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A THEORY OF NATURAL PHILOSOPHY
Latin-English Edition From Thi Tut Of Thk First Vinrtian Edition I’iriishid Undir hi
PfRSONAI SvPI RINTf NDf NCI Of TH! AUTHOR
In 1769 With A Short I iri Of Boscovich
Roger Joseph Boscovich

LIBRARY
UWTVERSITY OF CALIFORNIA DAVIS

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THEORIA
PHILOSOPHISE NATURALIS
REDACTA AD UNICAM LEGEM VIRIUM IN NATURA EXISTENTIUM,
A V C T0 RE
PaROGERIO josepho boscovich
Societatis Jesu, NUNC AB IPSO PERPOLITA, ET AUCTA,
Ac a plurimis praecedentium editionum mendis expurgata.
EDITIO VENETA PRIMA
IPSO AUCTORE PRAESENTE, ET CORRIGENTE.

MDCCLXIII.

" **

u «a «» «• ut> «• «• u» «» u» •

Ex Trp0CRAPHIA Remomdiniana.

SUPERIORyM P £ JI M ] S S ac PRIVILEGIO,

A THEORY OF
NATURAL PHILOSOPHY
PUT FORWARD AND EXPLAINED BY
ROGER JOSEPH BOSCOVICH, S.J.
LATIN—ENGLISH EDITION
FROM THE TEXT OF THE FIRST VENETIAN EDITION PUBLISHED UNDER THE PERSONAL SUPERINTENDENCE OF THE AUTHOR
IN 1763
WITH
A SHORT LIFE OF BOSCOVICH

CHICAGO

LONDON

OPEN COURT PUBLISHING COMPANY

1922

LIBRARY
UNIVERSITY OF CALIFORNIA DAVIS

PRINTED IN GREAT BRITAIN
>T
Butlii & Tanner, Fkome, 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 Fir at ^enna> because, as stated on the title-page, it was the first edition (revised EEf and enlarged) issued under the personal superintendence of the author.

tjCJL

In the English translation, an endeavour has been made to adhere as

V&T closely as possible to a literal rendering of the Latin ; except that the some­

what lengthy and complicated sentences have been broken up. This has

made necessary slight changes of meaning in several of the connecting words. This will be

noted especially with regard to the word “ adeoque ”, which Boscovich uses with a variety

of shades of meaning, from “ indeed ”, “ also ” or “ further ”, through ” thus ”, to a decided

“therefore”, which would have been more correctly rendered by “ ideoque ”, There is

only one phrase in English that can also take these various shades of meaning, viz., “ and so ” ;

and this phrase, for the use of which there is some justification in the word “ adeo ” itself,

has been usually employed.

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

something characteristic of the author and his century would have been lost.

My translation has had the advantage of a revision by Mr. A. O. Prickard,M.A., Fellow

of New College, Oxford, whose task has been very onerous, for he has had to watch not

only for flaws in the translation, but also for misprints in the Latin. These were necessarily

many; in the first place, there was only one original copy available, kindly loaned to me by

the authorities of the Cambridge University Library; and, as this copy could not leave my charge, a type-script had to be prepared from which the compositor worked, thus doub­

ling the chance of error. Secondly, there were a large number of misprints, and even

omissions of important words, in the original itself; for this no discredit can be assigned to

Boscovich; for, In the printer’s preface, we read that four presses were working at the

same time in order to take advantage of the author’s temporary presence in Venice. Further, owing to almost insurmountable difficulties, there have been many delays in the production

of the present edition, causing breaks of continuity in the work of the translator and reviser ;

which have not conduced to success. We trust, however, that no really serious faults remain.

The short life of Boscovich, which follows next after this preface, has been written by

Dr. Branislav Petronievic, Professor of Philosophy at the University of Belgrade. It is to

be regretted that, owing to want of space requiring the omission of several addenda to the

text of the Theoria itself, a large amount of interesting material collected by Professor

Petronievic has had to be left out.

The financial support necessary for the production of such a costly edition as the present

has been met mainly by the Government of the Kingdom of Serbs, Croats and Slovenes;

and the subsidiary expenses by some Jugo-Slavs interested in the publication.

After the £t Life, there follows an “ Introduction,” in which I nave 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
Br BRANISLAV PETRONIEVlC
TlE Slav world, being still in its infancy, has, despite a considerable number
n of scientific men, been unable to contribute as largely to general science as Itnhethoethefrolglorweaint gEuarrotipcelaen, Inaptriopnos.se Itot hdaess,crniebveerbtrhieflelysst,hdeemlifoensotfrathede Jugo-Slav, Boscoitvsiccha, pwahciotsye opfrinpcriopdaulcwinogrksicsiehnetirfeicpuwbolriskhsedoffotrhethehigsihxethst tvimaluee;. thAebfoirvset edition having aaplpl,eaarsedI ihnav1e75e8l,seawnhdeorethienrdsicinate1d75,*9,it1p76o3ss,es1s7e6s4,Caonpdern1i7c6u5s., LTohbeacphreevsseknit, text is from theMeednitdioelnjeovf, a1n76d3,Btohsecofviircsht .Venetian edition, revised and enlarged.
On his father’s side, the family of Boscovich is of purely Serbian origin, his grandfather, Bosko, having been an orthodox Serbian peasant of the village of Orakova in Herzegovina. His father, Nikola, was first a merchant in Novi Pazar (Old Serbia), but later settled in Dubrovnik (Ragusa, the famous republic in Southern Dalmatia), whither his father, Bosko, soon followed him, and where Nikola became a Roman Catholic. Pavica, Boscovich’s
mother, belonged to the Italian family of Betere, which for a century had been established in Dubrovnik and had become Slavonici zed—Bara Betere, Pavica’s father, having been a poet of some reputation in Ragusa.
Roger Joseph Boscovich (Rudjer Josif Boskovic’, in Serbo-Croatian) was born at Ragusa on September 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 J733 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 Philosophia Naturalis-, the publication was entrusted to a Jesuit, Father Scherffer, Boscovich having to leave Vienna, and the first edition appeared in 1758, followed by a second edition in the following year. With both of these editions, Boscovich was to some extent dissatisfied (see the remarks made by the printer who carried out the third edition at Venice, given in this volume on page 3); so a third edition was issued at Venice, revised, enlarged and rearranged under the author’s personal superintendence in 1763. The revision was so extensive that as the printer remarks, “ it ought to be considered in some measure as a first and original edition ” ; and as such it has been taken as the basis of the translation now published. The fourth and fifth editions followed in 1764 and 1765.
One of the most important tasks which Boscovich was commissioned to undertake was that of measuring an arc of the meridian in the Papal States. Boscovich had designed to take part in a Portuguese expedition to Brazil on a similar errand ; but he was per-
Slav Achievements in Advanced Science, London, 1917, vii

viii

A THEORY OF NATURAL PHILOSOPHY

suaded by Pope Benedict XIV, in 1750, to conduct, in collaboration with an English Jesuit,
Christopher Maire, the measurements in Italy. The results of their work were published, in 1755, by Boscovich, in a treatise, De Litteraria Expeditione per Pontificiam, &c.; this
was translated into French under the title of Voyage astronomique ei geographique dans rfiiat de I’Lglise, 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 Lun<z dejectibus, 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 purpiosc. 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 1779Shortly 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 be^un at Milan according to the plans of Boscovich ; and in 1770, Boscovich was appointed its director. Unfortunately, only two years later he was deprived of office by the Austrian Government which, in a controversy between Boscovich and another astronomer of the observatory, the Jesuit Lagrange, took the part of his opponent. The position of Boscovich was still further complicated by the disbanding of his company ; for, by the decree of Clement V, the Order of Jesus had been suppressed in 1773- In the same year Boscovich, now free for the second time, again visited raris, where he was cordially received in official circles. The French Government appointed him director of “ Optiquc Marine,” with an annual salary of 8,000 francs; and Boscovich became a French suniect. 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.” 4 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, dejeilner. 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 a ashing off some impromptu verses to well-known men or charming women. To the latter he paid no other attentions, for his austerity was always exemplary. . . . With such talents, it is not to be wondered at that he was everywhere appreciated and sought after. Ministers, princes and sovereigns all received him with the greatest distinction. M. de Lalande witnessed this in every part
of Italy where Boscovich accompanied him in 1765.” Boscovich was acquainted with several languages—Latin, Italian, French, as well as
his native Serbo-Croatian, which, despite his long absence from his country, he did not forget. Although he had studied in Italy and passed the greater part of his life there, he had never penetrated to the spirit of the language, as his Italian Biographer, Ricca, notices. His command of French was even more defective ; but in spite of this fact, French men of science urged him to write in French. English he did not understand,as he confessed in a letter to Priestley ; although he had picked up some words of polite conversation
during his stay in London. His correspondence was extensive. The greater part of it has been published in
the Memoirs de I’Academie Jougo-Slave of Zagrab, 1887 to 1912.

• Oratio in funere R. J. Boscovichii . . . a Bernardo Zamagna. • Voyage Aitronomique, p. 750; also 00 pp. 707 scq. * Journal del Sfavam, Fivricr, 179a, pp. 113-118.

INTRODUCTION
LTHOUGH the title to this work to a very large extent correctly describes
H 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
xi

xii

INTRODUCTION

Leibniz, was observed. It would appear that at this time (1745) Boscovich was concerned

mainly, if not solely, with the facts of the change of velocity, and not with the causes for

this change. The title of the dissertation, De Ciribus vivis, shows however that a secondary

consideration, of almost equal importance in the development of the Theory of Boscovich,

also held the field. The natural philosophy of Leibniz postulated monads, without parts,

extension or figure. In these features the monads of Leibniz were similar to the material

points of Boscovich; but Leibniz ascribed to his monads 1 perception and appetition in

addition to an equivalent of inertia. They are centres of force, and the force exerted is a

vis viva. Boscovich opposes this idea of a “ living,” or “ lively ” force ; and in this first

dissertation we may trace the first ideas of the formulation of his own material points.

Leibniz denies action at a distance; with Boscovich it is the fundamental characteristic of

a material point.

The principles developed in the work on collisions of bodies were applied to the problem

of the centre of oscillation. During the latter investigation Boscovich was led to a theorem

on the mutual forces between the bodies forming a system of three ; and from this theorem

there followed the natural explanation of a whole sequence of phenomena, mostly connected

with the idea of a statical moment ; and his initial intention was to have published a

dissertation on this theorem and deductions from it, as a specimen of the use and advantage

of his principles. But all this time these principles had been developing in two directions,

mathematically and philosophically, and by this time included the fundamental notions

of the law of forces for material points. The essay on the centre of oscillation grew in length

as it proceeded ; until, finally, Boscovich added to it all that he had already published on

the subject of his principles and other matters which, as he says, “ obtruded themselves on

his notice as he was writing.” The whole of this material he rearranged into a more logical

(but unfortunately for a study of development of ideas, non-chronological) order before

publication. As stated by Boscovich, in Art. 164, the whole of his Theory is contained in his statement

that: “ Matter is composed of perfectly indivisible, non-extended, discrete points.” To this

assertion is conjoined the axiom that no two material points can be in the same point of

space at the same time. As stated above, in opposition to Clerk Maxwell, this is no matter

of prejudice. Boscovich, in Art. 361, gives his own reasons for taking this axiom as part

of his theory. He lays it down that the number of material points is finite, whereas the

number of local points is an infinity of three dimensions; hence it is infinitely improbable,

i.e., impossible, that two material points, without the action of a directive mind, should

ever encounter one another, and thus be in the same place at the same time. He even goes

further ; he asserts elsewhere that no material point ever returns to any point of space in

which it has ever been before, or in which any other material point has ever been. Whether

his arguments are sound or not, the matter does not rest on a prejudgment formed from

experience of bodies of sensible size ; Boscovich has convinced himself by such arguments

of the truth of the principle of Impenetrability, and lays it down as axiomatic ; and upon

this, as one of his foundations, builds his complete theory. The consequence of this axiom

is immediately evident ; there can be no such thing as contact between any two material

points; two points cannot be contiguous or, as Boscovich states, no two points of matter

can be in mathematical contact. For, since material points have no

dimensions, if, to form an imagery of Boscovich’s argument, we take ACE

two little squares ABDC, CDFE to represent two points in mathema- r——i——

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 sec, there does not enter the Newtonian idea

of inertia as a characteristic of mass. The propensity is akin to the characteristic ascribed

to the monad by Leibniz ; with this difference, that it is not a symptom of activity, as with

Leibniz, but one of inactivity.

1 See Bertrand Russell, Philoiofby oj Leitniz.; 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 oy the mathematicians of the period of Boscovich, had many different meanings ;

or rather that its meaning was given by the descriptive adjective that was associated with it.

Thus we have vis viva (later associated with energy), vis mortua (the antithesis of vis viva,

as understood by Leibniz), vis acceleratrix (acceleration), vis matrix (the real equivalent

of force, since it varied with the mass directly), vis descensiva (moment of a weight hung at

one end of a lever), and so on. Newton even, in enunciating his law of universal gravitation,

apparently asserted nothing more than the fact of gravitation—a propensity for approach—

according to the inverse square of the distance : and Boscovich imitates him in this. The

mutual vires, ascribed by Boscovich to his pairs of points, are really accelerations, i.e.

tendencies for mutual approach or recession of the two points, depending on the distance

between the points at the time under consideration. Boscovich’s own words, as given in

Art. 9, are : “ Censeo igitur bina quaecunque materiie puncta determinari sque in aliis

distantiis ad mutuum accessum, in aliis ad recessum mutuum, quam ipsam determinationem

apello vim” The cause of this determination, or propensity, for approach or recession,

which in the case of bodies of sensible size is more correctly called “ force ” (vis matrix),

Boscovich does not seek to explain ; he merely postulates the propensities. The measures

of these propensities, i.e., the accelerations of the relative velocities, are the ordinates of

what is usually called his curve of forces. This is corroborated by the statement of Boscovich

that the areas under the arcs of his curve are proportional to squares of velocities ; which

is in accordance with the formula we should now use for the area under an “acceleration­

space ” graph (Area = ff.ds =

= f v.dv). See Note (I) to Art. 118, where it is

evident that the word vires, 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 arc 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 wc understand
such terms. The sole characteristic that has a finite measure is the relative acceleration produced by the simultaneous existence of two points of matter ; and this acceleration depends solely upon the distance between them. The Newtonian idea of mass is replaced by something totally different; it is a mere number, without “ dimension ” ; the “ mass ” of a body is simply the number of points that are combined to “ form ” the body.
Each of these points, if sufficiently close together, will exert on another point of matter,
at a relatively much greater distance from every point of the body, the same acceleration very approximately. Hence, if we have two small bodies A and B, situated at a distance s
from one another (the wording of this phrase postulates that the points of each body are very close together as compared with the distance between the bodies) : and if the number of points in A and B are respectively a and b, and / is the mutual acceleration between any pair of material points at a distance s from one another ; then, each point of A will give to
each point of B an acceleration /. Hence, the body A will give to each point of B, and therefore to the whole of B, an acceleration equal to 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 dc 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 innnite in Nature, or in extension, or a selfdetermined infinitely small.” The admission of infinite componibility is necessitated by his definition of the material point; since it has no parts, a fresh point can always be placed between any two points without being contiguous to either. Now, since he denies the existence of the infinite and the infinitely small, the attraction or repulsion between two points of matter (except at what he calls the limiting intervals) must be finite : hence, since the attractions of masses are all by observation finite, it follows that the number of points in a mass must be finite. To evade the difficulty thus raised, he appeals to the scale of integers, in which there is no infinite number : but, as he says, the scale of integers is a sequence of numbers increasing indefinitely, and having no last term. Thus, into any space, however small, there may be crowded an indefinitely great number of material points; this number can be still further increased to any extent; and yet the number of points finally obtained is always finite. It would, again, seem that the system of Boscovich was not a material system, but a system of relations; if it were not for the fact that he asserts, in Art. 7, that his view is that “ the Universe does not consist of vacuum interspersed amongst matter, but that matter is interspersed in a vacuum and floats in it.” The whole question is still further complicated by his remark, in Art. 393, that in the continual division of a body, “ as soon as we reach intervals less than the distance between two material points, further sections will cut empty intervals and not matter ”; and yet he has postulated that there is no minimum value to the interval between two material points. Leaving, however, this question of the philosophical standpoint of Boscovich to be decided by the reader, after a study of the supplements that follow the third part of the treatise, let us now consider the
curve of Boscovich. Boscovich, from experimental data, gives to his curve, when the interval is large, a
branch asymptotic to the axis of intervals ; it approximates to the “ hyperbola ” x*y— r, in which x represents the interval between two points, and y the vis corresponding to that interval, which wc have agreed to call an attraction, meaning thereby, not a force, but an

INTRODUCTION

xv

acceleration of the velocity of approach. For small intervals he has as yet no knowledge of the quality or quantity of his ordinates. In Supplement IV, he gives some very ingenious arguments against forces that are attractive at very small distances and increase indefinitely, such as would be the case where the law of forces was represented by an inverse power of the interval, or even where the force varied inversely as the interval. . For the inverse fourth or higher power, he shows that the attraction of a sphere upon a point on its surface would be less than the attraction of a part of itself on this point; for the inverse third power, he con­ siders orbital motion, which in this case is an equiangular spiral motion, and deduces that after a finite time the particle must be nowhere at all. Euler, considering this case, asserted that on approaching the centre of force the particle must be annihilated; Boscovich, with more justice, argues that this law of force must be impossible. For the inverse square law, the limiting case of an elliptic orbit, when the transverse velocity at the end of the major axis is decreased indefinitely, is taken ; this leads to rectilinear motion of the particle to the
centre of force and a return from it; which does not agree with the otherwise proved oscillation through the centre of force to an equal distance on either side.
Now it is to be observed that this supplement is quoted from his dissertation De Lege Virium in Natura existentium, which was published in 1755 ; also that in 1743 he had published a dissertation of which the full title is : De Motu Corporis attracti in centrum immobile viribus decrescentibus in ratione distantiarum reciproca duplicata in spatiis non resistentibus. Hence it is not too much to suppose that somewhere between 1741 and 1755 he had tried to find a means of overcoming this discrepancy ; and he was thus led to suppose that, in the case of rectilinear motion under an inverse square law, there was a departure from the law on near approach to the centre of force ; that the attraction was replaced by a repulsion increasing indefinitely as the distance decreased ; for this obviously would lead to an oscillation to the centre and back, and so come into agreement with the limiting case of the elliptic orbit. I therefore suggest that it was this consideration that led Boscovich to
the doctrine of Impenetrability. However, in the treatise itself, Boscovich postulates the axiom of Impenetrability as applying in general, and thence argues that the force at infinitely small distances must be repulsive and increasing indefinitely. Hence the ordinate to the curve near the origin must be drawn in the opposite direction to that of the ordinates for sensible distances, and the area under this branch of the curve must be indefinitely great. That is to say, the branch must be asymptotic to the axis of ordinates ; Boscovich however considers that this does not involve an infinite ordinate at the origin, because the interval
between two material points is never zero; or, vice versa, since the repulsion increases indefinitely for very small intervals, the velocity of relative approach, no matter how great, of two material points is always destroyed before actual contact; which necessitates a finite interval between two material points, and the impossibility of encounter under any circum­ stances : the interval however, since a velocity of mutual approach may be supposed to be of any magnitude, can have no minimum. Two points are said to be in physical contact, in opposition to mathematical contact, when they are so close together that this great mutual repulsion is sufficiently increased to prevent nearer approach.
Since Boscovich nas these two asymptotic branches, and he postulates Continuity, there must be a continuous curve, with a one-valued ordinate for any interval, to represent
the “ force ” at all other distances; hence the curve must cut the axis at some point in between, or the ordinate must become infinite. He does not lose sight of this latter possi­ bility, but apparently discards it for certain mechanical and physical reasons. Now, it is known that as the degree of a curve rises, the number of curves of that degree increases very rapidly; there is only one of the first degree, the conic sections of the second degree, while Newton had found over three-score curves with equations of the third degree, and nobody had tried to find all the curves of the fourth degree. Since his curve is not one of the known curves, Boscovich concludes that the degree of its equation is very high, even if it is not transcendent. But the higher the degree of a curve, the greater the number of possible intersections with a given straight line ; that is to say, it is highly probable that there are a
great many intersections of the curve with the axis; i.e., points giving zero action for material points situated at the corresponding distance from one another. Lastly, since the ordinate is one-valued, the equation of the curve, as stated in Supplement III, must be of the form P-Qy = o, where P and Q are functions of x alone. Thus we have a curve winding about the axis for intervals that are very small and developing finally into the hyperbola of the third degree for sensible intervals. This final branch, however, cannot be exactly this hyperbola ; for, Boscovich argues, if any finite arc of the curve ever coincided exactly with the hyperbola of the third degree, it would be a breach of continuity if it ever departed from it. Hence he concludes that the inverse square law is observed approximately only, even
at large distances. As stated above, the possibility of other asymptotes, parallel to the asymptote at the

rv i

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 innnite 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 centrobaryc 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. 230236; 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, jo long as there are portione of the
curve winding about the axij for very email intervals between the pointe. It is sufficient, for the next point, to draw the reader’s attention to Art. 266-278, on
collision, and to the articles which follow on the agreement between resolution and com­
position of forces as a working hypothesis. From what Boscovich says, it would appear that philosophers of his time were much perturbed over the idea that, when a force was resolved into two forces at a sufficiently obtuse angle, the force itself might be less than either of

INTRODUCTION

xvii

the resolutes. Boscovich points out that, in his Theory, there is no resolution, only com­ position; and therefore the difficulty does not arise. In this connection he adds that there are no signs in Nature of anything approaching the vires viva of Leibniz.
In Art. 294 we have Boscovich’s contribution to the controversy over the correct
measure of the “ quantity of motion ” ; but, as there is no attempt made to follow out the change in either the velocity or the square of the velocity, it cannot be said to lead to any­ thing conclusive. As a matter of fact, Boscovich uses the result to prove the non-existence of vires 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 philosophic al 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 = x4 as his variable, and P and Q as functions of z, in the equation P-Qy = o, referred to above. Note.—In this connection (p. 410, Art. 25, 1. 5), Boscovich has apparently made a slip over the negative sign : as the intention is clear, no attempt has been made to amend the Latin.
The second point is that Boscovich does not seem to have any idea of integrating between limits. He has to find the area, in Fig. I on p. 134, bounded by the axes, the curve and the ordinate ag; this he does by the use of the calculus in Note (1) on p. 141. He assumes that

xviii

INTRODUCTION

the equation of the curve is x"y" = i, and obtains the integral--f-l-* xy + A, where A is the

fi—m

constant of integration. He states that, if n is greater than mt 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

He says : “ The area is infinite, when n = m, because

this makes the divisor zero; and thus the area becomes still more infinite if n<wi.” Put

into symbols, the argument is : Since n-m<o, -2—> ao . The historically interesting n—m 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 <x to stand for “simple infinity,” the value of i/o, and hence of n/o), Wallis had justification for his error, if
indeed it was an error in his case ; for his exponents were characteristics of certain infinite
series, and he could make his own laws about these so that they suited the geometrical problems to which they were applied ; it was not necessary that they should obey the laws of inequality that were true for ordinary numbers. Boscovich’s mistake is, of course, that
of assuming that the constant is zero in every case ; and in this he is probably deceived by

using the formula—n——mxny—m-f- A, instead of——x-71"-"1 4 A, for the area. From the latter

it is easily seen that since the initial area is zero, we must have A = ”

If n is

m-n

equal to or greater than m, the constant A is indeed zero; but if n is less than m, the constant

is infinite. The persistence of this error for so long a time, from 1655 to 1758, during which

we have the writings of Newton, Leibniz, the Bcrnoullis 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 Ofera 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 Lobachevsld 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 fallowing important corrections, omissions, and alternative renderings; misprints involving a single letter or syllable only arc given at the end of the volume. p. 27,1. 8, for in one plane read in the same direction p. 47,1. 61, literally on which ... is exerted P' 49, 33« M just as ... is read so that . . . may be p. 53,1. 9, after a line add but not parts of the line itself p. 61, Art. 47, Alternative rendering; These instances make good the same point as water making its way through
the pores of a sponge did for impenetrability; p. 67, I. 5, far it is allowable for me read I am disposed ; unless in the original libet is taken to be a misprint for licet p. 73,1. 26, after nothing add in the strict meaning of the term p. 85,1, 27, after conjunction add of the same point of space p. 91, I. 25, Alternative rendering; and these properties might distinguish the points even in the view of the followers
of Leibniz I. 5 from bottom, Alternative rendering: Not to speak of the actual form of the leaves present in the seed p. H5,l. J5> after the left add but that the two outer elements do not touch each other 1. 28, far two little spheres read one b'ttlc sphere p. 117,1. 41, for precisely read abstractly p. 125,1. 29, for ignored read urged in reply p. 126,1. 6 from bottom, it is possible that acquirere is intended for acquiescere, with a corresponding change in the
translation p. 129, Art. 162, marg, note, far on what they may be founded read in what it Consists. p. 167, An. 214, 1. 2 of marg, note, transpose by and on
footnote, 1. 1, for be at read bisect it at p. 199, I. 24, for so that read just as P- I33> I- 4 from bottom, for base to the angle read base to the sine of the angle
last line, after vary insert inversely p. 307, 1. 5 from end, for motion, as (with fluids) takes place read motion from taking place p. 323, I- 39» hr the agitation will read the fluidity will P- 3451 1. 3Ji /or described read destroyed p. 357,1.44, for others read some, others of others
1. 5 from end, for fire read a fiery and insert a comma before substance
xix

THEORIA PHILOSOPHIZE NATURALIS

TYPOGRAPHIC
VENETUS
LECTORI
PUS, quod tibi offero, jam ab annis quinque Viennae editum, quo plausu exceptum sit per Europam, noveris sane, si Diaria publica perlegeris, inter qua si, ut omittam extera, consulas ea, quae in Bcrnensi pertinent ad initium anni 1761 ; videbis sane quo id loco haberi debeat. Systema continet Naturalis Philosophia omnino novum, quod jam ab ipso Auctore suo vulgo Bojcovichianum appellant. Id quidem in pluribus Academiis jam passim publice traditur, nec tantum in annuis thesibus, vei disserta­
tionibus impressis, ac propugnatis exponitur, sed & in pluribus clementaribus libris pro juventute instituenda editis adhibetur, exponitur, & a pluribus habetur pro archetypo. Verum qui omnem systematis compagem, arctissimum partium nexum mutuum, facun­ 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, necessc est.
Haec omnia me permoverant jam ab initio, ut novam Operis editionem curarem: accedebat illud, quod Viennensia exemplaria non ita facile extra Germaniam itura videbam, & quidem nunc etiam in reliquis omnibus Europa: 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 Viennae, quo se ad breve tempus contulerat, digestum, ac editum, Italicis potissimum typis, censebam, per univer­ sam Europam disseminandum. Et quidem editionem ipsam e Viennensi exemplari jam 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, qu® 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 praesentem 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 praeter correctores nostros omnem ipse etiam in corrigendo diligentiam adhibuit; quanquam is ipse haud quidem sibi ita fidit, ut nihil omnino effugisse censeat, cum ea sit humanae mentis conditio, ut in eadem re diu satis intente defigi non possit.
Haec idcirco ut prima quaedam, atque originaria editio haberi debet, quam qui cum Viennensi contulerit, videbit sane discrimen. E minoribus mutatiunculis multae pertinent ad expolienda, & declaranda plura loca; sunt tamen etiam nonnulla potissimum in pagin­ arum fine exigua additamenta, vel mutatiuncul® exiguae factae post typograpnicam constructionem idcirco tantummodo, ut lacunulae implerentur quae aliquando idcirco supererant, quod plures phylirae a diversis compositoribus simul adornabantur, & quatuor simul prxla sudabant; quod quidem ipso praesente fieri facile potuit, sine ulla pertur­ batione sententiarum, & ordinis.
2

THE PRINTER AT VENICE
TO
THE READER
OU will be well aware, if you have read the public journals, with what applause the work which I now offer to you has been received throughout Europe since its publication at Vienna five years ago. Not to mention others, if you refer to the numbers of the Berne Journal for the early part of the year 1761, you will not fail to see how highly it has been esteemed. It contains an entirely new system of Natural Philosophy, which is already commonly known as the Boscovichian theory, from the name of its author,
As a matter of fact, it is even now a subject of public instruction in several Universities in different parts ; it is expounded not only in yearly theses or dissertations, both printed & debated; but also in several elementary nooks 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.
AU 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 snort 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 iU-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

4

TYPOGRAPHUS VENETUS LECTORI

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

Supplementorum ordo mutatus est itidem ; quae enim fuerant 3, & 4, jam sunt X, & 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 controversiae occasionem exhibuit in Encyclopedico etiam dictionario attactum, in qua dissertatiuncula demonstrat Auctor non esse, cur ad vim exprimendam potentia quaepiam distantiae adhibeatur potius, quam functio.

Accesserunt per totum Opus notulae marginales, in quibus eorum, quz pertractantur argumenta exponuntur brevissima, quorum ope unico obtutu videri possint omnia, & in memoriam facile revocari.
Postremo loco ad calcem Operis additus est fusior catalogus eorum omnium, quae huc usque ab ipso Auctore sunt edita, quorum collectionem omnem expolitam, & correctam, ac eorum, quz 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 rcimpressione ejus poematis de Solis ac Lunae defectibus. Porro eam. omnium suorum Operum Collectionem, ubi ipse adornaverit, typis ego meis excudendam suscipiam, quam magnificentissime potero.

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

THE PRINTER AT VENICE TO THE READER

5

Among the more important alterations will be found a change in the order of numbering the paragraphs. Thus, Art. 82 is additional matter that is entirely new; that which was formerly Art. 261 is now broken up into five parts ; &, in the Appendix, following Art. 534, both some slight changes and also several additions have been made in the passages that relate to the Seat of the Soul.
The order of the Supplements has been altered also : those that were formerly num­ bered III and IV are now I and II respectively. This was done because they are reauired for use in this work before the others. To that which was formerly numbered J, 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 Encyelopeclique. 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
PRIM7E EDITIONIS VIENNENSIS
AD CELSISSIMUM TUNC PRINCIPEM ARCHIEPISCOPUM VIENNENSEM, NUNC PRAETEREA ET CARDINALEM EMINENTISSIMUM, ET EPISCOPUM VACCIENSEM CHRISTOPHORUM E COMITATIBUS DE MIGAZZI
ABIS 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 Philosophia; principia continentem. Novi ego quidem, quam totus in eo sis, ut, quam geris, personam sustineas, ac vigilantissimi sacrorum Antistitis partes agas. Videt utique Imperialis hsec Aula, videt universa Regalis Uros, & indenti 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 Romse dantem operam studiis cognoscerem, mihi fors obtigit, altissime jam insederat animo, id in omni reliquo amplissimorum munerum Tibi commissorum cursu h<esit 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 Romse in celeberrimo illo, quod Auditorum Rotse appellant, collegio toti Christiano orbi jus diceres, accesserat Hetrusca Imperialis Legatio apud Romanum Pontificem exercenda; cum repente Mechliniensi Archiepiscopo in amplissima illa administranda Ecclesia Adjutor datus, & destinatus Successor, possessione prae sta ntissimi muneris vixdum capta, ad Hispanicum Regem ab Augustissima Romanorum Imperatrice ad gravissima tractanda negotia Legatus es missus, in quibus cum summa utriusque Aula: 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 Hungarise Regnum, quanquam exterum hominem, non ut civem suum tantummodo, sed ut Parentem amantissimum 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 Austriacse 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, qua: ad sacra pertinent, curandis vel per Te ipsum usque adeo, ut ssepe, raro admodum per
6

AUTHOR’S EPISTLE DEDICATING
THE FIRST VIENNA EDITION
TO
CHRISTOPHER, COUNT DE MIGAZZI, THEN HIS HIGHNESS THE PRINCE ARCHBISHOP OF VIENNA, AND NOW ALSO IN ADDITION HIS EMINENCE THE CARDINAL, BISHOP OF VACZ
will pardon me, Most Noble Prince, if perchance I come to disturb at an inopportune moment the unremitting cares of your Holy Office, & offer you a volume so inconsiderable in size; one too that contains none of the inner mysteries of Religion, such as you administer from the highly exalted position to which you are ordained ; one that merely deals with the prin­ ciples of Natural Philosophy. I know full well how entirely your time is taken up with sustaining the reputation that you bear, & in performing the duties of a highly conscientious Prelate. This Imperial Court sees, nay, the whole of this Royal City sees, with what care, what toil, you exert yourself to carry out the duties of so great a sacred office, & stands wrapt with an overwhelming admiration. Of a truth, that well-known old saying, “ What you do, DO” which from your earliest youth, when chance first allowed me to make your acquaintance while you were studying in Rome, had already fixed itself deeply in your mind, has remained firmly implanted there during the whole of the remainder of a career in which duties of the highest importance have been committed to your care. Your strict observance of this maxim in particular, joined with those numerous talents so lavishly showered upon you by Nature, & those virtues which you have acquired for yourself by daily practice & unremitting toil, throughout your whole, career, forensic, courtly, & sacerdotal, has so to speak heaped upon your shoulders those unusually rapid advances in dignity that have been your lot. It has aroused the admiration of all, both peoples & princes alike, in every land; & at the same time it has earned for you their deep affection. The consequence was that one office after another, each ever more exalted & honourable than the preceding, has in a sense seized upon you & borne you away a captive. Whilst you were in Rome, giving judicial decisions to the whole Christian world in that famous College, the Rota of Auditors, there was added the duty of acting on the Tuscan Imperial Legation at the Court of the Roman Pontiff. Sud­ denly you were appointed coadjutor to the Archbishop of Malines in the administration of that great churcn, & his future successor. Hardly nad 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 priestnood, 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

8 EPISTOLA AUCTORIS DEDICATORIA PRIM.E EDITIONIS VIENNENSIS
hacc nostra tempora exemplo, & publico operatum, ac ipsa etiam Sacramenta administrantem videamus in templis, & Tua ipsius voce populos, e superiore loco doccntum audiamus, atque ad omne virtutum genus inflammantem.
Novi ego quidem hxc 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 tantulae libellum molis homini ad tantum culmen evecto porrigerem, ac Tuo vellem Nomine insignitum. Quod enim ad primum pertinet caput, non Theologicas tantum, sed Philosophicas etiam perquisitiones Christiano Antistite ego quidem dignissimas esse censeo, & universam Naturae contemplationem omnino arbitror cum Sacerdotii sanctitate penitus consentire. Mirum enim, quam belle ab ipsa consideratione Naturae ad caelestium rerum contemplationem disponitur animus, & ad ipsum Divinum tantae molis Conditorem assurgit, infinitam ejus Potentiam Sapientiam, Providentiam admiratus, quae erumpunt undique, & utique se produnt.
Est autem & illud, quod ad supremi sacrorum Moderatoris curam pertinet providere, ne in prima ingenuae juventutis institutione, quae semper a naturalibus studiis exordium ducit, prava teneris mentibus irrepant, ac perniciosa principia, quae sensim Religionem corrumpant, & vero etiam evertant penitus, ac eruant a fundamentis; quod quidem jam dudum tristi quodam Europx 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 Universae Physica: Theoriam, & novam potissimum Theoriam sistat, rem is quidem praestet aequissimam, 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 praestitero, si minus infiliciter me gessero, nemo sane me impudentiae arguat, quasi vilem aliquam, & tanto indignam fastigio rem offeram. Habetur in eo novum quoddam Universae Naturalis Philosophiae genus a receptis huc usque, usi­ tatisque plurimam discrepans, quanquam etiam ex iis, quae maxime omnium per hxc tempora celebrantur, casu quodam prxeipua quaeque mirum sane in modum compacta, atque inter se veluti coagmentata conjunguntur ibidem, uti sunt simplicia atque inextensa Leibnitianorum 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 quxdam ad arbitrium selecta, qux utcumque inter se componerem, atque compaginarem : sed omni prxjudicio 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, quae mihi & constitutionem elementorum materiae, & Mechanicae leges, & generales materiae ipsius proprietates, & praecipua corporum discrimina, sua applicatione ita exhibuit, ut eadem in iis omnibus ubique se prodat uniformis agendi ratio, non ex arbitrariis hypothesibus, & fictitiis commentationibus, sed ex sola continua ratio­ cinatione deducta. Ejusmodi autem est omnis, ut eas ubique vel definiat, vel adumbret combinationes elementorum, quae ad diversa praestanda phaenomena sunt adhibendae, ad quas combinationes Conditoris Supremi consilium, & immensa Mentis Divinx vis ubique requiritur, qux infinitos casus perspiciar, & 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 somew’hat succeeded, if I have not failed altogether, let no one accuse me of
presumption, as if I were offering some worthless thing, something unworthy of such dis­ tinguished honour. In it is contained a new kind of Universal Natural Philosophy, one that differs widely from any that are generally accepted & practised at the present time; although it so happens that the principal points of all the most distinguished theories of the
present day, interlocking and as it were cemented together in a truly marvellous way, are combined in it; so too are the simple unextended elements of the followers of Leibniz, as well as the Newtonian forces producing mutual approach at'some distances & mutual separation at others, usually called attractions and repulsions. I use the words “ it so
happens ” because I have not, in eagerness to make the whole consistent, selected one thing here and another there, just as it suited me for the purpose of making them agree & form a connected whole. On the contrary, I put on one side all prejudice, & started from fundamental principles that are incontestable, & indeed are those commonly accepted ; I used perfectly sound arguments, & by a continuous chain of deduction I arrived at a single, simple, continuous law for the forces that exist in Nature. The application of this law explained to me the constitution of the elements of matter, the laws of Mechanics, the general properties of matter itself, & the chief characteristics of bodies, in such a manner
that the same uniform method of action in all things disclosed itself at all points; being deduced, not from arbitrary hypotheses, and fictitibus explanations, but from a single con­ tinuous chain of reasoning. Moreover it is in all its parts of such a kind as defines, or suggests, in every case, the combinations of the elements that must be employed to produce different phenomena. For these combinations the wisdom of the Supreme Founder of the Universe, & the mighty power of a Divine Mind are absolutely necessary; naught but one that could survey the countless cases, select those most suitable for the purpose, and introduce them into the scheme of Nature.
This then is the argument of my work, in which I explain, prove & defend my theory ; then I apply it, in the first instance to Mechanics, & afterwards to Physics, & set forth the many advantages to be derived from it. Here, although the book is but small, I yet include the well-nigh daily meditations of the last thirteen years, carrying on my conclu-

io EPISTOLA AUCTORIS DEDICATORIA PRIM7E EDITIONIS VIENNENSIS
communibus Philosophorum consentio placitis, & ubi ea, quae habemus jam pro compertis, ex meis etiam deductionibus sponte fluunt, quod usque adeo voluminis molem contraxit. Dederam ego quidem dispersa dissertatiunculis variis TTieoriae meae quaedam velut specimina, quae inde & in Italia Professores publicos nonnullos adstipulatores est nacta, & jam ad exteras quoque gentes pervasit; sed ea nunc primum tota in unum compacta, Severo etiam plusquam duplo aucta, prodit in publicum, quem laborem postremo hoc mense, molestiori' bus negotiis, quae me Viennam adduxerant, & curis omnibus exsolutus suscepi, dum in Italiam rediturus opportunam itineri tempus inter assiduas nives opperior, sed omnem in eodem adornando, & ad communem mcdiocrum etiam Philosophorum captum accommo­ dando diligentiam adhibui.
Inde vero jam facile intelliges, cur ipsum laborem meum ad Te deferre, & Tuo nuncupare Nomini non dubitaverim. Ratio ex iis, quae proposui, est duplex : primo quidem ipsum argumenti genus, quod Christianum Antistitem non modo non dedecet, sed etiam apprime decet: tum ipsius argumenti vis, atque dignitas, quae nimirum confirmat, & erigit nimium fortasse impares, sed quantum fieri per me potuit, intentos conatus meos; nam quidquid eo in genere meditando assequi possum, totum ibidem adhibui, ut idcirco nihil arbitrer a mea tenuitate proferri posse te minus indignum, cui ut aliquem offerrem laborum meorum fructum quantumcunque, exposcebat sane, ac ingenti clamore quodam efflagitabat tanta erga me humanitas Tua, qua jam olim immerentem complexus Romae, hic etiam fovere pergis, nec in tanto dedignatus fastigio, omni benevolentiae significatione prosequeris. Accedit autem & illud, quod in hisce terris vix adhuc nota, vel etiam ignota penitus Theoria mea Patrocinio indiget, quod, si Tuo Nomine insignata prodeat in publicum, obtinebit sane validissimum, & secura vagabitur : Tu enim illam, parente velut hic orbatam suo, in dies nimirum discessuro, & quodammodo veluti posthumam post ipsum ejus discessura typis impressam, & in publicum prodeuntem tueberis, fovebisque.
Haec sunt, quae meum Tibi consilium probent, Princeps Celsissime : Tu, qua soles humanitate auctorem excipere, opus excipe, & si forte adhuc consilium ipsum Tibi visum fuerit improbandum; animum saltem aequus respice obsequentissimum Tibi, ac devinct­ issimum. Vale.
Dabam Vienna in Collegio Academico Soc. JESU Idibus Febr. MDCCLVIII.

AUTHOR’S EPISTLE DEDICATING THE FIRST VIENNA EDITION n

sions for the most part only up to the point where I finally agreed with the opinions com­ monly held amongst philosophers, or where theories, now accepted as established, are the natural results of my deductions also; & this has in some measure helped to diminish the size of the volume. I had already published some instances, so to speak, of my general theory in several short dissertations issued at odd times; & on that account the theory
has found some supporters amongst the university professors in Italy, & has already made
its way into foreign countries. But now for the first time is it published as a whole in a single volume, the matter being indeed more than doubled in amount. This work I have
carried out during the last month, being quit of the troublesome business that brought me to Vienna, and of all other cares ; whilst I wait for seasonable time for my return journey through the everlasting snow to Italy. I have however used my utmost endeavours in
preparing it, and adapting it to the ordinary intelligence of philosophers of only moderate
attainments. From this you will readily understand why I have not hesitated to bestow this book
of mine upon you, & to dedicate it to you. My reason, as can be seen from what I have said, was twofold; in the first place, the nature of my theme is one that is not only not
unsuitable, but is suitable in a high degree, for the consideration of a Christian priest; secondly, the power & dignity of the theme itself, which doubtless gives strength &
vigour to my efforts—perchance rather feeble, but, as far as in me lay, earnest. What­ ever in that respect I could gain by the exercise of thought, I have applied the whole of it
to this matter; & consequently I think that nothing less unworthy of you can be pro­ duced by my poor ability; & that I should offer to you some such fruit of my labours was surely required of me, & as it were clamorously demanded by your great kindness
to me; long ago in Rome you had enfolded'my unworthy self in it, & here now you continue to be my patron, & do not disdain, from your exalted position, to honour me with every mark of your goodwill. There is still a further consideration, namely, that my Theory is as yet almost, if not quite, unknown in these parts, & therefore needs a patron’s support; & this it will obtain most effectually, & will go on its way in security if it comes before the public franked with your name. For you will protect & cherish it,
on its publication here, bereaved as it were of that parent whose departure in truth draws nearer every day; nay rather posthumous, since it will be seen in print only after he has

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

University College of the Society of Jesus,

Vienna,

February

1758.

AD LECTOREM
EX EDITIONE VIENNENSI
BES, amice Lector, Philosophia Naturalis ‘Theoriam ex unica lege virium deductam, quam IA ubi jam oliin adumbraverim, vel etiam ex parte explica­ verim, y qua occasione nunc uberius pertractandum, atque augendam etiam, susceperim, invenies in ipso prima partis exordio. Libuit autem hoc opus dividere in partes tres, quarum prima continet explicationem Theoria ipsius, ac ejus analyticam deductionem, y vindicationem : secunda applicationem satis uberem ad Mechanicam ; tertia applicationem ad Physicam.
Porro illud inprimis curandum duxi, ut omnia, quam liceret, dilucide exponerentur, nec sublimiore Geometria, aut Calculo indigerent. Et quidem in prima, ac tertia parte non tantum nulla analytica, sed nec geometrica demonstrationes occurrunt, paucissimis quibusdam, quibus indigeo, rejectis in adnotatiunculas, quas in fine paginarum quarundam invenies. Quadam autem admodum pauca, qua majorem Algebra, IA Geometria cognitionem requirebant, vel erant complicatiora aliquando, y alibi a me jam edita, in fine operis apposui, qua Supplementorum appellavi nomine, ubi IA ea addidi, qua sentio de spatio, ac tempore, Theoria mea consentanea, ac edita itidem jam alibi. In secunda parte, ubi ad Mechanicam applicatur Theoria, a geome­ tricis, y aliquando etiam ab algebraicis demonstrationibus abstinere omnino non potui; sed ea ejusmodi sunt, ut vix unquam requirant aliud, quam Euclideam Geometriam, IA primas Trigonometria notiones maxime simplices, ac simplicem algorithmum.
In prima quidem parte occurrunt Figura geometrica complures, qua prima fronte vide­ buntur etiam complicata rem ipsam intimius non perspectanti ; verum ea nihil aliud exhibent, nisi imaginem quandam rerum, qua ipsis oculis per ejusmodi figuras sistuntur contemplanda. Ejusmodi est ipsa illa curva, qua legem virium exhibet. Invenio ego quidem inter omnia materia puncta vim quandam mutuam, qua a distantiis pendet, iA mutatis distantiis mutatur ita, ut in aliis attractiva sit, in aliis repulsiva, sed certa quadam, y continua lege. Leges ejusmodi variationis binarum quantitatum a se invicem pendentium, uti Jsic sunt distantia, A vis, exprimi possunt vel per analyticam formulam, vel per geometricam curvam ; sed illa prior expressio E? multo plures cognitiones requirit ad Algebram pertinentes, IA imaginationem non ita adjuvat, ut hac posterior, qua idcirco sum usus in ipsa prima operis parte, rejecta in Supplementa formula analytica, qua y curvam, y legem virium ab illa expressam exhibeat.
Porro huc res omnis reducitur. Habetur in recta indefinita, qua axis dicitur, punctum quoddam, a quo abscissa ipsius recta segmenta referunt distantias. Curva linea protenditur secundum rectam ipsam, circa quam etiam serpit, tA eandem in pluribus secat punctis: recta a fine segmentorum erecta perpendiculariter usque ad curvam, exprimunt vires, qua majores sunt, vel minores, prout ejusmodi recta sunt itidem majores, vel minores ; ac eadem ex attractivis migrant in repulsivis, vel vice versa, ubi illa ipsa perpendiculares recta directionem mutant, curva ab altera axis indefiniti plaga migrante ad alteram. Id quidem nullas requirit geometricas demonstrationes, sed meram cognitionem vocum quarundam, qua vel ad prima per­ tinent Geometria elementa, IA notissima sunt, vel ibi explicantur, ubi adhibentur. Notissima autem etiam est significatio vocis l\.sycnptovas, unde IA crus asymptoticum curva appellatur ; dicitur nimirum recta asymptotus cruris cujuspiam curva, cum ipsa recta in infinitum producta, ita ad curvilineum arcum productum itidem in infinitum semper accedit magis, ut distantia minuatur in infinitum, sed nusquam penitus evanescat, illis idcirco nunquam invicem con­ venientibus.
Consideratio porro attenta curva proposita in Fig. 1, y rationis, qua per illam exprimitur

THE PREFACE TO THE READER
THAT APPEARED IN THE VIENNA EDITION
YR Reader, you have before you a Theory of Natural Philosophy deduced from a single law of Forces. You will find in the opening paragraphs of the first section a statement as to where the Theory has been already
published in outline, & to a certain extent explained ; & also the occasion that led me to undertake a more detailed treatment & enlargement of it. For I have thought fit to divide the work into three parts; the first of these contains the exposition of the Theory itself, its analytical deduction & its demonstration ; the second a fairly full application to Mechanics ; & the third an application to Physics. The most important point, I decided, was for me to take the greatest care that every­ thing, as far as was possible, should be clearly explained, & that there should be no need for higher geometry or for the calculus. Thus, in the first part, as well as in the third, there are do 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. Ir 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. r, & of the way in which the relation
13

»4

AD LECTOREM EX EDITIONE VIENNENSI

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

In tertia parte supponuntur utique nonnulla, qua demonstrantur in secunda ; sed ea ipsa sunt admodum pauca, IA iis, qui geometricas demonstrationes fastidiunt, facile admodum exponi
possunt res ipsa ita, ut penitus etiam sine ullo Geometria adjumento percipiantur, quanquam sine iis ipsa demonstratio haberi non poterit; ut idcirco in eo differre debeat is, qui secundam partem attente legerit, W Geometriam calleat, ab eo, qui eam omittat, quod ille primus veritates in tertia parte adhibitis, ac ex secunda erutas, ad explicationem Physica, intuebitur per evi­ dentiam ex ipsis demonstrationibus haustam, hic secundus easdem quodammodo per fidem Geo­ metris adhibitam credet. Hujusmodi inprimis est illud, particulam compositam ex punctis
etiam homogeneis, praditis lege virium proposita, posse 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, W qua respectu diver­
sarum particularum diversa sint, fA diversa respectu partium diversarum ejusdem particula, ac aliam particulam alicubi etiam urgeant in latus, unde plurium phanomenorum explicatio in Physica sponte fluit.

Verum qui omnem Theoria, W deductionum compagem aliquanto altius inspexerit, ac diligentius perpenderit, videbit, ut spero, me in hoc perquisitionis genere multo ulterius
progressum esse, quam olim Newtonus ipse desideravit. Is enim in postremo Optica questione prolatis iis, qua per vim attractivam, & vim repulsivam, mutata distantia ipsi attractiva suc­ cedentem, explicari poterant, hac addidit ; “ Atque hae quidem omnia si ita sint, jam Natura
universa valde erit simplex, & consimilis sui, perficiens nimirum magnos omnes corporum calesiium motus attractione gravitatis, aua est mutua inter corpora illa omnia, U minores fere
omnes particularum suarum motus alia aliqua vi attrahente, fA repellente, qua est inter particulas illas mutua." Aliquanto autem inferius de primigeniis particulis agens sic habet: “ Porra videntur mihi ha particula primigenia non modo in se vim inertia habere, motusque leges passivas illas, qua ex vi ista necessario oriuntur ; verum etiam motum perpetuo accipere a certis principiis
actuosis, qualia nimirum sunt gravitas, W causa fermentationis, tA coharentia corporum. Atque hac quidem principia considero non ut occultas qualitates, qua ex specificis ferum formis oriri fingantur, sed ut universales Natura leges, quibus res ipsa sunt formata. Nam principia
quidem talia revera existere ostendunt phanomena Natura, licet ipsorum causa qua sint, nondum fuerit explicatum. Affirmare, singulas rerum species specificis praditas esse qualita­ tibus occultis, per quas eae vim certam in agendo habent, hoc utique est nihil dicere : at ex
phanomenis Natura duo, vel tria derivare generalia motus principia, iA deinde explicare, quemadmodum proprietates, IA actiones rerum corporearum omnium ex istis principiis conse­ quantur, id vero magnus esset factus in Philosophia progressus, etiamsi principiorum istorum causa nondum essent cognita. Quare motus principia supradicta proponere non dubito, cum per Naturam universam latissime pateant."

Hac ibi Newtonus, ubi is quidem magnos in Philosophia progressus facturum arbitratus
est eum, qui ad duo, vel tria generalia motus principia ex Natura phanomenis derivata phano­ menorum explicationem reduxerit, tA sua principia protulit, ex quibus inter se diversis eorum aliqua tantummodo explicari posse censuit. Quid igitur, ubi IA ea ipsa tria, IA alia pracipua quaque, ut ipsa etiam impenetrabilitas, fA impulsio reducantur ad principium unicum legitima ratiocinatione deductum ? At id per meam unicam, IA simplicem virium legem prastari, patebit sane consideranti operis totius Synopsim quandam, quam hic subjicio ; sed multo magis opus
ipsum diligentius pervolventi.

THE PRINTER AT VENICE
TO
THE READER
OU will be well aware, if you have read the public journals, with what applause the work which I now offer to you has been received throughout Europe since its publication at Vienna five years ago. Not to mention others, if you refer to the numbers of the Berne Journal for the early part of the year 1761, you will not fail to see how highly it has been esteemed. It contains an entirely new system of Natural Philosophy, which is already commonly known as the Boscovichian theory, from the name of its author,
As a matter of fact, it is even now a subject of public instruction in several Universities in different parts; it is expounded not only in yearly theses or dissertations, both printed & debated; but also in several elementary books issued for the instruction of the young it is
introduced, explained, & by many considered as their original. Any one, however, who wishes to obtain more detailed insight into the whole structure of the theory, the close relation that its several parts bear to one another, or its great fertility & wide scope for the purpose of deriving the whole of Nature, in her widest range, from a single simple law of forces; any one who wishes to make a deeper study of it must perforce study the work here offered.
All these considerations had from the first moved me to undertake a new edition of the work ; in addition, there was the fact that I perceived that it would be a matter of some difficulty for copies of the Vienna edition to pass beyond the confines of Germany—indeed, at the present time, no matter how diligently they are inquired for, they are to be found
on sale nowhere, or scarcely anywhere, in the rest of Europe, The system had its birth in Italy, & its outlines had already been sketched by the author in several dissertations pub­
lished here in our own land; though, as luck would have it, the system itself was finally put into shape and published at Vienna, whither he had gone for a short time. I therefore
thought it right that it should be disseminated throughout the whole of Europe, & that preferably as the product of an Italian press. I had in fact already commenced an edition founded on a copy of the Vienna edition, when it came to my knowledge that the author was greatly dissatisfied with the Vienna edition, taken in hand there after his departure ; that innumerable printer’s errors had crept in ; that many passages, especially those that contain Algebraical formulae, were ill-arranged and erroneous; lastly, that the author himself had in mind a complete revision, including certain alterations, to give a better finish to the work, together with certain additional matter.
That being the case, I was greatly desirous of obtaining a copy, revised & enlarged by himself; I also wanted to have him at hand whilst the edition was in progress, & that he should superintend the whole thing for himself. This, however, I was unable to procure during the last few years, in which he has been travelling through nearly the whole of Europe ; until at last he came here, a little while ago, as he returned home from his lengthy wanderings, & stayed here to assist me during the whole time that the edition was in hand. He, in addition to our regular proof-readers, himself also used every care in cor­ recting the proof; even then, however, he has not sufficient confidence in himself as to imagine that not the slightest thing has escaped him. For it is a characteristic of the human mind that it cannot concentrate long on the same subject with sufficient attention.
It follows that this ought to be considered in some measure as a first & original edition ; any one who compares it with that issued at Vienna will soon see the difference between them. Many of the minor alterations are made for the purpose of rendering certain passages more elegant & clear; there are, however, especially at the foot of a page, slight additions also, or slight changes made after the type was set up, merely for the purpose of filling up gaps that were left here & there—these gaps being due to the fact that several sheets were being set at the same time by different compositors, and four presses were kept hard at work together. As he was at hand, this could easily be done without causing any disturbance of the sentences or the pagination.
3

14

AD LECTOREM EX EDITIONE VIENNENSI

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

In tertia parte supponuntur utique nonnulla, qua demonstrantur in secunda ; sed ea ipsa
sunt admodum pauca, y iis, qui geometricas demonstrationes fastidiunt, facile admodum exponi possunt res ipsa ita, ut penitus etiam sine ullo Geometria adjumento percipiantur, quanquam sine iis ipsa demonstratio haberi non poterit; ut idcirco in eo differre debeat is, qui secundam partem attente legerit, iff Geometriam calleat, ab eo, qui eam omittat, quod ille primus veritates in tertia parte adhibitis, ac ex secunda erutas, ad explicationem Physica, intuebitur per evi­ dentiam ex ipsis demonstrationibus haustam, hic secundus easdem quodammodo per fidem Geo­ metris adhibitam credet. Hujusmodi inprimis est illud, particulam compositam ex punctis etiam homogeneis, praditis lege virium proposita, posse 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, y qua respectu diver­ sarum particularum diversa sint, y diversa respectu partium diversarum ejusdem particula,
ac aliam particulam alicubi etiam urgeant in latus, unde plurium phcenomenorum explicatio in Physica sponte fluit.

VeT.um omnem Theoria, y deductionum compagem aliquanto altius inspexerit, ac diligentius perpenderit, videbit, ut spero, me in hoc perquisitionis genere multo ulterius
progressum esse, quam olim Newtonus ipse desideravit. Is enim in postremo Optica questione prolatis iis, qua per vim attractivam, y vim retulsivam, mutata distantia ipsi attractiva suc­ cedentem, explicari poterant, hac addidit: “ Atque hac quidem omnia si ita sint, jam Natura universa valde erit simplex, y consimilis sui, perficiens nimirum magnos omnes corporum
calestium motus attractione gravitatis, qua est mutua inter corpora illa omnia, y minores fere
omnes particularum suarum motus alia aliqua vi attrahente, iff repellente, qua est inter particulas illas mutua.” Aliquanto autem inferius de primigeniis particulis agens sic habet: “ Porro
videntur mihi ha particula primigenia non modo in se vim inertia habere, motusque leges passivas illas, qua ex vi ista necessario oriuntur ; verum etiam motum perpetuo accipere a certis principiis
actuosis, qualia nimirum sunt gravitas, y causa fermentationis, iff coh arentia corporum. Atque hac quidem principia considero non ut occultas qualitates, qua ex specificis rerum formis oriri fingantur, sed ut universales Natura leges, quibus res ipsa sunt formata. Nam principia
quidem talia revera existere ostendunt phanomena Natura, licet ipsorum causa qua sint, nondum fuerit explicatum. Affirmare, singulas rerum species specificis praditas esse qualita­ tibus occultis, per quas eae vim certam in agendo habent, hoc utique est nihil dicere; at ex phanomenis Natura duo, vel tria derivare generalia motus principia, W deinde explicare,
quemadmodum proprietates, y actiones rerum corporearum omnium ex istis principiis conse­
quantur, id vero magnus esset factus in Philosophia progressus, etiamsi principiorum istorum causa nondum essent cognita. Quare motus principia supradicta proponere non dubito, cum per Naturam universam latissime pateant.”

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

THE PRINTER AT VENICE
TO
THE READER
OU will be well aware, if you have read the public journals, with what applause the work which I now offer to you has been received throughout Europe since its publication at Vienna five years ago. Not to mention others, if you refer to the numbers of the Berne Journal for the early part of the year 1761, you will not fail to see how highly it has been esteemed. It contains an entirely new system of Natural Philosophy, which is already commonly known as the Boscovichian theory, from the name of its author,
As a matter of fact, it is even now a subject of public instruction in several Universities in different parts; it is expounded not only in yearly theses or dissertations, both printed & debated ; but also in several elementary nooks 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

4

TYPOGRAPHUS VENETUS LECTORI

Inter mutationes occurret ordo numerorum mutatus in paragraphis: nam numerus 82 de novo accessit totus: deinde is, qui fuerat 261 discerptus est in 5 ; demum in Appendice post num. 534. factas sunt & mutatiuncul® nonnulla:, & additamenta plura in iis, quas
pertinent ad sedem animx.

Supplementorum ordo mutatus est itidem ; quae enim fuerant 3, & 4, jam sunt 1, & 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 controversia: occasionem exhibuit in Encyclopedico etiam dictionario attactum,
in qua disserta tiuncula demonstrat Auctor non esse, cur ad vim exprimendam potentia qusepiam distantiae adhibeatur potius, quam functio.

Accesserunt per totum Opus notulae marginales, in quibus eorum, qua: pertractantur argumenta exponuntur brevissima, quorum ope unico obtutu videri possint omnia, & in
memoriam facile revocari. Postremo loco ad calcem Operis additus est fusior catalogus eorum omnium, quae huc
usque ab ipso Auctore sunt edita, quorum collectionem omnem expolitam, & correctam,
ac eorum, quae 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 Lunae defectibus.
Porro eam omnium suorum Operum Collectionem, ubi ipse adornaverit, typis ego meis excudendam suscipiam, quam magnificentissime potero.

Haec erant, quas te monendum censui 5 tu laboribus nostris fruere, & vive felix.

THE PREFACE TO THE READER
THAT APPEARED IN THE VIENNA EDITION
v EAR Reader, you have before you a Theory of Natural Philosophy deduced A \ 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 1 published in outline, & to a certain extent explained ; & also the occasion > that led me to undertake a more detailed treatment & enlargement of it.
For I have thought fit to divide the work into three parts; the first of these contains the exposition of the Theory itself, its analytical deduction & its demonstration; the second a fairly full application to Mechanics ; & the third an application to Physics. The most important point, I decided, was for me to take the greatest care that every­ thing, as far as was possible, should be clearly explained, & that there should be no need for higher geometry or for the calculus. Thus, in the first part, as well as in the third, there are no proofs by analysis ; nor are there any by geometry, with the exception of a very few that are absolutely necessary, & even these you will find relegated to brief notes set at the foot of a page. I have also added some very few proofs, that required a knowledge of higher algebra & geometry, or were of a rather more complicated nature, all of which nave 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 easyanalytical 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 tnis. 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 1 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. t, & of the way in which the relation
13

4

AD LECTOREM EX EDITIONE VIENNENSI

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

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

Vtftem qui omnem Theoria, W deductionum compagem aliquanto altius inspexerit, ac diligentius perpenderit, videbit, ut spero, me in hoc perquisitionis genere multo ulterius
progressum esse, quam olim Newtonus tpse desideravit. Is enim in postremo Optica questione prolatis iis, qua per vim attractivam, cf vim repulsivam, mutata distantia ipsi attractiva suc­ cedentem, explicari poterant, hac addidit: “ Atque hac quidem omnia si ita sint, jam Natura
universa valde erit simplex, consimilis sui, perficiens nimirum magnos omnes corporum calestium motus attractione gravitatis, qua est mutua inter corpora illa omnia, y minores fere omnes particularum suarum motus alia aliqua vi attrahente, y repellente, qua est inter particulas illas mutua.” Aliquanto autem inferius de primigeniis particulis agens sic habet: “ Porro videntur mihi ba particula primigenia non modo in se vim inertia habere, motusque leges passivas illas, qua ex vi ista necessario oriuntur ; verum etiam motum perpetuo accipere a certis principiis actuosis, qualia nimirum sunt gravitas, y causa fermentationis, tsf coharentia corporum. Atque hac quidem principia considero non ut occultas qualitates, qua ex specificis rerum formis oriri fingantur, sed ut universales Natura leges, qutbus res ipsa sunt formata. Nam principia
quidem talia revera existere ostendunt phanomena Natura, licet ipsorum causa qua sint, nondum fuerit explicatum. Affirmare, singulas rerum species specificis praditas esse qualita­ tibus occultis, ter quas eae vim certam in agendo habent, hoc utique est nihil dicere ; at ex phanomenis Natura duo, vel tria derivare generalia motus principia, y deinde explicare,
quemadmodum proprietates, & actiones rerum corporearum omnium ex istis principiis conse­ quantur, id vero magnus esset factus in Philosophia progressus, etiamsi principiorum istorum causa nondum essent cognita. Quare motus principia supradicta proponere non dubito, cum per Naturam universam latissime pateant.”

Hac ibi Newtonus, ubi is quidem magnos in Philosophia progressus facturum arbitratus
est eum, qui ad duo, vel tria generalia motus principia ex Natura phanomenis derivata phano­ menorum explicationem reduxerit, sua principia protulit, ex quibus inter se diversis eorum aliqua tantummodo explicari posse censuit. Quid igitur, ubi y ea ipsa tria, U alia pracipua
quaque, ut ipsa etiam impenetrabilitas, y impulsio reducantur ad principium unicum legitima ratiocinatione deductum P At id per meam unicam, W simplicem vtrium legem prastari, patebit sane consideranti operis totius Synopsim quandam, 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 moderat» 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 faitli 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, cither 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 readdy 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 al] 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 iustance, 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 pauses 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 arc 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.

SYNOPSIS TOTIUS OPERIS

EX EDITIONE VIENNENSI

PARS I

"RIMIS sex numeris exhibeo, quando, & qua occasione Theoriam meam

i ] invenerim, ac ubi hucusque de ea egerim in dissertationibus jam editis, quid

2 ea commune habeat cum Leibnitiana, quid cum Newtoniana Theoria, in

47« quo ab utraque discrepet, & vero etiam utrique praestet: addo, quid

'

alibi promiserim pertinens ad aequilibrium, & oscillationis centrum, &

quemadmodum iis nunc inventis, ac ex unico simplicissimo, ac elegant-

issimo theoremate profluentibus omnino sponte, cum dissertatiunculam

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

molis evaserit tractatio.

Tum usque ad num. it expono Theoriam ipsam: materiam constantem punctis prorsus simplicibus, indivisibilibus, & inextensis, ac a se invicem distantibus, qme puncta habeant singula vim inertis, & praeterea vim activam mutuam pendentem a distantiis, ut nimirum, data distantia, detur & magnitudo, & directio vis ipsius, mutata autem distantia, mutetur vis ipsa, qus, imminuta distantia in infinitum, sit repulsiva, & quidem excrescens in infinitum : aucta autem distantia, minuatur, evanescat, mutetur in attractivam crescentem primo, tum decrescentem, evanescentem, abeuntem iterum in repulsivam, 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 binae elastri cuspides conantur ad es invicem accedere, vel a se invicem recedere, prout sunt plus justo distractae, vel con­
tractae.

Inde ad num. 16 ostendo, quo pacto id non sit aggregatum quoddam virium temere coalescentium, sed per unicam curvam continuam exponatur ope abscissarum exprimentium distantias, & ordinatarum exprimentium vires, cujus curvae ductum, & naturam expono, ac ostendo, in quo differat ab hyperbola illa gradus tertii, quae 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 19 illud, in collisione corporum debere vd haberi compenetrationem, vel violari legem continuitatis, velocitate mutata per saltum, si cum inxqualibus velocitatibus deveniant ad immediatum contactum, quse continuitatis lex cum (ut evinco) debeat omnino observari, illud infero, antequam ad contactum deveniant corpora, debere mutari eorum velocitates per vim quandam, quae sit par extinguendae velocitati, vel velocitatum differentia, cuivis utcunque magna.

A num. 19 ad 28 expendo effugium, quo ad eludendam argumenti mei vim utuntur ii, qui negant corpora dura, qua quidem responsione uti non possunt Newtoniani, & Corpusculares generaliter, qui dementares corporum particulas assumunt prorsus duras: qui autem
omnes utcunque parvas corporum particulas molles admittunt, vel elasticas, difficultatem non effugiunt, sed transferunt ad primas superficies, vel puncta, in quibus committeretur omnino saltus, & lex continuitatis violaretur : ibidem quendam verborum lusum evolvo, frustra adhibitum ad eludendam argumenti mei vim.

Series numerorum, quibus tractari incipiunt, quae aunt in textu. iG

SYNOPSIS OF THE WHOLE WORK
(FROM THE VIENNA EDITION)
PART I
the first six articles, I state the time at which I evolved my Theory, what i* led me to it, & where I have discussed it hitherto in essays already pub­ lished : also what it has in common with the theories of Leibniz and Newton ; in what it differs from either of these, & in what it is really superior to them both. In addition I state what I have published else­ where about equilibrium & the centre of oscillation ; & how, having found out that these matters followed quite easily from a single theorem of the most simple & elegant kind, I proposed to write a short essay thereon ; but when I set to work to deduce the matter from this principle, the discussion, quite unexpectedly to me, developed into a whole work of considerable magnitude. From this until Art. 11, I explain the Theory itself: that matter is unchangeable, 7 and consists of points that are perfectly simple, indivisible, of no extent, & separated from one another; that each of these points has a property of inertia, & in addition a mutual active force depending on the distance in such a way that, if the distance is given, both the magnitude & the direction of this force arc 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. 16 I show that it is not merely an aggregate of forces combined u 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 16 which I came upon a Theory of this kind, & from which I believe I have deduced the whole of it by a straightforward & perfectly rigorous chain of reasoning. I contend indeed, from here on until Art. 19, that, in the collision of solid bodies, either there must be compenetration, or the Law of Continuity must be violated by a sudden change of velocity, if the bodies come into immediate contact with unequal velocities. Now since the Law of Continuity must (as I prove that it must) be observed in every case, I infer that, before the bodies reach the point of actual contact, their velocities must be altered by some force which is capable of destroying the velocity, or the difference of the velocities, no matter how great that may be. From Art. 19 to Art. 28 I consider the artifice, adopted for the purpose of evading the 19 strength of my argument by those who deny the existence of hard bodies; as a matter of fact this cannot be used as an argument against me by the Newtonians, or the Corpuscularians 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 clastic, 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 number! are the numbers oi the articles, in which the matters given in the text arc first discussed.
x7

18

SYNOPSIS TOTIUS OPERIS

28

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

mei argumenti vis elidatur, affirmat quispiam, prima materia: elementa compenetrari, alter

dicuntur materia: puncta adhuc moveri ad se invicem, ubi localiter omnino quiescunt, &

contra primum effugium evinco impenetrabilitatem ex inductione ; contra secundum

expono aequi vocationem quandam in significatione vocis motui, cui sequivocationi totum

innititur.

3°

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

quam ego contemplatus sum, collisione corporum, conclusit, continuitatis legem violari,

cum ego eandem illaesam esse debere ratus ad totam devenerim Theoriam meam.

32

Hic igitur, ut mere 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,

quse nimirum excludat omnem saltum ab una magnitudine ad aliam sine transitu per

39 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 erutre passim per inductionem, donec demum ejus vim

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

48

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

probationem aggredior metaphysicam quandam, ex necessitate utriusque limitis in quanti­

tatibus realibus, vel seriebus quantitatum realium finitis, quae nimirum nec suo principio,

nec suo fine carere possunt. Ejus rationis vim ostendo in motu locali, & in Geometria

52 sequentibus duobus numeris: tum num. 52 expono difficultatem quandam, quae petitur ex eo, quod in momento temporis, in quo transitur a non esse ad esse, videatur juxta ejusmodi

Theoriam debere simul haberi ipsum esse, & non esse, quorum alterum ad finem praecedentis

seriei statuum pertinet, alterum ad sequentis initium, ac solutionem ipsius fuse evolvo,

Geometria etiam ad rem oculo ipsi sistendam vocata in auxilium.

63

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

principium applico ad excludendum saltum immediatum ab una velocitate ad aliam, sine

transitu per intermedias, quod & inductionem laederet pro continuitate amplissimam, &

induceret pro ipso momento temporis, in quo fieret saltus, binas velocitates, ultimam

nimirum seriei pra:cedentis, & primam novae, cum tamen duas simul velocitates idem mobile

habere omnino non possit. Id autem ut illustrem, & evincam, usque ad num. 72 considero

velocitatem ipsam, ubi potcntialem quandam, ut appello, velocitatem ab actuali secerno,

& multa, quae ad ipsarum naturam, ac mutationes pertinent, diligenter evolvo, nonnullis

etiam, quae inde contra meae Theoriae probationem objici possunt, dissolutis.

His expositis concludo jam illud ex ipsa continuitate, ubi corpus quodpiam velocius
movetur post aliud lentius, ad contactum immediatum cum illa velocitatum inaequalitate deveniri non posse, in quo scilicet contactu primo mutaretur vel utriusque velocitas, vel
alterius, per saltum, sed debere mutationem velocitatis incipere ante contactum ipsum. 73 Hinc num. 73 infero, debere haberi mutationis causam, quae appelletur vis : tum num. 74 74 hanc vim debere esse mutuam, & agere in partes contrarias, quod per inductionem evinco, 75 & inde infero num. 75, appellari posse repulsivam ejusmodi vim mutuam, ac ejus legem
exquirendam propono. In ejusmodi autem perquisitione usque ad num. 80 invenio illud, debere vim ipsam imminutis distantiis crescere in infinitum ita ut par sit extinguendre
velocitati utcunque magnae; 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 curvae per ordinatas suas exprimentis virium legem determino.

SYNOPSIS OF THE WHOLE WORK

*9

In the next articles, 28 & 29,1 refute a further pair of arguments advanced by others; 28 in the first of these, in order to evade my reasoning, someone states that there is compenetration 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 motiont an equivocation upon which the whole thing depends.
Then, in Art. 30, 31, I show in what respect I differ from Maclaurin, who, having 30 considered the same point as myself, came to the conclusion that in the collision of bodies the Law of Continuity was violated ; whereas I obtained the whole of my Theory from the assumption that this law must be unassailable.
At this point therefore, in order that the strength of my deductive reasoning might 32 be shown, I investigate the Law of Continuity; and from Art. 32 to Art. 38, I set forth its nature, & what is meant by a continuous change through all intermediate stages, such as to exclude any sudden change from any one magnitude to another except by a passage through intermediate stages; & I call in geometry as well to help my explanation of the matter. Then I investigate its truth first of all by induction; &, investigating the prin- 39 ciple of induction itself, as far as Art. 44,1 show whence the force of this principle is derived, & where it can be used. I give by way of illustration an example in which impenetrability
is derived entirely by induction ; & lastly I apply the force of the principle to demonstrate the Law of Continuity. In the articles that follow I consider certain cases of two kinds, 45 in which the Law of Continuity appears to be violated, but is not however really violated.
After this proof of the principle of continuity procured through induction, in Art. 48, 48 I undertake another proof of a metaphysical kind, depending upon the necessity of a limit on either side for either real quantities or for a finite series of real quantities; & indeed it is impossible that these limits should be lacking, either at the beginning or the end. I demonstrate the force of this reasoning in the case of local motion, & also in geometry, in the next two articles. Then in Art. 52 I explain a certain difficulty, which is derived from the 52 fact that, at the instant at which there is a passage from non-exitience to existence^ it appears according to a theory of this kind that we must have at the same time both exijtence and
■non-existence. For one of these belongs to the end of the antecedent series of states, & the other to the beginning of the consequent series. I consider fairly fully'the solution of this problem; and I call in geometry as well to assist in giving a visual representation of the
matter. In Art. 63, after summing up all that has been said about the Law of Continuity, I 63
apply the principle to exclude the possibility of any sudden change from one velocity to another, except by passing through intermediate velocities ; this would be contrary to the very full proof that I give for continuity, as it would lead to our having two velocities at the instant at which the change occurred. That is to say, there would be the final velocity
of the antecedent series, & the initial velocity of the consequent series ; in spite of the fact that it is quite impossible for a moving body to have two different velocities at the same
time. Moreover, in order to illustrate & prove the point, from here on to Art. 72, I consider velocity itself; and I distinguish between a potential velocity, as I call it, & an actual velocity ; I also investigate carefully many matters that relate to the nature of these velocities & to their changes. Further, I settle several difficulties that can be brought up in opposition to the proof of my Theory, in consequence.
This done, I then conclude from the principle of continuity that, when one body with a greater velocity follows after another body having a less velocity, it is impossible that there should ever be absolute contact with such an inequality of velocities ; that is to say, a case of the velocity of each, or of one or the other, of them being changed suddenly at the instant of contact. I assert on the other hand that the change in the velocities must begin before contact. Hence, in Art. 73,1 infer that there must be a cause for this change: 73 which is to be called “ force.” Then, in Art. 74, I prove that this force is a mutual one, & 74 that it acts in opposite directions; the proof is by induction. From this, in Art. 75, I 75 infer that such a mutual force may be said to be repulsive ; & I undertake the investigation of the law that governs it. Carrying on this investigation as far as Art. 80, I find that this force must increase indefinitely as the distance is diminished, in order that it may be capable of destroying any velocity, however great that velocity may be. Moreover, I find that, whilst the force must be indefinitely increased as the distance is indefinitely decreased, it must be on the contrary attractive at very great distances, as is the case for gravitation. Hence I infer that there must be a limit-point forming a boundary between attraction & repulsion; & then by degrees I find more, indeed very many more, of such limit-points, or points of transition from attraction to repulsion, & from repulsion to attraction ; & I determine the form of the entire curve, that expresses by its ordinates the law of these forces.

20

SYNOPSIS TOTIUS OPERIS

81

Eo usque virium legem deduco, ac definio; tum num. 81 eruo ex ipsa lege consti­

tutionem elementorum materiae, quae debent esse simplicia, ob repulsionem in minimis

distantiis in immensum auctam; nam ea, si forte ipsa elementa partibus constarent, nexum

omnem dissolveret. Usque ad num. 88 inquiro in illud, an haec elementa, ut simplicia esse

debent, ita etiam inextensa esse debeant, ac exposita illa, quam virtualcm extensionem

appellant, eandem excludo inductionis principio, & difficultatem evolvo tum eam, quae peti

possit ab exemplo ejus generis extensionis, quam in anima indivisibili, & simplice per aliquam

corporis partem divisibilem, & extensam passim admittunt : vel omnipraesentiae Dei: tum

eam, quae peti possit ab analogia cum quiete, in qua nimirum conjungi debeat unicum

spatii punctum cum serie continua momentorum temporis, uti in extensione virtuali unicum

momentum temporis cum serie continua punctorum spatii conjungeretur, ubi ostendo, nec

quietem omnimodam in Natura haberi usquam, nec adesse semper omnimodam inter

88 tempus, & spatium analogiam. Hic autem ingentem colligo ejusmodi determinationis

fructum, ostendens usque ad num. 91, quantum prosit simplicitas, indivisibilitas, inextensio

elementorum materiae, ob summotum transitum a vacuo continuo per saltum ad materiam

continuam, ac ob sublatum limitem densitatis, quae in ejusmodi Theoria ut minui in

infinitum potest, ita potest in infinitum etiam augeri, dum in communi, ubi ad contactum

deventum est, augeri ultra densitas nequaquam potest, potissimum vero ob sublatum omne

continuum coexistens, quo sublato & gravissimae difficultates plurima: evanescunt, &

infinitum actu existens habetur nullum, sed in possibilibus tantummodo remanet series

finitorum in infinitum producta.

91

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

homogenca, 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, 8c foliis; ac per inductionem, & analogiam demonstro, naturam nos ad homo-

geneitatem elementorum, non ad heterogeneitatem deducere.

100

Ea ad probationem Theoriae pertinent; qua absoluta, antequam inde fructus colli­

gantur multiplices, gradum hic facio ad evolvendas difficultates, quae vel objectae 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, quae ex ipsa ejusmodi virium lege deducuntur.

101

Primo quidem, ut iis etiam faciam satis, qui inani vocabulorum quorundam sono

perturbantur, a num. 101 ad 104 ostendo, vires hasce non esse quoddam occultarum

qualitatum genus, sed patentem sane Mechanismum, cum & idea earum sit admodum

distincta, & existentia, ac lex positive comprobata ; ad Mechanicam vero pertineat omnis

104

tractatio de Motibus, qui a datis viribus etiam sine immediato impulsu oriuntur. A num.

104 ad 106 ostendo, nullum committi saltum in transitu a repulsionibus ad attractiones,

106 & vice versa, cum nimirum per omnes intermedias quantitates is transitus fiat. Inde vero

ad objectiones gradum facio, quae totam curv«e formam impetunt. Ostendo nimirum usque

ad num. 116, non posse omnes repulsiones a minore attractione desumi; repulsiones ejusdem

esse seriei Cum attractionibus, a quibus differant tantummodo ut minus a majore, sive ut

negativum a positivo; ex ipsa curvarum natura, quae, quo altioris sunt gradus, eo in

pluribus punctis rectam secare possunt, & eo in immensum plures sunt numero; haberi

potius, ubi curva guacritur, quae vires exprimat, indicium pro curva ejus naturae, ut rectam

in plurimis punctis secet, adeoque plurimos sccum afferat virium transitus a rcpulsivis ad

attractivas, quam pro curva, quae nusquam axem secans attractiones solas, vel solas pro

distantiis omnibus repulsiones exhibeat: sed vires repulsivas, & multiplicitatem transituum

esse positive probatam, & deductam totam curvae formam, quam itidem ostendo, non esse

ex arcubus natura diversis temere coalescentem, sed omnino simplicem, atque eam ipsam

SYNOPSIS OF THE WHOLE WORK

21

So far I have been Occupied in deducing and settling the law of these forces. Next, in Art. 81,1 derive from this law the constitution of the elements of matter. These must be 81 quite simple, on account of the repulsion at very small distances being immensely great; for if by chance those elements were made up of parts, the repulsion would destroy all connections between them. Then, as far as Art. 88, I consider the point, as to whether these elements, as they must be simple, must therefore be also of no extent; &, having ex­ plained what is called “ virtual extension,” I reject it by the principle of induction. I then consider the difficulty which may be brought forward from an example of this kind of extension; such as is generally admitted in the ease 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 scries of points of space. I show that there can neither be perfect rest any- 88 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 91
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 Lcionizian 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 joo 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 tnat I have enunciated & proved, & finally against those indivisible, non-extended points that
are deduced from a law of forces of this kind. First of all then, in order that I may satisfy even those who are confused over the 101
empty sound of certain terms, I show, in Art. 101 to 104, that these forces are not some sort of mysterious qualities; but that they form a readily intelligible mechanism, since both the idea of them is perfectly distinct, as well as their existence, & in addition the law that governs them is demonstrated in a direct manner. To Mechanics belongs every dis­ cussion concerning motions that arise from given forces without any direct impulse. In Art. 104 to 106, I show that no sudden change takes place in passing from repulsions to 104 attractions or vice versa ; for this transition is made through every intermediate quantity. Then I pass on to consider the objections that are made against the whole form of my 106 curve. I show indeed, from here on to Art. 116, that all repulsions cannot be taken to come from a decreased attraction ; that repulsions belong to the self-same series as attrac­ tions, differing from them only as less does from more, or negative from positive. From the very nature of the curves (for which, the higher the degree, the more points there are in which they can intersect a right line, & vastly more such curves there are), I deduce that there is more reason for assuming a curve of the nature of mine (so that it may cut a right line in a large number of points, & thus give a large number of transitions of the forces from repulsions to attractions), than for assuming a curve that, since it docs not cut the axis anywhere, will represent attractions alone, or repulsions alone, at all distances. Further, I point out that repulsive forces, and a multiplicity of transitions are directly demonstrated, & the whole form of the curve is a matter of deduction; &I also show that it is not formed of a number of arcs differing in nature connected together haphazard ;

22

SYNOPSIS TOTIUS OPERIS

simplicitatem in Supplementis evidentissime demonstro, exhibens methodum, qua deveniri possit ad aequationem ejusmodi curvae simplicem, & uniformem ; licet, ut hic ostendo, ipsa illa lex virium possit mente resolvi in plures, quse per plurcs curvas exponantur, a quibus tamen omnibus illa reapse unica lex, per unicam illam continuam, & in se simplicem curvam
componatur.

I2i

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

duplicata distantiarum, quae nimirum in minimis distantiis attractionem requirit crescentem

in infinitum. Ostendo autem, ipsam non esse uspiam accurate in ejusmodi ratione, nisi

imaginarias resolutiones exhibeamus; nec vero ex Astronomia deduci ejusmodi legem

prorsus accurate servatam in ipsis Planctarum, & Cometarum distantiis, sed ad summum ita

124 proxime, ut differentia ab ea lege sit perquam exigua : ac a num. 124 expendo argumentum,

quod pro ejusmodi lege desumi possit ex eo, quod cuipiam visa sit omnium optima, &

idcirco electa ab Auctore Naturae, ubi ipsum Optimismi principium ad trutinam revoco, ac

excludo, & vero illud etiam evinco, non esse, cur omnium optima ejusmodi lex censeatur ;

in Supplementis vero ostendo, ad quae potius absurda deducet ejusmodi lex, & vero etiam

aliae plurcs attractionis, quae imminutis in infinitum distantiis excrescat in infinitum.

13X

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

sorum ideam non habeamus, quod nimirum eam haurire non possumus per sensus, quos

solae massae, & quidem grandiores, afficiunt, atque idcirco eandem nos ipsi debemus per

reflexionem efformare, quod quidem facile possumus. Ceterum illud ostendo, me non

inducere primum in Physicam puncta indivisibilia, & inextensa, cum co etiam Leibnitiana:

monades recidant, sed sublata extensione continua difficultatem auferre illam omnem, quze

jam olim contra Zenonicos objecta, nunquam est satis soluta, qua fit, ut extensio continua

ab inextensis effici omnino non possit.

[40

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

autem existentiam vel inde probari, quod continuitas se se ipsam destruat, & ex ea assumpta

probetur argumentis a me institutis hoc ipsum, prima elementa esse indivisibilia, & inextensa,

143 nec ullum haberi extensum continuum. A num. 143 ostendo, ubi continuitatem admittam,

nimirum in solis motibus; ac illud explico, quid mihi sit spatium, quid tempus, quorum

naturam in Supplementis multo uberius expono. Porro continuitatem ipsam ostendo a

natura in solis motibus obtineri accurate, in reliquis affectari quodammodo ; ubi & exempla

quaedam evolvo continuitatis primo aspectu violatae, in quibusdam proprietatibus luminis,

ac in aliis quibusdam casibus, in quibus quaedam crescunt per additionem partium, non (ut

ajunt) per intussumptionem.

153

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

evolvo, unde fiat, ut in ipsa idea corporis videatur incluui extensio continua, ubi in ipsam

idearum nostrarum originem inquiro, & quae inde praejudicia profluant, expono. Postremo

165 autem loco num. 165 innuo, qui fieri possit, ut puncta inextensa, & a se invicem distantia,

in massam coalescant, quantum libet, cohaerentem, & iis proprietatibus praeditam, quas in

corporibus experimur, quod tamen ad tertiam partem pertinet, ibi multo uberius pertrac­

tandum ; ac ibi quidem primam hanc partem absolvo.

PARS II

166

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

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

168 usque ad num. 172 in ipsos arcus inquiro, quorum alii attractivi, alii repulsivi, alii asym­

ptotic!, ubi casuum occurrit mira multitudo, & in quibusdam consectaria notatu digna, ut

& illud, cum ejus formae curva plurium asymptotorum esse possit, Mundorum prorsus

similium seriem posse oriri, quorum alter respectu alterius vices agat unius, & indissolubilis

SYNOPSIS OF THE WHOLE WORK

23

but that it is absolutely onc-fold. This one-fold character I demonstrate in the Supple­ ments in a very evident manner, giving a method by which a simple and uniform equation
may be obtained for a curve of this kind. Although, as I there point out, this law of forces may be mentally resolved into several, and these may be represented by several correspond­ ing curves, yet that law, actually unique, may be compounded from all of these together
by means of the unique, continuous & one-fold curve that I give. In Art. 121, I start to give a refutation of those objections that may be raised from 121
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 theplanets & the comets, but one merely that is at
most so very nearly correct, that the difference from the law of inverse squares is very slight. From Art. 124 onwards,! examine the value of the argument that can be drawn 124 in favour of a law of this sort from the view that, is some have thought, it is the best of all, & that on that account it was selected by the Founder of Nature. In connection with
this I examine the principle of Optimism, & I reject it; moreover I prove conclusively that there is no reason why this sort of law should be supposed to be the best of all. Fur­ ther in the Supplements, I show to what absurdities a law of this sort is more likely to lead ; & the same thing for other laws of an attraction that increases indefinitely as the distance is diminished indefinitely.
In Art. 131 I pass from forces to elements. I first of all show the reason why we may 131 not appreciate the idea of non-extended points; it is because we are unable to perceive them by means of the senses, which are only affected by masses, & these too must be of considerable size. Consequently we have to build up the idea by a process of reasoning; & this we can do without any difficulty. In addition, I point out that I am not the first
to introduce indivisible & non-extended points into physical science ; for the *£ monads ” of Leibniz practically come to the same thing. But I show that, by rejecting the idea of continuous extension, I remove the whole of the difficulty, which was raised against the disciples of Zeno in years gone by, & has never been answered satisfactorily; namely, the difficulty arising from the fact that by no possible means can continuous extension be made up from things of no extent.
In Art. 140 I show that the principle of induction yields no argument against these 140 indivisibles; rather their existence is demonstrated by that principle, for continuity is self-contradictory. On this assumption it may be proved, by arguments originated by
myself, that the primary elements are indivisible & non-extended, & that there does not
exist anything possessing the property of continuous extension. From Art. 143 onwards, j I point out the only connection in which I shall admit continuity, & that is in motion. I state the idea that I have with regard to space, & also time : the nature of these I explain
much more fully in the Supplements. Further, I show that continuity itself is really a property of motions only, & that in all other things it is more or less a false assumption.
Here I also consider some examples in which continuity at first sight appears to be violated, such as in some of the properties of light, & in certain other cases where things
increase by addition of parts, and not by intussumption, as it is termed. From Art. 153 onwards, I show how greatly these points of mine differ from object- 153
souls. I consider how it comes about that continuous extension seems to be included in the very idea of a body; & in this connection, I investigate the origin of our ideas
& I explain the prejudgments that arise therefrom. Finally, in Art. 165, I lightly 165 sketch what might happen to enable points that are of no extent, & at a distance from one another, to coalesce into a coherent mass of any size, endowed with those properties that we experience in bodies. This, however, belongs to the third part; & there it will be much more fully developed. This finishes the first part.

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

24

SYNOPSIS TOTIUS OPERIS

172 dementi. /Xd. num. 179 areas contemplor arcubus clausas, qux respondentes segmento axis cuicunquc, esse possunt magnitudine utcunque magnae, vel parvx, sunt autem mensura
i79 incrementi, vd decrementi quadrati velocitatum. Ad num. 189 inquiro in appulsus curvx ad axem, sive is ibi secetur ab eadem (quo casu habentur transitus vel a repulsione ad
attractionem, vd ab attractione ad repulsionem, quos dico limites, & quorum maximus est in tota mea Theoria usus), sive tangatur, & curva retro redeat, ubi etiam pro appulsibus considero recessus in infinitum per arcus asymptoticos, & qui transitus, sive limites, oriantur inde, vel in Natura admitti possint, evolvo.

189

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

primo quidem usque ad num. 204 ago de systemate duorum punctorum, ea pertractans,

qux 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

cohxsionis varia genera, fermentationes, conflagrationes, emissiones vaporum, proprietates

luminis, dasticitatem, mollitiem.

204

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

vires generaliter facile definiuntur data ipsorum positione quacunque : verum utcunque

data positione, & celeritate nondum a Geometris inventi sunt motus ita, ut generaliter pro

casibus omnibus absolvi calculus possit. Vires igitur, & variationem ingentem, quam

diversx pariunt combi nationes punctorum, utut tantummodo numero trium, persequor

209 usque ad num. 209. Hinc usque ad num. 214 quxdam evolvo, qux pertinent ad vires

ortas in singulis ex actione composita reliquorum duorum, & qux 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 Naturx

214 explicationem futurum sit usui, significo. Usque ad num. 221 ipsis etiam oculis contem­

plandum propono ingens discrimen in legibus virium, quibus bina puncta agunt in tertium,

sive id jaceat in recta, qua junguntur, sive in recta ipsi perpendiculari, & eorum intervallum

secante bifariam, constructis ex data primigenia curva curvis vires compositas exhibentibus :

221 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

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

223

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 quxdam exurgunt, qux multo generaliores redduntur inferius, ubi

in tribus etiam punctis tantummodo adumbrantur, qux pertinent ad virgas rigidas, flexiles,

elasticas, ac ad vectem, & ad alia plura, qux itidem inferius, ubi de massis, multo generaliora

228 fiunt. Demum usque ad num. 238 contemplor tria puncta posita non in directum, sive in

xquilibrio sint, sive in perimetro ellipsium quarundam, vel curvarum aliarum; in quibus

mira occurrit analogia limitum quorundam cum limitibus, quos habent bina puncta in axe

curvx primigenix ad se invicem, atque ibidem multo major varietas casuum indicatur pro

massis, & specimen applicationis exhibetur ad soliditatem, & liquationem per celerem

238 intestinum motum punctis impressum. Sequentibus autem binis numeris generalia quxdam

expono de systemate punctorum quatuor cum applicatione ad virgas solidas, rigidas, flexiles,

ac ordines particularum varios exhibeo per pyramides, quarum infimx ex punctis quatuor,

superiores ex quatuor pyramidibus singulx coalescant.

240

A num. 240 ad massas gradu facto usque a num. 264 considero, qux ad centrum gravi­

tatis pertinent, ac demonstro generaliter, in quavis massa esse aliquod, & esse unicum :

ostenao, quo pacto determinari generaliter possit, & quid in methodo, qux communiter

adhibetur, desit ad habendam demonstrationis vim, luculenter expono, & suppleo, ac

SYNOPSIS OF THE WHOLE WORK

25

to Art. 179, I consider the areas included by the arcs; these, corresponding to different 172

segments of the axis, may be of any magnitude whatever, either great or small; moreover they measure the increment or decrement in the squares of the velocities. Then, on as 179

far as Art. 189,1 investigate the approach of the curve to the axis; both when the former

is cut by the latter, in which case there are transitions from repulsion to attraction and

from attraction to repulsion, which I call ‘ limits,’ & use very largely in every part of my

Theory; & also when the former is touched by the latter, & the curve once again recedes

from the axis. I consider, too, as a case of approach, recession to infinity along an asymp­

totic arc; and I investigate what transitions, or limits, may arise from such a case, &

whether such arc 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 arc not so disturbed, I make an antici­

patory note of the great use to which this will be put in the third part, for the purpose

of explaining various kinds of cohesion, fermentations, conflagrations, emissions of vapours,

the properties of light, elasticity and flexibility.

There follows, from Art. 204 to Art. 239, the much more fruitful consideration of a 204

system of three points. The forces connected with them can in general be easily deter­

mined for any given positions of the points; but, when any position & velocity are given,

the motions have not yet been obtained by geometricians in such a form that the general

calculation can be performed for every possible case. So I proceed to consider the forces,

& the huge variation that different combinations of the points beget, although they are

only three in number, as far as Art. 209. From that, on to Art. 214, I consider certain 209

things that have to do with the forces that arise from the action, on each of the points, of

the other two together, & how these urge the third point not only to approach, or recede

from, themselves, but also in a direction at right angles; in this connection there comes

forth an analogy with solidity, & a truly immense difference between the several cases when

the distances arc very small, & the greatest conformity possible at very great distances

such as those at which gravity acts ; & I point out what great use will be made of this also

in explaining the constitution of Nature. Then up to Art. 221, I give ocular demonstra- 214

tions of the huge differences that there are in the laws of forces with which two points act upon a third, whether it lies in the right line joining them, or in the right line that is the

perpendicular which bisects the interval between them ; this I do by constructing, from

the primary curve, curves representing the composite forces. Then in the two articles 22i

that follow, I consider the case, a really important one, in which, by merely changing the

position of the two points, the third point, at any and the same definite interval situated

at the same distance from the middle point of the interval between the two points, will

be either continually attracted, or continually repelled, or neither attracted nor repelled;

& since a difference of this kind should hold to a much greater degree in masses, I point

out, in Art. 222, the great use that will be made of this also in Physics.

222

At this point then, in Art. 223, I pass from the forces derived from two points to the 223

consideration of a whole system of them; and, as far as Art. 228, I study three points

situated in a right line, from the mutual forces of which there arise certain relations, which

I return to later in much greater generality ; in this connection also are outlined, for three

points only, matters that have to do with rods, either rigid, flexible or elastic, and with

the lever, as well as many other things; these, too, are treated much more generally later

on, when I consider masses. Then right on to Art. 238, I consider three points that do 228

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 23^

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 240

the centre of gravity ; & I prove that in general there is one, & only one, in any given mass.

I show how it can in general be determined, & I set forth in clear terms the point that is

lacking in the usual method, when it comes to a question of rigorous proof; this deficiency

26

SYNOPSIS TOTIUS OPERIS

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

264 26$ 266 276 277. 278
279

Ejus tractionis fructus colligo plures : conservationem ejusdem quantitatis motuum in Mundo in eandem plagam num. 264, xqualitatem 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, qua: prima fronte virium resolutionem requirere videntur.

289

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

notjssimum, quo pacto in compositione decrescat vis, in resolutione crescat, sed in illa priore

conspirantium summa semper maneat, contrariis elisis; in hac posteriore concipiantur

tantummodo binae vires contrariae adjectae, quae consideratio nihil turbet phenomena ;

unde fiat, ut nihil inde pro virium vivarum Theoria deduci possit, cum sine iis explicentur

omnia, ubi plura itidem explico ex iis phaenomenis, quae pro ipsis viribus vivis afferri solent.

297

A num. 297 occasione inde arrepta aggredior quaedam, quae ad legem continuitatis pertinent, ubique in motibus sancte servatam, ac ostendo illud, idcirco in collisionibus

corporum, ac in motu reflexo, leges vulgo definitas, non nisi proxime tantummodo observari,

& usque ad num. 307 relationes varias persequor angulorum incidentia:, & 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,

3uid, si ingens, ac elementa profero, quae ad luminis reflexionem, & refractionem cxplican-

am, defimendamque ex Mechanica requiritur, relationem itidem vis absolutae ad relativam

in obliquo gravium descensu, & nonnulla, quae ad oscillationum accuratiorem Theoriam

necessaria sunt, prorsus elementaria, diligenter expono.

307

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

evolvo plura, qua: pertinent ad directionem virium in singulis compositarum e binis

reliquarum actionibus, ut illud, eas directiones vel esse inter se parallelas, vel, si utrinque

313 indefinite producantur, per quoddam commune punctum transire omnes: tum usque ad

321 theoremata aEa plura, qua: 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, distantia: ipsarum a massa tertia, sinus

anguli, quem singularum directio continet cum sua ejusmodi distantia, & massa: ipsius eam

habentis compositam vim, ad distantiam, sinum, massam alteram ; vires autem motrices

habere tantummodo priores rationes duas elisa tertia.

321

Eorum theorematum fructum colligo deducens inde usque ad num. 328, qua: ad

equilibrium pertinent divergentium utcumque virium, & ipsius aequilibrii centrum, ac

nisum centri in fulcrum, & qua: ad praeponderandam, Theoriam extendens ad casum etiam,

quo massae non in se invicem agant mutuo immediate, sed per intermedias alias, quae 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 aequilibrium, sed in ipso centro

debere sentiri vim a fulcro, vel sustinente puncto, aequalem summae ponderum totius

systematis, unde demum pateat ejus ratio, quod passim sine demonstratione assumitur,

nimirum systemate quiescente, & impedito omni partium motu per equilibrium, totam

massam concipi posse ut in centro gravitatis collectam.

SYNOPSIS OF THE WHOLE WORK

27

I supply, & I bring forward a certain example of the same sort, that deals with the multi­ plication of numbers, & to the composition of forces by the parallelogram law ; the latter I prove by another more general method, analogous to that which I use in the general
investigation for the centre of gravity. Then by its help I prove very expeditiously & with extreme rigour that well-known theorem of Newton, in which he affirmed that the
state of the centre of gravity is in no way altered by the internal mutual forces. I gather several good results from this method of treatment. In Art. 264, the con- 264
servation of the same quantity of motion in the Universe in one plane ; in Art. 265 the 265 equality of action and reaction amongst masses; then the collision of solid bodies, and the 266
communication of motions in direct impacts & the laws that govern them, & from that, 276 in Art. 276, oblique impacts; in Art, 277 I reduce the theory of these from resolution of 277
motions to compositions, & in the article that follows, Art. 278, I pass to impact on to a 278 fixed plane; from Art. 279 to Art. 289 I show that there can be no real resolution of forces 279
or of motions in Nature, but only a hypothetical one; & in this connection I consider & explain all sorts of cases, in which at first sight it would seem that there must be resolution.
From Art. 289 to Art. 297,1 state the laws for the composition & resolution of forces; 289 here also I give the explanation of that well-known fact, that force decreases in composition, increases in resolution, but always remains equal to the sum of the parts acting in the same ‘ direction as itself in the first, the rest being equal & opposite cancel one another; whilst
in the second, al] 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 297
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 thoroughness 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 arc produced indefinitely on both sides. Then, as pj
far as Art. 321, I make out several other theorems dealing with the Ratios of these same resultant forces to one another ; such as the following very simple & elegant theorem, that
the accelerating forces of two of the masses will always be in a ratio compounded of three reciprocal ratios; namely, that of the distance of either one of them from the third mass, that of the sine of the angle which the direction of each force makes with the corresponding distance of this kind, & that of the mass itself on which the force is acting, to the corre­ sponding distance, sine and mass for the other : also that the motive forces only have the first two ratios, that of the masses being omitted.
I then collect the results to be derived from these theorems, deriving from them, as far ^2I
as Art. 328, theorems relating to the equilibrium of forces diverging in any manner, Sc 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, Sc 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 Sc some other assumed mass, Sc from this 1 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 Sc all motions of its parts are pro­ hibited by equilibrium.

28

SYNOPSIS TOTIUS OPERIS

328

A num. 328 ad 347 deduco ex iisdem theorematis, qux pertinent ad centrum oscilla­

tionis quotcunque massarum, sive sint in eadem recta, sire in plano perpendiculari ad axem

rotationis ubicunque, qux Theoria per systema quatuor massarum, excolendum aliquanto

diligentius, uberius promoveri deberet & extendi ad generalem habendum solidorum nexum,

344 qua re indicata, centrum itidem percussionis inde evolvo, & ejus analogiam cum centro

oscillationis exhibeo.

347

Collecto ejusmodi fructu ex theorematis pertinentibus ad massas tres, innuo num. 347,

qux mihi communia sint cum ceteris omnibus, & cum Newtonianis potissimum, pertinentia

ad summas virium, quas habet punctum, vel massa attracta, vel repulsa a punctis singulis

348 alterius massx ; tum a num. 348 ad finem hujus partis, sive ad num. 358, expono quxdam,

qux 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, prxter alias curvas, exponi posse per Logisticam,

& jn fluidis gravitate nostra terrestri prxditis pressiones haberi debere ut altitudines ;

deinde vero attingo illa etiam, qux pertinent ad velocitatem fluidi erumpentis evase,&

expono, quid requiratur, ut ea sit xqualis velocitati, qux acquiretur cadendo per altitudinem

ipsam, quemadmodum videtur res obtingere in aqux cflluxu: quibus partim expositis,

partim indicatis, hanc secundam partem concludo.

PARS III

358

Num. 358 propono argumentum hujus tertix partis, in qua omnes c Theoria mea

36o generales materix proprietates deduco, & particulares plcrasquc : tum usque ad num. 371

ago aliquanto fusius de impenetrabilitate, quam duplicis generis agnosco in meis punctorum

inextensorum massis, ubi etiam de ea apparenti quadam compcnctratione ago, ac de luminis

transitu per substantias intimas sine vera compenetrationc, & mira quxdam phxnomcna

3/1 huc.pertinentia explico admodum expedite. Inde ad num, 375 de extensione ago, qux

mihi quidem in materia, & corporibus non est continua, sed adhuc eadem prxbet phxno-

menx sensibus, ac in communi sententia ; ubi etiam de Geometria ago, qux vim suam in

375 mea Theoria retinet omnem : tum ad num. 383 figurabilitatem persequor, ac molem,

massam, densitatem singillatim, in quibus omnibus sunt quxdam Theorix mex propria

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

388 digna continentur : tum usque ad num. 391 ago de xqualitate actionis, & reactionis, cujus

consectaria vires ipsas, quibus Theoria mea innititur, mirum in modum confirmant.

391 Succedit usque ad num. 398 divisibilitas, quam ego ita admitto, ut quxvis massa existens

numerum punctorum realium habeat finitum tantummodo, sed qui in data quavis mole

possit esse utcunque magnus; guamobrem divisibilitati in infinitum vulgo admissx sub­

stituo componibilitatem in innnitum, ipsi, quod ad Naturx phxnomcna explicanda

398 pertinet, prorsus xquivalentem. His evolutis addo num. 398 immutabilitatem primorum

materix elementorum, qux cum mihi sint simplicia prorsus, & inextensa, sunt utique

immutabilia, & ad exhibendam perennem phxnomenorum seriem aptissima.

399

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

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

406 non coalescant, quod gravitas generalis requirere videretur. Inde ad num. 419 ago de

cohxsionc, qui est itidem voluti 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 fibrx ante fractionem distendantur, vel contrahantur, & innuo, qux ad

cohxsionem pertinentia mihi cum reliquis Philosophis communia sint.

419

A colixsione gradum facio num. 419 ad particulas, qux ex punctis cohxrentibus

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

SYNOPSIS OF THE WHOLE WORK

29

From Art. 328 to Art. 347, I deduce from these same theorems, others that relate to 328 the centre of oscillation of any number of masses, whether they are in the same right line, or anywhere in a plane perpendicular to the axis of rotation ; this theory wants to be worked somewhat more carefully with a system of four bodies, to be gone into more fully, & to
be extended so as to include the general case of a system of solid bodies; having stated this, I evolve from it the centre of percussion, & I show the analogy between it & the centre 344 of oscillation.
I obtain all such results from theorems relating to three masses. After that, in Art. 347 347, 1 intimate the matters in which I agree with all others, & especially with the followers of Newton, concerning sums of forces, acting on a point, or an attracted or repelled mass, due to the separate points of another mass. Then, from Art. 348 to the end of this part, 348 i.e., as far as Art. 359, I expound certain theorems that belong to the theory of fluids; & first of all, theorems with regard to pressure, in connection with which I mention that one which was proved by Newton, namely, that, if the compression of a fluid is proportional to the compressing force, then the repulsive forces between the points are in the reciprocal
ratio of the distances, & conversely. Moreover, I show that, if the same force is insen­ sible, then the matter can be represented by the logistic & other curves ; also that in fluids subject to our terrestrial gravity pressures should be found proportional to the depths. After that, I touch upon those things that relate to the velocity of a fluid issuing from a
vessel; & I show what is necessary in order that this should be equal to the velocity which would be acquired by falling through the depth itself, just as it is seen to happen in the case of an efflux of water. These things in some part being explained, & in some part merely indicated, I bring this second part to an end.

PART III
In Art. 358, I state the theme of this third part; in it I derive all the general & most 35g of the special, properties of matter from my Theory. Then, as far as Art. 371, I deal some- 360 what more at length with the subject of impenetrability, which I remark is of a twofold kind in my masses of non-extended points; in this connection also, I deal with a certain
apparent case of compcnetrability, & the passage of light through the innermost parts of bodies without real compenetration; I also explain in a very summary manner several striking phenomena relating to the above. From here on to Art. 375, I deal with exten- 371
sion; this in my opinion is not continuous either in matter or in solid bodies, & yet it yields the same phenomena to the senses as does the usually accepted idea of it; here I also deal with geometry, which conserves all its power under my Theory. Then, as far 375 as Art. 383, I discuss ngurability, 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 383 here on to Art. 388 ;" then, as far as Art. 391, I deal with the equality of action & reaction, 388
& 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. 391 398; this principle I admit only to the extent that any existing mass may be made up of a number of real points that arc finite only, although in any given mass this finite number may be as great as you please. Hence for infinite divisibility, as commonly accepted, I substitute infinite multiplicity; which comes to exactly the same thing, as far as it is concerned with the explanation of the phenomena of Nature. Having considered these subjects I add, in Art. 398, that of the immutability of the primary elements of matter; 398 according to my idea, these are quite simple in composition, of no extent, they are every­ where unchangeable, & hence arc splendidly adapted for explaining a continually recurring
set of phenomena. From Art. 399 to Art. 406,1 derive gravity from my Theory of forces, as if it were a 399
particular branch on a common trunk ; in this connection also I explain how it can happen
that the fixed stars do not all coalesce into one mass, as would seem to be required under 406 universal gravitation. Then, as far as Art. 419, I deal with cohesion, which is also as it were another branch; I show that this is not dependent upon quiescence, nor on motion, that is the same for all parts, nor on the pressure of some fluid, nor on the idea that the attraction is greatest at actual contact, but on the limits between repulsion and attraction.
I propose, & solve, a general problem relating to this, namely, why masses, once broken, do not again stick together, why the fibres are stretched or contracted before fracture takes place; & I intimate which of my ideas relative to cohesion are the same as those held by other philosophers.
In Art. 419,1 pass on from cohesion to particles which are formed from a number of 4*9 cohering points; & I consider these as far as Art. 426, & investigate the various distinctions

jo

SYNOPSIS TOTIUS OPERIS

ostendo nimirum, quo pacto varias induere possint figuras quascunque, quarum tenacissime sint; possint autem data quavis figura discrepare plurimum in numero, & distributione
punctorum, unde & oriantur admodum inter se diversae vires unius particul® in aliam, ac itidem divers® in diversis partibus ejusdem particul® respectu diversarum partium, vel etiam respectu ejusdem partis particul® alterius, cum a solo numero, & distributione
punctorum pendeat illud, ut data particula datam aliam in datis earum distantiis, & superficiorum 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, quxeunque punctorum dispositio habeatur, & in aliis proprietatibus plerisquc debere esse admodum (uti observamus) diversas, qu® diversitas multo major in majoribus massis esse debeat.

A num. 426 ad 446 de solidis, & fluidis, quod discrimen itidem pertinet ad varia coh®sionum 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 ® quili bus ; illorum vero ex inaequalitate virium, & viribus quibusdam in latus, quibus certam positionem ad se invicem servare debeant. Varia autem distinguo fluidorum genera, & discrimen profero inter virgas rigidas, flexiles, elasticas, fragiles, ut & de viscositate, & humiditate ago, ac de organicis, & ad certas figuras determinatis corporibus, quorum efformatio nullam habet difficultatem, ubi una particula unam aliam possit in certis tantummodo superficiei partibus attrahere, & proinde cogere ad certam quandam positionem acquirendam respectu ipsius, & retinendam. Demonstro autem & illud, posse admodum facile ex certis particularum figuris, quarum ips® tcnacissim® sint, totum etiam Atomistarum, & Corpuscularium systema a mea Theoria repeti ita, ut id nihil sit aliud,
nisi unicus itidem hujus veluti trunci foecundissimi ramus e diversa cohxsionis ratione prorumpens. Demum ostendo, cur non qu®vis massa, utut constans ex homogeneis punctis, & circa se maxime in gyrum mobilibus, fluida sit; & fluidorum resistentiam quoque attingo, in ejus leges inquirens.

446

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

dc 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 ; h®c a limitum frequentia, atque ingenti vicinia, qua fiat, ut ex

uno ad alium delata limitem puncta, ibi quiescant itidem respective, ut prius. Tum vero

dc ductilibus, & mallcabilibus ago, ostendens, in quo a fragilibus discrepent : ostendo autem,

h®c omnia discrimina a densitate nullo modo pendere, ut nimirum corpus, quod

multo sit altero densius, possit tam multo majorem, quam multo minorem soliditatem, &

coh®sioncm habere, & qu®vis ex proprietatibus expositis ®que possit cum quavis vel majore,

vel minore densitate componi.

450

Num. 450 inquiro in vulgaria quatuor elementa ; tum a num. 451 ad num. 467 persequor

452 chemicas operationes ; num. 452 explicans dissolutionem, 453 pr®cipitationem, 454, & 455

commixtionem plurium substantiarum in unam : tum num. 456, & 457 liquationem binis

methodis, 458 volatilizationem, & effervescentiam, 461 emissionem effluviorum, qu® c massa

constanti debeat esse ad sensum constans, 462 ebullitionem cum variis evaporationum

generibus; 463 deflagrationem, & generationem aeris; 464 crystallizationem cum certis

figuris; ac demum ostendo illud num. 465, quo pacto possit fermentatio desinere ; & num.

466, quo pacto non omnia fermentescant cum omnibus.

467

A fermentatione num. 467 gradum facio ad ignem, qui mihi est fermentatio qu®dam

substanti® lucis cum sulphurea quadam substantia, ac plura inde consectaria deduco usque

471 ad num. 471 ; tum ab igne ad lumen ibidem transeo, cujus proprietates pnecipuas, ex

472 quibus omnia lucis ph®nomena oriuntur, propono num. 472, ac singulas a Theoria mea

deduco, & fuse explico usque ad num. 503, nimirum emissionem num. 473, celeritatem 474,

propagationem rcctilincam per media homogenea, & apparentem tantummodo compene-

trationem a num. 475 ad 483, pcllucidatcm, & opacitatem num. 483, reflexionem ad angulos

squales inde ad 484, refractionem ad 487, tenuitatem num. 487, calorem, & ingentes

intestinos motus allapsu tenuissim® lucis genitos, num. 488, actionem majorem corporum

oleosorum, & sulphurosorum in lumen num. 489 : tum num. 490 ostendo, nullam resist-

SYNOPSIS OF THE WHOLE WORK

3’

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 paints,
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 (wnich we observe to be the
case); & that this difference ought to be much greater in larger masses. From Art. 426 to Art. 446,1 consider solids & fluids, the difference between which is 42g
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 Corpuscularians, can be quite easily derived by my Theory, from the idea of particles of definite
shape, offering a high resistance to deformation; so that it comes to nothing else than another single branch of this so to speak most fertile trunk, breaking forth from it
on account of a different manner of cohesion. Lastly, I show the reason why it is that
not every mass, in spite of its being constantly made up of homogeneous points, & even these in a high degree capable of rotary motion about one another, is a fluid. I also touch
upon the resistance of fluids, & investigate the laws that govern it. From Art. 446 to Art. 450, I deal with those things that relate to the different kinds 446
of solidity, that is to say, with elastic bodies, & those that are soft. I attribute the nature
of the former to the existence of a large interval between the consecutive limits, on account of which it comes about that points that are far removed from their natural positions still feel the effects of the same kind of forces, & therefore return to their natural positions;
& that of the latter to the frequency & great closeness of the limits, on account of which it comes about that points that have been moved from one limit to another, remain there in relative rest as they were to start with. Then I deal with ductile and malleable solids,
pointing out how they differ from fragile solids. Moreover I show' that all these differ­
ences arc in no way dependent on density; so that, for instance, a body that is much more dense than another body may have either a much greater or a much less solidity and cohesion than another; in fact, any of the properties set forth may just as well be combined
with any density either greater or less. In Art. 450 I consider what are commonly called the “ four elements ” ; then from 450
Art. 451 to Art. 467, I treat of chemical operations; I explain solution in Art. 452, preci- 452 pitation in Art. 453, the mixture of several substances to form a single mass in Art. 454,
455> liquefaction by two methods in Art. 456, 457, volatilization & effervescence in Art.
458, emission of effluvia (which from a constant mass ought to be approximately constant) in.Art. 461, ebullition & various kinds of evaporation in Art. 462, deflagration & generation of gas in Art. 463, crystallization with definite forms of crystals in Art. 464 ; & lastly, I show, in Art. 465, how it is possible for fermentation to cease, & in Art. 466, how it is that any one thing does not ferment when mixed with any other thing.
From fermentation I pass on, in Art. 467, to fire, which I look upon as a fermentation 467
of some substance in light with some sulphureal substance; & from this I deduce several propositions, up to Art. 471. There I pass on from fire to light, the chief properties of 471 which, from which all the phenomena of light arise, I set forth in Art. 472 ; & I deduce 472 & fully explain each of them in turn as far as Art. 503. Thus, emission in Art. 473, velo­
city in Art. 474, rectilinear propagation in homogeneous media, & a campenetration that is merely apparent, from Art. 475 on to Art. 483, pellucidity & opacity in Art. 483, reflec­ tion at equal angles to Art. 484, & refraction to Art. 487, tenuity in Art. 487, heat & the
great internal motions arising from the smooth passage of the extremely tenuous light in Art. 488, the greater action of oleose & sulphurous bodies on light in Art. 489. Then I

32

SYNOPSIS TOTIUS OPERIS

entiam veram pati, ac num. 401 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 aequalia redeuntes intervalla, unde num. 497 vices illas a Ncwtono detectas facilioris reflexionis, & facilioris
transmissus eruo, & num. 498 illud, radios alios debere reflecti, alios transmitti in appulsu ad novum medium, & eo plurcs reflecti, quo obliquitas incidentiae sit major, ac num.
499 & 500 expono, unde discrimen in intervallis vicium, ex quo uno omnis naturalium colorum pendet Newtoniana Theoria. Demum num. 501 miram attingo crystalli Islandica: proprietatem, & ejusdem causam, ac num. 502 diffractioncm expono, quae est quaedam inchoata refractio, sive reflexio.

503

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

504 sapore, & odore, ac sequentibus tribus numeris de sono: tum aliis quator de tactu, ubi

507 etiam de frigore, & calore: deinde vero usque ad num. 514 de clectricitate, ubi totam

511 Franklinianam Theoriam ex meis principiis explico, eandem ad bina tantummodo reducens

principia, quae ex mea generali virium Theoria eodem fere pacto deducuntur, quo praecipi-

514 tationes, atque dissolutiones. Demum num. 514, ac 515 magnetismum persequor, tam

directionem explicans, quam attractionem magneticam.

516

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

ad finem usque generalem corporum complector naturam, & quid materia sit, quid forma,

qpise censeri debeant essentialia, quae accidentialia attributa, adeoque quid transformatio

sit, quid alteratio, singillatim persequor, & partem hanc tertiam Theoria: meae absolvo.

De Appendice ad Mctaphysicam pertinente innuam hic illud tantummodo, me ibi exponere de anima illud inpnmis, quantum spiritus a materia differat, quem nexum anima haoeat cum corpore, & quomodo in ipsum agat: tum de Deo, ipsius & existentiam me pluribus evincere, qu$ nexum habeant cum ipsa Theoria mea, & Sapientiam inprimis, ac Providentiam, ex qua gradum ad revelationem faciendum innuo tantummodo. Sed haec
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 5°3 next four articles, I consider touch, & in connection with it, cold & heat also. After that, 5°4as far as Art. $14, I deal with electricity ; here I explain the whole of the Franklin theory 5°7
by means of my principles; I reduce this theory to two principles only, & these are 51* derived from iny general 1'heory of forces in almost the same manner as I have already derived
precipitations & solutions. Finally, in Art. 514, 515, I investigate magnetism, explaining 51! both magnetic direction 8c attraction.
These things being expounded, all of which relate to special properties, I once more consider, in the articles from 516 to the end, the general nature of bodies, what matter is, 516 its form, what things ought to be considered as essential, & what as accidental, attributes; and also the nature of transformation and alteration are investigated, each in turn; & thus I bring to a close the third part of my Theory.
I will mention here but this one thing with regard to the appendix on Metaphysics; namely, that I there expound more especially how greatly different is the soul from matter, the connection between the soul & the body, & the manner of its action upon it. Then with regard to God, I prove that He must exist by many arguments that have a close con­ nection with this Theory of mine ; I especially mention, though but slightly, His Wisdom and Providence, from which there is but a step to be made towards revelation. But I think that I have, so to speak, given my preliminary foretaste quite sufficiently.

[,] PHILOSOPHIZE NATURALIS THEORIA

PARS I

Theorize expositio, analytica deductio, & vindicatio.

Cujuamodi systema*

IRIUM mutuarum Theoria, in quam incidi jam ab Anno 174.5, dum e

Theoria exhibeat.

notissimis principiis alia ex aliis consectaria eruerem, & ex qua ipsam

simplicium materiae elementorum constitutionem deduxi, systema

exhibet medium inter Leibnitianum, & Newtonianum, quod nimirum

& ex utroque habet plurimum, & ab utroque plurimum dissidet; at

utroque in immensum simplicius, proprietatibus corporum generalibus

sane omnibus, & [2] peculiaribus quibusque praecipuis per accuratissimas

demonstrationes deducendis est profecto mirum in modum idoneum.

In quo conveniat
cum systemate Newtoniano, & Leibnitiano.

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

In qua differat a

3. Distat autem a Leibnitiana Theoria longissime, tum quia nullam extensionem

prastet^”0 & ,pS' continuam admittit, quae 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 Naturse analogia

ducit, ac chemics resolutiones inprimis, tn 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 dcpraedicata, meo quidem judicio,

nihil omnino possunt.

in quo differat a

a Distat itidem a Newtoniano systemate quamplurimum, tum in eo, quod ea, qute

pristet.

Newtonus in ipsa postremo Quiestione Opticse conatus est explicare per tria principia, gravitatis, cohxsionis, fermentationis, immo & reliqua quamplurima, quae ab iis tribus

principiis omnino non pendent, per unicam explicat legem virium, expressam unica, & ex

pluribus inter se commixtis non composita algebraica formula, vel unica continua geometrica

curva : 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

HE following Theory of mutual forces, which I lit upon as far back as the year The kind of sys­

1745,

whilst

I

was

studying

various

propositions

arising

from

other

very

tem the presents.

Theory

well-known principles, & from whicn I have derived the very constitu­

tion of the simple elements of matter, presents a system that is midway

between that of Leibniz & that of Newton ; it has very much in common

with both, & differs very much from either ; &, as it is immensely more

simple than either, it is undoubtedly suitable in a marvellous degree for

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

means of the most rigdrous demonstrations.

2. It indeed holds to those simple & perfectly non-extended primary elements upon What there is in

which is founded the distances of the points

theory of from one

Leibniz; & also to the mutual forces, which vary as the another vary, the characteristic of the theory of Newton;

it common to the systems of New­
ton <ft Leibniz,

in addition, it deals not only with the kind of forces, employed by Newton, which oblige

the points to approach one another, & are commonly called attractions; but also it

considers forces of a kind that engender recession, & are called repulsions. Further, the

idea is introduced in such a manner that, where attraction ends, there, with a change of

distance, repulsion begins; this idea, as a matter of fact, was suggested by Newton in the

last of his ‘ Questions on Optics & he illustrated it by the example of the passage from

positive to negative, as used in algebraical 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, How it differs from, because it docs not admit the continuous extension that arises from the idea of consecutive, thco^oTLcihniz.

non-extended points touching one another; here, the difficulty raised in times gone by in

opposition to zicno, & 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

indisccrnibles, & 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, Haw it differs from,

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 formulas compounded together ; or

by a single continuous geometrical curve. For another thing, it admits forces that at very

small distances arc not positive or attractive, as Newton supposed, but negative or repul-

35

36

PHILOSOPHIA NATURALIS THEORIA

infinitum, quo distantia: in infinitum decrescant. Unde illud necessario consequitur, ut nec cohaesio a contactu immediato oriatur, quam ego quidem longe aliunde desumo; nec ullus
immediatus, &, ut illum appellare soleo, mathematicus materize contactus habeatur, quod simplicitatem, & inextensionem inducit elementorum, qua: ipse variarum figurarum voluit, & partibus a se invicem distinctis composita, quamvis ita cohsrentia, ut nulla Natura: vi dissolvi possit compages, & adhaesio labefactari, qux adhesio ipsi, respectu virium nobis cognitarum, est absolute infinita.

Ubi de ipsa ctum

5. Quae ad ejusmodi Theoriam pertinentia hucusque sunt edita, continentur disserta­

ante ; & missum.

quid

pro­

tionibus A. 1754,

meis, De viribus vivis, edita Anno 1745, De Lumine A. 1748, De Lege De Lege virium in natura existentium A. 1755, &e divisibilityte materia,

Continuitatis W principiis

corporum A. 1757, ac in meis Supplementis Stayanx Philosophiae versibus traditae, cujus primus

Tomus prodiit A- 1755 : eadem autem satis dilucide proposuit, & amplissimum ipsius per

omnem Physicam demonstravit usum vir e nostra Societate doctissimus Carolus Bcnvenutus

in sua Physica Generalis Synopsi edita Anno 1754. In ea Synopsi proposuit idem & meam

deductionem zequilibrii binarum massarum, viribus parallelis animatarum, qua: ex ipsa mea

Theoria per notissimam legem compositionis virium, & zequalitatis inter actionem, & reac­

tionem, fere sponte consequitur, cujus quidem in supplementis illis § 4. ad lib. 3. mentionem

feci, ubi & qux in dissertatione De centro Gravitatis edideram, paucis proposui; & de centro

oscillationis agens, protuli aliorum methodos pracipuas quasque, qua ipsius determinationem

a subsidiariis tantummodo principiis quibusdam repetunt. Ibidem autem de aequilibrii

centra agens illud affirmavi : In Natura nulla sunt rigida virga, inflexiles, Lf omni gravitate, ac inertia carentes, adeoque nec revera ulla leges pro iis condita ; W si ad genuina, & simpli­

cissima natura principia, res exigatur, invenietur, omnia pendere a compositione virium, quibus in

se invicem agunt particula materia ; a quibus nimirum viribus omnia Natura phanomena

proficiscuntur. Ibidem autem exhibitis aliorum methodis ad centrum oscillationis perti­

nentibus, promisi, me in quarto ejusdem Philosophia tomo ex genuinis principiis investiga­

turum, ut aequilibrii, sic itidem oscillationis centrum.

Qua occasione hoc

6. Porro cum nuper occasio se mihi praebuisset inquirendi in ipsum oscillationis centrum

de ipsa conscrip­ tum opus.

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 & aequilibrium, 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 Theoria meze 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 & qua: maxima

notatu digna erant, in memoratis dissertationibus ordine suo digessi omnia, & alia adjeci

quamplurima, qus vel olim animo conceperam, vel modo sese obtulerunt scribenti, ftomnem

hanc rerum farraginem animo pervolventi.

Prima elementa in­ 7. Prima elementa materia: mihi sunt puncta prorsus indivisibilia, & inextensa, quae in

divisibilia ines ten­ sa, nec contigua.

immenso vacuo ita dispersa sunt, quod quidem indefinite augeri

ut bina potest,

quavis a se invicem distent per aliquod & minui, sed penitus evanescere non

intervallum, potest, sine

conpenctrationc ipsorum punctorum : eorum enim contiguitatem nullam admitto possi­

bilem ; sed illud arbitror omnino certum, si distantia duorum materia: 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 arc concerned, is in his opinion unlimited.

5. What has already been published relating to this kind of Theory is contained in my When & where I

dissertations,

De

Viribus

vivis,

issued

in

1745,

De

Lumine,

1748,

De

Lege

Continuitatis,

have already dealt with this theory ;

1754, De Lege virium in natura existentium, 1755, De divisibilitate materice, W principiis * a promise that I

corporum, 1757, & *n m7 Supplements to the philosophy of Benedictus Stay, issued in verse, made.

of which the first volume was published in 1755. The same theory was set forth with

considerable lucidity, & its extremely wide utility in the matter of the whole of Physics

was demonstrated, by a learned mcmocr 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 tne 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 1 had published in my dissertation De

centro 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 :—

“Zw Nature there are no rods that are rigid, inflexible, totally devoid of weight & inertia;

W so, neither are there really any laws founded on them. If the matter is worked back to the genuine W simplest natural principles, it will be found that everything depends on the com­ position of the forces with which tne particles of matter act upon one another ; W from these

very forces, as a matter of fact, all phenomena of Nature take their origin.” Moreover, here
too, having stated the methods of others for the determination of the centre of oscillation, I promised that, in the fourth volume of the Philosophy, I would investigate by means of

genuine principles, such as I had used for the centre of equilibrium, the centre of oscillation as well

6. Now, lately I had occasion to investigate this centre of oscillation, deriving it from The occasion that

my own principles, at the request of Father Scherffer, a man of much learning, who teaches mathematics in this College of the Society. Whilst doing this, I happened to hit upon a

led to my
this work matter.

writing on the

really most simple & truly elegant theorem, from which the forces with which three

masses mutually act upon one another arc 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 csssay 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 arc in my opinion perfectly indivisible & non­ The primary ele­

extended

points;

they are so

scattered

in

an

immense vacuum

that

every two

of

them arc

ments are indivi­ sible, non extended

separated from one another by a definite interval; this interval can be indefinitely & they are not

increased or diminished, but can never vanish altogether without compenetration of the contiguous.

points themselves; for I do not admit as possible any immediate contact between them.

On the contrary I consider that it is a certainty that, if the distance between two points of matter should become absolutely nothing, then the very same indivisible point of space, according to the usual idea of it, must be occupied by both together, & we have true

38

PHILOSOPHIC NATURALIS

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

Eorum Inertiae vis cujusmodi.

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, prsecedentcm motum
cum mo-[5]-tu quem determinant vires mutusc, quas inter bina quaevis puncta agnosco a distantiis pendentes, & iis mutatis mutatas, juxta generalem quandam omnibus com­ munem legem. In ea determinatione stat illa, quam dicimus, inertiae vis, quae, an a libera pendeat Supremi Conditoris lege, an ab ipsa punctorum natura, an ab aliquo iis
adjecto, quodcunque, istud sit, ego quidem non quaero ; nec vero, si velim quaerere, in­ veniendi spem habeo ; quod idem sane censeo de ea virium lege, ad quam gradum jam facio.

Eorundem vires

9. Censeo igitur bina quaecunque materiae puncta determinari aeque in aliis distantiis

mutum i n aliis distantiis attrac­

ad

mutuum

accessum,

in

aliis

ad

recessum

mutuum,

quam

ipsam

determinationem

appello

tive, in aliis re­ vim, in priore casu attractivam, in posteriore rcpulsivam, eo nomine non agendi modum, sed

pulsive : ejusmodi

virium exempla.

ipsam

determinationem

exprimens,

undecunque

proveniat,

cujus

vero

magnitudo

mutatis

distantiis mutetur & ipsa secundum certam legem quandam, quae per geometricam lineam

curvam, vel algcbraicam formulam exponi possit, & oculis ipsis, uti moris est apud Mechanicos

repraesentari. Vis mutuae a distantia pendentis, & ea variata itidem variatae, atque ad omnes

in immensum & magnas, & parvas distantias pertinentis, habemus exemplum in ipsa

Newtoniana generali gravitate mutata In ratione reciproca duplicata distantiarum, qu®

idcirco numquam c positiva in negativam migrare potest, adeoque ab attractiva ad repul-

sivam, sive a determinatione ad accessum ad determinationem aa recessum nusquam migrat.

Verum in elastris inflexis habemus etiam imaginem ejusmodi vis mutuae variatae secundum

distantias, & a determinatione ad recessum migrantis in determinationem ad accessum, &

vice versa. Ibi enim si duae cuspides, compresso elastro, ad se invicem accedant, acquirunt

determinationem ad recessum, eo majorem, quo magis, compresso elastro, distantia

decrescit; aucta distantia cuspidum, vis ad recessum minuitur, donec in quadam distantia evanescat, & fiat prorsus nulla ; tum distantia adhuc aucta, incipit determinatio ad accessum,

quae perpetuo eo magis crescit, quo magis cuspides a se invicem recedunt: ac si e contrario

cuspidum distantia minuatur perpetuo; determinatio ad accessum itidem minuetur,

evanescet, & in determinationem ad recessum mutabitur. Ea determinatio oritur utique

non ab immediata cuspidum actione in se invicem, sed a natura, & forma totius intermediae laminae plicata:; sed hic physicam rei causam non moror, & solum persequor exemplum

determinationis ad accessum, & recessum, qua determinatio in aliis distantiis alium habeat

nisum, & migret etiam ab altera in alteram.

Virium earundem

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

lex.

majores in infinitum, quo distantiae ipsae minuuntur in infinitum, ita, ut pares sint extinguen-

[6]-d$ cuivis velocitati utcunque magna, 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 attractive, & ad sensum reciproce

(■) Id quidem respectu ejus /patii, in quo continemur nos, U omnia qux 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 ex hujusmodi determinatione sequi debeant ipsa materix puncta ; tum hxc mea erit quxdam non absoluta, sed respectiva inertix vis, quam ego quidem exposui W in dissertatione De Miris zstu W in Supplementis Stayanis Lib. I. § 1J ; ubi etiam illud occurrit, quam ob causam ejusmodi respectivam inertiam excogitarim, & quibus rationibus evinci putem, absolutam omnino demonstrari non posse i sed ea huc non pertinent.

A THEORY OF NATURAL PHILOSOPHY

39

compcnetration 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 same state

an of

attribute of these points I rest, or of uniform motion

admit an inherent propensity in a straight line, (a) in which

to remain in the they are initially

The nature of the
force of inertia that they possess.

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 arc subject to a determination The mutual forces

to approach one another at some distances, & in an equal degree recede from one another at

between them are attractive at some

other distances. This determination I call ‘force’; in the first case ‘attractive’, in the distances tc repul­

second case ‘ repulsive ’;

this term does not denote the mode of

action, but

the

propen­

sive at others; ex­ amples of forces of

sity itself, whatever its origin, of which the magnitude changes as the distances change; this kind.

this is in accordance with a certain definite law, which can be represented by a geometrical

curve or by an algebraical formula, & visualized in the manner customary with Mechanicians.

We have an example of a force dependent on distance, & varying with varying distance, &

pertaining to all distances cither 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 tnat 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.

to. Now the law of forces is of this kind tances, & become indefinitely greater & greater,

; the forces are as the distances

repulsive at very small dis­ arc diminished indefinitely,

The law of forces for the points.

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*1 are all still very^ minute : until, finally, when

we get to comparatively great distances, they begin to be continually attractive & approxi-

(a] Tbit indeed bolds true for that iface 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 soma 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 ; as indeed I explained both in the dissertation De Maris Aestu, and also in the Supplementi to Stay's Philosophy, boob J, section IJ. 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 al present.

4°

PHILOSOPHIC NATURALIS THEORIA

proportionales quadratis distantiarum, atque id vel utcunque augeantur distantia: etiam in

infinitum, vel saltem donec ad distantias deveniatur omnibus Planetarum, & Cometarum

distantiis longe majores.

Legi» limpiidua

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

corJam' temere inter se coagmentatis coalescens; at simplicissima, & prorsus incomposita esse potest,

expressa videlicet per unicam continuam curvam, vd simplicem Algcbraicam formulam, uti

innui superius. Hujusmodi curva linea est admodum apta ad sistendam oculis ipsis ejusmodi

legem, nec requirit Geometram, ut id prxstare possit: satis est, ut quis eam intueatur

tantummodo, Se in ipsa ut in imagine quadam solemus intueri depictas res qualescunque,

virium illarum indolem contempletur. In ejusmodi curva eae, quas Geometra: abscissas

dicunt, & sunt segmenta axis, ad quem ipsa refertur curva, exprimunt distantias binorum

punctorum a se invicem: illa: vero, qua: dicuntur ordinata:, ac sunt perpendiculares linea:

ab axe ad curvam ducta:, referunt vires: qua: quidem, ubi ad alteram jacent axis partem,

exhibent vires attractivas; ubi jacent ad alteram, rcpulsivas, & prout curva accedit ad axem,

vel recedit, minuuntur ipsa: etiam, vel augentur: ubi curva axem secat, & ab altera ejus

parte transit ad alteram, mutantibus directionem ordinatis, abeunt ex positivis in negativas,

vd vice versa : ubi autem arcus curvx 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 Geometra:, ibi vires ipsx in infinitum

excrescunt.

Forma curv* ips­

12. Ejusmodi curvam exhibui, & exposui in dissertationibus De viribus vivis a Num. 51,

ius.

De Lumine Num. 5, De Lege virium in Naturam existentium a Num. 68, & in sua Synopsi

Physica Generalis P. Benvenutus eandem protulit a Num. 108. En brevem quandemejus

ideam. In Fig. 1, Axis C'AC habet in puncto A asymptotum curvx rectilincam AB

indefinitam, circa quam habentur bini curvx rami hinc, & inde xquales, prorsus inter se, &

similes, quorum alter DEFGH1KLMNOPQRSTV 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 Vab eadem recedunt arcus reliqui omnes, quin uspiam recessus mutetur inaccessum.

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 oesinit in alterum

crus TprV, jacens ex parte opposita axis respectu primi cruris, quod alterum crus ipsum

habet axem pro asymptoto, & ad ipsum acccait ad sensum ita, ut distantix ab ipso sint in

ratione reciproca duplicata distantiarum a recta BA.

Abociu» exprunen-

13. Si ex quovis axis puncto o, b, d, erigatur usque ad curvam recta ipsi perpendicularis

dta»tznUuprimen*

5egmentura ax^s Au, Ab, Ad, dicitur abscissa, & refert distantiam duorum materix

te» vires.

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.

Natationes ordinatarum, virium iis expressarum.

14. Patet autem, in ea curvx forma ordinatam ag augeri abscissa Ac, minuatur pariter ultra quoscunque limites ; qux

ultra quoscunque limites, si 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 mn ad illam priorem, donec post evanescentiam, & directionis

mutationem factam in omnibus sectionibus I, L, N, P, R, fiant ordinatx 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

PHILOSOPHISE NATURALIS THEORIA

A THEORY OF NATURAL PHILOSOPHY

43

matcly inversely proportional to the squares of the distances. This holds good as the

distances are increased indefinitely to any extent, or at any rate until we get to distances

that are far greater than all the distances of the planets & comets.

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

of combining together several different laws in a haphazard sort of way; but it can be if the law can tie re

the simplest kind & not complicated in the slightest degree;

it

can

be

represented

for

presented by means of a continuous

instance by a single continuous curve, or by an algebraical formula, as I intimated above. curve.

A curve of this sort is perfectly adapted to the graphical representation of this sort of law,

& it does not require a knowledge of geometry to set it forth. It is sufficient for anyone

merely to glance at it, & in it, just as in a picture we are accustomed to view all manner of

things depicted, so will he perceive the nature of these forces. In a curve of this kind,

those lines, that geometricians call abscissae, namely, segments of the axis to which the

curve is referred, represent the distances of two points from one another; & those, which

we called ordinates, namely, lines drawn perpendicular to the axis to meet the curve, repre­

sent forces. These, when they lie on one side of the axis represent attractive forces, and,

when they lie on the other side, repulsive forces; & according as the curve approaches the

axis or recedes from it, they too are diminished or increased. When the curve cuts the

axis & passes from one side of it to the other, the direction of the ordinates being changed

in consequence, the forces pass from positive to negative or vice versa. When any arc of

the curve approaches ever more closely to some straight line perpendicular to the axis and

indefinitely produced, in such a manner that, even if this goes on beyond all limits, yet

the curve never quite reaches the line (such an arc is called asymptotic by geometricians),

then the forces themselves will increase indefinitely.

12. I set forth and explained a curve of this sort in my dissertations De Viribus vivis The form of the (Art. Jl), De Lumine (Art. 5), De lege virium in Natura existentium (Art. 68) ; and Father curve.

Bcnvenutus published the same thing in his Synopsis Physica Generalis (Art. 108). This

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

In Fig. 1 the axis C'AC has at the point A a straight line AB perpendicular to itself,

which is an asymptote to the curve ; there are two branches of the curve, one on each side

of AB. which are equal & similar to one another in every way. Of these, one, namely

DEFGH1KLMN0PQRSTV, 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 TpjV, lying on the opposite side of the axis with

respect to the first branch; and this second branch has the axis itself as its asymptote,

& approaches it approximately in such a manner that the distances from the axis are in

the inverse ratio of the squares of the distances from the straight line AB.

13. If from any point of the axis, such as a, b, or d, there is erected a straight line per­ The abscissa re­

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

present <fl the forces.

distances. ordinates

one another; the perpendicular, ag, br, or dh, is called the ordinate, Sc this represents the

force, which is repulsive or attractive, according as the ordinate lies with regard to the

axis on the side towards D, or on the opposite side.

14. Now it is clear that, in a curve of this form, the ordinate ag will be increased Change in the or­

beyond all bounds, if the abscissa Aa is in the same way diminished beyond all bounds; Sc if the latter is increased and becomes Ab, the ordinate will be diminished, Sc it will become

dinates ft the forces that they represent-

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, Sc 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 abscissa Ao, Av. Hence it is perfectly evident that, by a curve of this kind, we can

44

PHILOSOPHISE NATURALIS THEORIA

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

Discrimen hu us legis virium a gravitate N e w-

15. Hrec virium lex a exprimentis quae nimirum,

Ncwtoniana gravitate differt in ductu, & progressu curva: eam ut in fig. 2, apud Newtonum est hyperbola DV gradus tertii,

toniana .* ejus usus jacens tota citra axem, quem nuspiam

in Physica: ordo pertractandorum.

secat, jacentibus omni-[8]-bus ordinatis

B

w, op, bt, ag ex parte attractiva, ut

idcirco nulla habeatur mutatio e positivo

A

ad negativum, ex attractione in repulsi­

onem, vel vice versa ; cacterum utraque

per ductum exponitur curvae continua:

s

habentis duo crura infinita asymptotica

in ramis singulis utrinque in infinitum

firoductis. Ex hujusmodi autem virium

ege, & ex solis principiis Mechanicis

notissimis, nimirum quod ex pluribus

viribus, vel motibus componatur vis, vel

motus quidam ope parallclogrammorum,

quorum latera exprimant vires, vel mo­

F1C 2.

tus componentes, & quod vires ejusmodi

in punctis singulis, tempusculis singulis aequalibus, inducant velocitates, vel motus proportion­

ales sibi, omnes mihi profluunt generales, & praecipua: quaeque particulares proprietates cor­

porum,uti etiam supenus innui, nec ad singulares proprietates derivandas in genere affirmo, eas

haberi per diversam combinationem, sed combinationes ipsas evolvo, & geometrice demon­

stro, quae e quibus combinationibus phaenomena, & corporum species oriri debeant. Verum

antequam ea evolvo in parte secunda, & tertia, ostendam in hac prima, qua via, & quibus

positivis rationibus ad cam virium legem devenerim, & qua ratione illam elementorum

materia: simplicitatem eruerim, tum quae difficultatem aliquam videantur habere posse,

dissolvam.

Occasio invenienda

16. Cum anno 1745 Ciribus vivis dissertationem conscriberem, & omnia, qua: a

Theorue eratione

ex consid­ impulsus.

viribus

vivis

repetunt,

qui

Leibnitianam

tuentur

sententiam, & vero etiam

pierique

ex iis,

qui per solam velocitatem vires vivas metiuntur, repeterem immediate a sola velocitate

genita per potentiarum vires, qua: juxta communem omnium Mechanicorum sententiam

velocitates vel generant, vel utcunque inducunt proportionales sibi, & tempusculis, quibus

agunt, uti est gravitas, elasticitas, atque aliae vires ejusmodi; coepi aliquanto diligentius

inquirere in eam productionem velocitatis, quae per impulsum censetur fieri, ubi rota

velocitas momento temporis produci creditur ab iis, qui idcirco percussionis vim infinities

majorem esse censent viribus omnibus, quae pressionem solam momentis singulis exercent.

Statim illud mihi sese obtulit, alias pro percussionibus ejusmodi, quae nimirum momento

temporis finitam velocitatem inducant, actionum leges haberi debere.

Origo ejusdem ex

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

sus^miMdUri^um eum agendi modum submovendum esse a Natura, quae nimirum eandem ubique virium

lega Continuit&tis. legem, ac eandem agendi rationem adhibeat; impulsum nimirum immediatum alterius

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

finita: velocitatis facta momento temporis indivisibili, & hanc sine saltu quodam, & laesione

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

satis [9] 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.

Laesio legis Continu­

18. Concipiantur duo corpora aequalia, qua: moveantur in directum versus eandem

itatis necessaria, sl corpus velocius im­
mediate incurrat in

plagam, & gradus 12.

id, quod praecedit, habeat gradus velocitatis Si hoc posterius cum sua illa velocitate illaesa

6, id vero, quod ipsum persequitur deveniat ad immediatum contactum

minus velox

cum illo priore ; oportebit utique, ut ipso momento temporis, quo ad contactum devenerint,

illud posterius minuat velocitatem suam, & illud primus suam augeat, utrumque per saltum,

abeunte hoc a 12 ad 9, illo a 6 ad 9, sine ullo transitu per intermedios gradus 11, & 7 ; 10, &

8 ; 94, & 8i, &c. Neque enim fieri potest, ut per aliquam utcunque exiguam continui

A THEORY OF NATURAL PHILOSOPHY

45

represent the forces in question, which are initially repulsive & increase indefinitely as the

distances are diminished indefinitely, but which, as the distances increase, are first of all

diminished, then vanish, then become changed in direction & so attractive, again vanish,

& change their direction, & so on alternately ; until at length, at a distance comparatively

great they finally become attractive & are sensibly proportional to the inverse squares of

the distance. K. This law of forces differs from the law of gravitation enunciated by Newton in JJ.ffc™nco between

the construction & development of the curve that represents it; thus, the curve given in’ & Newton's law of Fig. 2, which is that according to Newton, is DV, a hyperbola of the third degree, lying ^v,t^1“p^yajCts a

altogether on one side of the axis, which it does not cut at any point ; all the ordinates, the order in which

such as vs, op, bt, ag lie on the side of the axis representing attractive forces, & there- ^et^enCCtS are t0

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

Eroduced to infinity on both sides. Now, from a law of forces of this kind, & with the

elp 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.

l6. In the year 174.5, I wa3 Putting together my dissertation De Viribus vivis, & had The occasion that derived everything that they who adhere to the idea of Leibniz, & the greater number of QHny^eorj^from

those who measure ‘ living forces ’ by means of velocity only, derive from these ‘ living the consideration forces as, I say I had derived everything directly & solely from the velocity generated by of ,mPulaivoactlQn’

the forces of those influences, which, according to the generally accepted view taken by

all Mechanicians, either generate, or in some way induce, velocities that are proportional

to themselves & the intervals of time during which they act; take, for instance, gravity,

elasticity, & other forces of the same kind. I then began to investigate somewhat more

carefully that production of velocity which is thought to arise through impulsive action,

in which the whole of the velocity is credited with being produced in an instant of time by

those, who think, because of that, that the force of percussion is infinitely greater than all

forces which merely exercise pressure for single instants. It immediately forced itself upon

me that, for percussions of this kind, which really induce a finite velocity in an instant of

time, laws for their actions must be obtained different from the rest.

17. However, when I considered the matter more thoroughly, it struck me that, if The cause of

we

employ

a

straightforward

method

of

argument,

such

a

mode

of

action

must

be

with­

the was

Investigation the opposition

drawn from Nature, which in every case adheres to one & the same law of forces, & the raised to the Law

same mode of action.

I

came

to

the

conclusion

that

really

immediate impulsive action

of

of Continuity by the idea of direct

one body on another, & immediate percussion, could not be obtained, without the pro­ impulse.

duction of a finite velocity taking place in an indivisible instant of time, & this would have

to be accomplished without any sudden change or violation of what is called the Law of

Continuity ; this law indeed I considered as existing in Nature, & that this could be shown

to be so by a sufficiently valid argument. The following is the line of argument that I

employed initially ; afterwards I made it clearer & confirmed it by further arguments &

fresh reflection.

j8. Suppose there are two equal bodies, moving in the same straight line & in the Violation of the

same direction ; & let the one that is in 6, & the one behind a degree represented

front have a degree by 12. If the latter,

of velocity represented by i.e., the body that was be­

Law of Continuity, if a body moving more swiftly comes

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

into actual con­ tact with another body moving more

very instant of time at which this contact happened, the hindermost body should diminish slowly.

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, 11 & 7, 10 & 8, 9$ & 8f, & so on. For it cannot possibly happen

46

PHILOSOPHIA NATURALIS THEORIA

temporis particulam ejusmodi mutatio fiat per intermedios gradus, durante contactu. Si enim aliquando alterum corpus jam habuit 7 gradus velocitatis, & alterum adhuc retinet
11; toto illo tempusculo, quod effluxit ab initio contactus, quando velocitates erant 12, & 6, ad id tempus, quo sunt 1 r, & 7, corpus secundum debuit moveri cum velocitate majore,
ejuam 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 compcnctrari debuit, quod cum ob impenetrabilitatem, quam in materia agnoscunt passim omnes Physici, & quam ipsi tri­ buendam omnino esse, facile evincitur, fieri omnino non possit; oportuit sane, in ipso
primo initio contactus, in ipso indivisibili momento temporis, quod inter tempus continuum praecedens contactum, & subsequens, est indivisibilis limes, ut punctum apud Geometras est limes indivisibilis inter duo continua: lineae segmenta, mutatio velocitatum facta fuerit
per saltum sine transitu per intermedias, Ixsa penitus illa continuitatis lege, quae itum ab una magnitudine ad aliam sine transitu per intermedias omnino vetat. Quod autem in
corporibus aequalibus diximus de transitu immediato utriusque ad 9 gradus velocitatis, recurrit utique in iisdem, vel in utcunque inaequalibus dc quovis alio transitu ad numeros 3uosvis. Nimirum ille posterioris corporis excessus graduum 6 momento temporis auferri
ebet, sive imminuta velocitate in ipso, sive aucta in priore, vel in altero imminuta utcunque, & aucta in altero, quod utique sine saltu, qui omissis infinitis intermediis velocitatibus habeatur, obtineri omnino non poterit.

Objectio petiu a

19. Sunt, qui difficultatem omnem submoveri posse censeant, dicendo, id quidem ita se

corporum. urorum habere debere, si corpora dura habeantur, qua: nimirum nullam compressionem sentiant,

nullam mutationem figura:; & quoniam hxc a multis excluduntur penitus a Natura ; dum

se duo globi contingunt, introcessione, [10] & compressione partium fieri posse, ut in ipsis

corporibus velocitas immutetur per omnes intermedios gradus transitu facto, & omnis

argumenti vis eludatur.

Ea nil non posse,
qui admittunt ele­ menta «olida, &

20. At inprimis ea cum plerisque veterum

responsione uti non possunt, quicunquc cum Newtono, & vero etiam Philosophorum prima elementa materix omnino dura admittunt, &

dura

solida, cum adhxsione infinita, & impossibilitate absoluta mutationis figurx. Nam in primis

clementis illis solidis, & duris, qux in anteriore adsunt sequentis corporis parte, & in prece­

dents posteriore, qux nimirum se mutuo immediate contingunt, redit omnis argumenti vis

prorsus illxsa.

Extensionem con­

21. Deinde vero illud omnino intelligi sane non potest, quo pacto corpora omnia partes

tinuam primos

requirere poros, &

aliquas

postremas

circa

superficiem

non

habeant

penitus

solidas, qux

idcirco

comprimi

parietes solidos, ac duros.

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 quamcunquc 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 determinatx crassitudinis paries usque ad

primum porum, poris utique carens ; vd quomodo, quod eodem recidit, nullus sit extimus,

& superficiei externx omnium proximus porus, qui nimirum si sit aliquis, parietem habeat

utique poris expertem, & compressionis incapacem, in quo omnis argumenti superioris vis

redit prorsus illxsa.

e«.o leps Con-

22. At ea etiam, utcunque penitus inintdlieibili, sententia admissa, redit omnis eadem

in primis superG- argumenti vis in ipsa prima, & ultima corporum se immediate contingentium supernae, vel mebus. vel punctis. sj quIIx conttnux superficies congruant, in lineis, vel punctis. Quidquid enim sit id, in quo

contactus fiat, debet utique esse aliquid, quod nimirum impenetrabilitati occasionem

prxstet, & cogat motum in sequente corpore minui, in prxccdcnte augeri ; id, quidquid est, in quo exeritur impenetratibilitatis vis, quo fit immeaiatus contactus, id sane velocitatem

mutare debet per saltum, sine transitu per intermedia, & in eo continuitatis lex abrumpi

A THEORY OF NATURAL PHILOSOPHY

47

that this kind of change is made by intermediate stages in some finite part, however small,

of continuous time, whilst the bodies remain in contact. For if at any time the one

body then had 7 degrees of velocity, the other would still retain 11 degrees; thus, during

the whole time that has passed since the beginning of contact, when the velocities were respectively 12 & 6, until tne time at which they are 11 & 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 oodies that arc unequal in any way, concerning any other passage to any numbers.

In fact, the excess of velocity in the hindmost body, amounting to 6 degrees, has to be got

rid of in an instant of time, whether by diminishing the velocity of this body, or by increasing

the velocity of the other, or by diminishing somehow the velocity of the one & increasing

that of the other ; & this cannot possibly be done in any case, without the sudden change

that is obtained by omitting the infinite number of intermediate velocities.

19, There are some people, who think that the whole difficulty can be removed by An objection de­

saying

that

this

is

just

as

it

should

be,

if

hard

bodies,

such

as

indeed

experience

no

com­

rived from denying the existence of

pression or alteration of shape, are dealt with ; whereas by many philosophers hard bodies hard bodies.

are altogether excluded from Nature ; & therefore, so long as two spheres touch one

another, it is possible, by introcession & compression of their parts, for it to happen that in

these bodies the velocity is changed, the passage being made through all intermediate stages ;

& thus the whole force of the argument will be evaded.

20. Now in the first place, this reply can not be used by anyone who, following New­ This reply cannot

ton,

&

indeed

many

of

the

ancient

philosophers

as

well,

admit

the

primary

elements

of

be made by those who admit solid &

matter to be absolutely hard & solid, possessing infinite adhesion & a definite shape that it hard elements.

is perfectly impossible to alter. For the whole force of my argument then applies quite

unimpaired to those solid and hard primary elements that are in the anterior part of the

body that is behind, & in the hindmost part of the body that is in front; & certainly these

parts touch one another immediately.

21. Next it is truly impossible to understand in the slightest degree how all bodies do Continuous exten­

not have some of

their last parts just near to the surface perfectly solid, & on that account

sion requires pri­ mary pores & walls

altogether incapable of being compressed. If matter is continuous, it may & must be sub­ bounding them,

ject to infinite divisibility; but actual division carried on indefinitely brings in its train solid & hard.

difficulties that are truly inextricable ; however, this infinite division is required by those who do not admit that there are any particles, no matter how small, in bodies that are

perfectly free from, & incapable of, compression. For they must admit the idea that every particle is marked off & divided up, by the action of interspersed pores, into many boundary

walls, so to speak, for these pores; & these walls again are distinct from the pores them­

selves. It is quite impossible to understand why it comes about that, in passing from

empty vacuum to solid matter, we are not then bound to encounter some continuous wall of

some definite inherent thickness from the surface to the first pore, this wall being everywhere devoid of pores ; nor why, which comes to the same thing in the end, there does not exist

a pore that is the last & nearest to the external surface ; this pore at least, if there were one,

certainly has a wall that is free from pores & incapable of compression; & here then the

whole force of the argument used above applies perfectly unimpaired.

22. Moreover, even if this idea is admitted, although it may be quite unintelligible, Violation of the

then

the

whole

force

of the

same

argument

applies to

the

first

or last surface of the

bodies

Law of Continuity takes place, at any

that are in immediate contact with one another; or, if there are no continuous surfaces rate, in prime sur­

congruent, then to the lines or points. For, whatever the manner may be in which contact faces or points.

takes place, there must be something in every case that certainly affords occasion for

impenetrability, & causes the motion of the body that follows to be diminished, & that of

the one in front to be increased. This, whatever it may be, from which the force of impene­

trability is derived, at the instant at which immediate contact is obtained, must certainly

change the velocity suddenly, & without any passage through intermediate stages; & by

48

PHILOSOPHIAE NATURALIS THEORIA

debet, atque labefactari, si ad ipsum immediatum contactum illo velocitatum discrimine deveniatur. Id vero est sane aliquid in quacunque e sententiis omnibus continuam extensionem tribuentibus materix. Est nimirum reaiis affectio quxdam corporis, videlicet ejus limes ultimus reaiis, superficies, reaiis superficiei limes linea, reaiis linex hmes punctum, qux affectiones utcunque in iis sententiis sint prorsus inseparabiles [it] ab ipso corpore, sunt tamen non utique intellectu confictx, sed reales, qux nimirum reales dimensiones aliquas habent, ut superficies binas, linea unam, ac rcalcm motum, & translationem cum ipso corpore, cujus idcirco in iis sententiis debent, esse affectiones quxdam, vel modi.

/.l.'j‘ •

23. Est, qui dicat, nullum in iis committi saltum idcirco, quod censendum sit, nullum

quz auper- habere motum, superficiem, lineam, punctum, qux massam habeant nullam. Motus, inquit,

& pu"tH* a Mechanicis habet pro mensura massam in velocitatem ductam : massa autem est super-

ficies baseos ducta m crassitudinem, sive altitudinem, ex. gr. m prismatis. (Juo 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 Hncam, punctun/ Quantitatem materix, & motum corporum metiuntur per massam ejusmodi, ac velocitatem, continua<]uema<lniodum in ipsa geometrica quantitate tria genera sunt quantitatum, corpus, vel quid"003’ ' ' soliaum, quod trinam dimensionem habet, superficies qux binas, linx, qux unicam, quibus
accedit linex limes punctum, omni dimensione, & extensione carens; sic etiam in Physica habetur in communi corpus tribus extensionis spccicbus prxditum ; superficies reaiis extimus corporis limes, prxdita binis; linea, limes reaiis superficiei, habens unicam; & ejusdem linex indivisibilis limes punctum. Utrobique alterum alterius est limes, non pars, & quatuor diversa genera constituunt. Superficies est nihil corporeum, sed non & ninil superficiale, quin immo partes habet, & augeri potest, & minui; & eodem pacto linea in ratione quidem superficiei est nihil, sed aliquid in ratione linex ; ac ipsum demum punctum est aliquid in suo genere, licet in ratione linex sit nihil.

Quo pacto nomen

25. Hinc autem in iis ipsis massa quxdam considerari potest duarum dimensionum, vel

Inotiit'debeat’con- unius, vel etiam nullius continux dimensionis, sed numeri punctorum tantummodo, uti

venire supwficic- quantitas ejus genere designetur ; quod si pro iis etiam usurpetur nomen massx generaliter, tus, line», punctis. motus quantitas definiri poterit per productum ex velocitate, & massa ; si vero massx nomen

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

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

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

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

quantitatum, solidi, superficiei, linex, punctorum ; ac ut altera harum erit ninil in alterius

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

ratione sui, non purum nihil.

Motum paisim tribui punctis; fore, ut in eo leda-

[12] 26. Et quidem ipsi Mechanici vulgo motum tribuunt & superficiebus & lineis, & punctis, ac centri gravitatis motum ubique nominant Physici, quod centrum utique punctum

t u t lex.

Continuitatis est aliquod, non corpus trina prxditum dimensione, quam iste ad motus rationem, & appellationem requirit, ludendo, ut ajebam, in verbis. Porro in ejusmodi motibus exti­

marum saltem superficiorum, vel linearum, vel punctorum, saltus omnino committi debet,

si ca ad contactum immediatum deveniant cum illo velocitatum discrimine, & continuitatis

lex violari.

Fore, ut ea Izdatur

27. Verum hac omni disquisitione omissa de notione motus, & massx, si factum ex

«altem in velocitate punctorum.

velocitate,

&

massa,

evanescente

una

e

tribus

dimensionibus,

evanescit;

remanet

utique

velocitas reliquarum dimensionum, qux remanet, si ex reapse remanent, uti quidem omnino

remanent in superficie, & ejus velocitatis mutatio haberi deberet per saltum, ac in ea violari

continuitatis lex jam toties memorata.

^aec Qu'^cm ’ta evidentia sunt, ut omnino dubitari non possit, quin continuitatis in minimis parti- lex infringi debeat, & saltus in Naturam induci, ubi cum velocitatis discrimine ad se invicem tn*'* ^US C°nru acce^ant 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 clementis 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 Objection derived

sidered that a surface, a line or a point, having no mass, cannot have any motion.

He may

from the terms mass and motion. which

say that motion has, according to Mechanicians, as its measure, the mass multiplied by the do not accord with

velocity; also mass is the surface of the base multiplied by the thickness or the altitude, surfaces & points.

as for instance in prisms. Hence the less the thickness, the less the mass & the motion ;

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

as well.

24. Now the man who reasons in this manner is first of all merely playing with words. Commencement of

Mass

is

commonly

called

quantity

of

matter,

&

the

motion

of

bodies

is

measured

by

mass

the answer to this ; a surface, or a line,

of this kind & the velocity. But, just as in a geometrical quantity there are three kinds of or a point, is some-

quantities, with one :

namely, a body or a solid having three dimensions, a surface with two, & a line to which is added the boundary of a line, a point, lacking dimensions altogether,

tiling real, if con­ tinuous extension is supposed to ex-

& of no extension. So also in Physics, a body is considered to be endowed with three ist.

species of extension ; a surface, the last real boundary of a body, to be endowed with two ;

a line, the real boundary of a surface, with one ; & the indivisible boundary of the line, to

be a point. In both subjects, the one is a boundary of the other, & not a part of it; &

they form four different kinds. There is nothing solid about a surface ; but that does not

mean that there is also nothing superficial about it; nay, it certainly has parts & can be

increased or diminished. In the same way a line is nothing indeed when compared with

a surface, but a definite something when compared with a line ; & lastly a point is a definite

something in its own class, although nothing in comparison with a line.

25. Hence also in these matters, a mass can be considered to be of two dimensions, or The manner in

of one, or even of no continuous this kind is indicated. Now, if

dimension, but only numbers for these also, the term mass

of points, just as is employed in a

quantity of generalized

which the term mass may, and the term motus is bound

sense,

we

shall

be

able

to

define

the

quantity

of

motion

by

the

product

of

the

velocity

&

to.apply to surfaces, lines, & points.

the mass. But if the term mass is only to be used in connection with a solid body, then

indeed the motion of a solid body will be measured by the mass multiplied by the velocity ;

but the motion of a surface, or a line, or any number of points will nave as their measure

the quantity of the surface, or line, or the number'of the points, multiplied by the velocity.

Motion at any rate will be ascribed in all these cases, & there will be four kinds of motion,

as there are four kinds of quantity, namely, for a solid, a surface, a line, or for points ; and, as each class of the latter will be as nothing compared with the class before it, but something in its own class, so the motion of the one will be as nothing compared with the motion of the other, but yet really something, & not entirely nothing, compared with those of

its own class. 26. Indeed, Mechanicians themselves commonly ascribe motion to surfaces, lines & Motion is ascribed
points, & Physicists universally speak of the motion of the centre of gravity; this centre is minatety8 the taw

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

isvio‘ y 0,08 **'

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 ft is at least a fact the velocity & the mass vanishes when one of the three dimensions vanish, there will still fo^thbyIatheSfciea

remain the velocity of the remaining dimensions; & this will persist so long as the dimen- of the velocity of sions persist, as they do persist undoubtedly in the case of a surface. Hence the change po^ts.

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 Of impenetrability approach one another with a difference of velocity & come into immediate contact, if only vcr? Pfr’

we are to ascribe impenetrability to bodies, as we really should. And this property too, tion. not in whole bodies only, but in any of the smallest particles of bodies, & in the elements as well, is recognized by Physicists universally. There was one, I must confess, who, after I

E

50

PHILOSOPHISE NATURALIS THEORIA

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

29' ^u,t “’^em e Leibnitianorum familia, qui post evulgatam Theoriam meam cenpro*1 mutatione* suerit, difficultatem ejusmodi amoveri posse diecnao, duas monades sibi etiam invicem

reUtote1 #tm°o '”i occurrentes cum velocitatibus quibuscunque oppositis squalibus, post ipsum contactum

localis.

pergere moveri sine locali progressione. Eam progressionem, ajebat, revera omnino nihil

esse, si a spatio percurso aestimetur, 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 itiaem ludit in voce motus, quam adhibet pro mutatione quacunque,

& actione, vel actionis modo. Motus localis, & velocitas motus ipsius, sunt ea, qu® ego quidem adhibeo, & qu® ibi abrumpuntur per saltum. Ea, ut evidentissime constat, erant

aliqua ante contactum, & post contactum mo-[i3]-mento temporis in eo casu abrumpuntur;

nec vero sunt nihil; licet spatium pure imaginarium sit nihil. Sunt realis affectio rei

mobilis fundata in ipsis modis localiter existendi, qui modi etiam relationes inducunt dis­

tantiarum reales utique. Quod duo corpora magis a se ipsis invicem distent, vel minus;

quod localiter celerius moveantur, vel lentius ; est aliquid non imaginarie tantummodo, sed

realiter diversum; in eo vero per immediatum contactum saltus utique induceretur in eo

casu, quo ego superius sum usus.

Qui continuiut»,

30. Et sane summus nostri ®vi Geometra, & Philosophus Mac-Laurinus, cum etiam ipse

legem summover- co]iisjonem 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. r, cap. 4. Et sane sunt alii nonnulli, qui ipsam con­

tinuitatis legem nequaquam admiserint, quos inter Maupcrtuisius, vir celeberrimus, ac de

Republica Litteraria optime meritus, absurdam etiam censuit, & quodammodo inexplica­

bilem. Eodem nimirum in nostris de corporum collisione contemplationibus devenimus

Mac-Laurinus, & ego, ut videamus 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.

Tbeoriz exortus,

31. Ast ego cum ipsam continuitatis legem aliquanto diligentius considerarim, &

ea lege, uti fieri fundamenta, quibus ea innititur, perpenderim, arbitratus sum, ipsam omnino e Natura

debet, retenta.

submoveri non posse, qua proinde retenta contactum ipsum immediatum submovendum

censui in collisionibus corporum, ac ea consectaria persecutus, qu® ex ipsa continuitate

servata sponte profluebant, directa ratiocinatione delatus sum ad eam, quam superius

exposui, virium mutuarum legem, qu® consectaria suo quxque ordine proferam, ubi ipsa,

qu® ad continuitatis legem retinendam argumenta me movent, attigero.

^id rit^discri1

32, Continuitatis lex, de qua hic agimus, in eo sita est, uti superius innui, ut quxvis

men inter status, quantitas, dum ab una magnitudine ad aliam migrat, debeat transire per omnes intermedias

& incrementa. ejusdem generis magnitudines. Solet etiam idem exprimi nominandi transitum per gradus

intermedios, quos quidem gradus Maupcrtuisius ita accepit, quasi vero qu®dam exigu®

accessiones fierent momento temporis, in quo quidem is censuit violari jam necessario legem

ipsam, qu® utcunque exiguo saltu utique violatur nihilo minus, quam maximo ; cum

nimi-[i4)-rum magnum, & parvum sint tantummodo respective ; & jure quidem id censuit ;

si nomine graduum incrementa magnitudinis cujuscunquc momentanea intelligerentur.

A THEORY OF NATURAL PHILOSOPHY

5«

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 compcnetration the velocities were changed gradually. But it can be easily proved that this is contrary to that induction & analogy, such as we have in Physics, one peculiarly adapted for the investigation of the general laws of Nature. What the power of this induction is, & where it can be used (one
of the cases is this very matter of extending impenetrability to the minute particles of a
body), I will set forth later.

29. There was also one of the followers of Leibniz who, after I had published my Objection to the

Theory,

expressed his opinion that

this kind of difficulty

could

be removed

by saying that

term mofux being used for a change;

two monads colliding with one another with any velocities that were equal & opposite refutation from the

would, after they came into contact, go on moving without any local progression.

He

reality of local mo­ tion.

added that that progression would indeed be absolutely nothing, if it were estimated by the

space passed over, since the space was nothing ; but the motion would go on & be destroyed

by degrees, because the energy with which they act upon one another, by mutual pressure,

would be gradually destroyed. He also is playing with the meaning of the term motus,

which he uses both for any change, & for action & mode of action. Local motion, & the

velocity of that motion are what I am dealing with, & these are here broken off suddenly.

These, it is perfectly evident,were something definite before contact, & after contact in

an instant of time in this case they are broken off. Not that they are nothing ; although

purely imaginary space is nothing. They are real conditions of the movable thing

depending on its modes of extension as regards position ; & these modes induce relations

between the distances that are certainly real. To account for the fact that two bodies

stand at a greater distance from one another, or at a less; or for the fact that they are

moved in position more quickly, or more slowly ; to account for this there must be some­

thing that is not altogether imaginary, but real & diverse. In this something there would

be induced, in the question under consideration, a sudden change through immediate

contact.

30. Indeed the finest geometrician & philosopher of our times, Maclaurin, after he too There are some who

had

considered

the

collision

of solid

bodies

&

observed

that

there is

nothing

which

could

would deny the Law of Continuity.

maintain & preserve the Law of Continuity in the collision of bodies accomplished by

immediate contact, thought that the Law of Continuity ought to be abandoned. He

asserted that, in general in the case of collision, the law was violated, publishing his idea in

the work that he wrote on the discoveries of Newton, bk. I, chap. 4. True, there are some

others too, who would not admit the Law of Continuity at all; & amongst these, Mauper-

tuis, a man of great reputation & the highest merit in the world of letters, thought it was

senseless, & in a measure inexplicable. Thus, Maclaurin came to the same conclusion as

myself with regard to our investigations on the collision of bodies ; for we both saw that, in

collision, immediate contact & impulsive action could not be reconciled with the Law of

Continuity. But, whereas he came to the conclusion that there could be no doubt about

the fact of impulsive action & immediate contact between the bodies, he impeached &

abrogated the Law of Continuity. Nor indeed do I know of anyone else before me, who

has had the courage to deny the existence of all immediate contact for any bodies whatever,

although there are some who would retain a thin layer of air, (that is to say, of another body),

in between the two in collision.

31. But I, after considering the Law of Continuity somewhat more carefully, & The origin of my

pondering

over

the fundamental

ideas

on which

it

depends, came

to

the

conclusion

that

Theory, retaining this Law, as should

it certainly could not be withdrawn altogether out of Nature. Hence, since it had to be be done.

retained, I came to the conclusion that immediate contact in the collision of solid bodies

must be got rid of; &, investigating the deductions that naturally sprang from the

conservation of continuity, I was led by straightforward reasoning to the law that I have set

forth above, namely, the law of mutual forces. These deductions, each set out in order,

I will bring forward when I come to touch upon those arguments that persuade me to

retain the Law of Continuity.

32. The Law of Continuity, as we here deal with it, consists in the idea that, as I intimated above, any quantity, in passing from one magnitude to another, must pass through

The nature of the Law of Continuity: distinction between

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

states ments.

&

incre­

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

52

PHILOSOPHl/E NATURALIS THEORIA

Verum id ita intcUigcndum est; ut singulis momentis singuli status respondeant; incre­

menta, vel decrementa non nisi continuis tempusculis.

Geometriae usus ad eam exponendam : momenta punctis,

fig.

33. Id sane admodum facile concipitur ope 3, ad quam referatur quxdam alia linea CDE.

Geometriae. Sit recta quxdam AB in Exprimat prior ex iis tempus, uti solet

tempera continua utique in ipsis horologiis circularis peripheria

lineis expressa.

ab indicis cuspide denotata tempus definire.

Quemadmodum in Geometria in lineis

fiuncta sunt indivisibiles limites continuarum

inex partium, non vero partes linex ipsius;

ita in tempore distinguenda: erunt partes

continui temporis respondentes ipsis linea:

partibus, continuae itidem & ipsa:, a mo­

mentis, qux sunt indivisibiles carum partium

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,

tempusculum appellabo.

Fluxus ordinatae

34.. Si jam a quovis puncto rectx AB, ut F, H, erigatur ordinata perpendicularis FG,

transeuntis per m ag nit u di nes

HI,

usque

ad

lineam

CD ;

ea poterit reprxsentare quantitatem quampiam continuo

omnes intermedias. variabilem. Cuicunque momento temporis F, H, respondebit sua ejus quantitatis magnitudo

FG, HI; momentis autem intermediis aliis K, M, alix magnitudines, KL, MN, respondebunt;

ac si puncto G ad I continua, & finita abeat pars linex 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 intermedix inter FG, HI, qux alicui puncto inter F, H intermedio non respondeat.

idem in quantitate

35. Quantitas illa variabilis per hanc variabilem ordinatam expressa mutatur juxta

VtkTTn continuitatis legem, quia a magnitudine FG, quam habet momento temporis F, ad magni-

voce gradui.

tudinem HI, qux respondet momento temporis H, transit per omnes intermedias magnitu­

dines KL, MN, respondentes intermediis momentis K, M, & momento cuivis respondet

determinata magnitudo. Quod, si assumatur tempusculum quoddam continuum KM

utcunque exiguum ita, ut inter puncta L, N arcus ipse LN non mutet recessum a recta AB

in accessum ; ducta LO ipsi parallela, habebitur quantitas NO, qux in schemate exhibito

est incrementum magnitudinis ejus quantitatis continuo variatx. Quo minor est ibi

temporis particula KM, eo minus est id ncrementum NO, 8c illa evanescente, ubi congruant

momenta K, M, hoc etiam evanescit. Potest quxvis magnitudo KL, MN appellari status

quidam variabilis illius quantitatis, & gradus nomine deberet potius in-[ 15]-tclligi 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 transcatur; quod quidem xquivocationibus omnibus occasionem exhibuit.

statu» singulo*

36. Sed omissis xquivocationibus ipsis, illud, quod ad rem facit, est accessio incremen-

momeat>*. incre- torUrn facta non momento temporis, sed tempusculo continuo, quod est particula continui

qu» parva tem- temporis. Utcunque exiguum sit incrementum ON, ipsi semper respondet tempusculum ra^ndcrtCOntiau15 quondam KM continuum. Nullum est in linea punctum M ita proximum puncto K, ut sit

primum post ipsum ; sed vel congruunt, vel intercipiunt lineolam continua bisectione per alia intermedia puncta perpetuo divisibilem in infinitum. Eodem pacto nullum est in tempore momentum ita proximum alteri prxccdcnti 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 prxcedenti statui, ut sit primus post ipsum accessu aliquo

momentaneo facto : sed differentia, qux inter ejusmodi status est, debetur intermedio continuo tempusculo; ac data lege variationis, sive natura linex ipsam exprimentis, & quacunque utcunque exigua accessione, inveniri potest tempusculum continuum, quo ea

accessio advenerit.

Transitu* sine *ai-

37’ Atque sic quidem intelligitur, quo pacto fieri possit transitus per intermedias

21 ^^aUva^^ni8 magn,tu<^’ncs omnes, per intermedios status, per gradus intermedios, quin ullus habeatur hUumfqwrf wmen saltus utcunque exiguus momento temporis factus. Notari illud potest tantummodo,

Q°n e*«LiVCtt id"m mutadonem neri alicubi per incrementa, ut ubi KL abit, in MN per NO; alicubi per reaii* status?* *m decrementa, ut ubi K'L' abeat in N'M' per O'N'; quin immo si linea CDE, qux legem

A THEORY OF NATURAL PHILOSOPHY

53

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

33. The idea can be very easily assimilated by the help of geometry.

Explanation by the

Let

Let AB be any straight line (Fig. 3), to which as axis let any other line CDE be referred. the first of them represent the time, in the same manner as it is customary to specify

use of geometry; instants represen­ ted by points, con­

the time in the case of circular clocks by marking off the periphery with the end of a pointer.

tinuous intervals of time by lines.

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 injtant 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 tcmpusculc, or small interval of time.

34. If now from any points F,H on the straight line AB there are erected at right angles The flux of the or­

to

it

ordinates

FG, HI, to

meet

the

line

CD

;

any of

these

ordinates

can

be

taken

to

repre­

dinate as it through all

passes inter­

sent a quantity that is continuously varying. To any instant of time F, or H, there will mediate values.

correspond its own magnitude of the quantity FG, or HI ; & to other intermediate instants

K, M, other magnitudes KL, MN will correspond. Now, if from the point G, there pro­

ceeds a continuous & finite part of the line CDE, it is very evident, & it can be rigorously

proved, that, no matter how the curve twists & turns, there is no intermediate point K,

to which some ordinate KL does not correspond ; &, conversely, there is no ordinate of

magnitude intermediate between FG & HI, to which there does not correspond a point

intermediate between F & H.

35. The variable quantity that is represented by this variable ordinate is altered in The

accordance with the Law of Continuity; for, from the magnitude FG, which it has at

quanti'/*»

the instant of time F, to the magnitude HI, which corresponds to the instant H, it passes represented through all intermediate magnitudes KL, MN, which correspond to tHe intermediate of’the'term step. *

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 Sing^ states cor-

increments is accomplished, not m an instant of time, but in a small interval of con- but increments

tinuous time, which is a part of continuous time. However small the increment ON may

5^allcJl0

be, there always corresponds to it some continuous interval KM. There is no point M ttauou*s time,

in the straight line AB so very close to the point K, that it is the next after it; but either

the points coincide, or they intercept between them a short length of line that is divisible

again & again indefinitely by repeated bisection at other points that are in between M &

K. In the same way, there is no instant of time that is so near to another instant that has

gone before it, that it is the next after it ; but either they arc 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 ^^®eseh^i^lout through all intermediate magnitudes, through intermediate states, or through intermediate fron/^positive 8 to

stages, without any sudden change being made, no matter how small, in an instant of time, negative through

It can merely be remarked that change in some places takes place by increments (as when

not” real

KL becomes MN by the addition of NO), in other places by decrements (as when K'L' nothing. hut a cer-

'

r

'

'

tain real state.

54

PH1LOSOPHLE NATURALIS THEORIA

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, qux, quo magis ex oppositae parte crescit,
eo minor censetur in ratione priore, quemadmodum in ratione possessionis, vel divitiarum, pergit perpetuo se habere pejus, qui iis omnibus, quae habebat, absumptis, xs alienum contrahit perpetuo majus. Et in Geometria quiaem 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 Conicis, 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-[l6]-vis ad negativa, ac vice versa ; quanquam, uti inferius innuam, id ipsum sit non nihilum revera in se ipso, sed reaiis 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.

Proponitur pro­

38. Exposita hoc pacto, & vindicata continuitatis lege, eam in Natura existere plerique

banda existeutia legis Continnitatj.

Philosophi

arbitrantur,

contradicentibus

nonnullis,

uti

supra

innui.

Ego, cum in eam

primo inquirerem, censui, eandem omitti omnino non posse ; si eam, quam habemus unicam,

Naturae 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. 119; in quibus etiam locis, prout diversis occasionibus

conscripta sunt, repetuntur non nulla.

Ejus probatio ab inductione satis ampla.

39. Longum hic esset singula inde excerpere in ordinem redacta : satis erit exscribere dissertationis De lege Continuitatis numerum 138. Post inductionem petitam prxcedentc

numero a Geometria, qua: 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, qux utique a distantiis

pendet particularum in aliam, nisi transeundo per intermedias; fit gradus in eo numero ad

motuum velocitates, & ductus, qux 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 servatur etiam in eo, quod motus omnes in lineis continuis

fiunt nusquam abruptis. Plurimos ejusmodi motus videmus. Planetx, & cometx in lineis

continuis cursum peragunt suum, & omnes rctrogradationes fiunt paullatim, ac in stationibus

semper exiguus quidem motus, sed tamen habetur semper, atque hinc etiam dies paullatim

per auroram venit, per vespertinum crepusculum abit, Solis diameter non per saltum, sed continuo motu supra horizontem ascendit, vel descendit. Gravia itidem oblique projecta

in lineis itidem pariter continuis motus exercent suos, nimirum in parabolis, seclusa aeris resistentia, vel, ea considerata, in orbibus ad hyperbolas potius accedentibus, & quidem

semper cum aliqua exigua obliquitate projiciuntur, cum infinities infinitam improbabilitatem

habeat motus accurate verticalis inter infinities infinitas inclinationes, licet exiguas, & sub sensum non cadentes, fortuito obvenient, qui quidem motus in hypothesi Telluris motx a

parabolicis plurimum distant, & curvam continuam exhibent etiam pro casu projectionis

accurate verticalis, quo, quiescente penitus Tellure, & nulla ventorum vi deflectente motum,

haberetur [17) ascensus rcctilineus, vel descensus. Immo omnes alii motus a gravitate

pendentes, omnes ab elasticitate, a vi magnetica, continuitatem itidem servant; cum eam

servent vires illx ipsx, quibus gignuntur. Nam gravitas, cum decrescat in ratione reciproca

duplicata distantiarum, & distantix 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 acris

compressi. In iis, & omnibus ejusmodi viribus, & motibus, quos gignunt, continuitas habetur

semper, tam in lineis qux describuntur, quam in velocitatibus, qux 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 G'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 MTJ' 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 formula, the passage being made not only through nothing, but also through infinity; such I have discussed, the one in a dissertation added to my Conic Sections, the other in my Algebra (§ 14), & both
of them together in my essay De Lege Continuitatis; but in Physics, where no quantity ever increases to an infinite extent, the second case has no place ; hence, unless the passage is made through the value nothing, there is no passage from positive to negative, or vice versa. Although, as I point out below, this nothing is not really nothing in itself, but a
certain real state; & it may be considered as nothing only in a certain sense. In the same sense, too, negatives, which are true states, are positive in themselves, although, as they
belong to the first set in a certain negative way, they are called negative.

38. Thus explained & defended, the Law of Continuity is considered by most philoso­ I propose to prove

phers to exist in Nature, though there are some who deny it, as I mentioned above.

I,

the existence of the Law of Continuity.

when first I investigated the matter, considered that it was absolutely impossible that it

should be left out of account, if we have regard to the unparalleled analogy that there is

with Nature & to the power of induction; & by the help of this induction I endeavoured

to prove the law in several of the dissertations that I have mentioned, & Benvenutus also

used the same form of proof in his Synopsis (Art. 119). In these too, as they were written

on several different occasions, there are some repetitions.

39. It would take too long to extract & arrange in order here each of the passages in Proof by induction

these essays ; it After induction

will be sufficient if I give Art. 138 of the dissertation De Lege Continuitatis. derived in the preceding article from geometry, in which there is no sudden

sufficient purpose.

for

the

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 arc 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 arc

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 wc 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

$6

PHILOSOPHIA NATURALIS THEORIA

& 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 continuae occurrant, vel primi generis, quas Natura videtur affectare in spinis, vel secundi generis, quas videtur affectare in avium unguibus, & rostro, in quibus tamen manente in ipsa cuspide unica tangente continuitatem servari videbimus infra. Infinitum esset singula persequi, in quibus continuitas in Natura observatur. Satius est generaliter provocare aa exhibendum casum in Natura, in quo
continuitas non servetur, qui omnino exhiberi non poterit.”

Duplex inductionis

40. Inductio amplissima tum ex hisce motibus, ac velocitatibus, tum ex aliis pluribus

vtaihabcatinductio exemplis, quo: habemus in Natura, in quibus ea ubique, quantum observando licet depre-

incompteu.

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 134, & totum 135, disserta­

tionis De Lege Continuitatis. Sic autem habent ibidem : “ Inprimis ubi generales Naturae

leges investigantur, inductio vim habet maximam, & ad earum inventionem vix alia ulla

superest via. Ejus ope extensionem, figurabiUtem, 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-[l8]-rae legibus stabiliendis locum habere non potest. Habet locum laxior

quaedam inductio, quae, ut adhiberi possit, debet esse ejusmodi, ut inprimis in omnibus iis

casibus, qui ad trutinam ita revocari possunt, ut deprehendi debeat, an ea lex observetur,

eadem in iis omnibus inveniatur, & ii non exiguo numero sint; in reliquis vero, si quae prima

fronte contraria videantur, re accuratius perspecta, cum illa lege possint omnia conciliari;

licet, an eo potissimum pacto concilientur, immediate innotescere, nequaquam possit. Si

eae conditiones habeantur ; inductio ad legem stabiliendam censeri debet idonea. Sic quia

videmus corpora tam multa, quae habemus prae manibus, aliis corporibus resistere, ne in

eorum locum adveniant, & loco cedere, si resistendo sint imparia, potius, quam eodem

perstare simul; impenetrabilitatem corporum admittimus; nec obest, quod quaedam

corpora videamus intra alia, licet durissima, insinuari, ut oleum in marmora, lumen in

crystalla, & gemmas. Videmus enim hoc phaenomenum facile conciliari cum ipsa impene-

trabilitate, dicendo, per vacuos corporum poros ea corpora permeare. (Num. 135).

Praeterea, quaecunque proprietates absolutae, nimirum quae relationem non habent ad

nostros sensus, deteguntur generaliter in massis sensibilibus corporum, easdem ad quascunque

utcunque exiguas particulas debemus transferre ; nisi positiva aliqua ratio obstet, & nisi sint

ejusmodi, quae pendeant a ratione totius, seu multitudinis, contradistincta a ratione partis.

Primum evincitur ex eo, quod magna, & parva sunt respectiva, ac insensibilia dicuntur ea,

quae respectu nostrae molis, & nostrorum sensuum sunt exigua. Quare ubi agitur de

proprietatibus absolutis non respectivis, quaecunque communia videmus in iis, qux 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 analogiae

laesio, 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, quae si juxta

A THEORY OF NATURAL PHILOSOPHY

57

bodies, & in a thousand other things of the same kind, where the changes of velocity occur gradually, & the path is not retraced before the velocity has been diminished through all

degrees. All these things most strictly preserve continuity. Hence it follows that no

sharp angles are met with in natural motions, but in every case a change of direction occurs gradually; neither do perfect angles occur in bodies themselves, for, however fine an edge

or point in them may seem, one can usually detect curvature by the help of the microscope

if nothing else. We have this gradual change of direction also in the beds of rivers, in the

leaves, boughs & branches of trees, & stones of all kinds; unless, in some cases perchance,

there may be continuous pointed ends, either of the first kind, which Nature is seen to

affect in thorns, or of the second kind, which she is seen to do in the claws & the beak of

birds ; in these, however, we shall see below that continuity is still preserved, since we are

left with a single tangent at the extreme end. It would take far too long to mention every single thing in which Nature preserves the Law of Continuity ; it is more than sufficient

to make a general statement challenging the production of a single case in Nature, in which

continuity is not preserved ; for it is absolutely impossible for any such case to be brought

forward.”

40. The effect of the very complete induction from such motions as these & velocities, Induction of a two­

as well as from a large number of other examples, such as we have in Nature, where Nature in every case, as far as can be gathered from direct observation, maintains continuity or

fold kind : when & why incomplete induction has vali­

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

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

I may here quote part of Art. 134 & the whole of Art. 135 from my dissertation De Lege Continuitatis. The passage runs thus: “ Especially when we investigate the general laws

of Nature, induction has very great power; & there is scarcely any other method beside it for the discovery of these laws. By its assistance, even the ancient philosophers attributed

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

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

universal gravitation. Now, induction should take account of every single case that can possibly happen, before it can have the force of demonstration ; such induction as this has no

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

type; in order that this kind of induction may be employed, it must be of such a nature that in all those cases particularly, which can be examined in a manner that is bound to

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

them the same result is arrived at; & that these cases are not merely a few. Moreover, in the other cases, if those which at first sight appeared to be contradictory, on further &

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

whether they can be made to agree in this way better than in any other whatever, it is impossible to know directly anyhow. If such conditions obtain, then it must be considered

that the induction is adapted to establishing the law. Thus, as we see that so many of the bodies around us try to prevent other bodies from occupying the position which they

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

than that both should occupy the same place at the same time, therefore we admit the impenetrability of bodies. Nor is there anything against the idea in the fact that we see

certain bodies penetrating into the innermost parts of others, although the latter arc very hard bodies; such as oil into marble, & light into crystals & gems. For we see that this

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

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

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

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

attribute these same properties also to all small parts whatsoever, no matter how small they may be. That is to say, unless some positive reason prevents this ; such as that they are of such a nature that they depend on argument having to do with a body as a whole, or with a group of particles, in contradistinction to an argument dealing with a part only.

The proof comes in the first place from the fact that great & small are relative terms, & those things are called insensible which are very small with respect to our own size & with

regard to our senses. Therefore, when we consider absolute, & not relative, properties, whatever we perceive to be common to those contained within the limits that are sensible

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

For these limits, with regard to such matters as arc 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

PHILOSOPHI/E NATURALIS THEORIA

quasdam prudentes regulas fiat, successum habere solet. Cum id indicium fallere possit; fieri potest, ut committatur error, sed contra ipsum errorem habebitur praesumptio, ut etiam in jure appellant, donec positiva ratione evincatur oppositum. Hinc addendum fuit, nisi ratio positiva obstet. Sic contra hasce regulas peccaret, qui diceret, corpora quidem magna compcnetrari, ac replicari, & inertia carere non posse, compcnctrari tamen posse, vel replicari, vel sine inertia esse exiguas eorum partes. At si proprietas sit respcctiva, respectu nostrorum sensuum, ex [19] 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 materiz, 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 figurabitis, & extensi, ut habeat aliquid, quod ab alio distet, adeoque,
ut habeat partes ; hinc eae proprietates, licet in quovis aggregato particularum materia:, sive in quavis sensibili massa, inveniantur, non debent inductionis vi transferri ad particulas quascunque.”

Et impenetrate i- 41. Ex his patet, & impenetrabilita tem, & continuitatis legem per ejusmodi inductionis uiutem evtndjper genus abunde probari, atque evinci, & illam quidem ad quascunque utcunque exiguas inductionem : ad particulas corporum, hanc ad gradus utcunque exiguos momento temporis adjectas debere ipwn quid requira- cxtcnfjj_ 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, cam 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.

Ejiu applicatio ad

41. Id ipsum fit, ubi per inductionem impenetrabilitas corporum accipitur pro generali

unpcnetrabiiiutem, jege ]\Jatur3E. Nam impcnetrabilitatcm ipsam magnorum corporum observamus in exemplis

sane innumeris tot corporum, quae 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 praesto est conciliatio phaenomeni cum impenetra-

bilitate, petita ab eo, quod illa corpora, in quae se ejusmodi substantiae insinuant, poros

habeant, quos eae permeent. Et quidem haec conciliatio exemplum habet manifestissimum

in spongia, quae per poros ingentes aqua immissa imbuitur. Poros marmorum illorum, &

multo magis vitrorum, non videmus, ac multo minus videre possumus illud, non insinuari

eas substantias nisi per poros. Hoc satis est reliquae inductionis vi, ut dicere debeamus, eo

fjotissimum pacto se rem habere, & ne ibi quidem violari generalem utique impenetrabilitatis

egem.

Similis ad coatinu- . t20! 43. Eodem igitur pacto in lege ipsa continuitatis agendum est. Illa tam ampla

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

quibus ea"videatur casibus, in quibus determinare immediate per observationes non possumus, an eadem

i»)»

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

violari videatur; ineunda est ratio aliqua, qua ipsum phamomenum 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 saepe & 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, altnough 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 Bothimpenetra-

of Continuity can be proved by a kind of induction of this type ; & the former must be

dTm'o'n^

extended to all particles of bodies, no matter how small, & the latter to all additional steps, strated by induc-

however small, made in an instant of time. Now, in the first place, to use this kind of

p£-

induction, it is required that the property, for the proof of which it is to be used, must be pose-

observed in a very large number of cases ; for otherwise the probability would be very

small. Also it is required that no case should be observed, in which it can be proved that

it is violated. It is not necessary that, in those cases in which at first sight it is feared that

there may be a failure of the property, that it should be directly proved that there is no

failure. It is sufficient if in those cases some reason can be obtained which will make the

observation agree with the property; & all the more so, if in other cases an example of

reconciliation can be obtained, & it can be positively proved that sometimes reconciliation

can be obtained in that way.

42. This is just what does happen, when the impenetrability of solid bodies is accepted Application of in­

as a law of Nature through inductive large bodies in innumerable examples

reasoning. For we observe this impenetrability of of the many bodies that wc consider. There are

duction to trabiJity.

impeno-

indeed also cases, in which one would think that it was violated, such as when oil penetrates

wood and marble, & works its way through them, or when light passes through glasses &

gems. But we have ready a means of making these phenomena agree with impenetrability,

derived from the fact that those bodies, into which substances of this kind work their way,

possess pores which they can permeate. There is a very evident example of this recon­

ciliation in a sponge, which is saturated with water introduced into it by means of huge

pores. We do not see the pores of the marble, still less those of glass ; & far less can we see

that these substances do not penetrate except by pores. It satisfies the general force of

induction if we can say that the matter can be explained in this way better than in any

other, & that in this case there is absolutely no contradiction of the general law of impene­

trability.

43. In the same way, then, we must deal with the Law of Continuity. The full Similar application induction that we possess should lead us to admit in general this law even in those cases in which it is impossible for us to determine directly by observation whether the same law which there seems holds good, as for instance in the collision of bodies. Also, if there are some cases in which to l“ v’o,atlon-
the law at first sight seems to be violated, some method must be followed, through which each phenomenon can be reconciled with the law, as is in every case possible. I brought forward several cases of this kind in the dissertations I have mentioned, some of which pertained to geometrical continuity, & others to physical continuity. I will not delay over the first of these : for geometrical continuity is not necessary for the defence of the physical variety ; I used it as an example in confirmation of a wider induction. The latter, as well as very frequently the former, reduces to two classes; & the first of these classes is that class
in which a sudden change seems to have been made on account of our having omitted the intermediate quantities with a jump. I give a geometrical illustration, and then add one in physics.

6o

PHILOSOPHI/E NATURALIS THEORIA

Exemplum geome­ tricum primi gene­ ris, ubi nos inter­

44. In axe curvae cujusdam in fig. erigantur ordinatae AB, CD, EF, GH.

4. sumantur segmenta AC, CE, EG aequalia, & Area: BACD, DCEF, FEGH videntur continua:

medias magnitu­ dines omittimus.

cujusdam serici termini ita, ut ab illa BACD transeatur, & tamen secunda a prima, ut

ad

DCEF,

&

inde

ad

FEGH immediate

& tertia a secunda, differunt per quanti­

tates finitas : si enim capiantur CI, EK

jequales BA, DC, & arcus BD transferatur

in IK ; arca DIK.F erit incrementum se­

cundae supra primam, quod videtur imme­

diate advenire totum absque eo, quod

unquam habitum sit ejus dimidium, vel

quxvis alia pars incrementi ipsius ; ut idcirco

a prima ad secundam magnitudinem area:

itum sit sine transitu per intermedias. At

ibi omittuntur a nobis termini intermedii,

qui continuitatem servant ; si cnimar aqualis AC motu continuo feratur ita, ut incipiendo

ab AC desinat in CE ; magnitudo areae BACD per omnes intermedias bacd abit in magnitu­

dinem DCEF sine ullo saltu, & sine ulla violatione continuitatis.

Quando id accidat

45. Id sane ubique accidit, ubi initium secunda magnitudinis aliquo intervallo distat

exempla dierum,

physica & oscilla­

ab

initio

prima

;

sive statim veniat

post ejus

finem, sive quavis

alia lege

ab

ea

disjungatur.

tionum consequen­ Sic in physicis, si diem concipiamus intervallum temporis ab occasu ad occasum, vel etiam

tium.

ab ortu ad occasum, dies praccdcns a sequenti quibusdam anni temporibus differt per plura

secunda, ubi videtur fieri saltus sine ullo intermedio dic, qui minus differat. At seriem

quidem continuam ii dies nequaquam constituunt. Concipiatur parallelus integer Telluris, in quo sunt continuo ductu disposita loca oinnia, qua 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 pra-[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 prae edente; sed &

initium & finis ejus finito intervallo temporis distat a praecedentis initio, & fine, ac intermedii

termini continua serie fluente a prima oscillatione ad secundam essent ii, qui haberentur, si

primae, & secundae oscillationis arcu in aqualem 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.

Exempla secundi

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

generis, ubi mutatio sit celerrime, sed

&

tamen

peragitur

tempore

successivo, sed

perbrevi.

Sunt, qui objiciant pro violatione

non momento tem­ continuitatis casum, quo quisquam manu lapidem tenens, ipsi statim det velocitatem

poris.

quandam finitam : alius objicit aquae e vase effluentis, foramine constituto aliquanto infra

superficiem ipsius aquae, velocitatem oriri momento temporis finitam. At in priore casu

admodum evidens est, momento temporis velocitatem finitam nequaquam produci. Tempore

opus est, utcunque brevissimo, ad excursum spirituum per nervos, & musculos, ad fibrarum

tensionem, & alia ejusmodi : ac idcirco ut velocitatem aliquam sensibilem demus lapidi,

manum retrahimus, & ipsum aliquandiu, perpetuo accelerantes, retinemus. Sic etiam, ubi

tormentum bellicum exploditur, videtur momento temporis emitti globus, ac totam

celeritatem acquirere; at id successive fieri, patet vel inde, quod debeat inflammari tota

massa pulveris pyrii, & dilatari aer, ut elasticitate sua globum acceleret, quod quidem fit

omnino per omnes gradus. Successionem multo etiam melius videmus in globo, qui ab

elastro sini relicto propellatur: quo clasticitas est major, eo citius, sed nunquam momento

temporis velocitas m globum inducitur.

Applicatio ipsorum

47. Hac exempla illud praestant, quod aqua per poros spongiae ingressa respectu

ad

inipcnctrabilitatis,. ut.ea responsione uti possimus in aliis casibus omnibus, in quibus accessio

b

aliqua magnitudinis videtur fieri tota momento temporis ; ut nimirum dicamus fieri tempore

A THEORY OF NATURAL PHILOSOPHY

61

44. In the axis of any curve (Fig. 4) let there be taken the segments AC, CE, EG equal

to one another ; & let the ordinates AB, CD, EF, GH be erected. The areas BACD, DCEF, kind, where wC

FEGH seem to be terms of some continuous series such that we can pass directly from BACD

"<^mediate

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 DI KF 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

bad until it reaches the magnitude of the area DCEF without any sudden change, & hence

without any breach of continuity. 45. Indeed this always happens when the beginning of the second magnitude is distant when this win
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 case of consecutive look upon the day as the interval of time between sunset & sunset, or even between sunrise ^1\^o“asecutive

& 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 scries. 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 isthelast 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 thesudden

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 endof the second differs from the beginning & the end of the first bya finite inter­

val of time ; & the intermediate terms in a continuously varying series from the first oscillation to the second would be those that would be obtained, if the arcs of the first & second oscilla­

tions were each divided into the same number of equal parts, & the path traversed (or the time spent in traversing the path) is taken between the ends of all these proportional paths; such as that between the third or fourth part of the first arc & the third or fourth part

of the second arc. This argument can be easily transferred so as to apply to all cases of this

kind ; & in such cases it can always be directly proved that there is no breach of continuity.

46. The second class of cases is that in which something seems to have been done in an Examples of the instant of time, but still it is really done in a continuous, but very short, interval of time. ““"hd

There are some who bring forward, as an objection in favour of a breach of continuity, the is veryrapid, but 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 time, from a vessel, where, if an opening is made below the level of the surface of the water, a

finite velocity is produced in an instant of time. But in the first case it is perfectly clear

that a finite velocity is in no wise produced in an instant of time. For there is need of

time, although this is exceedingly short, for the passage of cerebral impulses through

the nerves and muscles, for the tension of the fibres, and other things of that sort; and

therefore, in order to give a definite sensible velocity to the stone, we draw back the hand,

and then retain the stone in it for some time as we continually increase its velocity forwards.

So too when an engine of war is exploded, the ball seems to be driven forth and to acquire the whole of its speed in an instant of time. But that it is done continuously is clear, if

only from the fact that the whole mass of the gunpowder has to be inflamed and the gas has to be expanded in order that it may accelerate the ball by its elasticity ; and this latter certainly takes place by degrees. The continuous nature of this is far better seen in the

case of a ball propelled by releasing a spring; here the stronger the elasticity, the greater

the speed; but in no case is the speed imparted to the ball in an instant of time. 47. These examples are superior to that f water entering through the pores of a sponge, Application of

which we employed in the matter of impenetrability ; so that we can make use of this reply

particularly

in all other cases in which some addition to a magnitude seems to have taken place entirely in to thaflowof water

an instant of time. Thus, without doubt we may say that it takes place in an exceedingly from » v«sei.

6z

PHILOSOPHI/E NATURALIS THEORIA

brevissimo, utique per omnes intermedias magnitudines, ac illaesa penitus lege continuitatis. Hinc & in aquae 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, neccsse 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 quierimus, 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 aquae orta, oriri utique non potest, nisi per accessiones continuas tempusculo admodum parvo, sed non omnino nullo : nam pressio tempore indiget, ut velocitatem progignat, in communi omnium sententia.

Transitus ad meta-

48. Illsesa igitur esse debet continuitatis lex, nec ad eam evertendam contra inductionem,

phyricam probati­ onem : limes in

tam

uberem

quidquam

poterunt

casus

allati

hucusque,

vel

iis

similes.

At ejusdem con­

continuis unicus, tinuitatis aliam metaphysicam rationem adinveni, & proposui in dissertatione De Lege

ut in Geometria. Continuitati!, petitam ab ipsa continuitatis natura, in qua quod Aristoteles ipse olim

notaverat, communis esse debet limes, qui praecedentia cum consequentibus conjungit, qui

idcirco etiam indivisibilis est in ea ratione, in qua est limes. Sic superficies duo solida

dirimens & crassitudine caret, & est unica, in qua immediatus ab una parte fit transitus ad

aliam ; linea dirimens binas superficiei continuae partes latitudine caret; punctum continuae

lineae segmenta discriminans, dimensione omni ; nec duo sunt puncta contigua, quorum

alterum sit finis prioris segmenti, alterum initium sequentis, cum duo contigua indivisibilia,

& inextensa haberi non possint sine compcnetratione, &. coalescentia quadam in unum.

Idem in tempore

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

A in quavii serie
continua: eviden­ tius in quibusdam.

continuum indivisibilis

praecedens, & continuo subsequens unicum habeatur momentum, quod sit 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,

quae continuo tempore duret, haberi debet series quaedam magnitudinum ejusmodi, ut

momento temporis cuivis respondeat sua, quae praecedentem cum consequente conjungat,

& ab illa per aliquam determinatam magnitudinem differat. Quin immo in illo quantitatum

genere, in quo [23] binae magnitudines simul haberi non possunt, id ipsum multo evidentius

conficitur, nempe nullum haberi posse saltum immediatum ab una ad alteram. Nam illo

momento temporis, quo deberet saltus fieri, & abrumpi series accessu aliquo momentaneo,

deberent haberi duae magnitudines, postrema seriei praecedentis, & prima serici 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; 8c ex alia duos simul ejusmodi status habere non potest.

Inde cur motus lo­ cata non fiat, nisi
per lineam contin­

sane

50. Id quidem satis patebit notissimum, sed cujus ratio

in ipso non ita

locali facile

motu, in quo habetur aliunde redditur, inde

phaenomenum omnibus autem patentissima est,

uam.

Corpus a quovis loco ad alium quemvis devenire utique potest motu continuo per lineas

quascunque utcunque contortas, & in immensum productas quaquaversum, quae numero

infinities infinitae sunt : sed omnino debet per continuam aliquam abire, & nullibi inter­

ruptam. En inde rationem ejus rei admodum manifestam. Si alicubi linea motus abrum­

peretur ; vel momentum temporis, quo esset in primo puncto posterioris lineae, esset

posterius eo momento, quo esset in puncto postremo anterioris, vel esset idem, vel anterius ?

In primo, & tertio casu inter ea momenta intercederet tempus aliquod continuum divisibile

in infinitum per alia momenta intermedia, cum bina momenta temporis, in eo sensu accepta,

in quo ego hic ea accipio, contigua esse non possint, uti superius exposui. Quamobrem in

A THEORY OF NATURAL PHILOSOPHY

6?

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 Passing to ameta-
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 in theca» of conmetaphysical one, in favour of this continuity, & published it in my dissertation De Lege ^ometry.'np‘ ** m

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 tnat it is a boundary. In the same way, a surface of separation of two solids is also without thickness & is single, & in it there is immediate passage from one side to the other ; the line of separation of two parts of a continuous surface lacks any breadth ; a point determining

segments of a continuous line has no dimension at all; nor are there two contiguous points, one of which is the end of the first segment, & the other the beginning of the next; for two contiguous indivisibles, of no extent, cannot possibly be considered to exist, unless
there is compenetration & a coalescence into one.

49. In the same way, this should also happen with regard to time, namely, that between similarly for time

a preceding continuous time & the next following there should be a single instant, which

monTVvi*

is the indivisible boundary of either. There cannot be two instants, as we intimated above, dentin some than contiguous to one another ; but between one instant & another there must always intervene “ 0,hCT3

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 dear in the case of local motion, in regard to which Hence the reason the phenomenon is perfectly well known to all; the reason for it, however, is not so easily '^r^n °» derived from any other source, whilst it follows most clearly from this idea. A body can continuous line,

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

<54

PHILOSOPHIZE 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 ; in terio haberetur

replicatio non tantum respectu eorum binorum momentorum, sed omnium etiam inter­

mediorum, in quibus nimirum omnibus id corpus esset in binis locis. Cum igitur corpus

existens nec nullibi esse possit, nec simul in locis pluribus; illa vix mutatio, & ille saltus

haberi omnino non possunt.

Illustratio ejus

51. Idem ope Geometnx magis adhuc oculis ipsis subjicitur. Exponantur per rectam

argumenti es Ceo metria: ratiocina­

AB

tempora,

ac

per

ordinatas

ad

lineas

CD,

EF,

abruptas

alicubi,

diversi

status

rei

cujuspiam.

tione metaphysics, Ductis ordinatis DG, EH, vel punctum H jaceret post G, ut in Fig. 5 ; vel cum ipso

plurtlms exemplis. congrueret, ut in 6 ; vel ipsum prxcederct, ut in 7. In primo casu nulla responderet

ordinata omnibus punctis rectae GH ; in secundo binx responderent GD, & HE eidem puncto

G; in tertio vero binx HI, & HE puncto H, binx GD, GK puncto G, & binx LM, LN

Fio. 5.

Fig. 6.

Fio. 7.

puncto cuivis intermedio L; nam ordinata est relatio quxdam distantix, quam habet punctum curvx cum puncto axis sibi respondente, adeoque ubi jacent in recta eadem perpendiculari axi bina curvarum puncta, habentur binx ordinatx respondentes eidem puncto axis. Quamobrcm 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, qux 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 rcali serie accidit; ut hic linea finita sine
puncto primo, & postremo, quod sit ejus limes, & superficies sine linea esse non potest; unde nt, ut in casu figurx 6 binx ordinatx 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 co 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 dux simul haberentur distantix, vel dux densitates,

quod utique sine replicatione haberi non potest; caloris itidem, & frigoris mutatio in

thermometris, ponderis atmosphxrx mutatio in barometris, non fit per saltum, quia binx

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; qux quidem theoria innumeris casibus

pariter aptari potest.

Objectio ab

&

52. Contra hoc argumentum videtur primo aspectu adesse aliquid, quod ipsum pforsus

«xm ttu coftjun- evertat, & tamen ipsi illustrando idoneum est maxime. Videtur nimirum inde erui,

jini'hiiat!

impossibilem esse & creationem rei cujuspiam, & interitum. Si enim conjungendus est

eju» solutio.

postremus terminus prxcedentis seriei cum primo sequentis in ipso transitu a non esse ad esse, vel vice versa, aebebit utrumque conjungi, ac idem simul erit, & non erit, quod est absurdum. Responsio in promptu est. Seriei finitx rcalis, & existentis, reales itidem, &
existentes termini esse debent; non vero nihili, quod nullas proprietates habet, quas exigat, Hinc si rcalium statuum serici altera series realium itidem statuum succedat, qux non sit communi termino conjuncta ; bini eodem momento debebuntur status,- qui nimirum
sint bini limites earundem. At quoniam non este 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 ali.im mutetur duplam, duraturam itidem per alteram sequentem horam ; momento temporis, [25] quod horas dirimit, binx debebunt esse densitates simul,
nimirum & simplex, & dupla, qux sunt reales binarum realium serjerum 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.

Dlustration of this

Let

times

be

represented

by

the

straight

line

AB,

& diverse

states

of

any

thing

by

argument from geo­ metry ; the line of

ordinates drawn to meet the lines CD, EF, which are discontinuous at some point. If the reasoning being

ordinates

DG,

EH

arc

drawn,

either

the

point

H

will fall after

the

point

G,

as

in

Fig.

5 ;

metaphysical, with several examples.

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 Sc 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, Sc 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 togetheroitxisunct

exemplify it. It seems that from this argument it follows that both the creation of any ^e’J$Xe«fCT«tia»

thing,& its destruction,are impossible. For, if the last term of a series that precedes is to or annihilation; 4

be connected with the first term of the series that follows,.then in the passage from a state *t» ®oiut»cui.

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 Sc existent must themselves be seal & 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 Sc directly cut off by fact of existence. Wherefore, at the first &

at the last instant of that continuous interval of time, duringwhich 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, Sc 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, Sc these are real terms of two real series.

F

66

PHILOSOPHIA NATURALIS THEORIA

Unde huc transfer­ enda solutio Ipsa.

53. Id ipsum luculenter exposui,

in ac

dissertatione De lege virium in Natura existentium satis, geometricis figuris illustravi, adjectis nonnullis, qux eodem

ni fallor, recidunt,

& qua: in applicatione ad rem, de qua agimus, & in cujus gratiam hxc 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.

Solutio petita ex

54. “ Sit in fig. 8 circulus GMM'm, qui referatur ad datam rectam AB per ordinatas

geometrico plo.

exem­ HM ipsi rectae perpendiculares;

uti itidem perpendiculares sint binae tangentes EGF,

E'G'F'. Concipiantur igitur recta quaedam indefinita ipsi rectx AB perpendicularis, motu

quodam continuo delata ab A ad B. Ubi ea habuerit, positionem quamcumque CD, qux

praecedat tangentem EF, vel C'D', qua: consequatur tangentem E'F'; ordinata ad circulum

nulla erit, sive erit impossibilis, & ut Geometra:

loquuntur, imaginaria. Ubicunque autem ea sit

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

D

D

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

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

H'ttz'. Ordinata quidem ipsa respondet soli inter­

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

momentum E est limes inter tempus praecedens

continuum AE, quo ordinata non est, & tempus continuum EE' subsequens, quo ordinata est ; punc­ tum E' est limes inter tempus prxcedens EE\ quo

M M

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

igitur quaedam ordinata: est tempus EE'; ortus

habetur in E, interitus in E'.

autem in

ipso ortu, & interitu ? Habetur-ne quoddam esse

ordinatx, an non esse l Habetur utique esse, nimi­ rum EG, vel E'G', non autem non esse. Oritur

Ftc. 8.

tota finitae magnitudinis ordinata EG, interit tota finitae magnitudinis E'G', nec tamen

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

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

ultimus terminus seriei praecedentis sine primo termino seriei sequentis.”

Solutio ex moia-

55. 11 Quare autem id ipsum accidat, si metaphysica consideratione rem perpendimus,

physica atione.

consider­ statim patebit. Nimirum veri nihili nulls sunt verae proprietates : entis reaiis verx, & reales proprietates sunt. Quaevis reaiis 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 prxcedens ordinatae nullius, ultimum

terminum non [26] habet, series consequens non habet primum : series reaiis 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.”

Illustratio ulterior

56. “ Atque id quidem manifestum fit magis: si consideremus seriem aliquam

geometrica.

praecedentem realem, quam exprimant ordinatx ad lineam continuam PLg, qux 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 binx ordinatx 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; qux si non interjaceat; jam illa puncta in unicum coalescunt. Quare non potest

deesse nisi lineola aliqua gL ita, ut terminus seriei prxcedcntis sit in aliquo momento C

prxccdcnte momentum E, & disjuncto ab eo per tempus quoddam continuum, in cujus

temporis momentis omnibus ordinata sit nulla.”

Applicatio ad crea­

57. “ Patet igitur discrimen inter transitum a vero nihilo, nimirum a quantitate

tionem. A la ti onem

annihi­ imaginaria, ad esse, nihili non habetur;

& transitum ab una magnitudine ad aliam. In primo casu terminus habetur terminus uterque seriei veram habentis existentiam, & potest

?uantitas, cujus ea est series, oriri, vel occidere quantitate finita, ac per se excludere non esse.

n secundo casu necessario haberi debet utriusque seriei terminus, alterius nimirum postre­ mus, alterius primus. Quamobrcm etiam in creatione, & in annihilatione potest quantitas

oriri, vd interire magnitudine finita, & primum, ac ultimum esse erit quoadam esse, quod

secum non conjunget una non esse. Contra vero ubi magnitudo reaiis 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 S to be borrowed,

some additions that reduced to the same thing. These will appear below, as an application to the matter in question ; for the sake of which all these things relating to the Law of

Continuity have been adduced. It is allowable for me to quote in this connection the whole of nine articles from that dissertation, beginning with Art. 8 ; but I will here change the numbering of the articles, & of the diagrams as well, so that they may agree

with those already given. 54. “In Fig. 8, let GMM'm be a circle, referred to a given straight line AB as axis, by
means of ordinates HM drawn perpendicular to that straight line ; also let the two tan-

pie®*01118 **

gents EGF, E'G'F' be perpendiculars to the axis. Now suppose that an unlimited straight line perpendicular to the axis AB is carried with a continuous motion from A to B. When it reaches some such position as CD preceding the tangent EF, or as C'D' subsequent to the tangent E'F', there will be no ordinate to the circle, or it will be impossible &, as the

geometricians call it, imaginary. Also, wherever it falls between the two tangents EGF, E'G'F', as at HI or HT, it will meet the circle in two points, M, m or M', m’; & for the value of the ordinate there will be obtained HM & Htm, or H'M' & H'wi'. 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 Sc 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 scries without the last term of the preceding series ; Sc 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 "deration,

the properties of a real absolute entity are also real. Any real series must have a real beginning & end, or a first term Sc a last. TTiat which does not exist can have no true property; & on that account docs 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 Sc a last term. The real terms of this series of themselves exclude the term of no value, since the fact of existence of itself excludes non-existence.”
56. “ This indeed will be still more evident, if we consider some preceding series of

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, Sc that end must also be a point; hence that point would be before Sc contiguous to the point g ; Sc 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; Sc if such a line does not intervene, then those points must coalesce into one. Hence nothing can be absent, except it be a short length of line gL, so that the end of the series that precedes occurs’ at some instant, C, preceding the instant E, Sc 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, Sc 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 scries & 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 ; Sc this will not associate with itself a state of non-existence. But, on the other hand, where a real magnitude is bound

68

PHILOSOPHISE NATURALIS THEORIA

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

°id, VquSI

58. “At hic illud etiam notandum est; quoniam ad ortum, & interitum considerandum

e»t aliquid. ’

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 praecedentibus, licet alio

nomine insignitus.”

Ordinatam nullam. r2yi

« Sit in Fig. Q. Linea AB, ut prius, ad quam linea quaedam PL deveniat in G

nullam existentium (pertinet punctum G ad lineam rL, k ad AB continuatas, & sibi occurrentes ibidem), & sive ti<meni<:Ompenetra Per8at ul*1"3 ipsam in GM', sive retro resiliat per GM'. Recta CD habebit ordinatam CL,

qu® evanescet, ubi puncto C abeunte in E, ipsa CD abibit in EF, tum in positione ulteriori

rectas perpendicularis HI, vel abibit in nega­

tivam HM, vel retro positiva regredietur

in HM'. Ubi linea altera cum altera coit,

& punctum E alterius cum alterius puncto

G congreditur, ordinata CL videtur abire in

nihilum ita, ut nihilum, quemadmodum &

supra innuimus, sit limes quidam inter seriem

ordinatarum positivarum CL, & negativarum

HM; vel positivarum CL, & iterum posi­

tivarum HM'. Sed, si res altius considere­

tur ad metaphysicum conceptum reducta,

in situ EF non habetur verum nihilum.

In situ CD, HI habetur distantia quaedam

punctorum C, L ; H, M: in situ EF

habetur eorundem punctorum compene-

tratio. Distantia est relatio quaedam

binorum modorum, quibus bina puncta

existunt; compenetratio itidem est relatio

qu® compenetratio est aliquid reale ejusdem prorsus generis, cujus est distantia, constituta

nimirum per binos reales existendi modos.”

'"genus

&>• “ Totum discrimen est in vocabulis, qu® nos imposuimus. Bini locales existendi

eam distantiam modi infinitas numero relationes possunt constituere, alii alias. Hae omnes inter se & nullam, & aliquam, differunt, & tamen simul etiam plurimum conveniunt; nam reales sunt, & in quodam genere

congruunt, quod nimirum sint relationes ort® a binis localibus existendi modis. Diversa

vero habent nomina ad arbitrarium instituta, cum ali® ex ejusmodi relationibus, ut CL,

dicantur distanti® positiv®, relatio EG dicatur compenetratio, relationes HM dicantur

distanti® negativ®. Sed quoniam, ut a decem palmis distanti® demptis 5, relinquuntur 5,

ita demptis aliis 5, habetur nihil (non quidem verum nihil, sed nihil in ratione distanti® a

nobis ita appellat®, cum remaneat compenetratio); ablatis autem aliis quinque, remanent

quinque palmi distanti® negativ® ; ista omnia realia sunt, & ad idem genus pertinent; cum

eodem prorsus modo inter se differant distantia palmorum 10 a distantia palmorum 5, h®c

a distantia nulla, sed reali, qu® compenetrationem importat, & h®c a distantia negativa

palmorum 5. Nam ex prima illa quantitate eodem modo devenitur ad hascc posteriores per

continuam ablationem palmorum 5. Eodem autem pacto infinitas ellipses, ab infinitis

hyperbolis unica interjecta parabola discriminat, qu® 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.”

Alia, nihil, quid

quae videntur & sunt ali­
1 discrimen

[28] 61. “Et quidem eodem eodem modo locali existendi;

pacto status’quidam status quidam realis

realis est quies, sive perseverantia in est velocitas nulla puncti existentis.

inter radicem ima­ ginariam, & zero.

nimirum determinatio perseverandi in est vis nulla, nimirum determinatio

eodem loco; status quidam realis puncti existentis retinendi pr®cedentem velocitatem, & ita porro;

plurimum h®c discrepant a vero non esse. Casus ordinat® respondentis line® EF in fig. 9,

differt plurimum a casu ordinat® circuli respondentis line® CD figura 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 prxditum, quod ipsis etiam solutionibus

problematum, & constructionibus veras sufficit, & reales determinationes; cum realis, non

imaginaria sit radix equationis cujuspiam, qu® sit = o, sive nihilo ®qualis.”

A THEORY OF NATURAL PHILOSOPHY

69

to pass suddenly from one quantity to another, then at the instant in which the sudden

change is accomplished, both terms must be obtained. Hence, our argument on

metaphysical grounds in favour of the exclusion of a sudden change from creation or

annihilation, or production & destruction, remains quite unimpaired.”

58. “ In this connection the following point must be noted. As we have used geometrical Sometimes what is ideas for the consideration of production & destruction, it seems also that sometimes £^*^notbiag

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. “InFig. 9, let AB be a line, as before, which some line PL reaches at G (where the When the ordinate

point G belongs to the line PL, & E to the line AB, both being produced to meet one

another at this point) ; & suppose that PL either goes on beyond the point as GM, or between two exts-

recoils along GM’. Then the straight line CD will contain the ordinate CL, which will tS^g

iTcom^

vanish when, as the point C gets to E, CD attains the position EF; & after that, in the penetration,

further position of tne 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

com penetration 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. Th^ ‘n0^ Two local modes of existence can constitute an infinite number of relations, some of one kind1 of °serieT'of

sort & some of another. All of these differ from one another, & yet agree with one ^meq>Ud^t^eM another in a high degree; for they are real & to a certain extent identical, since indeed 901,18

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,sowhen five more are taken away,there is nothing

left (& yet not really nothing, but nothing in comparison with what we usually call

distance ; for compenetration is left). Again, if we take away another five, there remain

five palms of negative distance. All of these are real & belong to the same class; for

they differ amongst themselves in exactly the same way, namely, the distance of ten palms from the distance of five palms, the latter from ‘ no ’ distance (which however is something

real that denotes compenetration), & this again from a negative distance of five palms.

For starting with the first quantity, the others that follow are obtained in the same manner,

by a continual subtraction of five palms. In a similar manner a single intermediate

parabola discriminates between an infinite number of ellipses & an infinite number of

hyperbolas; & this single curve receives a special name, whilst under the one term we include

an infinite number of them that to a certain extent are all different from one another,

although one that is considerably elongated may be very different from another that is

less elongated.” 61. “ In the same way, rest, i.e., a perseverance in the same mode of local existence, Other things that
is some real state ; so is ‘ no ’ velocity a real state of an existent point, namely, a propensity ^™ye°t^aJeSiy

to remain in the same place ; so also is ‘ no ’ force a real state of an existent point, namely, something ; di»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 & «ero.

to the line EF in Fig. 9 differs altogether from the case of the ordinate of the circle

corresponding to the line CD in Fig. 8. In the first there exist two points, but there is

compenetration of these points; in the other case, the second point cannot possibly exist.

When, in the solution of problems, we arrive at a quantity of the first kind, the problem

receives a special sort of solution ; but when the result is a quantity of the second kind,

the problem turns out to be incapable of solution. So much indeed that, in this second case,

there is obtained a true nothing that lacks every real property; in the first case, we get

something endowed with real properties, which also supplies true & real values to the

solutions & constructions of the problems. For the root of any equation that — o, or is

equal to nothing, is something that is real, & is not an imaginary thing.”

7°

PHILOSOPHISE NATURALIS THEORIA

Cjndusio > rv «Ui>tionc cins objec­ tionis.

62. “ Firmum igitur manebit semper, & continuo tempore finito duret, debere habere

stabile, seriem realem quamcunque, qux & primum principium, & ultimum finem

realem, sine ullo absurdo, & sine conjunctione sui esse cum non esse, si forte duret eo solo

tempore: dum si praecedenti etiam exstitit tempore, habere debet & ultimum terminum

serici praecedentis, & primum sequentis, qui debent esse unicus indivisibilis communis limes,

ut momentum est unicus indivisibilis limes inter tempus continuum praecedens, & subsequens.

Sed haec de ortu, & interitu jam satis.”

aSShiuitatis,ead

63. Ut igitur contrahamus jam vela, continuitatis lex & inductione, & mctaphysico

coiiiskuicm corpo- argumento abunde nititur, qux idcirco etiam in velocitatis communicatione retineri omnino

run‘‘

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, qux

videri possent mutatx per saltum. Quod autem pertinet ad metaphysicum argumentum, si

toto tempore ante contactum subsequentis corporis superficies antecedens habuit 12 gradus

velocitatis, & sequenti 9, saltu facto momentaneo ipso initio contactus; in ipso momento ea

tempora dirimente debuisset habere & 12, & 9 simul, quod est absurdum. Duas enim

velocitates simul habere corpus non potest, quod ipsum aliquanto diligentius demonstrabo.

Duo velocitatum genera, potenthlis. A actualis.

64. Velocitatis nomen, enim significare velocitatem

uti passim usurpatur a Mechanicis, xquivocum est; potest actualem, qux nimirum est relatio quxdam in motu xquabili

spatii percursi divisi per tempus, quo percurritur ; & potest significare [29] quandam, quam

apto Scholiasticorum vocabulo potentialcm appello, qux nimirum est determinatio, ad

actualem, sive determinatio, quam habet mobile, si nulla vis mutationem inducat, percur­

rendi motu xquabili determinatum quoddam spatium quovis determinato tempore, qux

quidem duo & in dissertatione De Viribus Vivis, & in Stayanis Supplementis distinxi, distinctione utique .necessaria ad xquivocationcs evitandas. Prima haberi non potest

momento temporis, sed requirit tempus continuum, quo motus fiat, & quidem etiam motum

xqOabilem requirit ad accuratam sui mensuram ; secunda habetur etiam momento quovis

determinata ; & hanc alteram intelligunt utique Mechanici, cum scalas geometricas effor-

mant pro motibus qaibuscunque difformibus, sive abscissa exprimente tempus, & ordinata

velocitatem, utcunque etiam variatam, area exprimat spatium : sive abscissa exprimente itidem tempus, & ordinata vim, arca exprimat velocitatem jam genitam, quod itidem in aliis

ejusmodi scalis, & formulis algebraicis fit passim, hac potentiali velocitate usurpata, qux sit tantummodo determinatio ad actualem, quam quidem ipsam intclligo, ubi in collisione

corporum eam nego mutari posse per saltum ex hoc posteriore argumento.

Biius velocitates tum actuales, tum potentiales simul

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,

haberi non posse, nc detur, vel exiga­ tur compenetratio.

in omnibus sit duplex,

sequentibus occupare alterum pro altera

duo puncta velocitate

ejusdem spatii, determinanda,

ut nimirum spatium percursum 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 auobis 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 potentialcm 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, sea ex componuntur in unicam

ita, ut singulx constituant statum mobilis, qui status respectu dispositionum, quas eo

momento, in quo tum est, habet ipsum mobile, complectentium omnes circumstantias

prxteritas, & prxsentes, est tantummodo conditionatus, non absolutus; nimirum ut con­

tineant dctermi-[3o]-nationem, quam ex omnibus prxtcritis, & prxsentibus 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, £^0®* button
which lasts for some finite continuous time, is bound to have a first beginning & a final of this difficulty, end, without any absurdity coming in, & without any linking up of its existence with a state of non-existence, if perchance it lasts for that interval of time only. But if it existed at a previous time as well, it must have both a last term of the preceding series & a first term of the subsequent series ; just as an instant is a single indivisible boundary between the continuous time that precedes & that which follows. But what I have said about production & destruction is already quite enough.”
63. But, to come back at last to our point, the Law of Continuity is solidly founded Application of the both on induction & on metaphysical reasoning ; & on that account it should be retained 'the in every case of communication of velocity. So that indeed there can never be any passing solid bodies,
from one velocity to another except through all intermediate velocities, & then without any sudden change. We have employed induction for actual motions & velocities in Art. 39 & solved difficulties with regard to velocities in Art. 46, 47, in cases in which they might seem to be subject to sudden changes. As regards metaphysical argument, if in the whole time before contact the anterior surface of the body that follows had 12 degrees of velocity & in the subsequent time had 9, a sudden change being made at the instant of first
contact; then at the instant that separates the two times, the body would be bound to have 12 degrees of velocity, & 9, atone & the same time. This is absurd ; for a body cannot at the same time have two velocities, as I will now demonstrate somewhat more carefully.
64. The term velocity, as it is used in general by Mechanicians is equivocal. For it Two kinds of velomay mean actual velocity, that is to say, a certain relation in uniform motion given by a1cttu’a)P°teiltial &
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 J* b im^ssibie time in the same moving body. For, then it would be necessary that the moving body, Live two velocities”
which at the beginning of a certain time occupied a certain given point of space, should at citJierti2Jc‘^ss °r all times afterwards occupy two points of that space ; so that the space traversed would be C°given, or JjTare
twofold, the one space being determined by the one velocity & the other by the other. forocJh Thus an actual replication would be required; & this we can clearly prove in a perfectly ^LetraUon. C°m
simple way from the principle of induction. Because, for instance, we never see the same movable body departing from the same place in two directions, nor being in two places at the same time in such a way that it is clear to us that it is in both. Again, it can be easily proved that it is also impossible that there should be two potential velocities at the same time. For potential velocity is the propensity that the body has, at the end of any given
continuous time, for existing at a certain given point of space that has a given distance from that point of space, which the moving body occupied at the instant of time in which it is said to have the prescribed potential velocity. Wherefore to have at one & the same time two potential velocities is the same thing as being prescribed to occupy at the same
instant of time two points of space; each of which has its own distinct distance from that point of space that the body occupied at the start; & this is the same thing as prescribing that there should be replication at all subsequent instants of time. It is commonly said that a movable body acquires from different causes .several velocities simultaneously; but these velocities are compounded into one in such a way that each produces a state of the moving body; & this state, with regard to the dispositions that it has at that instant (these include all circumstances both past & present), is only conditional, not absolute. That is to say, each involves the propensity which the body, on account of all past & present circumstances, would have for occupying that prescribed point of space at that particular

72

PHILOSOPHIAE NATURALIS THEORIA

temporis; nisi aliunde ejusmodi determinatio per conjunctionem alterius causie, qua: tum agat, vel jam 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 eaedem, vel aliae causae agentes sequentibus momentis non mutent determinationem, & punctum illud loci, ad quod revera deveniri deinde debet dato illo momento temporis, & actu devenitur; si ipsae nihil aliud agant. Porro patet ejusmodi status ex omnibus 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 ejus momenti, & omnium praeteritarum ; nam toto praecedenti tempore habita fuisset reaiis series statuum cum illa priore, & toto sequenti deberet haberi cum illa posteriore, adeoque eo momento, simul utraque, cum neutra series reaiis sine reali suo termino stare possit.

Quovis momento

66. Praeterea corporis, vel puncti existentis potest utique nulla esse velocitas actualis,

punctum debere

existens habere

saltem accurate talis;

si nimirum difformem habeat motum, quod ipsum etiam semper in

statum realem ex Natura accidit, ut demonstrari posse arbitror, sed huc non pertinet; at semper utique

genere velocitatis potential is.

haberi

debet

aliqua

velocitas

potentials,

vel

saltem

aliquis

status,

qui

licet

alio

vocabulo

appellari soleat, & dici velocitas nulla, est tamen non nihilum quoddam, sed reaiis status,

nimirum determinatio ad quietem, quanquam hanc ipsam, ut & quietem, ego quidem

arbitrer in Natura reapse haberi nullam, argumentis, qua: in Stayanis Supplementis exposui

in binis paragraphis de spatio, ac tempore, quos hic addam in fine inter nonnulla, qua: hic

etiam supplementa appellabo, & occurrent primo, ac secundo loco. Sed id ipsum itidem

nequaquam huc pertinet. Iis etiam penitus praetermissis, eruitur e reliquis, qua: diximus,

admisso etiam ut existente, vel possibili in Natura motu uniformi, & quiete, utramque

velocitatem habere conditiones necessarias ad [31] hoc, ut secundum argumentum pro

continuitatis lege superius allatum vim habeat suam, nec ab una velocitate ad alteram abiri

possit sine transitu per intermedias.

Non posse mom­ ento temporis tran­ sit! ab una veloci-

67. Patet autem, hinc velocitatem totam corporis,

illud evinci, nec interire momento temporis posse, vel puncti non simul intereuntis, vel orientis, nec huc

nec oriri transferri

tate ad aliam, posse, quod de creatione, & morte diximus; cum nimirum ipsa velocitas nulla corporis, vel

demonstratur, vindicatur.

&

puncti

existentis, sit

non

purum

nihil,

ut

monui, sed reaiis

quidam status,

qui

simul

cum

alio reali statu determinatae illius intereuntis, vel orientis velocitatis deberet conjungi; unde

etiam fit, ut nullum effugium haberi possit contra superiora argumenta, dicendo, quando a

12 gradibus velocitatis transitur ad 9, durare utique priores 9, & interire reliquos tres, in

quo nullum absurdum sit, cum 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 12 gradus velocitatis non sunt quid

compositum e duodecim rebus inter se distinctis, atque disjunctis, quarum 9 manere possint,

3 interire, sed sunt unica determinatio ad existendum in punctis spatii distantibus certo

intervallo, ut 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 quaidam pars ordinata: HE communis ipsi usque ad D, sed

sunt duae ordinatae, quarum prima constitit in relatione distantiae, puncti curvae D a puncto

axis G, secunda in relatione puncti curvae E a puncto axis H, quod est ibi idem, ac punctum G.

A THEORY OF NATURAL PHILOSOPHY

73

instant of time ; were it not for the fact that that particular propensity is for other reasons

altered by the conjunction of another cause, which acts at the time, or has already done so ;

& then another propensity, which is termed compound, will take the place of the former.

But the absolute propensity, which arises from the combination of all the past & present

circumstances of the moving body for that instant, is but a single propensity for existing at

any prescribed instant of subsequent time in a certain prescribed point of space; & this

state is absolute for all past & present circumstances, although it may be conditional for

future circumstances. That is to say, if the same or other causes, acting during subsequent

instants, do not change that propensity, & the point of space t 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 8c 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, 8c 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

8c the same time, since neither real senes can stand good without each having its own

real end term.

66. Again, it is at least possible that the actual velocity of a body, or of an existing At any initant an

point, may be nothing; that is to say, if the motion is non-uniform.

Now,

this

always

existing point must have a read state

is the case in Nature ; as I think can be proved, but it does not concern us at present. But, arising from a kind

at

any

rate,

it

is

bound

to

have

some

potential

velocity,

or

at

least

some

state,

which,

of potential city.

velo­

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 8c time ; 8c 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 8c 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, 8c that no passage from one velocity to another can be made

except through intermediate stages.

67. Further, it is quite clear that from this it can be rigorously proved that the whole Rigorous proof (hat

velocity of

a

body cannot

perish

or arise

in

an

instant of

time,

nor

for

a

point

that

does

it is impossible to pass from one velo­

not perish or arise along with it; nor can our arguments with regard to production 8c city to another in

destruction be made to refer to this. For, since that ‘ no ’ velocity of a body, or of an an instant of time.

existing point, is not absolutely nothing, as I remarked, but is some real state ; 8c 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 8c 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 arc 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, wnich is here the

74

PHILOSOPHIAE NATURALIS THEORIA

Relationem distantia: punctorum D, & G constituunt duo reales modi existendi ipsorum, relationem distantiae punctorum D. & E duo reales modi existendi ipsorum, & relationem
distantia punctorum H, & E duo reales modi existendi ipsorum. Haec ultima relatio constat duobus modis realibus tantummodo pertinentibus ad puncta E, & H, vel G, & summa priorum constat modis realibus omnium trium, E, D, G. Sed nos indefinite con­ cipimus possibilitatem omnium modorum rcalium intermediorum, ut infra dicemus, in qua pnecisiva, & indefinita idea stat mihi idea spatii continui; & intermedii modi possibiles inter G, & D sunt pars intermediorum inter E, & H. Praeterea omissis etiam hisce omnibus ipse ille saltus a velocitate finita ad nullam, vel a nulla ad finitam, haberi non potest.

Cur adhibita col­ 68. Atque hinc ego quidem potuissem etiam adhibere duos globos aequales, qui sibi

lisio pergentium in eandem plagam pro Theoria deducenda.

invicem occurrant cum velocitatibus aequalibus, quae 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 12 velocitatis gradibus

assequatur alterum praecedentem cum 6; ut nimirum abeundo ad velocitatem aliam

quamcunque haberetur saltus ab una velocitate ad aliam, in-quo evidentius esset absurdum.

Quo pacto mutata

69. Jam vero in hisce casibus utique haberi deberet’ saltus quidam, & violatio legis

velocitate poten­ tial! per saltum, non mutetur per

continuitatis, non cum velocitatum

quidem in discrimine

velocitate actuali, sed in potential!, si aliquo determinato quocunquc. In

ad contactum deveniretur velocitate actuali, si eam

saltum actualis. metiamur spatio, quod conficitur, diviso per tempus, transitus utique fieret per omnes

intermedias, quod sic facile ostenditur ope Geometris. In fig. io designent AB, BC bina

tempora ante & post contactum, & momento quolibet H sit velocitas potcntialis illa major

HI, quae xquetur velocitati primae AD; quovis autem momento Q posterioris temporis sit

velocitas potentialis minor QR, qua: xquetur

velocitati cuidam datae CG. Assumpto quovis

tempore HK determinata: magnitudinis, arca

IHKL divisa per tempus HK, sive recta HI,

exhibebit velocitatem actualem. Moveatur

tempus HK versus B, & donec K adveniat ad

B, semper eadem habebitur velocitatis men-

।

sura ; eo autem progresso in O ultra B, sed adhuc

।

H existente in M citra B, spatium illi tern-

।

pori respondens componetur ex binis MNEB,

।

BFPO, quorum summa si dividatur per MO ;

I

jam nec erit MN aequalis priori AD, nec BF, -—----------- --------- 1 I 1 I____________ ipsa minor per datam quantitatem FE; sed AH K MB OQ S C

facile demonstrari potest (t>), capta VE aequali

Fig- IO-

IL, vel HK, sive MO, & ducta recta VF, qua: 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.

irrcguiantas alia 70. Patet igitur in ea consideratione a velocitate actuali praecedente HI ad sequentem j?ai!sTv’d^?tetis?ct‘ Q? 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 aliae, atque alia: velocitatis actualis

mensurx ob-[33]-vcniant, secus ac accidit in motu semper aequabili, quam ipsam ob causam,

velocitatis actualis in motu difformi nulla est revera mensura accurata, quod supra innui

sed ejus idea praecisa, ac distincta aequabilitatem motus requirit, & idcirco Mechanici in

difformibus motibus ad actualem velocitatem determinandam adhibere solent spatiolum

infinitesimo tempusculo percursum, in quo ipso motum habent pro aequabili.

(b) Si enim producatur OP usque ad NE in T, erit ET = VN, ob VE = MO —NT. Esi autem PE : VN: : EF : NX ; quare VN X EF — VE X NX, sive posito EX pro VN, fj MO pro VE, erit ET xEF t=MO X NX. Totum MNTO est MO X MN, pars FETP estx ET x EF. Quate residuus gnomon NMOPFE est MOX(MN-NX), sive est MO X MX, quo diviso per MO baletur 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 8c 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 ; 8c 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 the sama direction

destroyed in an instant of time. But, in order to avoid the very considerations just stated “ crn^oy^ with regard to the passage from a real state to another real state (when we pass from a E^Syihoory.

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.

t

69. Now, at least in such cases as these, there is bound to be some sudden change & t^Ucre" we"»

a breach of the Law of Continuity, not indeed in the actual velocity, but in the potential sudden change in

velocity, if the collision occurs with any given difference of velocities whatever. In the

might

actual velocity, measured by the space traversed divided by the time, the change will at any not te a sudden

rate be through all intermediate stages ; 8c this can easily be shown to be 50 by the aid of

*et’

Geometry.

In Fig. 10 let AB, BC represent two intervals of time, respectively before 8c after

contact; 8c at any instant let the potential velocity be the greater velocity HI, equal to the

first velocity AD; 8c at any instant Q of the time subsequent to contact let the potential

velocity be the less velocity QR, equal to some ^iven velocity CG. If any prescribed interval

of time HK be taken, the area IHKL dividea 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 cither MN (which is equal

to the first ADk or BF (which is less than MN by the given quantity FE). But it can

easily be provea ( ) that, if VE is taken equal to IL, or HK, or MO, 8c the straight line

VF is drawn to cut MN in X; then the quotient obtained by the division will be MX.

This holds until, when the whole of the interval of time has passed beyond B into the

position QS, the area QRTS divided by the time QS now represents a constant velocity

equal to QR.

70. From the foregoing reasoning it is therefore clear that the change from the a further inegu-

preceding actual velocity HI to the subsequent velocity QR is made through all intermediate ^Jytl^

velocities such as MX, which will be determined by the continuous straight line VF. There velocity,

is, however, some irregularity arising from the fact that the actual velocity XM must turn

out to be different for different magnitudes of the assumed interval of time HK. For,

according as this is taken to be greater or less, so the point V is removed to a greater or

less distance from E; 8c 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

8c determined by the quotient of the space traversed divided by the time taken, far other

8c 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

arc 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 8c 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) Fir »’/ OP br produced te mttt NE in T, iben ET = PN ; fir PE «= MO = NJ". Moreover PE: PN=EF: NX ; and therefore PN.EF*=PE.NX. Hence, replacing PN by EF", and PE by MO, we have ErEF—MO.NX. Now, th whole MNF"O - MOWN, and the part FEF"P= EFJIF. Hence the remainder (the gnomon NMOPFE) MO.[MN — NX) ■= MO.MX ; and tbit, on diviiitn by MO, will give MX.