, & ustionem
provenire ab earum
viri b u s internis,
quod ipsum proba-
tur hic.
Densiora, agere in
lumen fortius : sed
sulphurosa, & ole-
osa pari densitate
plus : cur id ipsum.
Lumen in progressu
nullam pati resist-
entiam, p o siti ve
probatur.
Unde lux in phos-
phoris quibusdam.
Cur in majore obli-
quitate plus lum-
inis reflectatur.
Diversam refrangi-
bilitatem non pen-
dere a, sola diversa,
celeritate particu-
larum luminis, sed
etiam a diverso
earum textu induc-
ente vim diversam.
motum per se se non imprimunt : qui autem censent, & fluvios retardari orienti Soli
contrarios, & Terræ motus fieri ex impulsu radiorum Solis, ii sane nunquam per legitima
Mechanicæ principia inquisiverunt in luminis tenuitatem.
488. Solis particulis tenuissimis corporum imprimunt motum radii, ex quo per internas
vires aucto Oritur calor, & quidem in opacis corporibus multo facilius, ubi tantæ sunt
reflexionum, & refractionum internæ vicissitudines : exiguo motu impresso paucis particulis,
reliqua internæ mutuæ vires agunt juxta ea, quæ diximus num. 467. Sic ubi radiis solaribus
speculo collectis comburuntur aliqua, alia calcinantur [228] etiam ; omnes illi motus ab
internis utique viribus oriuntur, non ab impulsione radiorum. Regulus antimonii ita
calcinatus auget aliquando pondus decima sui parte. Sunt, qui id tribuant massæ radiorum
ibi collectæ. Si ad ita esset ; debuisset citissime abire illa substantia cum parte decima
velocitatis amissæ a lumine, sive citius, quam binis arteriæ pulsibus ultra Lunam fugere.
Quamobrem alia debet esse ejus phænomeni causa, qua de re fusius egi in mea dissertatione
De Luminis Tenuitate. -
489. Quoniam lumen in sulphuris particulas agit validissime, nam sulphurosæ, &
oleosæ substantiæ facillime accenduntur ; eæ contra in lumen validissime agunt. Sub-
stantiæ generaliter eo magis agunt in lumen, quo densiores sunt, & attractionum summa
prævalet, 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 suæ densitatis. Ego sane arbitror, uti monui num.
467, ipsum ignem nihil esse aliud, nisi fermentationem ingentem lucis cum sulphurea
substantia.
490. Lumen per media homogenea progredi motu liberrimo, & sine ulla resistentia
medii, per quod propagetur, eruitur etiam ex illo, quod velocitas parallela maneat constans,
uti assumpsimus num. 3o2, quod assumptum si non sit verum, manentibus ceteris ; ratio
sinus incidentiæ ad sinum anguli refracti non esset constans : sed idem eruitur etiam ex
eo, quod ubi radius ex aere abivit in vitrum, tum 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
majorem : nam velocitatem haberet minorem, quæ semel amissa non recuperatur per
hoc, quod resistentia minuatur, & eadem vis mobile minori velocitate motum magis
detorquet a directione sui motus.
49I. Posteaquam lux intra opaca corpora tam multis, tam variis erravit ambagibus
aliqua saltem sui parte deveniet iterum ad superficiales particulas, & avolabit. Inde omnino
ortum habebit lux illa tam multorum phosphororum, quæ 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, quæ num. 472 proposuimus.
492. Primo quidem illud facile perspicitur, ex Theoria, quam exposuimus, cur, ubi
radius incidit cum majore inclinatione ad superficiem, major luminis pars reflectatur.
Et quidem in dissertatione, quam superiore anno die I2 Novembris legit [229] Bouguerius
in Academiæ 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 æque fortem, ac in Mercurio ut nimirum reflectantur duo trientes, dum in
incidentia perpendiculari 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, quæ agunt intra illa duo plana, eo
facilius, & in pluribus particulis totam velocitatem perpendicularem elident, & reflex-
ionem determinabunt.
493. Verum id quidem jam supponit, non in omnes lucis particulas eandem exerceri
vim, sed in iis discrimen haberi aliquod. Ejusmodi discrimina diligenter evolvam.
Inprimis discrimen aliquod haberi debet ex ipso textu particularum luminis, ex quo pendeat
constans discrimen proprietatum quarundam, ut illud imprimis diversæ radiorum refran-
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;
& from this, when increased by internal forces, arises heat, & this all the more easily in the
case of opaque bodies, where there are such a number of internal alternations of reflections
& refractions. If a slight motion is impressed on but a few particles, the internal mutual
forces do all the rest, as we stated in Art. 467. Thus, when some substances are set on
fire by solar rays collected by a mirror, while some are even reduced to powder, all the
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
substances are very easily set on fire, these on the other hand act very strongly on light.
In general, substances have the greater action on light, the denser they are ; & the sum
of the attractions will be stronger when the ray is refracted as it passes through each of
the planes. For this reason, in general, when a ray passes from a less dense to a more dense
medium, refraction takes place with approach to the normal, & when from a more dense
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,
without suffering any resistance from the medium through which it is propagated, is proved
by the fact that the parallel component of the velocity remains unaltered. We made this
assumption in Art. 302; & if the assumption is not true, other things being unaltered,
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,
at some part at least it will once more arrive at the superficial particles of the bodies & fly
off. This 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 the proportion of light reflected is greater, when the ray falls on the surface with greater
inclination to it. Indeed, in a dissertation, read on November 12th of last year by Bouguer
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, destro
the whole of the perpendicular velocity, & thus determine reflection. -
493. But this supposes that the same force is not exerted on all particles of light, but
that even for them there is some difference. I will carefully discuss these differences. First
of all, there is bound to be some difference owing to the structure of the particles of light ;
& upon this will depend a constant difference in some of its properties, such as that of the
different refrangibilities of rays, in particular. The fact that the same ray is refracted by
There is a very
slight motion given
to the particles of
bodies by light;
heat & combustion
arise from their
internal forces, as
is here proved.
Denser substances
act more strongly
on light ; but sul-
ph u rous & oily
Substances more
So than others of
equal density; the
reason for this.
Positive de m on -
stration that light
does not suffer any
resistance in its pro-
gressive motion.
The source of the
light in certain
phosphorous bodies.
Why at greater
obliquity there is
more of the light
reflected.
Different refrangi-
bility does not
depend on different
velocities of the
particles of light
alone, but also on
their different
structure which
induces different
forces,
348 PHILOSOPHIÆ NATURALIS THEORIA
Ex eadem refrac-
tione radiorum ejus-
dem coloris emis-
sorum ab omnibus
lucidis corporibus
evinci eandem in
iis celeritatem, &
textum.
Vices facilioris re-
flexionis &c., oriri
a contractione, &
expansione particu-
larum in progressu
in d u c e n t e dis-
crimen.
etiam inclinatione incidentiæ, id quidem provenit a diversa natura substantiæ 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 provenire, atque id præstitiin secunda parte meæ dissertationis
De Lumine : quanquam etiam radii diversæ refrangibilitatis debeant habere omnino
diversam quoque celeritatem ; nam si ante ingressum in medium refringens habuissent
æqualem ; T jam in illo inæqualem haberent, cum velocitas præcedens ad velocitatem
sequentem sit in ratione reciproca sinus incidentiæ ad sinum anguli refracti : & hæc ratio
in Tradiis diversæ refrangibilitatis sit omnino diversa. Quare provenit etiam a vi diversa,
quæ 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.
494. 'At quoniam experimentis constat, radios ejusdem coloris eandem refractionem
pati ab eodem corpore, sive a stellis fixis provenerint, sive a Sole, sive a nostris ignibus, sive
etiam a naturalibus, vel artificialibus phosphoris, nam ea omnia eodem telescopio æque
distincta videntur : manifesto patet, omnes radios ejusdem coloris pertinentes ad omnia
ejusmodi lucida corpora eadem velocitate esse præditos, & eadem [230] dispositione punc-
torum : neque enim probabile est, (& fortasse nec fieri id potest), celeritatem diversam
a diversa vi compensari ubique accurate ita, ut semper eadem habeatur refractio per
ejusmodi compensationem.
495. Sed oportet invenire aliud discrimen inter diversas constitutiones particularum
pertinentium ad radios ejusdem refrangibilitatis ad explicandas vices faciliores reflexionis, &
facilioris transmissus ; ac inde mihi prodibit etiam ratio phænomeni radiorum, qui in reflex-
ione, & refractione irregularitur disperguntur, & ratio discriminis inter eos, qui reflectuntur
potius, quam refringantur, ex quo etiam fit, ut in majore inclinatione reflectantur plures.
Newtonus plures innuit in Optica sua hypotheses ad rem utcunque adumbrandam, quarum
tamen nullam absolute amplectitur : ego utar hic causa, quam adhibui in illa dissertatione
De Lumine parte secunda, quæ causa & existit & rei explicandæ 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, quæ sociis relictis processissent, nisi mutuæ vires, acceleratis lentioribus,
ea retardassent, unde necessario oriri debuit particulæ progredientis Oscillatio quædam,
in qua Oscillatione particula ipsa debuit jam produci non nihil, jam contrahi : & quoniam
In limitibus ejus
velut oscillationis
diutius perstare
formam : in diver-
sa parte ejusdem
v1r1um s u m m a, m
esse diversam.
dum per medium homogeneum particula progreditur, inæqualitas summæ actionum in
punctis singulis debet esse ad sensum nulla ; durabit eadem per ipsum medium homogeneum
reciprocatio contractionis, ac productionis particulæ, quæ quidem productio, & contractio
poterit esse satis exigua ; si nimirum nexus punctorum sit satis validus : sed semper erit
aliqua, & potest itidem esse non ita parva, nec vero debet esse eadem in particulis diversi
teXtUS.
496. Porro in ea reciprocatione figuræ habebuntur limites quidam productionis maxi-
mæ, & maximæ contractionis, in quibus juxta communem admodum indolem maximorum,
& minimorum diutissime perdurabitur, motu reliquo, ubi jam inde discessum fuerit ad
distantiam sensibilem cum ingenti celeritate peracto, uti in pendulorum oscillationibus
videmus, pondus in extremis Oscillationum limitibus quasi hærere diutius, in reliquis vero
locis celerrime prætervolare : ac in alio virium genere diverso a gravitate constanti, illa
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 æqualia temporum
intervalla, ut æquales pendulorum oscillationes sunt æque diuturnæ, ac idcirco dum
particula progreditur per medium homogeneum, recurrent illæ ipsæ binæ dispositiones
post æqua-[231]-lia intervalla spatiorum pendentia a constanti velocitate particulæ, &
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
refraction by the same body, whether they come from the fixed stars, or from the
Sun, or from our fires, or even from natural or artificial phosphorous substances, for they
all appear equally distinct when viewed with the same telescope; it is clearly evident that
all rays of the same colour pertaining to such light-giving bodies are endowed with the
same velocities, & the same distribution of their points. For, it is very improbable, not to
say impossible, that a difference in velocity should be everywhere exactly balanced by a
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
particles belonging to rays of the same reirangibility, to account for the fits of easier reflection
& easier transmission. From it I shall obtain also the reason for the phenomenon of rays that
are irregularly scattered in reflection & refraction; & the reason for the difference between
those that are reflected in preference to being refracted, from which also it comes about
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,
corresponding to maximum extension & maximum contraction; & in these, according to
a universal property of all maxima & minima, there will be quite a long pause ; whereas,
the rest of the motion, after a departure from them has taken place to a sensible distance,
is accomplished with a great velocity. Thus, we see in the oscillations of pendulums that
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
From the equality
of refraction of
rays of the colour
coming from all
bodies emitting
light it is proved
that for such rays
there is the same
velocity & struc-
ture.
Fits of easier re-
flection, &c., are
due to contraction
& expansion of the
particles, which
induce a difference
in the progressive
motion.
At the boundaries
of this oscillation,
So to speak, the
particle will pre-
serve its shape
longer; & the sum
of the forces at
different parts will
be different.
35O PHILOSOPHIÆ NATURALIS THEORIA
Inde binæ disposi-
tiones constantes
vicium in maxima
particularum parte
appellente in iis
limitibus : in parte
exigua appellente
inter eos dispersio.
lU n d e discrimen
rati o n i s luminis
reflexi ad trans-
missum.
a constanti tempore, quo particulæ cujusvis oscillatio durat. Demum summa virium,
quam novum medium, ad quod accedit particula, exercet in omnia particulæ puncta,
non erit sane eadem in diversis illis oscillantis particulæ dispositionibus.
497. Hisce omnibus rite consideratis, concipiatur jam ille fere continuus affluxus
particularum etiam homogenearum ad superficiem duo heterogenea media dirimentem.
Multo maximus numerus adveniet in altera ex binis illis oppositis dispositionibus, non
quidem in medio ipsius, sed prope ipsam, & admodum exiguus erit numerus earum, quæ
adveniunt cum dispositione satis remota ab illis extremis. Quæ in hisce intermediis
adveniunt, mutabunt utique dispositiones suas in progressu inter illa duo plana, inter
quæ agit vis motum particulæ perturbans, ita, ut in datis ab utrovis plano distantiis vires
ad diversas particulas pertinentes, sint admodum diversæ inter se. Quare illæ, quæ retro
regredientur, non eandem ad sensum recuperabunt in regressu velocitatem perpendicu-
larem, quam habuerunt in accessu, adeoque non reflectentur in angulo reflexionis æquali
ad sensum angulo incidentiæ, & illæ, quæ superabunt intervallum illud omne, in appulsu
ad planum ulterius, aliæ aliam summam virium expertæ, habebunt admodum diversa
inter se incrementa, vel decrementa velocitatum perpendicularium, & proinde in admodum
diversis angulis egredientur dispersæ. At quæ advenient cum binis illis dispositionibus
contrariis, habebunt duo genera virium, quarum singula pertinebunt constanter ad classes
singulas, cum quarum uno idcirco facilius in illo continuo curvaturæ flexu devenietur
ad positionem illis planis parallelam, sive ad extinctionem velocitatis perpendicularis
cum altero difficilius : adeoque habebuntur in binis illis dispositionibus oppositis binæ
vices, altera facilioris, altera difficilioris reflexionis, adeoque facilioris transitus, quæ quidem
regredientur post æqualia spatiorum intervalla, quanquam ita, ut summa facilitas in media
dispositione sita sit, a qua quæ minus, vel magis in appulsu discedunt, magis e contrario,
vel minus de illa 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 æque inflexam circa
suum axem, & inde reliqua omnia, quæ ad vices, & earum consectaria pertinent, luculen-
tissime explicantur. -
498. Porro hinc & illud patet, qui fieri possit, ut e radiis homogeneis ad eandem
superficiem advenientibus alii transmittantur, & alii reflectantur, prout nimirum advenerint
in altera e binis dispositionibus : & quoniam non omnes, qui cum altera ex extremis illis
dispositionibus adveniunt, adve-[232]-niunt prorsus in media dispositione, fieri utique
poterit, ut ratio reflexorum ad transmissos sit admodum diversa in diversis circumstantiis,
nimirum diversi mediorum discriminis, vel diversæ inclinationis in accessu : ubi enim
inæqualitas virium est minor, vel major perpendicularis velocitas per illam extinguenda
Unde discrimen in
intervallis vicium.
ad habendam reflexionem, non reflectentur, nisi illæ particulæ, quæ advenerint in dispositione
illi mediæ quamproxima, adeoque multo pauciores quam ubi vel inæqualitas virium est
major, vel velocitas perpendicularis est minor, unde fiet, ut quemadmodum experimur,
quo minus est mediorum discrimen, vel major incidentiæ angulus, eo minor radiorum
copia reflectetur : ubi & illud notandum maxime, quod ubi in continuo flexo curvaturæ
viæ particulæ cujusvis, quæ via jam in alteram plagam est cava, jam in alteram, prout
prævalent 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
quæ determinat velocitatem perpendicularem totalem, debeat esse ad sensum eadem,
nimirum nihil mutata ad sensum ab exigua illa differentia virium, quam peperit exiguum
dispositionis discrimen a media dispositione. *a
499. Atque hoc pacto satis luculenter jam explicatum est discrimen inter binas vices,
sed superest exponendum, unde discrimen intervalli vicium, quod proposuimus num. 472.
Quod diversi colorati radii diversa habeant intervalla, nil mirum est : nam & diversæ
A THEORY OF NATURAL PHILOSOPHY 35 I
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
continuous flow of even homogeneous particles towards the surface of separation of two
unlike media. By far the greater number of them will arrive at the surface in one or other
of those two opposite dispositions; not indeed exactly so, but very nearly so. A very few
of them will reach the surface with a disposition considerably removed from those extremes.
Those that do arrive in these intermediate states, will in all cases change their dispositions
in their passage between the two planes, between which the force disturbing the motion
of the particle acts; & in such a manner that at any given distance from either plane the
forces pertaining to different particles will be altogether different. Therefore, those which
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
homogeneous rays reaching the same surface, some are transmitted & others are reflected,
according as they reach it in one or other of two dispositions. Since, of those particles
which do [not] reach the surface with one of the two extreme dispositions, not all reach it
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
fits; but we have still to explain the source of the difference in the intervals between the
fits, which we propounded in Art. 472. There is nothing wonderful in the fact that
differently coloured rays should have different intervals. For, different velocities require
Hence, we have the
two constant dis-
positions yielding
fits, with the greater
proportion of par-
ticles, which are
Striking in those
limiting states; &
for the few that
Strike in states
intermediate be-
tween them, we
have dispersion.
The cause of the
difference in the
ratio of the amount
of light reflected to
that which is trans-
mitted.
The cause of the
difference in the
intervals between
Successive fits.
352 PHILOSOPHIAE NATURALIS THEORIA
Discrimen id non
posse definiri, nisi
per observationes :
non pendere a sola,
velocitate.
Quod de crystallo
Islandica Newtonus
prodidit, id in hac
Theoria n u 1 1 a. m
habere difficulta-
tem.
Diffractionem esse
inchoatam reflexi-
onem, vel refractio-
116. IIl.
De sapore, & odore:
multorum error in
ratione densitatis
odoris propagati.
velocitates diversa requirunt intervalla spatii inter vices oppositas, quando etiam eæ vices
redeant æqualibus temporis intervallis, & diversus particularum heterogenearum textus
requirit diversa Oscillationum tempora. Quod in diversis mediis particulæ ejusdem generis
habeant diversa intervalla, itidem facile colligitur ex diversa velocitate, quam in iis haberi
post refractionem Ostendimus num. 493 ; sed præterea in ipsa mediorum mutatione
inæqualis 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 Tviæ particulæ
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 colligitür. -
500. Si possemus nosse peculiares constitutiones particula-[233]-rum ad diversos
coloratos radios pertinentium, Ordinem, & numerum, ac vires, & velocitates punctorum
singulorum ; tum mediorum constitutionem suam in singulis, ac satis Geometriæ, 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,
præstitit, qui determinatis per observationem singulis, mira inde consectaria deduxit,
& Naturæ phænomena explicavit, uti multo luculentius videre est in illa 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 incidentiæ ad sinum anguli refracti : ac tam 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 hæc innuisse
S1t SatlS.
5oI. Illud etiam innuam tantummodo, quod Newtonus in Opticis Quæstionibus
exponit, esse miram quandam crystalli Islandicæ proprietatem, quæ radium quemvis, dum
refringit, discerpit in duos, & alium usitato modo refringit, alium inusitato quodam, ubi &
certæ quædam 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 particulæ diversas dispositiones punctorum, & vires, qua ipsa diversitate usi sumus
superius ad explicandam solidorum cohæsionem, & organicam formam, ac certas figuras
tot corporum, quæ illas vel affectant constanter, vel etiam acquirunt.
5o2. Remanet demum diffractio luminis explicanda, quam itidem num. 472 proposue-
ramus. Ea est quædam velut inchoata reflexio, & refractio. Dum radius advenit ad eam
distantiam a corpore diversæ naturæ ab eo, per quod progreditur, quæ virium inæqualitatem
inducit, incurvat viam vel accedendo, vel recedendo, & directionem mutat. Si corporis
superficies ibi esset satis ampla, vel reflecteretur ad angulos æquales, vel immergeretur
intra novum illud medium, & refrin-[234]-geretur ; at quoniam acies ibidem progressum
superficiei interrumpit ; progreditur quidem radius aciem ipsam evitans & circa illam
prætervolat ; sed egressus ex illa 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 præclarissima comperta virium, quibus cælestia corpora motus
peragunt suos & quibus particulæ luminis reflectuntur, refringuntur, diffringuntur. Sed
de luce, & coloribus jam satis.
5o3. Post ipsam lucem, quæ oculos percellit, & visionem parit, ac ideam colorum
excitat, pronum est delabi ad sensus ceteros, in quibus multo minus immorabimur, cum
circa eos multo minora habeamus comperta, quæ determinatam physicam explicationem
ferant. Saporis sensus excitatur in palato a salibus. De angulosa illorum forma jam
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.
590. If we could know the particular constitutions of particles for differently coloured
rays, the order, number, forces & velocities of each point, & the constitution of each medium
for each ray, and if we had a sufficiency of geometry, imagination & intelligence to solve
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
end of his Optics, there is a wonderful property of Iceland Spar ; namely, that when it
refracts a ray of light it divides it into two, refracting one part according to the normal
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.
This is, so to speak, an incomplete reflection or refraction. When a ray of light attains
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
naturally come to the other senses; over these I will spend far less time, since we have
far less knowledge of them, such as will help us to give a definite physical explanation.
The sense of taste is excited in the palate by salts. I have already spoken of the
*. A A
This difference can-
not be definitely
given, unless by
observation ; it
does not depend on
velocity alone.
That which Newton
recorded concerning
Iceland Spar pre-
sents no difficulty
in my Theory.
Diffraction is in-
complete reflection
or refraction.
Concerning taste &
smell ; the error of
many people with
regard to the ratio
of the density of a
propagated odour,
354 PHILOSOPHIÆ NATURALIS THEORIA -
De sono difficultas
in determinandis
undis excitatis in
fluido elastico.
Quo pacto oriantur
undæ in serie con-
tinua, punctorum se
invicem repellen-
tium.
diximus num. 464, quæ ad diversum excitandum motum in papillis palati abunde sufficit ;
licet etiam dum dissolvuntur, vires varias pro varia punctorum dispositione exercere debeant,
quæ saporum discrimen inducant. Odor est quidam tenuis vapor ex odoriferis corporibus
emissus, cujus rei indicia sunt sane multa, nec omnino assentiri possum illi, qui odorem
etiam, ut sonum, in tremore medii cujusdam interpositi censet consistere. Porro quæ
evaporationum sit causa, explicavimus abunde num. 462. Illud unum hic innuam, errare
illos, uti pluribus ostendi in prima parte meæ dissertationis De Lumine, qui multi sane
sunt, & præstantes 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, quæ a dato puncto diffunduntur in
sphæram, sed quæ diffunduntur cum uniformi celeritate, ut lumen. Si enim concipiantur
orbes concentrici tenuissimi datæ crassitudinis ; ii erunt ut superficies, adeoque ut quadrata
distantiarum a communi centro, ac densitas materiæ erit in ratione ipsorum reciproca :
si massa sit eadem : ut ea in ulterioribus Orbibus sit eadem, ac in citerioribus ; oportet
sane, tota materia, quæ erat in citerioribus ipsis, progrediatur ad ulteriores orbes motu
uniformi, quo fiet, ut, appellente ad citeriorem superficiem Orbis ulterioris particula, quæ
ad citeriorem citerioris appulerat, appellat simul ad ulteriorem ulterioris quæ 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
imminutæ, & impeditæ a progressu partis vaporum, quæ compensatio accurata est admodum
improbabilis ; non habebitur densitas reciproce proportionalis Orbibus, sive eorum super-
ficiebus, vel distantiarum quadratis.
[235] 504. Sonus geometricas determinationes admittit plures, & quod pertinet ad
vibrationes chordæ elasticæ, vel campani æris, vel motum impressum aeri per tibias, &
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 propagato idea soni excitatur, res est multo opero-
sior, & pendet plurimum ab ipsa medii constitutione : ac si accurate solvi debeat problema,
quo quæratur ex data medii fluidielasticitate propagatio undarum, & ratio inter Oscillationum
celeritates, a qua multipliciter variata pendent omnes toni, & consonantiæ, ac dissonantiæ,
& 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, & determinationi resistentiæ fluidorum est admodum affinis, cum qua motum in
fluido propagatum communem habet. Exhibebo hic tantummodo simplicissimi casus
undas, ut appareat, qua via ineundam censeam in mea Theoria ejusmodi investigationem.
5o5. Sit in recta linea disposita series punctorum ad data intervalla æqualia a se invicem
distantium, quorum bina quæque sibi proxima se repellant viribus, quæ crescant imminutis
distantiis, & dentur ipsæ. Concipiatur autem ea series utraque parte in infinitum producta,
& uni ex ejus punctis concipiatur externa vi celerrime agente in ipsum multo magis, quam
agant puncta in se invicem, brevissimo tempusculo impressa velocitas quædam finita in
ejusdem rectæ 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 mutuæ, quæ 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 illa parie, & 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
vibrations of an elastic cord or bell-metal, or the motion given to the air by flutes &
trumpets, all belong to the science of Mechanics; & for them my Theory is in agreement
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
intervals of distance from one another ; & of these let any two consecutive points repel one
another with forces, which increase as the distance decreases, & suppose that the magnitudes
of these forces are also given. Also suppose that this series is continued on either side to
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;
Sound ; difficulty
in determining the
waves excited in an
elastic fluid.
The manner in
which waves may
arise in a continu-
ous series of points
repelling one an-
Other.
356 - PHILOSOPHIÆ NATURALIS THEORIA
Solutio difficultatis
pertinentis ad pro-
pagationem rectili-
neam diversorum
SOnOrum admodum
fa cilis in h a c
Theoria.
De calore & frigore :
materiæ c i e n t i s
calorem expansio
orta ab elasticitate :
fixatio ejusdem, &
velocitas ut tor-
rentis cujusdam.
Post tempusculum utcunque exiguum omnia puncta æquilibrium amittent, & motum
habebunt aliquem. Interea cessante actione vis impellentis punctum primum incipiet
ipsum retar-[236]-dari vi repulsiva secundi dexteri prævalente supra vim secundi sinistri,
sed adhuc progredietur, & accedet ad secundum, ac ipsum accelerabit : verum post aliquod
tempus retardatio continua puncti impulsi, & acceleratio secundi reducent illa ad veloci-
tatem eandem : tum 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 :
tum habebitur retardatio, tum motus iterum reflexus. Dum id punctum iterum incipit
regredi versus dexteram, erit aliquod e dexteris, quod tunc 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. Undæ amplitudinem determinabit distantia duorum punctorum, quæ simul eunt
& simul redeunt, ac celeritatem propagationis soni tempus, quod requiritur ad unam
Oscillationem puncti impulsi, & distantia a se invicem punctorum, quæ simul cum eo eunt,
& redeunt ; & quod ad dexteram accidit ad sinistram. Sed & ea perquisitio est longe
altioris indaginis, quam ut hic 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. • .
5o6. Interea illud unum adjiciam, in mea Theoria admodum facile solvi difficultatem,
quam Eulerus objecit Mairanio, explicanti propagationem diversorum sonorum, a quibus
diversi toni pendent, per diversa genera particularum elasticarum, quæ habentur in aere,
quorum singula singulis sonis inserviant, ut diversi sunt colorati radii cum diverso constanti
refrangibilitatis gradu, & colore. Eulerus illud objicit, uti tam multa sunt sonorum
genera, quæ 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 plano non nisi sex tantummodo alii globi in gyrum
possint consistere. Difficultas in mea Theoria nulla est, cum particulæ aliæ in alias non
agant per immediatum contactum, sed in aliqua distantia, quæ 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
tum eos, qui pendent a viribus mutuis inter duos globulos, a quibus proveniunt undæ,
tum 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
instrümentorum chordis cernimus, quarum una percussâ sonant & reliquæ. Ubique
libertas motuum, & dispositionis, quæ sublato immediato impulsu, & accurata continuitate
in corporum textu, acquiritur ad explicandam naturam, est perquam idonea, & opportuna.
507. Quod pertinet ad tactiles proprietates, quid sit solidum, fluidum, rigidum, molle
elasticum, flexile, fragile, grave, abunde explicavimus : quid lævigatum, quid asperum,
per se patet. Caloris causam repono in motu vehementi intestino particularum igneæ,
vel sulphureæ substantiæ fermentescentis potissimum cum particulis luminis, & qua ratione
id fieri possit, exposuimus. Frigus haberi potest per ipsum defectum ejusmodi substantiæ,
vel defectum motus in ipsa. Haberi possunt etiam particulæ, quæ frigus cieant actione
sua, ut nitrosæ, 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
of the difficulty, which Euler brought forward in opposition to Mairan ; the latter tried
to explain the propagation of the different sounds, upon which different musical tones
depend, by the presence of different kinds of elastic particles in the air; each kind of
particle was of service to the corresponding sound, just as there are differently coloured
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,
rigid, soft, elastic, flexible, fragile & heavy bodies; what a smooth, or a rough, body is,
is self-evident. I consider the cause of heat to consist of a vigorous internal motion
of the particles of fire, or of a sulphurous substance fermenting more especially with
particles of light; & I have shown the mode in which this may take place. Cold may
be produced by a lack of this substance, or by a lack of motion in it. Also there may be
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
The solution of the
difficulty with re-
spect to the rectili-
near propagation of
different so unds
comes quite easily
from my Theory.
Heat & cold ; the
expansion of the
matter producing
heat arises from
elasticity ; fixation
of the same, & a
velocity as of a
torrent.
358 PHILOSOPHIÆ NATURALIS THEORIA
Imago in aeris fixa-
tione, & affluxu.
Attractio, quæ po-
t e S t intestinum
motum sistere, &
fixare : communi-
catio ad æqualem
saturitatem p o s t
partem fixatam :
saturitatis varia
discrimina.
Quæ a diffusione
a d æqualitatem
Consequantur potis-
simum respectu re-
frige r a t i o nis,
& conglaciationis.
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 accéssum
corporis frigefacti 9b solum ejusmodi substantiæ defectum. Eâ enim, dum fermentat,
& in suo naturali yolatilizationis statu permanet, nititur elasticitate sua ipsa ad expansionem,
per quam, si in aliquo medio conclusa sit, utcunque inerte respectu ipsius, ad æqualitatem
per ipsum diffunditur, unde fit, ut si uno in loco dematur aliqua éjus pars, statim illuc
ex aliis tantum devolet, quantum ad illam æqualitatem requiritur. Hinc nimirum, si
in aere libero cesset fermentantis ejusmodi substantiæ quantitas, vel per imminutam con-
tinuationem impulsuum ad continuandum motum, ut imminuta radiorum Solis copia per
hyemem, ac in locis remotioribus ab Æquatore, vel per accessum ingentis copiæ particularüm
sistentium ejusdem substantiæ motum, unde fit, ut in climatis etiam non multum ab
Æquatore distantibus ingentia pluribus in locis habeantur frigora, & glacies per nitrosorum,
effluyiorum copiam ; e corporibus omnibus expositis aeri perpetuo erumpet magna copia
ejusdem fermentescentis ibi adhuc, & elasticæ materiæ igneæ ; & ea corpora remanebùnt
admodum frigida per solam imminutionem ejus materiæ, quibus si manüm admoveamus,
ingens illico ex ipsa manu particularum earundem multitudo avolabit transfusa illuc, ut
res ad æqualitatem redu-[238]-catur, & tam ipsa cessatio illius intestini motus, qua immuta-
bitur status fibrarum organici corporis, quam ipse rapidus ejus substantiæ in aliam
irrumpentis torrens, eam poterit, quam adeo molestam experimur, frigoris sensationem,
€XC1tare. -
508. Torrentis ejusmodi ideam habemus 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 fenestræ, & tabulata, ac omnes portæ, quæ
cubiculorum mutuam communicationem impediunt, repente dissiliunt, & ipsi parietes
nonnunquam evertuntur, ac corruunt, quemadmodum Romæ ante aliquot observavimus
annos, & in dissertatione De Turbine superius memorata, quam tum edidi, pluribus exposui.
509. Verum hæc sola substantiæ hujusce fermentantis expansiva vis non est satis ad
rem explicandam, sed requiritur etiam certa vis mutua, qua ejusmodi substantia in
alias quasdam attrahatur magis, in alias minus, quod qui fieri possit, vidimus, ubi de
dissolutione, & præcipitatione egimus : & ejusmodi attractio potest esse ita valida, ut
motum ipsum intestinum prorsus impediat appressione ipsa, ac fixationem ejus substantiæ
inducat, quæ 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. Hic 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 æque, vel
etiam minus trahat, adhuc tamen cum utraque in minoribus distantiis trahat plus, in
majoribus minus, illa habeat ejus substantiæ multo minus etiam pro ratione attractionis
suæ, quam altera ; nam in hoc secundo casu, adhuc ab hac posteriore avellerentur particulæ
affusæ ipsius particulis ad distantias aliquanto majores, & affunderentur particulis prioris
substantiæ, donec in utravis substantia haberetur æqualis saturitas, si ejus partes inter
se conferantur, & æqualis itidem attractiva vis particularum substantiæ igneæ maxime
remotarum a particulis utriusque substantiæ, quibus ea affunditur : sed copia ipsius
substantiæ igneæ 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 altitudo ejusmodi veluti marium in altera esse major, minor in
altera, & in iisdem distantiis possit in altera haberi ob vim majorem densitas major sub-
stantiæ ipsius igneæ affusæ, quam in altera. Ex hisce quidem principiis, ac diversis
combinationibus, mirum sa-[239]-ne, quam multa deduci possint ad explicationem Naturæ
perquam idoneis.
5 Io. Sic etiam ex hac diffusione ad ejusmodi æqualitatem eandem inter diversas
ejusdem substantiæ partes, sed admodum diversam inter substantias diversas, facile intelli-
gitur, qui fiat, ut manus in hyeme exposita libero aeri minus sentiat frigoris, quam solido
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
aere 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
some part of space is suddenly reduced to fixation in large quantities, air will rush in violently
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
explain thoroughly what happens; we require also a certain mutual force, due to which
the substance is attracted more by some bodies & less by others; & the manner in which
this can happen was explained when we dealt with solution & precipitation. Such an
attraction may be so powerful as to prevent that internal motion altogether by its pressure,
& lead to fixation of the substance; but if this is fairly small, it will indeed allow some
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. -
51O. Thus, from the principle of such diffusion tending to establish the same equality
between different parts of the same substance, but an equality that is quite different for
different substances, it is easily seen how it comes about that in winter the hand when exposed
to the open air, feels the cold less than when exposed to a solid body of sufficient density,
such as marble, which has previously been exposed to the same cold air for a long time;
& 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,
An illustration
from the fixation
& inflow of air.
The at traction
which can stop &
fix internal motion;
motion shared So
as to give equality
of saturation after
a part is fixed ;
different kinds of
Saturation.
The consequences
of this diffusion
tending to estab-
lish equality ;
especially in the
matter of refrigera-
tion & congelation.
36o PHILOSOPHIÆ NATURALIS THEORIA
Eodem pacto expli-
cari & electricita-
tem : P r in cipia
E ra n k 1 i n i an æ
theoriæ Electrici-
tatis.
eodem tempore admodum diversa copia igneæ substantiæ, quæ calorem in manu fovebat.
Atque hic quidem & analogiæ sunt quædam cum iis, quæ de refractione diximus : nam
plerumque corpora, quæ plus habent materiæ, nisi oleosa, & sulphorosa sint, majorem
habent vim refractivam, pro ratione densitatis suæ, & corpora itidem communiter, quo
densiora sunt, eo citius manum admotam calore spoliant, quæ 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 admixtæ substantiæ plus habet, vel minus. Sic fieri posset,
ut aer ejusmodi substantiam igneam respueret, sed ob heterogenea corpora, quæ sustinet, inter
quæ inprimis est aqua in vapores elevata, surripiat nonnihil ; ubi autem in ipso volitantes
particulæ, quæ 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 æqualitatem intra idem corpus fieri utique
debet, ut ubi altius infra Terræ 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, quæ omnia se mutuo compensant saltem intra annum, antequam
sensibilis differentia haberi possit in profundioribus locis : ac ex diversa vi, quam diversæ
substantiæ exercent in ejusmodi substantiam igneam, provenire debet & illud, quod
experimenta evincunt, ut nimirum nec eodem tempore æque frigescant diversæ substantiæ
aeri libero expositæ, nec 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.
5 II. Patet autem ex iisdem principiis repeti posse explica-[24o]-tionem etiam præci-
puorum omnium ex Electricitatis phænomenis, quorum Theoriam a Franklino mira sane
sagacitate inventam in America & exornavit plurimum, & confirmavit, ac promovit Taurini
P. Beccaria vir doctissimus opere egregio ea de re edito ante hos aliquot annos. Juxta
ejusmodi Theoriam huc omnia reducuntur : esse quoddam fluidum electricum, quod in
aliis substantiis & per superficiem, & per interna ipsarum viscera possit pervadere, per
alias motum non habeat, licet saltem harum aliquæ ingentem contineant ejusdem substantiæ
copiam sibi firmissime adhærentem, nec sine frictione, & motu intestino effundendam,
quarum priora sint per communicationem electrica, posteriora vero electrica natura sua :
in prioribus illis diffundi statim id fluidum ad æqualitatem in singulis ; licet alia majorem,
alia minorem ceteris paribus copiam ejusdem poscant ad quandam sibi veluti connaturalem
Saturitatem : hinc e duobus ejusmodi corporibus, quæ respectu naturæ suæ non eundem
habeant saturitatis gradum, esse alterum respectu alterius electricum per excessum, &
alterum per defectum, quæ ubi admoveantur ad eam distantiam, in qua particulæ circa
ipsa corpora diffusæ, & iis utcunque adhærentes ad modum atmosphærarum quarundam,
possint agere aliæ in alias, e corpore electrico per excessum fluere illico ejusmodi fluidum
in corpus electricum per defectum, donec ad respectivam æqualitatem deventum sit, in
quo effluxu & substantiæ ipsæ, quæ fluidum dant, & recipiunt, simul ad se invicem accedant,
si satis leves sint, vel libere pendeant, & si motus coacervatæ materiæ sit vehemens, explo-
siones habeantur, & scintillæ, & vero etiam fulgurationes, tonitrua, & fulmina. Hinc
nimirum facile repetuntur omnia consueta electricitatis phænomena, præter Batavicum
experimentum phialæ, quod multo generalius est, & in Frankliniano plano æque habet
locum. Id enim phænomenum 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 adverso
respondente æqualis 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, transfuso statim fluido ex altera superficie in alteram, obtinetur æqualitas, & 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
of all the chief phenomena in electricity; the theory of these, discovered by Franklin in
America with truly marvellous sagacity, has been greatly embellished & confirmed, & even
further developed at Turin by Fr. Beccaria, a most learned man, in his excellent work
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.
Electricity can also
be explained in the
same way; Frank-
lin's principles of
the theory of elec-
tricity.
362 PHILOSOPHIÆ NATURALIS THEORIA
Eorum explicatio
in hac Theoria.
Quod videatur esse
discrimen inter
Imateriam e 1 e c-
tricam, & igneam.
IDe magnetica, vi :
directionem, & ejus
variationem p e n-
dere ab attractione,
& mutatione mass-
a ru m ingentium
attrahentium.
512. Hujusmodi Theoriæ ea pars, quæ continet respectivam [241] illam saturitatem,
conspirat cum iis, quæ 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 particulæ ipsum fluidum,
quanquam viribus inæqualibus, ad se trahunt, ad se invicem accedere, ac facile itidem
patet, cur aer humidus, in quo ob admixtas aquæ particulas vidimus citius manum frigescere,
electricis phænomenis 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 difficilior,
explicatione tamen sua non caret. Fieri utique potest, ut in certis corporibus ingens sit
ejus substantiæ copia ob attractionem ingentem, & ad exiguas distantias pertinentem,
congesta, quæ in aliquanto majore distantia in repulsionem transeat, sed attractioni non
prævalentem. Hæc repulsio cum illa copia materiæ potest esse in causa, ne per ejusmodi
substantias transire possit is vapor, & ne per ipsam superficiem excurrat, nec vero ad eam
accedat satis ; nisi alterius substantiæ adjunctæ actio simul superveniat, & adjuvet. Tum
vero ubi lamina sit tenuis, potest repulsio, quam exercent particulæ 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 hærent 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 : tum vero ipsa prævaleat. Ipsa autem prævalente,
effluet ex ulteriore superficie ejus fluidi pars novum illud corpus admotum ingressa, ac
ex ejus partis remotione, cessante parte vis repulsivæ, quam nimirum id, quod effluit,
exercebat in particulas citerioris superficiei, ipsi citeriori superficiei adhæreat jam idcirco
major copia fluidi electrici admota per aquam, vel aurum, donec 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 æqualitatem 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 substantiæ per se electricæ, utut spoliata magna parte sui fluidi, non poterit
rapere partem satis magnam fluidi alteri parti affusi, & appressi.
5 I 3. Hæc quidem an eo modo se habeant, definire non licet [242] nisi & illud ostendatur
simul, rem aliter se habere non posse. Sed illud jam patet, Theoriam meam, servato semper
eodem agendi modo, suggerere ideam earum etiam dispositionum materiæ, quæ possint
maxime omnium ardua, & composita explicare Naturæ phænomena, ac corporum discrimina.
Illud unum hic 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, & intestino
motu, quamobrem etiam comburat, & calefaciat, & dilatet, ac rarefaciat substantias, altera
ad fermentescendum apta sit, sed sine ulla, saltem tanta agitatione, quantam fermentatio,
inducit Orta ex collisione ingenti mutua, vel ex aliarum admixtione substantiarum, quæ
sint ad fermentandum idoneæ.
5 I4. Quod ad magneticam vim pertinet, adnotabo illud tantummodo, ejus phænomena
omnia reduci ad solam attractionem certarum substantiarum ad se invicem. Nam directio,
ad quam & inclinatio, & declinatio reducitur, repeti utique potest ab attractione ipsa sola.
Videmus acum magneticam inclinari statim prope fodinas ferri, intra quas idcirco nullus
est pyxidis magneticæ usus. Si ingens adesset in ipsis polis, & in iis solis, massa ferrea ;
omnes acus magneticæ dirigerentur ad polos ipsos : sed quoniam ubique terrarum fodinæ
ferreæ habentur, si circa polos eædem sint in multo majore copia, quam alibi ; dirigentur
utique acus polos versus, sed cum aliqua deviatione in reliquas massas per totam Tellurem
dispersas, quæ nunquam poterit certum superare graduum numerum ; nisi plus æquo 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
what we have said with respect to the fiery substance, when we gave a full explanation of
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,
Expla nation of
the se matters in
my Theory.
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
be shown at the same time that it is impossible for them to be otherwise. But this fact
is clear, that my Theory, always maintaining the same mode of action, suggests also the
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
phenomena with regard to it reduce to a mere attraction of certain substances for one
another. For direction, to which both inclination & declination can be reduced, can always
be derived from attraction alone. We notice that a magnetic needle is immediately inclined
near iron mines; & therefore within these a magnetic compass-box is of no service. If
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
The manner in
which electric mat-
ter seems to differ
from fire.
Magnetic force ; its
direction & varia-
tion depends upon
the attraction &
alteration in the
large masses at-
tracting.
364 PHILOSOPHIÆ NATURALIS THEORIA
Attract i o n e m, &
polos cohærere cum
hac Theoria : diffi-
cultas de distantia,
ad quam vi s e a.
extenditur: conjec-
tu ra, de solutione
ipsius.
Quid materia, & .
unde ejus vire s:
tria diversa prin-
cipia, a quibus
provenire possunt.
EHomogeneitas ele-
mentorum. Si ea,
non ad mittatu r,
quanto plures com-
binationes per di-
versas leges virium:
formam substantia-
lem, & accidentia,
posse Peripateticos,
si velint, agnoscere
in ipsis punctis.
eorum locorum positionem ad omnes ejusmodi massas, & vero etiam variabitur, cum fodinæ
ferri & destruantur in dies novæ, & generentur, ac augeantur, & minuantur in horas.
Variatio intra unum diem exigua erit, cum eæ mutationes in fodinis intra unum diem
exiguæ sint : procedente tempore evadet major, eritque omnino irregularis ; si mutationes,
quæ in fodinis accidunt, sint etiam ipsæ irregulares. -
5 I 5. Quod autem ad attractionem pertinet eam in particulis haberi posse patet, &
ab earum textu debere pendere : plurima autem sunt magnetismi phænomena, quæ
Ostendant, mutata dispositione particularum generari magneticam vim, vel destrui, &
multo frequentius intendi, vel remitti, cujus rei exempla passim occurrunt apud eos, qui
de magneticis agunt. Poli autem ex altera parte attractivi, ex altera repulsivi, qui haben-
tur in magnetismo itidem, cohærent 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 effluviorum
intermedium genus, quod tenui-[243]-tate sua effugerit huc usque observantium oculos,
& quod per intermedias vires suas connectat etiam massas remotas, si forte ex sola diversa
combinatione punctorum habentium vires ab eadem illa mea curva expressas id etiam
phænomenon provenire non possit. Sed ad hæc omnia rite evolvenda, & illustranda singu-
lares tractatus, & longæ perquisitiones requirerentur ; hic mihi satis est indicasse
ingentem Theoriæ meæ fœcunditatem, & usum in difficillimis quibuscunque Physicæ etiam
particularis partibus pertractandis. -
516. Superest, ut postremo loco dicamus hic aliquid de alterationibus, & transforma-
tionibus corporum. Pro materia mihi sunt puncta indivisibilia, inextensa, prædita vi
inertiæ, & viribus mutuis expressis per simplicem continuam curvam habentem deter-
minatas illas proprietates, quas expressi a num. I I7, & quæ per æquationem quoque
algebraicam definiri potest. An hæc virium lex sit intrinseca, & essentialis ipsis indivisi-
bilibus punctis ; an sit quiddam substantiale, vel accidentale ipsis superadditum,
quemadmodum sunt Peripateticorum formæ substantiales, vel accidentales ; an sit libera
lex Auctoris Naturæ, qui motus ipsos secundum legem a se pro arbitrio constitutam dirigat :
illud non quæro, nec vero inveniri potest per phænomena, quæ eadem sunt in omnibus
iis sententiis. Tertia est causarum occasionalium ad gustum Cartesianorum, secunda
Peripateticis inservire potest, qui in quovis puncto possunt agnoscere materiam, tum 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, & quæ 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, & Newtoniani passim, videntur agnoscere pro primariis materiæ
proprietatibus in ipsa ejus essentia sitis. Potest utique hæc mea Theoria adhiberi in
omnibus hisce philosophandi generibus, & suo cujusque peculiari cogitandi modo aptari
potest. -
517. Hæc materia mihi est prorsus homogenea, quod pertinet ad legem virium, &
argumenta, quæ habeo pro homogeneitate, exposui num. 92. Siqua occurrent Naturæ
phænomena, quæ per unicum materiæ genus explicari non possint ; poterunt adhiberi
plura genera punctorum cum pluribus legibus inter se diversis, atque id ita, ut tot leges
sint, quot sunt binaria generum, & præterea, quot sunt ipsa genera, ut illarum singulæ
exprimant vires mutuas inter puncta pertinentia ad bina singulorum binariorum genera,
& harum singulæ vires mutuas inter puncta pertinentia ad idem genus, singulæ pro generibus
singulis. Porro inde mirum sane, quanto major [244] combinationum numerus oriretur,
& quanto facilius explicarentur omnia phænomena. Possent autem illæ leges exponi
per curvas quasdam, quarum aliquæ haberent aliquid commune, ut asymptoticum impene-
trabilitatis arcum, & arcum gravitatis, ac aliæ ab aliis possent distare magis, ut habeantur
quædam genera, & quædam differentiæ, quæ corporum elementa in certas classes
distribuerent ; & hic Peripateticis, si velint, occasio daretur admittendi materiam ubique
homogeneam, ac formas substantiales diversas, quæ accidentalem virium formam diversam
exigant, & vero etiam plures accidentales formas, quæ 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
it must depend upon their structure. Moreover, there are very many phenomena of
magnetism, which will show that magnetic force is generated by changing the disposition
of the particles, or is destroyed, or more frequently is augmented or abated; examples
of this everywhere come under the observation of those who study magnets. Moreover,
poles that are attractive on one side & repulsive on the other, which are also had in magnetism,
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 I 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-
formations of bodies. To me, matter is nothing but indivisible points, that are non-extended,
endowed with a force of inertia, & also mutual forces represented by a simple continuous
curve having those definite properties which I stated in Art. I 17; these can also be defined
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
forces, & the arguments which I have in favour of homogeneity, I have stated in Art. 92.
If there are any phenomena of Nature, which cannot be explained by a single kind of matter,
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
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
of each pair, & each of the latter the mutual forces between points belonging to the same
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.
Attraction, & the
poles, are consis-
tent with my
Theory; difficulty
over the distance
to which the force
extends ; conjec-
ture as to the solu-
tion of this problem.
The nature of mat-
ter, & the source of
its forces ; three
different principles
from which they
may arise.
Homogeneity of the
elements. If this
is not admitted,
there will be all the
more combinations
through different
laws of forces ; &
the Peripatetics can,
if they wish, admit
Substantial form &
circumstances into
these points.
366 PHILOSOPHIÆ NATURALIS THEORIA
Mira variatas con-
Sectariorum : possi-
bilitas quotlibuerit
Mundorum in eo-
dem spatio cum ap-
parenti compene-
tratione, sine ulla,
notitia unius cujus-
vis in aliis.
518. Posset autem admitti vis in quibusdam generibus nulla, & tunc substantia unius
ex iis generibus liberrime permearet per substantiam alterius sine ullo occursu, qui in
numero finito punctorum indivisibilium nullus haberetur, adeoque transiret cum impene-
trabilitate reali, & compenetratione apparente : ac posset unum genus esse colligatum
cum alio per legem virium, quam habeant cum tertio, sine ulla lege virium mutua inter
ipsa, 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, nec alter ullam alterius
notitiam posset unquam acquirere. Mirum sane, quam multæ aliæ in casibus illius nexus
cujuspiam duorum generum cum tertio combinationes haberi possint ad explicanda Naturæ
phænomena : sed argumenta, quæ 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 hic 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
distantiæ a punctis in nostro existentibus, nos prorsus ignoramus, nihil enim eo pertinens
omnino ex Naturæ phænomenis colligere possumus, & nimis est audax, qui eorum omnium,
Formam in homo-
geneitatis supposi-
tione esse numerum,
& dispositionem
punctorum, quæ
sunt radix omnium
proprietatum : quæ
dici possint formæ
specificæ : u n d e
a lter a, t i o n e s &
transformationes.
Discrimen inter
transformationem,
& alterationem.
quæ condidit Divinus Naturæ Fabricator limitem ponat suam sentiendi, & vero etiam
cogitandi vim. -
5I9. Sed redeundo ad meam homogeneorum elementorum Theoriam, singulares
corporum formæ erunt combinatio punctorum homogeneorum, quæ habetur a distantiis
& positionibus, ac præter solam combinationem velocitas, & directio motus punctorum
singulorum ; pro individuis vero corporum massis accedit punctorum numerus. Dato
numero & dispositione punctorum in data massa, datur radix omnium proprietatum, quas
habet eadem massa in se, & omnium relationum, [245] quas eadem habere debet cum
aliis massis, quas nimirum determinabunt numeri, & combinationes, ac motus earum,
& datur radix omnium mutationum, quæ ipsi possunt accidere. Quoniam vero sunt
quædam combinationes peculiares, quæ exhibent quasdam peculiares proprietates con-
stantes, quas determinavimus, & exposuimus, nimirum suæ pro cohæsione, & variis solidi-
tatum gradibus, suæ pro fluiditate, suæ pro elasticitate, suæ pro mollitie, suæ pro certis
acquirendis figuris, suæ pro certis habendis oscillationibus, quæ & per se, & per vires sibi
affixas diversos sapores pariant, & diversos ordores, & colorum diversas constantes proprie-
tates exhibeant, sunt autem aliæ combinationes, quæ inducunt motus, & mutationes non
permanentes, uti est omne fermentationum genus ; possunt a primis illis constantium
proprietatum combinationibus desumi specificæ corporum formæ, & differentiæ, & per
hasce posteriores habebuntur alterationes, & transformationes.
520. Inter illas autem proprietates constantes possunt seligi quædam, quæ magis
constantes sint, & quæ non pendeant a permixtione aliarum particularum, vel etiam, quæ
si amittantur, facile, & prompte acquirantur, & illæ haberi pro essentialibus illi speciei,
quibus constanter mutatis habeatur transformatio, iisdem vero manentibus, habeatur
tantummodo alteratio. Sic si fluidi particulæ 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 igneæ, quam
in poris continet, vel aucta quantitate materiæ igneæ, quam in poris continet, vel aucto
motu ejusdem, vel etiam aucta aliqua suarum partium oscillatione, ac dicetur calefactione
nova alteratum tantummodo : & aquæ massa, quæ post ebullitionem redit ad priorem
formam, erit per ipsam ebullitionem alterata, non transformata : figuræ itidem mutatio,
ubi ex cera, vel metallo diversa fiunt opera, alterationem quandam inducet. At ubi mutatur
ille textus, qui habebatur in particulis, atque id mutatione constanti, & quæ longe alia
phænomena præbeat ; tum 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 arcte inter se cohæreant novo
nexu earum particulæ, & novum mixtum efforment, ubi mixti particulæ separatæ per
solutionem nexus ipsius, quod accidit in putrefactione, & in fermentationibus plurimis,
novam singulæ 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
the substance of one of these classes will pasº perfectly freely through the substance of
another without any collisions; for, with a finite number of indivisible points, there would
not be any ; & thus the substance would pass through with real impenetrability & apparent
compenetration. Also it would be possible for one kind to be bound up with another by
means of a law of forces, which they have with a third, without any mutual law of forces
between themselves, or these two kinds might have no connection with any third. In this
latter case there might be a large number of material & sensible universes existing in the same
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 combinations 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
gathered from the phenomena of Nature; & it would be great presumption for any one to
fix 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
bodies will consist of a combination of homogeneous points, which comes from their distances
& positions, &, in addition to combination alone, the velocity & direction of the motion
of each of the points; also for individual masses of bodies there is to be added the number
of points that form them. Given the number & disposition of the points in a given mass,
the basis of all its properties, which are inherent in the mass, is given; & also that of all
the relations that the same mass must have with other masses; that is to say, those determined
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
constant than others; such as do not depend upon admixture with other particles, & also
such as, if they should be lost, would be easily & quickly acquired. These properties could
be considered to be essential to the species ; & 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, -
Wonderful variety
of consequences ;
the possibility of
any number of
Universes, occupy-
ing the same space
with apparent Com-
penetration, with-
out any indication
of the presence of
any one of them in
the others.
Form in the hypo-
thesis of homogene-
ity is the number &
disposition of the
points; these con-
stitute the basis of
all properties; what
may be said about
specific form; hence,
alterations & trans-
formations.
Distinction between
transformation &
alteration.
368 PHILOSOPHIÆ NATURALIS THEORIA
Qu id requireretur
a d inspiciendam
formam intimam,
unde liceret a priori
reducere massas ad
genera, & species :
quid præstandum,
cum id non liceat.
Videri, nos nun-
quam posse devenire
a d cognoscendam
intimam substan-
tiam, & essentiam,
ac discrimina speci- .
fica.
Quid tamen præs-
t a r i possit circa
generales proprie-
tates, & generalia,
principia : id esse
hic præstitum.
Quo pacto interea
species distingua-
1 mllS.
52I. Si possemus inspicere intimam particularum constitutionem, & textum, ac
distinguere a se invicem particulas Ordinum gradatim altiorum a punctis elementaribus
ad hæc nostra corpora ; fortasse inveniremus aliqua particularum genera [246] ita suæ
formæ tenacia, ut in omnibus permutationibus ea nunquam corrumpantur, sed mutentur
quorundam altiorum Ordinum particulæ per solam mutationem compositionis, quam
habent a diversa dispositione particularum constantium ordinis inferioris ; liceret multo
certius dividere corpora in suas species, & distinguere elementa quædam, quæ haberi
possent pro simplicibus, & inalterabilibus vi Naturæ, tum compositiones mixtorum
specificas, & essentiales ab accidentalibus proprietatibus discernere. Sed quoniam in
intimum ejusmodi textum penetrare nondum licet ; eas proprietates debemus diligenter
notare, quæ ab illo intimo textu proveniunt, & nostris sensibus sunt perviæ, quæ quidem
omnes consistunt in viribus, motu, & mutatione dispositionis massularum grandiuscularum,
quæ sensibus se nostris objiciunt, & constanter habitas, vel facile, & brevi recuperatas
distinguere a transitoriis, vel facile, & constanter amissas, & ex illarum aggregato distinguere
species, hasce vero habere pro accidentalibus. -
522. Verum quod ad omne hoc argumentum pertinet, non erit abs re, si postremo
loco huc transferam ex Stayana Recentiore Philosophia, ac meis in eam adnotationibus,
illud, quod habeo ad versum 547 libri i : “ Quamvis intrinsecam corporum naturam intueri
non liceat, non esse adjiciendum, affirmat, Naturæ investigandæ studium : posse ex externis
illis proprietatibus plures detegi in dies : ad ipsum summæ laudi esse : ideam sane, quam
habemus confusam substantiæ eas habentis proprietates, proprietatibus ipsis auctis exten-
dimus. Rem illustrat aptissimo exemplo ejus substantiæ, 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,
& præcipitari immisso sale. Porro & aliæ supererunt plurimæ ejusmodi proprietates olim
fortasse detegendæ : quo plures detegimus eo plus ad confusam illam naturæ auri
cognitionem accedimus : a clara, atque intima ipsius naturæ contemplatione adhuc absumus.
Idem, quod in hoc vidimus peculiari corpore, de corporis in genere natura affirmat.
Investigandas proprietates, quibus detectis illum intimum proprietatum fontem attingi
nunquam posse : nil nisi inania proferri vocabula, ubi intimæ proprietates investigantur.”
523. Hæc ego quidem ex illo : tum meam hanc ipsam Theoriam respiciens, quam
& ipse libro IO exposuit nondum edito, sic persequor : “ Quid autem, si partim observatione
partim ratiocinatione adhibita, constaret demum, materiam homogeneam esse, ac omne
discrimen inter corpora prove-[247]-nire a forma, nexu, viribus, & motibus particularum,
quæ sint intima origo sensibilium omnium proprietatum. Ea nostros sensus non alia
effugiunt ratione, nisi ob nimis exiguam particularum molem : nec nostræ 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.”
524. Demum ibidem illud addo, quod pertinet ad genera, & species : “ Interea specificas
naturas æstimamus, & distinguimus a collectione illa externarum proprietatum, in quo
plurimum confert ordo, quo deteguntur. Si quædam collectio, quæ sola innotuerat,
inveniatur simul cum nova quadam proprietate conjuncta, in aliis fere æquali numero
cum alia diversa ; eam, 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 illa nova : tum, nova illa jam in naturæ ideam admissa, hanc substantiam
ea carentem ab ejusmodi natura arcemus, nec ipsi id nomen tribuimus. Si nunc inveniretur
massa, quæ 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,
& distinguish particles from one another & separate them into classes, step by step of higher
orders, from elementary points up to our own bodies; then, perhaps, we should find some
classes of particles to be so tenacious of their form that in all changes they would never
be broken down; but the particles of higher orders would be changed by mere-change
of the composition that they have owing to a different disposition of the particles of a lower
order from which they are formed. It would then be possible to divide with far greater
certainty bodies into their species, & to distinguish certain elements which could be taken
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
place, I here quote from Stay's Recentior Philosophia, & my notes thereon, that which I
have written on verse 547 of Book I. “Although we cannot peer into the intrinsic nature
of bodies, the endeavour to investigate Nature, he states, must not be abandoned. Many
things can be detected daily from those external properties. This is worthy of all praise;
we truly extend the idea, which we have in a confused form of a substance possessing these
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
he also explained in Book Io, not yet published, I went on thus –“But what if, partly
by observation & partly by using deduction, it should finally be established that matter
is homogeneous, & that all distinction between bodies comes from form, connection, forces,
& motions of the particles, such as may be the fundamental origin of all sensible properties :
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 Io, 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.”
524. Lastly, I add this in the same connection, relating to classes & species —“Amongst
other things, we estimate specific natures, & distinguish them from the collection of external
properties; & in this the order in which they are detected is of special assistance. If any
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 regia, we should say that it was not gold. If at the beginning it was
- B B
What is required
to enable us to look
into the innermost
constitution, in
order that from it
we might be able
from first principles
to reduce matter to
classes & species;
what is to be done,
since such a thing is
impossible.
It is thus to be seen
that we can never
arrive at a full
knowledge of the
innermost & essen-
tial substance, or
the distinction be-
tween species.
What may, how-
ever, be accom-
plished with regard
to general properties
& general prin-
ciples, has been
done in this work.
The manner,
a mongst other
things, in which we
sh all distinguish
between species.
37o PHILOSOPHIÆ NATURALIS THEORIA
eam non esse aurum diceremus. Si initio compertum esset, alias ejusmodi massas solvi,
alias non solvi per aquam regiam, sed per alium liquorem, & utrumque in æquali fere earum
massarum numero notatum esset, putatum fuisset, binas esse auri species, quarum altera
alterius liquoris ope solveretur.”
Hæc ego ibi ; unde adhuc magis patet, quid specificæ formæ sint, & inde, quid sit
transformatio. Sed de his omnibus jam satis.
A THEORY OF NATURAL PHILOSOPHY 37I
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.
Argumentum hujus
Appendicis, & cur
sit addita.
Discrimen inter ani-
mam & corpus: in
hoc omnia peragi
per distantias lo-
cales, motus, ac
vires inducentes
motum localem.
In anima IlOS
experiri sensationes,
& cogitationes, ac
volitiones: Vim
esse in nobis inna-
tam, qua videamus
harum discrimina, &
relationem quam
habent ad substan-
tias, a quibus
procedunt essential-
iter diversas.
Duo genera actuum
vitalium, quae in
nobis perspicimus,
sensationes, & cogi-
tationes ac voliti-
Ones, quas possumus
etiam sine corpore
exercere,
[248] A P P E N D I X
AD METAPHYSICAM PERTINENS
DE ANIMA ET DEO
525. Quæ pertinent ad discrimen animæ a materia, & ad modum, quo anima in corpus
agit, rejecta Leibnitianorum harmonia præstabilita, persecutus jam sum in parte prima
a num. I 53. Hic primum & id ipsum discrimen evolvam magis, & addam de ipsius animæ,
& ejus actuum vi, ac natura, nonnulla, quæ cum eodem operis argumento arctissime con-
nectuntur : tum ad eum colligendum, qui semper maximus esse debet omnium philosophic-
arum meditationum fructus, nimirum ad ipsum potentissimum, ac sapientissium Auctorem
Naturæ conscendam.
526. Imprimis hic iterum patet, quantum discrimen sit inter corpus, & animam, ac
inter ea, quæ corporeæ materiæ tribuimus, & quæ in nostra spirituali substantia experimur.
Ibi omnia perfecimus tantummodo per distantias locales, & motus, ac per vires, quæ nihil
aliud sunt, nisi determinationes ad motus locales, sive ad mutandas, vel conservandas locales
distantias certa lege necessaria, & a nulla materiæ ipsius libera determinatione pendentes.
Nec vero ullas ego repræsentativas vires in ipsa materia agnosco, quarum nomine haud
scio, an ii ipsi, qui utuntur, satis norint, quid intelligant, nec ullum aliud genus virium,
aut actionum ipsi tribuo, præter illud unum, quod respicit localem motum, & accessus
mutuos, ac recessus. -
527. At in ea nostra substantia, qua vivimus, nos quidem intimo sensu, & reflexione,
duplex aliud operationum genus experimur, & agnoscimus, quarum alterum dicimus
sensationem, alterum cogitationem, & volitionem. Profecto idea, quam de illis habemus
intimam, & prorsus experimentalem, est longe diversa ab idea, quam habemus, localis
distantiæ, & motus. Et quidem illud mihi, ut in prima parte innui, omnino persuasum
est, inesse animis nostris vim quandam, qua ipsas nostras ideas, & illos, non locales, sed
animasticos motus, quos in nobis ipsis inspicimus, intime cognoscamus, & non solum similes
a dissimilibus possimus discernere, 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 oririjin 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 varios : 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, quæ scripserint, & typis
vulgaverint, perlegant, qui eorum rationes voce expressas aure excipiant, & victi demum
se dedant.
528. Et vero ex motibus quibusdam localibus in nostro corpore factis per impulsum
ab externis corporibus, vel per se etiam eo modo, quo ab externis fierent, ac delatis ad
cerebrum (in eo enim alicubi videtur debere esse saltem præcipua sedes animæ, ad quam
nimirum tot nervorum fibræ pertingunt idcirco, ut impulsiones propagatæ, vel per succum
372
A P P E N D I X
RELATING TO METAPHYSICS
THE MIND AND GOD
525. What relates to the distinction between the mind & matter, & the manner in
which the mind acts on the body, I have already investigated in the First part, from Art.
153 on, after rejecting the pre-established harmony of the followers of Leibniz. Here I will
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
body & the mind, & between those things that we term corporeal matter & those which
we feel in our spiritual substance. In Art. I53, we did everything by the sole means of
local distances & motions, & by forces that are nothing else but propensities to local motions,
or propensities to change, or preserve, local distances in accordance with a certain necessary
law; & these do not depend on any free determination of the matter, itself. But I do not
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
sense & thought, another twofold class of operations; one of which we call sensation, &
the other thought or will. Without any doubt, the idea which we have within us, which
is altogether the result of experience, of the former, is far different to that which we have
of local distance & motion. Indeed I am quite of the opinion, as I remarked in the First
Part, that there is in our minds a certain force, by means of which we obtain full cognition
of our very ideas & those non-local, but mental, motions that we observe in our own selves;
& we can distinguish between like & unlike, as we assuredly do, when after the idea
of a horse that has been seen there presents itself the idea of a fish, & we say that this is
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
The theme of this
appendix; & the
reason for adding .
it.
Distinction be -
tween the mind &
the body : in this
everything is ac-
complished by local
distances & mo-
tions, & forces indu-
cing local motions.
In the mind we
feel sensations,
thoughts & pur-
pose ; force is in- .
nate within us,
by which we see the
differences between
these things, & the
relation that they
bear to essentially
different sub-
stances, from which
they proceed.
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
bodies, or even self-produced by the manner in which they come from without, & these
are carried to the brain. For in the brain, somewhere, it seems that the seat of the mind
must be situated; & that is why so many nerve-fibres extend to it, so that the impulses
can be carried to it, propagated either by a volatile juice or by rigid fibres in all directions,
Two kinds of vital
acts which we per-
ceive in ourselves,
sensation S., &
thought or will,
which we can exer-
cise even without
the body
373
374 PHILOSOPHIÆ NATURALIS THEORIA
Si ea brutis con-
veniant, qua n to
imperfectiora in iis
esse debeant, & quid
de voce spíritus
Discrimen inter
motus, a quibus
id ea excitatür, &
id ea m ipsam :
quatuor acceptiones
vocis color.
Commercium a ni-
mae cum corpore
continere tria legum
genera : quae sint
priora duo.
In altero ex iis
nexus inter animam,
& corpus necessar-
ius, in altero liber :
exponuntur ambo.
volatilem, vel per rigidas fibras quaquaversus deferri possint, & inde imperium in universum
exerceri corpus) exurgunt motus quidam non locales in animo, nec vero liberi, & ideæ
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. Præter
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, præter 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
illa nec vivere possint, nec agere ; dum nostras animas etiam a corpore separatas credimus
posse eosdem æque cogitationis, & volitionis actus exercere.
[25o] 529. Porro ex his, qui cogitationem, & voluntatem brutis attribuunt, alii
utrique generi applicant nomen spiritus, sed distinguunt diversa spirituum genera, alii
vocem spiritualis substantiæ tribuunt illis solis, quæ cogitare, & velle possint etiam sine
ullo nexu cum corpore & sine ulla materiæ organica dispositione, & motu, qui necessarius
est brutis, ut vivant. Atque id quidem admodum facile revocari potest ad litem de nomine,
& ad ideam, quæ affigatur huic voci spiritus, vel spiritualis, cujus vocis latina vis originaria
non nisi tenuem flatum significat : nec magna erit in vocum usurpatione difficultas ;
dummodo bene distinguantur a se invicem materia expers omni & sentiendi, & cogitandi,
ac volendi vi, a viventibus sensu præditis ; & 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 dissoluto per organici corporis corruptionem nexu cum ipso corpore,
prorsus dispereant.
530. Ceterum longe aliud profecto est & tenuitas lamellæ, quæ determinat hunc
potius, quam illum coloratum radium ad reflexionem, ut ad oculos nostros deveniat, in
quo sensu adhibet coloris nomen vulgus, & opifices ; & dispositio punctorum componentium
particulam luminis, quæ certum ipsi conciliat refrangibilitatis 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, saltem satis arcto analogiæ, & omnimodæ similitudinis genere, est idea illa,
quæ 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.
53 I. Porro commercium illud inter animam, & corpus, quod unionem appellamus,
tria habet inter se diversa legum genera, quarum bina sunt prorsus diversa ab ea etiam,
quæ habetur inter materiæ puncta, tertium in aliquo genere convenit cum ipsa, sed ita
longe in aliis plurimis ab ea distat, ut a materiali mechanismo penitus remotum sit. Priores
sunt in ordine ad motus locales organici nostri corporis, vel potius ejus partis, sive ea sit
fluidum quoddam tenuissimum, sive sint solidæ fibræ ; & ad motus non locales, sed
animasticös nostri a-[251]-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 animæ positionem ad ipsas : ubi enim læsione quadam satis magna
organici corporis ea mutua positio partium turbatur, ejusmodi legum observantia cessat :
nec vero ea locum habere potest, si anima procul distet a corpore extra ipsum sita.
532. Sunt autem ejusmodi legum duo genera : alterum genus est illud, cujus nexus
est necessarius, alterum, cujus nexus est liber : habemus enim & liberos, & necessarios
motus, & sæpe fit, ut aliquis apoplexia ictus amittat omnem, saltem respectu aliquorum
membrorum, facultatem liberi motus ; at necessarios, non eos tantum, qui ad nutritionem
pertinent, & a sola machina pendent, sed & eos, quibus excitantur sensationes, retineat.
APPENDIX \ 37;
& 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
to either class the term “spirit,” but distinguish between two different kinds of spirits;
others attribute the name of spiritual substance to those only that can think & will without
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,
which determines one coloured ray of light rather than another to be reflected, so that it
comes to the eyes, in which sense ordinary people & craftsmen use the term colour; &
the disposition of the points forming a particle of light, to which corresponds a definite
degree of refrangibility, & in certain circumstances a definite interval between the fits of
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. -
531. Now, the intercourse between the mind & the body, which we term union, has
three kinds of laws different from one another ; & of these, two are also quite different
also from that which obtains between points of matter; while the third in some sort agrees
with it, but is so far different from it in very many other ways that it is altogether remote
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.
532. Moreover, of such laws there are two kinds; the one kind is that in which the
connection is necessary, while in the other the connection is free. For, we have both
necessary & free motions; & it often happens that one who is stricken with apoplexy loses
all power of free motions, at least with respect to some of his limbs; while he retains the
necessary motions, not only those which relate to nutrition, & depend solely upon a mechanism,
If these powers are
to be credited to
the brutes, they
must be much more
imperfect in them ;
the term “spirit.”
Distinction b e-
tween the motion
by which an idea
is excited & the idea
itself; four accepta-
tions of the term
colour.
The intercourse of
the mind with the
body contains
three kinds of laws ;
the nature of the
first two.
In one of these, the
connection between
the mind & the
body is of a
necessary nature,
in the other it is
free ; explanation
of each of them.
376 PHILOSOPHIÆ NATURALIS THEORIA
Tertium genus in
quo conveniat cum
nexu mutuo inter
puncta materiæ, &
in quo ab eo pluri-
mum differat.
Ubi sit sedes animæ,
ex puris phænomen-
is.sciri non posse.
Demonstratur id
ipsum producendo,
quid oporteret nosse
ad resolv e m da m
ejusmodi quæstio-
nem ex phænomen-
1S.
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, quæ est primum instrumentum liberorum
motuum, nulli autem sint motus locales partis ullius nostri corporis, nullæ ideæ nostræ
mentis, quæ animum certa lege determinent ad hunc potius, quam illum voluntatis liberum
actum ; licet fieri possit, ut certa lege ad id inclinent, & actus alios aliis faciliores reddant,
manente tamen semper in animo, in ipsa illa ejus facultate, quam dicimus voluntatem,
potestate liberrima eligendi illud etiam, contra quod inclinatur, & efficiendi, ut ex mera
sua determinatione præponderet 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, quæ leges,
quam multæ sint, quam variæ, & an singula genera ad unicam aliquam satis generalem
reduci possint, id vero nobis quidem saltem huc usque est penitus inaccessum.
533. Tertium legum genus convenit cum' lege mutua punctorum in hoc genere, quod
ad motum localem pertinet animæ ipsius, ac certam ejus positionem ad corpus, & ad certam
organorum dispositionem. Durante nimirum dispositione, a qua pendet vita, anima
necessario debet mutare locum, dum locum mutat corpus, atque id ipsum quodam necessario
nexu, non libero : si enim præceps gravitate sua corpus ruit, si ab alio repente impellitur,
si vehitur navi, si ex ipsius ani-[252]-mæ voluntate progreditur, moveri utique cum ipso
debet necessario & anima, ac illam eandem respectivam sedem tenere, & corpus comitari
ubique. Dissoluto autem eo nexu organicorum instrumentorum, abit, illico, & a corpore,
jam suis inepto usibus, discedit. At in eo hæc virium lex localem motum animæ respiciens
plurimum differt a viribus materiæ, quod nec in infinitum protenditur, sed ad certam
quandam satis exiguam distantiam, nec illam habet tantam reciprocationem determinationis
ad accessum, & recessum cum tot illis limitibus, vel saltem nullum earum rerum habemus
indicium. Fortasse nec in minimis distantiis a quovis materiæ puncto determinationem
ullam habet ad recessum, cum potius ipsa compenetrari cum materia posse videatur : nam
ex phænomenis nec illud certo colligi posse arbitror, an cum ullo materiæ puncto com-
penetretur. Deinde nec hujusmodi vires habet perennes, & immutabiles, pereunt enim
destructa organizatione corporis, nec eas habet, cum suis similibus, nimirum cum aliis
animabus, cum quibus idcirco nec impenetrabilitatem habet, nec illos, nexus cohæsionum,
ex quibus materiæ sensibilitas oritur. Atque ex iis tam multis discriminibus, & tam
insignibus, satis luculenter patet, quam longe hæc etiam lex pertinens ad unionem animæ
cum corpore a materiali mechanismo distet, & penitus remota sit.
534. Ubi sit animæ sedes, ex puris phænomenis certo no$$e omnin9 non possumus : an
nimirum ea sit præsens certo cuidam punctorum numero, & toti spatio intermedio habens
virtualem illam extensionem, quam num. 84 in primis materiæ elementis rejecimus, an
compenetretur cum uno aliquo puncto materiæ, cui unita secum ferat.& necessarios illos,
& liberos nexus, ut vel illud punctum cum aliis etiam legibus agat in alia puncta quædam,
vel ut, enatis certis quibusdam in eo motibus, cætera fiant per virium legem toti materiæ
communem ; an ipsa existat in unico puncto spatii, quod a nullo materiæ punctQ occupetur,
& inde nexum habeat cum certis punctis, respectu quorum habeat omnes illas motuum
localium, & animasticorum leges, quas diximus ; id sane ex puris Naturæ phænomenis, &
vero etiam, ut arbitror, ex reflexione, & meditatione quavis, quæ fiat circa ipsa phænomena,
nunquam nobis innotescet. -
535. Nam ad id determinandum ex phænomenis utcunque consideratis, oporteret
nosse, an ea phænomena possint haberi eadem quovis ex iis modis, an potius reguiratur
aliquis ex iis determinatus ut conjunctio, localis etiam, animæ 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
pertains to local motion of the mind itself, to a definite position which it has with regard
to the body, & to the definite arrangement of the organs. Thus, while the arrangement
persists, upon which life depends, the mind must of necessity change its position, as the
body changes its position, & that on account of some connection of the necessary type,
& not a free connection. For, if the body rushes headlong through its own gravity, or
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
position of the seat of the mind. That is to say, we cannot ascertain whether it is present
in any definite number of points, & has such a virtual extension through the whole of the
intermediate space, as, in Art. 84, we rejected in the case of the primary elements of
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
to these phenomena.
535. For, in order to determine it from any consideration of phenomena in any way,
it would be necessary to know whether these phenomena could happen in any of these
ways, or rather some particular one of them is required, determined as a conjunction, also
The points in
which the third
kind of law agrees
with the mutual
connection b e-
tween points of
matter : and those
in which it is most
different from it.
It is not possible
from phenomena.
alone to determine
the position of the
seat of the mind.
This is proved by
setting forth what
would have to be
known in order to
obtain a solution of
this problem from
phenomena.
378 PHILOSOPHIÆ NATURALIS THEORIA
Ealsitas plurium
opinionum de ejus
sede : non probari,
eam non extendi
per totum corpus.
Conclusio pro ignor-
atione : ubi & quo-
modo possit esse.
Nunquam produciab
anima motum, nisi
æqualem in partes
oppositas : quid inde
consequatur.
Transitus ad Auc-
t o r e m N a, turae,
cujus perfectiones in
hac Theoria elucent
maxime.
vel etiam cum toto corpore. Ad id autem cognoscendum oporteret distinctam habere
notitiam earum legum, quas secum trahit conjunctio animæ cum corpore, & totius
dispositionis punctorum omnium, quæ corpus constituunt, ac legis virium mutuarum
inter materiæ puncta, tum etiam ha-[253]-bere tantam Geometriæ vim, quanta opus est
ad determinandos omnes motus, qui ex sola mechanica distributione eorundem punctorum
oriri possint. Iis omnibus opus esset ad videndum, an ex motibus, quos anima imperio
suæ voluntatis, vel necessitate suæ naturæ induceret in unicum punctum, vel in aliqua
determinata puncta, consequi deinde possent per solam legem virium communem punctis
materiæ 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 illa omnia nobis incognita sunt,
nec ad illud adeo sublime Geometriæ genus adspirare nobis licet, qui nondum penitus
determinare potuimus motus omnes trium etiam massularum, quæ certis viribus in se
InV1cem agant. -
536. Fuerunt, qui animam concluserint intra certam aliquam exiguam corporis nostri
particulam, ut Cartesius intra glandulam pinealem : at deinde compertum est, ea parte
Sola non contineri: nam ea parte dempta, vita superfuit : sic sine pineali glandula aliquando
vitam perdurasse, compertum jam est, ut animalia aliqua etiam sine cerebro vitam produx-
erunt. Alii diffusionem animæ 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 præsens :
inde inferri posse arbitrantur, quotiescumque digitorum sentimus dolorem, illam sentiri
sine præsentia animæ in digitis. At ea ratio nihil evincit : fieri enim posset, ut ad habendum
prima vice sensum, quem in digitorum dolore experimur, requireretur præsentia animæ
in ipsis digitis, sine qua ejus doloris idea primo excitari non possit, possit autem efformata
semel per ejusmodi præsentiam excitari iterum sine ipsa per eos motus nervorum, qui
cum motu fibrarum digiti in primo illo sensu conjuncti fuerant : præterquam quod adhuc
remanet definiendum illud, an ad nutritionem requiratur præsentis animæ impulsus aliquis,
an ea per solum mechanismum obtineri possit tota sine ulla animæ operatione.
537. Hæc omnia abunde ostendunt, phænomenis rite consultis nihil satis certo definiri
posse circa animæ sedem, nec 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 materiæ elementa
non nisi in singulis spatii punctis existant singula singulis momentis temporis, anima autem
licet itidem sim-[254]-plex, adhuc tamen simul existet in punctis spatii infinitis conjungens
cum unico momento temporis seriem continuam punctorum spatii, cui toti simul erit
præsens per illam extensionem virtualem, ut & Deus per infinitam Immensitatem suam
præsens est punctis infinitis spatii (& ille quidem omnibus omnino), sive in iis materia
S1t, S1Ve S1nt vacua.
538. Et hæc quidem de sede animæ : illud autem postremo loco addendum hic censeo
de legibus omnibus constituentibus ejus conjunctionem cum corpore, quod est observa-
tionibus conforme, quod diximus num. 74, & 387, nunquam ab anima produci motum in
uno materiæ puncto, quin in alio aliquo æqualis motus in partem contrariam producatur,
unde fit, ut nec liberi, nec necessarii materiæ motus ab animabus nostris Orti perturbent
actionis, & reactionis æqualitatem, conservationem ejusdem status centri communis
gravitatis, & conservationem ejusdem quantitatis motus in Mundo in eandem plagam
computar1.
539. Hæc quidem de anima : jam quod pertinet ad ipsum Divinum Naturæ Opificem,
in hac Theoria elucet maxime & necessitas ipsum omnino admittendi, & Summa ipsius,
atque infinita Potentia, Sapientia, Providentia, quæ venerationem a nobis demississimam,
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
the body; for instance, Descartes suggested the pineal gland. But, later, it was discovered
that it could not be contained in that part alone; for, if that part were removed, life still
went on. It has been already discovered that life endured for some time without the
pineal gland, just as some animals produced life even without a brain. Others argued
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; & since 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
seat of the mind from a due consideration of phenomena; nor that its diffusion through-
out any great part of the body, or even throughout the whole body, is excluded. But if
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
be added here in the last place, concerning all the laws governing its conjunction with the
body, that which is in conformity with the observations that I made in Art. 74 & Art.
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,
there shines forth very clearly in my Theory, not only the necessity of admitting His existence
in every way, but also His excellent & infinite Power, Wisdom, & Foresight; which demand
from us the most humble veneration, along with a grateful heart, & loving affection. The
Falsity of several
opinions as to the
seat of the mind :
it cannot be proved
that it does not
extend throughout
the whole of the
body.
Conclusion that the
seat of the mind is
unknown ; where
& in what manner
it may be.
Motion can never
be produced by the
mind, unless it is
equal in opposite
directions ; what
follows from this.
Transition to the
Author of Nature,
the perfections of
Whom shine forth
very clearly in this
Theory.
38o PHILOSOPHIÆ NATURALIS THEORIA
Error tribuentium
Mundi originem
ca sui fortuito :
CaSl1m eSSe vOCem
vanam sine re.
Numerum combina-
tionum in terminis
etiam numero finitis
esse infinitum : si
rite omnia expen-
dantur.
Cujus ordinis infini-
tus sit: nimirum
altissimi, & in im-
IménSUlIm. altioris
numero momento-
rum temporis in tota
aeternitate.
& simul gratum animum, atque amorem exposcant : ac vanissima illorum somnia corruunt
penitus, qui Mundum vel casu quodam fortuito putant, vel fatali quadam necessitate
potuisse condi, vel per se ipsum existere ab æterno suis necessariis legibus consistentem.
540. Et primo quidem quod ad casum pertinet, sic ratiocinantur : finiti terminorum
numeri combinationes numero finitas habent, combinationes autem per totam infinitam
aeternitatem debent extitisse numero infinitæ ; etiamsi nomine combinationum assumamus
totam seriem pertinentem ad quotcunque millenos annos. Quamobrem in fortuita
atomorum agitatione, si omnia se æqualiter habuerint, ut in longa fortuitorum serie semper
accidit, debuit quævis ex ipsis redire infinitis vicibus, adeoque infinities major est prob-
abilitas pro reditu hujus individuæ combinationis, quam habemus, quocunque finito
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
quædam, quia causas, a quibus eorum existentia determinatur, ignoramus.
54I. Sed eo omisso, falsissimum est, numerum combinationum esse finitum in terminis
numero finitis : si omnia, quæ ad Mundi constitutionem necessaria sunt, perpendantur.
Est quidem finitus numerus combinationum, si nomine combinationis assumatur tantum-
modo ordo quidam, quo alii termini post alios jacent : hinc ultro agnosco illud : si omnes
litteræ, quæ [255] Virgilii poema componunt, versentur temere in sacco aliquo, tum
extrahantur, & ordinentur omnes litteræ, aliæ post alias, atque ejusmodi operatio continuetur
in infinitum, redituram & ipsam combinationem Virgilianam numero vicium quenvis
determinatum numerum superante. At ad Mundi constitutionem habetur inprimis
dispositio punctorum materiæ in spatio patente in longum, latum, & profundum : porro
rectæ in uno plano sunt infinitæ, plana in spatio sunt infinita, & pro quavis recta in quovis
plano infinita sunt curvarum genera, quæ cum eadem ex dato puncto directione oriantur,
in quarum singularum classibus infinities plures sunt, quæ per datum punctorum numerum
non transeant. Quare ubi seligenda sit curva, quæ transeat per omnia materiæ puncta,
jam habemus infinitum saltem ordinis tertii. Præterea, determinata ejusmodi curva,
potest variari in infinitum distantia puncti cujusvis a sibi proximo : quamobrem numerus
dispositionum possibilium pro quovis puncto materiæ adhuc ceteris manentibus est infinitus,
adeoque is numerus ex omnium mutationibus possibilibus est infinitus ordinis expositi a
numero punctorum aucto saltem 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 plano, & plana infinita in spatio.
Quare cum constitutio Mundi, & sequentium phænomenorum series pendeat ab ipsa velo-
citate, & directione motus ; numerus, qui exprimit gradum infiniti, ad quem assurgit
numerus casuum diversorum, debet multiplicari ter per numerum punctorum materiæ.
542. Est igitur numerus casuum diversorum non finitus, sed infinitus ordinis expositi
a quarta potentia numeri punctorum aucta saltem ternario, atque id etiam determinata
curva virium, quæ 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, quæ concipiatur utrinque in infinitum
producta. At huic infinito est analogum infinitum momentorum temporis in tota utraque
æternitate, cum unicam dimensionem habeat tempus. Igitur numerus combinationum
est infinitus ordinis in immensum altioris ordine infiniti momentorum temporis, adeoque
In ipso immenso
combinationum nu-
1mero in immensum
plures esse combina-
tiones inordinatas,
quam ordinatas.
non solum non omnes combinationes non debent redire infinities : sed ratio numeri earum,
quæ non redeunt, est infinita ordinis altissimi, quam nimirum exponit quarta potentia
numeri punctorum aucta saltem binario, vel, si libeat variare virium leges, saltem ternario.
Quamobrem ruit futile ejusmodi, atque inane argumentum.
543. Sed inde etiam illud eruitur, in immenso isto com-[256]-binationum numero
infinities esse plures pro quovis genere combinationes inordinatas, quæ exhibeant incertum
chaos, & massam temere volitantium punctorum, quam quæ exhibeant Mundum ordinatum,
& certis constantem perpetuis legibus. Sic ex. gr. ad efformandas particulas, quæ 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
combinations of a finite number of terms are finite in number ; but the combinations
throughout the whole of infinite eternity must have been infinite in number, even if we
assume that what is understood by the name of combinations is the whole series pertaining
to so many thousands of years. Hence, in a fortuitous agitation of the atoms, if all cases
happen equally, as is always the case in a long series of fortuitous, things, one of them is
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
The error made by
those who consider
that the Universe.
was produced by
fortuitous chance ;
‘ch a n c e ' is an
empty phrase with-
out a thing to cor-
respond to it.
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
combinations from a finite number of terms is finite, if all things that are necessary to
the constitution of the Universe are considered. The number-of combinations is indeed
finite, if by the term combination there is implied merely a certain order, in which
some of the terms follow the others. I readily acknowledge this much ; that, if all
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 in 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 in 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
by the fourth power of the number of points increased threefold at least ; & that is so, even
if there is a definite curve of forces which also can be varied in an infinity of ways. Hence
the number of relative combinations necessary to the formation of the Universe is not
finite for any given instant of time; but it is infinite, of an exceedingly high order with
respect to an infinity of the kind to which belongs the infinity of the number of points
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,
there will be, for any kind, infinitely more irregular combinations, such as represent indefinite
chaos & a mass of points flying about haphazard, than there are of those that exhibit the
regular combinations of the Universe, which follow definite & everlasting laws. For instance,
in order to form particles which continually maintain their form, there is required their
The number of
comb in a tions
amongst terms that
are even finite in
number are infinite,
if they are all right-
ly considered.
The Order of the
infinity ; it is ex-
ceeding ly high,
immensely higher
than the number of
instants of time in
the whole of eter-
nity.
In this immense
number of combin-
ations even, there
3, I e immensely
more of them that
are irregular than
there are regular.
382 PHILOSOPHIÆ NATURALIS THEORIA
INon determinari ab
homine individuum:
sed eo determinante
intra limites, ad quos
cognitio pertingit,
reliquam in d e ter-
minationem vinci
ab Ente in infinitum
libero.
Hunc ordinem non
posse dici per se
necessarium : prima
impugnatio a nullo
nexu, qui videtur
haberi inter distan-
tiam, & vim, quæ
idcirco liberum de-
terminantem requi-
runt.
Secunda a numero
punctorum finito,
qui determinantem
voluntatem poscit.
quorum numerus debet esse infinities minor, quam numerus punctorum sitorum extra
ipsos : nam intersectiones curvæ 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 æque per se possibilibus seligat unam ex ordinatis ; infinities
probabilius est, infinitate ordinis admodum elevati, obventuram inordinatam combinationum
seriem, & chaos, non ordinatam, & Mundum, quem cernimus, & admiramur. Atque ad
vincendam determinate eam infinitam improbabilitatem, requiritur infinita vis Conditoris
Supremi seligentis unam ex iis infinitis.
544. Nec vero iJlud objici potest, etiam hominem, qui statuam aliquam effingat,
finita vi eligere illam individuam formam, quam illi dat, inter infinitas, quæ haberi possunt.
Nam imprimis ille eam individuam non eligit, sed determinat modo admodum confuso
figuram quandam, & individua illa oritur ex Naturæ legibus, & Mundi constitutione illa
individua, quam naturæ Opifex Infinitus infinitam indeterminationem superans deter-
minavit, per quam ab ejus voluntatis actu oriuntur illi certi motus in ejus brachiis, & ab
hisce motus instrumentorum. Quin etiam in genere idcirco tam multi Philosophi
determinationem ad individuum, & determinationem ad omnes illos gradus, ad quos
cognitio creati determinantis non pertingit, rejecerunt in Deum infinita cognoscendi, &
discernendi vi præditum, necessaria ad determinandum unum individuum casum ex
infinitis ad idem genus pertinentibus ; cum creatæ 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 præ
aliis per se, aut per fortuitam eventualitatem prodire omnino non potest.
545. Sed nec dici potest, hunc ipsum ordinem necessarium esse, & æternum ac per
se subsistere, casu quovis sequente determinato a proxime præcedente, & a lege virium
intrinseca, & necessaria iis individuis punctis, & non aliis. Nam contra hoc ipsum miserum
sane effugium quamplurima sunt, quæ opponi possunt. Inprimis admodum difficile est,
ut homo sibi serio persuadeat, hanc unam virium legem, quam habet hoc individuum
punctum respectu hujus individui puncti, fuisse possibilem, & necessariam, ut nimirum
in hac individua [257] distantia se potius attrahant, quam repellant, & se attrahant tanta
potius attractione, quam alia. Nulla appalet sane connexio inter distantiam tantam,
& tantam talis speciei vim, ut ibi non potuerit esse alia quævis, & 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. -
546. Præterea cum & infinitum, & infinite parvum in se determinatum, & in se tale,
in creatis sit impossibile (quod de infinito in extensione demonstravi (!) pluribus in locis,
nec una tantum demonstratione, ut in dissertatione De Natura, & usu infinitorum, &5
infimite parvorum, ac in dissertatione adjecta meis Sectionum Conicarum Elementis, Element.
tom. 3); finitus est numerus punctorum materiæ, vel saltem in communi etiam sententia
finita est materiæ existentis massa, quæ finitum spatium occupare debet, & non in infinitum
/ B
ICM, sicut BCD est pars ACD, nec MCG, & HCG sunt unum, &
(t) Em unam ex ejusmodi demonstrationibus. Sit in fig. 7I spatium
a C versus AE infinitum, & in eo angulus rectilineus ACE bifariam
sectus per rectam CD. Sit autem GH parallela CA, quæ occurrat CD
in H, ac producatur ita, ut HF fiat dupla GH, ducaturque CF, & omnes
CA, CB, CD, CE in infinitum producantur. Imprimis totum $patium
infinitum ECD debet esse æquale infinito 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 & summa omnium FHhf dupla summæ omnium HGgh, nimirum
tota area infinita BCD dupla infinitæ 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 centro C ; sector
ICM erit æqualis GCM, & triangulum FCH duplum GCH. Donec
sumus in quantitatibus finitis, res bene procedit, qui a FCH non est pars
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, quæ 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
but a finite force selects that individual form which he gives to it, from among an infinite
number which are possible. For, first of all, the man does not select that individual form;
he determines in a very confused way a certain shape, & that individual form arises from
the laws of Nature, & from that individual constitution of the Universe which the
Infinite Founder of Nature, overcoming the infinite lack of determination, has determined;
through which, by an act of his will, arise those definite motions in the arms of the man,
& from these the motions of his tools. Moreover, in general, on this account, so man
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-
sustained, any one case following the one next before it & determined by it, & by a law of
forces that is intrinsic & necessary to those individual points & to no others. For against this
really worthless subterfuge there are very many arguments that can be brought forward.
First of all, it is very difficult to see how a man can seriously persuade himself that one
particular law of forces, which one particular point has with regard to another particular
point, should be possible & necessary, so that, for instance, at one particular distance the
points should attract one another rather than repel one another, & attract one another
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,
is impossible in created things; as I proved concerning the infinite in extension (t) in
several places, & with more than one proof, for instance, in the dissertation De Natura,
{5 usu infinitorum, & infinite parvorum, & in a dissertation added to my Sectionum Comicarum
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 ; & 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, & let it be produced so that HF is double GH; join CF, 85 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, since FH is double HG. In the same way, if other parallels like ghfare
drawn, h0f will be double hCo & 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, & therefore 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, 85 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 &
HCG one & 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. -
The individual is
not determined by
man ; but, when it
has been determined
within the limits to
which man's know-
ledge attains, the
rest that is unde-
termined is over-
come by a Being
who is infinitely
free,
This regularity
cannot be said to
be necessary in
itself ; first, be-
cause of the ap-
parent absence of
a ny connection
between distance
& force ; the latter
therefore requires
a free determina-
tion.
Second argument
derived from the
finite number of
points, which re-
quires a determin-
ing will.
384 PHILOSOPHIÆ NATURALIS THEORIA
Tertia ab æternitate,
per quam durassent
motus, cum linea
necessario infinita :
ejus impOSsibilitas.
Validissima ab im-
possibilitate seriei
infinitæ terminorum,
in qua alii ab aliis
determinentur ad
existendum sine ex-
trinseco determin-
ante; ea hic demon-
stratur.
In quo hoc argu-
mentum differat a
communi adhibente
impossibilitatem se-
riei contingentium
sine ente necessariQ,
protendi. Porro cur hic sit potius numerus punctorum, hæc potius massæ quantitas in
Natura, quam alia ; nulla sane ratio esse potest, nisi arbitrium entis infinita determinativa
potentia præditi, & nemo sanus sibi facile serio persuadebit, in quodam determinato numero
punctorum haberi necessitatem existentiæ potius, quam in alio quovis.
547. Accedit illud, quod si Mundus cum hisce legibus fuisset ab æterno ; extitissent
jam motus æterni, & lineæ a singulis punctis descriptæ debuissent fuisse jam in infinitum
productæ: 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, & illa impossibilitas pertinere debet ad omne genus linearum,
quæ in infinitum productæ sint. Potest utique motus continuari in infinitum per
æternitatem futuram, quia si aliquando coepit, nunquam habebitur momentum temporis,
in quo jam fuerit existentia infinitæ lineæ : secus vero, si per æternitatem præcedentem
jam extiterit : nec in eo futuram æternitatem cum præterita prorsus analogam esse
censeo, ut illud indefinitum futuræ non sit verum quoddam infinitum præteritæ. Quod
si linea infinita non fuerit, & quies est infinities adhuc improbabilior, quam regressus pro
unico temporis momento ad idem spatii punctum, ac multo magis æterna quies : utique
nec motum habuit æternum materia, nec existere potuit ab æterno, 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 æternitatis in utramque partem indefinitæ illud posset seligere individuum momentum,
in quo materiam crearet, ac ex omnibus infinitis illis possibilibus statibus, & quidem tam
sublimi infinitatis gradu, seligere illum 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.
548. Verum hisce omnibus etiam omissis, est illud a determinatione itidem necessaria
repetitum, in quavis Theoria validissimum, sed adhuc magis in mea, in qua omnia phænomena
pendent a curva virium, & inertiæ vi. Nimirum materia licet ponatur ejusmodi, ut habeat
necessariam, & sibi essentialem vim inertiæ, & virium activarum legem ; adhuc ut quovis
dato tempore posteriore habeat determinatum statum, quem habet, debet determinari
ad ipsum a statu præcedenti, qui si fuisset diversus, diversus esset & subsequens ; neque
enim lapis, qui sequenti tempore est in Tellure, ibi esset ; si immediate antecedenti fuisset
in Luna. Quare status ille, qui habetur tempore sequenti, nec a se ipso, nec a materia,
nec ab ullo ente materiali tum existente, habet determinationem ad existendum, & pro-
prietates, quas habet materia perennes, indifferentiam per se continent, nec ullam
determinationem inducunt. Determinationem igitur, quam habet ille status ad existendum,
accipit a statu præcedenti. Porro status præcedens non potest determinare sequentem,
nisi quatenus ipse determinate existit. Ipse autem nullam itidem in se habet determin-
ationem ad existendum, sed illam accipit a præcedente. [259] Ergo nihil habemus adhuc
in ipso secundum se considerato determinationis ad existendum pro postremo illo statu.
Quod de secundo diximus, dicendum de tertio præcedente, qui determinationem debet
accipere a quarto, adeoque in se nullam habet determinationem pro existentia sui, nec
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, illum Guidonis Grandi, utut summi
Geometræ, paralogismum fuisse, quo ex expressione seriei parallelæ ortæ per divisionem
I ę • • • • ę • «• ç»
I -+ I intulit summam infinitorum zero esse revera æqualem dimidio. Non potest igitur
illa series per se determinare existentiam cujuscunque certi sui termini, adeoque nec tota
ipsa potest determinate existere, nisi ab ente extra ipsam posito determinetur.
*\
549. Hoc quidem argumento jam ab annis multis uti soleo, quod & cum aliis pluribus
communicavi, neque id ab usitato argumento, quo rejicitur series contingentium infinita
sine ente extrinseco dante existentiam seriei toti, in alio differt, 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 in any other.
547. In addition, if the Universe had gone on with these laws from eternity, then
already there would have been eternal motions, & straight lines described by the several
points would already have been produced to infinity. For they do not re-enter themselves,
except by the will of a Being who overcomes the infinite improbability; since I have
proved above in several places that it is infinitely more improbable that any point should
return at some time to the same place as it had occupied at some other instant, than that
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
in any Theory, derived also from a necessity for determination; but especially strong in
my Theory, where all phenomena depend on a curve of forces, & the force of inertia. Thus,
although matter may be assumed to be of such a nature as to have a necessary & essential
force of inertia & a law of active forces; yet, in order that at any subsequent time it may
have the determinate state, which it actually has, it must be determined to that state, from
the state just preceding ; & if this preceding state had been different, the subsequent state
would also have been different. For a stone, which at a subsequent instant is on the Earth,
would not have been there at the instant, if at the instant immediately preceding it had been,
on the Moon. Hence the state which occurs at the subsequent instant, neither of itself,
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 ififinite series of states, in each of which we have absolutely nothing Ior 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 Guido Grandi, although a very eminent geometer, enunciated a fallacy when, from
an expression of a parallel series derived by division of I by (1 + 1), he deduced that the sum
of an infinite number of zeros was really equal to #. Therefore, that series of states
cannot determine the existence of any particular one of its terms, & so neither can the whole
of it exist determinately, unless it be determined by a Being situated without itself.
549. I have employed this argument for many years past, & I have communicated it
to several others; it does not differ from the usual argument employed, which denies
the possibility of an infinite series of contingents without an outside Being giving existence
C C
A third argument
derived from eter-
nity, during which
motions have last-
ed, because a line is
necessarily infinite ;
the impossibility of
this.
A very strong
argument derived
from the impossi-
bility of an infinite
series of terms, in
which the deter-
mination to exist
of one comes from
that of another
without something
determining exist-
ence from without ;
proof of the impos-
sibility here given.
I n w h a t t his
argument differs
from the usual one,
depending upon the
impossibility of a
series of events
without a necessary
being.
386 PHILOSOPHIÆ NATURALIS THEORIA
Attributa, quæ Ens
extrinsecum habere
debet.
Infinita improba-
bilitas, quæ hic
occurrit, a quo uno
vinci possit: nimi-
rum a solo infinite
libero.
Quanta sapientia
opus fuerit a d
seligendum numer-
u m, & ordine m
punctorum, & legem
virium.
Congeries eorum,
quæ evincunt in eli-
gente potentiam,
sapientiam, provi-
dentiam, immensas,
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 præstiti, ne eludatur argumentum
dicendo, in tota serie haberi determinationem ad ipsam totam, cum pro quovis termino
habeatur determinatio intra eandem seriem, nimirum in termino præcedente. Illa
reductione ad vim determinativam existentiæ postremi quæsitam per omnem seriem,
devenitur ad seriem nihilorum respectu ipsius quorum summa adhuc est nihilum.
550. Jam vero hoc ens extrinsecum seriei ipsi, quod hanc seriem elegit præ seriebus
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 cæco agendi modo,
fatalismo, & necessitate, sit infinitus, qui idcirco certitudinem inducit.
55 I. Atque hic notandum & illud, pro quovis indivi-[26o]-duo statu respondente
cuivis momento temporis, & multo magis pro quavis individua serie respondente cuivis
continuo tempori, improbabilitas determinatæ ipsius existentiæ est infinita, & nos
deberemus esse certi de ejus non existentia, nisi determinaretur ab infinito determinantes
& nisi ejus determinationis notitiam nos haberemus. Sic si in urna sint nomina centum,
& unum, & agatur de uno determinato, an extractum inde prodierit, centuplo major est
improbabilitas ipsi contraria : si mille, & unum, millecupla : si numerus sit infinitus ;
improbabilitas erit infinita, quæ in certitudinem transit : sed si quis viderit extrac-
tionem, & nobis nunciet ; tota improbabilitas illa repente corruit. Verum & in hoc
exemplo individua illa determinatio a creato agente non habebitur inter infinitas possi-
biles, nisi ex legibus ab infinito determinante jam determinatis in Natura, & ab ejusdem
determinatione ad individuum, uti paullo ante dicebamus de individuæ figuræ electione
pro statua. - *
552. Porro qui aliquanto diligentius perpenderit vel illa pauca, quæ adnotavimus
necessaria in distributione punctorum ad efformanda diversa particularum genera, quæ
exhibeant diversa corpora, videbit sane quanta sapientia, & potentia sit opus ad ea omnia
perspicienda, eligenda, præstanda. Quid vero, ubi cogitet, quanta altissimorum Proble-
matum indeterminatio occurrat in infinito illo combinationum possibilium numero, &
quanta cognitione opus fuerit ad eligendas illas potissimum, quæ necessariæ erant ad hanc
usque adeo inter se connexorum phænomenorum seriem exhibendam ? Cogitet, quid
una lux præstare 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, quæ illos determinatos colores exhiberent sine
ceterarum & alterationum, & transformationum jactura, disponendæ oculorum partes,
ut imago pingeretur in fundo, & propagaretur ad cerebrum, ac simul nutritioni daretur
locus, & alia ejusmodi præstanda 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, & eam irrigat, ac foecundat, sua mole
ædificia 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 æternitatem rediturum.
553. Sed quid ego hæc singularia persequor ? quanta Geometria opus fuit ad eas com-
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
to all others of the infinite number in the same class, must have infinite determinative &
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 series 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
given instant of time, & much more for any particular series corresponding to a given
continuous time, the improbability of a self-determined existence is infinite ; & we ought
to be certain of its non-existence, unless it were determined by an infinite determinator, &
we had evidence of the determination. Thus, if in an urn there are a hundred & one names,
& it is a question with regard to one determined name, whether it has been drawn from
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
mentioned as necessary in the arrangement of the points for the formation of the different
kinds of particles, which different bodies exhibit, he must perceive how great the wisdom &
power must needs be, to comprehend, select & establish all these things. What then,
when he considers how great an indeterminateness in problems of very high degree occurs
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
Earth, & impel him with ever so slight a force, or even but blow upon him with his breath,
& drive him from intercourse with all humanity away to infinity, nevermore to return
throughout all eternity.
553. But why do I enumerate these separate things : Consider how much geometry
The necessary
attributes of the
external Being.
The sort of Being
who could over-
come the infinite
improbability
which here occurs :
it could be accom-
plished only by
One Who is infi-
nitely free.
How great the
w is do m would
have to be to select
the number &
arrangement of the
points, & the law of
forces.
Groups of points,
which p r ove
conclusively theim-
measurability of the
power, wisdom and
foresight employed
in selection.
388 PHILSOPHIÆ NATURALIS THEORIA
Quid prospiciendum
fuerit pro nostra,
existentia, & nostris
commodis: quantum
ipsi inde simus ob-
stricti.
Mundum non esse
possibilium optim-
um, cum in possibi-
libus nullus terminus
sit u ltimus: nec
officere sapientiæ,
ac bonitati infinitæ,
quod non fecerit, nec
potentiæ, quod non
potuerit id facere.
Quam multa pessima
consectaria se cu m
trahat sent e ntia
Mundi perfectissimi.
binationes inveniendas, quæ tot organica nobis corpora exhiberent, tot arbores, & flores
educerent, tot brutis animantibus, & hominibus tam multa vitæ instrumenta submini-
starent ? Pro fronde unica efformanda quanta cognitione opus fuit, & providentia, ut
motus omnes per tot sæcula perdurantes, & cum omnibus aliis motibus tam arcte connexi
illas individuas materiæ particulas eo adducerent, ut illam demum, illo determinato tempore
frondem illius determinatæ curvaturæ producerent ? quid autem hoc ipsum respéctu
eorum, ad quæ nulli nostri sensus pervadunt, quæ longissime supra telescopiorum, & infra
microscopiorum potestatem latent ? Quid `respectu eorum, quæ 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 humanæ
mentis captum omnem longissime superantes, ignorare potest, qui penitus mente cæcutit,
yel 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. -
554. Sed in hac tanta eligentis, ac omnia providentis Supremi Conditoris sapientia,
atque exsequentis potentia, quam admirari debemus perpetuo, & venerari, illud adhuc
magis cogitandum est nobis, quantum inde in nostros etiam usus promanarit, quos utique
respexit ille, qui videt omnia, & fines sibi istos omnes constituit, qui per ea omnia & nostræ
ipsi existentiæ viam stravit, ac nos præ infinitis aliis hominibus, qui existere utique poterant,
elegit ab ipso Mundi exordio, motus omnes, ad horum, quibus utimur, organorum forma-
tionem disposuit, præter ea tam multa quæ 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 Naturæ, sed omnes eos
animo sibi constitutos habuisse fines, ad quos conducunt media, quæ videmus adhibita.
555. Haud ego quidem Leibnitianis, & aliis quibuscunque [262] Optimismi defen-
soribus assentior, qui Mundum hunc, in quo vivimus, & cujus pars sumus, omnium
perfectissimum esse arbitrantur, ac Deum faciunt natura sua determinatum ad id creandum
quod perfectissimum sit, ac eo ordine, qui perfectissimus sit. Id sane nec fieri posse arbitror :
cum nimirum in quovis possibilium genere seriem agnoscam finitorum tantummodo,
quanquam in infinitum productam, ut num. 90 exposui, in qua, ut in distantiis duorum
punctorum nulla est minima, nulla maxima ; ita ibidem nulla sit perfectionis maximæ,
nulla minimæ, sed quavis finita perfectione utcunque magna, vel parva, sit alia perfectio
major, vel minor : unde fit, ut quancunque seligat Naturæ Auctor, necessario debeat
alias majores omittere : nec vero ejus potentiæ illud officit, quod creare non possit optimum,
aut maximum, ut nec 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 : nec officit aut sapientiæ, aut bonitati
infinitæ, quod optimum non seligat, ubi optimum est nullum.
556. Ex alia parte determinatio illa ad optimum, & libertatem Divinam tollit, &
contingentiam rerum omnium, cum, quæ existunt, necessaria fiant, quæ non existunt,
evadant impossibilia ; ac præterea nobis quodammodo in illa hypothesi debemus, quod
existimus, non illi. Qui enim potuit non existere id, quod habuit pro sua existentia
rationem prævalentem, quam Naturæ Auctor cum viderit, non potuerit non sequi, nec
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, & essentiali
merito prævalenti ? Distinguendum est inter hæc duo : unum esse alio melius, & esse
melius creare potius unum, quam aliud. Illud primum habetur ubique, hoc secundum
nusquam, sed æque 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 for 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
Supreme Founder, & the power of carrying it out, there is still another thing for us to
consider ; namely, how much proceeds from it to meet the needs of us, who are all under
the care of Him Who sees all things, & has imposed on Himself the accomplishment of all
those purposes; Who has smoothed the path of our existence with them all, & from the
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 necessary 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 firmest 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
of Optimism, who consider that this Universe, in which we live & of which we are part,
is the most perfect of all; & who thus make God determined by His own nature for the
creation of that which is the most perfect, & in that order which is the most perfect. In
truth, I think that such a thing would be impossible; for, I recognize, in any kind of
possibles, a series of finites only, although prolonged to infinity, as I explained in Art. 90 ;
& in this series, just as in the case of the distances between two points, there is no greatest
or least, here also there is no case of greatest or of least perfection; but, for any finite
perfection, however great or small, there is another perfection that is greater or smaller.
Hence it comes about that, whatever the Author of Nature should select, He would have
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
freedom of God, & the contingency of all things ; for, those things which exist become
necessary, & those that do not are impossible. Besides, on that hypothesis, we should be
under some sort of obligation to ourselves, & not to Him, for the fact that we exist. For
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 —(1) 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
How our existence
and our conveni-
ences would have
to be provided
for ; what a debt
we are therefore
under to Him.
The Universe is not
‘the best of all
possibles ': for
amongst possibles
there is no last
term : nor is it an
argument against
infinite wisdom & .
goodness, because
He did not make it
So; nor against His
power, because
He was unable to
make it so.
The number of
gross imperfections
involved in the idea
of a most perfect
|Universe.
390 PHILOSOPHIÆ NATURALIS THEORIA
Media tamen idonea
necessario eligi ab
ipso Auctore Natu-
ræ ad fines sibi
propositos : quan-
tum illi debeamus.
Deduci nos inde ad
revelationem, quæ
ta men hu c non
pertineat, ad opus
nimirum pure philo-
$ophicum.
Divinæ libertatis exercitium infinities perfectius est quavis perfectione creata, quæ idcirco
nullum potest offerre Divinæ libertati meritum determinativum ad se creandum.
557. Cum ea infinita libertate Divina componitur tamen illud, quod ad sapientiam
pertinet, ut ad eos fines, quos sibi pro liberrimo suo arbitrio præfixit Deus, media semper
apta debeat seligere, quæ finem propositum frustrari non sinant. Porro hæc media etiam
in nostrum bonum selegit plurima, dum totam Naturam conderet, quod quem a nobis
exigat beneficiorum memorem, & gratum animum, quem etiam tan-[263]-tæ beneficentiae
respondentem amorem cum ingenti illa admiratione, & veneratione conjunctum, nemo
non videt.
558. Superest & illud innuendum, neminem sanæ mentis hominem dubitare posse,
quin, qui tantam in ordinanda Natura providentiam ostendit, tantam erga nos in nobis
seligendis, in consulendo nostris & indigentiis, & commodis beneficentiam, illud etiam
præstare voluerit, ut cum adeo imbecilla sit, & hebes mens nostra, & ad ipsius cognitionem
per sese vix quidquam possit, se ipse nobis per aliquam revelationem voluerit multo uberius
præbere cognoscendum, colendum, amandum ; quo ubi devenerimus, quæ inter tam
multas falso jactatas absurdissimas revelationes unica vera sit perspiciemus utique admodum
facile. Sed ea jam Philosophiæ 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,
God, to those purposes which he of His own unfettered will had designed, was always
bound to select suitable means, such as would not allow these purposes to be frustrated.
Further, He has selected these means for the most part suitable for our welfare, whilst he
founded the whole of Nature; & this demands from us a remembrance of His favours & a
thankful heart, nay, even a love that shall correspond to such great beneficence together
with a mighty wonder & admiration, as every one will see.
558. It now remains but to mention that there is no man of sound mind who could
possibly doubt that One, Who has shown such great foresight in the arrangement of Nature,
such great beneficence towards us in selecting us, & in looking after both our needs & our
comforts, would not also wish to accomplish this also ; namely that, since our mind is so
weak & dull that it can scarcely of itself arrive at any sort of knowledge about Him, He
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.
Fit means, however,
had necessarily to
be selected by the
Author of Nature
Himself to carry
out the purposes
He has designed
for Himself ; how
much We Owe to
Him.
We are thus led to
revelation, which
however does not
come within the
scope of such a
work as this, which
is purely philo-
Sophical.
Argumentum : quæ
spatii attributa.
Necessario ab omni-
bus admitti debere
reales modos exist-
endi locales & tem-
porarios.
Quocunque is modus
nomine appelletur.
Modi reales, qui sint
reale s p at iu m, &
tempus.
Eorum n a tura, &
relationes.
Contiguitas puncto-
rum spatii impossi-
|bilis.
[264] SUPPLEMENTA
§ I
De Spatio, ac Tempore (e)
I. Ego materiæ extensionem prorsus continuam non admitto, sed eam constituo
punctis prorsus indivisibilibus, & inextensis a se invicem disjunctis aliquo intervallo, &
connexis per vires quasdam jam attractivas, jam repulsivas pendentes a mutuis ipsorum
distantiis. Videndum hic, quid mihi sit in hac sententia spatium, ac tempus, quomodo
utrumque dici possit continuum, divisibile in infinitum, æternum, immensum, immobile,
necessarium, licet neutrum, ut in ipsa adnotatione Ostendi, suam habeat naturam realem
ejusmodi proprietatibus præditam. -
2. Inprimis illud mihi videtur evidens, tam eos, qui spatium admittunt absolutum,
natura sua reali, continuum, æternum, immensum, tam eos, qui cum Leibnitianis, &
Cartesianis ponunt spatium ipsum in ordine, quem habent inter se res, quæ existunt, præter
ipsas res, quæ existunt, debere admittere modum aliquem non pure imaginarium, sed realem
existendi, per quem ibi sint, ubi sunt, & qui existat tum, 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, quæ ordinem illum habent, quotiescunque res illæ existerent, eodem semper
existerent ordine illo, nec proinde unquam locum mutarent. Atque id, quod de loco
dixi, dicendum pariter de tempore.
3. Necessario igitur admittendus est realis aliquis existendi modus, per quem res est
ibi, ubi est, & tum, cum est. Sive is modus 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.
4. Ego igitur pro singulis materiæ punctis, ut de his [265] loquar, e quibus ad res etiam
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 tum, cum est. Hi reales 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
1maginarium.
5. Modi illi reales singuli & oriuntur, ac pereunt, & indivisibiles prorsus mihi sunt, ac
inextensi, & immobiles, ac in suo ordine immutabiles. Ii & sua ipsorum loca sunt realia,
ac tempora, & punctorum, ad quæ pertinent. Fundamentum præbent realis relationis
distantiæ, sive localis inter duo puncta, sive temporariæ inter duos eventus. Nec aliud
est in se, quod illam determinatam distantiam habeant illa duo materiæ puncta, quam quod
illos determinatos habeant existendi modos, quos necessario mutent, ubi eam mutent
distantiam. Eos modos, qui in ordine ad locum sunt, dico puncta loci realia, qui in ordine
ad tempus, momenta, quæ partibus carent singula, ac omni illa quidem extensione, hæc
duratione, utraque divisibilitate destituuntur.
6. Porro punctum materiæ prorsus indivisibile, & inextensum, alteri puncto materiæ
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) Hic, & sequens paragraphus habentur in Supplementis tomi I. Philosophiae Recentioris Benedicti Stay, § 6, $ 7.
392
SUPPLEMENTS
§ I
Of Space and Time (*)
I. I do not admit perfectly continuous extension of matter; I consider it to be made
up of perfectly indivisible points, which are non-extended, set apart from one another by
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
is of its own real nature continuous, eternal & immense, but also those who, following Leibniz
& Descartes, consider space itself to be the relative arrangement which exists amongst
things that exist, over and above these existent things; it seems to me, I say, that all must
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
to which a thing is where it is, & exists then, when it does exist. Whether this mode is
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
can be easily transferred to immaterial things), I admit two real kinds of modes of existence,
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
indivisible, non-extended, immovable & unvarying in their order. They, as well as the
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 modes 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 thé latter lack duration; both are
indivisible.
6. Further, a point of matter that is perfectly indivisible & non-extended cannot be
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 & the following section are to be found in the Philosophiae Recentior, by Benedict Stay, Vol. I, § 6, 7.
The theme ; what
are the attributes
of space 2
Real local and
temporal modes of
existence must of
necessity be ad-
mitted by every
Olle.
The name by which
this mode is known
is immaterial.
Real modes ; what
real space & real
time may be.
Their nature &
relations.
Contiguity of
points of space is
impossible.
393
394 PHILOSOPHIÆ NATURALIS THEORIA
Posse binis punctis
addi alia in directum
ad distantias æqua-
les, interseri alia
in infinitum.
Existentia puncta
spatii semper fore
finita numero, & in
finitis distantiis: in
possibilibus nullum
finem.
Quomodo inde spa-
tium infinitum, con-
tinuum,necessarium
aeternum, immobile
p er cognitionem
præcisivam.
In momentis eadem,
quæ in punctis:
post primum nullum
Secundum, aut ulti-
mum : sed in tem-
pOre unica dimensio,
in spatio triplex.
possunt ex parte coire tantummodo, & ex parte altera se contingere, ex altera mutuo
aversari. Præjudicium 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.
Globulos reipsa nobis confingimus, nec abstrahimus animum ab extensione illa, & partibus,
quas voce, & ore secludimus.
7. Porro ubi bina materiæ puncta a se invicem distant, semper aliud materiæ punctum
potest collocari in directum ultra utrumque ad eandem distantiam, & alterum ultra hoc,
& ita porro, ut patet, sine ullo fine. Potest itidem inter utrumque collocari in medio aliud
punctum, quod neutrum continget: si enim alterum contigeret, utrumque contingeret,
adeoque cum utroque congrueret, & illa etiam congruerent, non distarent, contra hypo-
thesim. Dividi igitur poterit illud intervallum in partes duas, ac eodem argumento illa
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. Hinc ultra, & inter bina loci puncta realia quæcunque alia loci puncta realia possibilia
sunt, quæ ab iis recedant, vel ad ipsa accedant sine ullo limite determinato, & divisibilitas realis
intervalli inter duo puncta in infinitum est, ut ita dicam, interseribilitas punctorum realium
sine ullo fine. Quotiescunque illa puncta loci realia interposita fuerint, interpositis punctis
materiæ realibus, finitus erit eorum numerus, finitus intervallorum numerus illo priore
interceptorum, & ipsi simul æqualium : at numerus ejusmodi partium possibilium finem
habebit nullum. Illorum singulorum magnitudo certa erit, ac finita : horum magnitudo
minuetur ultra quoscunque limites, sine ullo determinato hiatu, qui adjectis novis inter-
mediis punctis imminui adhuc non possit ; licet nec possit actuali divisione, sive inter-
positione exhauriri.
9. Hinc vero dum concipimus possibilia hæc loci puncta, spatii infinitatem, &
continuitatem habemus, cum divisibilitate in infinitum. In existentibus limes est semper
certus, certus punctorum numerus, certus intervallorum : in possibilibus nullus est finis.
Possibilium abstracta cognitio, excludens limitem a possibili augmento intervalli, &
diminutione, ac hiatu, infinitatem lineæ imaginaræ, & continuitatem constituit, quæ
partes actu existentes non habet, sed tantummodo possibiles. Cumque ea possibilitas &
æterna sit, & necessaria, ab æterno enim, & necessario verum fuit, posse illa puncta cum
illis modis existere ; spatium hujusmodi imaginarium continuum, infinitum, simul etiam
æternum fuit, & necessarium, sed non est aliquid existens, sed aliquid tantummodo potens
existere, & a nobis indefinite conceptum : immobilitas autem ipsius spatii a singulorum
punctorum immobilitate orietur. -
Io. Atque hæc omnia, quæ hucusque de loci punctis sunt dicta, ad temporis momenta
eodem modo admodum facile transferuntur, inter quæ ingens quædam habetur analogia.
Nam & punctum a puncto, & momentum a momento quovis determinato certam distantiam
habet, nisi coeunt, qua major, & minor haberi alia potest sine ullo limite. In quovis
intervallo spatii imaginarii, ac temporis adest primum punctum, vel momentum, & ultimum,
secundum vero, & penultimum habetur nullum : quoyis 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. Nec punctum continuæ lineæ,
nec momentum continui temporis, pars est, sed limes & terminus. Linea continua, &
tempus continuum generari intelligentur non repetitione puncti, vel momenti, sed ductu
continuo, in quo intervalla alia aliorum sint partes, non ipsa puncta, vel momenta, quæ
continuo ducuntur. Illud unicum erit [267] discrimen, quod hic ductus in spatio fieri
poterit, non in unica directione tantum per lineam, sed in infinitis per planum,.quod
concipietur ductu continuo in latus lineæ jam conceptæ, & iterum in infinitis per solidum,
quod concipietur ductu continuo plani jam concepti, in tempore autem unicus ductus
durationis habebitur, quod idcirco soli lineæ 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
point of matter can always be placed in the same straight line with them, on the far side
of either, at an equal distance; & another beyond that, & so on without end, as is evident.
Also another point can be placed halfway between the two points, so as to touch neither
of them ; for, if it touched either of them it would touch them both, & thus would coincide
with both ; hence the two points would coincide with one another & could not be separate
points, which is contrary to the hypothesis. Therefore that interval can be divided into
two parts ; & therefore, by the same argument, those two can be divided into four others,
& 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
real points of position possible; & these recede from them & approach them respectively,
without any determinate limit. There will be a real divisibility to an infinite extent of
the interval between two points, or, if I may call it so, an endless “insertibility’ of real
points. However often such real points of position are interpolated, by real points of
matter being interposed, their number will always be finite, the number of intervals
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
infinity of space, & continuity, together with infinite divisibility. With existing things
there is always a definite limit, a definite number of points, a definite number of intervals;
with possibles, there is none that is finite. The abstract concept of possibles, excluding
as it does a limit due to a possible increase of the interval, a decrease or a gap, gives us the
infinity of an imaginary line, & continuity; such a line has not actually any existing parts,
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.
IO. Everything, that has so far been said with regard to points of position, can quite
easily in the same way be applied to instants of time; & indeed there is a very great analogy
of a sort between the two. For, a point from a given point, or an instant from a given
instant, has a definite distance, unless they coincide ; & another distance can be found
either greater or less than the first, without any limit whatever. In any interval of imaginary
space or time, there is a first point or instant, & a last; but there is no second, or last but
one. For, if any particular one is supposed to be the second, then, since it does not coincide
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
Given two points,
it is possible to add
others in the same
straight line at
e qual distances
apart ; & it is
possible to insert
O the r s between
the m ; to a ny
extent in either
Ca,Se.
The number of
points existing in
space will always
be finite, & the
distances between
them finite ; there
is no end to the
possible cases.
Hence, the manner
in which we arrive
at space that is
finite, continuous,
necessary, eternal
& immovable, by
means of an abstract
concept.
The same things
hold for instants of
time; as for points,
after the first
there is no second
Or last ; in time,
however, there is
but one dimension,
while sin space
there are three.
396 PHILOSOPHIAE NATURALIS THEORIA
Quodvis punctum
materiæ habere in-
tegrum spatium, ac
tempus imaginarium
suum : quid sit com-
penetratio.
Elura, momenta,
ejusdem puncti non
pOSSe coexistere.
Combinationes quat-
uor spatii, & tem-
poris pro unico
puncto m at eriæ
quatuor pro binis
maxime notabiles:
idea, singularis spatii
alterius alibi positi.
habetur triplex in longum, latum, & profundum, temporis habetur unica in longum, vel
diuturnum tantummodo. In triplici tamen spatii, & unico temporis genere, punctum,
ac momentum erit principium quoddam, a quo ductu illo suo hæc ipsa generata intelligentur.
II. Illud jam hic diligenter notandum : non solum ubi duo puncta materiæ existunt,
& aliquam distantiam habent, existere duos modos, qui relationis illius distantiæ funda-
mentum præbeant, & sint bina diversa puncta loci realia, quorum possibilitas a nobis
concepta, exhibeat bina puncta spatii imaginarii, adeoque infinitis numero possibilibus
materiæ punctis respondere infinitos numero possibiles existendi modos, sed cuivis puncto
materiæ respondere itidem infinitos possibiles existendi modos, qui sint omnia ipsius puncti
possibilia loca. Hæc omnia satis sunt ad totum spatium imaginarium habendum, & quodvis
materiæ punctum habet suum spatium imaginarium immobile, infinitum, continuum,
quæ tamen omnia spatia pertinentia ad omnia puncta sibi invicem congruunt, & habentur
pro unico. Nam si assumatur unum punctum reale loci ad unum materiæ punctum
pertinens, & conferatur cum omnibus punctis realibus loci pertinentibus ad aliud punctum
materiæ ; est unum inter hæc posteriora, quod si cum illo priore coexistat, relationem
inducet distantiæ nullius, quam compenetrationem appellamus. Unde patet punctorum,
quæ 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 determinatæ distantiæ, & positionis. Porro illa loci puncta, quæ
nullius distantiæ relationem inducunt, pro eodem accipimus, & quenvis ex infinitis hujus-
nmodi punctis ad infinita puncta materiæ 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 primo, poterit secundum collocari ita, ut habeat eandem illam
distantiam, & positionem, respectu tertii, quam habebat primum. Igitur modus hic, quo
existit, pro eodem habetur, ac modus, quo existebat illud primum, & si hi bini modi simul
existerent, nullius distantiæ relationem inducerent inter primum, ac secundum : & hæc
pariter, quæ hic de spatii punctis dicta sunt, æque temporis momentis conveniunt.
[268] I2. An autem possint simul existere, id vero pertinet ad relationem, quam
habent puncta loci cum momentis temporis, sive spectetur unicum materiæ punctum,
sive plura. Inprimis plura momenta ejusdem puncti materiæ coexistere non possunt, sed
alia necessario post alia, sic itidem bina puncta localia ejusdem puncti materiæ conjungi
non possunt, sed alia jacere debent extra alia, atque id ipsum ex eorum natura, & ut
ajunt, essentia.
I 3. Deinde considerentur conjunctiones variæ punctorum loci, & momentorum.
Quodvis punctum materiæ, si existit, conjungit aliquod punctum spatii cum aliquo
momento temporis. Nam necessario alicubi existit, & aliquando existit ; ac si solum
existat, semper suum habet, & localem, & temporarium existendi modum, per quod, si
aliud quodpiam existat, quod suos itidem habebit modus, distantiæ & localis, & temporis
relationem ad ipsum acquiret. Id saltem omnino accidet, si omnium, quæ 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, quæ aliud, ut ita dicam, infinitum spatium
habeant, quod a nostro itidem infinito 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 distantiæ inducat. Atque id ipsum de tempore etiam
dici posset extra omne nostrum æternum tempus collocato. At id menti, ipsum conanti
concipere, vim summam infert, ac a cogitatione directa admitti vel nullo modo potest,
vel saltem vix potest. Quamobrem iis rebus, vel rerum spatiis, & temporibus, quæ ad nos
nihil pertinere possent, prorsus omissis, agamus de notris hisce. Si igitur primo idem
punctum materiæ 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, quæ
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.
II. Now here there is one thing that must be carefully noted. Not only when two
points of matter exist, & have a distance from one another, do two modes exist which give
the foundation of the relation of this distance; & there are two different real points of
position, the possibility of which, as conceived by us, will yield two points of imaginary
space; & thus, to the infinite number of possible points of matter there will correspond
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. -
I2. Now, whether they can coexist is a question that pertains to the relation between
points of position & instants of time, whether we consider a single point of matter or
several of them. In the first place, several instants of time belonging to the same point
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
& instants of time. Any point of matter, if it exists, connects together some point of
space & some instant of time; for it is bound to exist somewhere & sometime. Even if
it exists alone, it always has its own mode of existence, both local & temporal; & by this
fact, if any other point of matter exists, having its own modes also, it will acquire a relation
of distance, both local & temporal, with respect to the first. This at least will certainly
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,
Every point of
matter is possessed
of the whole of
imaginary Space,
& time ; the
nature of Com-
penetration.
Several inst a nt s
belonging to the
same point cannot
coexist.
Four combinations
of space & time for
a single point of
m a t t er ; four
worth considering
for two points;
extraordinary idea
of another space
situated elsewhere.
398 PHILOSOPHIÆ NATURALIS THEORIA
IRelationes ea ru m
ad se invicem : quæ,
& quomodo possi-
biles.
Quietem, & regres-
sum ad eundem
locum in Natura
esse in infinitum
im probabiles, &
inde ingens analogia.
momentis temporis. Si tertio idem punctum materiæ conjungat idem momentum tem-
poris cum pluribus punctis loci a se invicem distantibus aliquo intervallo ; habebitur illa,
quam dicimus replicationem. Si quarto id conjungat cum serie continua punctorum loci
aliquo intervallo continuo contentorum, habebitur quædam quam plures Peripatetici
admiserunt, virtualem appellantes extensionem, qua indivisibilis, & partibus omnino
destituta materiæ particula spatium divisibile occuparet. Sunt aliæ quatuor combina-
tiones, ubi plura materiæ pun-[269]-cta considerentur. Nimirum quinto si conjungant
idem momentum temporis cum pluribus punctis loci, in quo sita est coexistentia. Sexto
si conjungantidem punctum spatii cum diversis momentis temporis, quod fieret in successivo
appulsu diversorum punctorum materiæ ad eundem locum. 'Septimo si conjungant idem
momentum temporis cum eodem puncto spatii, in quo sita esset compenetratio. Octavo
si nec momentum ullum, nec punctum spatii commune habeant, quod haberetur, si nec
coexisterent, nec ea loca occuparent, quæ ab aliis occupata fuissent aliquando.
I4. Ex hisce octo casibus primo respondet tertius, secundo quartus, quinto sextus,
septimo octavus. Tertium casum, nimirum replicationem, communitur censent naturaliter
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 præsens, & 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. Eam sententiam amplectendam esse non censeo idcirco, quod ubicunque
materiam loca distincta occupantem sensu percipimus, separabilem etiam, ingenti saltem
adhibita vi, videmus ; sejunctis partibus, quæ distabant: nec vero alio ullo argumento
excludimus a Natura replicationem, nisi quia nullam materiæ partem, quantum sensu
percipere possumus, videmus, bina simul occupare loca. Virtualis illa extensio materiæ
infinities ulterius progreditur ultra simplicem replicationem.
I 5. Si secundus casus quietis, & primus casus regressus ad eundem locum naturaliter
haberi possent, esset is quidem defectus quidam analogiæ inter spatium, & tempus. At
mihi videor probare illud posse, neutrum unquam in Natura contingere, adeoque naturaliter
haberi non posse. Id autem evinco hoc argumento. Sit punctum materiæ quodam
momento in quodam spatii puncto, & pro quovis alio momento ignorantes, ubi sit, quæramus,
quanto probabilius sit, ipsum alibi esses, quam ibidem. Tanto erit probabilius illud,
quam hoc, quanto plura sunt alia spatii puncta, quam illud unicum. Hæc in quavis linea
sunt infinita, infinitus in quovis plano linearum numerus, infinitus in toto spatio planorum
numerus. Quare numerus aliorum punctorum est infinitus tertii generis, adeoque illa
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 saltem 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 infinito etiam tempore est infinitus ejusdem ordinis,
cujus est numerus linearum, in quovis plano. Quare improbabilitas casus, quo determin-
atum quodpiam materiæ 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 materiæ punctis, quæ numero
finita sunt, decrescit in ratione finita ejus numeri ad unitatem (quod secus accidit in com-
muni sententia, in qua punctorum materiæ 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 sixth to the fifth, the eighth to the seventh. The third case, namely replication, is
usually considered to be naturally impossible. Many think that the fourth case holds
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
The relations of
these cases to one
another ; which of
them are possible,
& how.
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
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;
& so they cannot be obtained naturally ; this is my argument. If a point of matter at
any instant of time is at a certain point of space, & we do not know where it is at some
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
Rest & return to
the same position
are infinitely impro-
bable in Nature ;
hence arises a very
great analogy be-
tween them. .
4OO PHILOSOPHIÆ NATURALIS THEORIA
Nullum pu n c tu m
materiæ a d v e nire
ad ullum punctum
spati i, in qu o
aliquando fu erit
aliu d punctu m
quodvis. In sola
coexistentia respon-
dente huic adventui
lædi analogiam.
Qui ca s u s sint
p os sibile s per
Divinam Omnipo-
tentia, m : u s u s
superioris . theore-
matis in Impenetra-
bilitate.
adhuc remanet infinita improbabilitas regressus puncti materiæ cujusvis indefinite, ad
punctum loci quodvis, occupatum quovis mgmento præcedenti indefinite, regressus inquam,
habendi quovis indefinite momento sequenti temporis, qui regressus idcirco sine ullo erroris
metu debet excludi, cum infinitam improbabilitatem in relativam quandam impossibilitatem
migrare censendum sit : quæ quidem Theoria communi sententiæ applicari non potest.
Quamobrem eo pacto, patet, in mea materiæ punctorum Theoria e Natüra tolli & quietem,
quam etiam supra exclusimus, & vero etiam regressum ad idem loci punctum, in quO
semel ipsum punctum materiæ extitit : unde fit, ut omnes illi primi quatuor casus exclúd-
antur ex Natura, & in iis accurata temporis, & spatii servetur analogia.
I6. Quin imo si quæratur, an aliquod materiæ punctum occupare debeat quopiam
momento punctum loci, quod alio momento aliquo aliud materiæ punctum occupavit; adhuc
improbabilitas erit infinities infinita. Nam numerus punctorum materiæ existentium est
finitus, adeoque si pro regressu puncti cujusvis ad puncta loci a se occupata adhibeatur
regressus ad puncta occupata a quovis alio, numerus casuum crescit in ratione unitatis ad
numerum punctorum finitum utique, nimirum in ratione finita tantummodo. Hinc '
improbabilitas appulsus alicujus puncti materiæ 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-[271]-riæ 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 materiæ puncta idem
momentum temporis cum diversis punctis loci conjungant, non modo possibilis est, sed
etiam necessarius pro omnibus materiæ punctis, coexistentibus nimirum : fieri enim non
potest, ut septimus, & octavus excludantur ; nisi continuo ob id ipsum includatur quintus
ille, ut consideranti patebit facile. Quamobrem in eo analogia deficit, quod possint plura
materiæ puncta conjungere diversa puncta spatii cum eodem momento temporis, qui est
hic casus quintus, non autem possit idem punctum spatii, cum pluribus momentis temporis,
qui est casus tertius, quem defectum necessario inducit exclusio septimi, & octavi, quorum
altero incluso, excludi posset hic quintus, ut si possent materiæ puncta, quæ simul creata
sunt, nec pereunt, non coexistere, tum enim idem momentum cum diversis loci punctis
nequaquam conjungeretur.
17. Ex illis 7 casibus videntur omnino 6 per Divinam Omnipotentiam possibiles,
dempta nimirum virtuali illa materiæ extensione, de qua dubium esse poterit, quia deberet
simul existere numerus absolute infinitus punctorum illorum loci realium, quod impossibile
est ; si infinitum numero actu existens repugnat in modis ipsis. Quoniam autem possunt
omnia existere alia post alia puncta loci in quavis 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 quæs-
tionem definire non ausim. Illud unum moneo, sententiam hanc meam de spatii natura,
& continuitate præcipuas omnes difficultates, quibus premuntur reliquæ, peni-[272]-tus
evitare, & ad omnia, quæ huc pertinent, explicanda commodissimam esse. Tum illud
addo, excluso appulsu puncti cujusvis materiæ ad punctum loci, ad quod punctum quodvis
materiæ quovis momento appellit, & inde compenetratione, veram impenetrabilitatem
materiæ necessario consequi, quod in decimo nobis libro (?) plurimum proderit. Nimirum
4.
(b) Hic casus nusquam itidem haberetur ; si duratio non esset quid continenter permanens, sed loco ipsius admit-
teretur quaedam existentia, ut ita dicam, saltitans, nimirum si quodvis materiæ punctum (& idem potest transferri ad
quævis creata entia) existeret tantum in momentis indivisibilibus 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 materiae ad punctum spatii, in quo aliud quodvis punctum umquam fuerit. In eodem nullum haberetur
reale continuum me in motu quidem ; diversæ celeritates multo melius explicarentur : multo magis pateret, quomodo
vita insecti brevissima po$sit aequivalere vitae cuivis longi$$imæ, per eumdem nimirum numerum existentiarum inter
extrema momenta. Verum &5 exclusio cujusvis coexistentiae abriperet secum omnes prorsus influxus phy$icos immediatos,
ac determinationes, &5 deberet haberi continua reproductio, immo creatio nova perpetua, &5 alia ejusmodi, quae admitti,
non possunt, haberentur.
(c) Stayamae nimirum philosophiae, in quo Auctor elegantissimus, & doctissimus hanc meam Philosophiam exponit.
Hunc ejus theorematis fructum jam cepimus hic supra, ubi in ipso opere de impenetrabilitate egimus, &5 de apparenti
compenetratione, quæ sine viribus mutuis haberetur a num. 36o.
SUPPLEMENT I 40 I
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.
16. Finally, if we seek to find whether any point of matter is bound to occupy at some
instant a point of position which was occupied by some other point of matter at some
other instant, still the improbability will be infinitely infinite. For the number of existing
points of matter is finite ; & thus, if instead of the return of any point to points of position
occupied by itself we consider the return to points that have been occupied by another,
the number of cases increases in the ratio of unity to a number of points that is in every
case finite, that is to say, in a finite ratio only. Hence, the improbability of the arrival
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.
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,
that is to say, omitting the virtual extension of matter, about which there may be possibly
some doubt; for in this case there must exist at the same time an absolutely infinite number
of those real points of position ; & this is impossible, if an existing thing that is infinite
in number is contradictory in the modes. Moreover, since all points of position can exist
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 usin 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 85 last instants.
Indeed the exclusion of any coexistence would carry away with it all immediate physical influence altogether, 85 deter-
minations ; indeed, a continually fresh creation, {5 other inadmissible things of that sort, would be obtained.
(c) The reference is to Stay’s “Philosophy,” in which that most refined 85 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, & the apparent compenetration that would result, if there were no mutual forces.
D D
Therefore in this point the analogy fails, namely, in that several points
No point of matter
can come into any
point of space that
was once occupied
by any other point :
it is only in
Coexistence, which
corresponds to this
that the analogy
is broken,
Which of the cases
a re poss i b le
through D i vine
Omnipotence ; use
of a theorem given
above on impene-
trability.
4O2 PHILOSOPHIAE NATURALIS THEORIA
nisi vires repulsivae prohiberent; liberrime massa quaevis 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 repulsivae 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 inaequalitas, ex circumjacentium punctorum
inaequali distantia Orta, ac liberrimus habetur progressus in omnes plagas sine ullo occursus
periculo, quod summam, & unicam difficultatem propagationis 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.
INos ne c m o d o s
existendi lo ca 1 e s
posse a b s o 1 u t e
c og n o s cere, nec
absolute distantias,
& magnitudines.
Motum communem
nobis, & Mundo
non posse a nobis
cogn o s ci, ne c s i
i p se in quavis
ratione augeatur,
vel minuatur totus.
Mutata positione
nostra, & omnium,
quæ videmus, non
mutari n o stra, s
ideas, & idcirco nos
motum nec nobis
a d s cribere, n e c
reliquis.
[273] § II
De Spatio, & Tempore, ut a mobis cognoscumtur
18. Diximus in superiore Supplemento de spatio, ac tempore, ut sunt in se ipsis :
superest, ut illud attingam, quod pertinet ad ipsa, ut cognoscuntur. Nos nequaquam
immediate cognoscimus per sensus illos existendi modos reales, nec discernere possumus
alios ab aliis. Sentimus quidem a discrimine idearum, quæ per sensus excitantur in animo,
relationem determinatam distantiæ, & positionis, quæ e binis quibusque localibus existendi
modis exoritur, sed eadem idea oriri potest ex innumeris modorum, sive punctorum realium
loci binariis, quæ inducant relationes æqualium distantiarum, & similium positionum
tam inter se, quam ad nostra organa, & ad reliqua circumjacentia corpora. Nam bina
materiæ puncta, quæ alicubi datam habent distantiam, & positionem inductam a binis
quibusdam existendi modis, alibi possunt per alios binos existendi modos habere relationem
distantiæ æqualis, & positionis similis, distantiis nimirum ipsis existentibus parallelis.
Si illa puncta, & nos, & omnia circumjacentia corpora mutent loca realia, ita tamen, ut
omnes distantiæ æquales maneant, & prioribus parallelæ ; nos easdem prorsus habebimus
ideas, quin imo easdem ideas habebimus ; si manentibus distantiarum magnitudinibus,
directiones omnes in æquali angulo converterentur, adeoque æque ad se invicem inclinarentur
ac prius. Et si minuerentur etiam distantiæ illæ omnes, manentibus angulis, & manente
illarum ratione ad se invicem, vires autem ex ea distantiarum mutatione non mutarentur,
rite mutata virium scala illa, nimirum curva illa linea, per cujus ordinatas ipsæ vires
exprimuntur ; nullam nos in nostris ideis mutationem haberemus.
I9. Hinc autem consequitur illud, si totus hic Mundus nobis conspicuus motu parallelo
promoveatur in plagam quamvis, & simul in quovis angulo convertatur, nos illum motum,
& conversionem sentire non posse. Sic si cubiculi, in quo sumus, & camporum, ac montium
tractus omnis motu aliquo Telluris communi ad sensum simul convertatur ; motum ejusmodi
sentire non possumus : ideæ enim eædem 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.
20. Ubi vel objecta externa, vel nostra organa mutant illos suos existendi modos ita,
ut prior illa æqualitas, [274] vel similitudo non maneat, tum vero mutantur ideæ, &
mutationis habetur sensus, sed ideæ eædem omnino sunt, sive objecta externa mutationem
subeant, sive nostra organa, sive utrumque inæqualiter. Semper ideæ nostræ differentiam
novi status a priore referent, non absolutam mutationem, quæ sub sensus non cadit. Sic
sive astra circa Terram moveantur, sive Terra motu contrario circa se ipsam nobiscum ;
eædem sunt ideæ, idem sensus. Mutationes absolutas nunquam sentire possumus, discrimen
a priori forma sentimus. Cum autem nihil adest, quod nos de nostrorum organorum
mutatione commoneat ; tum vero nos ipsos pro immotis habemus communi præjudicio
habendi pro nullis in se, quæ 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, & montes, ac ipsam undam moveri arbitratur.
404
§ II
Of Space & Time, as we know them
18. We have spoken, in the preceding Supplement, of Space & Time, as they are in
themselves; it remains for us to say a few words on matters that pertain to them, in so
far as they come within our knowledge. We can in no direct way obtain a knowledge
through the senses of those real modes of existence, nor can we discern one of them from
another. We do indeed perceive, by a difference of ideas excited in the mind by means
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
parallel motion in any direction, & at the same time rotated through any angle, we could
never be aware of the motion or the rotation. Similarly, if the whole region containing
the room in which we are, the plains & the hills, were simultaneously turned round by some
approximately common motion of the Earth, we should not be aware of such a motion;
for practically the same ideas would be excited in the mind. Moreover, it might be the
case that the whole Universe within our sight should daily contract or expand, while the
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
in such a way that that first equality or similitude does not remain constant, then indeed
the ideas are altered, & there is a feeling of change ; but the ideas are the same exactly,
whether the external objects suffer the change, or our organs, or both of them unequally.
In every case our ideas refer to the difference between the new state & the old, & not to
the absolute change, which does not come within the scope of our senses. Thus, whether
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
ImOt1OIl.
We cannot obtain
an absolute know-
ledge of local modes
of existence ; nor
yet of absolute dis-
tances or magni-
tudes.
The motion, if any,
Common to us &
the Universe could
not come within
our knowledge ;
nor could we know
it, if it were in-
creased in a ny
ratio, or diminished,
as a whole.
Since, if our posi-
tion & that of
everything we see
is changed, our ideas
are not changed ;
therefore we can
ascribe no motion
to ourselves or to
anything else.
405
406 PHILOSOPHIAE NATURALIS THEORIA
Quomodo judice-
mus de æqualitate
duorum, ex æqual-
itate cum tertio :
nunquam haberi
congrue ntia m in
longitudine, ut nec
in tempore, sed in-
ferri a causis.
Conclusio : discri-
men vulgi a Philo-
sophisin judicando.
Licet translata, de-
cempeda, mutentur
modi, qui intervalli
relationem consti-
tuunt ; tamen inter-
valla æqualia haberi
p r o e o d e m ex
causis.
Eadem ad tempus
transferenda, sed
in eo etiam vulgo
notum esse, inter-
vallum t e m p or-
arium non posse
transferri idem pro
comparatione duo-
rum : errari ab eo
circa spatium.
2 I. 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, quae sunt æqualia eidem, sunt
aequalia inter se, ex quo deducitur hoc aliud, ad ipsum pertinens, quae sunt aeque multipla,
vel submutipla alterius, sunt itidem inter $e æqualia, & hoc alio, quae congruant, æqualia sunt.
Assumimus ligneam, vel ferream decempedam, quam uni intervallo semel, vel centies
applicatam si inveniamus congruentem, tum alteri intervallo applicatam itidem semel,
vel centies itidem congruentem, illa intervalla æqualia 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 decempedæ
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 æqualium distantiarum relationes ; terminus compara-
tionis non erit idem, adhuc tamen æqualis erit, & æqualitas mensuratorum intervallorum
rite colligetur. Longitudinem decempedæ in priore situ per illos priores reales modos
constitutæ, cum longitudine in posteriore situ constituta per hosce posteriores, immediate
inter se conferre nihilo magis possumus, quam illa ipsa intervalla, quæ mensurando conferi-
mus. Sed quia nullam in translatione mutationem sentimus, quæ 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 æqualem, vel similem statum rediret. Exigua tamen aliqua
mutatio habetur omnino idcirco, quod vires, quæ illa materiæ 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 interjectis,
& omnis penitus compressionis, ac dilatationis est incapax, quæ quidem dilatatio, &
compressio saltem exigua in omni translatione omnino habetur. Nos tamen mensuram
illam pro eadem habemus, cum, ut monui, nullam mutationem sentiamus.
22. Ex his omnibus consequitur, nos absolutas distantias nec immediate cognoscere
omnino posse, nec per terminum communem inter se comparare, sed æstimare 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 violatæ notabili mutatione æqualitatis causam agnoscant,
mutationem ipsam pro æqualiter facta habent.
23. Porro licet, ubi puncta materiæ locum mutant, ut in decempeda translata, mutetur
revera distantia, mutatis iis modis realibus, quæ ipsam constituunt ; tamen si mutatio ita
fiat, ut posterior illa distantia æqualis prorsus priori sit, ipsam appellabimus eandem, & nihil
mutatam ita, ut eorundem terminorum æquales distantiæ dicantur distantia eadem, &
magnitudo dicatur eadem, quæ per eas æquales distantias definitur, ut itidem ejusdem
directionis nomine intelligantur binæ etiam directiones parallelæ ; nec mutari distantiam,
vel directionem dicemus in sequentibus, nisi distantiæ magnitudo, vel parallelismus mutetur.
24. Quæ 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 æquabilem. Multa, quæ huc perti-
nent, & quæ ad idearum ipsarum naturam, & successionem spectant, diximus in notis.
Unum hic addo, in mensura temporis, ne vulgus quidem censere ab uno tempore ad aliud
tempus eandem temporis mensuram transferri. Videt aliam esse, sed æqualem supponit ob
motum suppositum æqualem. In mensura locali æque 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-
tudo, ut alia duratio substituitur, quæ priori illi æqualis censetur, nimirum nova realia
SUPPLEMENT II 4O7
21. Again, it is to be observed first of all that from this principle of the unchangeability
of those things, of which we cannot perceive the change through our senses, there comes
forth the method that we use for comparing the magnitudes of intervals with one another;
here, that, which is taken as a measure, is assumed to be unchangeable. Also we make
use of the axiom, things that are equal to the ſame thing are equal to one another; & from this
is deduced another one pertaining to the same thing, namely, things that are equal multiples,
or ſubmultiples, of each, are also equal to one another; & also this, things that coincide are
equal. We take a wooden or iron ten-foot rod; & if we find that this is congruent with
one given interval when applied to it either once or a hundred times, & also congruent to
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
of absolute distances; & we cannot compare them with one another by a common standard.
We have to estimate magnitudes by the ideas through which we recognize them ; & to
take as common standards those measures which ordinary people think suffer no change.
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
of the ten-foot rod, the position of the points of matter is altered, those real modes which
constitute the distance being altered; nevertheless if the change takes place in such a way
that the second distance is exactly equal to the first, we shall call it the same, & say that it is
altered in no way, so that the equal distances between the same ends will be said to be the
same distance & the magnitude will be said to be the same ; & this is defined by means of
these equal distances, just as also two parallel directions will be also included under the name
of the same direction. In what follows we shall say that the distance is not changed, or
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
can be applied to time; in this also we have no definite & constant measurement. We
obtain all that is possible from motion ; but we cannot get a motion that is perfectly uniform.
We have remarked on many things that belong to this subject, & bear upon the nature &
succession of these ideas, in our notes. I will but add here, that, in the measurement of
time, not even ordinary people think that the same standard measure of time can be translated
from one time to another time. They see that it is another, consider that it is an equal,
on account of some assumed uniform motion. Just as with the measurement of time,
so in my theory with the measurement of space it is impossible to transfer a fixed length
from its place to some other, just as it is impossible to transfer a fixed interval of time,
so that it can be used for the purpose of comparing two of them by means of a third. In
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
The manner in
which we are to
judge of the equal-
ity of two things
from their equality
with a third ; there
never can be con-
gruence in length,
any more than there
can be in time ; the
matter is to be
in ferred from
CallSCS.
Conclusion reached;
the difference be-
tween ordinary
people & philo-
sophers in the
matter of judgment.
Although, when
the ten-foot rod is
moved in position,
those modes that
constitute the rela-
tions of the interval
are also altered, yet
equal intervals are
reckoned as same
for the reasons
stated.
The same obser-
v at ions a pply
equally to Time ;
but in it, it is well
known, even to
ordinary people,
that the same
temporal interval
cannot be translated
for the purpose of
comparing two
intervals; it is be-
cause of this that
they fall into error
with regard to
Space.
4o8 PHILOSOPHIÆ NATURALIS THEORIA
punctorum ejusdem decempedæ 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 saltem haberi posse per mensuram perfecte solidam, & continuam,
in tempore tantummodo æqualem : ego vero utrobique æqualem 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.
Denominatio, a c
præparatio.
Assumptio cujus-
dam valoris ad rem
idonei.
Formula, continens
æquationem quæ-
situm.
Aequationem fore
simplicem non re-
solubilem in plures.
IExhibituram da-
tum numerum in-
tersectionum curvæ,
in datis punctis.
Singulas ordinatas
responsuras singu-
lis abscissis.
[277] § III
So/utio ama!ytica 'Problematis determimamtis maturam Legis
'Ojrium (é)
25. Ut hasce conditiones impleamus, formulam inveniemus algebraicam, quæ ipsam
continebit legem nostram, sed hic elementa communia vulgaris Cartesianæ algebræ 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
negativo, & 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 f. Numerus eorundem valorum dicatur m. His positis ponatur
z" + az** + bz* + cz** &c . . . + f = P. Si ponatur P = o, patet æquationis ejus
omnes radices fore reales, & positivas, nimirum sola illa quadrata quantitatum AE, AG,
AI &c, qui erunt valores ipsius 2. ; adeoque cum ob xx = z, sit x = + /z, patet, valores
ac fore tam AE, AG, AI positivas, quam AE', AG', &c negativas.
26. Deinde sumatur quæcunque quantitas data per z, & constantes quomodocunque,
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 quæcunque formula z' -+- gz* -+- hz* &c -+- l, quæ 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. ;
27. Si jam fiat P — Qy = o ; dico, hanc æquationem satisfacere reliquis omnibus
hujus curvæ conditionibus, & rite determinato valore Q, posse infinitis modis satisfieri
etiam postremæ conditioni expositæ sexto loco.
[278] 28. Nam inprimis, quoniam valores P, & Q positi = o, nullam habent radicem
communem, nullum habebunt divisorem communem. Hinc hæc æquatio non potest per
divisionem reduci ad binas, adeoque non est composita ex binis æquationibus, sed simplex,
& proinde simplicem quandam curvam continuam exhibet, quæ ex aliis non componitur.
Quod erat primum.
29. Deinde curva hujusmodi secabit axem C'AC in iis omnibus, & solis punctis, E,
G, I, &c, E', G', &c. Nam ea secabit axem C'AC solum in iis punctis, in quibus y = o,
& 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
æquationis 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 æquationum P = o, & Q = o. Quare
erit y = o, & curva axem secabit. Quod erat secundum.
ę ç_» P. ę w»
3o. Præterea cum sit P — Qx = o, erit y = #, ; determinata autem utcunque abscissa
Q.
æ, habebitur determinata quædam z, adeoque & P, Q erunt unicæ, & determinatæ. Erit
igitur etiam y unica, & determinata ; ac proinde respondebunt singulis abscissis z singulæ
tantum ordinatæ y. Quod erat tertium.
(d) Haec solutio excerpta est ex dissertatione De Lege Virium in Natura existentium. Accedit iis, quae inde
sunt eruta, scholium 3 primo adjectum in hac editione Veneta prima. Ipsum problema hic solvendum habetur in ipso
hoc Opere parte I num. I 17, ac ejus conditiones mum. I I8.
410
§ III
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
our law; to do so, we shall take it that the first principles of the ordinary Cartesian algebra
are known ; for, without that, the thing can in no way be accomplished. Suppose that
y is the ordinate, & the abscissa, & let wº = 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 f; suppose that the number of these value
is m. Then suppose P to stand for -
z" -- azº-, -- bg”---|- cz”-3 + + f.
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 w” = z, & therefore & = + V z, it is evident that the values of x will be
AE, AG, AI, positive, & AE’, AG', &c., negative. ISee Fig. I.]
26. Next, assume some quantity that is given by 2, & constants, in any manner, so
long as it has not got any common measure with P, nor vanishes when z vanishes; also,
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
2." + gzº- + h2-2 + + l
(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)
of we multiply the whole by z. Call the product Q. t
27. If now we put P – Qy = 0, I say that this equation will satisfy all the remaining
conditions of the curve; & if Q is correctly determined, it can satisfy in an infinite number
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
common root, they cannot have a common divisor. Hence this equation cannot by division
be reduced to two ; & therefore it is not a composite equation formed from two equations,
but is simple. Hence, it will represent some simple continuous curve, which is not made
up of others. This was the first condition. -
29. Next, this curve will cut the axis CAC in all those points, & in them only, such
as E, G, I, &c., E', G', &c. For it will cut the axis CAC in those points only, for which
y = o, & it will cut it in all of them. Further, when y = o, we have also Qy = o & there-
fore, since P – Qy = 0, we have P = 0. Now this happens only at those points for which
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 = 0, Q = 0. Hence
y = o, & the curve will cut the axis. This was the second condition.
30. Further, since P – Qy = 0, it follows that y = P/Q ; hence, for any determinate
abscissa w, there will be a determinate z; & thus P & Q will be uniquely determinate.
Therefore also y will be uniquely determinate; hence, to each abscissa & there will correspond
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, 85 the conditions in Art. II8.
Statement, & pre-
paration.
Assumption of
some value suitable
to the matter.
Formula containing
the equation re-
quired.
The equation will
be simple, that is,
it cannot be re-
solved into several
others.
It will represent
a given number
of intersections of
the curve at given
points.
To each abscissa
there will corre-
spond one ordinate
& one only.
411
4 I 2 PHILOSOPHIAE NATURALIS THEORIA
Abscissis hinc inde
aequalibus respon-
Suras aequales ordi-
nataS.
Primum arcum
fore crus aSymp-
toticum cum area
infinita.
Post eas condi-
ti o ne s remanere
indeterminationem
parem cuicunque
accessui ad quasvis
curvas in punctis
datis quibusvis.
Quid requirature
ut transeat per quae
vis earum puncta.
Quomodo id prae-
standum.
Progressus ulterior.
Conclusio, è cohaer-
entia cum omnibus
praecedentibus con-
ditionibus.
I n de
o scula,
quivis.
contactus,
3,CCGSSU1S
3I. Rursus sive x assumatur positiva, sive negativa, dummodo ejusdem longitudinis
sit, semper valor z = xx eritidem ; ac proinde valores tam P, quam Q erunt semperiidem.
uare semper eadem y. Sumptis igitur abscissis z aequalibus hinc, & inde ab A, altera
positiva, altera negativa, respondebunt ordinata aequales. Quod erat quartum.
32. Si autem x minuatur in infinitum, sive ea positiva sit, sive negativa; semper z
minuetur in infinitum, & evadet infinitesima ordinis secundi. Quare in valore P decrescent
in infinitum omnes termini praeter f, quia omnes praeter eum multiplicantur per z, adeoque
valor P erit adhuc finitus. Valor autem Q, qui habet formulam ductam in z totam,
Q
infinitum ita, ut evadat infinita ordinis secundi. Quare curva habebit pro asymptoto
rectam AB, & area BAED excrescet in infinitum, & si ordinata y assumantur ad partes AB,
& exprimant vires repulsivas, arcus asymptoticus ED jacebit ad partes ipsas AB. Quod
erat quintum.
minuetur in infinitum, eritgue infinitesimus ordinis secundi. Igitur ſy augebitur in
[279] 33. Patet igitur, utcunque assumpto Q cum datis conditionibus, satisfieri primis
quinque conditionibus curvae. Jam vero potest valor Q variari infinitis modis ita, ut
adhuc impleat semper conditiones, cum quibus assumptus est. Ac proinde arcus curvae
intercepti intersectionibus poterunt infinitis modis variari ita, ut primae quinque ipsius
curvae conditiones impleantur; unde fit, ut possint etiam variariita, ut sextam conditionem
impleant.
34. Si enim dentur quotcunque, & quicunque arcus, quarumcunque curvarum, modo
sintejusmodi, 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 Pita, ut
curva per omnia eiusmodi puncta transeat, & idem poterit infinitis modis variari ita, ut
adhuc semper curva transeat per eadem illa puncta.
35. Sit enim numerus punctorum assumptorum quicunque - r, & a singulis ejusmodi
punctis demittantur rectae parallelae AB usque ad axem C'AC, quae debent esse ordinata
curvae quaesitae, & singulae abscissae ab A usque ad eiusmodi ordinatas dicantur MI, M2,
M3, &C, singulae autem ordinata N’I, N 2, N 3, &c. Assumatur autem quaedam quantitas
Az + Bz + Cz º + . . . -- Gz, quae ponatur = R. Tum alia assumatur
quantitas T ejusmodi, ut evanescente z evanescat quivis ejus terminus, & ut nullus sit
divisor communis valoris P, & valoris R + T, quod facile fiet, cum innotescant omnes
divisores quantitatis P. Ponatur autem Q = R + T, & jam aequatio ad curvam erit
P – R y – Ty = 0. Ponantur in hac aequatione successive MI, M2, M3, 8cc, pro x, &
N’I, N 2, 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, M2, M3, 8cc, NI, N2, N3, &c, ac valores arbitrarios, qui in T sunt
coefficientes ipsius z.
36. Per illas aequationes numero radmodum facile 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 aequationibus omnibus sequen-
tibus, quo pacto habebuntur aequationes r – I. Hae autem eiecto valore B reducentur
ad r – 2, & ita porro, donec ad unicam ventum fuerit, in qua determinato valore G, per
ipsum ordine retrogrado determinabuntur valores omnes praecedentes, singuli in singulis
aequationibus.
37. Determinatis hoc pacto valoribus A, B, C, . . . G [28ol in aequatione P – Ry
– Ty = o, sive P – Qy = o, patet positis successive pro e valoribus MI, M2, M3, 8cc,
debere valores ordinata y esse successive NI, N2, N3, &c; ac proinde debere curvam
transire per data illa puncta in datis illis curvis : & tamen valor Q adhuc habebit
omnes conditiones praecedentes. 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. «
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 aequalibus binis N; jam curva quaesita ibidem
SUPPLEMENT III 4. I3
31. Again, whether x is taken positive or negative, so long as its length is the same,
the value of z, or x*, will be the same. Hence the values of both P & Q will be the same.
Hence y will always be the same for either. Hence, if equal abscissae w are taken one on
either side of A, the one positive & the other negative, the corresponding ordinates will
be equal. This was the fourth condition.
32. Now, if x is diminished indefinitely, whether it is positive or negative, z will be
also diminished indefinitely, & will become an infinitesimal of the second order. Hence,
every term in the value of P, except f, will diminish indefinitely; for each of them except
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
first five conditions for our curve will be satisfied. Now, the value of Q can be varied
in an infinite number of ways, such that it will still fulfil the conditions under which it
was assumed. Then the arcs of the curves intercepted between the intersections with
the axis could be varied in an infinite number of ways, such that the first five conditions
for the curve are satisfied. Hence it follows that they can be varied also, in such a way
that the sixth condition is satisfied.
34. Now, if any number of arcs of any kind, belonging to any curves, are given; so
long as these are such that they continually recede from the asymptote AB, & therefore
such that no straight line parallel to this asymptote will cut any of them in more than one
point; & if in these arcs there are taken any number of points, no matter how close they
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,
let a straight line be drawn parallel to AB, to meet the axis CAC; these must be ordinates
of the curve required. Let the several abscissae measured from A to these ordinates be
M., M., Ms, &c.; & let the corresponding ordinates be N1, Nº, Ns, &c. Then assume
some quantity Az' + B z^* + Cz' + . . . -- 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 M, M., M.
&c. for x, & Nº, Nº, N, &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 M1, M2, Ms, &c.,
N1, Na, Ns, &c., & the arbitrary values which appear as the coefficients of z in the expression
- - - - -
36. From these equations, r in number, the values of A, B, C, . . . , G, which are also
f 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 — 2 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 = 0, or
P – Qy = 0, having been thus found, it is clear that, if the values M., M., Ms, &c., are
substituted for w in succession, the values of y will be N., N, Ns, &c. Hence, the curve
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,
are supposed to approach one another on the same side of the axis beyond all limits, &
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
TO
Scissae,
e qual a b-
therefore,
there will corre-
Spond equal or-
dinates, on either
side of the origin.
The first arc will
be an asymptotic
branch with an
infinite area.
After these con-
ditions have been
fulfilled, there still
remains an equal
indetermination
for approach to
any given curves
at any given points.
The conditions
that it should pass
through given
points of these
Cllr VeS.
How this can be
managed. -
Further progress.
Conclusion ; agree-
ment with a 11
the preceding
conditions.
He n ce contacts,
O sculations, &
approach of any
kind.
4I4 PHILOSOPHIÆ NATURALIS THEORIA
Adhuc indetermina-
tio relicta pro infini-
tis modis.
E o s s e & axe m
contingere, Osculari,
&c.
Po s se contingere
simul, & secare.
Quid congruentia
in te r se ctio nu m
plurium.
Posse axem secari
in quibuscunque
angulis, & a quavis
m a g n i t u d i n e
2.ICllllIIl.
D e m o n stratio :
limitatio necessaria.
Quid possint arcus
curvarum assump-
t a ru m : O m n i a
posse & inventam.
Conditio necessaria,
ex hujus curvæ
natura.
Corol. I includi in
corol. 2.
Quid ubivis etiam
extra limites.
Demonstratio
dem.
ea,-
tanget arcum curvæ datæ : & si tria ejusmodi puncta congruant, eam osculabitur : quin
immo illud præstari poterit, ut coeant quot libuerit puncta, ubi libuerit, & habeantur
Oscula ordinis cujus libuerit, & ut libuerit sibi invicem proxima, arcu curvæ datæ 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.
39. Cum vero adhuc infinitis modis variari possit valor T ; infinitis modis idem
præstari poterit : ac proinde infinitis modis inveniri poterit curva simplex datis conditionibus
satisfaciens. Q.E.F. -
40. Coroll. I. Curva poterit contingere axem C'AC in quot libuerit punctis, &
contingere simul, ac secare in iisdem, ac proinde eum osculari quocunque osculi genere.
Nam si binæ quævis e distantiis limitum fiant æquales ; 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, tum non transiret in attractionem, sed iterum cresceret repulsio ipsa per
arcum LM. Idem autem accideret attractioni, si coeuntibus punctis LN, evanesceret
arcus repulsivus LMN.
4I. Si autem tria puncta coirent, ut LNP ; curva contingeret simul axem C'AC, &
ab eodem simul secaretur, ac proinde haberet in eodem puncto contactus flexum
contrarium. Haberetur autem ibidem transitus ab attractione ad repulsionem, vel vice
versa, adeoque verus limes.
42. 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.
43. Coroll. 2. In iis limitibus, in quibus curva secat axem C'AC, potest ipsa curva
secare eundem in quibuscunque angulis ita tamen, ut angulus, quem efficit ad partes A
arcus curvæ in perpetuo recessu ab asymptoto 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
infinitum.
44. Nam pro illis punctis datis in arcubus curvarum quarumcunque, quas curva
inventa 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 curvæ, ut tNy debet ab asymptoto
recedere, non poterit punctum ullum t præcedens limitem N jacere ultra rectam axi perpen-
dicularem erectam ex N, vel punctum y sequens ipsum N jacere citra ; ac proinde non
poterit angulus ANt, quem efficit ad partes A arcus tN in perpetuo recessu ab asymptoto
appellens ad axem C'AC, esse major recto.
45. Possunt autem arcus curvarum assumptarum in iisdem punctis aut axem, aut
rectam axi perpendicularem contingere, aut osculari, quocunque contactus, aut osculi
genere, ut nimirum sit radius Osculi magnitudinis cujuscunque, & vel utcunque evanescens,
vel utcunque abiens in infinitum. Quare idem accidere poterit ut innuimus, & arcui
curvæ inventæ, quæ ad eos arcus potest accedere, quantum libuerit, & eos contingere, vel
osculari quocunque Osculi genere in iis ipsis punctis.
46. Solum si curva inventa tetigerit in ipso limite rectam axi C'AC perpendicularem,
debebit simul ibidem eandem secare ; cum debeat semper recedere ab asymptoto, adeoque
debebit ibidem habere flexum contrarium.
47. Scholium I. Corollarium I est casus particularis hujus corollarii secundi, ut patet :
sed libuit ipsum seorsum diversa methodo, & faciliore prius eruere.
48. Coroll. 3. Arcus curvæ etiam extra limites potest habere tangentem in quovis
angulo inclinatam ad axem, vel ei parallelam, vel perpendicularem cum iisdem contactuum,
& osculorum conditionibus, quæ habentur in corollario 2.
49. Demonstratio est prorsus eadem : nam arcus curvarum dati, ad quos arcus
curvæ inventæ potest accedere ubicunque, quantum libuerit, possunt habere ejusmodi
conditiones,
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
be brought about in an infinite number of ways. Hence, in an infinite number of ways,
a simple curve can be found satisfying the given conditions. Q . E. F.
40. Cor. I. The curve may touch the axis CAC in any desired number of points;
or at the same time touch & cut it at the same points; & hence it may osculate the axis
with any kind of osculation. For, if any two of the distances for the limit-points become
equal, the curve will touch the straight line CºA, 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
time touch the axis CAC & intersect it; thus, at that point of contact there would be
contrary flexure. Also, there would be there a passage from attraction to repulsion, or
vice versa, & therefore a true limit-point.
42. In the same way, four points may coincide, or five, or any number. If the number
of points that coincide is even, there will be touching contact; if the number is odd, there
will be contact & intersection at the same time. The greater the number of the points
that coincide, the more the curve will approach to coincidence with the axis CAC 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 CAC, the curve
may cut it at any angle ; but in such a way that the angle, which the arc of the curve,
in its continuous recession from the asymptote, makes with the direction of A as it comes
up to the axis CAC, 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.
. For, we may take as our chosen points in the arcs of any curves, which the curve
of forces is found to touch or to osculate with an osculation of any order, from which
the value of R is determined, arcs of any curves cutting the axis CAC at any angles. Except
that, since the arc of the curve, such as thy, 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 ANt, which
it makes with the direction of A, as the arc in continually recedes from the asymptote,
as it comes up to the axis CAC, cannot be greater than a right angle.
45. Again, the arcs of the assumed curves may, at these points either touch the axis
or the straight line perpendicular to the axis, or they may osculate, the contact or the
Osculation being of any order; that is, the radius of Osculation may be of any magnitude
whatever, either vanishing or becoming infinite, in any way. Hence, as I said, this may
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
perpendicular to the axis CAC, it must at the same time cut it at that point; for the
curve must always recede from the asymptote, & thus is bound to have contrary flexure
at the point.
47. Scholium I. The first corollary is a particular case of the second, as is evident.
But I preferred to take it first, with an independent proof by a different & an easier
method.
48. Cor. 3. Even beyond the limit-points, the arc of the curve can have a tangent
inclined at any angle to the axis, or parallel to it, or perpendicular to it; with the same
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
which the arc of the curve that is found can be made to approximate as closely as desired,
may have the conditions stated.
There is still left
indetermination in
countless ways.
It is possible also
for the curve to
touch the axis, or
to OSculate it, etc.
It is possible that
there may be
simultaneous con-
tact & section of
the axis.
The result of the
coincidence of
Several intersec-
tions.
The axis may be
cut by the curve
at any angle, & by
arcs of any size.
Demonstration ;
necessary limita-
tion.
What the arcs of
the assumed curves
may be ; the same
properties may all
be possessed by the
curve found.
Necessary condi-
tion, arising from
the nature of the
Cllr.Ve.
The first corollary
is included in the
Second.
What happens also
at any point beyond
the limit-points.
Proof the same as
before,
4I6 PHILOSOPHIÆ NATURALIS THEORIA
Mutationem ab-
scissæ posse habere
ad mutationem or-
dinatæ relationem
quancunque.
IDemonstratur pro
ratione finita.
Itidem pro quovis
infinit e s i m oru m
ordine.
Relationem ejus.
modi p e n d e re a
positione tangentis.
Quid, ubi abscissa,
terminetur in lim-
ite.
Posse arcus utcun-
que recedere ab
3,X€.
IDemonstratio.
Posse haberi postre-
mum crus asymp-
toticum,& alia crura
asymptotica.
IRatio præstandi
primum.
ERatio præstandi &
reliquum.
Legem virium hic
exhiberi per func-
tionem distantiæ,
alios multos censere
p r æ f e r e n d a, m
unicam potentiam:
cur id.
Qua occasione hæc
qu æ stio fuerit
agitata in Parisiensi
Academia.
59. Coroll. 5. Mutata abscissa per quodcunque intervallum datum, potest ordinata
mutaj Per aliud quodcunque datum utcunque minus, vel majus ipsa mütätione abscissæ,
& ut-[282]-cunque majus quantitate quacunque data ; ac si differéntia abscissæ sit infini?
tesima, & dicatur ordinis primi ; poterit differentia ordinatæ esse ordinis cujuscunque,
vel utcunque inferioris, vel intermedii, inter quantitates finitas, & quantitaíes ordinis
pr1m1. -
51. Patet primum ex eo, quod, ubi determinatur valor R, potest curva transire per
quotcunque, & quæcunque puncta, adeoque per puncta, ex quibus ductæ ordinatæ $int
utcunque inter se proximæ, & utcunque inæquales.
52. Patet secundum : quia in curvis, ad quas accedit arcus curvæ inventæ vel quas
osculatur quocunque osculi genere, potest differentia abscissæ ad differentiam ordinatæ
esse pro diversa, curvarum natura in datis earum punctis in quavis ratione, quantitatis
infinitesimæ ordinis cujuscunque ad infinitesimam cujuscunque àlterius.
53. Scholium 2. Illud notandum, ubicunque fuerit tangens curvæ inventæ inclinata
in angulo finito ad axem, fore differentiam abscissæ ejusdem ordinis, ac est differentia
ordinatæ : ubi tangens fuerit parallela axi, fore differéntiam ordinatæ ordinis inferioris,
quam sit differentia abscissæ, & vice versa, ubi tangens fuerit perpendicularis axi.
54. Præterea notandum : si abscissa fuerit ipsa distantia limitis, quæ vel augeatur, vel
minuatur utcunque ; differentia ordinatæ erit ipsa ordinata integra ; cum nimirum in
limite ordinata sit nihilo æqualis.
55. Coroll. 5. Arcus repulsionum, vel attractionum intercepti binis limitibus
quibuscunque, possunt recedere ab axe, quantum libuerit, adeoque fieri potest, ut alii
propiores asymptoto recedant minus, quam alii remotiores, vel ut quodam ordine eo minus
recedant ab axe, quo sunt remotiores ab a$ymptoto, vel ut post aliquot arcus minus
recedentes aliquis arcus longissime recedat.
56. Omnia manifesto consequuntur ex eo, quod curva possit transire per quævis
data puncta.
57. Coroll. 6. Potest curva ipsum axem C'AC habere pro asymptoto ad partes C^,
& C ita, ut arcus asymptoticus sit vel repulsivus, vel attractivus ; & potest arcus quivis
binis limitibus quibuscunque interceptus abire in infinitum, ac habere pro asymptoto
rectam axi perpendicularem, utcunque proximam utrilibet limiti, vel ab eo remotam.
58. Nam si concipiatur, binos postremos limites coire, abeuntibus binis intersectionibus
in contactum, tum concipiatur, ipsam distantiam contactus excrescere in infinitum ; jam
axis æquivalet rectæ 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.
[285] 59. Eodem pacto si concipiatur, quamvis ordinatam respondentem puncto cuilibet,
per quod debet transire curva, abire in infinitum ; jam arcus curvæ abibit in infinitum, &
erit ejus asymptotus in illa ipsa ordinata in infinitum excrescens.
6o. Scholium 3. Ope formulæ exhibentis curvam propositam habetur lex virium
expressa per functionem quandam distantiæ constantem plurimis terminis, immo per
æquationem commiscentem abscissam, & ordinatam, ac utriusque potentias inter se, &
cum rectis datis, non per solam ipsius distantiæ potentiam. Sunt, qui censeant, expre$-
sionem per solam potentiam debere præferri expressioni per functionem aliam, quia haec sit
simplicior, quam illa, & quia in illa præter distantias debeant haberi aliquæ aliæ parametri,
quæ non sint solæ distantiæ ; dum in formula— exprimente x distantias, distantiæ solæ
2%
rem conficiant, videatur autem vis debere pendere a solis distantiis, potissimum si sit
quædam essentialis proprietas materiæ : præterea addunt, nullam fore rationem suffi-
cientem, cur una potius, quam alia parameter expressionem virium deberet ingredi, si
parametri sint admiscendæ. ę ę «» «• «*
61. Hæc agitata sunt potissimum ante hos aliquot annos in Academia Parisiensi,
cum censeretur, motum Apogei Lunaris observatum non cohærere cum grayitate decrescente
in 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
changed by any other given interval however much the latter may be smaller or greater
than the change of the abscissa, or however much greater than any given quantity it may
be. Further, if the difference in the abscissa is infinitesimal, & we call it an infinitesimal
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,
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
found approximates, or which it osculates with any order of osculation, the difference of
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
has been found is inclined at a finite angle to the axis, the difference of the abscissa is of
the same order as the difference of the ordinate ; when the tangent is parallel to the axis,
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,
& this is either increased or diminished in any way, the difference of the ordinate will be
the whole ordinate itself, for at the limit-point itself the ordinate is indeed equal to zero.
55. Cor. 5. The arcs of repulsion or attraction, which are intercepted between any
pair of limit-points, may recede from the axis to any extent; & thus, it may happen that
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
any given points.
57. Cor. 6. The curve may have the axis CAC as an asymptote in the directions
of C' & C, in such a manner that the asymptotic arc is either repulsive or attractive ; also any
arc intercepted between a pair of limit-points may go off to infinity, & have for an
asymptote a straight line perpendicular to the axis, however near or far from either limit-
point. 4
58. For, if we suppose that the last two limit-points coincide, as the two intersections
coincide & become a point where the curve touches the axis; & then suppose that the
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,
through which the curve has to pass, should go off to infinity; then the arc of the curve
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,
the law of forces is obtained expressed as a definite function of the distance with many
terms; or rather, by means of an equation involving the abscissa & the ordinate, & powers
of these, along with given straight lines, & not by a single power of the distance. There
are some who think that representation by means of a single power is to be preferred to
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 1/xm, 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. u
61. This question came in for a large amount of discussion a number of years ago in
the Academy of Paris. For, it was thought that the motion of the lunar apogee, as observed,
did not agree with the idea of gravity decreasing in the inverse duplicate ratio of the distances.
They considered that an expression for gravity should be employed, in which it was represented
E E
The change of the
abscissa may bear
any ratio to the
change of the ordi-
nate.
Proof finite
ratios.
for
The same for any
order of infinitesi.
mals.
This relation de-
pends on the posi-
tion of the tan-
gent.
The case of an
abscissa terminat-
ing at one of the
limit-points.
The arcs may re.
cede from the axis
to any extent.
Proof of this state-
ment.
The curve can
have its last branch
asymptotic, & also
other asymptotic
branches.
of the
of this
The proof
first part
Statement.
Of the
of the
The proof
remainder
Statement.
The law of forces
is here represented
by a function of the
distance ; many
others think that a
single power of the
distance is prefer-
able ; their reasons.
The occasion on
which this question
was discussed in the
Academy of Paris.
4I8 PHILOSOPHIÆ NATURALIS THEORIA
Occasionem substi-
tuendi tum functio-
nem cessasse, sed
rationes c o ntra
allata s n u 1 1 a m
habere vim : curvas
O m ne s uniformis
naturæ esse in se
æque simplices.
Esse æque simpli-
cem re latio n e m
O r d i n at a ru m ad
abscissas : termino-
rum multitudinem
pro ea, exprimenda
oriri a nostro cog-
noscendi modo.
Origo ejus modi ab
intuitione, quam
habemus nos hom-
ines naturæ solius
rectæ, a d q u a s
omnes curvas refer-
imus.
Aliud me nt iu m
genus a d expri-
mendam relationem
potentiæ necessario
adhibiturum æqua-
lem, vel majorem
farraginem.
Sola etiam poten-
tiæ ex pressione
includi etiam apud
nos homines para-
metros plures: para-
meter in unitate
arbitraria, & affix- %
ione certæ vis ad
certam distantiam,
quævis.
� o 4 b ¢ e • • • «• • •
expressa per binomium z8 + 32* cujus pars prior in magnls, pars posterior in exiguis
distantiis respectu sociæ partis evanesceret ad sensum, sed illa prior in distantia Lunæ a
Terra adhuc turbaret hánc posteriorem, quantum satis erat åd eam præstandam rem.
Atque eam ipsam binomii expressionem adhibuerant jam plures Physici ad deducendam
simul ex eadem formula gravitatem, & majores minimiarum particulàrum attractiones, ac
multo validiorem cohæsionem, ut innuimus num. 121 : atqué hæ difficultates in Parisiensi
Encyclopædia inculcantur ad vocem Attractio, Tomo I tùm edito.
62. Paullo post, correctis calculis innotuit, motum Apogei lunaris ea composita formula
Inon indigere : at rationes contra id propositæ, quæ multo magis contra meam virium
legem pugnarent, meo quidem judicio nullam habent vim. Nam in primis quod ad
simplicitatem pertinet, hic habent locum ea omnia, quæ dicta sunt in ipso opere num. I 16
de simplicitate curvarum. Formula exprimens solam potentiam quandam distantiæ
designatæ per abscissam exprimit ordinatam ad locum geometricum pertinentem ad
familiam, quam exhibet [284] y = x", qui quidem locus est Parabola quædam ; si m
sit numerus positivus, nec sit unitas : recta ; si sit unitas, vel zero : quædam Hyperbola ;
si sit numerus negativus : formula autem continens functionem aliam quamvis exprimit
ordinatam ad aliam curvam, quæ erit continua, & simplex, si illa formula per divisionem non
possit discerpi in alias plures. Omnes autem ejusmodi curvæ sunt æque simplices in se,
& aliæ aliis sunt magis affines, aliæ 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 æque simplices sint omnes illæ, quæ ductum uniformem
habent, & naturam ubique constantem.
63. Hinc ipsa ordinata ad quamvis naturæ uniformis curvam est quidam terminus
simplicissimæ relationis cujusdam, quam habet ordinata ad abscissam, cui termino impositum
est generale nomen functionis continens sub se omnia functionum genera, ut etiam quam-
cunque solam potentiam, & si haberemus nomina ad ejusmodi functiones denominandas
singillatim ; haberet nomen suum quævis ex ipsis, ut habet quadratum, cubus, potestas
Si omnia curvarum genera, omnes ejusmodi relationes nostra mens intueretur
immediate in se ipsis ; nulla indigeremus terminorum farragine, nec multitudine signorum
ad cognoscendam, & enuntiandam ejusmodi functionem, vel ejus relationem ad abscissam.
64. Verum nos, quibus uti monui recta linea est omnium locorum geometricorum
simplicissima, omnia referimus ad rectam, & idcirco etiam ad ea, quæ oriuntur ex recta,
ut est quadratum, quod fit ducendo perpendiculariter rectam super aliam rectam æqualem,
& cubus, qui fit ducendo quadratum eodem pacto per aliam rectam primæ radici æqualem,
quibus & sua signa dedimus ope exponentium, & universalizando exponentes efformavimus
nobis ideas jam non geometricas superiorum potentiarum, nec integrarum tantummodo,
& positivarum, sed etiam fractionariarum, & negativarum : & vero etiam, abstrahendo
semper magis, irrationalium. Ad hasce potentias, & ad producta, quæ simili ductu conci-
piuntur genita, reducimus cæteras functiones omnes per relationem, quam habent ad
ejusmodi potentias, & producta earum cum rectis datis, ac ad eam reductionem, sive ad
expressionem illarum functionum per hasce potentias, & per hæc producta, indigemus
terminis jam paucioribus, jam pluribus, & quandoque etiam, ut in functionibus transcendent-
alibus, serie terminorum infinita, quæ ad valorem, vel naturam functionis propositæ accedat
semper magis, utut in hisce casibus eam nunquam ac-[285]-curate attingat : habemus
autem pro magis, vel minus compositis eas, quæ pluribus, vel paucioribus terminis indigent,
sive quæ ad solas potentias relationem habent propiorem.
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
aliam potentiam, contemplaretur illam eandem relationem, sed inverse assumptam ita, ut
incipiendo a functione ipsa per eam, & per similes ejus functiones, ac functionum citeriorum
functiones ulteriores, addendo, ac subtrahendo deveniret demum ad quæsitam. Relatio
potentiæ ad functionem, & nexus mutuus compositionem habet, & multitudinem terminorum
inducit : uterque relationis terminus est in se æque simplex.
66. Quod pertinet ad parametros, quas dicitur includere functio, non autem potentia
distantiæ, non est verum id ipsum, quod potentia parametros non includat. Formula
â, includit unitatem ipsam, quæ non est aliquid in se determinatum, sed potest exprimere
magnitudinem quamcunque. Et quidem ea species includit omnes species Hyperbolarum,
SUPPLEMENT III 4I9
by the formula of two terms, aſ w8 + b/w”; 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 Encyclopædia Parisiensis under the
heading Attraction, in Vol. I published at that time. p
62. Shortly afterwards, the calculations were corrected & it was found that the motion
of the lunar apogee did not necessitate this compound formula. But the arguments brought
forward against it, which were still more in opposition to this Theory of mine with regard
to the law of forces, have no weight, at any rate in my eyes. For, in the first place, as
regards simplicity, all those things held good in this case, which I stated in this work,
Art. II6, with regard to simplicity of curves. A formula in terms of a single power of
the distance represented by an abscissa expresses the ordinate of a geometrical locus belonging
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.
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
simple relation that the ordinate has to the abscissa. To this term there is given the
general name, function; this name includes every kind of function, for instance, even a
single power. If we had names to denote such functions singly, each of them would have
its own name, just as a square, a cube, or any other power. If our minds were capable of
viewing all kinds of curves, & all such relations in themselves, at a glance, then there would
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
loci, refer all curves to a straight line, and therefore also to all those things that arise from a
straight line; such as a square, which is formed by moving a straight line perpendicular
to another straight line which is equal to it; & a cube, which is formed by moving the
square in the same way all along another straight line equal to its prime root. To these
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
line, it would consider a single function of that curve to be the most simple of all; &, to
express a square or another power, it would consider the self-same relation, inversely taken,
so that, beginning with the function, through it & like functions of it, & of higher functions
of these lower functions, by addition & subtraction, the mind would finally arrive at the
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.
66. As regards the introduction of parameters, which they say are included in a function
but not in a power of the distance, it is not true that a power does not include a
parameter. The formula 1/xm includes unity itself; & this is not something that is
self-determinate, but something that can express any magnitude. Indeed, that
species of formula includes all species of hyperbolas, &, if the exponent m is given
The reasons for
substituting the
formula for the
function, which
then existed, have
ceased to exist ;
but the arguments
brought forward
against it have no
weight ; all curves
that are uniform
are in themselves
equally simple.
The relation be.
tween the ordinate
& the abscissa is
equally simple ; the
number of terms
used to express
this relation arises
from our way of
knowing it.
The origin of this
method comes from
the intuition which
we men have of the
nature of a straight
line alone, to which
we refer all curves.
Another type of
mind, to express
the relation of a
power would neces.
Sarily have to use
an equal or greater
medley of terms.
Even in the single
expression of a
power, we men
include severa. I
p a r a meters ; a
parameter in the
arbitrary unity, &
the combination of
a certain force with
some certain dis-
tance,
42O PHILOSOPHIÆ NATURALIS THEORIA
IParameter in ex-
ponente potentiæ.
Non esse, cur vis
debeat pendere a
sola distantia
etiam, si vis sit
essentialis proprie-
tas materiæ.
Etiam si vis de-
beat pendere a solis
distantiis, Ordina-
tas quoque in se,
data, curva, pendere
a solis abscissis.
Parametros ipsas
esse ** distantias'':
€ 2. S functionem
esse ingressas, quod
in datis distantiis
debuerit haberi vis
data, vel nulla.
Argumentum con-
trarium a defectu
rationis sufficientis.
Si vis sit essentialis
materiæ ; rationem
taliu m parame-
trorum esse ipsam
eju s natura m :
cur hoc genus mate-
riæ existat, ration-
em esse arbitrium
Creatoris : idem,
si ea non sit essen-
tialis. -
Præter arbitrium
retorsio in poten-
tia : rationem utro-
bique esse fines,
quos sibi ipse pro-
posuerit, qui pos-
sunt esse nobis
ignoti.
ac definito exponente m, exprimit unicam quidem earum speciem, sed quæ continet infinitas
numero individuas Hyperbolas, quarum quælibet suam parametrum diversam habet pro
diversitate unitatis assumptæ. Potest quidem quævis ex iis Hyperbolis ad arbitrium
assumi ad exprimendam vim decrescentem in ea ratione reciproca ; sed adhuc in ipsa
expressione includitur quædam parameter, quæ determinet certam vim a certa ordinata
exprimendam, sive certam vim certæ distantiæ respondentem, qua semel determinata
remanent determinatæ reliquæ omnes, sed ipsa infinitis modis determinari potest, stante
expressione facta per ordinatas ejusdem curvæ, sive per eandem potentiæ formulam.
Ejusmodi primus nexus a sola distantia utique non pendet.
67. Accedit autem alia quasi parameter in exponente potentiæ : illius numeri m
determinatio utique non pendet a distantia, nec distantiam aliquam exprimit.
68. Sed nec illud video, cur etiam si dicatur vis esse proprietas quædam materiæ
essentialis, ea debeat necessario pendere a solis distantiis. Si esset quædam virtus, quæ a
materiæ puncto quovis egressa progrederetur motu uniformi, & rectilineo ad omnes circum
distantias : tum quidem diffusio ejus virtutis per orbes majores æque crassos fieret in
ratione reciproca duplicata distantiarum, & a distantiis solis penderet ; quanquam ne tum
quidem ab iis penitus solis, sed ab iis, & exponente secundæ potentiæ, 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 solæ distantiæ ingredi formulam functionis exprimentis vim.
69. Verum admisso etiam, quod necessario vis debeat pendere a solis distantiis, nihil
habetur contra expressionem factam per functionem quandam. Nam ipsa functio per se
immediate pendet a distantia, & est ordinata quædam ad curvam quandam certæ naturæ,
respondens abscissæ datæ cuilibet sua. Parametri inducuntur ex eo, quod illius relationem
ad abscissam exprimere debeamus per potentias abscissæ, & potentiarum producta cum
aliis rectis ; sed in se, uti supra diximus, ejusdem est naturæ & illa functio, ac potentia
quævis, & illa, ut hæc, ordinatam immediate simplicem exhibet respondentem abscissæ
ad curvam quandam uniformis, & in se simplicis curvæ.
70. Præterea ipsæ illæ parametri, quæ formulam functionis ingrediuntur, possunt
esse certæ quædam distantiæ & assumi debere ad hoc, ut illis datis distantiis illæ datæ, &
non aliæ vires respondeant. Sic ubi quæsita est formula, quæ exprimeret æquationem
ad curvam quæsitam, assumpsimus quasdam distantias, in quibus curva secaret axem,
nimirum in quibus, evanescente vi haberentur limites, & earum distantiarum valores ingressi
sunt formulam inventam, ut quædam parametri. Possunt igitur ipsæ parametri esse
distantiæ quædam ; 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.
7I. Reliquum est, ut dicamus aliquid de Ratione Sufficienti, quæ dicitur parametros
excludere, cum non sit ratio, cur aliæ præ aliis parametri seligantur.
72. Inprimis si vis est in ipsa natura materiæ ; nulla ratio ulterior requiri potest
præter eam ipsam naturam, quæ determinet hanc potius, quam aliam vim pro hac potius,
quam pro illa distantia, adeoque hanc potius, quam aliam parametrum. Quæri ad summum
poterit, cur elegerit Naturæ Auctor eam potissimum materiam, quæ eam legem virium
haberet essentialem, quam aliam : ubi ego quidem, qui summam in Auctore Naturæ
libertatem agnosco, censeo, ut in aliis omnibus, nihil aliud requiri pro ratione sufficienti
electionis, quam ipsam liberam determinationem Divinæ voluntatis, a cujus arbitrio
pendeat tum, quod hanc potius, quam aliam eligat rem, quam condat, tum quod ea re
hanc in se naturam habente, ubi jam condita fuerit, utatur ad hoc potius, quam ad illud
ex tam multis, ad quæ natura quævis a tanti Artificis manu adhibita potest esse idonea.
Atque hæc responsio [287] æque valet, si vis non est ipsi materiæ essentialis, sed libera
Auctoris lege sancita, quo casu ipse pro libero arbitrio suo hanc huic materiæ potuit legem
dare præ aliis electam. -
73. At si ratio etiam exhiberi debeat, quæ Auctorem Naturæ potuerit impellere ad
seligendam materiam hac potissimum præditam essentiali virium lege, vel ad seligendam
pro hac materia hanc legem virium ; quæri primo potest, cur hunc potius exponentem
potentiæ elegerit, & hanc parametrum in unitate inclusam, sive in quadam determinata
SUPPLEMENT III 42I
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
the power; the determination of the number m at any rate does not depend on the distance,
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
matter, it should of necessity depend on distances alone. If it were some virtue, which
proceeded from any point of matter & progressed with uniform motion in a straight line
to all distances round; then indeed the diffusion of this virtue through greater spheres
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
alone; still there is nothing against the expression being formed of some function. For
the function in itself depends directly upon distance, & is an ordinate to some curve of
known nature, corresponding to its own given abscissa, which may be anything you please.
Parameters are induced by the fact that we have to express the relation of the ordinate
to the abscissa by means of powers of the abscissa, & the products of these powers with
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
known distances; & they have to be assumed for the purpose of ensuring that to these
given distances those given forces, & not others, correspond. So, when we seek a formula
to express the equation to the curve required, we assume certain distances in which the
curve shall cut the axis; that is to say, distances for which, as the force vanishes, we shall
obtain limit-points; & the values of these distances have entered the formula we have
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
said to exclude parameters, because there is no reason why some parameters should be
chosen in preference to others.
72. First of all, if force is an essential property of matter, there is no need for any
other reason beside that of the very nature of matter, to determine that this, rather
than another, force should correspond to this, rather than to another, distance; & therefore
this parameter, rather than any other. It may be asked, & we can go no further, why
the Architect of Nature chose this matter in particular, such as should have this essential
law of forces, & no other. In that case, I, who believe in the Supreme freedom of the
Architect of Nature, think, as in all other things, that there is nothing else required for the
sufficient reason for His choice beyond the free determination of the Divine will. Upon
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
give this law to this matter, having chosen it in preference to all other laws.
73. Now, if we have also to give the reason which might have forced the Author of
Nature to select in particular this matter possessed of this essential law of forces, or to
select for this matter this law of forces especially; it may first be asked why He should have
preference for this exponent of the power, this parameter that is included in the unity,
There is a para-
meter in the expo-
nent of the power.
There is no reason
why it should
depend on the
distance alone, if
force is an essential
property of matter.
Even if the force
did depend on the
distances a 1 on e,
the ordinates also,
in themselves, de-
pend on the ab-
scissae alone, for
any given curve.
The parameters
themselves are dis-
tances : they have
come into the
function, because
at given distances
there must be a
given force or none
at all.
The argument
against it from a
defect of Sufficient
Tea,SO11.
If force is a n
essential property
of m atter, the
reason for such
parameters is the
very nature of
matter ; why Such
matter exists is due
to the will of the
Creator ; the same
thing if force is not
an essential.
There is something
beyond will in the
limitation of his
power; the reason in
both cases is the aim
that He set before
Himself ; & this we
may not know.
422 PHILOSOPHIÆ NATURALIS THEORIA
JEvolutio finis
s iu s : necessitas
habendi h u m c
nexum ab Algebra
humana non ex-
primi bile m, nisi
per functionem, ad
solvendum creati-
onis problema, pro
hac cOrpOrum con
stitutione, & mo-
tuum serie.
Id non potuisse
solvi per solam
potentiam : legem
quadrati distantiæ
non esse p er-
fectissimam.
Conclusio contra
necessitatem, vel
convenientiam sol-
ius potentiæ.
ip-
distantia quandam determinatam vim. Quod de iis dicitur, applicari poterit parametris
reliquis functionis cujusvis. Ut ille exponens, illa unitas, ille nexus potuit habere aliquid,
quod cæteris præstaret ad eos obtinendos fines, quos sibi Naturæ Auctor præscripsit ; sic
etiam aliquid ejusmodi habere poterant reliquæ omnes quotcunque, & qualescunque
parametri.
74. Deinde rem ipsam diligenter consideranti facile patebit, ad obtinendos fines,
quos sibi Naturæ Auctor debuit proponere, non fuisse aptam solam potentiam quandam
distantiæ pro lege virium, sed debuisse assumi functionem, quæ ubi exprimi deberet per
nostram humanum Algebram, alias quoque parametros admisceret. Si ex. gr. voluisset
per eandem vim & motum Planetarum ad sensum ellipticum cum Kepleriano nexu inter
quadrata temporum periodicorum, & cubos distantiarum mediarum, & cohæsionem per
contactum, nulla sola potentia ad utrumque præstandum finem fuisset satis, quem finem
«• «• • 4. � © *• • *. e-»
obtinuisset illa, formula z8 + %, At nec ea formula potuit ipsi sufficere, si vera est Theoria
2% -
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.
Cohæsio punctorum se invicem repellentium in minimis distantiis, & attrahentium in
majoribus haberi non potuit sine intersectione curvæ cum axe, quæ intersectio sine para-
metro aliqua non obtinetur. Verum ad omnem hanc phænomenorum seriem obtinendam
multo pluribus, uti ostensum est suo loco, intersectionibus curvæ, & flexibus tam variis
opus erat, quæ sine plurimis parametris obtineri non poterant. Consideretur elevatissimum
inversum problema affine alteri, cujus mentio est facta num. 547, quo quæratur numerus
punctorum, & lex virium mutuarum communis omnibus necessaria ab habendam ope
cujusdam primæ combinationis, hanc omnem tam diuturnam, tam variam phænomenorum
seriem, cujus perquam exiguam particulam nos homines intuemur, & statim patebit eleva-
tissimum debere esse, & respectu habito ad nostros exprimendi modos complicatissimum
genus curvæ ad ejusmodi problematis solutionem ne-[288]-cessarium ; quod tamen problema
certas quasdam parametros in singulis saltem solutionibus suis, quæ numero fortasse infinito
sunt, involveret, sola unica potentia ad tanti problematis solutionem inepta.
75.?Debuit igitur Naturæ Auctor, qui hanc sibi potissimum Phænomenorum seriem
proposuit, parametros quasdam seligere, & quidem plures, nec potuit solam unicam pro
legé virium exprimenda distantiæ potentiam adhibere : ubi & illud præterea ad rem eandem
confirmandam recolendum, quod a num. I24, dictum est de ratione reciproca duplicata
distantiarum, quam vidimus non esse omnium perfectissimam, nec omnino eligendam,
& illud, quod sequenti horum Supplementorum paragrapho exhibetur contra vires in
minimis distantiis attractivas & excrescentes in infinitum, ad quas sola potentia demum
deducit, * - _.
76. Atque hoc demum pacto, videtur mihi, dissoluta penitus omnis illa difficultas,
quæ proposita fuerat, nec ulla esse ratio, cur sola potentia quædam distantiæ anteferri
debuérit "functioni utcunque, si nostrum exprimendi modum spectemus, complicatissimæ.
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/x" + b/w”. 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. I24 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
The evolution
O f this a im ;
the necessity for
this connection
which is not
expressible by
human algebra,
unless by a func-
tion, to solve the
problems of crea-
tion for this consti-
tution of bodies, &
Series of motions.
It could not be
solved by a single
power ; the law of
the squares of the
distances is not the
most perfect.
Conclusion against
the necessity or the
convenience of a
distance should be preferred to a function, no matter how complicated it may be, if single power.
regard is paid to our methods of expressing it.
IPrima difficultas
ex eo, quod ubi
conatus deberet
esse maximus in
appulsu, debeat
esse nullus, vel
irritus.
Secunda, si ratio
sit reciproca, dis-
tantiæ, a viabsolute
infinita, ad quam
deveniri deberet.
T e rtia, ex eo,
quod, si sit major
quam simplex, de-
beat in contactu
deveniri etiam ad
velocitatem infini-
tam.
Alia, absurda : si
ratio sit duplicata,
regressus a centro :
saltus ab accelera-
tione crescente ad
nullam in ingressu
in superficiem sphæ-
ricam.
ERegressus a centro
simul, & procursus
ultra, ad eandem
[289] § IV
Comtra vires im mimimis distamtiis attractivas, & excrescemtes im
infimitum (e)
77. At præterea contra solam attractionem plures habentur difficultates, quæ per
gradus crescunt. Nam inprimis si eæ imminutis utcunque distantiis agant, augent velóci-
tatem usque ad contactum, ad quem ubi deventum est, incrementum velocitatis ibi per
saltum abrumpitur, & ubi maxima est, ibi perpetuo incassum nituntur partes ad ulteriorem
effectum habendum, & necessario irritos conatus edunt.
78. Quod si in infinitum imminuta distantia, crescant in aliqua ratione distantiarum
reciproca ; multæ itidem difficultates habentur, quæ nostrum oppositam sententiam
confirmant. Inprimis in ea hypothesi virium deveniri potest ad contactum, in quo vis,
sublata omni distantia, debet augeri in infinitum magis, quam esset in aliqua distantia.
Porro nos putamus accurate demonstrari, nullas quantitates existere posse, quæ in se
infinitæ sint, aut infinite parvæ. Hinc autem statim habemus absurdum, quod nimirum
si vires in aliqua distantia aliquid sunt, in contactu debeant esse absolute infinitæ.
79. Augetur difficultas, si debeat ratio reciproca esse major, quam simplex (ut ad gravi-
tatem requiritur reciproca duplicata, ad cohæsionem adhuc major) & ad bina puncta
pertineat. Nam illa puncta in ipso congressu devenient ad velocitatem absolute infinitam.
Velocitas autem absolute infinita est impossibilis, cum ea requirat spatium finitum percursum
momento temporis, adeoque replicationem, sive extensionem simultaneam per spatium
finitum divisibile, & quovis finito tempore requirat spatium infinitum, quod cum inter
bina puncta interjacere non possit, requireret ex natura sua, ut punctum ejusmodi
velocitatem adeptum nusquam esset.
80. Accedunt plurima absurda, ad quæ ejusmodi leges nos deducunt. Tendat punctum
aliquod in fig. 72 in centrum F in ratione reciproca duplicata distantiarum, & ex A pro-
jiciatur directione AB perpendiculari ad AF, cum velocitate satis exigua : describet
Ellipsim ACDE, cujus focus erit F, & semper regredietur ad A. Decrescat velocitas AB per
gradus, donec demum evanescat. Semper magis arctatur Ellipsis, & vertex D accedit ad
focum F, in quem demum recidit abeunte Ellipsi in rectam AF. Videtur igitur id [29o]
punctum sibi relictum debere descendere ad F, tum post acquisitam ibi infinitam veloci-
tatem, eam sine ulla contraria vi convertere in oppositam, & retro regredi. At si id punctum
tendat in omnia puncta superficiei sphericæ, vel globi EGCH in eadem illa ratione ;
demonstratum est a Newtono, debere per AG descendere motu accelerato eodem modo,
quo acceleraretur, si omnia ejusmodi puncta superficiei, vel sphæræ compenetrarentur in
F: abrupta vero lege accelerationis in G, debere per GH ferri motu æquabili, viribus
omnibus per contrarias actiones elisis, tum per HI tantundem procurrere motu retardato,
adeoque perpetuam Oscillationem peragere, velocitatis mutatione bis in singulis oscillation-
ibus per saltum interrupta.
8I. In eo jam absurdum quoddam videtur esse : sed id quidem multo magis crescit ;
si consideretur, quid debeat accidere, ubi tota sphærica superficies, vel tota sphæra abeat
in unicum punctum F. Tum itidem corpus sibi relictum, deveniet ad centrum cum
infinita velocitate, sed procurret ulterius usque ad I, dum prius, ubi Ellipsis evanescebat,
debebat redire retro. Nos quidem pluribus in locis alibi demonstravimus, in prima
distantiam, v e l
saltus in tanto
procursu, s 1 n e
præviis minoribus.
(e) Haec excerpta sunt ex eadem dissertatione De Lege Virium in Natura existentium a mum. 59.
* 24
SUPPLEMENT IV 425
FIG. 72.

426
PHILOSOPHIAE NATURALIS THEORIA
FIG. 72.

§ IV
Arguments against forces that are attractive at very sma//
distances and increase indefinitely ()
77. Besides, there are many difficulties in the way of attraction alone, which increase
by degrees. For, first of all, if these act at diminished distances of any sort, they will
increase the velocity right up to the moment of contact: & when contact is attained,
the increment of the velocity will then be suddenly broken off; & when this is greatest,
the parts will continually strive in vain to produce a further effect, & the efforts will
necessarily turn out to be fruitless. © ©
'78. But if, when the distances are infinitely diminished, the forces increase according
to some ratio that is inversely as the distances, many difficulties will again be had, which
confirm our opposite opinion. On that hypothesis of forces especially, contact may be
attained, in which, as all distance is taken away, the force is bound to be increased
infinitely more than it would be at a distance of some amount. Further, I think that it
is rigorously proved that no quantities can possibly exist, such as are infinite in themselves
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. . º
79. The difficulty is increased, if the inverse ratio is greater than a simple ratio (as
for gravity we require the inverse square, & for cohesion one that is still greater); & it
has to do with a pair of points. For these points on collision will attain a yelocity that is
absolutely infinite. . But such an absolutely infinite velocity is impossible, since it requires
that a finite space should be passed over in an instant of time, that is, replication, or simul-
taneous extension through finite divisible space; & for any finite time it would require
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.
72, let any point tend towards a centre F in the inverse ratio of the squares of the distances,
& suppose it to be projected from the point A in a direction, AB, perpendicular to AF,
with a fairly small velocity. Then it will describe the ellipse ACDE, of which F is the
focus; & it will always return to A. Now let the velocity AB decrease by degrees, until
finally it vanishes. Then, the ellipse will continually, become more & more pointed, &
the vertex D will approach the focus F, & will coincide with It when the ellipse becomes
the straight line AF. It seems therefore that the point, if left to itself would fall towards
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
to follow. For, let us consider what will necessarily happen when the whole of the spherical
surface, or the whole of the sphere, becomes but a single point at F. Then indeed, the
body if left to itself would arrive at the centre with infinite velocity; but it would pass
through it & beyond as far as I, whereas in the former case when the ellipse vanished, it
had to return to its original position. Indeed, in many places elsewhere, I have proved
(e) These paragraphs are quoted from the same dissertation De Lege Virium in Natura existentium, starting with
Art. 59.
The first difficulty
arises from the fact
that, when the
effort should be
greatest on ap-
proach, it is bound
to be either nothing
Or to have no effect.
The second diffi-
culty arises from
the fact that, if
the ratio is in-
versely as the
distance, we must
COIme to a force
that is absolutely
infinite.
A third difficulty
from the fact that,
if the inverse ratio
is greater than a
simple one, we are
bound also to have
On COnta, Ct an
infinite velocity.
Other absurdities ;
if the ratio is the
square of the dis-
tance, there will be
return from the
centre ; a sudden
change from an
acceleration that is
increasing to one
that is nothing on
entering a spherical
Surface.
Sim ult a n e o us
return from the
Centre, & motion
beyond it to an
equal distance ; or
a sudden change in
this great motion,
without smaller
preparatory mo-
tions.
427
428 PHILOSOPHIÆ NATURALIS THEORIA
Si ratio sit tripli-
tata pejus : annihi-
latio puncti i n
appulsu ad cen-
trum.
Pejus in potentiis
altioribus : præpa-
ratio* ad demon-
strandum absur-
dum.
Partem fore majo-
rem toto.
determinatione latere errorem, cum Ellipsi evanescente, nullæ jam adsint omnes vires,
quæ agunt per arcum situm ultra F ad partes D, quæ priorem velocitatem debebant extin-
guere, & novam producere ipsi æqualem. Verum adhuc habetur saltus quidam, cui &
Natura, & Geometria ubique repugnat. Nam donec utcunque parva est velócitas, 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, quæ sit in ratione reciproca duplicata distantiarum, debeat e centro regredi retro ;
tum saltus habetur similis, ubi prius in sphæricam superficiem vel sphæram tendat, quæ
paullatim abeat in centrum. Donec enim aderit superficies illa, vel sphæra, habebitur
semper is procursus, qui abrumpetur in illo appulsu totius superficiei ad centrum, quin
habeantur prius minores procursus.
82. Hæc quidem in ratione reciproca duplicata distantiarum : in reciproca triplicata
habentur etiam graviora. Nam si cum debita quadam velocitate projiciatur per rectam
AB fig. 73 continentem angulum acutum cum AP, mobile, quod urgeatur in P vi crescente
in ratione reciproca triplicata distantiarum ; demonstratur in Mechanica, ipsum debere
percurrere curvam ACDEFGH, quæ vocatur spiralis logarithmica, quæ hanc habet pro-
prietatem, ut quævis recta, ut PF, ducta ad quodvis ejus punctum, contineat cum recta
ipsam ibidem tangente angulum æqualem angulo PAB, unde illud consequitur, ut ea quidem
ex una parte infinitis spiris cir-[29I]-cumvolvatur circa punctum P, nec tamen in ipsum
unquam desinat : si autem ducatur ex P recta perpendicularis ad AP, quæ tangenti AB
occurrat in B, tota spiralis ACDEFGH in infinitum continuata ad mensuram longitudinis
AB accedat ultra quoscunque limites, nec unquam ei æqualis 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 illa spiralis, quæ 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 formulæ ana-
lyticæ exhibent ejus locum post id tempus impossibilem, sive, ut dicimus, imaginarium ;
quo quidem argumento Eulerus in sua Mechanica affirmavit illud, debere id mobile in
appulsu ad centrum virium annihilari. Quanto satius fuisset inferre, eam legem virium
impossibilem esse ?
83. Quanto autem majora absurda in ulterioribus potentiis, quibus vires alligatæ
sint, consequentur ? Sit globus in fig. 74 ABE, & intra ipsum alius Abe, qui priorem
contingat in A, ac in omnia utriusque puncta agant vires decrescentes in ratione reciproca
quadruplicata distantiarum, vel majore, & quæratur ratio vis puncti constituti in concursu
A utriusque superficiei. Concipiatur uterque resolutus in pyramides infinite arctas, quæ .
prodeant ex communi puncto A, ut BAD, bAd. In singulis autem pyramidulis divisis
in partes totis proportionales sint particulæ MN, mm similes, & similiter positæ. Quantitas
materiæ in MN, ad quantitatem in mm 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 materiæ directe, & ut quarta potestas distantiarum reciproce, quæ
itidem distantiæ sunt, ut radii sphærarum ; erit vis in partem MN, ad vim in partem mm
directe, ut tertia potestas radii majoris ad tertiam minoris, & reciproce, ut quarta potestas
ipsius. Quare manebit ratio simplex reciproca radiorum.
84. Minor erit igitur actio singularum particularum homologarum MN, quam mm,
in ipsa ratione radiorum, adeoque punctum A minus trahetur a tota sphæra ABE, quam
a sphæra Abe, quod est absurdum, cum attractio in eam sphæram minorem debeat esse pars
SUPPLEMENT IV 429
FIG. 74.

430 PHILOSOPHIAE NATURALIS THEORIA
FIG. 74.

SUPPLEMENT IV 43I
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
the distances; for the inverse ratio of the cubes, we have even more serious difficulties.
For, if a body is projected along AB, in Fig. 73, making an acute angle with AP, with a
certain suitable velocity, & it is attracted towards P with a force increasing in the inverse
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
the forces may be connected In Fig. 74, let ABE be a sphere, & within it let there be
another one Abe, touching the former at A ; & suppose that on all points of each of them
there act forces which decrease in the inverse ratio of the fourth powers of the distances,
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
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
If the ratio is the
triplicate, it is still
worse ; annihila-
tion of the point on
arrival at the
Centre.
Still w or se for
higher powers;
prep a ration for
demonstrating an
absurdity.
The part greater
than the whole,
432 PHILOSOPHIÆ NATURALIS THEORIA
O mnia a b s u r d a.
CeSSare, s i i n
minimis distantiis
habeatur repulsio,
q u æ appulsum
impediat.
attractionis in sphæram majorem, quæ continet minorem, cum magna materiæ parte sita
extra ipsam usque ad superficiem sphæræ 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 R", posito R pro radio, & m pro quovis
numero ternarium superante, erit attractio sphæræ eodem argumento reciproce, ut R**,
quæ eo majorem indicat vim in sphæram minorem respectu majoris ipsam continentis,
quo numerus m est major.
85. Hoc quidem pacto inveniuntur plurima absurda in variis generibus attractionum
quæ si repulsiones, in minimis distantiis habeantur pares extinguendæ velocitati cuilibet
utcunque magnæ, cessant illico omnia, cum eæ repulsiones mutuum accessum usque ad
concursum penitus impediant. Inde autem manifesto iterum consequitur, repulsiones
in minimis distantiis præferendas potius esse attractioni, ex quarum variis generibus tam
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 R*-*; & 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,
1 © 11 di fä e l t t d t l it h 1 ities cease to be, if
repulsions at very small distances, sufficiently great to destroy any velocity however large, thºſe is repulsion
all these absurdities would cease to be immediately, for these repulsions would prevent iº º
mutual approach up to the point of actual contact. Hence it once again manifestly follows º: ºw.
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.
FF
Propositio pro b-
lematis de aequili-
lirio punctorum
quatuor, quorum
bina e xtrema
ha b ea nt quaS-
cunque massa, cum
viribus externissibi
proportionalibus, &
altera e mediis vim
a fulcro.
Vis ex bimis ex-
tremis in alterum
e mediis.
Vis, quam debet
habere illud pri-
mum, composita e
quatuor : enum-
e ra ti o virium
pertinentium a d
omnia puncta.
[293] S V
De AEquilibrio binarum massarum connexarum invicem per bina
alia puncta (l )
86. Continetur autem, quod pertinet ad momentum in vecte, & ad aequilibrium,
sequentis problematis solutione. Sit in fig. 75 quivis numerus punctorum materiae in A,
qui dicatur A, in D quivis alius, qui dicatur D, & puncta ea omnia secundum directiones
AZ, DX parallelas rectae data CF sollicitentur simulviribus, quae sint aequales inter omnia
puncta sita in A, itidem inter omnia sita in D, licet vires in A sint utcunque diversa a
viribus in D. Sint autem in C, & B bina puncta, quae in se invicem, & in illa puncta sita
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, 8 quantitas, ac quaeritur directio vis,
qua fulcrum urgeri debet a puncto C.
87. 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 ea 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 repellata 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 quovis 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 ductae 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 aequalis LD ductae in D.
88. Porro vis composita ex illis binis, quibus urgetur punctum B, elidi debet ab actione
mutua inter ipsum, & C; quare debebit habere directionem rectae BC in casu, quem exhibet
figura, in quo C jacet in angulo ABD : nam si angulus ABD hiatum obverteret ad partes
oppositas, ut C jaceret extra angulum ; ea haberet directionem CB, & reliqua omnis
demonstratio redireteodem. Punctum autem C ob actionem, 8 reactionem aequales
debebit habere vim aequalem, 8 contrarium illi, quam exercet B, adeoque vim aequalem,
& eiusdem 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) Excerpta hac sunt ex Synopsi Physica Generalis P. Caroli Benvenuti Soc. fesu, num. 146, cui hanc solutionem
ibi imprimendam tradideram,
434
SUPPLEMENT V 435
FIG. 75.

436
THEORIA
PHILOSOPHIAE
NATURALIS
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
the solution of the following problem. In Fig. 75, let there be any number of points of
matter at the point A, & let the number be called A ; similarly, any other number at D,
called D ; & suppose that all these are at the same time under the action of forces along
the directions AZ, DX parallel to the given straight line CF, & that these forces are equal
to one another for all the points situated at A, & also for all the points situated at D,
although the forces at A may be altogether different from those at D. Also, at C & B, let
there be two points, which act mutually upon one another & upon the points situated at
A & D. Suppose that by such actions the whole of the action of the forces on A & D has
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
respectively. To cancel these, we must have acting at these points forces AG & DK,
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 L.D.
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
actin 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,
must be cancelled by the mutual action between it & C ; hence, this must be in the
direction of the straight line BC, in the case given by the figure, where C lies within the
angle ABD ; for, if the angle ABD should turn its opening in the other direction, so that
C should lie outside the angle, then the force would have the direction CB, & all the rest
of the proof would come to the same thing. Now, the point C, on account of the equality of
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 Fr. Carolus Benvenuto, S.j., Art. 146, to which
author I gave this solution to print in that work.
Enunciation of the
problem of the
equilibrium of
four points ; of
which the two
outside points have
any masses with
external forces
proportional to
them; & one of
the inner points is
subject to a force
from a fulcrum.
The force from the
two extremeS On
either of the means.
The force, which
that first point, C,
must have, is
composed out of
four ; 6 Illi Iſler-
ation of the forces
pertaining to all
the points.
437
438 PHILOSOPHIAE NATURALIS THEORIA
Constructio praepar-
atoria pro solutione.
igitur quodvis punctum A binas vires AI, AH, quodvis punctum D binas vires DM, DL,
punctum B binas vires, quarum altera dirigetur ad A, & aequabitur 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 ductae 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, huc reducuntur:
Quodvis punctum A habebit vires binas . o o . AI, AH
Quodvis punctum D vires binas e o e o . DM, DL
Punctum B binas . º o ga º . A X HA, D X DL
Punctum C quatuor . A X IA, D X MD, A X HA, D X LD
89. 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, & parallela expriment vires illas
Vires su b no va
expressione inde
resultante.
Vis in fulcrum cui
aequalis.
Proportio, quae
vectem exhibet.
tertiam, & quartam puncti C. Producantur autem DC, AC donec 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.
90. In primis ob singula latera singulis lateribus parallela erunt similia triangula IAG,
CTR, & triangula MDK, CON. Quare eritut IG, sive AH, ad CR, sive NB, velA 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 illa exprimet summam omnium virium
AZ Omnium punctorum in A, ha cvim 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, 8 OC
aequalis MD ductae in D, adeoque illa exprimet summam omnium virium DX omnium
punctorum in D, ha cvim illam secundam puncti C, nimirum D X DM. Quare jam
erunt
Summa virium parallelarum in A . o o e, o TR
Summa virium parallelarum in D . dº - o e NO
Binae vires in B o º º º o o to . BN, BR
Quatuor vires in C. de so o o . CT, OC, RC, NC
9I. Jam vero patet, ex tertia RC, & prima CT componi vim RT aequalem summae
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 vires parallela in A, & D, & aequetur earum
summae, nimirum urgeri eodem modo, quo urgeretur, si omnia illa puncta, quae sunt in
D, & A, cum his viribus essent in C, & fulcrum per se ipsa immediate urgerent.
92. Praeterea ob parallelismum itidem omnium laterum similia erunt triangula 1?
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 : : NO . 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 tam 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, qua habetur hic post aquilibrium trium massarum num. 326.
SUPPLEMENT V 439
FIG. 75.

440
PHILOSOPHIAE NATURALIS THEORIA
FIG. 75.

SUPPLEMENT V - 44. I
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 © © º AI, AH
Any point at D, two forces . © gº © º DM, DL
The point B, two . º tº e o . 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
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,
CTR are similar, & so also are the triangles MDK, CON. Hence, as IG, or AH, is to CR,
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,
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 e © © º TR
The sum of the parallel forces on D e e º © NO
The two forces on B tº e e º & º . BN, BR
The four forces on C . © . CT, OC, RC, NC
9I. And now it is plain that, from the third force RC, & the first, CT, we have a
resultant force RT which is equal to the sum of the parallel forces at A ; & from the fourth,
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
are similar —(1) CNO, DPC; (2) CNQ, PDE; (3) CPR, VCN; (4) CRS, VNQ ; (5)
CVA, TCR; (6) WAF, 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 = WA : 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 (8).
(g) Moreover, the application to the lever is similar to that given in this work, after the equilibrium of three points
in Art. 326.
Construction neces-
sary for the solu-
tion.
The forces that
result from this
new method of
representation.
The force on the
fulcrum : to what it
is equal.
T he proportion,
which represents
the law of the lever.
Occasio, * & argu-
mentum epistolæ.
Translatio Theoriæ
centri oscillationis
a massis jacentibus
intra idem planum,
ad ubicunque posi-
tas affirmata ; in
opere, hic demon-
Stranda.
Viribu s triu m
massarum in eodem
plano, i n quo
jacent, translatis
ad aliud, rem ob-
tineri.
Si massæ sint
quatuor,reducendas
omnes ad planum
per p e n d icu la re
re ctæ jungenti
duas : inde transi-
tus ad massas
quotcunque.
[297] § VI
EPISTOLA AUCTORIS AD P. CAROLUM SCHERFFER
SOCIETATIS JESU
93. In meo discessu Vienna reliqui apud Reverentiam Vestram imprimendum opus,
cujus conscribendi occasionem præbuit Systema trium massarum, quafum vires mutuæ
Theoremata exhibuerunt & elegantia, & fœcunda, pertinentia tam ad directionem, quam
ad rationem virium compositarum e binis in massis singulis. Ex iis Theorematis evolvi
nonnulla, quæ in ipso primo inventionis æstu, & scriptionis fervore quodam, atque impetu
se se obtulerunt. Sunt autem & alia, potissimum nonnulla ad centrum percussionis
pertinentia ibi attactum potius, quam pertractatum, quæ mihi deinde occurrerunt & in
itinere, & hic in Hetruria, ubi me negotia mihi commissa detinuerunt hucusque, quæ
quidem ad Reverentiam Vestram transmittenda censui, ut si forte satis mature advenerint,
ad calcem operis addi possint ; pertinent enim ad complementum eorum, quæ ibidem
exposui, & ad alias sublimiores, ac utilissimas perquisitiones viam sternunt.
94. Inprimis ego quidem ibi consideravi directiones virium in eodem illo plano, in
quo jacent tres massæ, & idcirco ubi Theoremata applicavi ad centrum æquilibrii, &
Oscillationis pro pluribus etiam massis, restrinxi Theoriam ad casum, in quo omnes massæ
jaceant in eodem plano perpendiculari ad axem conversionis. In nonnullis Scholiis tantum-
modo innui, posse rem transferri ad massas, utcunque dispersas, si eæ reducantur ad id
planum per rectas perpendiculares plano eidem ; sed ejus applicationis per ejusmodi
reductionem nullam exhibui demonstrationem, & affirmavi, requiri systema quatuor
massarum ad rem generaliter pertractandam.
95. At admodum facile demonstratur ejusmodi reductionem rite fieri, & sine nova
peculiari Theoria massarum quatuor generalis habetur applicatio tenui extensione Theoriæ
massarum trium. Nimirum si concipiatur planum quodvis, & vires singulæ resolvantur
in duas, alteram perpendicularem plano ipsi, alteram parallelam ; priorum summa elidetur,
cum oriantur e viribus mutuis contrariis, & æqualibus, quæ ad quamcunque datam
directionem redactæ æquales itidem remanent, & con-[298]-trariæ, evanescente (b) summa :
posteriores autem componentur eodem prorsus pacto, quo componerentur ; si massæ per
illas perpendiculares vires reducerentur ad illud planum, & in eo essent, ibique vires haberent
æquales redactas ad directionem ejusdem plani, quarum oppositio & æqualitas redderet
eandem figuram, & eadem Theoremata, quæ in opere demonstrata sunt pro viribus jacentibus
in eodem plano, in quo sunt massæ. Porro hæc consideratio extendet Theoriam æquilibrii,
& 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.
96. Quod si sint quatuor massæ, & concipiatur planum perpendiculare rectæ transeunti
per binas ex iis, ac fiat resolutio eadem, quæ superius ; res iterum eodem recidet : nam
illæ binæ massæ ita in illud planum projectæ, coalescent in massam unicam, & vires ad
(h) Haec tum quidem in hac epistola. Addi potest illud, ubi nulla externa vis in ea directione agens, &$ in
contraria applicetur diversis partibus ipsius systematis, debere vim hujusmodi in singulis etiam ipsius systematis punctis
esse nullam. Nam per mutuum nexum impeditur mutatio positionis mutuæ, que utique induceretur, si in aliquibus tantum-
modo ejus partibus remaneret vis externis viribus non impedita. Porro ubi agitur de centro oscillationis, & percussionis,
ac etiam de æquilibrio, nulla supponitur vis, externa agens secundum directionem axis rotationis, seu conversionis. Quare
in iis casibus, pro quibus haec theoria hic extenditur, $atis est considerare reliquas illas vires, quæ agunt $ecundum
directionem plani perpendicularis eidem axi, quod hic praestatur in iis, quae consequuntur.
442
§ VI
A LETTER FROM THE AUTHOR
TO
FR. CAROLUS SCHERFFER, S.J.
93. When I departed from Vienna, I left with Your Reverence to be printed a work,
which I had written as an outcome of the consideration of a system of three masses; the
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
plane as that in which the masses were situated ; &, therefore, when I applied the theorems
to the centre of equilibrium & Oscillation even for several masses, I restricted the Theory
to the case in which all the masses were lying in the same plane, perpendicular to the axis
of rotation. Only in some notes did I mention that the matter could be developed for
masses that were disposed in any manner, if these were reduced to that plane by perpendi-
culars to the plane. But I gave no demonstration of this application by means of such a
reduction ; & I asserted that the consideration of a system of four masses would be necessary
before the matter could be dealt with thoroughly, & in general.
95. But it is quite easily proved that such a reduction can be correctly made ; & a
general application, without any special fresh theory for four masses, can take place, with
a very slight extension of the theory for three masses. Thus, if any plane is taken & each
force is resolved into two forces, of which one is perpendicular & the other parallel to the
plane; then the sum of all the first will be eliminated, since they arise from mutual forces
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 (h). 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,
96. Now if there are four masses, & a plane is taken perpendicular to the straight
line joining any two of them, & the same resolution is made as in the preceding paragraph ;
then, the matter will again come to the same thing. For, those two masses, being thus
thrown into the same plane, will coalesce into a single mass ; & the forces belonging to
º (h) This is what I said in the letter. To it may be added the point that, when no external force is applied acting
in one direction on one part, & the opposite direction on another part, of the system, this kind of force must also be zero
for each of the points of the system. For, a change of mutual position is prevented by the mutual connection ; 85 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 exiºma
forces. Further, when dealing with the centre of oscillation, 85 of percussion, & 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 ; 85 this is done in what follows.
The occasion for,
& the contents of,
the letter.
The transition to
the theory of the
centre of Oscilla-
tion from the case
of masses all lying
in the same plane
to masses lying
anywhere, merely
asserted in the
work itself, is here
to be proved.
The forces for
three masses in the
same plane as that
in which they lie,
being transferred
to another, the
thing is done.
If there are four
masses, they are all
to be reduced to a
plane perpendicular
to the straight line
joining two of
them ; hence the
transition to any
number of masses.
443
444 PHILOSOPHIÆ NATURALIS THEORIA
Applicatio ad
centri oscillationis
generalem deter-
minationem.
Aliud utile corol-
larium pertinens ad
centrum os cilla,-
tionis.
Ejus demonstratio.
Usus pro longi-
tudine p e n d u 1 i
co m p o sit o is o-
chroni fa cil i u s
invenienda.
reliquas binas massas pertinentes habebunt ad se invicem eas rationes, quæ pro systemate
trium massarum deductæ $unt. Hinc ubi systema massarum utcumqué dispersarum
conyerti debet circa axem aliquem, sive de æquilibrii centro agatur, sive de centro oscilla-
tionis, sive de centro 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.
97. Sic ex. gr. ubi agitur de centro oscillationis, quæ pro massis existentibus in unico
plano perpendiculari ad axem rotationis proposui, ac demonstravi respectu puncti suspen-
sionis, centri gravitatis, traducentur ad massas quascunque, utcunque dispersas respectu
axis, & respectu rectæ parallelæ axi ductæ 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 centro oscillationis erit æquale summæ omnium productorum, quæ
habentur, si massæ singulæ ducantur in quadrata suarum distantiarum perpendicularium
ab axe gravitatis, divisæ per summam massarum. Si enim omnes massæ reducantur ad
unicum planum perpendiculare axi conversionis, abit is totus axis in punctum suspensionis,
totus axis gravitatis in centrum gravitatis, & singulæ distantiæ perpendiculares ab iis
axibus evadunt distantiæ ab iis punctis : unde patet generalem Theoriam reddi omnem
per solam applicationem systematis massarum trium rite adhibitam.
98. Quod ad centrum Oscillationis pertinet, erui potest aliud Corollarium, præter illa,
quæ proposui, quod summo sæpe usui esse potest : est autem ejusmodi. Si plurium partium
systematis compositarum ex massis quotcumque, utcumque dispersis inventa fuerint seorsim
centra gravitatis, &5 centra oscillationis respondentia dato 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, & dividemdo 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 centro Oscillationis, quæ
respondet figuræ 63 exprimenti unicam massam A ex pluribus quotcunque, quæ 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,
& formula est PQ = A × AP* + B × BP* -|- &c.
M × GP
99. Nam ex ejusmodi formula est M x GP × PQ = A × AP* + B × BP* &c.
Quare si singularum partium massæ M ducantur in suas binas distantias GP, PQ ; habetur
in singulis summa omnium A × AP* + B × BP* &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 × PQ
pertinentium ad singulas systematis partes, uti in hoc novo Corollario enunciatur.
ioo. Usus hujus Corollarii facile patebit. Pendeat ex. gr. globus aliquis suspensus
per filum quoddam. Pro globo jam constat centrum gravitatis esse in ipso centro globi,
& constat [3oo] itidem, ac e superioribus etiam Theorematis facile deducitur, centrum
oscillationis jacere infra centrum globi, per $ tertiæ proportionalis post distantiam, puncti
suspensionis a centro globi, & radiüm ; pro filo autem cgnsiderato ut recta quadam habetur
cerítrum gravitatis in medio ipso filo, & centrum oscillationis, suspensiqne facta per fili
extremum est in fine secundi Ttrientis 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
results which I enunciated for masses existing in a single plane perpendicular to the axis
of rotation, and proved, with respect to the point of suspension & the centre of gravity, may
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
the one that I have enunciated ; & this has often been of great service to me; it is as fol-
lows. If, for two or more parts of a system composed of any number of masses, situated in any
manner, the centres of gravity, 85 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, {5 the perpendicular distance
of the centre of oscillation from the same axis ; 85 dividing the sum of these products by the
mass of the whole system, 85 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
PQ = * * AP* + B X BP* -- &c.
M × GP
99. Thus, from the formula given, we have
M × GP × PQ = A X AP* + B X. BP2 + &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 AP* + B X BP* + &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 × GP × PQ belonging to the several parts of the system, as we have
stated in this new corollary.
IOo. The use of this corollary will be easily seen. For example, suppose we have a
sphere suspended by a thin rod. For a sphere, it is well-known that the centre of gravity
is at the centre of the sphere ; and it is also well-known, & indeed it can be easily deduced
from the theorems given above, that the centre of oscillation lies below the centre of the
sphere, at a distance from it equal to two-fifths of the third proportional to the distance of
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
Application to the
general determina-
tion of the centre
of oscillation.
Another use ful
corollary pertain-
ing to the centre of
Oscillation.
Demonstration of
this corollary.
Its use in providing
an easy determina-
tion of the length
of a pendulum
isochronous with a
given composite
pendulum.
446 PHILOSOPHIÆ NATURALIS THEORIA
Calculus & formula,
pro pendulo globi
pendentis e filo.
Non licere hic con-
cipere massas sing-
ulas ut collectas
in suis centris Oscil-
lationis, aut gravi-
tatis, aut aliis
intermediis docu-
mentum utile.
Transitus ad cen-
trum percussionis :
ejus notiones haberi
posse plures.
Initium a notione
adhibita in Opere :
centri gravitatis
Status COnSerVatuS
in motu libero.
Inde erui, in sys-
temate translato
cum rotatione, fore
rectam Cum eO
connexam immo-
lbile m q u o vis
tempusculo suam ;
quæ facile definiri
possit.
generali facillime deducitur. Inde centrum oscillationis commune globi, & fili nullo
negotio definietur per corollarium superius. .
IOI. Sit Longitudo fili a, massa seu pondus b, radius globi r, massa seu pondus p : erit .
distantia centri gravitatis fili ab axe conversionis erit $a, distantia centri oscillationis
ejusdem §a. Quare productum illud pertinens ad filum erit $ a*b. Pro globo erit
distantia centri gravitatis a -+- r, quæ ponatur = m ; Distantia centri oscillationis erit
^r • •
m -+- 3 X m' Quare productum pertinens ad globum erit m°p -+- 3 rrp. Horum summa
est m*p -+- 3 rrp + $ a*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 + 3 ab. Quare formula pro centro Oscillationis utriusque
simul, erit
m*p + $ rrp + 3a*b
mp + 3ab -
Io2. Hic autem notandum illud, ad centrum oscillationis commune habendum non
licere singularem partium massas concipere, ut collectas in suis singulas aut centris oscilla-
tionis, aut centris gravitatis. In primo casu numerator colligeretur ex summa omnium
productorum, quæ fierent ducendo singulas massas in quadrata distantiarum centri
oscillationis sui ; in secundo in quadrata distantiarum sui centri gravitatis. In illo nimirum
haberetur plus justo, in hoc minus. Sed nec possunt concipi ut collectæ in aliquo puncto
intermedio, cujus distantia sit media continue proportionalis inter illas distantias ; nam
in eo casu numerator maneret idem, at denöminator 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 infinitesimæ ut massæ compenetratæ in unico suo puncto æquivaleat.
[3oI] Io3. Quod attinet ad centrum percussionis, id attigi tantummodo determinando
punctum systematis massarum jacentium in recta quadam, & libere gyrantis, cujus puncti
impedito motu sistitur motus totius systematis. Porro æque facile determinatur centrum
percussionis in eo sensu acceptum pro quovis systemate massarum utcunque dispositarum,
& res itidem facile perficitur, si aliæ diversæ etiam centri percussionis ideæ adhibeantur.
Rem hic paullo diligentius persequar.
Io4. Inprimis ut agamus de eadem centri percussionis notione, moveatur libere
systema quodcunque ita inter se connexum, ut ejus partes mutare non possint distantias a se
invicem. ' Centrum gravitatis totius systematis vel quiescet, vel movebitur uniformiter
in directum, cum per theorema inventum a Newtono, & a me demonstratum in ipso Opere
num. 25o, actiones mutuæ non turbent statum ipsius : systema autem totum sibi relictum
vel movebitur motu eodem parallelo, vel convertetur motu æquali circa axem datum
transeuntem per ipsum centrum gravitatis, & vel quiescentem cum ipso centro, vel ejusdem
uniformi mofu párallelo delatum simul, quod itidem demonstrari potest haud difficulter.
Io5. Inde autem colligitur illud, in motu totius systematis composito ex motu uniformi
in directum, & ex rotatione circulari circa axem itidem translatum haberi semper rectam
quandam pertinentem ad systema, nimirum cum eo connexam, pro quovis tempusculo
suam, quæ illo tempusculo maneat immota, & circa quam, ut circa quendam axem immotum
converfatur eo tempusculo totum systema. Concipiatur enim planum quodvis transiens
per axem rotationis circularis, & in eo plano sit recta quævis axi parallela ; ea conyertetur
çirca axem velocitate eo majore, quo magis ab ipso distat. Erit igitur aliqua distantia
ejus rectæ ejusmodi, ut velocitas conversionis æquetur ibi velocitati, quam habet centrum
gravitatis cum axe translato ; & in altero e binis appulsibus ipsius rectæ parallelæ 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.
IOI. 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 of its centre of oscillation will be ša. 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 + r ; call this m. Then the distance of the centre of oscillation will be
m + 3 × r". Hence, the product for the sphere will be mºp + 3rºp. The sum of these is
7/1
m*p + 3rºp + 4a2b. 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 + £ab. Hence the
formula for the centre of oscillation for both together will be
m*p + #rºp + , a”b.
mp + £ab
IO2. 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.
IO3. 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.
IO4. 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.
IO5. 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
Calculation giving
the formula for a
pendulum formed
of a sphere hanging
at the end of a thin
rod.
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 ; sever a 1
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 PHILOSOPHIÆ NATURALIS THEORIA
cum systemate ad planum perpendiculare ei plano, quod axis uniformiter progrediens
describit, ejus rectæ motus circularis fiet contrarius motui axis ipsius, adeoque motui, quo
ipsa axem comitatur, cui cum ibi & æqualis sit, motu altero per alterum eliso, ea recta
quiescet illo tempusculo, & systema totum motu composito gyrabit circa ipsam. Nec
erit difficile dato motu centri gravitatis, & binarum massarum non jacentium in eodem
plano transeunte per axem rotationis, invenire positionem axis, & hujus rectæ immotæ pro
quovis dato momento temporis.
Ę. Io6. Quæratur jam in ejusmodi systemate punctum aliquod, cujus motus, si per aliquam
S «• &• • ę ©» ¢ © « » � «•
ííííííí*° vim externam impediatur, debeat mutuis actionibus sisti motus totius systematis, quod
punctum, si uspiam fuerit, dicatur centrum percussionis. Concipiantur autem massæ
omnes translatæ per rectas parallelas rectæ [302] illi manenti immotæ tempusculo, quo
motus sistitur, quam rectam hic appellabimus axem rotationis, in planum ipsi perpendiculare
transiens per centrum gravitatis, & in figura 64 exprimatur id planum ipso plano 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 rectæ PQ agentem directione Aa, alterum ipsi parallelum agentem
directione PG, ac velocitas absoluta puncta Q dicatur V.
- - • • • PAX V ( . • a ea- e
Definit loci- � �
$i; tìoIuií Io7. Erit PQ. PA :: V. -pö-, quæ erit velocitas absoluta massae A. Erit autem
relativarum cujus-
VlS ImaSSae. PA . Pa : : á × V . ά × V, quæ erit velocitas secundum directionem Aa, &
PA Aa � e • ©,
PA . Aa : : PQ × V . PQ × V, quæ erit velocitas secundum directionem PG.
Nam in compositione, & resolutione motuum, si rectæ 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
Pa Aa
j< A × V, & jK A × V.
PQ * ~ PQ -
IE t • Aa • &• ę
va n e s c e n t 1 a Io8. Jam vero summa ÅX A × V est zero, cum ob naturam centri gravitatis
summæ determi- PQ
nans problema.
summa omnium Aa × A sit æqualis zero, & po*** quantitas data. Quare si per vim
externam applicatam cuidam puncto Q, & mutuas actiones sistatur summa omnium motuum
Éx A × V, sistetur totus systematis motus, reliqua summa elisa per solas vires mutuas,
quarum nimirum summa est itidem zero.
Inventio summæ Io9. Ut habeatur id ipsum punctum Q, concipiatur quævis massa A connexa cum eo,
É. æquandæ & .„„, 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.
Hæc igitur posterior summa erit investiganda, & ponenda = o.
galculus, & formula [3o3] 1 Io. Porro posito radio = I, est ex Theoremate trium massarum ut PX PQ × 1
derivata. ää ÄX AQ × sim`QAa, sive ut P × PQ ad A × Qa, ita actio in A perpendicularis ad
Pa A × Qa X P4«v.
PQ —j<V ad actionem in P secundum eandem directionem, quæ evadit P × PQa
PQ
• - 2
nimirum ob Qa = PQ — Pa, erit actio in P = A × re; !; Q2 A × Pa × V. Cum
harum summa debat æquari zero demptis communibus æquabuntur positiva
V
P × PQ2*
negativis, nimirum posita pro characteristica summæ, habebitur fAX PQ×Pa=fAX Pa*,
sive PQ = %£, vel ob f. A × Pa = M × PG, posito ut prius M pro summa
• _ f. A × Pa*
massarum, fiet PQ = TNTSZTPG 3 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
is prevented by some external force, the motion of the whole system is thereby checked by
mutual actions; this point, if there is one, will be called the centre of percussion. Suppose
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.
Enunciation of a
problem & prepar-
ation for the
Solution.
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
v = V × PA/PQ. Similarly, since we have PA : Pa = V × PA/PQ: V × Pa / PQ;
therefore V × Pa/PQ will be the velocity in the direction Aa. Also, since we have
PA: Aa-V × PA/PQ : V × Aa/PQ; hence, V × Aa/PQ will be the velocity in the
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
Pa Aa
equal to po ’ A X V, and #5 × A X V.
Aa
Io8. Now, the sum of all such as PO × A × V is equal to zero, since, on account
of 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 force applied at any point Q,
Pa tº
Fo’s A × 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,
or with masses supposed to be situated at these points; then the sum of all the motions,
which are derived from the connection for Q, when this motion is destroyed for every A,
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.
IIo. Now, if the radius is made the unit, then, from the theorem for three masses,
we have the ratio of P × PQ X I to A X AQ X sin()Aa, or P × PQ to A X Qa, equal
& the mutual actions, the sum of all the motions
to the ratio of the action at A perpendicular to PQ (which is equal to º V) to the action
at P in the same direction; & therefore the latter is equal to A º *x V, that
o tº g A × PQ X Pa — A X Pa”
is to say, since Qa = PQ — Pa, the action at P = P × PQ2 × V.
Since the sum of all of these has to be equated to zero, on cancelling the common factor
V/(P × PQ?), the positives will be equal to the negatives; hence, using the symbol ſ
as the characteristic of a sum, we have ſ. A X PQ X Pa = ſ. A X Pa”; that is,
PQ = ſ. A X Pa”/(ſ. A X Pa). Now, if as before we put M for the sum of all the masses,
then ſ. A × Pa- M × PG, & we have PQ = ſ. A × Pa”/(M × 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.
G G
Determination of
t he absolute
velocity, & also the
relative velocities,
of any mass.
Evanescence of this
sum which deter-
mines the problem.
The determination
of the sum which
is to be equated to
ZëIO.
The calculation,
and the formula
obtained.
45o
> PHILOSOPHIÆ NATURALIS THEORIA
Theorema erutum
ex formula.
Deductio casu s,
quo jaceant omnes
massæ in eodem
plano.
Si qua massa sit
extra : discrimem
centri oscillationis,
a centro percus-
S1O 1) 1 S.
Formulæ deductae
pro pluribus aliis
theorematis.
Theorema de posi-
tione centri gravi-
tatis.
T h e ore m a. d e
duarum distanti-
arum producto.
Corollarium in de
deductum.
Axe rotationis ab-
eunte in infinitum,
centrum percussio-
nis abire in cent-
rum gravitatis.
Si axis rotationis
transeat per cen-
tru m gravitatis,
motum sisti non
posse.
Centri percussionis
positio notabilis.
f. A × PG2
III. Corollarium I. . Quoniam aP æquatur distantiæ perpendiculari A a plano trans-
eunte per P perpendiculari ad rectam PG, habebitur hujusmodi Theorema. Distantia
centri percu$$ionis ab axe rotationis in recta ipsi axi perpendiculari transeunte per centrum
gragitatis habebitur, ducendo singulas massas in quadrata suarum distantiarum perpendicularium
a plano perpendiculari eidem rectae transeunte per axem ipsum rotationis, ac dividendo summam
omnium eju$modi productorum per factum ex summa massarum in distantiam perpendicularem
centri gravitatis communis ab eodem plano.(')
[304] I I2. Corollarium II. Si massæ jaceant in eodem unico plano quovis transeunte
per axem ; A, & a congruunt, adeoque distantiæ Pa sunt ipsæ distantiæ ab axe. Quamobrem
in hoc casu formula hæc inventa pro centro percussionis congruit prorsus cum formula
inventa pro centro Oscillationis, & ea duo centra sunt idem punctum, si axis rotationis sit
idem, adeoque in eo casu transferenda sunt ad centrum percussionis, quaecumque pro centro
oscillationis $unt demonstrata.
II3. Corollarium III. Si aliqua massa jaceat extra ejusmodi planum pertinens ad aliam
quampiam ; erit ibi Pa minor, quam PA, adeoque centrum percussionis distabit mimus ab
axe rotationis, quam distet centrum oscillationis.
J. A X Pa* habetur Pa* = PG2 -|-
M × GP
Ga* — 2PQ × Ga. Porro J. A × 2PQ × Ga evanescit ob evanescentem f. A × Ga, &
*-* f. A× Ga* _ f. A × Ga*
$i ej- est PG. Quare fit PQ = PG +%£j;;, & GQ =%e^;.
deducuntur sequentia Theoremata affinia similibus pertinentibus ad centrum oscillationis
deductis in ipso opere.
II4. Corollarium IV. In formula generali PG =
Inde autem
II 5. Si impre$sio ad sistendum motum fiat in recta perpendiculari axi rotationis transeunte
per centrum gravitatis, centrum gravitatis jacet inter centrum percussionis, & axem rotationis.
Nam PQ evasit major quam PG. -
I I6. Productum sub binis distamtiis illius ab his est constans, ubi axis rotationis sit im
eodem plano quovis transeunte per centrum gravitatis cum eadem directione in quacumque distantia
_f. AX Ga* _. _f. A × Ga*
Nam ob GQ = MIXTPG erit GQ × PG = —Rt-—.
ab ip$o centro gravitatis.
I I7. In eo casu punctum axis pertinens ad id planum, & centrum percussionis recipro-
cantur ; cum nimirum productum sub binis eorum distantiis a constanti centro gravitatis sit
c011j t411J. »
II8. Abeunte axe rotationis in infinitum, ubi nimirum totum systema movetur tantummodo
motu parallelo, centrum percussionis abit in centrum gravitatis. Nam altera e binis distantiis
excrescente in infinitum, debet altera evanescere. Porro is casus accidit semper etiam,
ubi omnes massæ abeunt in unum punctum, quod erit tum ipsum gravitatis centrum to-
[305]-tius systematis, & progredietur sine rotatione ante percussionem.
II9. Abeunte axe rotationis in centrum gravitatis, nimirum quiescente ipso gravitatis
centro, centrum percussionis abit in infinitum, nec ulla percussione applicata unico puncto motus
sisti potest. Nam e contrario altera distantia evanescente, altera abit in infinitum.
I2o. Corollarium V. Centrum percussionis debet jacere in recta perpendiculari ad axem
rotationis transeunte per centrum gravitatis. Id evincitur per quartum e superioribus Theo-
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 utcumque dispersarum satis
esse singulas massas reducere ad rectam transeuntem per centrum gravitatis, &5 perpendicularem axi rotationis per rectas
ipsi axi perpendiculares, &$ invenire massarum ita reductarum centrum oscillationis, habito puncto rotationis pro puncto
suspensionis ; id enim erit ipsum centrum percussionis quaesitum. Nam distantiae ab ip$o plano perpendiculari illi rectæ,
quarum distantiarum fit mentio in hoc corollario, manent eædem in ejusmodi translatione massarum, è ¢ evadunt, distamtiae
a puncto suspensionis. Theorema autem post substitutionem di$tantiarum a puncto suspensionis pro iis ipsis distantiis ab
illo plano exhibet ipsam formulam di$tantiæ centri oscillationis a puncto suspensionis, quae habetur num. 334. Hinc autem
consequitur generalis reciprocatio puncti rotationis, &5 centri percussionis, ac alia plura in $equentibus deducta multo
immediatius deducuntur e proprietatibus centri oscillationis jam demonstratis.
LETTER TO FR. SCHERFFER 45I
111. Collorary I. Since aP is equal to the perpendicular distance of..A from a Plane
passing through P perpendicular to the straight line PG, we have the following theorem.
The distance of the centre of percussion from the axis of rotation in a straightline 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 axi; of rotation, & perpen-
dicular to the straight line ; & 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.() º º
112. Corollary II. If the masses lie in any the same single plane passing through the
axis, A & a coincide, & therefore the distances Pa become the distances of the masses from
the axis. Hence, in this case, the formula here found for the centre of percussion agrees
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
Pa will be less than PA; hence, the centre of percussion will be at a less distance from the axis
of rotation than the centre of oscillation.
II4. Corollary IV. In the general formula PQ = ſ.AX Paº
T M × GP
-- Ga” — 2 PQ X Ga. Also, the sum ſ.A X 2 PQ X Ga vanishes, sinceſ.A X Ga vanishes ;
& ſ.A X PGA/(M × PG)= PG. Hence we have
-*== ſ. A X Ga” _ſ.A X Ga”
PQ = PG+º, & GQ =*-ī;.
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
perpendicular to the axis of rotation & passing through the centre of gravity, the centre of gravity
will lie between the centre of percussion & the axis of rotation. For PQ is greater than PG.
116. The product of the two distances of the former from the two latter is constant, when
the axis of rotation is in any the same plane passing through the centre of gravity, the direction
of measurement being the same for any distance from the centre of gravity. For, since
2 2
CQ = **ś, therefore GQ X PG – A Ge. - •
117. In that case, the point on the axis corresponding to the plane & the centre of
percussion will be interchangeable ; for, the product of their two distances from a constant centre
of gravity is constant. -
118. If the axis of rotation goes off to infinity, that is to say, when the whole system is trans-
lated with simply a parallel motion, the centre of percussion will become coincident with the centre
of gravity. For, if one of the two distances increases indefinitely, the other must become
evanescent. Also, this will always happen, when all the masses coincide at a single point;
this point will then be the centre of gravity of the whole system, & it will be moving without
rotation before percussion.
119. If the axis of rotation passes through the centre of gravity, the centre of percussion
passes off to infinity, 85 the motion cannot be checked by any blow applied at a single point.
For, on the contrary, when the finite distance vanishes, the other distance must become
infinite. - -
120. Corollary W. The centre of percussion must lie in the straight line perpendicular
to the axis of rotation & passing through the centre of gravity. This is proved by the fourth
of the theorems given above. The method of solution of the problem that was employed
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
we have Pa” = PGA
(i) It 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
85 perpendicular to the axis of rotation, by means of straight lines perpendicular to the axis ; 85 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 & 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 & the centre
of percussion ; & many other things deduced in what follows can be more easily derived from the properties of the centre
of oscillation already proved.
A Theorem derived
from this formula.
Deduction of the
case in which all
the masses lie in
the Same plane,
If any mass does
not lie within the
plane, there is a
distinction between
the centre of oscilla-
tion & the centre of
percussion.
Formulae deduced
for several other
theorems.
Theorem concern-
ing the position of
the centre of
gravity.
Theorem concern-
ing the product
of the two dis-
tances.
Corollary derived
from this.
If the axis of
rotation goes off to
infinity, the centre
of percussion will
become coincident
with the centre of
gravity.
If the axis of
rotation passes
through the centre
of gravity, the
motion cannot be
checked.
A noteworthy
position of the
centre of percus-
SIOIl.
452 PHILOSOPHIÆ NATURALIS THEORIA
Impactus in cen-
trum percussionis
qui sit.
Demonstratio pri-
mæ partis.
Demonstratio Se-
cundæ.
Impressio ubi fieri
possit extra cen-
trum percuss1on1s
cum eodem effectu.
communi-
catus quovis im-
pactu systemati
quiescenti.
Motus
axi reducantur per rectas ipsi axi parallelas & massæ omnes, & ipsum centrum gravitatis
commune, adeoque inde non haberetur unicum centrum percussionis, sed series eorum
continua parallela axi ipsi, quæ 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
massæ respectu rectarum perpendicularium axi priori in eo jacentium rotationem nullam
habent, cum distantiam ab eo plano non mutent, sed ferantur secundum ejus directionem,
adeoque respectu omnium directionum priori axi perpendicularium jacentium in eo plano
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, nimirum 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, & eam
vincat, debet centrum percussionis jacere in plano perpendiculari ad axem transeunte per
centrum gravitatis, & debent in solutione problematis omnes massæ reduci ad id ipsum
planum, ut præstitimus, non ad aliud quodpiam ipsi parallelum : ac eo pacto habebitur
æquilibrium massarum, hinc & inde positarum, quarum ductarum in suas distantias ab
eodem plano summæ hinc, & inde acceptæ æquabuntur inter se. Porro eo plano ad solu-
tionem adhibito, patet ex ipsa solutione, centrum percussionis jacere in recta perpendiculari
axi ducta per centrum gravitatis : jacet enim in recta, quæ a centro gravitatis ducitur ad
illud punctum in quo axis id planum secat, quæ recta ipsi axi perpendicularis toti illi plano
perpendicularis esse debet.
12 I. Corollarium VI. Impactus in centro percussionis in corpus externa vi ejus motum
sistens est idem, qui esset, $i $ingulae massae incurrerent in ipsum cum suis velocitatibus respecti-
[3o6]-vis redactis ad directionem perpendicularem plano transeunti per axem rotationis, &
centrum gravitatis, sive si ma$$arum summa in ipsum incurreret directione, & velocitate motus,
qua fertur centrum gravitatis. -
I 22. Patet primum, quia debet in Q haberi vis contraria directioni illius motus
perpendicularis plano transeunti per axem, & PG, par extinguendis omnibus omnium
massarum velocitatibus ad eam directionem redactis, quæ vis itidem requireretur, si omnes
massæ eo immediate devenirent cum ejusmodi velocitatibus.
- ® . Pa
I 23. Patet secundum ex eo, quod velocitas illa pro massa A sit PO × V, adeoque
Q
A × Pa M × PG PG
ΡQ TPOT × V. Est autem pö × V,
velocitas puncti G, quod punctum movetur solo motu perpendiculari ad PG, adeoque si
massa totalis M incurrat in Q cum directione, & celeritate, qua fertur centrum gravitatis G,
faciet impressionem eandem. •
mOtUS × V, quorum motuum summa est
124. Corollarium VII. Potest motus sisti impressione facta etiam extra rectam PG,
seu extra planum transiens per axem rotationis, &5 centrum gravitatis, nimirum si impressio
fiat in quodvis punctum rectæ eidem plano perpendicularis, & transeuntis per Q, directione
rectae ipsius. Nam per nexum inter id punctum, & Q statim impressio per eam rectam
transfertur ab eo puncto ad ipsum Q.
I 25. Corollarium VIII. Contra vero si imprimatur dato cuidam puncto systematis
quiescentis vis quaedam motrix ; invenietur facile motus inde communicandus ipsi systemati.
Nam ejusmodi motus erit is, qui contrario æquali 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 massæ
totius systematis, quæ ad eam, quam potest imprimere massæ cuivis, est ut hæc 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, præterea habebitur conversio, cujus axis, & celeritas sic invenietur.
Per centrum gravitatis G agatur planum perpendiculare rectæ, secundum quam fit impactus,
& notetur punctum Q, in quo eidem plano occurrit eadem recta. Per ipsum punctum G
ducatur in eo plano recta perpendicularis ad QG, quæ erit axis quæsitus. Per punctum Q
concipiatur alterum planum perpendiculare rectæ GQ, ca-[307]-piantur omnes distantiæ
perpendiculares omnium massarum A ab ejusmodi plano, æquales 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.
12.I. Corollary VI. The impact at the centre of percussion on a body by an external
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
{5 velocity of motion, with which the centre of gravity is moving.
I22. The first part is evident, because there must be at Q a force opposite in direction
to the motion perpendicular to the plane passing through the axis & PG, capable of destroying
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.
I23. The second part is evident from the fact that the velocity for the mass A is
º × V ; & thus, the motion # ** × V ; & the sum of these motions is
Mºe X V. But; × 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. -
I24. Corollary VII. The motion may be checked even by a blow applied without the
straight line PG, or without the plane passing through the axis of rotation & the centre of gravity;
that is, if it is applied at any point of a straight line perpendicular to the same plane, 85 passing
through Q, in the direction of this straight line. For, through the connection between that
i. & 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
given point of a system at rest, it is easy to find the motion thereby communicated to the system.
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.
The nature of the
impact a t the
centre of percus-
Sion.
Proof of the first
part.
Proof of the second
part.
When the blow can
be applied beyond
the centre of per-
cussion with the
same effect.
Motion communi-
cated by a blow to
a system at rest.
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 PHILOSOPHIÆ NATURALIS THEORIA
Demonstratio.
Aditus ad perquisi-
tiones ulteriores
m O t u impresso
Systemati moto.
Transitus ad aliam
notionem ejus cen-
tri.
Pr o b ] e m a, conti-
nens hanc ideam.
Solutio : formulæ
continentes motum
massæ in quam
incidit, & suum
reliquum.
Casus particulares,
ad quos applicari
potest.
singularum quadrata ducantur in suas massas, & factorum summa dividatur per summam
massarum, tum in recta GQ producta capiantur GP æqualis ; ei quoto diviso per ipsam
QG, & celeritas puncti P revolventis circa axem inventum in circulo, cujus radius GP, erit
æqualis celeritati inventæ centri gravitatis, directio autem motus contraria eidem. Unde
habetur directio, & celeritas motus punctorum reliquorum systematis.
I 26. 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 præmissam centri percussionis sisteretur impressione contraria, & æquali impressioni
datæ. -
I27. Scholium. Hoc postremo corollario definitur motus vi externa impressus
systemati quiescenti. Quod si jam systema habuerit aliquem motum progressivum, &
circularem, novus motus externa vi inductus juxta corollarium ipsum componendus erit
cum priore, quod, quo pacto fieri debeat, hic non inquiram, ubi centrum percussionis
persequor tantummodo. Ea perquisitio ex iisdem principiis perfici potest, & ejus ope
patet, aperiri aditum ad inquirendas etiam mutationes, quæ ab inæquali actione Solis,
& Lunæ in partes supra globi formam extantes inducuntur in diurnum motum, adeoque
ad definiendam ex genuinis principiis præcessionem æquinoctiorum, & nutationem axis :
sed ea investigatio peculiarem tractionem requirit.
I 28. Interea gradum hic faciam ad aliam notionem quandam centri percussionis,
nihilo minus, imo etiam magis aptam ipsi nomini. Ad eam perquisitionem sic progrediar.
I 29. Problema. Si systema datum gyrans data velocitate circa axem datum externa vi
immotum incurrat in dato suo puncto in massam datam, delatam velocitate data in directione
motus puncti ejusdem, quam massam debeat abripere secum ; quaeritur velocitas, quam ei ma$$æ
imprimet, & ipsum systema retinebit post impactum.
I 3o Concipiatur totum systema projectum in planum perpendiculare axi rotationis
transiens per centrum gravitatis G, in quo plano punctum conversionis sit P, massa autem
in recta PG in Q. Velocitas puncti cujusvis systematis, quod distet ab axe per intervallum
= I, ante incursum sit = a, velocitas ab eodem amissa sit = x, adeoque velocitas post
impactum = a — x, velocitas autem massæ Q ante impactum sit = PQ × b. Erit ut I
ad AP, ita x ad velocitatem amissam a massa A, quæ erit AP × x. Erit autem ut I ad
a — x ita PQ ad velocitatem residuam in puncto systematis Q, quæ fiet PQ × (a — x),
& ea erit itidem velocitas massæ Q post [308] impactum, adeoque massa Q acquiret veloci-
tatem PQ × (a — b — x), sive pósito Ia — b = c, habebitur PQ × (c — x). Porro ex
mutuo nexu massæ A cum P, & Q erit Q × PQ ad A × AP, ut effectus ad velocitatem
A × AP2
Q × QP
tuum provenientium e massis omnibus erit æqualis ;;i; :;ie in Q. Nimirum
f. A × AP* J . _ ;„ /. A × AP* + Q × QR* «« — OP &
*;;-:; x x = QP × e- OP ×s, sive Q × QP X % Q. X £,
Q × QP2
{% = ×c. Dato autem x datur a — x, & is valor ductus in distantiam
f. A × AP* -+- Q × QP* 9
punëti cujusvis systematis, vel etiam massæ Q, exhibebit velocitatem quæsitam. Q.E.F.
pertinens in A = AP × x ad effectum in Q = X %. Summa horum effec-
131. Scholium. Formula habet locum etiam pro casu, quo massa Q quiescat, vel quo
feratui contra motum systematis, dummodo in primo casu fiat b = 0, & c = 4; ac in secundo
valor b mutetur in négativum, adeoque sit g = a + b, Posset etiaig facile applicari ad
casum, quo in conflictu ageret. elasticas perfecta vel imperfecta. Determinatio tradita
exhiberet partem effectus in collisione facti tempore amissæ figuræ, 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
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
a system at rest is determined. But if now the system should have some motion, progressive
& circular, the new motion induced by the external force in accordance with the corollary
will have to be compounded with what it already has. I do not inquire here, how this will
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
is no less, nay it is even more, fit to have that name given to it. To this investigation
I proceed in the following manner.
I29. Problem. If a given system, gyrating with given velocity about a given axis, not
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,
& retained by the system after impact.
I30. Suppose that the whole system is projected on a plane perpendicular to the axis
of rotation passing through the centre of gravity G : in this plane let the point of rotation
be P, & let the mass be in the straight line PG at Q. Let the velocity of any point of the
system, whose distance from the axis is unity, before the impact be a, & let the velocity
lost by it be w; & thus, the velocity after impact will be a-w. Also let the velocity of the
mass at Q before impact be PQ X b. Then, as I is to AP so is a to the velocity lost by the
mass at A, which will therefore be APX w. Also, as I is to a -& so is PQ to the velocity
that remains in the point Q of the system ; & therefore this is PQ X (a—w); this will
also be the velocity of the mass Q after impact. , Hence, the mass Q will acquire a velocity
PQ X (a – b – 3); or, if we put a - b = c, it will be PQ X (c — wy. Further, from
the mutual connection between the mass A & P & Q, we shall have the ratio of Q X PQ to
A × AP equal to that of the effect pertaining to the velocity at A, which is equal to
A X AP2
Q_x OPT X % The sum of
these effects, arising from all the masses, will be equal to the velocity acquired at Q. That
is to say, we have
AP × w, to the effect at Q, which is therefore equal to
—ºx <= OPX – OPX w.
o, ſº QP 4 w = QP × c ;
O X QP2
and x = X C.
ſ. A X AP* + Q × QP2
But, if we are given x, we are also given a – & ; 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.
I31. Scholium. The formula holds good even when the mass Q is at rest, or when
it moves in the opposite direction to the system ; so long as, in the first case, b is made
equal to zero, or c = a ; & in the second case, the value of b is changed from positive to
negative, so that c = a + 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
Demonstration.
The way is open
for further Investi-
gations when mo-
tion is impressed on
a moving System.
Passing on to an-
other idea of this
Centre.
Problem embody-
ing the idea.
Solution ; formulae
containing t he
motion of the mass
with which it col-
lides, a n d the
motion left in it-
self.
Particular cases to
which it can be
applied.
456 PHILOSOPHIÆ NATURALIS THEORIA
Ejusdem ulterior
extensio.
Eelatio ad cen-
trum oscillationis.
Expressio veloci-
tatis in massa, sim-
plicior ope illius.
Ubi colligendum es-
set totum systema
ad eandem veloci-
t at e m im pri-
mendam massae.
In quot, & quibus
distantiis ab axe
massa eandem ex
impactu velocita-
tem acquireret : ubi
maximam.
tempori totus collisionis usque ad finem recuperatæ figuræ colligitur facile, duplicando
priorem, vel augendo in ratione data uti fit in col]isionibus.
I 32. 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 S1 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 plano 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, & quævis 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. -
I33. Corollarium I. Si distantia centri oscillationis totius systematis ab axe P dicatur
R, distantia centri gravitatis G, massa tota M, habebitur -
- Q × PQ2 c _ M × G × R
*= stata jèíósz po. x , & [309] -*j*jj*-+ 1
f. A × AP2
Patet ex eo, quod ex natura centri oscillationis habetur R = adeoque
f. A × AP* = M X G × R.
I 34. Corollarium II. Velocitas acquisita a massa Q erit
M × G ?
M × G × R × PQ
M × G × R-|- Q × PQ2
Q × PQ2 × c), quod
MχG>zTRIEQXPQ, ^ % %
reductum ad eundem denominatorem elisis terminis contrariis eo redit.
X 6.
Est enim ea velocitas PQ × (c — x), sive PQ (c —
I 35. Corollarium III. Si manente velocitate circulari systematis tota ejus massa
concipiatur collecta in unico puncto jacente inter centra gravitatis, & oscillationis, cujus
distantia a puncto conversionis sit media geometrice proportionalis inter distantias reli-
quorum punctorum, vel in eadem distantia ex parte opposita ; velocitas eadem imprimeretur
novæ massæ in quovis puncto sitæ. Tunc enim abiret in illud punctum utrumque centrum,
& valor G × R esset idem, ac prius, nimirum æqualis quadrato ejus distantiæ ab axe, quod
quadratum est positivum etiam, si distantia accepta ex parte opposita fiat negativa.
I 36. Corollarium IV. Si capiatur hinc, vel inde in PG segmentum, quod ad distantiam
ejus puncti ab axe sit in subduplicata ratione massæ totius systematis ad massam Q ; ipsa
massa Q in quatuor distantiis ab axe, binis hinc, & binis inde, quarum binarum producta
æquentur singula quadrato ejus segmenti, acquiret velocitatem in omnibus eandem
magnitudine, licet in binis directionis contrariæ, & ea fiet maxima, ubi ipsa massa sit in
fine ejus segmenti ex parte axis ultralibet. Erit enim velocitas acquisita directe ut
M × G × R × PQ M × G × R
M × G × R -|- Q × PQ* Q.
endo illud segmentum = == T, cujus quadratum T* debet esse =
X c, & pon-
× c, vel dividendo per constantem
*! x G x R, erit
Q.
«-
directe ut φάor adeoque reciproce ut Ę. + PQ. Is autem [3Io] valor manet
idem, si pro PQ ponantur bini valores, quorum productum æquatur T*, migrante tantum-
- 2
modo altera binomii parte in alteram Si enim alter valor sit m, erit alter m * & posito
2 2 2
illo pro PQ : habetur £ -+- m, posito hoc habetur £;; -|- £. sive m -+- £. Sed cum
© e ¢ 'T2 . • � �
eæ distantiæ abeunt ad partes oppositas, fiunt — m, & ;, migrante in negativum etiam
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. o
132. The formula also holds good for the case in which the new mass does not lie at
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
the axis P is denoted by R, the distance of the centre of gravity by G., & the total mass by
Q X PQ2 & M × G. K. --
M × G X R + Q × QP2 w Q X PQ? º
evident from the fact that, from the nature of the centre of oscillation, we have
R=ſ***, 3, thus . Ax AP = M × G. R.
M × G.
I34. Corollary II. The velocity acquired by the mass Q will be
M × G X R × PQ
M × G X R + Q X PQ2
for, this is the velocity PQ (c — wº, or PQ (c
It is
M, then we have x = X. ( ;
-
X C ;
=- ------- Q X PQ2
M × G. × R -- Q X PQ2
and this, when reduced to the same denominator, comes to that which was given, after
cancelling terms of opposite sign.
I35. Corollary III. If, while the circular velocity remained unaltered, the whole mass
of the system is supposed to be collected at a single point lying between the centres of
gravity & 9scillation, the distance of which from the point of rotation is a geometrical mean
between the distances of the other points, or at the same distance on the other side of the
Point of rotation ; then, the same velocity would be impressed on the new mass situated
at any Point. For, in that case, each centre would coincide with that point, & the value
-X c);
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. -
I36. If, on one side or the other, in PG a segment is taken, which is to the distance
of the point from the axis in the subduplicate ratio of the whole mass of the system to the
mass Q3 then, the mass Q, if placed at one of four distances from the axis, two on one side
& two on the other, so that the products for each pair should be equal to the square of the
segment, would at each distance acquire a velocity of the same magnitude although in
opposite directions for the two pairs. Also this velocity would be greatest, when the mass
was placed at the end of the segment on either side of the axis. For, the velocity acquired
M × G. × R × PQ
M × G X R + Q X PQ2
× c, and denoting the segment by + T, of which the square, Tº, must be
PQ
equal to O T2 -EPO2:
T2
3S PO + PQ. Now, this value remains the same, if for PQ we substitute either of the
varies directly as
M × G X R
Q
X c ; dividing this by the constant
X G × R, the velocity will vary directly as & therefore, inversely
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
T*/m ; &, if the former is substituted for PQ, we get Tº/m + m ; or, if the latter, we
have Tºm/Tº H-Tº/m, i.e., m + Tº/m. But, when these distances are taken on the
opposite side, they become – m & – Tº/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
Further extension
of this idea.
Relation to the
centre of oscilla
tion.
A simpler expres-
sion for the velo-
city in the mass
by its help.
The point in which
the whole system
would have to be
Collected in order
to impress the
same velocity on
the mass.
The number of
points, and their
distances from the
axis, for which the
maSS would ac-
quire the same
velocities from the
impact ; where the
velocity would be
greatest.
458 PHILOSOPHIÆ NATURALIS THEORIA
Demonstratio de-
terminationis maX-
mi.
Maximi determina-
tio per calculum
differentialem.
IDuæ aliæ accep-
tiones centri per
cussionis, & ejus
determinatio e x
superioribus.
A quo ita consid-
eratum, & pro
particulari casu
determinatum.
Hic generalius, &
alit e r determin-
atum ad foecundi-
t at e m Theoriæ
ostendendam.
valore formulæ, quod ostendit directionem motus contrariam priori, systemate nimirum
hinc, & inde ab axe in partibus oppositis habente directiones motuum oppositas.
§+ PQ est
finita, & evadit infinita facto PQ tam infinito, 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 quem hinc & inde valores
æquales sunt. Ibi igitur id habetur maximum.
I 37. Quoniam autem assumpto quovis valore finito pro PQ, formula
I 38. Scholium 2. Libuit sine calculo differentiali invenire illud maximum, quod ope
- 2
calculi ipsius admodum facile definitur. Ponantur T = t, & PQ = z. Fiet formula £+z,
0 • d e •
& differentiando — '£ + dz = o, sive — t* -+- z* = o, vel z* = t*, & z = -E t, sive
Z,
PQ = + T, ut in corollario 4 inventum est.
I 39. Licebit autem jam ex postremis duobus corollariis deducere alias duas notiones
centri percussionis, cum suis eorundem determinationibus. Potest primo appellari centrum
percussionis illud punctum, in quo tota systematis massa collecta eandem velocitatem
imprimeret massæ eidem incurrendo in eam eodem suo puncto cum eadem velocitate, quæ
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 massæ, in [3II] quam incurritur. In
hac acceptione centrum percussionis itidem eleganter determinatur per corollarium
quartum, mutando eam distantiam in ratione subduplicata massæ, in quam incurritur,
ad massam totius systematis. -
I40. In hoc secundo sensu acceptum, & investigatum esse centrum percussionis a
summo Geometra Celeberrimo Pisano Professore Perrellio, nuper mihi significavit Vir
itidem Doctissimus, & geometra insignis Eques Mozzius, qui & suam mihi ejus centri
determinationem exhibuit pro casu systematis continentis unicam massam in rectilinea
virga inflexili. -
I4I. Libuit rem longe alia methodo hic erutam generaliter, & cum superioribus
omnibus conspirantem, ac ex iis sponte propemodum profluentem proponere, ut innotescat
mira sane fœcunditas Theorematis simplicissimi pertinentis ad rationem virium
compositarum in systemate massarum trium. Sed de his omnibus jam satis.
DABAM FLORENTIAE, I7 junii, I758.
EINIS.
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 Tº/PQ + 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 2. 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 tº/z + z. Differentiating, we have
— t dz/z* + dz = 0, or — tº + z* = 0, or z* = t”; & z = + t, or PQ = + T, as
was found in corollary IV. r
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, Signor 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,
17th june, 1758.
THE END,
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.
I N D E X
PARS I
Introductio
Expositio Theoriæ . • t• � � £• ę • •
Occasio inveniendæ, & ordo, ac analytica deductio inventæ Theoriæ
Lex continuitatis quid sit
Ejus probatio ab inductione ; vis inductionis
Ejusdem probatio metaphysica
Ejus applicatio ad excludendum immediatum contactum
Deductio legis virium, & determinatio curvæ eam exprimentis
Primorum elementorum materiæ indivisibilitas, & inextensio
Eorundem homogeneitas • �
Objectiones, contra vires in genere, & contra hanc virium legem
Objectiones contra hanc constitutionem primorum elementorum materiæ
PARS II
Applicatio Theoriæ ad Mechanicam
Argumentum hujus partis
Consideratio curvæ virium
De arcubus
De areis . ç • «• e. • � • �
De appulsibus ad axem, & recessibus in infinitum, ubi de limitibus virium .
De combinationibus punctorum, & primo quidem de systemate punctorum duorum
De systemate punctorum trium
De systemate punctorum quatuor . • • ts & • r • • - ©
De massis, & primo quidem de centro gravitatis, ubi etiam de viribus quotcunque generaliter
componendis
De æqualitate actionis, ac reactionis . ę ę
De collisionibus corporum, & incursu in planum immobile .
Exclusio veræ virium resolutionis «• ç � t. * .
De compositione, & imaginaria resolutione virium, ubi aliquid etiam de Viribus vivis
De continuitate servata in variis motibus, ubi quædam de collisionibus, de reflexionibus, &
refractionibus motuum -
De systemate trium massarum - • ©
Theoremata pertinentia ad directiones virium compositarum in singulis
Theoremata pertinentia ad ipsarum virium magnitudines
Centrum æquilibrii, & vis in fulcrum inde. ę
Momenta pro machinis, & omnia vectium genera inde itidem
Centrum itidem oscillationis
Centrum etiam percussionis . • • A* * * ę *
Multa huic Theoriæ communia cum aliis hic tantummodo indicata
De fluidorum pressione
De velocitate fluidi erumpentis
PARS III
Applicatio Theoriæ ad Physicam
Argumentum hujus partis
Impenetrabilitas � «• •» ω • • e
Extensio cujusmodi sit in hac Theoria, ubi de Geometria
Figurabilitas, ubi de mole, massa, densitate -
Mobilitas, & continuitas motuum
Æqualitas actionis, & reactionis &• • e • è •
Divisibilitas quæ sit : componibilitas æquivalens divisibilitati in infinitum
Pag.
I3
I6
22,
28
33
37
4I
45
59
77
77
77
79
82
86
92
I IO
III
I24
I 25
I 32
I 36
I 39
I43
I43
I45
I48
I 5o
I 52
I 57
I 58
I 59
I62
I72
I75
178
I79
Num.
I6
32
39
48
63
73
8I
9I
IOO
I3I
I 66
167
I 68
I72
I79
I89
2o4
238
24O
265
266
279
289
297
307
3o8
3I3
32 I
325
328
344
347
348
354
358
36o
37I
375
383
388
39I
460
gº I N D E X
PART I
Introduction
Statement of the theory e e º e e © º e . . .
What led to its discovery, the order, & the analytical deduction of the theory found
What the Law of Continuity is º e e
Proof of this law by induction; the power of induction
Metaphysical proof of the same thing º e e º & º e
Its application for the purpose of eliminating the idea of immediate contact
Deduction of the law of forces & the determination of the curve representing this law
Indivisibility & non-extension of the primary elements of matter
Their homogeneity . © º tº tº e e e º -
Objections against forces in general, & against this particular law of forces
Objections against this particular constitution of the primary elements of matter
PART II
Application of the Theory to Mechanics
The theme of the second part .
Consideration of the curve of forces .
The arcs
The areas º e e g e e º tº * º º g º
Approach towards & recession to an infinite distance from the axis; the limits of the forces
Combinations of points; firstly, a system of two points -
System of three points
System of four points e • * : * e º e e º e te
Masses; firstly, the centre of gravity; & the general composition of any number of forces
Equality of action & reaction . e º º
Collision of solid bodies, & impact on a fixed plane
Exclusion of the idea of real resolution of forces o e º e
Composition, & hypothetical resolution, of forces; remarks on “living forces.” º
Continuity observed in various motions; also some remarks upon collisions, reflections &
refractions of motions
System of three masses e e e • º o º © e tº
Theorems relating to the directions of the resultant forces on each of the systems
Theorems relating to the magnitude of these forces o º
The centre of equilibrium, & the force on the fulcrum derived from it
Theory of moments for machines, & hence all kinds of levers also
The centre of oscillation .
The centre of percussion . e e º º e o e
Many points of this Theory common to others, merely mentioned
Pressure of fluids e e
Velocity of a fluid issuing from a vessel
PART III
Application of the Theory to Physics
The theme of the third part
Impenetrability © © e e e e e e
What kind of extension is admitted in this Theory; geometry .
Figurability; also volume, mass & density.
Mobility, & continuity of motions
Equality of action & reaction g º tº º e e
What divisibility is; componibility equivalent to infinite divisibility
* •
Page Art.
35 I
37 7
45 I6
5 I 32
55 39
63 48
7I 63
77 73
83 81
89 91
95 IOO
III I31
135 I66
I35 I67
135 168
I39 I72
I43 I79
I49 I89
I55 204
187 238
I89 24o
203 265
205 266
2I3 279
223 289
227 297
237 307
237 308
24I 313
247 32I
249 325
25I 328
257 344
259 347
259 348
263 354
267 358
267 360
273 371
275 375
281 383
283 388
285 391
461
462 INDEX
Immutabilitas primorum materiae elementorum .
Gravitas
Cohaesio . º º
Discrimen inter particulas.
Soliditas, & fluiditas º º º
Virgae rigidae, flexiles, elastica, fragiles
Viscositas.
Certa: quorundam corporum figurae .
De fluidorum resistentia
De elasticis, & mollibus .
Ductilitas, & Malleabilitas
Densitas indifferens ad omnes proprietates
Vulgaria 4 elementa quid sint .
De operationibus chemicis singillatim
De natura ignis º º o º º º i º º
De lumine, ubi de omnibus ejus proprietatibus, ac de Phosphoris
De sapore, & odore .
De sono º º g
De tactu, ubi de frigore, & calore
De electricitate, ubi de analogia, & differentia materia electrica, & ignea .
De Magnetismo d o º i º
Quid sit materia, forma, corruptio, alteratio
APPENDIX
Ad M etaphysicam pertinens
De Anima
De DEO.
SUPPLEMENTA
S I De Spatio, & Tempore º º
S II De Spatio, ac Tempore, ut a nobis cognoscuntur
S III Solutio analytica Problematis determinantis naturam legis virium
S IV Contra vires in minimis distantiis attractivas, & excrescentes in infinitum
S V De AEquilibrio binarum massarum connexarum invicem per bina alia puncta
S VI Epistola ad P. Scherffer .
NOI RIFORMATORI
Dello Studio di Padova.
. 289
Pag.
I8I
I82
185
I9I
I94
I99
200
2OO
203
2O4
205
2o6
206
207
2I5
217
234
235
237
239
242
243
248
254
264
273
277
293
297
Num.
398
399
406
4 I9
426
436
438
439
442
446
448
449
45O
45I
467
472
5O3
504
507
5II
5I4
516
526
539
I8
25
77
86
93
VENDO veduto per la Fede di Revisione, ed Approvazione del P. F. Gio. Paolo Zapparella,
Inquisitor Generale del Santo Officio di Venizia, del Libro intitolato Philosophie Naturalis Theoria
redacta ad unicam legem virium in natura existentium, Auctore P. Rogerio fosepho Boscovich (5 e 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 alle 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 o © º e º º . 287 398
Gravity . . . . . . . . . . . . . . . 287 399
Cohesion. º * e © º º e e & º e e º . 291 406
Distinction between particles . º e o º e e e e e . 299 419
Solidity & fluidity © ſe © º © e e e e º . 303 426
Rigid, flexible, elastic & fragile rods . º º e e º º e e . 309 436
Viscosity . te e º © º e * º * § © © . 3 II 438
Definite shapes of certain bodies . e Q º tº º e e e . 3 II 439
Resistance of fluids . º © º e º º © º e e º . 3 IS 442
Elastic, & soft, bodies . º g & e • º º e e e . 317 446
Ductility & malleability . º e tº e tº º º º e e . 317 448
Density unrelated to all other properties . e e & e ë © ſe . 3 I7 449
The so-called “four elements * ſº tº e © º o e * © . 319 450
Chemical operations, each in turn . * º e º e tº e e . 3I9 45I
The nature of fire . e . . te e e º © e e e . 329 467
Light, its properties, & light-giving bodies . º º e e . . . e . 33 I 472
Taste and smell . © © e º e º º e to © • . 353 503
Sound . e º º º g e e º tº * e © º . 355 504
Touch ; cold & heat © o e e & º º e e º º . 357 507
Electricity; the resemblances & differences between electric matter & fire . º . 361 5 II
Magnetism . • © º e e º © e e c º e . 363 514
Matter, form, corruption, alteration . s e º º º & e e . 365 516
APPENDIX
Relating to Metaphysics
The Soul 373 526
GOD 379 539
SUPPLEMENTS
I On Space & Time º e 393 I
II. On Space & Time, as we know them º º e º º - 0. 405 18
III Analytical solution of the problem to determine the nature of the law of forces . 4II 25
IV Arguments against forces that are attractive 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
E, 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 Philosophiae Naturalis Theoria 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. E.Mo, Procurator, Censor.
Z. ALVISE Mocenico, Censor.
; : *
Registered in Book, p. 47, no. 383.
September 18th, 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 BOSCOVICH, SJ.
impressorum usque ad initium anni 1763.
Opera, 6 opuscula justa molis.
Sopra il Turbine, che la notte tra gli II, e 12 Giugno del 1749 danneggiò una gran parte di Roma. Dis-
sertazione del P. Ruggiero Giuseppe Boscovich della Comp. di Gesù. In Roma appresso Nicolò,
e Marco Pagliarini, in 8.
Elementorum Matheseos tomi tres, in 4. Prodierunt anno 1752 sub titulo, Elementorum Matheseos ad
usum studiosa juventutis, tomi primi pars prima complectens Geometriam planam, Arithmeticam
vulgarem, Geometriam Solidorum, & Trigonometriam cum planam, tum sphaericam. Pars altera,
in qua Algebrae finita elementa traduntur. Romae: excudebat Generosus Salomoni. Iis binis tomis
sine nova eorum impressione mutatus est titulus anno 1754 in hunc, Elementorum Universa, Matheseos
Auctore P. Rogerio Josepho Boscovich Soc. Jesu Publico Matheseos Professore Tomus I continens
&c. Tomus II continens &c, 85 adjectus est sequens.
Tomus III continens Sectionum Conicarum Elementa nova quadam methodo concinnata, 8 Disserta-
tionem de Transformatione locorum Geometricorum, ubi de Continuitatis lege, ac de quibusdam
Infiniti mysteriis: Typis iisdem eiusdem Generosi Salomoni omnes in 8. Extat eorundem impressio
Veneta anni 1758, sed typorum mendis deformatissima.
De Litteraria Expeditione per Pontificiam ditionem ad dimetiendos duos Meridiani gradus, & corrigendam
mappam geographicam, jussu, & auspiciis Benedicti XIV. P.M. suscepta Patribus Soc. Jesu Chris-
tophoro Maire, & Rogerio Josepho Boscovich, Romae 1755. In Typographio 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 Inaequalitatibus, quas Saturnus, & Jupiter sibi mutuo videntur inducere, prasertim circa tempus
conjunctionis. Opusculum ad Parisiensem Academiam transmissum, & nunc primum editum.
Auctore P. Rogerio Josepho Boscovich Soc. Jesu ; Romae ; ex Typographia Generosi Salomoni, in 8.
Philosophiae Naturalis Theoria redacta ad unicam legem virium in Natura existentium Auctore P. Rogerio
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 hic exhibetur : Epistola habetur in eſus
Supplementis.
Admotationes in aliorum Opera.
Caroli Noceti e Societate Jesu de Iride, 8; Aurora Boreali Carmina . . . cum notis Josephi Rogerii Bosco-
vich ex eadem Societate. Romae: excudebant Nicolaus, & Marcus Palearini, in 4. Perperam
nomen Josephi antepositum est ibi nomini Rogerii.
Philosophiae Recentioris a Benedicto Stay in Romano Archigymnasio Publico Eloquentiae Professore . . .
cum adnotationibus, & Supplementis P. Rogerii Josephi Boscovich S.J. in Collegio Rom. Publici
Matheseos Professoris. Tomus I. Romae : Typis, & sumptibus Nicolai, & Marci Palearini, in 8.
Dua ejus editiones prodierunt simul.
Tomus II Romae : Typis, & Sumptibus Nicolai, & Marci Palearini, in 8. In singulis hisce voluminibus
ea, qua ad P. Boscovich pertinent, efficerent per se ipsa justum volumen. In solis primi Stayani tomi
supplementis occurrunt 39 ipsius Dissertationes de variis argumentis pertinentibus potissimum ad Meta-
physicam (5 Mechanicam.
Dissertationes impressa pro exercitationibus annuis, 85 publice propugnata : omnes
º in 4.
De Maculis Solaribus, Exercitatio Astronomica habita in Collegio Romano Soc. Jesu. Romae: ex Typo-
graphia Komarek.
De Mercurii novissimo infra Solem transitu. Dissertatio habita in Seminario Romano. Romae, Typis
Antonii de Rubeis. . .
Constructio Geometrica Trigonometriae sphaericae. Romae, ex Typographia Komarek. Hujus titulus vel
est hic ipse, vel parum ab hoc differt.
De Aurora Boreali Dissertatio habita in Seminario Romano. Romae : Typis Antonii de Rubeis. Eadem
eodem anno edita fuit etiam typis Komarek.
De Novo Telescopii usu ad objecta caelestia determinanda. Dissertatio habenda a PP. Soc. Jesu in Collegio
Romano. Romae, ex Typographia Komarek. Extat recusa sine ulla mutatione in Actis Lipsiensibus
ad annum 174O.
sir-
FH H
Annus primae
edition.
I749
I752
I754
I755
1756
1758
I747
I755
176o
1736
1737
I738
I739
Annus primae
editiom.
I74O
1741
I742
I743
I744
I745
I746
I747
I748
I749
I75 I
I753
I754
I755
De Veterum argumentis pro Telluris sphæricitate. Dissertatio habita in Seminario Romano Soc. Jesu.
Romæ : Typis Antonii de Rubeis. - -
Dissertatio de Telluris Figura habita in Seminario Romano Soc. Jesu. Romæ : Typis Antonii de Rubeis.
Eadem prodiit in 8, anno 1744 in opere, cui titulus Memorie &c. In Lucca per li Salani, e Giuntini,
& in titulo additur : nunc primum aucta, & illustrata ab ipsomet Auctore ; sed ea editio scatet typorum
erroribus, ut & reliqua inferius nominanda in eadem collectione inserta.
De Circulis Osculatoribus. Dissertatio habenda a PP. Societatis Jesu in Collegio Romano. Romæ :
ex Typographia Komarek.
De Motu corporum projectorum in spatio non resistente. Dissertatio habita in Seminario Romano Soc.
Jesu, Romæ : Typis Antonii de Rubeis. -
De Natura, & usu infinitorum, & infinite parvorum. Dissertatio habita in Collegio Romano Soc. Jesu.
Romæ : ex Typographia Komarek. -
De Inæqualitate gravitatis in diversis Terræ locis. Dissertatio habita in Seminario Romano Soc. Jesu. .
Romæ : Typis Aritonii de Rubeis.
De Annuis Fixarum aberrationibus. Dissertatio habita in Collegio Romano Societatis Jesu. Romæ :
ex Typographia Komarek. -
De Observationibus Astronomicis, & quo pertingat earundem certitudo. Dissertatio habita in Seminario
Romano Soc. Jesu. Romæ : Typis Antonii de Rubeis.
Disquisitio in Universam Astronomiam publicæ Disputationi proposita in Collegio Romano Soc. Jesu.
Romæ : ex Typographia Komarek. -
De Motu Corporis attracti in centrum immobile viribus decrescentibus in ratione distantiarum reciproca
duplicata in spatiis non resistentibus. Dissertatio habita in Collegio Romano. Romæ : Typis
Komarek. Eadem prodiit anno 1747 sine ulla mutatione in Commentariis Acad. Bononiensis Tom. II.
par. III. a.
Nova methodus adhibendi phasium observationes in Eclipsibus Lunaribus ad exercendam Geometriam,
& promovendam Astronomiam. Dissertatio habita in Collegio Romano. Romæ : ex Typographia
Komarek. Eadem prodiit in 8, anno 1747 cum exigua mutatione, vel additament0 in Opere $uperius
Mmemorato, cui titulus Memorie &c. In Lucca per li Salani, e Giuntini.
De Viribus Vivis. Dissertatio habita in Collegio Romano Soc. Jesu. Romæ : Typis Komarek. Eadem
prodiit ammo I747 sine ulla mutatione in Commentariis Acad. Bonom. To. II. par. III, §5 in Germania
pluribus vicibus est recusa. - •
De Cometis. Dissertatio habita a PP. Soc. Jesu in Collegio Rom. Romæ : ex Typographia Komarek.
De Aestu Maris. Dissertatio habita a PP. Soc. Jesu in Collegio Romano. Romæ : ex Typographia
Κomarek. Ea est Dissertationis pars I ; secunda pars mumquam prodiit. Quæ pro illa fuerant des-
timata, habentur in Opere De Expeditione Litteraria, & in supplementis Philosophiae Stayamae tomo II.
Dissertationis de Lumine pars prima publice propugnata in Seminario Romano Soc. Jesu. Romæ : Typis
Antonii de Rubeis. <•
Dissertationis de Lumine pars secunda publice propugnata a PP. Soc. Jesu in Collegio Romano. Romæ :
ex Typographia Komarek. - -
De Determinanda Orbita Planetæ ope Catoptricæ, ex datis vi, celeritate, & directione motus in dato
puncto. Exercitatio habita a PP. Soc. Jesu in Collegio Romano. Romæ : ex Typographia Komarek.
De Centro Gravitatis. Dissertatio habita in Collegio Romano Soc. Jesu. Romæ : ex Typographia
Komarek. Eadem paullo post prodiit iterum cum sequenti titulo, & additamento. De Centro Gravitatis.
Dissertatio publice propugnata in Collegio Romano Soc. Jesu Auctore P. Rogerio Josepho Boscovich
Societatis ejusdem. Editio altera. Accedit Disquisitio in centrum Magnitudinis, qua quædam in
ea Dissertatione proposita, atque alia iis affinia demonstrantur. Romæ, Typis, & sumptibus Nicolai,
& Marci Palearini.
De Lunæ Atmosphæra. Dissertatio habita a PP. Soc. Jesu in Collegio Romano. Romæ : ex Typographia
Generosi Salomoni. Multa eorumdem typorum exemplaria prodierunt paullo post cum nomine Auctoris
in ipso titulo, &5 cum exigua unius loci mutatione.
De Continuitatis Lege, & Consectariis pertinentibus ad prima materiæ elementa, eorumque vires. Disser-
tatio habita a PP. Societatis Jesu in Collegio Romano. Romæ : ex Typographia Generosi Salomoni.
De Lege virium in Natura existentium. Dissertatio habita a PP. Soc. Jesu in Collegio Romano. Romæ :
Typis Generosi Salomoni. -
De Lentibus, & Telescopiis dioptricis. Dissertatio habita in Seminario Romano. Romæ : ex Typographia
Antonii de Rubeis. -
Plures ex hisce Dissertationibus prodierunt etiam iisdem typis, sed cum alio titulo, habente non locum, ubi
sunt habitae, vel propugnatae, sed tantummodo nomen Auctoris. In hac postrema mutatæ sunt binae paginae,
posteaquam plurima exemplaria fuerant distracta. In prioribus tribus sunt pauca quaedam mutata, vel
addita a P. Horatio Burgundio adhuc Professore Matheseos in Collegio Romano, qui fuerat ejus Prae-
ceptor ; sed eo jam ad Dissertationes ejusmodi conscribendas utebatur. -
Eæ omnes, quae pertinent ad Seminarium Romanum, habent in ipso titulo adscripta nomina Nobilium Com-
victorum, qui illas propugnarunt, & sub eorum nomine referuntur plures ex iis in Actis Lipsiensibus.
Multa pertinentia ad ipsum P. Boscovich habentur in binis Dissertationibus, quarum tituli, Synopsis Physicæ
Generalis, & De Lumine, quarum utraque est edita Romæ anno 1754, Typis Antonii de Rubeis, in 4.
Id ibidem testatur earumdem 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 im admota-
tionibus ad Carmem P. Noceti, inserta in quadam Dissertatione impressa Romæ circa annum 1756, &
Annus primae
publice propugnata, cujus Auctor est P. Lunardi Soc. fesu, qui affirmat ibidem, se eandem acceptam ab
ipso P. Boscovich proponere effusdem verbis.
Subjiciemus jam bina opuscula Italica, qua communi nomine PP.º Le Seur, jacquier, ac suo conscripsit
ipse P. Boscovich. Utrumque est sine loco impressionis, 5 nomine Typographi, impresserunt autem
Palearini Fratres Romae jussu Praesulis, qui tum curabat Fabricam S. 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
per ordine di Nostro Signore Benedetto XIV, in 4. In fine opusculi habentursubscripta omnium tria
7l07714, 71 (2.
Riflessioni de PP. Tomaso Le Seur, Francesco Jacquier dell' Ordine dei Minimi, e Ruggerio Giuseppe
Boscovich della Comp. di Gesù sopra alcune difficoltà 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 iis, quas Itali vocant Scritture, pro quadam lite Ecclesiae S. Agnetis
Romanae, pertinentes ad aquarum cursum Romae edita anno 1757.
Inserta.
Nunc faciemus gradum ad inserta in Publicis Academiarum monumentis, in diariis, in collectionibus, 5 in
privatorum Auctorum Operibus.
In Monumentis Acad. Bononiensis.
Praeter reimpressionem binarum Dissertationum in To. II, de quibus supra, habetur in To. IV De Litteraria
Expeditione per Pontificiam ditionem. Est Synopsis amplioris Operis, ac habenturplura eſus 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 rarità,
che si sono tra le rovine della medesima ritrovate. Luogo di Vitruvio illustrato. Ibi ejus schedias-
matis Auctor profert, uti ipse profitetur, qua singillatim audierat ab ipso P. Boscovich.
Dimostrazione facile di una principale proprietà delle Sezioni Coniche, la quale non dipende da altri
Teoremi conici, e disegno di un nuovo metodo di trattare questa dottrina.
Dissertazione della Tenuità della Luca Solare, Del P. Ruggiero Gius. Boscovich Matematico del Collegio
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 Gesù.
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
Gius. Boscovich della Comp. di Gesù.
Parte seconda &c. -
Soluzione Geometrica di un Problema spettante l'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 Gesù. -
Dialogi Pastorali V sull'Aurora Boreale del P. Ruggiero Gius. Boscovich della Comp. di Gesù.
Dimostrazione di un metodo dato dall'Eulero per dividere una frazione razionale in più frazioni più semplici
con delle altre riflessioni sulla stessa materia.
Lettera del P. Ruggiero Gius. Boscovich della Comp. di Gesù al Sig. Ab. Angelo Bandini in risposta
alla lettera del Sig. Ernesto Freeman sopra L'Obelisco d'Augusto. Nomen Freeman est fictitium,
Auctorem denotans Neapoli latentem, 5 aliis Operibus satis notum. Extat eadem etiam in folio.
Altera de eodem Obelisco admodum prolixa Epistola, Italice, 85 Latine scripta ad eundem Bandinium suo
nomine ab ipso P. Boscovich habetur in eſusdem Bandinii Opere, cui titulus, De Obelisco Caesaris Augusti
e Campi Martii ruderibus nuper eruto. Commentarius Auctore Angelo Maria Bandinio. Romae
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 Valentium in eſus discessu ab Urbe
eam Epistolam conscripsit, ac eius 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,
e raccolte dal P. Ruggiero Gius. Boscovich della Comp. di Gesù 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, Praeter binas dissertationes, de quibus supra, habetur.
Problema Mechanicum de solido maxima attractionis solutum a P. Rogerio Josepho Boscovich Soc. Jesu
Publico Professore Matheseos in Collegio Romano: Tomo I.
De Materiae divisibilitate, 8 Principiis corporum. Dissertatio conscripta jam ab anno 1748, & nunc
primum edita. Auctore P. Rogerio Jos. Boscovich Soc. Jesu, To. IV.
Omnium horum quatuor Opusculorum habentur etiam exemplaria seorsum impressa,
edition.
I742
I743
1757
I757
1746
I747
I748
I749
I75o
I753
I743
1757
Annus primae
edition.
I745
I752
1758
I753
1757
1758
176o
In editione Elementorum Geometria Patris Tacqueti facta Roma sumptibus Venantii Monaldini, Typis
Hieronymi Mainardi, in 8. habetur Trigonometria sphaerica P. Rogerii Josephi Boscovich, qua deinde
adhue magis expolita prodiit Tomo I. ejus Elementorum Matheseos. Habetur praterea ibidem Trac-
tatus De Cycloide, 8 Logistica, qui etiam seorsum impressus est iisdem typis.
In Opere Comitis Zoannis Baptista Soardi, cui titulus Nuovi instrumenti &c. in Brescia dalle stampe di
Gio Battista Rizzardi, in 4, habentur bina epistola Italica ipsius P. Boscovich de Curvis quibusdami,
cum figuris, & demonstrationibus.
In Optica Abbatis De la Caille latine reddita a P. Carolo Scherffer Soc. Jesu, 5 impressa Vienna in Austria
habetur schediasma Patris Boscovich de Micrometro objectivo.
In potremo tomo Commentar Academia Parisiensis in Historia, & in uno e tomis Correspondentium
eiusdem Academia, creditur esse breve aliquid pertinens ad ipsum P. Boscovich. Est aliquid etiam in
diario Gallico Journal des avans, 5 fortasse in Anglicanis Transactionibus, aidue alibi insertum hisce
itinerum annis.
Poetica,
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 Arcadia
Custos illustrium Poetarum Arcadum effigies formandas jaculorum ludi substitnerat. Romae in 8.
Extat eadem iisdem Typis etiam in Collectione tum impressa omnium, qua ea occasione sunt recitata.
Stanislai Poloniae Regis, Lotharingiae, ac Barri Ducis, & inter Arcades Euthimii Aliphiraei, dum eius effigies
in publico Arcadum Coetu erigeretur, Apotheosis. Auctore P. Rogerio Josepho Boscovich Soc.
Jesu inter Arcades Numenio Anigreo. Romae ex Typographia Generosi Salomoni, in 8. Est
poema versu heroico. Idem autem recusum fuit Nancei cum versione Gallica Domini Cogolin,
Pro Benedicto XIV. P.M. Soteria. Est itidem poema Heroicum eiusdem 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 Pontifice convalescente.
In Nuptiis Joannis Corrarii, & Andrianae Pisauriae e nobilissimis Venetae Reip. Senatoriis familiis. Carmen
P. Rogerii Jos. Boscovich S.J. Publici in Romano Collegio Matheseos Professoris. Romae : ex
Typographio Palladis: excudebant Nicolaus, & Marcus Palearini, in 4.
De Solis, ac Lunae defectibus libri V P. Rogerii Josephi Boscovich Societatis Jesu ad Regiam Societatem
Londinensem, Londini 1760. in 4. Non habetur nomen Typographi, qui impressit, sed Bibliopolarum
quorum sumptibus est impressum : deest hic eius editionis exemplar, ex quo ea nomina correcte describantur.
Idem recusum fuit anno 176I Venetiis apud Zattam in 8°. cum exiguo additamento in fine, 5 cum hoc
catalogo, quem inde huc derivavimus. Habentur in adnotationibus bina Epigrammata cum zersionibus
Italicis, sive Sonetti.
Est (5 aliud eſus poema Heroicum anno 1756 impressum Vienna in Austria in collectione carminum facta
occasione inaugurationis novarum Academia Viennensis adium. -
Sunt 5 epigrammata nonnulla in Collectionibus Arcadum, inter qua unum pro recuperata valetudine Johannis
V Lusitania Regis, 3 unum pro Rege tum utriusque Sicilia, 3 nunc Hispania, ac pro Regina eſus
conjuge.
Extant etiam pauca admodum exemplaria unius ex illis, quas in Italia appellamus Cantatine, impressa Viterbii
anno 175o pro Visitatione B. Mariae Virginis, in qua sex, quas dicimus Ariette, profanae ad sacrum
argumentum transferenda erant, mamente Musica, 3 inter se connectenda.
p.
.
p.
:
ERRATA
2,
. II, for ac omnem read ad omnem
. 5, for has been read should be
. 18, for Venetisis read Venetiis
. 9 from bottom, for exceres read exerces
. 4 from bottom, for eocatum read evocatum
. 18 from bottom, after despatched add to the
- Court of Spain
. 13 from bottom, for befits read befit
. I, for publico read publice
. I 3, for utique read ubique
. 28, for infiliciter read infeliciter
8, for opportunam read opportunum
. 9, for mediocrum read mediocrium
. I3, for aliquando read aliquanto
. IO from bottom, for repulsivis read repulsivas
. I 3, for adhibitis read adhibitas
1. 24, for postremo read postrema
. 2, for alter read altera
. 15, after vero etiam insert leges
. I7, for acquiretur read acquireretur
1. 28, for -menae read -mena
1. 22, for Naturam read Natura
1. 23, for quandem read quandam
1
l
2
2
87Ö4.3
253.
I
O
2
. 29, for recidit read recedit
. 32, for postguam read post quam
. 34, for many read most
. 18, for linae read lineae
1. 29, for genere read generis
. 26, for deferendam read deserendam
l. 31, for viderimus read videremus
1. 46, for nominandi read nominando
52, ll. 5, 6 of marginal note to $7, for nihilmu read
nihilum.
54, l. I, for exhibit read exhibet
1. 3, for oppositae read opposita
1. I2, for sit read fit
55, 1. 4, after & add then
56, 1. 3, for servat read servant
58, 1. 32, for crederit read crederet
60, l. 3, marg. note, Art. 46, for sit read fit
64, l. 2, for terio read tertio
65, l. 57, for fact read by the fact
66, l. 9, for concipiantur read concipiatur
1. I6, for ordinate read ordinatae
67, 1. 48, for before & read previously
68, l. II, for in GM' read in GM
71, 1.46, for and this read and that
nowhere
72, 1. I, for ejusmodi read hujusmodi
1. 4 from bottom, for potissimuim
potissimum
74, 1. 38, for illo read illa
76, l. 3 from bottom, for devenirent read devenerint
81, 1.42, for is read ought to be
82, l. 5, marg. note, Art. 82, for se read sed
86, Art. 89, in marg. note, for densitatis read densitas
88, 1. II, for adi read ad
1. 16, for reliquent read relinquent
90, 1.30, for diversimodo read diversimode
1. 34, for distantia read distantiae 3.
Art. 95, marg. note, for de- read dif-
92, 1.33, for apparent read apparerent
94, 1. 22, for incurrant read incurrunt
95, Art. Io9, l. I, for are read is ; and in marg. note
insert in between and and what
96, l. 8, for potissimo read potissimum
1. 16, for praecedentum read praecedentem
marg. note, Art. IoS, for transire read transiri
pſ
.
. IO7, l.
99, l. 40, insert a comma after locus
. IOO, marg. note, Art. II2, for recte read recta
l. 32, for ellipis read ellipsis
. IO6, marg. note, Art. I25, for perfectionum read
perfectionum
23, for off they are read away they go
1. IS from bottom, for have read has
. 199, l. 27, after tantummodo add admitto
. IIo, l. 17, for expandantur read expendantur
1. 24, for a read &
bottom line, for distinctis read distinctas
. II2, 1. 27, for veteram read veterem
. II.3, 1.
5 from bottom, for because read that
1. 4 from bottom, after change add is excluded by
. II5, 1.33, for and read et
marg. note, Art. I39, add at end impugned
. I 18, 1. 7 from bottom for ali read alia
. 122, 1. 26, for justmodi read ejusmodi
. I25, 1. 29, for ignored read urged in reply
. I28, 1.31, for ea read ea:
. 129, 1. 16, for Principii read Principiis
. 139, l. 8, for arm E read arm ED
footnote, 1. 5, for DP read OP
. I40, l. 34, insert cum before directione
1. 4, footnote, for ut in read ut n
. 148, 1. Io, for Expositas read Expositis, for curva read
CUTV2 iſl
. I56, l. I, for a que read atque
1. 7, for caculo read calculo
1.39, for Tam read Tum
. 158, Art. 209, marg. note, add at end Legum multitudo
& varietas
footnote, l. II from bottom of page, for obvenerit
read obveniret
. 160, footnote, 1. I, for sit read fit
1. I2, for ed read sed
. 161, footnote, 1. 20, after segment add DR
. 162, 1. 7 from bottom, for reflexionis read reflex-
;
97, l. 9 from bottom, for quite enough read better
this is found
read
;p:
. 167, 1.40, for ae read da
. I68, 1. 8, from bottom for 27C.—'AC read 27 C’AC
. 171, 1.4 from bottom for GL, or LI read GI, or IL
. 172, l. 34, for compositas read compositis
. 175, 1. I3, for 30 read 27
. 208, last line, for
iones
1. 39, insert a comma after approximately
. 176, 1. 7, for delatam read delatum
. 178, marg. note, Art. 230, 1, 5, for foco read focos
. 188, 1.31, for summa read summa:
. 195, l. 19, insert a comma after point P
. 197, 1.35, for sum of the (at end of line) read sums of the
. 198, Fig. 40, insert F where AE cuts CD -
. 199, 1.35, for ceases read cease
1. 37, after all, & insert I assume
. 202, l. 6, for summa read summam
Art. 264, 1.3, for quacunque read quancunque and
in marg. note for corallarium read corol-
larium
. 205, 1. 21, for recessions read recession
. 206, l. 3, for globis read globus
l. 2 from bottom, insert motu before quodam
m –– n m + n
Il III
read in each case
. 209, l. II, for (2CQ — 2Cq) read (2CQ — 2CQ)
. 2 Io. l. 8, for quiescat read quiescit
. 211, 1. 25, the denominator (Q-H q) should be (Q -- q)*
. 215, 1. 5, for BP read BO
. 223, 1. 26, for 50,61,62 read 50,51,52
. 227, 1. 21 from bottom, for to read of
. 228, 1. 5, from bottom, for Angulum read Angulus
ERRATA
p. 233, l. 8, for in volute read evolute
bottom line, after vary insert inversely
. 241, marg. note, Art. 313, add at end This is very soon
proved -
. 242, last line, for denominator AD read BD
. 247, l. I5 from bottom, for A & B read B & C
. 248, 1. 24, for conversione read conversionem
. 250, 1.39, for justa read juxta
. 252, l. 5 from bottom, for gravitas read gravitatis
256, 1. Io, for quaesitum read quaesitam
. 270, marg, note, Art. 366, 1.5, for magnas read magna
. 278, l. 7, for varior read rarior
280, l. I2 from bottom, for tranctanda read tractanda
284, l. I5, for sit read fit
. 286, l. 20, for multuplicetur read multiplicetur
. 288, 1. 21, for Solum read Solem
. 292, l. I, for quietam read quietem
l. 3, for sit, read fit
1. 25, for ullo, read ulla deleting the comma
1. 26, for illa read ille
. 304, l. I2 from bottom, for propre read prope
. 310, 1. I5 from bottom, for haebebunt read ha-rebunt
3I4, l. 2, insert & before mutandam
319, 1. I8 from bottom, for some repel read and repel
. 22, for pertinet read pertinent
. 25, for ethers read others
. I6, for aequabilis read a qualibus
. 21, for Benvenuti read Benvenutus
l. 3 from bottom, for qui a read quia
. 35, insert utcunque after circunquaque
, 346, l. 19, for sit read fit
i
#.
. 348, l. 28, for irregularitur read irregulariter
. 350, l. 13 from bottom, for flexo read flexu
. 355, l. I4, for with read to
. 356, l. 9, after porro aliud insert post aliud
1. 19, after accidit insert idem accidit
. 366,1. 32, for ordores read odores
. 394, 1.31, for imaginara read imaginaria
. 396, l. 19 from bottom, after solum add etiam
1. I4 from bottom, for sunt read sint
. 398, 1. 20, after est insert tota
l. 35, for esses read esse
p. 400, 1.33, after omnino insert Saltem
p. 406, l. 6, for congruant read congruunt
p. 410, l. 8, for bz” read bz”
Art 27, marg. note, for quaesitum read quaesitam
p. 412, l. II, insert positiva before assum antur
p. 422, 1. 22, for ab read ad
p. 434, Art 86, marg. note, for massa read massas
l. 5 from bottom, for contrarium read contrariam
p. 444, 1. II, insert & before centri
p. 448, 1. 17, for PQ read PG
Art. Ioy, marg. note, for absoluutaº read absolutaº
1. 4 from bottom, after posita insert ſ
p. 454, l. 2 from bottom, for elasticas read elasticitas
p;:
T2 T2
p. 456, l. 5 from bottom for PQ? read;
t?dz
t dz
. 458, l. II, for tº read — tº
p. 458, l. II, for ZZ read -z,
p. 459, l. 13, for —tdz/z” read —tºdz/z”
p. 464, l. 4, for Discrimen read Discrimina
1. Io from bottom, for Venizia read Venezia
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