LIBRARY CWWVERSITY OF CALIFORNIA DAVIS TH E O R I A PHILOSOPHISE NATURALIS REDACTA AD UNICAM LEGEM VIRIUM IN NATURA EXISTENTIUM, A V C T O P^ROGERIO JOSEPHO BOSCOVICH SOCIETATIS J S U, NUNC AB IPSO PERPOLITA, ET AUCTA, Ac a plurimis praeccclcntium edifionum mendis expurgata. EDITIO VENETA PR1MA IPSO fUCTORE PRJESENTE, ET CORRIGENTE. V E N E T I I S, MDCCLXIII. ^> M ?> 4 ft *<?> 4 V0> <k v * ^ Ex TTPOCRAPHIA REMOWDINIANA. r^IO^Z/M PZllMII.yi/, ac P R IV 1LE G 1 0, A THEORY OF NATURAL PHILOSOPHY PUT FORWARD AND EXPLAINED BY ROGER JOSEPH BOSCOVICH, S.J. LATIN ENGLISH EDITION FROM THE TEXT OF THE FIRST VENETIAN EDITION PUBLISHED UNDER THE PERSONAL SUPERINTENDENCE OF THE AUTHOR IN 1763 WITH A SHORT LIFE OF BOSCOVICH CHICAGO LONDON OPEN COURT PUBLISHING COMPANY 1922 LIBRARY UNIVERSITY OF CALIFORNIA DAVIS PRINTED IN GREAT BRITAIN BY BUTLER & TANNER, FROME, ENGLAND Copyright PREFACE HE text presented in this volume is that of the Venetian edition of 1763. This edition was chosen in preference to the first edition of 1758, published at Vienna, because, as stated on the title-page, it was the first edition (revised and enlarged) issued under the personal superintendence of the author. In the English translation, an endeavour has been made to adhere as closely as possible to a literal rendering of the Latin ; except that the some- what lengthy and complicated sentences have been broken up. This has made necessary slight changes of meaning in several of the connecting words. This will be noted especially with regard to the word " adeoque ", which Boscovich uses with a variety of shades of meaning, from " indeed ", " also " or " further ", through " thus ", to a decided " therefore ", which would have been more correctly rendered by " ideoque ". There is only one phrase in English that can also take these various shades of meaning, viz., " and so " ; and this phrase, for the use of which there is some justification in the word " adeo " itself, has been usually employed. The punctuation of the Latin is that of the author. It is often misleading to a modern reader and even irrational ; but to have recast it would have been an onerous task and something characteristic of the author and his century would have been lost. My translation has had the advantage of a revision by Mr. A. O. Prickard, M.A., Fellow of New College, Oxford, whose task has been very onerous, for he has had to watch not only for flaws in the translation, but also for misprints in the Latin. These were necessarily many ; in the first place, there was only one original copy available, kindly loaned to me by the authorities of the Cambridge University Library ; and, as this copy could not leave my charge, a type-script had to be prepared from which the compositor worked, thus doub- ling the chance of error. Secondly, there were a large number of misprints, and even omissions of important words, in the original itself ; for this no discredit can be assigned to Boscovich ; for, in the printer's preface, we read that four presses were working at the same time in order to take advantage of the author's temporary presence in Venice. Further, owing to almost insurmountable difficulties, there have been many delays in the production of the present edition, causing breaks of continuity in the work of the translator and reviser ; which have not conduced to success. We trust, however, that no really serious faults remain. The short life of Boscovich, which follows next after this preface, has been written by Dr. Branislav Petronievic, Professor of Philosophy at the University of Belgrade. It is to be regretted that, owing to want of space requiring the omission of several addenda to the text of the Theoria itself, a large amount of interesting material collected by Professor Petronievic has had to be left out. The financial support necessary for the production of such a costly edition as the present has been met mainly by the Government of the Kingdom of Serbs, Croats and Slovenes ; and the subsidiary expenses by some Jugo-Slavs interested in the publication. After the " Life," there follows an " Introduction," in which I have discussed the ideas of Boscovich, as far as they may be gathered from the text of the Tbeoria alone ; this also has been cut down, those parts which are clearly presented to the reader in Boscovich's own Synopsis having been omitted. It is a matter of profound regret to everyone that this discussion comes from my pen instead of, as was originally arranged, from that of the late Philip E. P. Jourdain, the well-known mathematical logician ; whose untimely death threw into my far less capable hands the responsible duties of editorship. I desire to thank the authorities of the Cambridge University Library, who time after time over a period of five years have forwarded to me the original text of this work of Boscovich. Great credit is also due to the staff of Messrs. Butler & Tanner, Frome, for the care and skill with which they have carried out their share of the work ; and my special thanks for the unfailing painstaking courtesy accorded to my demands, which were frequently not in agreement with trade custom. J. M. CHILD. MANCHESTER UNIVERSITY, December, 1921. LIFE OF ROGER JOSEPH BOSCOVICH By BRANISLAV PETRONIEVIC' ]HE Slav world, being still in its infancy, has, despite a considerable number of scientific men, been unable to contribute as largely to general science as the other great European nations. It has, nevertheless, demonstrated its capacity of producing scientific works of the highest value. Above all, as I have elsewhere indicated," it possesses Copernicus, Lobachevski, Mendeljev, and Boscovich. In the following article, I propose to describe briefly the life of the Jugo-Slav, Boscovich, whose principal work is here published for the sixth time ; the first edition having appeared in 1758, and others in 1759, 1763, 1764, and 1765. The present text is from the edition of 1763, the first Venetian edition, revised and enlarged. On his father's side, the family of Boscovich is of purely Serbian origin, his grandfather, Bosko, having been an orthodox Serbian peasant of the village of Orakova in Herzegovina. His father, Nikola, was first a merchant in Novi Pazar (Old Serbia), but later settled in Dubrovnik (Ragusa, the famous republic in Southern Dalmatia), whither his father, Bosko, soon followed him, and where Nikola became a Roman Catholic. Pavica, Boscovich's mother, belonged to the Italian family of Betere, which for a century had been established in Dubrovnik and had become Slavonicized Bara Betere, Pavica's father, having been a poet of some reputation in Ragusa. Roger Joseph Boscovich (Rudjer Josif Boskovic', in Serbo-Croatian) was born at Ragusa on September i8th, 1711, and was one of the younger members of a large family. He received his primary and secondary education at the Jesuit College of his native town ; in 1725 he became a member of the Jesuit order and was sent to Rome, where from 1728 to 1733 he studied philosophy, physics and mathematics in the Collegium Romanum. From 1733 to 1738 he taught rhetoric and grammar in various Jesuit schools ; he became Professor of mathematics in the Collegium Romanum, continuing at the same time his studies in theology, until in 1744 he became a priest and a member of his order. In 1736, Boscovich began his literary activity with the first fragment, " De Maculis Solaribus," of a scientific poem, " De Solis ac Lunse Defectibus " ; and almost every succeeding year he published at least one treatise upon some scientific or philosophic problem. His reputation as a mathematician was already established when he was commissioned by Pope Benedict XIV to examine with two other mathematicians the causes of the weakness in the cupola of St. Peter's at Rome. Shortly after, the same Pope commissioned him to consider various other problems, such as the drainage of the Pontine marshes, the regulariza- tion of the Tiber, and so on. In 1756, he was sent by the republic of Lucca to Vienna as arbiter in a dispute between Lucca and Tuscany. During this stay in Vienna, Boscovich was commanded by the Empress Maria Theresa to examine the building of the Imperial Library at Vienna and the cupola of the cathedral at Milan. But this stay in Vienna, which lasted until 1758, had still more important consequences ; for Boscovich found time there to finish his principal work, Theoria Philosophies Naturalis ; the publication was entrusted to a Jesuit, Father Scherffer, Boscovich having to leave Vienna, and the first edition appeared in 1758, followed by a second edition in the following year. With both of these editions, Boscovich was to some extent dissatisfied (see the remarks made by the printer who carried out the third edition at Venice, given in this volume on page 3) ; so a third edition was issued at Venice, revised, enlarged and rearranged under the author's personal superintendence in 1763. The revision was so extensive that as the printer remarks, " it ought to be considered in some measure as a first and original edition " ; and as such it has been taken as the basis of the translation now published. The fourth and fifth editions followed in 1764 and 1765. One of the most important tasks which Boscovich was commissioned to undertake was that of measuring an arc of the meridian in the Papal States. Boscovich had designed to take part in a Portuguese expedition to Brazil on a similar errand ; but he was per- " Slav Achievements in Advanced Science, London, 1917. vii viii A THEORY OF NATURAL PHILOSOPHY suaded by Pope Benedict XIV, in 1750, to conduct, in collaboration with an English Jesuit, Christopher Maire, the measurements in Italy. The results of their work were published, in 1755, by Boscovich, in a treatise, De Litter aria Expedition^ -per Pontificiam, &c. ; this was translated into French under the title of Voyage astronomique et geograpbique dans VEtat de VEglise, in 1770. By the numerous scientific treatises and dissertations which he had published up to 1759, and by his principal work, Boscovich had acquired so high a reputation in Italy, nay in Europe at large, that the membership of numerous academies and learned societies had already been conferred upon him. In 1760, Boscovich, who hitherto had been bound to Italy by his professorship at Rome, decided to leave that country. In this year we find him at Paris, where he had gone as the travelling companion of the Marquis Romagnosi. Although in the previous year the Jesuit order had been expelled from France, Boscovich had been received on the strength of his great scientific reputation. Despite this, he did not feel easy in Paris ; and the same year we find him in London, on a mission to vindicate the character of his native place, the suspicions of the British Government, that Ragusa was being used by France to fit out ships of war, having been aroused ; this mission he carried out successfully. In London he was warmly welcomed, and was made a member of the Royal Society. Here he published his work, De Solis ac Lunce defectibus, dedicating it to the Royal Society. Later, he was commissioned by the Royal Society to proceed to Cali- fornia to observe the transit of Venus ; but, as he was unwilling to go, the Society sent him to Constantinople for the same purpose. He did not, however, arrive in time to make the observation ; and, when he did arrive, he fell ill and was forced to remain at Constantinople for seven months. He left that city in company with the English ambas- sador, Porter, and, after a journey through Thrace, Bulgaria, and Moldavia, he arrived finally at Warsaw, in Poland ; here he remained for a time as the guest of the family of PoniatowsM. In 1762, he returned from Warsaw to Rome by way of Silesia and Austria. The first part of this long journey has been described by Boscovich himself in his Giornale di un viaggio da Constantinopoli in Polonia the original of which was not published until 1784, although a French translation had appeared in 1772, and a German translation in 1779. Shortly after his return to Rome, Boscovich was appointed to a chair at the University of Pavia ; but his stay there was not of long duration. Already, in 1764, the building of the observatory of Brera had been begun at Milan according to the plans of Boscovich ; and in 1770, Boscovich was appointed its director. Unfortunately, only two years later he was deprived of office by the Austrian Government which, in a controversy between Boscovich and another astronomer of the observatory, the Jesuit Lagrange, took the part of his opponent. The position of Boscovich was still further complicated by the disbanding of his company ; for, by the decree of Clement V, the Order of Jesus had been suppressed in 1773. In the same year Boscovich, now free for the second time, again visited Paris, where he was cordially received in official circles. The French Government appointed him director of " Optique Marine," with an annual salary of 8,000 francs ; and Boscovich became a French subject. But, as an ex- Jesuit, he was not welcomed in all scientific circles. The celebrated d'Alembert was his declared enemy ; on the other hand, the famous astronomer, Lalande, was his devoted friend and admirer. Particularly, in his controversy with Rochon on the priority of the discovery of the micrometer, and again in the dispute with Laplace about priority in the invention of a method for determining the orbits of comets, did the enmity felt in these scientific circles show itself. In Paris, in 1779, Boscovich published a new edition of his poem on eclipses, translated into French and annotated, under the title, Les Eclipses, dedicating the edition to the King, Louis XV. During this second stay in Paris, Boscovich had prepared a whole series of new works, which he hoped would have been published at the Royal Press. But, as the American War of Independence was imminent, he was forced, in 1782, to take two years' leave of absence, and return to Italy. He went to the house of his publisher at Bassano ; and here, in 1 785^ were published five volumes of his optical and astronomical works, Opera pertinentia ad opticam et astronomiam. Boscovich had planned to return through Italy from Bassano to Paris ; indeed, he left Bassano for Venice, Rome, Florence, and came to Milan. Here he was detained by illness and he was obliged to ask the French Government to extend his leave, a request that was willingly granted. His health, however, became worse ; and to it was added a melancholia. He died on February I3th, 1787. The great loss which Science sustained by his death has been fitly commemorated in the eulogium by his friend Lalande in the French Academy, of which he was a member ; and also in that of Francesco Ricca at Milan, and so on. But it is his native town, his beloved Ragusa, which has most fitly celebrated the death of the greatest of her sons A THEORY OF NATURAL PHILOSOPHY ix in the eulogium of the poet, Bernardo Zamagna. " This magnificent tribute from his native town was entirely deserved by Boscovich, both for his scientific works, and for his love and work for his country. Boscovich had left his native country when a boy, and returned to it only once after- wards, when, in 1747, he passed the summer there, from June 20th to October 1st ; but he often intended to return. In a letter, dated May 3rd, 1774, he seeks to secure a pension as a member of the Jesuit College of Ragusa ; he writes : " I always hope at last to find my true peace in my own country and, if God permit me, to pass my old age there in quietness." Although Boscovich has written nothing in his own language, he understood it per- fectly ; as is shown by the correspondence with his sister, by certain passages in his Italian letters, and also by his Giornale (p. 31 ; p. 59 of the French edition). In a dispute with d'Alembert, who had called him an Italian, he said : " we will notice here in the first place that our author is a Dalmatian, and from Ragusa, not Italian ; and that is the reason why Marucelli, in a recent work on Italian authors, has made no mention of him." * That his feeling of Slav nationality was strong is proved by the tributes he pays to his native town and native land in his dedicatory epistle to Louis XV. Boscovich was at once philosopher, astronomer, physicist, mathematician, historian, engineer, architect, and poet. In addition, he was a diplomatist and a man of the world ; and yet a good Catholic and a devoted member of the Jesuit order. His friend, Lalande, has thus sketched his appearance and his character : " Father Boscovich was of great stature ; he had a noble expression, and his disposition was obliging. He accommodated himself with ease to the foibles of the great, with whom he came into frequent contact. But his temper was a trifle hasty and irascible, even to his friends at least his manner gave that impression but this solitary defect was compensated by all those qualities which make up a great man. . . . He possessed so strong a constitution that it seemed likely that he would have lived much longer than he actually did ; but his appetite was large, and his belief in the strength of his constitution hindered him from paying sufficient attention to the danger which always results from this." From other sources we learn that Boscovich had only one meal daily, dejeuner. Of his ability as a poet, Lalande says : " He was himself a poet like his brother, who was also a Jesuit. . . . Boscovich wrote verse in Latin only, but he composed with extreme ease. He hardly ever found himself in company without dashing off some impromptu verses to well-known men or charming women. To the latter he paid no other attentions, for his austerity was always exemplary. . . . With such talents, it is not to be wondered at that he was everywhere appreciated and sought after. Ministers, princes and sovereigns all received him with the greatest distinction. M. de Lalande witnessed this in every part of Italy where Boscovich accompanied him in 1765." Boscovich was acquainted with several languages Latin, Italian, French, as well as his native Serbo-Croatian, which, despite his long absence from his country, he did not forget. Although he had studied in Italy and passed the greater part of his life there, he had never penetrated to the spirit of the language, as his Italian biographer, Ricca, notices. His command of French was even more defective ; but in spite of this fact, French men of science urged him to write in French. English he did not understand, as he confessed in a letter to Priestley ; although he had picked up some words of polite conversation during his stay in London. His correspondence was extensive. The greater part of it has been published in the Memoirs de VAcademie Jougo-Slave of Zagrab, 1887 to 1912. " Oratio in funere R. J. Boscovichii ... a Bernardo Zamagna. * Voyage Astronomique, p. 750 ; also on pp. 707 seq. Journal des Sfavans, Fevrier, 1792, pp. 113-118. INTRODUCTION ALTHOUGH the title to this work to a very large extent correctly describes the contents, yet the argument leans less towards the explanation of a theory than it does towards the logical exposition of the results that must follow from the acceptance of certain fundamental assumptions, more or less generally admitted by natural philosophers of the time. The most important of these assumptions is the doctrine of Continuity, as enunciated by Leibniz. This doctrine may be shortly stated in the words : " Every- thing takes place by degrees " ; or, in the phrase usually employed by Boscovich : " Nothing happens -per saltum." The second assumption is the axiom of Impenetrability ; that is to say, Boscovich admits as axiomatic that no two material points can occupy the same spatial, or local, point simultaneously. Clerk Maxwell has characterized this assumption as " an unwarrantable concession to the vulgar opinion." He considered that this axiom is a prejudice, or prejudgment, founded on experience of bodies of sensible size. This opinion of Maxwell cannot however be accepted without dissection into two main heads. The criticism of the axiom itself would appear to carry greater weight against Boscovich than against other philosophers ; but the assertion that it is a prejudice is hardly warranted. For, Boscovich, in accepting the truth of the axiom, has no experience on which to found his acceptance. His material points have absolutely no magnitude ; they are Euclidean points, " having no parts." There is, therefore, no reason for assuming, by a sort of induction (and Boscovich never makes an induction without expressing the reason why such induction can be made), that two material points cannot occupy the same local point simultaneously ; that is to say, there cannot have been a prejudice in favour of the acceptance of this axiom, derived from experience of bodies of sensible size ; for, since the material points are non- extended, they do not occupy space, and cannot therefore exclude another point from occupying the same space. Perhaps, we should say the reason is not the same as that which makes it impossible for bodies of sensible size. The acceptance of the axiom by Boscovich is purely theoretical ; in fact, it constitutes practically the whole of the theory of Boscovich. On the other hand, for this very reason, there are no readily apparent grounds for the acceptance of the axiom ; and no serious arguments can be adduced in its favour ; Boscovich 's own line of argument, founded on the idea that infinite improbability comes to the same thing as impossibility, is given in Art. 361. Later, I will suggest the probable source from which Boscovich derived his idea of impenetrability as applying to points of matter, as distinct from impenetrability for bodies of sensible size. Boscovich's own idea of the merit of his work seems to have been chiefly that it met the requirements which, in the opinion of Newton, would constitute " a mighty advance in philosophy." These requirements were the " derivation, from the phenomena of Nature, of two or three general principles ; and the explanation of the manner in which the properties and actions of all corporeal things follow from these principles, even if the causes of those principles had not at the time been discovered." Boscovich claims in his preface to the first edition (Vienna, 1758) that he has gone far beyond these requirements ; in that he has reduced all the principles of Newton to a single principle namely, that given by his Law of Forces. The occasion that led to the writing of this work was a request, made by Father Scherffer, who eventually took charge of the first Vienna edition during the absence of Boscovich ; he suggested to Boscovich the investigation of the centre of oscillation. Boscovich applied to this investigation the principles which, as he himself states, " he lit upon so far back as the year 1745." Of these principles he had already given some indication in the dissertations De Viribus vivis (published in 1745), De Lege Firium in Natura existentium (1755), and others. While engaged on the former dissertation, he investigated the production and destruction of velocity in the case of impulsive action, such as occurs in direct collision. In this, where it is to be noted that bodies of sensible size are under consideration, Boscovich was led to the study of the distortion and recovery of shape which occurs on impact ; he came to the conclusion that, owing to this distortion and recovery of shape, there was produced by the impact a continuous retardation of the relative velocity during the whole time of impact, which was finite ; in other words, the Law of Continuity, as enunciated by XI xii INTRODUCTION Leibniz, was observed. It would appear that at this time (1745) Boscovich was concerned mainly, if not solely, with the facts of the change of velocity, and not with the causes for this change. The title of the dissertation, De Firibus vivis, shows however that a secondary consideration, of almost equal importance in the development of the Theory of Boscovich, also held the field. The natural philosophy of Leibniz postulated monads, without parts, extension or figure. In these features the monads of Leibniz were similar to the material points of Boscovich ; but Leibniz ascribed to his monads 1 perception and appetition in addition to an equivalent of inertia. They are centres of force, and the force exerted is a vis viva. Boscovich opposes this idea of a " living," or " lively " force ; and in this first dissertation we may trace the first ideas of the formulation of his own material points. Leibniz denies action at a distance ; with Boscovich it is the fundamental characteristic of a material point. The principles developed in the work on collisions of bodies were applied to the problem of the centre of oscillation. During the latter investigation Boscovich was led to a theorem on the mutual forces between the bodies forming a system of three ; and from this theorem there followed the natural explanation of a whole sequence of phenomena, mostly connected with the idea of a statical moment ; and his initial intention was to have published a dissertation on this theorem and deductions from it, as a specimen of the use and advantage of his principles. But all this time these principles had been developing in two directions, mathematically and philosophically, and by this time included the fundamental notions of the law of forces for material points. The essay on the centre of oscillation grew in length as it proceeded ; until, finally, Boscovich added to it all that he had already published on the subject of his principles and other matters which, as he says, " obtruded themselves on his notice as he was writing." The whole of this material he rearranged into a more logical (but unfortunately for a study of development of ideas, non-chronological) order before publication. As stated by Boscovich, in Art. 164, the whole of his Theory is contained in his statement that : " Matter is composed of perfectly indivisible, non-extended, discrete points." To this assertion is conjoined the axiom that no two material points can be in the same point of space at the same time. As stated above, in opposition to Clerk Maxwell, this is no matter of prejudice. Boscovich, in Art. 361, gives his own reasons for taking this axiom as part of his theory. He lays it down that the number of material points is finite, whereas the number of local points is an infinity of three dimensions ; hence it is infinitely improbable, i.e., impossible, that two material points, without the action of a directive mind, should ever encounter one another, and thus be in the same place at the same time. He even goes further ; he asserts elsewhere that no material point ever returns to any point of space in which it has ever been before, or in which any other material point has ever been. Whether his arguments are sound or not, the matter does not rest on a prejudgment formed from experience of bodies of sensible size ; Boscovich has convinced himself by such arguments of the truth of the principle of Impenetrability, and lays it down as axiomatic ; and upon this, as one of his foundations, builds his complete theory. The consequence of this axiom is immediately evident ; there can be no such thing as contact between any two material points ; two points cannot be contiguous or, as Boscovich states, no two points of matter can be in mathematical contact. For, since material points have no dimensions, if, to form an imagery of Boscovich's argument, we take two little squares ABDC, CDFE to represent two points in mathema- tical contact along the side CD, then CD must also coincide with AB, and EF with CD ; that is the points which we have supposed to be contiguous must also be coincident. This is contrary to the axiom of Impenetrability ; and hence material points must be separated always O U Ir by a finite interval, no matter how small. This finite interval however has no minimum ; nor has it, on the other hand, on account of the infinity of space, any maximum, except under certain hypothetical circumstances which may possibly exist. Lastly, these points of matter float, so to speak, in an absolute void. Every material point is exactly like every other material point ; each is postulated to have an inherent propensity (determinatio) to remain in a state of rest or uniform motion in a straight line, whichever of these is supposed to be its initial state, so long as the point is not subject to some external influence. Thus it is endowed with an equivalent of inertia as formulated by Newton ; but as we shall see, there does not enter the Newtonian idea of inertia as a characteristic of mass. The propensity is akin to the characteristic ascribed to the monad by Leibniz ; with this difference, that it is not a symptom of activity, as with Leibniz, but one of inactivity. 1 See Bertrand Russell, Philosophy of Leibniz ; especially p. 91 for connection between Boscovich and Leibniz. INTRODUCTION xiii Further, according to Boscovich, there is a mutual vis between every pair of points, the magnitude of which depends only on the distance between them. At first sight, there would seem to be an incongruity in this supposition ; for, since a point has no magnitude, it cannot have any mass, considered as " quantity of matter " ; and therefore, if the slightest " force " (according to the ordinary acceptation of the term) existed between two points, there would be an infinite acceleration or retardation of each point relative to the other. If, on the other hand, we consider with Clerk Maxwell that each point of matter has a definite small mass, this mass must be finite, no matter how small, and not infinitesimal. For the mass of a point is the whole mass of a body, divided by the number of points of matter composing that body, which are all exactly similar ; and this number Boscovich asserts is finite. It follows immediately that the density of a material point must be infinite, since the volume is an infinitesimal of the third order, if not of an infinite order, i.e., zero. Now, infinite density, if not to all of us, to Boscovich at least is unimaginable. Clerk Maxwell, in ascribing mass to a Boscovichian point of matter, seems to have been obsessed by a prejudice, that very prejudice which obsesses most scientists of the present day, namely, that there can be no force without mass. He understood that Boscovich ascribed to each pair of points a mutual attraction or repulsion ; and, in consequence, prejudiced by Newton's Laws of Motion, he ascribed mass to a material point of Boscovich. This apparent incongruity, however, disappears when it is remembered that the word vis, as used by the mathematicians of the period of Boscovich, had many different meanings ; or rather that its meaning was given by the descriptive adjective that was associated with it. Thus we have vis viva (later associated with energy), vis mortua (the antithesis of vis viva, as understood by Leibniz), vis acceleratrix (acceleration), vis matrix (the real equivalent of force, since it varied with the mass directly), vis descensiva (moment of a weight hung at one end of a lever), and so on. Newton even, in enunciating his law of universal gravitation, apparently asserted nothing more than the fact of gravitation a propensity for approach according to the inverse square of the distance : and Boscovich imitates him in this. The mutual vires, ascribed by Boscovich to his pairs of points, are really accelerations, i.e. tendencies for mutual approach or recession of the two points, depending on the distance between the points at the time under consideration. Boscovich's own words, as given in Art. 9, are : " Censeo igitur bina quaecunque materise puncta determinari asque in aliis distantiis ad mutuum accessum, in aliis ad recessum mutuum, quam ipsam determinationem apello vim." The cause of this determination, or propensity, for approach or recession, which in the case of bodies of sensible size is more correctly called " force " (vis matrix), Boscovich does not seek to explain ; he merely postulates the propensities. The measures of these propensities, i.e., the accelerations of the relative velocities, are the ordinates of what is usually called his curve of forces. This is corroborated by the statement of Boscovich that the areas under the arcs of his curve are proportional to squares of velocities ; which is in accordance with the formula we should now use for the area under an " acceleration- space " graph (Area = J f.ds = j-r-ds = I v.dv). See Note (f) to Art. 118, where it is evident that the word vires, translated " forces," strictly means " accelerations ; " seejalso Art.64- Thus it would appear that in the Theory of Boscovich we have something totally different from the monads of Leibniz, which are truly centres of force. Again, although there are some points of similarity with the ideas of Newton, more especially in the postulation of an acceleration of the relative velocity of every pair of points of matter due to and depending upon the relative distance between them, without any endeavour to explain this acceleration or gravitation ; yet the Theory of Boscovich differs from that of Newton in being purely kinematical. His material point is defined to be without parts, i.e., it has no volume ; as such it can have no mass, and can exert no force, as we understand such terms. The sole characteristic that has a finite measure is the relative acceleration produced by the simultaneous existence of two points of matter ; and this acceleration depends solely upon the distance between them. The Newtonian idea of mass is replaced by something totally different ; it is a mere number, without " dimension " ; the " mass " of a body is simply the number of points that are combined to " form " the body. Each of these points, if sufficiently close together, will exert on another point of matter, at a relatively much greater distance from every point of the body, the same acceleration very approximately. Hence, if we have two small bodies A and B, situated at a distance s from one another (the wording of this phrase postulates that the points of each body are very close together as compared with the distance between the bodies) : and if the number of points in A and B are respectively a and b, and / is the mutual acceleration between any pair of material points at a distance s from one another ; then, each point of A will give to each point of B an acceleration /. Hence, the body A will give to each point of B, and therefore to the whole of B, an acceleration equal to a/. Similarly the body B will give to xiv INTRODUCTION the body A an acceleration equal to bf. Similarly, if we placed a third body, C, at a distance j from A and B, the body A would give the body C an acceleration equal to af, and the body B would give the body C an acceleration equal to bf. That is, the accelerations given to a standard body C are proportional to the " number of points " in the bodies producing these accelerations ; thus, numerically, the " mass " of Boscovich comes to the same thing as the " mass " of Newton. Further, the acceleration given by C to the bodies A and B is the same for either, namely, cf ; from which it follows that all bodies have their velocities of fall towards the earth equally accelerated, apart from the resistance of the air ; and so on. But the term " force," as the cause of acceleration is not applied by Boscovich to material points ; nor is it used in the Newtonian sense at all. When Boscovich investigates the attraction of " bodies," he introduces the idea of a cause, but then only more or less as a convenient phrase. Although, as a philosopher, Boscovich denies that there is any possibility of a fortuitous circumstance (and here indeed we may admit a prejudice derived from experience ; for he states that what we call fortuitous is merely something for which we, in our limited intelligence, can assign no cause), yet with him the existent thing is motion and not force. The latter word is merely a convenient phrase to describe the " product " of " mass " and " acceleration." To sum up, it would seem that the curve of Boscovich is an acceleration-interval graph ; and it is a mistake to refer to his cosmic system as a system of " force-centres." His material points have zero volume, zero mass, and exert zero force. In fact, if one material point alone existed outside the mind, and there were no material point forming part of the mind, then this single external point could in no way be perceived. In other words, a single point would give no sense-datum apart from another point ; and thus single points might be considered as not perceptible in themselves, but as becoming so in relation to other material points. This seems to be the logical deduction from the strict sense of the definition given by Boscovich ; what Boscovich himself thought is given in the supplements that follow the third part of the treatise. Nevertheless, the phraseology of " attraction " and " repulsion " is so much more convenient than that of " acceleration of the velocity of approach " and " acceleration of the velocity of recession," that it will be used in what follows : as it has been used throughout the translation of the treatise. There is still another point to be considered before we take up the study of the Boscovich curve ; namely, whether we are to consider Boscovich as, consciously or unconsciously, an atomist in the strict sense of the word. The practical test for this question would seem to be simply whether the divisibility of matter was considered to be limited or unlimited. Boscovich himself appears to be uncertain of his ground, hardly knowing which point of view is the logical outcome of his definition of a material point. For, in Art. 394, he denies infinite divisibility ; but he admits infinite componibility. The denial of infinite divisibility is necessitated by his denial of " anything infinite in Nature, or in extension, or a self- determined infinitely small." The admission of infinite componibility is necessitated by his definition of the material point ; since it has no parts, a fresh point can always be placed between any two points without being contiguous to either. Now, since he denies the existence of the infinite and the infinitely small, the attraction or repulsion between two points of matter (except at what he calls the limiting intervals) must be finite : hence, since the attractions of masses are all by observation finite, it follows that the number of points in a mass must be finite. To evade the difficulty thus raised, he appeals to the scale of integers, in which there is no infinite number : but, as he says, the scale of integers is a sequence of numbers increasing indefinitely, and having no last term. Thus, into any space, however small, there may be crowded an indefinitely great number of material points ; this number can be still further increased to any extent ; and yet the number of points finally obtained is always finite. It would, again, seem that the system of Boscovich was not a material system, but a system of relations ; if it were not for the fact that he asserts, in Art. 7, that his view is that " the Universe does not consist of vacuum interspersed amongst matter, but that matter is interspersed in a vacuum and floats in it." The whole question is still further complicated by his remark, in Art. 393, that in the continual division of a body, " as soon as we reach intervals less than the distance between two material points, further sections will cut empty intervals and not matter " ; and yet he has postulated that there is no minimum value to the interval between two material points. Leaving, however, this question of the philosophical standpoint of Boscovich to be decided by the reader, after a study of the supplements that follow the third part of the treatise, let us now consider the curve of Boscovich. Boscovich, from experimental data, gives to his curve, when the interval is large, a branch asymptotic to the axis of intervals ; it approximates to the " hyperbola " x*y c, in which x represents the interval between two points, and y the vis corresponding to that interval, which we have agreed to call an attraction, meaning thereby, not a force, but an INTRODUCTION xv acceleration of the velocity of approach. For small intervals he has as yet no knowledge of the quality or quantity of his ordinates. In Supplement IV, he gives some very ingenious arguments against forces that are attractive at very small distances and increase indefinitely, such as would be the case where the law of forces was represented by an inverse power of the interval, or even where the force varied inversely as the interval. For the inverse fourth or higher power, he shows that the attraction of a sphere upon a point on its surface would be less than the attraction of a part of itself on this point ; for the inverse third power, he con- siders orbital motion, which in this case is an equiangular spiral motion, and deduces that after a finite time the particle must be nowhere at all. Euler, considering this case, asserted that on approaching the centre of force the particle must be annihilated ; Boscovich, with more justice, argues that this law of force must be impossible. For the inverse square law, the limiting case of an elliptic orbit, when the transverse velocity at the end of the major axis is decreased indefinitely, is taken ; this leads to rectilinear motion of the particle to the centre of force and a return from it ; which does not agree with the otherwise proved oscillation through the centre of force to an equal distance on either side. Now it is to be observed that this supplement is quoted from his dissertation De Lege Firium in Natura existentium, which was published in 1755 ; also that in 1743 he had published a dissertation of which the full title is : De Motu Corporis attracti in centrum immobile viribus decrescentibus in ratione distantiarum reciproca duplicata in spatiis non resistentibus. Hence it is not too much to suppose that somewhere between 1741 and 1755 he had tried to find a means of overcoming this discrepancy ; and he was thus led to suppose that, in the case of rectilinear motion under an inverse square law, there was a departure from the law on near approach to the centre of force ; that the attraction was replaced by a repulsion increasing indefinitely as the distance decreased ; for this obviously would lead to an oscillation to the centre and back, and so come into agreement with the limiting case of the elliptic orbit. I therefore suggest that it was this consideration that led Boscovich to the doctrine of Impenetrability. However, in the treatise itself, Boscovich postulates the axiom of Impenetrability as applying in general, and thence argues that the force at infinitely small distances must be repulsive and increasing indefinitely. Hence the ordinate to the curve near the origin must be drawn in the opposite direction to that of the ordinates for sensible distances, and the area under this branch of the curve must be indefinitely great. That is to say, the branch must be asymptotic to the axis of ordinates ; Boscovich however considers that this does not involve an infinite ordinate at the origin, because the interval between two material points is never zero ; or, vice versa, since the repulsion increases indefinitely for very small intervals, the velocity of relative approach, no matter how great, of two material points is always destroyed before actual contact ; which necessitates a finite interval between two material points, and the impossibility of encounter under any circum- stances : the interval however, since a velocity of mutual approach may be supposed to be of any magnitude, can have no minimum. Two points are said to be in physical contact, in opposition to mathematical contact, when they are so close together that this great mutual repulsion is sufficiently increased to prevent nearer approach. Since Boscovich has these two asymptotic branches, and he postulates Continuity, there must be a continuous curve, with a one-valued ordinate for any interval, to represent the " force " at all other distances ; hence the curve must cut the axis at some point in between, or the ordinate must become infinite. He does not lose sight of this latter possi- bility, but apparently discards it for certain mechanical and physical reasons. Now, it is known that as the degree of a curve rises, the number of curves of that degree increases very rapidly ; there is only one of the first degree, the conic sections of the second degree, while Newton had found over three-score curves with equations of the third degree, and nobody had tried to find all the curves of the fourth degree. Since his curve is not one of the known curves, Boscovich concludes that the degree of its equation is very high, even if it is not transcendent. But the higher the degree of a curve, the greater the number of possible intersections with a given straight line ; that is to say, it is highly probable that there are a great many intersections of the curve with the axis ; i.e., points giving zero action for material points situated "at the corresponding distance from one another. Lastly, since the ordinate is one-valued, the equation of the curve, as stated in Supplement III, must be of the form P-Qy = o, where P and Q are functions of x alone. Thus we have a curve winding about the axis for intervals that are very small and developing finally into the hyperbola of the third degree for sensible intervals. This final branch, however, cannot be exactly this hyperbola ; for, Boscovich argues, if any finite arc of the curve ever coincided exactly with the hyperbola of the third degree, it would be a breach of continuity if it ever departed from it. Hence he concludes that the inverse square law is observed approximately only, even at large distances. As stated above, the possibility of other asymptotes, parallel to the asymptote at the INTRODUCTION origin, is not lost sight of. The consequence of one occurring at a very small distance from the origin is discussed in full. Boscovich, however, takes great pains to show that all the phenomena discussed can be explained on the assumption of a number of points of inter- section of his curve with the axis, combined with different characteristics of the arcs that lie between these points of intersection. There is, however, one suggestion that is very interesting, especially in relation to recent statements of Einstein and Weyl. Suppose that beyond the distances of the solar system, for which the inverse square law obtains approxi- mately at least, the curve of forces, after touching the axis (as it may do, since it does not coincide exactly with the hyperbola of the third degree), goes off to infinity in the positive direction ; or suppose that, after cutting the axis (as again it may do, for the reason given above), it once more begins to wind round the axis and finally has an asymptotic attractive branch. Then it is evident that the universe in which we live is a self-contained cosmic system ; for no point within it can ever get beyond the distance of this further asymptote. If in addition, beyond this further asymptote, the curve had an asymptotic repulsive branch and went on as a sort of replica of the curve already obtained, then no point outside our universe could ever enter within it. Thus there is a possibility of infinite space being filled with a succession of cosmic systems, each of which would never interfere with any other ; indeed, a mind existing in any one of these universes could never perceive the existence of any other universe except that in which it existed. Thus space might be in reality infinite, and yet never could be perceived except as finite. The use Boscovich makes of his curve, the ingenuity of his explanations and their logic, the strength or weakness of his attacks on the theories of other philosophers, are left to the consideration of the reader of the text. It may, however, be useful to point out certain matters which seem more than usually interesting. Boscovich points out that no philosopher has attempted to prove the existence of a centre of gravity. It would appear especially that he is, somehow or other, aware of the mistake made by Leibniz in his early days (a mistake corrected by Huygens according to the statement of Leibniz), and of the use Leibniz later made of the principle of moments ; Boscovich has apparently considered the work of Pascal and others, especially Guldinus ;, it looks almost as if (again, somehow or other) he had seen some description of " The Method " of Archimedes. For he proceeds to define the centre of gravity geometrically, and to prove that there is always a centre of gravity, or rather a geometrical centroid ; whereas, even for a triangle, there is no centre of magnitude, with which Leibniz seems to have confused a centroid before his conversation with Huygens. This existence proof, and the deductions from it, are necessary foundations for the centro- baryc analysis of Leibniz. The argument is shortly as follows : Take a plane outside, say to the right of, all the points of all the bodies under consideration ; find the sum of all the distances of all the points from this plane ; divide this sum by the number of points ; draw a plane to the left of and parallel to the chosen plane, at a distance from it equal to the quotient just found. Then, observing algebraic sign, this is a plane such that the sum of the distances of all the points from it is zero ; i.e., the sum of the distances of all the points on one side of this plane is equal arithmetically to the sum of the distances of all the points on the other side. Find a similar plane of equal distances in another direction ; this intersects the first plane in a straight line. A third similar plane cuts this straight line in a point ; this point is the centroid ; it has the unique property that all planes through it are planes of equal distances. If some of the points are conglomerated to form a particle, the sum of the distances for each of the points is equal to the distance of the particle multiplied by the number of points in the particle, i.e., by the mass of the particle. Hence follows the theorem for the statical moment for lines and planes or other surfaces, as well as for solids that have weight. Another interesting point, in relation to recent work, is the subject-matter of Art. 230- 236 ; where it is shown that, due solely to the mutual forces exerted on a third point by two points separated by a proper interval, there is a series of orbits, approximately confocal ellipses, in which the third point is in a state of steady motion ; these orbits are alternately stable and stable. If the steady motion in a stable orbit is disturbed, by a sufficiently great difference of the velocity being induced by the action of a fourth point passing sufficiently near the third point, this third point will leave its orbit and immediately take up another stable orbit, after some initial oscillation about it. This elegant little theorem does not depend in any way on the exact form of the curve of forces, so long as there are portions of the curve winding about the axis for very small intervals between the points. It is sufficient, for the next point, to draw the reader's attention to Art. 266-278, on collision, and to the articles which follow on the agreement between .resolution and com- position of forces as a working hypothesis. From what Boscovich says, it would appear that philosophers of his time were much perturbed over the idea that, when a force was resolved into two forces at a sufficiently obtuse angle, the force itself might be less than either of INTRODUCTION xvii the resolutes. Boscovich points out that, in his Theory, there is no resolution, only com- position ; and therefore the difficulty does not arise. In this connection he adds that there are no signs in Nature of anything approaching the vires viva of Leibniz. In Art. 294 we have Boscovich's contribution to the controversy over the correct measure of the " quantity of motion " ; but, as there is no attempt made to follow out the change in either the velocity or the square of the velocity, it cannot be said to lead to any- thing conclusive. As a matter of fact, Boscovich uses the result to prove the non-existence of vires vivce. In Art. 298-306 we have a mechanical exposition of reflection and refraction of light. This comes under the section on Mechanics, because with Boscovich light is matter moving with a very high velocity, and therefore reflection is a case of impact, in that it depends upon the destruction of the whole of the perpendicular velocity upon entering the " surface " of a denser medium, the surface being that part of space in front of the physical surface of the medium in which the particles of light are near enough to the denser medium to feel the influence of the last repulsive asymptotic branch of the curve of forces. If this perpendicular velocity is not all destroyed, the particle enters the medium, and is refracted ; in which case, the existence of a sine law is demonstrated. It is to be noted that the " fits " of alternate attraction and repulsion, postulated by Newton, follow as a natural consequence of the winding portion of the curve of Boscovich. In Art. 328-346 we have a discussion of the centre of oscillation, and the centre of percussion is investigated as well for masses in a plane perpendicular to the axis of rotation, and masses lying in a straight line, where each mass is connected with the different centres. Boscovich deduces from his theory the theorems, amongst others, that the centres of suspen- sion and oscillation are interchangeable, and that the distance between them is equal to the distance of the centre of percussion from the axis of rotation ; he also gives a rule for finding the simple equivalent pendulum. The work is completed in a letter to Fr. Scherffer, which is appended at the end of this volume. In the third section, which deals with the application of the Theory to Physics, we naturally do not look for much that is of value. But, in Art. 505, Boscovich evidently has the correct notion that sound is a longitudinal vibration of the air or some other medium ; and he is able to give an explanation of the propagation of the disturbance purely by means of the mutual forces between the particles of the medium. In Art. 507 he certainly states that the cause of heat is a " vigorous internal motion " ; but this motion is that of the " particles of fire," if it is a motion ; an alternative reason is however given, namely, that it may be a " fermentation of a sulphurous substance with particles of light." " Cold is a lack of this substance, or of a motion of it." No attention will be called to this part of the work, beyond an expression of admiration for the great ingenuity of a large part of it. There is a metaphysical appendix on the seat of the mind, and its nature, and on the existence and attributes of GOD. This is followed by two short discussions of a philosophical nature on Space and Time. Boscovich does not look on either of these as being in themselves existent ; his entities are modes of existence, temporal and local. These three sections are full of interest for the modern philosophical reader. Supplement V is a theoretical proof, purely derived from the theory of mutual actions between points of matter, of the law of the lever ; this is well worth study. There are two points of historical interest beyond the study of the work of Boscovich that can be gathered from this volume. The first is that at this time it would appear that the nature of negative numbers and quantities was not yet fully understood. Boscovich, to make his curve more symmetrical, continues it to the left of the origin as a reflection in the axis of ordinates. It is obvious, however, that, if distances to the left of the origin stand for intervals measured in the opposite direction to the ordinary (remembering that of the two points under consideration one is supposed to be at the origin), then the force just the other side of the axis of ordinates must be repulsive ; but the repulsion is in the opposite direction to the ordinary way of measuring it, and therefore should appear on the curve represented by an ordinate of attraction. Thus, the curve of Boscovich, if completed, should have point symmetry about the origin, and not line symmetry about the axis of ordinates. Boscovich, however, avoids this difficulty, intentionally or unintentionally, when showing how the equation to the curve may be obtained, by taking z = x* as his variable, and P and Q as functions of z, in the equation P-Qy = o, referred to above. Note. In this connection (p. 410, Art. 25, 1. 5), Boscovich has apparently made a slip over the negative sign : as the intention is clear, no attempt has been made to amend the Latin. The second point is that Boscovich does not seem to have any idea of integrating between limits. He has to find the area, in Fig. I on p. 134, bounded by the axes, the curve and the ordinate ag ; this he does by the use of the calculus in Note (1) on p. 141. He assumes that xviii INTRODUCTION gt the equation of the curve is x m y n = I, and obtains the integral - - xy + A, where A is the nm constant of integration. He states that, if n is greater than m, A = o, being the initial area at the origin. He is then faced with the necessity of making the area infinite when n = m, and still more infinite when n<jn. He says : " The area is infinite, when n = m, because this makes the divisor zero ; and thus the area becomes still more infinite if n<^m." Put into symbols, the argument is : Since -OT<O, >- > oo . The historically interesting n m o point about this is that it represents the persistance of an error originally made by Wallis in his Ariihmetica Infinitorum (it was Wallis who invented the sign oc to stand for " simple infinity," the value of i/o, and hence of /o). Wallis had justification for his error, if indeed it was an error in his case ; for his exponents were characteristics of certain infinite series, and he could make his own laws about these so that they suited the geometrical problems to which they were applied ; it was not necessary that they should obey the laws of inequality that were true for ordinary numbers. Boscovich's mistake is, of course, that of assuming that the constant is zero in every case ; and in this he is probably deceived by using the formula xy -f- A, instead of ^ B/( "-* l) -}- A, for the area. From the latter nm n m it is easily seen that since the initial area is zero, we must have A = o w/( "~ m) . If n is m n equal to or greater than m, the constant A is indeed zero ; but if n is less than m, the constant is infinite. The persistence of this error for so long a time, from 1655 to 1 75%> during which we have the writings of Newton, Leibniz, the Bernoullis and others on the calculus, seems to lend corroboration to a doubt as to whether the integral sign was properly understood as a summation between limits, and that this sum could be expressed as the difference of two values of the same function of those limits. It appears to me that this point is one of very great importance in the history of the development of mathematical thought. Some idea of how prolific Boscovich was as an author may be gathered from the catalogue of his writings appended at the end of this volume. This catalogue has been taken from the end of the original first Venetian edition, and brings the list up to the date of its publication, 1763. It was felt to be an impossible task to make this list complete up to the time of the death of Boscovich ; and an incomplete continuation did not seem desirable. Mention must however be made of one other work of Boscovich at least ; namely, a work in five quarto volumes, published in 1785, under the title of Opera pertinentia ad Opticam et Astronomiam. Finally, in order to bring out the versatility of the genius of Boscovich, we may mention just a few of his discoveries in science, which seem to call for special attention. In astro- nomical science, he speaks of the use of a telescope filled with liquid for the purpose of measuring the aberration of light ; he invented a prismatic micrometer contemporaneously with Rochon and Maskelyne. He gave methods for determining the orbit of a comet from three observations, and for the equator of the sun from three observations of a " spot " ; he carried out some investigations on the orbit of Uranus, and considered the rings of Saturn. In what was then the subsidiary science of optics, he invented a prism with a variable angle for measuring the refraction and dispersion of different kinds of glass ; and put forward a theory of achromatism for the objectives and oculars of the telescope. In mechanics and geodesy, he was apparently the first to solve the problem of the " body of greatest attraction " ; he successfully attacked the question of the earth's density ; and perfected the apparatus and advanced the theory of the measurement of the meridian. In mathematical theory, he seems to have recognized, before Lobachevski and Bolyai, the impossibility of a proof of Euclid's " parallel postulate " ; and considered the theory of the logarithms of negative numbers. J. M. C. N.B. The page numbers on the left-hand pages of the index are the pages of the original Latin Edition of 1763 ; they correspond with the clarendon numbers inserted throughout the Latin text of this edition. CORRIGENDA Attention is called to the following important corrections, omissions, and alternative renderings ; misprints involving a single letter or syllable only are given at the end of the volume. p. 27, 1. 8, for in one plane read in the same direction p. 47, 1. 62, literally on which ... is exerted p. 49, 1. 33, for just as ... is read so that . . . may be P- S3> 1- 9> a f ter a li ne a dd but not parts of the line itself p. 61, Art. 47, Alternative rendering: These instances make good the same point as water making its way through the pores of a sponge did for impenetrability ; p. 67, 1. 5, for it is allowable for me read I am disposed ; unless in the original libet is taken to be a misprint for licet p. 73, 1. 26, after nothing add in the strict meaning of the term p. 85, 1. 27, after conjunction add of the same point of space p. 91, 1. 25, Alternative rendering : and these properties might distinguish the points even in the view of the followers of Leibniz 1. 5 from bottom, Alternative rendering : Not to speak of the actual form of the leaves present in the seed p. 115, 1. 25, after the left add but that the two outer elements do not touch each other 1. 28, for two little spheres read one little sphere p. 117, 1. 41, for precisely read abstractly p. 125, 1. 29, for ignored read urged in reply p. 126, 1. 6 from bottom, it is -possible that acquirere is intended for acquiescere, with a corresponding change in the translation p. 129, Art. 162, marg. note, for on what they may be founded read in what it consists, p. 167, Art. 214, 1. 2 of marg. note, transpose by and on footnote, 1. I, for be at read bisect it at p. 199, 1. 24, for so that read just as p. 233, 1. 4 from bottom, for base to the angle read base to the sine of the angle last line, after vary insert inversely p. 307, 1. 5 from end, for motion, as (with fluids) takes place read motion from taking place p. 323, 1. 39, for the agitation will read the fluidity will P- 345 1- 3 2 > f 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 PHILOSOPHIC NATURALIS TYPOGRAPHUS VENETUS LECTORI PUS, quod tibi offero, jam ab annis quinque Viennse editum, quo plausu exceptum sit per Europam, noveris sane, si Diaria publica perlegeris, inter quse si, ut omittam caetera, consulas ea, quae in Bernensi pertinent ad initium anni 1761 ; videbis sane quo id loco haberi debeat. Systema continet Naturalis Philosophise omnino novum, quod jam ab ipso Auctore suo vulgo Boscovichianum appellant. Id quidem in pluribus Academiis jam passim publice traditur, nee tantum in annuis thesibus, vel disserta- tionibus impressis, ac propugnatis exponitur, sed & in pluribus elementaribus libris pro juventute instituenda editis adhibetur, exponitur, & a pluribus habetur pro archetype. Verum qui omnem systematis compagem, arctissimum partium nexum mutuum, fcecun- ditatem summam, ac usum amplissimum ac omnem, quam late patet, Naturam ex unica simplici lege virium derivandam intimius velit conspicere, ac contemplari, hoc Opus consulat, necesse est. Haec omnia me permoverant jam ab initio, ut novam Operis editionem curarem : accedebat illud, quod Viennensia exemplaria non ita facile extra Germaniam itura videbam, & quidem nunc etiam in reliquis omnibus Europse partibus, utut expetita, aut nuspiam venalia prostant, aut vix uspiam : systema vero in Italia natum, ac ab Auctore suo pluribus hie apud nos jam dissertationibus adumbratum, & casu quodam Viennae, quo se ad breve tempus contulerat, digestum, ac editum, Italicis potissimum typis, censebam, per univer- sam Europam disseminandum. Et quidem editionem ipsam e Viennensi exemplari jam turn inchoaveram ; cum illud mihi constitit, Viennensem editionem ipsi Auctori, post cujus discessum suscepta ibi fuerat, summopere displicere : innumera obrepsisse typorum menda : esse autem multa, inprimis ea, quas Algebraicas formulas continent, admodum inordinata, & corrupta : ipsum eorum omnium correctionem meditari, cum nonnullis mutationibus, quibus Opus perpolitum redderetur magis, & vero etiam additamentis. Illud ergo summopere desideravi, ut exemplar acquirerem ab ipso correctum, & auctum ac ipsum edition! praesentem haberem, & curantem omnia per sese. At id quidem per hosce annos obtinere non licuit, eo universam fere Europam peragrante ; donee demum ex tarn longa peregrinatione redux hue nuper se contulit, & toto adstitit editionis tempore, ac praeter correctores nostros omnem ipse etiam in corrigendo diligentiam adhibuit ; quanquam is ipse haud quidem sibi ita fidit, ut nihil omnino effugisse censeat, cum ea sit humanas mentis conditio, ut in eadem re diu satis intente defigi non possit. Haec idcirco ut prima quaedam, atque originaria editio haberi debet, quam qui cum Viennensi contulerit, videbit sane discrimen. E minoribus mutatiunculis multae pertinent ad expolienda, & declaranda plura loca ; sunt tamen etiam nonnulla potissimum in pagin- arum fine exigua additamenta, vel mutatiunculas exiguae factae post typographicam constructionem idcirco tantummodo, ut lacunulae implerentur quae aliquando idcirco supererant, quod plures ph'ylirae a diversis compositoribus simul adornabantur, & quatuor simul praela sudabant; quod quidem ipso praesente fieri facile potuit, sine ulla pertur- batione sententiarum, & ordinis. THE PRINTER AT VENICE TO THE READER ! OU will be well aware, if you have read the public journals, with what applause the work which I now offer to you has been received throughout Europe since its publication at Vienna five years ago. Not to mention others, if you refer to the numbers of the Berne Journal for the early part of the year 1761, you will not fail to see how highly it has been esteemed. It contains an entirely new system of Natural Philosophy, which is already commonly known as the Boscovichian theory, from the name of its author, As a matter of fact, it is even now a subject of public instruction in several Universities in different parts ; it is expounded not only in yearly theses or dissertations, both printed & debated ; but also in several elementary books issued for the instruction of the young it is introduced, explained, & by many considered as their original. Any one, however, who wishes to obtain more detailed insight into the whole structure of the theory, the close relation that its several parts bear to one another, or its great fertility & wide scope for the purpose of deriving the whole of Nature, in her widest range, from a single simple law of forces ; any one who wishes to make a deeper study of it must perforce study the work here offered. All these considerations had from the first moved me to undertake a new edition of the work ; in addition, there was the fact that I perceived that it would be a matter of some difficulty for copies of the Vienna edition to pass beyond the confines of Germany indeed, at the present time, no matter how diligently they are inquired for, they are to be found on sale nowhere, or scarcely anywhere, in the rest of Europe. The system had its birth in Italy, & its outlines had already been sketched by the author in several dissertations pub- lished here in our own land ; though, as luck would have it, the system itself was finally put into shape and published at Vienna, whither he had gone for a short time. I therefore thought it right that it should be disseminated throughout the whole of Europe, & that preferably as the product of an Italian press. I had in fact already commenced an edition founded on a copy of the Vienna edition, when it came to my knowledge that the author was greatly dissatisfied with the Vienna edition, taken in hand there after his departure ; that innumerable printer's errors had crept in ; that many passages, especially those that contain Algebraical formulae, were ill-arranged and erroneous ; lastly, that the author himself had in mind a complete revision, including certain alterations, to give a better finish to the work, together with certain additional matter. That being the case, I was greatly desirous of obtaining a copy, revised & enlarged by himself ; I also wanted to have him at hand whilst the edition was in progress, & that he should superintend the whole thing for himself. This, however, I was unable to procure during the last few years, in which he has been travelling through nearly the whole of Europe ; until at last he came here, a little while ago, as he returned home from his lengthy wanderings, & stayed here to assist me during the whole time that the edition was in hand. He, in addition to our regular proof-readers, himself also used every care in cor- recting the proof ; even then, however, he has not sufficient confidence in himself as to imagine that not the slightest thing has escaped him. For it is a characteristic of the human mind that it cannot concentrate long on the same subject with sufficient attention. It follows that this ought to be considered in some measure as a first & original edition ; any one who compares it with that issued at Vienna will soon see the difference between them. Many of the minor alterations are made for the purpose of rendering certain passages more elegant & clear ; there are, however, especially at the foot of a page, slight additions also, or slight changes made after the type was set up, merely for the purpose of filling up gaps that were left here & there these gaps being due to the fact that several sheets were being set at the same time by different compositors, and four presses were kept hard at work together. As he was at hand, this could easily be done without causing any disturbance of the sentences or the pagination. 4 TYPOGRAPHUS VENETUS LECTORI Inter mutationes occurret ordo numerorum mutatus in paragraphis : nam numerus 82 de novo accessit totus : deinde is, qui fuerat 261 discerptus est in 5 ; demum in Appendice post num. 534 factse sunt & mutatiunculse nonnullae, & additamenta plura in iis, quse pertinent ad sedem animse. Supplementorum ordo mutatus est itidem ; quse enim fuerant 3, & 4, jam sunt I, & 2 : nam eorum usus in ipso Opere ante alia occurrit. Illi autem, quod prius fuerat primum, nunc autem est tertium, accessit in fine Scholium tertium, quod pluribus numeris complec- titur dissertatiunculam integram de argumento, quod ante aliquot annos in Parisiensi Academia controversise occasionem exhibuit in Encyclopedico etiam dictionario attactum, in qua dissertatiuncula demonstrat Auctor non esse, cur ad vim exprimendam potentia quaepiam distantiae adhibeatur potius, quam functio. Accesserunt per totum Opus notulae marginales, in quibus eorum, quae pertractantur argumenta exponuntur brevissima, quorum ope unico obtutu videri possint omnia, & in memoriam facile revocari. Postremo loco ad calcem Operis additus est fusior catalogus eorum omnium, quse hue usque ab ipso Auctore sunt edita, quorum collectionem omnem expolitam, & correctam, ac eorum, quse nondum absoluta sunt, continuationem meditatur, aggressurus illico post suum regressum in Urbem Romam, quo properat. Hie catalogus impressus fuit Venetisis ante hosce duos annos in reimpressione ejus poematis de Solis ac Lunae defectibus. Porro earn, omnium suorum Operum Collectionem, ubi ipse adornaverit, typis ego meis excudendam suscipiam, quam magnificentissime potero. Haec erant, quae te monendum censui ; tu laboribus nostris fruere, & vive felix. THE PRINTER AT VENICE TO THE READER 5 Among the more Important alterations will be found a change in the order of numbering the paragraphs. Thus, Art. 82 is additional matter that is entirely new ; that which was formerly Art. 261 is now broken up into five parts ; &, in the Appendix, following Art. 534, both some slight changes and also several additions have been made in the passages that relate to the Seat of the Soul. The order of the Supplements has been altered also : those that were formerly num- bered III and IV are now I and II respectively. This was done because they are required for use in this work before the others. To that which was formerly numbered I, but is now III, there has been added a third scholium, consisting of several articles that between them give a short but complete dissertation on that point which, several years ago caused a controversy in the University of Paris, the same point being also discussed in the Dictionnaire Encydopedique. In this dissertation the author shows that there is no reason why any one power of the distance should be employed to express the force, in preference to a function. Short marginal summaries have been inserted throughout the work, in which the arguments dealt with are given in brief ; by the help of these, the whole matter may be taken in at a glance and recalled to mind with ease. Lastly, at the end of the work, a somewhat full catalogue of the whole of the author's publications up to the present time has been added. Of these publications the author intends to make a full collection, revised and corrected, together with a continuation of those that are not yet finished ; this he proposes to do after his return to Rome, for which city he is preparing to set out. This catalogue was printed in Venice a couple of years ago in connection with a reprint of his essay in verse on the eclipses of the Sun and Moon. Later, when his revision of them is complete, I propose to undertake the printing of this complete collection of his works from my own type, with all the sumptuousness at my command. Such were the matters that I thought ought to be brought to your notice. May you enjoy the fruit of our labours, & live in happiness. EPISTOLA AUCTORIS DEDICATORIA EDITIONIS VIENNENSIS AD CELSISSIMUM TUNC PRINCIPEM ARCHIEPISCOPUM VIENNENSEM, NUNC PR^TEREA ET CARDINALEM EMINENTISSIMUM, ET EPISCOPUM VACCIENSEM CHRISTOPHORUM E COMITATIBUS DE MIGAZZI IA.BIS veniam, Princeps Celsissime, si forte inter assiduas sacri regirninis curas importunus interpellator advenio, & libellum Tibi offero mole tenuem, nee arcana Religionis mysteria, quam in isto tanto constitutus fastigio adminis- tras, sed Naturalis Philosophise principia continentem. Novi ego quidem, quam totus in eo sis, ut, quam geris, personam sustineas, ac vigilantissimi sacrorum Antistitis partes agas. Videt utique Imperialis haec Aula, videt universa Regalis Urbs, & ingenti admiratione defixa obstupescit, qua dili- gentia, quo labore tanti Sacerdotii munus obire pergas. Vetus nimirum illud celeberrimum age, quod agis, quod ab ipsa Tibi juventute, cum primum, ut Te Romas dantem operam studiis cognoscerem, mihi fors obtigit, altissime jam insederat animo, id in omni reliquo amplissimorum munerum Tibi commissorum cursu haesit firmissime, atque idipsum inprimis adjectum tarn multis & dotibus, quas a Natura uberrime congestas habes, & virtutibus, quas tute diuturna Tibi exercitatione, atque assiduo labore comparasti, sanc- tissime observatum inter tarn varias forenses, Aulicas, Sacerdotales occupationes, istos Tibi tarn celeres dignitatum gradus quodammodo veluti coacervavit, & omnium una tarn populorum, quam Principum admirationem excitavit ubique, conciliavit amorem ; unde illud est factum, ut ab aliis alia Te, sublimiora semper, atque honorificentiora munera quodammodo velut avulsum, atque abstractum rapuerint. Dum Romse in celeberrimo illo, quod Auditorum Rotae appellant, collegio toti Christiano orbi jus diceres, accesserat Hetrusca Imperialis Legatio apud Romanum Pontificem exercenda ; cum repente Mech- liniensi Archiepiscopo in amplissima ilia administranda Ecclesia Adjutor datus, & destinatus Successor, possessione prsestantissimi muneris vixdum capta, ad Hispanicum Regem ab Augustissima Romanorum Imperatrice ad gravissima tractanda negotia Legatus es missus, in quibus cum summa utriusque Aulae approbatione versatum per annos quinque ditissima Vacciensis Ecclesia adepta est ; atque ibi dum post tantos Aularum strepitus ea, qua Christianum Antistitem decet, & animi moderatione, & demissione quadam, atque in omne hominum genus charitate, & singular! cura, ac diligentia Religionem administras, & sacrorum exceres curam ; non ea tantum urbs, atque ditio, sed universum Hungariae Regnum, quanquam exterum hominem, non ut civem suum tantummodo, sed ut Parentem aman- tissimum habuit, quern adhuc ereptum sibi dolet, & angitur ; dum scilicet minore, quam unius anni intervallo ab Ipsa Augustissima Imperatrice ad Regalem hanc Urbem, tot Imperatorum sedem, ac Austriacae Dominationis caput, dignum tantis dotibus explicandis theatrum, eocatum videt, atque in hac Celsissima Archiepiscopali Sede, accedente Romani Pontificis Auctoritate collocatum ; in qua Tu quidem personam itidem, quam agis, diligen- tissime sustinens, totus es in gravissimis Sacerdotii Tui expediendis negotiis, in iis omnibus, quae ad sacra pertinent, curandis vel per Te ipsum usque adeo, ut saepe, raro admodum per AUTHOR'S EPISTLE DEDICATING THE FIRST VIENNA EDITION TO CHRISTOPHER, COUNT DE MIGAZZI, THEN HIS HIGHNESS THE PRINCE ARCHBISHOP OF VIENNA, AND NOW ALSO IN ADDITION HIS EMINENCE THE CARDINAL, BISHOP OF VACZ OU will pardon me, Most Noble Prince, if perchance I come to disturb at an inopportune moment the unremitting cares of your Holy Office, & offer you a volume so inconsiderable in size ; one too that contains none of the inner mysteries of Religion, such as you administer from the highly exalted position to which you are ordained ; one that merely deals with the prin- ciples of Natural Philosophy. I know full well how entirely your time is taken up with sustaining the reputation that you bear, & in performing the duties of a highly conscientious Prelate. This Imperial Court sees, nay, the whole of this Royal City sees, with what care, what toil, you exert yourself to carry out the duties of so great a sacred office, & stands wrapt with an overwhelming admiration. Of a truth, that well-known old saying, " What you do, DO," which from your earliest youth, when chance first allowed me to make your acquaintance while you were studying in Rome, had already fixed itself deeply in your mind, has remained firmly implanted there during the whole of the remainder of a career in which duties of the highest importance have been committed to your care. Your strict observance of this maxim in particular, joined with those numerous talents so lavishly showered upon you by Nature, & those virtues which you have acquired for yourself by daily practice & unremitting toil, throughout your whole career, forensic, courtly, & sacerdotal, has so to speak heaped upon your shoulders those unusually rapid advances in dignity that have been your lot. It has aroused the admiration of all, both peoples & princes alike, in every land ; & at the same time it has earned for you their deep affection. The consequence was that one office after another, each ever more exalted & honourable than the preceding, has in a sense seized upon you & borne you away a captive. Whilst you were in Rome, giving judicial decisions to the whole Christian world in that famous College, the Rota of Auditors, there was added the duty of acting on the Tuscan Imperial Legation at the Court of the Roman Pontiff. Sud- denly you were appointed coadjutor to the Archbishop of Malines in the administration of that great church, & his future successor. Hardly had you entered upon the duties of that most distinguished appointment, than you were despatched by the August Empress of the Romans as Legate on a mission of the greatest importance. You occupied yourself on this mission for the space of five years, to the entire approbation of both Courts, & then the wealthy church of Vacz obtained your services. Whilst there, the great distractions of a life at Court being left behind, you administer the offices of religion & discharge the sacred rights with that moderation of spirit & humility that befits a Christian prelate, in charity towards the whole race of mankind, with a singularly attentive care. So that not only that city & the district in its see, but the whole realm of Hungary as well, has looked upon you, though of foreign race, as one of her own citizens ; nay, rather as a well beloved father, whom she still mourns & sorrows for, now that you have been taken from her. For, after less than a year had passed, she sees you recalled by the August Empress herself to this Imperial City, the seat of a long line of Emperors, & the capital of the Dominions of Austria, a worthy stage for the display of your great talents ; she sees you appointed, under the auspices of the authority of the Roman Pontiff, to this exalted Archiepiscopal see. Here too, sustaining with the utmost diligence the part you play so well, you throw your- self heart and soul into the business of discharging the weighty duties of your priesthood, or in attending to all those things that deal with the sacred rites with your own hands : so much so that we often see you officiating, & even administering the Sacraments, in our 8 EPISTOLA AUCTORIS DEDICATORIA PRI1VLE EDITIONIS VIENNENSIS haec nostra tempora exemplo, & publico operatum, ac ipsa etiam Sacramenta administrantem videamus in templis, & Tua ipsius voce populos, e superiore loco docentum audiamus, atque ad omne virtutum genus inflammantem. Novi ego quidem haec omnia ; novi hanc indolem, hanc animi constitutionem ; nee sum tamen inde absterritus, ne, inter gravissimas istas Tuas Sacerdotales curas, Philosophicas hasce meditationes meas, Tibi sisterem, ac tantulae libellum molis homini ad tantum culmen evecto porrigerem, ac Tuo vellem Nomine insignitum. Quod enim ad primum pertinet caput, non Theologicas tantum, sed Philosophicas etiam perquisitiones Christiano Antistite ego quidem dignissimas esse censeo, & universam Naturae contemplationem omnino arbitror cum Sacerdotii sanctitate penitus consentire. Mirum enim, quam belle ab ipsa consideratione Naturae ad caslestium rerum contemplationem disponitur animus, & ad ipsum Divinum tantae molis Conditorem assurgit, infinitam ejus Potentiam Sapientiam, Providentiam admiratus, quae erumpunt undique, & utique se produnt. Est autem & illud, quod ad supremi sacrorum Moderatoris curam pertinet providere, ne in prima ingenuae juventutis institutione, quae semper a naturalibus studiis exordium ducit, prava teneris mentibus irrepant, ac perniciosa principia, quae sensim Religionem corrumpant, & vero etiam evertant penitus, ac eruant a fundamentis ; quod quidem jam dudum tristi quodam Europae fato passim evenire cernimus, gliscente in dies malo, ut fucatis quibusdam, profecto perniciosissimis, imbuti principiis juvenes, turn demum sibi sapere videantur, cum & omnem animo religionem, & Deum ipsum sapientissimum Mundi Fabricatorem, atque Moderatorem sibi mente excusserint. Quamobrem qui veluti ad tribunal tanti Sacerdotum Principis Universae Physicae Theoriam, & novam potissimum Theoriam sistat, rem is quidem praestet sequissimam, nee alienum quidpiam ab ejus munere Sacerdotali offerat, sed cum eodem apprime consentiens. Nee vero exigua libelli moles deterrere me debuit, ne cum eo ad tantum Principem accederem. Est ille quidem satis tenuis libellus, at non & tenuem quoque rem continet. Argumentum pertractat sublime admodum, & nobile, in quo illustrando omnem ego quidem industriam coUocavi, ubi si quid praestitero, si minus infiliclter me gessero, nemo sane me impudentiae arguat, quasi vilem aliquam, & tanto indignam fastigio rem offeram. Habetur in eo novum quoddam Universae Naturalis Philosophiae genus a receptis hue usque, usi- tatisque plurimam discrepans, quanquam etiam ex iis, quae maxime omnium per haec tempora celebrantur, casu quodam praecipua quasque mirum sane in modum compacta, atque inter se veluti coagmentata conjunguntur ibidem, uti sunt simplicia atque inextensa Leibnitian- orum elementa, cum Newtoni viribus inducentibus in aliis distantiis accessum mutuum, in aliis mutuum recessum, quas vulgo attractiones, & repulsiones appellant : casu, inquam : neque enim ego conciliandi studio hinc, & inde decerpsi quaedam ad arbitrium selecta, quae utcumque inter se componerem, atque compaginarem : sed omni praejudicio seposito, a principiis exorsus inconcussis, & vero etiam receptis communiter, legitima ratiocinatione usus, & continue conclusionum nexu deveni ad legem virium in Natura existentium unicam, simplicem , continuam, quae mihi & constitutionem elementorum materiae, & Mechanicae leges, & generales materiae ipsius proprietates, & praecipua corporum discrimina, sua applicatione ita exhibuit, ut eadem in iis omnibus ubique se prodat uniformis agendi ratio, non ex arbitrariis hypothesibus, & fictitiis commentationibus, sed ex sola continua ratio- cinatione deducta. Ejusmodi autem est omnis, ut eas ubique vel definiat, vel adumbret combinationes elementorum, quae ad diversa prasstanda phaenomena sunt adhibendas, ad quas combinationes Conditoris Supremi consilium, & immensa Mentis Divinae vis ubique requiritur, quae infinites casus perspiciat, & ad rem aptissimos seligat, ac in Naturam inducat. Id mihi quidem argumentum est operis, in quo Theoriam meam expono, comprobo, vindico : turn ad Mechanicam primum, deinde ad Physicam applico, & uberrimos usus expono, ubi brevi quidem libello, sed admodum diuturnas annorum jam tredecim medita- tiones complector meas, eo plerumque tantummodo rem deducens, ubi demum cum AUTHOR'S EPISTLE DEDICATING THE FIRST VIENNA EDITION 9 churches (a somewhat unusual thing at the present time), and also hear you with your own voice exhorting the people from your episcopal throne, & inciting them to virtue of every kind. I am well aware of all this ; I know full well the extent of your genius, & your con- stitution of mind ; & yet I am not afraid on that account of putting into your hands, amongst all those weighty duties of your priestly office, these philosophical meditations of mine ; nor of offering a volume so inconsiderable in bulk to one who has attained to such heights of eminence ; nor of desiring that it should bear the hall-mark of your name. With regard to the first of these heads, I think that not only theological but also philosophical investigations are quite suitable matters for consideration by a Christian prelate ; & in my opinion, a contemplation of all the works of Nature is in complete accord with the sanctity of the priesthood. For it is marvellous how exceedingly prone the mind becomes to pass from a contemplation of Nature herself to the contemplation of celestial, things, & to give honour to the Divine Founder of such a mighty structure, lost in astonishment at His infinite Power & Wisdom & Providence, which break forth & disclose themselves in all directions & in all things. There is also this further point, that it is part of the duty of a religious superior to take care that, in the earliest training of ingenuous youth, which always takes its start from the study of the wonders of Nature, improper ideas do not insinuate themselves into tender minds ; or such pernicious principles as may gradually corrupt the belief in things Divine, nay, even destroy it altogether, & uproot it from its very foundations. This is what we have seen for a long time taking place, by some unhappy decree of adverse fate, all over Europe ; and, as the canker spreads at an ever increasing rate, young men, who have been made to imbibe principles that counterfeit the truth but are actually most pernicious doc- trines, do not think that they have attained to wisdom until they have banished from their minds all thoughts of religion and of God, the All- wise Founder and Supreme Head of the Universe. Hence, one who so to speak sets before the judgment-seat of such a prince of the priesthood as yourself a theory of general Physical Science, & more especially one that is new, is doing nothing but what is absolutely correct. Nor would he be offering him anything inconsistent with his priestly office, but on the contrary one that is in complete harmony with it. Nor, secondly, should the inconsiderable size of my little book deter me from approach- ing with it so great a prince. It is true that the volume of the book is not very great, but the matter that it contains is not unimportant as well. The theory it develops is a strik- ingly sublime and noble idea ; & I have done my very best to explain it properly. If in this I have somewhat succeeded, if I have not failed altogether, let no one accuse me of presumption, as if I were offering some worthless thing, something unworthy of such dis- tinguished honour. In it is contained a new kind of Universal Natural Philosophy, one that differs widely from any that are generally accepted & practised at the present time ; although it so happens that the principal points of all the most distinguished theories of the present day, interlocking and as it were cemented together in a truly marvellous way, are combined in it ; so too are the simple unextended elements of the followers of Leibniz, as well as the Newtonian forces producing mutual approach at 'some distances & mutual separation at others, usually called attractions and repulsions. I use the words " it so happens " because I have not, in eagerness to make the whole consistent, selected one thing here and another there, just as it suited me for the purpose of making them agree & form a connected whole. On the contrary, I put on one side all prejudice, & started from fundamental principles that are incontestable, & indeed are those commonly accepted ; I used perfectly sound arguments, & by a continuous chain of deduction I arrived at a single, simple, continuous law for the forces that exist in Nature. The application of this law explained to me the constitution of the elements of matter, the laws of Mechanics, the general properties of matter itself, & the chief characteristics of bodies, in such a manner that the same uniform method of action in all things disclosed itself at all points ; being deduced, not from arbitrary hypotheses, and fictitibus explanations, but from a single con- tinuous chain of reasoning. Moreover it is in all its parts of such a kind as defines, or suggests, in every case, the combinations of the elements that must be employed to produce different phenomena. For these combinations the wisdom of the Supreme Founder of the Universe, & the mighty power of a Divine Mind are absolutely necessary ; naught but one that could survey the countless cases, select those most suitable for the purpose, and introduce them into the scheme of Nature. This then is the argument of my work, in which I explain, prove & defend my theory ; then I apply it, in the first instance to Mechanics, & afterwards to Physics, & set forth the many advantages to be derived from it. Here, although the book is but small, I yet include the well-nigh daily meditations of the last thirteen years, carrying on my conclu- io EPISTOLA AUCTORIS DEDICATORIA PRIM. EDITIONIS VIENNENSIS communibus Philosophorum consentio placitis, & ubi ea, quae habemus jam pro compertis, ex meis etiam deductionibus sponte fluunt, quod usque adeo voluminis molem contraxit. Dederam ego quidem dispersa dissertatiunculis variis Theorise meae qusedam velut specimina, quae inde & in Italia Professores publicos nonnullos adstipulatores est nacta, & jam ad exteras quoque gentes pervasit ; sed ea nunc primum tota in unum compacta, & vero etiam plusquam duplo aucta, prodit in publicum, quern laborem postremo hoc mense, molestiori- bus negotiis, quae me Viennam adduxerant, & curis omnibus exsolutus suscepi, dum in Italiam rediturus opportunam itineri tempus inter assiduas nives opperior, sed omnem in eodem adornando, & ad communem mediocrum etiam Philosophorum captum accommo- dando diligentiam adhibui. Inde vero jam facile intelliges, cur ipsum laborem meum ad Te deferre, & Tuo nuncupare Nomini non dubitaverim. Ratio ex iis, quae proposui, est duplex : primo quidem ipsum argumenti genus, quod Christianum Antistitem non modo non dedecet, sed etiam apprime decet : turn ipsius argumenti vis, atque dignitas, quae nimirum confirmat, & erigit nimium fortasse impares, sed quantum fieri per me potuit, intentos conatus meos ; nam quidquid eo in genere meditando assequi possum, totum ibidem adhibui, ut idcirco nihil arbitrer a mea tenuitate proferri posse te minus indignum, cui ut aliquem offerrem laborum meorum fructum quantumcunque, exposcebat sane, ac ingenti clamore quodam efnagitabat tanta erga me humanitas Tua, qua jam olim immerentem complexus Romae, hie etiam fovere pergis, nee in tanto dedignatus fastigio, omni benevolentiae significatione prosequeris. Accedit autem & illud, quod in hisce terris vix adhuc nota, vel etiam ignota penitus Theoria mea Patrocinio indiget, quod, si Tuo Nomine insignata prodeat in publicum, obtinebit sane validissimum, & secura vagabitur : Tu enim illam, parente velut hie orbatam suo, in dies nimirum discessuro, & quodammodo veluti posthumam post ipsum ejus discessum typis impressam, & in publicum prodeuntem tueberis, fovebisque. Haec sunt, quae meum Tibi consilium probent, Princeps Celsissime : Tu, qua soles humanitate auctorem excipere, opus excipe, & si forte adhuc consilium ipsum Tibi visum fuerit improbandum ; animum saltern aequus respice obsequentissimum Tibi, ac devinct- issimum. Vale. Dabam Viennce in Collegia Academico Soc. JESU Idibus Febr. MDCCLFIIL AUTHOR'S EPISTLE DEDICATING THE FIRST VIENNA EDITION 11 sions for the most part only up to the point where I finally agreed with the opinions com- monly held amongst philosophers, or where theories, now accepted as established, are the natural results of my deductions also ; & this has in some measure helped to diminish the size of the volume. I had already published some instances, so to speak, of my general theory in several short dissertations issued at odd times ; & on that account the theory has found some supporters amongst the university professors in Italy, & has already made its way into foreign countries. But now for the first time is it published as a whole in a single volume, the matter being indeed more than doubled in amount. This work I have carried out during the last month, being quit of the troublesome business that brought me to Vienna, and of all other cares ; whilst I wait for seasonable time for my return journey through the everlasting snow to Italy. I have however used my utmost endeavours in preparing it, and adapting it to the ordinary intelligence of philosophers of only moderate attainments. From this you will readily understand why I have not hesitated to bestow this book of mine upon you, & to dedicate it to you. My reason, as can be seen from what I have said, was twofold ; in the first place, the nature of my theme is one that is not only not unsuitable, but is suitable in a high degree, for the consideration of a Christian priest ; secondly, the power & dignity of the theme itself, which doubtless gives strength & vigour to my efforts perchance rather feeble, but, as far as in me lay, earnest. What- ever in that respect I could gain by the exercise of thought, I have applied the whole of it to this matter ; & consequently I think that nothing less unworthy of you can be pro- duced by my poor ability ; & that I should offer to you some such fruit of my labours was surely required of me, & as it were clamorously demanded by your great kindness to me ; long ago in Rome you had enfolded my unworthy self in it, & here now you continue to be my patron, & do not disdain, from your exalted position, to honour me with every mark of your goodwill. There is still a further consideration, namely, that my Theory is as yet almost, if not quite, unknown in these parts, & therefore needs a patron's support ; & this it will obtain most effectually, & will go on its way in security if it comes before the public franked with your name. For you will protect & cherish it, on its publication here, bereaved as it were of that parent whose departure in truth draws nearer every day ; nay rather posthumous, since it will be seen in print only after he has gone. Such are my grounds for hoping that you will approve my idea, most High Prince. I beg you to receive the work with the same kindness as you used to show to its author ; &, if perchance the idea itself should fail to meet with your approval, at least regard favourably the intentions of your most humble & devoted servant. Farewell. University College of the Society of Jesus, VIENNA, February i$th, 1758. AD LECTOREM EX EDITIONS VIENNENSI amice Lector, Philosophic Naturalis Theoriam ex unica lege virium deductam, quam & ubi jam olim adumbraverim, vel etiam ex parte explica- verim, y qua occasione nunc uberius pertractandum, atque augendam etiam, susceperim, invenies in ipso -primes partis exordia. Libuit autem hoc opus dividere in partes tres, quarum prima continet explicationem Theories ipsius, ac ejus analyticam deductionem, & vindicationem : secunda applicationem- satis uberem ad Mechanicam ; tertia applicationem ad Physicam. Porro illud inprimis curandum duxi, ut omnia, quam liceret, dilucide exponerentur, nee sublimiore Geometria, aut Calculo indigerent. Et quidem in prima, ac tertia parte non tantum nullcs analyticee, sed nee geometries demonstrations occurrunt, paucissimis qiiibusdam, quibus indigeo, rejectis in adnotatiunculas, quas in fine paginarum quarundam invenies. Queedam autem admodum pauca, quce majorem Algebra, & Geometries cognitionem requirebant, vel erant complicatiora aliquando, & alibi a me jam edita, in fine operis apposui, quce Supplementorum appellavi nomine, ubi W ea addidi, quce sentio de spatio, ac tempore, Theories mece consentanea, ac edita itidem jam alibi. In secunda parte, ubi ad Mechanicam applicatur Theoria,a geome- tricis, W aliquando etiam ab algebraicis demonstrationibus abstinere omnino non potui ; sed ece ejusmodi sunt, ut vix unquam requirant aliud, quam Euclideam Geometriam, & primas Trigonometries notiones maxime simplices, ac simplicem algorithmum. In prima quidem parte occurrunt Figures geometricce complures, quce prima fronte vide- buntur etiam complicate? rem ipsam intimius non perspectanti ; verum ece nihil aliud exhibent, nisi imaginem quandam rerum, quce ipsis oculis per ejusmodi figuras sistuntur contemplandce. Ejusmodi est ipsa ilia curva, quce legem virium exhibet. Invenio ego quidem inter omnia materice puncta vim quandam mutuam, quce a distantiis pendet, 5" mutatis distantiis mutatur ita, ut in aliis attractiva sit, in aliis repulsiva, sed certa quadam, y continua lege. Leges ejusmodi variationis binarum quantitatum a se invicem pendentium, uti Jiic sunt distantia, y vis, exprimi possunt vel per analyticam formulam, vel per geometricam curvam ; sed ilia prior expressio & multo plures cognitiones requirit ad Algebram pertinentes, & imaginationem non ita adjuvat, ut heec posterior, qua idcirco sum usus in ipsa prima operis parte, rejecta in Supplementa formula analytica, quce y curvam, & legem virium ab ilia expressam exhibeat. Porro hue res omnis reducitur. Habetur in recta indefinita, quce axis dicitur, punctum quoddam, a quo abscissa ipsius rectce segmenta referunt distantias. Curva linea protenditur secundum rectam ipsam, circa quam etiam serpit, y eandem in pluribus secat punctis : rectce a fine segmentorum erectce perpendiculariter usque ad curvam, exprimunt vires, quce majores sunt, vel minores, prout ejusmodi rectce sunt itidem majores, vel minores ; ac eesdem ex attrac- tivis migrant in repulsivis, vel vice versa, ubi illce ipsce perpendiculares rectce directionem mutant, curva ab alter a axis indefiniti plaga migrante ad alter am. Id quidem nullas requirit geometricas demonstrations, sed meram cognitionem vocum quarundam, quce vel ad prima per- tinent Geometries elementa, y notissimce sunt, vel ibi explicantur, ubi adhibentur. Notissima autem etiam est significatio vocis Asymptotus, unde & crus asymptoticum curvce appellatur ; dicitur nimirum recta asymptotus cruris cujuspiam curvce, cum ipsa recta in infinitum producta, ita ad curvilineum arcum productum itidem in infinitum semper accedit magis, ut distantia minuatur in infinitum, sed nusquam penitus evanescat, illis idcirco nunquam invicem con- venientibus. Consider atio porro attenta curvce propositce in Fig. I, &rationis, qua per illam exprimitur 12 THE PREFACE TO THE READER THAT APPEARED IN THE VIENNA EDITION EAR Reader, you have before you a Theory of Natural Philosophy deduced from a single law of Forces. You will find in the opening paragraphs of the first section a statement as to where the Theory has been already published in outline, & to a certain extent explained ; & also the occasion that led me to undertake a more detailed treatment & enlargement of it. For I have thought fit to divide the work into three parts ; the first of these contains the exposition of the Theory itself, its analytical deduction & its demonstration ; the second a fairly full application to Mechanics ; & the third an application to Physics. The most important point, I decided, was for me to take the greatest care that every- thing, as far as was possible, should be clearly explained, & that there should be no need for higher geometry or for the calculus. Thus, in the first part, as well as in the third, there are no proofs by analysis ; nor are there any by geometry, with the exception of a very few that are absolutely necessary, & even these you will find relegated to brief notes set at the foot of a page. I have also added some very few proofs, that required a knowledge of higher algebra & geometry, or were of a rather more complicated nature, all of which have been already published elsewhere, at the end of the work ; I have collected these under the heading Supplements ; & in them I have included my views on Space & Time, which are in accord with my main Theory, & also have been already published elsewhere. In the second part, where the Theory is applied to Mechanics, I have not been able to do without geometrical proofs altogether ; & even in some cases I have had to give algebraical proofs. But these are of such a simple kind that they scarcely ever require anything more than Euclidean geometry, the first and most elementary ideas of trigonometry, and easy analytical calculations. It is true that in the first part there are to be found a good many geometrical diagrams, which at first sight, before the text is considered more closely, will appear to be rather complicated. But these present nothing else but a kind of image of the subjects treated, which by means of these diagrams are set before the eyes for contemplation. The very curve that represents the law of forces is an instance of this. I find that between all points of matter there is a mutual force depending on the distance between them, & changing as this distance changes ; so that it is sometimes attractive, & sometimes repulsive, but always follows a definite continuous law. Laws of variation of this kind between two quantities depending upon one another, as distance & force do in this instance, may be represented either by an analytical formula or by a geometrical curve ; but the former method of representation requires far more knowledge of algebraical processes, & does not assist the imagination in the way that the latter does. Hence I have employed the latter method in the first part of the work, & relegated to the Supplements the analytical formula which represents the curve, & the law of forces which the curve exhibits. The whole matter reduces to this. In a straight line of indefinite length, which is called the axis, a fixed point is taken ; & segments of the straight line cut off from this point represent the distances. A curve is drawn following the general direction of this straight line, & winding about it, so as to cut it in several places. Then perpendiculars that are drawn from the ends of the segments to meet the curve represent the forces ; these forces are greater or less, according as such perpendiculars are greater or less ; & they pass from attractive forces to repulsive, and vice versa, whenever these perpendiculars change their direction, as the curve passes from one side of the axis of indefinite length to the other side of it. Now this requires no geometrical proof, but only a knowledge of certain terms, which either belong to the first elementary principles of "geometry, & are thoroughly well known, or are such as can be defined when they are used. The term Asymptote is well known, and from the same idea we speak of the branch of a curve as being asymptotic ; thus a straight line is said to be the asymptote to any branch of a curve when, if the straight line is indefinitely produced, it approaches nearer and nearer to the curvilinear arc which is also prolonged indefinitely in such manner that the distance between them becomes indefinitely diminished, but never altogether vanishes, so that the straight line & the curve never really meet. A careful consideration of the curve given in Fig. I, & of the way in which the relation 14 AD LECTOREM EX EDITIONE VIENNENSI nexus inter vires, y distantias, est utique admodum necessaria ad intelligendam Theoriam ipsam, cujus ea est prcecipua qucedam veluti clavis, sine qua omnino incassum tentarentur cetera ; sed y ejusmodi est, ut tironum, & sane etiam mediocrium, immo etiam longe infra mediocritatem collocatorum, captum non excedat, potissimum si viva accedat Professoris vox mediocriter etiam versati in Mechanica, cujus ope, pro certo habeo, rem ita patentem omnibus reddi posse, ut ii etiam, qui Geometric penitus ignari sunt, paucorum admodum explicatione vocabulorum accidente, earn ipsis oculis intueantur omnino perspicuam. In tertia parte supponuntur utique nonnulla, quce demonstrantur in secunda ; sed ea ipsa sunt admodum pauca, & Us, qui geometricas demonstrationes fastidiunt, facile admodum exponi possunt res ipsce ita, ut penitus etiam sine ullo Geometries adjumento percipiantur, quanquam sine Us ipsa demonstratio baberi non poterit ; ut idcirco in eo differre debeat is, qui secundam partem attente legerit, & Geometriam calleat, ab eo, qui earn omittat, quod ille primus veritates in tertia parte adhibitis, ac ex secunda erutas, ad, explicationem Physicce, intuebitur per evi- dentiam ex ipsis demonstrationibus haustam, hie secundus easdem quodammodo per fidem Geo- metris adhibitam credet. Hujusmodi inprimis est illud, particulam compositam ex punctis etiam homogeneis, prceditis lege virium proposita, posse per solam diversam ipsorum punctorum dispositionem aliam particulam per certum intervallum vel perpetuo attrahere, vel perpetuo repellere, vel nihil in earn agere, atque id ipsum viribus admodum diversis, y quce respectu diver- sarum particularum diver see sint, & diver see respectu partium diver sarum ejusdem particulce, ac aliam particulam alicubi etiam urgeant in latus, unde plurium phcenomenorum explicatio in Physica sponte fluit. Verum qui omnem Theories, y deductionum compagem aliquanto altius inspexerit, ac diligentius perpenderit, videbit, ut spero, me in hoc perquisitionis genere multo ulterius progressum esse, quam olim Newtonus ipse desideravit. Is enim in postremo Opticce questione prolatis Us, quce per vim attractivam, & vim repulsivam, mutata distantia ipsi attractive suc- cedentem, explicari poterant, hcec addidit : " Atque hcec quidem omnia si ita sint, jam Natura universa valde erit simplex, y consimilis sui, perficiens nimirum magnos omnes corporum ccelestium motus attractione gravitatis, quce est mutua inter corpora ilia omnia, & minores fere omnes particularum suarum motus alia aliqua vi attrahente, & repellente, qua est inter particulas illas mutua" Aliquanto autem inferius de primigeniis particulis agens sic habet : " Porro videntur mihi hce particulce primigenice non modo in se vim inertice habere, motusque leges passivas illas, quce ex vi ista necessario oriuntur ; verum etiam motum perpetuo accipere a certis principiis actuosis, qualia nimirum sunt gravitas, ff causa fermentationis, & cohcerentia corporum. Atque hcec quidem principia considero non ut occultas qualitates, quce ex specificis rerum formis oriri fingantur, sed ut universales Naturce leges, quibus res ipsce sunt formatce. Nam principia quidem talia revera existere ostendunt phenomena Naturce, licet ipsorum causce quce sint, nondum fuerit explicatum. Affirmare, singulas rerum species specificis prceditas esse qualita- tibus occultis, per quas eae vim certam in agenda habent, hoc utique est nihil dicere : at ex phcenomenis Naturce duo, vel tria derivare generalia motus principia, & deinde explicare, quemadmodum proprietates, & actiones rerum corporearum omnium ex istis principiis conse- quantur, id vero magnus esset factus in Philosophia progressus, etiamsi principiorum istorum causce nondum essent cognitce. Quare motus principia supradicta proponere non dubito, cum per Naturam universam latissime pateant" Hcec ibi Newtonus, ubi is quidem magnos in Philosophia progressus facturum arbitratus est eum, qui ad duo, vel tria generalia motus principia ex Naturce phcenomenis derivata pheeno- menorum explicationem reduxerit, & sua principia protulit, ex quibus inter se diversis eorum aliqua tantummodo explicari posse censuit. Quid igitur, ubi tf? ea ipsa tria, & alia prcecipua quceque, ut ipsa etiam impenetrabilitas, y impulsio reducantur ad principium unicum legitima ratiocinatione deductum ? At id -per meam unicam, & simplicem virium legemprcestari, patebit sane consideranti operis totius Synopsim quandam, quam hie subjicio ; sed multo magis opus ipsum diligentius pervolventi. THE PRINTER AT VENICE TO THE READER \ OU will be well aware, if you have read the public journals, with what applause the work which I now offer to you has been received throughout Europe since its publication at Vienna five years ago. Not to mention others, if you refer to the numbers of the Berne Journal for the early part of the year 1761, you will not fail to see how highly it has been esteemed. It contains an entirely new system of Natural Philosophy, which is already commonly known as the Boscovichian theory, from the name of its author, As a matter of fact, it is even now a subject of public instruction in several Universities in different parts ; it is expounded not only in yearly theses or dissertations, both printed & debated ; but also in several elementary books issued for the instruction of the young it is introduced, explained, & by many considered as their original. Any one, however, who wishes to obtain more detailed insight into the whole structure of the theory, the close relation that its several parts bear to one another, or its great fertility & wide scope for the purpose of deriving the whole of Nature, in her widest range, from a single simple law of forces ; any one who wishes to make a deeper study of it must perforce study the work here offered. All these considerations had from the first moved me to undertake a new edition of the work ; in addition, there was the fact that I perceived that it would be a matter of some difficulty for copies of the Vienna edition to pass beyond the confines of Germany indeed, at the present time, no matter how diligently they are inquired for, they are to be found on sale nowhere, or scarcely anywhere, in the rest of Europe. The system had its birth in Italy, & its outlines had already been sketched by the author in several dissertations pub- lished here in our own land ; though, as luck would have it, the system itself was finally put into shape and published at Vienna, whither he had gone for a short time. I therefore thought it right that it should be disseminated throughout the whole of Europe, & that preferably as the product of an Italian press. I had in fact already commenced an edition founded on a copy of the Vienna edition, when it came to my knowledge that the author was greatly dissatisfied with the Vienna edition, taken in hand there after his departure ; that innumerable printer's errors had crept in ; that many passages, especially those that contain Algebraical formulae, were ill-arranged and erroneous ; lastly, that the author himself had in mind a complete revision, including certain alterations, to give a better finish to the work, together with certain additional matter. That being the case, I was greatly desirous of obtaining a copy, revised & enlarged by himself ; I also wanted to have him at hand whilst the edition was in progress, & that he should superintend the whole thing for himself. This, however, I was unable to procure during the last few years, in which he has been travelling through nearly the whole of Europe ; until at last he came here, a little while ago, as he returned home from his lengthy wanderings, & stayed here to assist me during the whole time that the edition was in hand. He, in addition to our regular proof-readers, himself also used every care in cor- recting the proof ; even then, however, he has not sufficient confidence in himself as to imagine that not the slightest thing has escaped him. For it is a characteristic of the human mind that it cannot concentrate long on the same subject with sufficient attention. It follows that this ought to be considered in some measure as a first & original edition ; any one who compares it with that issued at Vienna will soon see the difference between them. Many of the minor alterations are made for the purpose of rendering certain passages more elegant & clear ; there are, however, especially at the foot of a page, slight additions also, or slight changes made after the type was set up, merely for the purpose of filling up gaps that were left here & there these gaps being due to the fact that several sheets were being set at the same time by different compositors, and four presses were kept hard at work together. As he was at hand, this could easily be done without causing any disturbance of the sentences or the pagination. 14 AD LECTOREM EX EDITIONE VIENNENSI nexus inter vires, & distantias, est utique admodum necessaria ad intelligendam Theoriam ipsam, cujus ea est prcecipua queedam veluti clavis, sine qua omnino incassum tentarentur cetera ; sed y ejusmodi est, ut tironum, & sane etiam mediocrium, immo etiam longe infra mediocritatem collocatorum, captum non excedat, potissimum si viva accedat Professoris vox mediocriter etiam versati in Mechanics, cujus ope, pro certo habeo, rem ita patentem omnibus reddi posse, ut ii etiam, qui Geometric? penitus ignari sunt, paucorum admodum explicatione vocabulorum accidente, earn ipsis oculis intueantur omnino perspicuam, In tertia parte supponuntur utique nonnulla, que? demonstrantur in secunda ; sed ea ipsa sunt admodum pauca, & Us, qui geometricas demonstrationes fastidiunt, facile admodum exponi possunt res ipsee ita, ut penitus etiam sine ullo Geometric adjumento percipiantur, quanquam sine Us ipsa demonstratio haberi non poterit ; ut idcirco in eo differre debeat is, qui secundam partem attente legerit, y Geometriam calleat, ab eo, qui earn omittat, quod ille primus veritates in tertia parte adhibitis, ac ex secunda erutas, ad explicationem Physics, intuebitur per evi- dentiam ex ipsis demonstrationibus baustam, hie secundus easdem quodammodo per fidem Geo- metris adhibitam credet. Hujusmodi inprimis est illud, particulam compositam ex punctis etiam bomogeneis, preeditis lege virium proposita, posse per solam diversam ipsorum punctorum dispositionem aliam particulam per cerium intervallum vel perpetuo attrahere, vel perpetuo repellere, vel nihil in earn agere, atque id ipsum viribus admodum diversis, y que? respectu diver- sarum particularum diver see sint, y diverse? respectu partium diver sarum ejusdem particulce, ac aliam particulam alicubi etiam urgeant in latus, unde plurium pheenomenorum explicatio in Physica sponte ftuit. Ferum qui omnem Theorie?, y deductionum compagem aliquanto altius inspexerit, ac diligentius perpenderit, videbit, ut spero, me in hoc perquisitionis genere multo ulterius progressum esse, quam olim Newtonus ipse desideravit. Is enim in postremo Opticce questione prolatis Us, qua per vim attractivam, y vim repulsivam, mutata distantia ipsi attractive? suc- cedentem, explicari poterant, he?c addidit : " Atque he?c quidem omnia si ita sint, jam Natura universa valde erit simplex, y consimilis sui, perficiens nimirum magnos omnes corporum ccelestium motus attractione gravitatis, quee est mutua inter corpora ilia omnia, y minores fere omnes particularum suarum motus alia aliqua vi attrabente, y repellente, quiz est inter particulas illas mutua." Aliquanto autem inferius de primigeniis particulis agens sic habet : " Porro videntur mihi he? particule? primigeniee non modo in se vim inertice habere, motusque leges passivas illas, que? ex vi ista necessario oriuntur ; verum etiam motum perpetuo accipere a certis principiis actuosis, qualia nimirum sunt gravitas, y causa fermentationis, y cohcerentia corporum. Atque heec quidem principia considero non ut occultas qualitates, que? ex specificis rerum formis oriri fingantur, sed ut universales Nature? leges, quibus res ipse? sunt formates. Nam principia quidem talia revera existere ostendunt phenomena Nature?, licet ipsorum cause? que? sint, nondum fuerit explicatum. Affirmare, singulas rerum species specificis preeditas esse qualita- tibus occultis, per quas eae vim certam in agenda habent, hoc utique est nihil dicere : at ex phcenomenis Nature? duo, vel tria derivare generalia motus principia, y deinde explicare, quemadmodum proprietates, y actiones rerum corporearum omnium ex istis principiis conse- quantur, id vero magnus esset factus in Philosophia progressus, etiamsi principiorum istorum cause? nondum essent cognite?. Quare motus principia supradicta proponere non dubito, cum per Naturam universam latissime pateant" Hc?c ibi Newtonus, ubi is quidem magnos in Philosophia progressus facturum arbitratus est eum, qui ad duo, vel tria generalia motus principia ex Nature? pheenomenis derivata phe?no- menorum explicationem reduxerit, y sua principia protulit, ex quibus inter se diversis eorum aliqua tantummodo explicari posse censuit. Quid igitur, ubi y ea ipsa tria, y alia preecipua quczque, ut ipsa etiam impenetrabilitas, y impulsio reducantur ad principium unicum legitima ratiocinatione deductum ? At id per meam unicam, y simplicem virium legempr<zstari,patebit sane consideranti operis totius Synopsim quandam, quam hie subjicio ; sed multo magis opus ipsum diligentius pervolventi. THE PRINTER AT VENICE TO THE READER |JOU will be well aware, if you have read the public journals, with what applause the work which I now offer to you has been received throughout Europe since its publication at Vienna five years ago. Not to mention others, if you refer to the numbers of the Berne Journal for the early part of the year 1761, you will not fail to see how highly it has been esteemed. It contains an entirely new system of Natural Philosophy, which is already commonly known as the Boscovicbian theory, from the name of its author, As a matter of fact, it is even now a subject of public instruction in several Universities in different parts ; it is expounded not only in yearly theses or dissertations, both printed & debated ; but also in several elementary books issued for the instruction of the young it is introduced, explained, & by many considered as their original. Any one, however, who wishes to obtain more detailed insight into the whole structure of the theory, the close relation that its several parts bear to one another, or its great fertility & wide scope for the purpose of deriving the whole of Nature, in her widest range, from a single simple law of forces ; any one who wishes to make a deeper study of it must perforce study the work here offered. All these considerations had from the first moved me to undertake a new edition of the work ; in addition, there was the fact that I perceived that it would be a matter of some difficulty for copies of the Vienna edition to pass beyond the confines of Germany indeed, at the present time, no matter how diligently they are inquired for, they are to be found on sale nowhere, or scarcely anywhere, in the rest of Europe. The system had its birth in Italy, & its outlines had already been sketched by the author in several dissertations pub- lished here in our own land ; though, as luck would have it, the system itself was finally put into shape and published at Vienna, whither he had gone for a short time. I therefore thought it right that it should be disseminated throughout the whole of Europe, & that preferably as the product of an Italian press. I had in fact already commenced an edition founded on a copy of the Vienna edition, when it came to my knowledge that the author was greatly dissatisfied with the Vienna edition, taken in hand there after his departure ; that innumerable printer's errors had crept in ; that many passages, especially those that contain Algebraical formulae, were ill-arranged and erroneous ; lastly, that the author himself had in mind a complete revision, including certain alterations, to give a better finish to the work, together with certain additional matter. That being the case, I was greatly desirous of obtaining a copy, revised & enlarged by himself ; I also wanted to have him at hand whilst the edition was in progress, & that he should superintend the whole thing for himself. This, however, I was unable to procure during the last few years, in which he has been travelling through nearly the whole of Europe ; until at last he came here, a little while ago, as he returned home from his lengthy wanderings, & stayed here to assist me during the whole time that the edition was in hand. He, in addition to our regular proof-readers, himself also used every care in cor- recting the proof ; even then, however, he has not sufficient confidence in himself as to imagine that not the slightest thing has escaped him. For it is a characteristic of the human mind that it cannot concentrate long on the same subject with sufficient attention. It follows that this ought to be considered in some measure as a first & original edition ; any one who compares it with that issued at Vienna will soon see the difference between them. Many of the minor alterations are made for the purpose of rendering certain passages more elegant & clear ; there are, however, especially at the foot of a page, slight additions also, or slight changes made after the type was set up, merely for the purpose of filling up gaps that were left here & there these gaps being due to the fact that several sheets were being set at the same time by different compositors, and four presses were kept hard at work together. As he was at hand, this could easily be done without causing any disturbance of the sentences or the pagination. 4 TYPOGRAPHUS VENETUS LECTORI Inter mutationes occurret ordo numerorum mutatus in paragraphis : nam numerus 82 de novo accessit totus : deinde is, qui fuerat 261 discerptus est in 5 ; demum in Appendice post num. 534 factae sunt & mutatiunculae nonnullae, & additamenta plura in iis, quae pertinent ad sedem animse. Supplementorum ordo mutatus est itidem ; quae enim fuerant 3, & 4, jam sunt i, & 2 : nam eorum usus in ipso Opere ante alia occurrit. UK autem, quod prius fuerat primum, nunc autem est tertium, accessit in fine Scholium tertium, quod pluribus numeris complec- titur dissertatiunculam integrant de argumento, quod ante aliquot annos in Parisiensi Academia controversiae occasionem exhibuit in Encyclopedico etiam dictionario attactum, in qua dissertatiuncula demonstrat Auctor non esse, cur ad vim exprimendam potentia quaepiam distantice adhibeatur potius, quam functio. Accesserunt per totum Opus notulae marginales, in quibus eorum, quae pertractantur argumenta exponuntur brevissima, quorum ope unico obtutu videri possint omnia, & in memoriam facile revocari. Postremo loco ad calcem Operis additus est fusior catalogus eorum omnium, quae hue usque ab ipso Auctore sunt edita, quorum collectionem omnem expolitam, & correctam, ac eorum, quse nondum absoluta sunt, continuationem meditatur, aggressurus illico post suum regressum in Urbem Romam, quo properat. Hie catalogus impressus fuit Venetisis ante hosce duos annos in reimpressione ejus poematis de Solis ac Lunae defectibus. Porro earn omnium suorum Operum Collectionem, ubi ipse adornaverit, typis ego meis excudendam suscipiam, quam magnificentissime potero. Haec erant, quae te monendum censui ; tu laboribus nostris fruere, & vive felix. THE PREFACE TO THE READER THAT APPEARED IN THE VIENNA EDITION Reader, you have before you a Theory of Natural Philosophy deduced from a single law of Forces. You will find in the opening paragraphs of the first section a statement as to where the Theory has been already published in outline, & to a certain extent explained ; & also the occasion that led me to undertake a more detailed treatment & enlargement of it. For I have thought fit to divide the work into three parts ; the first of these contains the exposition of the Theory itself, its analytical deduction & its demonstration ; the second a fairly full application to Mechanics ; & the third an application to Physics. The most important point, I decided, was for me to take the greatest care that every- thing, as far as was possible, should be clearly explained, & that there should be no need for higher geometry or for the calculus. Thus, in the first part, as well as in the third, there are no proofs by analysis ; nor are there any by geometry, with the exception of a very few that are absolutely necessary, & even these you will find relegated to brief notes set at the foot of a page. I have also added some very few proofs, that required a knowledge of higher algebra & geometry, or were of a rather more complicated nature, all of which have been already published elsewhere, at the end of the work ; I have collected these under the heading Supplements ; & in them I have included my views on Space & Time, which are in accord with my main Theory, & also have been already published elsewhere. In the second part, where the Theory is applied to Mechanics, I have not been able to do without geometrical proofs altogether ; & even in some cases I have had to give algebraical proofs. But these are of such a simple kind that they scarcely ever require anything more than Euclidean geometry, the first and most elementary ideas of trigonometry, and easy analytical calculations. It is true that in the first part there are to be found a good many geometrical diagrams, which at first sight, before the text is considered more closely, will appear to be rather complicated. But these present nothing else but a kind of image of the subjects treated, which by means of these diagrams are set before the eyes for contemplation. The very curve that represents the law of forces is an instance of this. I find that between all points of matter there is a mutual force depending on the distance between them, & changing as this distance changes ; so that it is sometimes attractive, & sometimes repulsive, but always follows a definite continuous law. Laws of variation of this kind between two quantities depending upon one another, as distance & force do in this instance, may be represented either by an analytical formula or by a geometrical curve ; but the former method of representation requires far more knowledge of algebraical processes, & does not assist the imagination in the way that the latter does. Hence I have employed the latter method in the first part of the work, & relegated to the Supplements the analytical formula which represents the curve, & the law of forces which the curve exhibits. The whole matter reduces to this. In a straight line of indefinite length, which is called the axis, a fixed point is taken ; & segments of the straight line cut off from this point represent the distances. A curve is drawn following the general direction of this straight line, & winding about it, so as to cut it in several places. Then perpendiculars that are drawn from the ends of the segments to meet the curve represent the forces ; these forces are greater or less, according as such perpendiculars are greater or less ; & they pass from attractive forces to repulsive, and vice versa, whenever these perpendiculars change their direction, as the curve passes from one side of the axis of indefinite length to the other side of it. Now this requires no geometrical proof, but only a knowledge of certain terms, which either belong to the first elementary principles of geometry, & are thoroughly well known, or are such as can be defined when they are used. The term Asymptote is well known, and from the same idea we speak of the branch of a curve as being asymptotic ; thus a straight line is said to be the asymptote to any branch of a curve when, if the straight line is indefinitely produced, it approaches nearer and nearer to the curvilinear arc which is also prolonged indefinitely in such manner that the distance between them becomes indefinitely diminished, but never altogether vanishes, so that the straight line & the curve never really meet. A careful consideration of the curve given in Fig. I, & of the way in which the relation 13 i 4 AD LECTOREM EX EDITIONE VIENNENSI nexus inter vires, & distantias, est utique ad.rn.odum necessaria ad intelligendam Theoriam ipsam, cujus ea est prescipua qucsdam veluti clavis, sine qua omnino incassum tentarentur cetera ; sea y ejusmodi est, ut tironum, & sane etiam mediocrium, immo etiam longe infra mediocritatem collocatorum, captum non excedat, potissimum si viva accedat Professoris vox mediocriter etiam versati in Mechanica, cujus ope, pro certo habeo, rem ita patentem omnibus reddi posse, ut ii etiam, qui Geometries penitus ignari sunt, paucorum admodum explicatione vocabulorum accidente, earn ipsis oculis intueantur omnino perspicuam. In tertia parte supponuntur utique nonnulla, ques demonstrantur in secunda ; sed ea ipsa sunt admodum pauca, & Us, qui geometricas demonstrationes fastidiunt, facile admodum exponi possunt res ipsez ita, ut penitus etiam sine ullo Geometries adjumento percipiantur, quanquam sine Us ipsa demonstratio haberi non poterit ; ut idcirco in eo differre debeat is, qui secundam partem attente legerit, y Geometriam calleat, ab eo, qui earn omittat, quod ille primus veritates in tertia parte adbibitis, ac ex secunda erutas, ad explicationem Physices, intuebitur per evi- dentiam ex ipsis demonstrationibus haustam, hie secundus easdem quodammodo per fidem Geo- metris adhibitam credet. Hujusmodi inprimis est illud, particulam compositam ex punctis etiam homogeneis, presditis lege virium proposita, posse per solam diversam ipsorum punctorum dispositionem aliam particulam per certum intervallum vel perpetuo attrahere, vel perpetuo repellere, vel nihil in earn agere, atque id ipsum viribus admodum diversis, y qua respectu diver- sarum particularum diver see sint, & diver see respectu partium diver sarum ejusdem particules, ac aliam particulam alicubi etiam urgeant in latus, unde plurium phesnomenorum explicatio in Physica sponte ftuit. Verum qui omnem Theories, y deductionum compagem aliquanto altius inspexerit, ac diligentius perpenderit, videbit, ut spero, me in hoc perquisitionis genere multo ulterius progressum esse, quam olim Newtonus ipse desideravit. Is enim in postremo Optices questione prolatis Us, ques per vim attractivam, & vim repulsivam, mutata distantia ipsi attractives suc- cedentem, explicari poterant, hesc addidit : " Atque h<sc quidem omnia si ita sint, jam Natura universa valde erit simplex, y consimilis sui, perficiens nimirum magnos omnes corporum ceslestium motus attractione gravitatis, qucs est mutua inter corpora ilia omnia, & minores fere omnes particularum suarum motus alia aliqua vi attrahente, y repellente, ques est inter particulas illas mutua." Aliquanto autem inferius de primigeniis particulis agens sic habet : " Porro videntur mihi hce particules primigenics non modo in se vim inerties habere, motusque leges passivas illas, ques ex vi ista necessario oriuntur ; verum etiam motum perpetuo accipere a certis principiis actuosis, qualia nimirum sunt gravitas, y causa fermentationis, y cohesrentia corporum. Atque hesc quidem principia considero non ut occultas qualitates, ques ex specificis rerum formis oriri fingantur, sed ut universales Natures leges, quibus res ipscs sunt formates. Nam principia quidem talia revera existere ostendunt phesnomena Natures, licet ipsorum causes ques sint, nondum fuerit explicatum. Affirmare, singulas rerum species specificis presditas esse qualita- tibus occultis, per quas eae vim certam in agenda habent, hoc utique est nihil dicere : at ex phesnomenis Natures duo, vel tria derivare generalia motus principia, y deinde explicare, quemadmodum proprietates, y actiones rerum corporearum omnium ex istis principiis conse- quantur, id vero magnus esset factus in Philosophia progressus, etiamsi principiorum istorum causes nondum essent cognites. Quare motus principia supradicta proponere non dubito, cum per Naturam universam latissime pateant." Hcsc ibi Newtonus, ubi is quidem magnos in Philosophia progressus facturum arbitratus est eum, qui ad duo, vel tria generalia motus principia ex Natures phesnomenis derivata phesno- menorum explicationem reduxerit, y sua principia protulit, ex quibus inter se diversis eorum aliqua tantummodo explicari posse censuit. Quid igitur, ubi y ea ipsa tria, y alia prcscipua quesque, ut ipsa etiam impenetrabilitas, y impulsio reducantur ad principium unicum legitima ratiocinatione deductum ? At id per meam unicam, y simplicem virium legem presstari, patebit sane consideranti operis totius Synopsim quandam, quam hie subjicio ; sed multo magis opus ipsum diligentius pervolventi. PREFACE TO READER THAT APPEARED IN THE VIENNA EDITION 15 between the forces & the distances is represented by it, is absolutely necessary for the under- standing of the Theory itself, to which it is as it were the chief key, without which it would be quite useless to try to pass on to the rest. But it is of such a nature that it does not go beyond the capacity of beginners, not even of those of very moderate ability, or of classes even far below the level of mediocrity ; especially if they have the additional assistance of a teacher's voice, even though he is only moderately familiar with Mechanics. By his help, I am sure, the subject can be made clear to every one, so that those of them that are quite ignorant of geometry, given the explanation of but a few terms, may get a perfectly good idea of the subject by ocular demonstration. In the third part, some of the theorems that have been proved in the second part are certainly assumed, but there are very few such ; &, for those who do not care for geo- metrical proofs, the facts in question can be quite easily stated in such a manner that they can be completely understood without any assistance from geometry, although no real demonstration is possible without them. There is thus bound to be a difference between the reader who has gone carefully through the second part, & who is well versed in geo- metry, & him who omits the second part ; in that the former will regard the facts, that have been proved in the second part, & are now employed in the third part for the ex- planation of Physics, through the evidence derived from the demonstrations of these facts, whilst the second will credit these same facts through the mere faith that he has in geome- tricians. A specially good instance of this is the fact, that a particle composed of points quite homogeneous, subject to a law of forces as stated, may, merely by altering the arrange- ment of those points, either continually attract, or continually repel, or have no effect at all upon, another particle situated at a known distance from it ; & this too, with forces that differ widely, both in respect of different particles & in respect of different parts of the same particle ; & may even urge another particle in a direction at right angles to the line join- ing the two, a fact that readily gives a perfectly natural explanation of many physical phenomena. Anyone who shall have studied somewhat closely the whole system of my Theory, & what I deduce from it, will see, I hope, that I have advanced in this kind of investigation much further than Newton himself even thought open to his desires. For he, in the last of his " Questions " in his Opticks, after stating the facts that could be explained by means of an attractive force, & a repulsive force that takes the place of the attractive force when the distance is altered, has added these words : " Now if all these things are as stated, then the whole of Nature must be exceedingly simple in design, & similar in all its parts, accom- plishing all the mighty motions of the heavenly bodies, as it does, by the attraction of gravity, which is a mutual force between any two bodies of the whole system ; and Nature accomplishes nearly all the smaller motions of their particles by some other force of attrac- tion or repulsion, which is mutual between any two of those particles." Farther on, when he is speaking about elementary particles, he says : " Moreover, it appears to me that these elementary particles not only possess an essential property of inertia, & laws of motion, though only passive, which are the necessary consequences of this property ; but they also constantly acquire motion from the influence of certain active principles such as, for instance, gravity, the cause of fermentation, & the cohesion of solids. I do not consider these principles to be certain mysterious qualities feigned as arising from characteristic forms of things, but as universal laws of Nature, by the influence of which these very things have been created. For the phenomena of Nature show that these principles do indeed exist, although their nature has not yet been elucidated. To assert that each & every species is endowed with a mysterious property characteristic to it, due to which it has a definite mode in action, is really equivalent to saying nothing at all. On the other hand, to derive from the phenomena of Nature two or three general principles, & then to explain how the pro- perties & actions of all corporate things follow from those principles, this would indeed be a mighty advance in philosophy, even if the causes of those principles had not at the time been discovered. For these reasons I do not hesitate in bringing forward the principles of motion given above, since they are clearly to be perceived throughout the whole range of Nature." These are the words of Newton, & therein he states his opinion that he indeed will have made great strides in philosophy who shall have reduced the explanation of phenomena to two or three general principles derived from the phenomena of Nature ; & he brought forward his own principles, themselves differing from one another, by which he thought that some only of the phenomena could be explained. What then if not only the three he mentions, but also other important principles, such as impenetrability & impul- sive force, be reduced to a single principle, deduced by a process of rigorous argument ! It will be quite clear that this is exactly what is done by my single simple law of forces, to anyone who studies a kind of synopsis of the whole work, which I add below ; but it will be iar more clear to him who studies the whole work with some earnestness, SYNOPSIS TOTIUS OPERIS EX EDITIONE VIENNENSI PARS I sex numeris exhibeo, quando, & qua occasione Theoriam meam invenerim, ac ubi hucusque de ea egerim in dissertationibus jam editis, quid ea commune habeat cum Leibnitiana, quid cum Newtoniana Theoria, in quo ab utraque discrepet, & vero etiam utrique praestet : addo, quid alibi promiserim pertinens ad aequilibrium, & oscillationis centrum, & quemadmodum iis nunc inventis, ac ex unico simplicissimo, ac elegant- issimo theoremate profluentibus omnino sponte, cum dissertatiunculam brevem meditarer, jam eo consilio rem aggressus ; repente mihi in opus integrum justse molis evaserit tractatio. 7 Turn usque ad num. II expono Theoriam ipsam : materiam constantem punctis prorsus simplicibus, indivisibilibus, & inextensis, ac a se invicem distantibus, quae puncta habeant singula vim inertiae, & praeterea vim activam mutuam pendentem a distantiis, ut nimirum, data distantia, detur & magnitude, & directio vis ipsius, mutata autem distantia, mutetur vis ipsa, quae, imminuta distantia in infinitum, sit repulsiva, & quidem excrescens in infinitum : aucta autem distantia, minuatur, evanescat, mutetur in attrac- tivam crescentem primo, turn decrescentem, evanescentem, abeuntem iterum in repul- sivam, idque per multas vices, donee demum in majoribus distantiis abeat in attractivam decrescentem ad sensum in ratione reciproca duplicata distantiarum ; quern nexum virium cum distantiis, & vero etiam earum transitum a positivis ad negativas, sive a repulsivis ad attractivas, vel vice versa, oculis ipsis propono in vi, qua binae elastri cuspides conantur ad es invicem accedere, vel a se invicem recedere, prout sunt plus justo distractae, vel con- tractae. II Inde ad num. 16 ostendo, quo pacto id non sit aggregatum quoddam virium temere coalescentium, sed per unicam curvam continuam exponatur ope abscissarum exprimentium distantias, & ordinatarum exprimentium vires, cujus curvae ductum, & naturam expono, ac ostendo, in quo differat ab hyperbola ilia gradus tertii, quae Newtonianum gravitatem exprimit : ac demum ibidem & argumentum, & divisionem propono operis totius. 1 6 Hisce expositis gradum facio ad exponendam totam illam analysim, qua ego ad ejusmodi Theoriam deveni, & ex qua ipsam arbitror directa, & solidissima ratiocinatione deduci totam. Contendo nimirum usque ad numerum 19 illud, in collisione corporum debere vel haberi compenetrationem, vel violari legem continuitatis, velocitate mutata per saltum, si cum inaequalibus velocitatibus deveniant ad immediatum contactum, quae continuitatis lex cum (ut evinco) debeat omnino observari, illud infero, antequam ad contactum deveniant corpora, debere mutari eorum velocitates per vim quandam, quae sit par extinguendse velocitati, vel velocitatum differentiae, cuivis utcunque magnae. 19 A num. 19 ad 28 expendo effugium, quo ad eludendam argumenti mei vim utuntur ii, qui negant corpora dura, qua quidem responsione uti non possunt Newtoniani, & Corpus- culares generaliter, qui elementares corporum particulas assumunt prorsus duras : qui autem omnes utcunque parvas corporum particulas molles admittunt, vel elasticas, difficultatem non effugiunt, sed transferunt ad primas superficies, vel puncta, in quibus committeretur omnino saltus, & lex continuitatis violaretur : ibidem quendam verborum lusum evolvo, frustra adhibitum ad eludendam argumenti mei vim. * Series numerorum, quibus tractari incipiunt, quae sunt in textu, 16 SYNOPSIS OF THE WHOLE WORK (FROM THE VIENNA EDITION) PART I N the first six articles, I state the time at which I evolved my Theory, what i * led me to it, & where I have discussed it hitherto in essays already pub- lished : also what it has in common with the theories of Leibniz and Newton ; in what it differs from either of these, & in what it is really superior to them both. In addition I state what I have published else- where about equilibrium & the centre of oscillation ; & how, having found out that these matters followed quite easily from a single theorem of the most simple & elegant kind, I proposed to write a short essay thereon ; but when I set to work to deduce the matter from this principle, the discussion, quite unexpectedly to me, developed into a whole work of considerable magnitude. From this .until Art. II, I explain the Theory itself : that matter is unchangeable, 7 and consists of points that are perfectly simple, indivisible, of no extent, & separated from one another ; that each of these points has a property of inertia, & in addition a mutual active force depending on the distance in such a way that, if the distance is given, both the magnitude & the direction of this force are given ; but if the distance is altered, so also is the force altered ; & if the distance is diminished indefinitely, the force is repulsive, & in fact also increases indefinitely ; whilst if the distance is increased, the force will be dimin- ished, vanish, be changed to an attractive force that first of all increases, then decreases, vanishes, is again turned into a repulsive force, & so on many times over ; until at greater distances it finally becomes an attractive force that decreases approximately in the inverse ratio of the squares of the distances. This connection between the forces & the distances, & their passing from positive to negative, or from repulsive to attractive, & conversely, I illustrate by the force with which the two ends of a spring strive to approach towards, or recede from, one another, according as they are pulled apart, or drawn together, by more than the natural amount. From here on to Art. 1 6 I show that it is not merely an aggregate of forces combined n haphazard, but that it is represented by a single continuous curve, by means of abscissse representing the distances & ordinates representing the forces. I expound the construction & nature of this curve ; & I show how it differs from the hyperbola of the third degree which represents Newtonian gravitation. Finally, here too I set forth the scope of the whole work & the nature of the parts into which it is divided. These statements having been made, I start to expound the whole of the analysis, by 16 which I came upon a Theory of this kind, & from which I believe I have deduced the whole of it by a straightforward & perfectly rigorous chain of reasoning. I contend indeed, from here on until Art. 19, that, in the collision of solid bodies, either there must be compene- tration, or the Law of Continuity must be violated by a sudden change of velocity, if the bodies come into immediate contact with unequal velocities. Now since the Law of Continuity must (as I prove that it must) be observed in every case, I infer that, before the bodies reach the point of actual contact, their velocities must be altered by some force which is capable of destroying the velocity, or the difference of the velocities, no matter how great that may be. From Art. 19 to Art. 28 I consider the artifice, adopted for the purpose of evading the 19 strength of my argument by those who deny the existence of hard bodies ; as a matter of fact this cannot be used as an argument against me by the Newtonians, or the Corpuscular- ians in general, for they assume that the elementary particles of solids are perfectly hard. Moreover, those who admit that all the particles of solids, however small they may be, are soft or elastic, yet do not escape the difficulty, but transfer it to prime surfaces, or points ; & here a sudden change would be made & the Law of Continuity violated. In the same connection I consider a certain verbal quibble, used in a vain attempt to foil the force of my reasoning. * These numbers are the numbers of the articles, in which the matters given in the text are first discussed. 17 C 1 8 SYNOPSIS TOTIUS OPERIS 28 Sequentibus num. 28 & 29 binas alias responsiones rejicio aliorum, quarum altera, ut mei argument! vis elidatur, affirmat quispiam, prima materiae elementa compenetrari, alter dicuntur materiae puncta adhuc moveri ad se invicem, ubi localiter omnino quiescunt, & contra primum effugium evinco impenetrabilitatem ex inductione ; contra secundum expono aequivocationem quandam in significatione vocis motus, cui aequivocationi totum innititur. 30 Hinc num. 30, & 31 ostendo, in quo a Mac-Laurino dissentiam, qui considerata eadem, quam ego contemplatus sum, collisione corporum, conclusit, continuitatis legem violari, cum ego eandem illaesam esse debere ratus ad totam devenerim Theoriam meam. 32 Hie igitur, ut meae deductionis vim exponam, in ipsam continuitatis legem inquire, ac a num. 32 ad 38 expono, quid ipsa sit, quid mutatio continua per gradus omnes intermedios, quae nimirum excludat omnem saltum ab una magnitudine ad aliam sine transitu per 39 intermedias, ac Geometriam etiam ad explicationem rei in subsidium advoco : turn earn probo primum ex inductione, ac in ipsum inductionis principium inquirens usque ad num. 44, exhibeo, unde habeatur ejusdem principii vis, ac ubi id adhiberi possit, rem ipsam illustrans exemplo impenetrabilitatis erutae passim per inductionem, donee demum ejus vim 45 applicem ad legem continuitatis demonstrandam : ac sequentibus numeris casus evolvo quosdam binarum classium, in quibus continuitatis lex videtur laedi nee tamen laeditur. 48 Post probationem principii continuitatis petitam ab inductione, aliam num. 48 ejus probationem aggredior metaphysicam quandam, ex necessitate utriusque limitis in quanti- tatibus realibus, vel seriebus quantitatum realium finitis, quae nimirum nee suo principio, nee suo fine carere possunt. Ejus rationis vim ostendo in motu locali, & in Geometria 52 sequentibus duobus numeris : turn num. 52 expono difficultatem quandam, quas petitur ex eo, quod in momento temporis, in quo transitur a non esse ad esse, videatur juxta ejusmodi Theoriam debere simul haberi ipsum esse, & non esse, quorum alterum ad finem praecedentis seriei statuum pertinet, alterum ad sequentis initium, ac solutionem ipsius fuse evolvo, Geometria etiam ad rem oculo ipsi sistendam vocata in auxilium. 63 Num. 63, post epilogum eorum omnium, quae de lege continuitatis sunt dicta, id principium applico ad excludendum saltum immediatum ab una velocitate ad aliam, sine transitu per intermedias, quod & inductionem laederet pro continuitate amplissimam, & induceret pro ipso momento temporis, in quo fieret saltus, binas velocitates, ultimam nimirum seriei praecedentis, & primam novas, cum tamen duas simul velocitates idem mobile habere omnino non possit. Id autem ut illustrem, & evincam, usque ad num. 72 considero velocitatem ipsam, ubi potentialem quandam, ut appello, velocitatem ab actuali secerno, & multa, quae ad ipsarum naturam, ac mutationes pertinent, diligenter evolvo, nonnullis etiam, quae inde contra meae Theoriae probationem objici possunt, dissolutis. His expositis conclude jam illud ex ipsa continuitate, ubi corpus quodpiam velocius movetur post aliud lentius, ad contactum immediatum cum ilia velocitatum inaequalitate deveniri non posse, in quo scilicet contactu primo mutaretur vel utriusque velocitas, vel alterius, per saltum, sed debere mutationem velocitatis incipere ante contactum ipsum. 73 Hinc num. 73 infero, debere haberi mutationis causam, quae appelletur vis : turn num. 74 74 hanc vim debere esse mutuam, & agere in partes contrarias, quod per inductionem evinco, 75 & inde infero num. 75, appellari posse repulsivam ejusmodi vim mutuam, ac ejus legem exquirendam propono. In ejusmodi autem perquisitione usque ad num. 80 invenio illud, debere vim ipsam imminutis distantiis crescere in infinitum ita ut par sit extinguendae velocitati utcunque magnse ; turn & illud, imminutis in infinitum etiam distantiis, debere in infinitum augeri, in maximis autem debere esse e contrario attractivam, uti est gravitas : inde vero colligo limitem inter attractionem, & repulsionem : turn sensim plures, ac etiam plurimos ejusmodi limites invenio, sive transitus ab attractione ad repulsionem, & vice versa, ac formam totius curvae per ordinatas suas exprimentis virium legem determino. SYNOPSIS OF THE WHOLE WORK 19 In the next articles, 28 & 29, I refute a further pair of arguments advanced by others ; 28 in the first of these, in order to evade my reasoning, someone states that there is compene- tration of the primary elements of matter ; in the second, the points of matter are said to be moved with regard to one another, even when they are absolutely at rest as regards position. In reply to the first artifice, I prove the principle of impenetrability by induc- tion ; & in reply to the second, I expose an equivocation in the meaning of the term motion, an equivocation upon which the whole thing depends. Then, in Art. 30, 31, I show in what respect I differ from Maclaurin, who, having 30 considered the same point as myself, came to the conclusion that in the collision of bodies the Law of Continuity was violated ; whereas I obtained the whole of my Theory from the assumption that this law must be unassailable. At this point therefore, in order that the strength of my deductive reasoning might 32 be shown, I investigate the Law of Continuity ; and from Art. 32 to Art. 38, I set forth its nature, & what is meant by a continuous change through all intermediate stages, such as to exclude any sudden change from any one magnitude to another except by a passage through intermediate stages ; & I call in geometry as well to help my explanation of the matter. Then I investigate its truth first of all by induction ; &, investigating the prin- 39 ciple of induction itself, as far as Art. 44, 1 show whence the force of this principle is derived, & where it can be used. I give by way of illustration an example in which impenetrability is derived entirely by induction ; & lastly I apply the force of the principle to demonstrate the Law of Continuity. In the articles that follow I consider certain cases of two kinds, 45 in which the Law of Continuity appears to be violated, but is not however really violated. After this proof of the principle of continuity procured through induction, in Art. 48, 48 I undertake another proof of a metaphysical kind, depending upon the necessity of a limit on either side for either real quantities or for a finite series of real quantities ; & indeed it is impossible that these limits should be lacking, either at the beginning or the end. I demonstrate the force of this reasoning in the case of local motion, & also in geometry, in the next two articles. Then in Art. 52 I explain a certain difficulty, which is derived from the S 2 fact that, at the instant at which there is a passage from non-existence to existence, it appears according to a theory of this kind that we must have at the same time both existence and non-existence. For one of these belongs to the end of the antecedent series of states, & the other to the beginning of the consequent series. I consider fairly fully' the solution of this problem ; and I call in geometry as well to assist in giving a visual representation of the matter. In Art. 63, after summing up all that has been said about the Law of Continuity, I 63 apply the principle to exclude the possibility of any sudden change from one velocity to another, except by passing through intermediate velocities ; this would be contrary to the very full proof that I give for continuity, as it would lead to our having two velocities at the instant at which the change occurred. That is to say, there would be the final velocity of the antecedent series, & the initial velocity of the consequent series ; in spite of the fact that it is quite impossible for a moving body to have two different velocities at the same time. Moreover, in order to illustrate & prove the point, from here on to Art. 72, I consider velocity itself ; and I distinguish between a potential velocity, as I call it, & an actual velocity ; I also investigate carefully many matters that relate to the nature of these velocities & to their changes. Further, I settle several difficulties that can be brought up in opposition to the proof of my Theory, in consequence. This done, I then conclude from the principle of continuity that, when one body with a greater velocity follows after another body having a less velocity, it is impossible that there should ever be absolute contact with such an inequality of velocities ; that is to say, a case of the velocity of each, or of one or the other, of them being changed suddenly at the instant of contact. I assert on the other hand that the change in the velocities must begin before contact. Hence, in Art. 73, I infer that there must be a cause for this change : 73 which is to be called " force." Then, in Art. 74, I prove that this force is a mutual one, & 74 that it acts in opposite directions ; the proof is by induction. From this, in Art. 75, I 75 infer that such a mutual force may be said to be repulsive ; & I undertake the investigation of the law that governs it. Carrying on this investigation as far as Art. 80, I find that this force must increase indefinitely as the distance is diminished, in order that it may be capable of destroying any velocity, however great that velocity may be. Moreover, I find that, whilst the force must be indefinitely increased as the distance is indefinitely decreased, it must be on the contrary attractive at very great distances, as is the case for gravitation. Hence I infer that there must be a limit-point forming a boundary between attraction & repulsion ; & then by degrees I find more, indeed very many more, of such limit-points, or points of transition from attraction to repulsion, & from repulsion to attraction ; & I determine the form of the entire curve, that expresses by its ordinates the law of these forces. 20 SYNOPSIS TOTIUS OPERIS 8 1 Eo usque virium legem deduce, ac definio ; turn num. 81 eruo ex ipsa lege consti- tutionem elementorum materiae, quae debent esse simplicia, ob repulsionem in minimis distantiis in immensum auctam ; nam ea, si forte ipsa elementa partibus constarent, nexum omnem dissolveret. Usque ad num. 88 inquire in illud, an hasc elementa, ut simplicia esse debent, ita etiam inextensa esse debeant, ac exposita ilia, quam virtualem extensionem appellant, eandem exclude inductionis principio, & difficultatem evolvo turn earn, quae peti possit ab exemplo ejus generis extensionis, quam in anima indivisibili, & simplice per aliquam corporis partem divisibilem, & extensam passim admittunt : vel omnipraesentiae Dei : turn earn, quae peti possit ab analogia cum quiete, in qua nimirum conjungi debeat unicum spatii punctum cum serie continua momentorum temporis, uti in extensione virtuali unicum momentum temporis cum serie continua punctorum spatii conjungeretur, ubi ostendo, nee quietem omnimodam in Natura haberi usquam, nee adesse semper omnimodam inter 88 tempus, & spatium analogiam. Hie autem ingentem colligo ejusmodi determinationis fructum, ostendens usque ad num. 91, quantum prosit simplicitas, indivisibilitas, inextensio elementorum materiae, ob summotum transitum a vacuo continue per saltum ad materiam continuam, ac ob sublatum limitem densitatis, quae in ejusmodi Theoria ut minui in infinitum potest, ita potest in infinitum etiam augeri, dum in communi, ubi ad contactum deventum est, augeri ultra densitas nequaquam potest, potissimum vero ob sublatum omne continuum coexistens, quo sublato & gravissimae difficultates plurimse evanescunt, & infinitum actu existens habetur nullum, sed in possibilibus tantummodo remanet series finitorum in infinitum producta. 91 His definitis, inquire usque ad num. 99 in illud, an ejusmodi elementa sint censenda homogenea, an heterogenea : ac primo quidem argumentum pro homogeneitate saltern in eo, quod pertinet ad totam virium legem, invenio in homogenietate tanta primi cruris repulsivi in minimis distantiis, ex quo pendet impenetrabilitas, & postremi attractivi, quo gravitas exhibetur, in quibus omnis materia est penitus homogenea. Ostendo autem, nihil contra ejusmodi homogenietatem evinci ex principio Leibnitiano indiscernibilium, nihil ex inductione, & ostendo, unde tantum proveniat discrimen in compositis massulis, ut in frondibus, & foliis ; ac per inductionem, & analogiam demonstro, naturam nos ad homo- geneitatem elementorum, non ad heterogeneitatem deducere. 100 Ea ad probationem Theoriae pertinent ; qua absoluta, antequam inde fructus colli- gantur multiplices, gradum hie facio ad evolvendas difficultates, quae vel objectae jam sunt, vel objici posse videntur mihi, primo quidem contra vires in genere, turn contra meam hanc expositam, comprobatamque virium legem, ac demum contra puncta ilia indivisibilia, & inextensa, quae ex ipsa ejusmodi virium lege deducuntur. 101 Primo quidem, ut iis etiam faciam satis, qui inani vocabulorum quorundam sono perturbantur, a num. 101 ad 104 ostendo, vires hasce non esse quoddam occultarum qualitatum genus, sed patentem sane Mechanismum, cum & idea earum sit admodum distincta, & existentia, ac lex positive comprobata ; ad Mechanicam vero pertineat omnis 104 tractatio de Motibus, qui a datis viribus etiam sine immediate impulsu oriuntur. A num. 104 ad 106 ostendo, nullum committi saltum in transitu a repulsionibus ad attractiones, 1 06 & vice versa, cum nimirum per omnes inter medias quantitates is transitus fiat. Inde vero ad objectiones gradum facio, quae totam curvas formam impetunt. Ostendo nimirum usque ad num. 116, non posse omnes repulsiones a minore attractione desumi ; repulsiones ejusdem esse seriei cum attractionibus, a quibus differant tantummodo ut minus a majore, sive ut negativum a positivo ; ex ipsa curvarum natura, quae, quo altioris sunt gradus, eo in pluribus punctis rectam secare possunt, & eo in immensum plures sunt numero ; haberi potius, ubi curva quaeritur, quae vires exprimat, indicium pro curva ejus naturae, ut rectam in plurimis punctis secet, adeoque plurimos secum afferat virium transitus a repulsivis ad attractivas, quam pro curva, quae nusquam axem secans attractiones solas, vel solas pro distantiis omnibus repulsiones exhibeat : sed vires repulsivas, & multiplicitatem transituum esse positive probatam, & deductam totam curvas formam, quam itidem ostendo, non esse ex arcubus natura diversis temere coalescentem, sed omnino simplicem, atque earn ipsam SYNOPSIS OF THE WHOLE WORK 21 So far I have been occupied in deducing and settling the law of these forces. Next, in Art. 8r, I derive from this law the constitution of the elements of matter. These must be 81 quite simple, on account of the repulsion at very small distances being immensely great ; for if by chance those elements were made up of parts, the repulsion would destroy all connections between them. Then, as far as Art. 88, I consider the point, as to whether these elements, as they must be simple, must therefore be also of no extent ; &, having ex- plained what is called " virtual extension," I reject it by the principle of induction. I then consider the difficulty which may be brought forward from an example of this kind of extension ; such as is generally admitted in the case of the indivisible and one-fold soul pervading a divisible & extended portion of the body, or in the case of the omnipresence of GOD. Next I consider the difficulty that may be brought forward from an analogy with rest ; for here in truth one point of space must be connected with a continuous series of instants of time, just as in virtual extension a single instant of time would be connected with a continuous series of points of space. I show that there can neither be perfect rest any- gg where in Nature, nor can there be at all times a perfect analogy between time and space. In this connection, I also gather a large harvest from such a conclusion as this ; showing, as far as Art. 91, the great advantage of simplicity, indivisibility, & non-extension in the elements of matter. For they do away with the idea of a passage from a continuous vacuum to continuous matter through a sudden change. Also they render unnecessary any limit to density : this, in a Theory like mine, can be just as well increased to an indefinite extent, as it can be indefinitely decreased : whilst in the ordinary theory, as soon as contact takes place, the density cannot in any way be further increased. But, most especially, they do away with the idea of everything continuous coexisting ; & when this is done away with, the majority of the greatest difficulties vanish. Further, nothing infinite is found actually existing ; the only thing possible that remains is a series of finite things produced inde- finitely. These things being settled, I investigate, as far as Art. 99, the point as to whether QJ elements of this kind are to be considered as being homogeneous or heterogeneous. I find my first evidence in favour of homogeneity at least as far as the complete law of forces is concerned in the equally great homogeneity of the first repulsive branch of my curve of forces for very small distances, upon which depends impenetrability, & of the last attrac- tive branch, by which gravity is represented. Moreover I show that there is nothing that can be proved in opposition to homogeneity such as this, that can be derived from either the Leibnizian principle of " indiscernibles," or by induction. I also show whence arise those differences, that are so great amongst small composite bodies, such as we see in boughs & leaves ; & I prove, by induction & analogy, that the very nature of things leads us to homogeneity, & not to heterogeneity, for the elements of matter. These matters are all connected with the proof of my Theory. Having accomplished IO o this, before I start to gather the manifold fruits to be derived from it, I proceed to consider the objections to my theory, such as either have been already raised or seem to me capable of being raised ; first against forces in general, secondly against the law of forces that I have enunciated & proved, & finally against those indivisible, non-extended points that are deduced from a law of forces of this kind. First of all then, in order that I may satisfy even those who are confused over the 101 empty sound of certain terms, I show, in Art. 101 to 104, that these forces are not some sort of mysterious qualities ; but that they form a readily intelligible mechanism, since both the idea of them is perfectly distinct, as well as their existence, & in addition the law that governs them is demonstrated in a direct manner. To Mechanics belongs every dis- cussion concerning motions that arise from given forces without any direct impulse. In Art. 104 to 106, I show that no sudden change takes place in passing from repulsions to 104 attractions or vice versa ; for this transition is made through every intermediate quantity. Then I pass on to consider the objections that are made against the whole form of my 106 curve. I show indeed, from here on to Art. 116, that all repulsions cannot be taken to come from a decreased attraction ; that repulsions belong to the self-same series as attrac- tions, differing from them only as less does from more, or negative from positive. From the very nature of the curves (for which, the higher the degree, the more points there are in which they can intersect a right line, & vastly more such curves there are), I deduce that there is more reason for assuming a curve of the nature of mine (so that it may cut a right line in a large number of points, & thus give a large number of transitions of the forces from repulsions to attractions), than for assuming a curve that, since it does not cut the axis anywhere, will represent attractions alone, or repulsions alone, at all distances. Further, I point out that repulsive forces, and a multiplicity of transitions are directly demonstrated, & the whole form of the curve is a matter of deduction ; & I also show that it is not formed of a number of arcs differing in nature connected together haphazard ; 22 SYNOPSIS TOTIUS OPERIS simplicitatem in Supplementis cvidentissime demonstro, exhibens methodum, qua deveniri possit ad aequationem ejusmodi curvse simplicem, & uniformem ; licet, ut hie ostendo, ipsa ilia lex virium possit mente resolvi in plures, quae per plures curvas exponantur, a quibus tamen omnibus ilia reapse unica lex, per unicam illam continuant, & in se simplicem curvam componatur. 121 A num. 121 refello, quae objici possunt a lege gravitatis decrescentis in ratione reciproca duplicata distantiarum, quae nimirum in minimis distantiis attractionem requirit crescentem in infinitum. Ostendo autem, ipsam non esse uspiam accurate in ejusmodi ratione, nisi imaginarias resolutiones exhibeamus ; nee vero ex Astronomia deduci ejusmodi legem prorsus accurate servatam in ipsis Planetarum, & Cometarum distantiis, sed ad summum ita 124 proxime, ut differentia ab ea lege sit perquam exigua : ac a num. 124 expendo argumentum, quod pro ejusmodi lege desumi possit ex eo, quod cuipiam visa sit omnium optima, & idcirco electa ab Auctore Naturae, ubi ipsum Optimismi principium ad trutinam revoco, ac exclude, & vero illud etiam evinco, non esse, cur omnium optima ejusmodi lex censeatur : in Supplementis vero ostendo, ad qua; potius absurda deducet ejusmodi lex, & vero etiam aliae plures attractionis, quae imminutis in infinitum distantiis excrescat in infinitum. 131 Num. 131 a viribus transeo ad elementa, & primum ostendo, cur punctorum inexten- sorum ideam non habeamus, quod nimirum earn haurire non possumus per sensus, quos solae massae, & quidem grandiores, afficiunt, atque idcirco eandem nos ipsi debemus per reflexionem efformare, quod quidem facile possumus. Ceterum illud ostendo, me non inducere primum in Physicam puncta indivisibilia, & inextensa, cum eo etiam Leibnitianae monades recidant, sed sublata extensione continua difficultatem auferre illam omnem, quae jam olim contra Zenonicos objecta, nunquam est satis soluta, qua fit, ut extensio continua ab inextensis effici omnino non possit. 140 Num. 140 ostendo, inductionis principium contra ipsa nullam habere vim, ipsorum autem existentiam vel inde probari, quod continuitas se se ipsam destruat, & ex ea assumpta probetur argumentis a me institutis hoc ipsum, prima elementa esse indivisibilia, & inextensa, 143 nee ullum haberi extensum continuum. A num. 143 ostendo, ubi continuitatem admittam, nimirum in solis motibus ; ac illud explico, quid mihi sit spatium, quid tempus, quorum naturam in Supplementis multo uberius expono. Porro continuitatem ipsam ostendo a natura in solis motibus obtineri accurate, in reliquis affectari quodammodo ; ubi & exempla quaedam evolvo continuitatis primo aspectu violatae, in quibusdam proprietatibus luminis, ac in aliis quibusdam casibus, in quibus quaedam crescunt per additionem partium, non (ut ajunt) per intussumptionem. \ 153 A num. 153 ostendo, quantum haec mea puncta a spiritibus differant ; ac illud etiam evolvo, unde fiat, ut in ipsa idea corporis videatur includi extensio continua, ubi in ipsam idearum nostrarum originem inquire, & quae inde praejudicia profluant, expono. Postremo 165 autem loco num. 165 innuo, qui fieri possit, ut puncta inextensa, & a se invicem distantia, in massam coalescant, quantum libet, cohaerentem, & iis proprietatibus praeditam, quas in corporibus experimur, quod tamen ad tertiam partem pertinet, ibi multo uberius pertrac- tandum ; ac ibi quidem primam hanc partem absolve. PARS II 166 Num. 166 hujus partis argumentum propono ; sequenti vero 167, quae potissimum in curva virium consideranda sint, enuncio. Eorum considerationem aggressus, primo quidem 1 68 usque ad num. 172 in ipsos arcus inquire, quorum alii attractivi, alii repulsivi, alii asym- ptotici, ubi casuum occurrit mira multitudo, & in quibusdam consectaria notatu digna, ut & illud, cum ejus formae curva plurium asymptotorum esse possit, Mundorum prorsus similium seriem posse oriri, quorum alter respectu alterius vices agat unius, & indissolubilis SYNOPSIS OF THE WHOLE WORK 23 but that it is absolutely one-fold. This one-fold character I demonstrate in the Supple- ments in a very evident manner, giving a method by which a simple and uniform equation may be obtained for a curve of this kind. Although, as I there point out, this law of forces may be mentally resolved into several, and these may be represented by several correspond- ing curves, yet that law, actually unique, may be compounded from all of these together by means of the unique, continuous & one-fold curve that I give. In Art. 121, I start to give a refutation of those objections that may be raised from I2 i a consideration of the fact that the law of gravitation, decreasing in the inverse duplicate ratio of the distances, demands that there should be an attraction at very small distances, & that it should increase indefinitely. However, I show that the law is nowhere exactly in conformity with a ratio of this sort, unless we add explanations that are merely imaginative ; nor, I assert, can a law of this kind be deduced from astronomy, that is followed with per- fect accuracy even at the distances of the planets & the comets, but one merely that is at most so very nearly correct, that the difference from the law of inverse squares is very slight. From Art. 124 onwards,! examine the value of the argument that can be drawn 124 in favour of a law of this sort from the view that, as some have thought, it is the best of all, & that on that account it was selected by the Founder of Nature. In connection with this I examine the principle of Optimism, & I reject it ; moreover I prove conclusively that there is no reason why this sort of law should be supposed to be the best of all. Fur- ther in the Supplements, I show to what absurdities a law of this sort is more likely to lead ; & the same thing for other laws of an attraction that increases indefinitely as the distance is diminished indefinitely. In Art. 131 I pass from forces to elements. I first of all show the reason why we may 1*1 not appreciate the idea of non-extended points ; it is because we are unable to perceive them by means of the senses, which are only affected by masses, & these too must be of considerable size. Consequently we have to build up the idea by a process of reasoning ; & this we can do without any difficulty. In addition, I point out that I am not the first to introduce indivisible & non-extended points into physical science ; for the " monads " of Leibniz practically come to the same thing. But I show that, by rejecting the idea of continuous extension, I remove the whole of the difficulty, which was raised against the disciples of Zeno in years gone by, & has never been answered satisfactorily ; namely, the difficulty arising from the fact that by no possible means can continuous extension be made up from things of no extent. In Art. 140 I show that the principle of induction yields no argument against these 140 indivisibles ; rather their existence is demonstrated by that principle, for continuity is self-contradictory. On this assumption it may be proved, by arguments originated by myself, that the primary elements are indivisible & non-extended, & that there does not exist anything possessing the property of continuous extension. From Art. 143 onwards, j ., I point out the only connection in which I shall admit continuity, & that is in motion. I state the idea that I have with regard to space, & also time : the nature of these I explain much more fully in the Supplements. Further, I show that continuity itself is really a property of motions only, & that in all other things it is more or less a false assumption. Here I also consider some examples in which continuity at first sight appears to be violated, such as in some of the properties of light, & in certain other cases where things increase by addition of parts, and not by intussumption, as it is termed. From Art. 153 onwards, I show how greatly these points of mine differ from object- 153 souls. I consider how it comes about that continuous extension seems to be included in the very idea of a body ; & in this connection, I investigate the origin of our ideas & I explain the prejudgments that arise therefrom. Finally, in Art. 165, I lightly 165 sketch what might happen to enable points that are of no extent, & at a distance from one another, to coalesce into a coherent mass of any size, endowed with those properties that we experience in bodies. This, however, belongs to the third part ; & there it will be much more fully developed. This finishes the first part. PART II In Art. 1 66 I state the theme of this second part ; and in Art. 167 I declare what 166 matters are to be considered more especially in connection with the curve of forces. Com- ing to the consideration of these matters, I first of all, as far as Art. 172, investigate the 168 arcs of the curve, some of which are attractive, some repulsive and some asymptotic. Here a marvellous number of different cases present themselves, & to some of them there are noteworthy corollaries ; such as that, since a curve of this kind is capable of possessing a considerable number of asymptotes, there can arise a series of perfectly similar cosmi, each of which will act upon all the others as a single inviolate elementary system. From Art. 172 24 SYNOPSIS TOTIUS OPERIS 172 element!. Ad. num. 179 areas contemplor arcubus clausas, quae respondentes segmento axis cuicunque, esse possunt magnitudine utcunque magnae, vel parvae, sunt autem mensura 179 incrementi, vel decrement! quadrat! velocitatum. Ad num. 189 inquire in appulsus curvse ad axem, sive is ibi secetur ab eadem (quo casu habentur transitus vel a repulsione ad attractionem, vel ab attractione ad repulsionem, quos dico limites, & quorum maximus est in tota mea Theoria usus), sive tangatur, & curva retro redeat, ubi etiam pro appulsibus considero recessus in infinitum per arcus asymptoticos, & qui transitus, sive limites, oriantur inde, vel in Natura admitti possint, evolvo. 189 Num. 189 a consideratione curvae ad punctorum combinationem gradum facio, ac primo quidem usque ad num. 204 ago de systemate duorum punctorum, ea pertractans, quas pertinent ad eorum vires mutuas, & motus, sive sibi relinquantur, sive projiciantur utcunque, ubi & conjunctione ipsorum exposita in distantiis limitum, & oscillationibus variis, sive nullam externam punctorum aliorum actionem sentiant, sive perturbentur ab eadem, illud innuo in antecessum, quanto id usui futurum sit in parte tertia ad exponenda cohaesionis varia genera, fermentationes, conflagrationes, emissiones vaporum, proprietates luminis, elasticitatem, mollitiem. 204 Succedit a Num. 204 ad 239 multo uberior consideratio trium punctorum, quorum vires generaliter facile definiuntur data ipsorum positione quacunque : verum utcunque data positione, & celeritate nondum a Geometris inventi sunt motus ita, ut generaliter pro casibus omnibus absolvi calculus possit. Vires igitur, & variationem ingentem, quam diversae pariunt combinationes punctorum, utut tantummodo numero trium, persequor 209 usque ad num. 209. Hinc usque ad num. 214 quaedam evolvo, quae pertinent ad vires ortas in singulis ex actione composita reliquorum duorum, & quae tertium punctum non ad accessum urgeant, vel recessum tantummodo respectu eorundem, sed & in latus, ubi & soliditatis imago prodit, & ingens sane discrimen in distantiis particularum perquam exiguis ac summa in maximis, in quibus gravitas agit, conformitas, quod quanto itidem ad Naturae 214 explicationem futurum sit usui, significo. Usque ad num. 221 ipsis etiam oculis contem- plandum propono ingens discrimen in legibus virium, quibus bina puncta agunt in tertium, sive id jaceat in recta, qua junguntur, sive in recta ipsi perpendiculari, & eorum intervallum secante bifariam, constructis ex data primigenia curva curvis vires compositas exhibentibus : 221 turn sequentibus binis numeris casum evolvo notatu dignissimum, in quo mutata sola positione binorum punctorum, punctum tertium per idem quoddam intervallum, situm in eadem distantia a medio eorum intervallo, vel perpetuo attrahitur, vel perpetuo repellitur, vel nee attrahitur, nee repellitur ; cujusmodi discrimen cum in massis haberi debeat multo 222 majus, illud indico, num. 222, quantus inde itidem in Physicam usus proveniat. 223 Hie jam num. 223 a viribus binorum punctorum transeo ad considerandum totum ipsorum systema, & usque ad num. 228 contemplor tria puncta in directum sita, ex quorum mutuis viribus relationes quaedam exurgunt, quas multo generaliores redduntur inferius, ubi in tribus etiam punctis tantummodo adumbrantur, quae pertinent ad virgas rigidas, flexiles, elasticas, ac ad vectem, & ad alia plura, quae itidem inferius, ubi de massis, multo generaliora 228 fiunt. Demum usque ad num. 238 contemplor tria puncta posita non in directum, sive in aequilibrio sint, sive in perimetro ellipsium quarundam, vel curvarum aliarum ; in quibus mira occurrit analogia limitum quorundam cum limitibus, quos habent bina puncta in axe curvae primigeniae ad se invicem, atque ibidem multo major varietas casuum indicatur pro massis, & specimen applicationis exhibetur ad soliditatem, & liquationem per celerem 238 intestinum motum punctis impressum. Sequentibus autem binis numeris generalia quaedam expono de systemate punctorum quatuor cum applicatione ad virgas solidas, rigidas, flexiles, ac ordines particularum varies exhibeo per pyramides, quarum infimae ex punctis quatuor, superiores ex quatuor pyramidibus singulae coalescant. 24 A num. 240 ad massas gradu facto usque a num. 264 considero, quae ad centrum gravi- tatis pertinent, ac demonstro generaliter, in quavis massa esse aliquod, & esse unicum : ostendo, quo pacto determinari generaliter possit, & quid in methodo, quae communiter adhibetur, desit ad habendam demonstrationis vim, luculenter expono, & suppleo, ac SYNOPSIS OF THE WHOLE WORK 25 to Art. 179, I consider the areas included by the arcs; these, corresponding to different 172 segments of the axis, may be of any magnitude whatever, either great or small ; moreover they measure the increment or decrement in the squares of the velocities. Then, on as 179 far as Art. 189, 1 investigate the approach of the curve to the axis ; both when the former is cut by the latter, in which case there are transitions from repulsion to attraction and from attraction to repulsion, which I call ' limits,' & use very largely in every part of my Theory ; & also when the former is touched by the latter, & the curve once again recedes from the axis. I consider, too, as a case of approach, recession to infinity along an asymp- totic arc ; and I investigate what transitions, or limits, may arise from such a case, & whether such are admissible in Nature. In Art. 189, I pass on from the consideration of the curve to combinations of points. l %9 First, as far as Art. 204, I deal with a system of two points. I work out those things that concern their mutual forces, and motions, whether they are left to themselves or pro- jected in any manner whatever. Here also, having explained the connection between these motions & the distances of the limits, & different cases of oscillations, whether they are affected by external action of other points, or are not so disturbed, I make an antici- patory note of the great use to which this will be put in the third part, for the purpose of explaining various kinds of cohesion, fermentations, conflagrations, emissions of vapours, the properties of light, elasticity and flexibility. There follows, from Art. 204 to Art. 239, the much more fruitful consideration of a 204 system of three points. The forces connected with them can in general be easily deter- mined for any given positions of the points ; but, when any position & velocity are given, the motions have not yet been obtained by geometricians in such a form that the general calculation can be performed for every possible case. So I proceed to consider the forces, & the huge variation that different combinations of the points beget, although they are only three in number, as far as Art. 209. From that, on to Art. 214, I consider certain 209 things that have to do with the forces that arise from the action, on each of the points, of the other two together, & how these urge the third point not only to approach, or recede from, themselves, but also in a direction at right angles ; in this connection there comes forth an analogy with solidity, & a truly immense difference between the several cases when the distances are very small, & the greatest conformity possible at very great distances such as those at which gravity acts ; & I point out what great use will be made of this also in explaining the constitution of Nature. Then up to Art. 221, I give ocular demonstra- 214 tions of the huge differences that there are in the laws of forces with which two points act upon a third, whether it lies in the right line joining them, or in the right line that is the perpendicular which bisects the interval between them ; this I do by constructing, from the primary curve, curves representing the composite forces. Then in the two articles 221 that follow, I consider the case, a really important one, in which, by merely changing the position of the two points, the third point, at any and the same definite interval situated at the same distance from the middle point of the interval between the two points, will be either continually attracted, or continually repelled, or neither attracted nor repelled ; & since a difference of this kind should hold to a much greater degree in masses, I point out, in Art. 222, the great use that will be made of this also in Physics. 222 At this point then, in Art. 223, I pass from the forces derived from two points to the 22 3 consideration of a whole system of them ; and, as far as Art. 228, I study three points situated in a right line, from the mutual forces of which there arise certain relations, which I return to later in much greater generality ; in this connection also are outlined, for three points only, matters that have to do with rods, either rigid, flexible or elastic, and with the lever, as well as many other things ; these, too, are treated much more generally later on, when I consider masses. Then right on to Art. 238, I consider three points that do not lie in a right line, whether they are in equilibrium, or moving in the perimeters of certain ellipses or other curves. Here we come across a marvellous analogy between certain limits and the limits which two points lying on the axis of the primary curve have with respect to each other ; & here also a much greater variety of cases for masses is shown, & an example is given of the application to solidity, & liquefaction, on account of a quick internal motion being impressed on the points of the body. Moreover, in the two articles that then follow, I state some general propositions with regard to a system of four points, together with their application to solid rods, both rigid and flexible ; I also give an illus- tration of various classes of particles by means of pyramids, each of which is formed of four points in the most simple case, & of four of such pyramids in the more complicated cases. From Art. 240 as far as Art. 264, I pass on to masses & consider matters pertaining to 2 4 the centre of gravity ; & I prove that in general there is one, & only one, in any given mass. I show how it can in general be determined, & I set forth in clear terms the point that is lacking in the usual method, when it comes to a question of rigorous proof ; this deficiency 26 SYNOPSIS TOTIUS OPERIS exemplum profero quoddam ejusdem generis, quod ad numerorum pertinet multiplica- tionem, & ad virium compositionem per parallelogramma, quam alia methodo generaliore exhibeo analoga illi ipsi, qua generaliter in centrum gravitatis inquire : turn vero ejusdem ope demonstro admodum expedite, & accuratissime celebre illud Newtoni theorema de statu centri gravitatis per mutuas internas vires numquam turbato. 264 Ejus tractionis fructus colligo plures : conservationem ejusdem quantitatis motuum in 265 Mundo in eandem plagam num. 264, sequalitatem actionis, & reactionis in massis num. 265, 266 collisionem corporum, & communicationem motus in congressibus directis cum eorum 276 legibus, inde num. 276 congressus obliques, quorum Theoriam a resolutione motuum reduce 277, 278 ad compositionem num. 277, quod sequent! numero 278 transfero ad incursum etiam in 270 planum immobile ; ac a num. 279 ad 289 ostendo nullam haberi in Natura veram virium, aut motuum resolutionem, sed imaginariam tantummodo, ubi omnia evolvo, & explico casuum genera, quae prima fronte virium resolutionem requirere videntur. 289 A num. 289 ad 297 leges expono compositionis virium, & resolutionis, ubi & illud notissimum, quo pacto in compositione decrescat vis, in resolutione crescat, sed in ilia priore conspirantium summa semper maneat, contrariis elisis ; in hac posteriore concipiantur tantummodo binae vires contrarise adjectas, quse consideratio nihil turbet phenomena ; unde fiat, ut nihil inde pro virium vivarum Theoria deduci possit, cum sine iis explicentur omnia, ubi plura itidem explico ex iis phsenomenis, quse pro ipsis viribus vivis afferri solent. 2Q7 A num. 297 occasione inde arrepta aggredior qusedam, quae ad legem continuitatis pertinent, ubique in motibus sancte servatam, ac ostendo illud, idcirco in collisionibus corporum, ac in motu reflexo, leges vulgo definitas, non nisi proxime tantummodo observari, & usque ad num. 307 relationes varias persequor angulorum incidentisa, & reflexionis, sive vires constanter in accessu attrahant, vel repellant constanter, sive jam attrahant, jam repellant : ubi & illud considero, quid accidat, si scabrities superficiei agentis exigua sit, quid, si ingens, ac elementa profero, quae ad luminis reflexionem, & refractionem explican- dam, definiendamque ex Mechanica requiritur, relationem itidem vis absolutae ad relativam in obliquo gravium descensu, & nonnulla, quae ad oscillationum accuratiorem Theoriam necessaria sunt, prorsus elementaria, diligenter expono. 307 A num. 307 inquire in trium massarum systema, ubi usque ad num. 313 theoremata evolvo plura, quae pertinent ad directionem virium in singulis compositarum e binis reliquarum actionibus, ut illud, eas directiones vel esse inter se parallelas, vel, si utrinque 313 indefinite producantur, per quoddam commune punctum transire omnes : turn usque ad 321 theoremata alia plura, quae pertinent ad earumdem compositarum virium rationem ad se invicem, ut illud & simplex, & elegans, binarum massarum vires acceleratrices esse semper in ratione composita ex tribus reciprocis rationibus, distantise ipsarum a massa tertia, sinus anguli, quern singularum directio continet cum sua ejusmodi distantia, & massae ipsius earn habentis compositam vim, ad distantiam, sinum, massam alteram ; vires autem motrices habere tantummodo priores rationes duas elisa tertia. 321 Eorum theorematum fructum colligo deducens inde usque ad num. 328, quae ad aequilibrium pertinent divergentium utcumque virium, & ipsius aequilibrii centrum, ac nisum centri in fulcrum, & quae ad prseponderantiam, Theoriam extendens ad casum etiam, quo massae non in se invicem agant mutuo immediate, sed per intermedias alias, quse nexum concilient, & virgarum nectentium suppleant vices, ac ad massas etiam quotcunque, quarum singulas cum centro conversionis, & alia quavis assumpta massa connexas concipio, unde principium momenti deduce pro machinis omnibus : turn omnium vectium genera evolvo, ut & illud, facta suspensione per centrum gravitatis haberi aequilibrium, sed in ipso centro debere sentiri vim a fulcro, vel sustinente puncto, sequalem summae ponderum totius systematis, unde demum pateat ejus ratio, quod passim sine demonstratione assumitur, nimirum systemate quiescente, & impedito omni partium motu per aequilibrium, totam massam concipi posse ut in centro gravitatis collectam. SYNOPSIS OF THE WHOLE WORK 27 I supply, & I bring forward a certain example of the same sort, that deals with the multi- plication of numbers, & to the composition of forces by the parallelogram law ; the latter I prove by another more general method, analogous to that which I use in the general investigation for the centre of gravity. Then by its help I prove very expeditiously & with extreme rigour that well-known theorem of Newton, in which he affirmed that the state of the centre of gravity is in no way altered by the internal mutual forces. I gather several good results from this method of treatment. In Art. 264, the con- 264 servation of the same quantity of motion in the Universe in one plane ; in Art. 265 the 265 equality of action and reaction amongst masses ; then the collision of solid bodies, and the 266 communication of motions in direct impacts & the laws that govern them, & from that, 276 in Art. 276, oblique impacts ; in Art. 277 I reduce the theory of these from resolution of 277 motions to compositions, & in the article that follows, Art. 278, I pass to impact on to a 278 fixed plane; from Art. 279 to Art. 289 I show that there can be no real resolution of forces 279 or of motions in Nature, but only a hypothetical one ; & in this connection I consider & explain all sorts of cases, in which at first sight it would seem that there must be resolution. From Art. 289 to Art. 297, 1 state the laws for the composition & resolution of forces ; 289 here also I give the explanation of that well-known fact, that force decreases in composition, increases in resolution, but always remains equal to the sum of the parts acting in the same direction as itself in the first, the rest being equal & opposite cancel one another ; whilst in the second, all that is done is to suppose that two equal & opposite forces are added on, which supposition has no effect on the phenomena. Thus it comes about that nothing can be deduced from this in favour of the Theory of living forces, since everything can be explained without them ; in the same connection, I explain also many of the phenomena, which are usually brought forward as evidence in favour of these ' living forces.' In Art. 297, I seize the opportunity offered by the results just mentioned to attack 207 certain matters that relate to the law of continuity, which in all cases of motion is strictly observed ; & I show that, in the collision of solid bodies, & in reflected motion, the laws, as usually stated, are therefore only approximately followed. From this, as far as Art. 307, I make out the various relations between the angles of incidence & reflection, whether the forces, as the bodies approach one another, continually attract, or continually repel, or attract at one time & repel at another. I also consider what will happen if the roughness of the acting surface is very slight, & what if it is very great. I also state the first principles, derived from mechanics, that are required for the explanation & determination of the reflection & refraction of light ; also the relation of the absolute to the relative force in the oblique descent of heavy bodies ; & some theorems that are requisite for the more accurate theory of oscillations ; these, though quite elementary, I explain with great care. From Art. 307 onwards, I investigate the system of three bodies ; in this connection, as far as Art. 313, I evolve several theorems dealing with the direction of the forces on each one of the three compounded from the combined actions of the other two ; such as the theorem, that these directions are either all parallel to one another, or all pass through some one common point, when they are produced indefinitely on both sides. Then, as ^j, far as Art. 321, I make out several other theorems dealing with the ratios of these same resultant forces to one another ; such as the following very simple & elegant theorem, that the accelerating forces of two of the masses will always be in a ratio compounded of three reciprocal ratios ; namely, that of the distance of either one of them from the third mass, that of the sine of the angle which the direction of each force makes with the corresponding distance of this kind, & that of the mass itself on which the force is acting, to the corre- sponding distance, sine and mass for the other : also that the motive forces only have the first two ratios, that of the masses being omitted. I then collect the results to be derived from these theorems, deriving from them, as far , 2I as Art. 328, theorems relating to the equilibrium of forces diverging in any manner, & the centre of equilibrium, & the pressure of the centre on a fulcrum. I extend the theorem relating to preponderance to the case also, in which the masses do not mutually act upon one another in a direct manner, but through others intermediate between them, which connect them together, & supply the place of rods joining them ; and also to any number of masses, each of which I suppose to be connected with the centre of rotation & some other assumed mass, & from this I derive the principles of moments for all machines. Then I consider all the different kinds of levers ; one of the theorems that I obtain is, that, if a lever is suspended from the centre of gravity, then there is equilibrium ; but a force should be felt in this centre from the fulcrum or sustaining point, equal to the sum of the weights of the whole system ; from which there follows most clearly the reason, which is every- where assumed without proof, why the whole mass can be supposed to be collected at its centre of gravity, so long as the system is in a state of rest & all motions of its parts are pro- hibited by equilibrium. 28 SYNOPSIS TOTIUS OPERIS 328 A num. 328 ad 347 deduce ex iisdem theorematis, quae pertinent ad centrum oscilla- tionis quotcunque massarum, sive sint in eadem recta, sive in piano perpendiculari ad axem rotationis ubicunque, quse Theoria per systema quatuor massarum, excolendum aliquanto diligentius, uberius promoveri deberet & extendi ad generalem habendum solidorum nexum, 344 qua re indicata, centrum itidem percussionis inde evolve, & ejus analogiam cum centre oscillationis exhibeo. 347 Collecto ejusmodi fructu ex theorematis pertinentibus ad massas tres, innuo num. 347, quae mihi communia sint cum ceteris omnibus, & cum Newtonianis potissimum, pertinentia ad summas virium, quas habet punctum, vel massa attracta, vel repulsa a punctis singulis 348 alterius massae ; turn a num. 348 ad finem hujus partis, sive ad num. 358, expono quasdam, quae pertinent ad fluidorum Theoriam, & primo quidem ad pressionem, ubi illud innuo demonstratum a Newtono, si compressio fluidi sit proportionalis vi comprimenti, vires repulsivas punctorum esse in ratione reciproca distantiarum, ac vice versa : ostendo autem illud, si eadem vis sit insensibilis, rem, praeter alias curvas, exponi posse per Logisticam, & in fluidis gravitate nostra terrestri prseditis pressiones haberi debere ut altitudines ; deinde vero attingo ilia etiam, quae pertinent ad velocitatem fluidi erumpentis e vase, & expono, quid requiratur, ut ea sit sequalis velocitati, quae acquiretur cadendo per altitudinem ipsam, quemadmodum videtur res obtingere in aquae efHuxu : quibus partim expositis, partim indicatis, hanc secundam partem conclude. PARS III 358 Num. 358 propono argumentum hujus tertise partis, in qua omnes e Theoria mea 360 generales materis proprietates deduce, & particulares plerasque : turn usque ad num. 371 ago aliquanto fusius de impenetrabilitate, quam duplicis generis agnosco in meis punctorum inextensorum massis, ubi etiam de ea apparenti quadam compenetratione ago, ac de luminis trarlsitu per substantias intimas sine vera compenetratione, & mira quaedam phenomena 371 hue pertinentia explico admodum expedite. Inde ad num. 375 de extensione ago, quae mihi quidem in materia, & corporibus non est continua, sed adhuc eadem praebet phaeno- menae sensibus, ac in communi sententia ; ubi etiam de Geometria ago, quae vim suam in 375 mea Theoria retinet omnem : turn ad num. 383 figurabilitatem perseqUor, ac molem, massam, densitatem singillatim, in quibus omnibus sunt quaedam Theoriae meae propria 383 scitu non indigna. De Mobilitate, & Motuum Continuitate, usque ad num. 388 notatu 388 digna continentur : turn usque ad num. 391 ago de aequalitate actionis, & reactionis, cujus consectaria vires ipsas, quibus Theoria mea innititur, mirum in modum conformant. Succedit usque ad num. 398 divisibilitas, quam ego ita admitto, ut quaevis massa existens numerum punctorum realium habeat finitum tantummodo, sed qui in data quavis mole possit esse utcunque magnus ; quamobrem divisibilitati in infinitum vulgo admissae sub- stituo componibilitatem in infinitum, ipsi, quod ad Naturae phenomena explicanda 398 pertinet, prorsus aequivalentem. His evolutis addo num. 398 immutabilitatem primorum materiae elementorum, quse cum mihi sint simplicia prorsus, & inextensa, sunt utique immutabilia, & ad exhibendam perennem phasnomenorum seriem aptissima. 399 A num. 399 ad 406 gravitatem deduco ex mea virium Theoria, tanquam ramum quendam e communi trunco, ubi & illud expono, qui fieri possit, ut fixae in unicam massam 406 non coalescant, quod gravitas generalis requirere videretur. Inde ad num. 419 ago de cohaesione, qui est itidem veluti alter quidam ramus, quam ostendo, nee in quiete con- sistere, nee in motu conspirante, nee in pressione fluidi cujuspiam, nee in attractione maxima in contactu, sed in limitibus inter repulsionem, & attractionem ; ubi & problema generale propono quoddam hue pertinens, & illud explico, cur massa fracta non iterum coalescat, cur fibrae ante fractionem distendantur, vel contrahantur, & innuo, quae ad cohaesionem pertinentia mihi cum reliquis Philosophis communia sint. 419 A cohacsione gradum facio num. 419 ad particulas, quae ex punctis cohaerentibus efformantur, de quibus .ago usque ad num. 426. & varia persequor earum discrimina : SYNOPSIS OF THE WHOLE WORK 29 From Art. 328 to Art. 347, I deduce from these same theorems, others that relate to 328 the centre of oscillation of any number of masses, whether they are in the same right line, or anywhere in a plane perpendicular to the axis of rotation ; this theory wants to be worked somewhat more carefully with a system of four bodies, to be gone into more fully, & to be extended so as to include the general case of a system of solid bodies ; having stated this, I evolve from it the centre of percussion, & I show the analogy between it & the centre 344 of oscillation. I obtain all such results from theorems relating to three masses. After that, in Art. 347 347, I intimate the matters in which I agree with all others, & especially with the followers of Newton, concerning sums of forces, acting on a point, or an attracted or repelled mass, due to the separate points of another mass. Then, from Art. 348 to the end of this part, 348 i.e., as far as Art. 359, I expound certain theorems that belong to the theory of fluids ; & first of all, theorems with regard to pressure, in connection with which I mention that one which was proved by Newton, namely, that, if the compression of a fluid is proportional to the compressing force, then the repulsive forces between the points are in the reciprocal ratio of the distances, & conversely. Moreover, I show that, if the same force is insen- sible, then the matter can be represented by the logistic & other curves ; also that in fluids subject to our terrestrial gravity pressures should be found proportional to the depths. After that, I touch upon those things that relate to the velocity of a fluid issuing from a vessel ; & I show what is necessary in order that this should be equal to the velocity which would be acquired by falling through the depth itself, just as it is seen to happen in the case of an efflux of water. These things in some part being explained, & in some part merely indicated, I bring this second part to an end. PART III In Art. 358, I state the theme of this third part ; in it I derive all the general & most 358 of the special, properties of matter from my Theory. Then, as far as Art. 371, I deal some- 360 what more at length with the subject of impenetrability, which I remark is of a twofold kind in my masses of non-extended points ; in this connection also, I deal with a certain apparent case of compenetrability, & the passage of light through the innermost parts of bodies without real compenetration ; I also explain in a very summary manner several striking phenomena relating to the above. From here on to Art. 375, I deal with exten- 371 sion ; this in my opinion is not continuous either in matter or in solid bodies, & yet it yields the same phenomena to the senses as does the usually accepted idea of it ; here I also deal with geometry, which conserves all its power under my Theory. Then, as far 375 as Art. 383, I discuss figurability, volume, mass & density, each in turn ; in all of these subjects there are certain special points of my Theory that are not unworthy of investi- gation. Important theorems on mobility & continuity of motions are to be found from here on to Art. 388 ; then, as far as Art. 391, I deal with the equality of action & reaction, & my conclusions with regard to the subject corroborate in a wonderful way the hypothesis of those forces, upon which my Theory depends. Then follows divisibility, as far as Art. 39 1 398 ; this principle I admit only to the extent that any existing mass may be made up of a number of real points that are finite only, although in any given mass this finite number may be as great as you please. Hence for infinite divisibility, as commonly accepted, I substitute infinite multiplicity ; which comes to exactly the same thing, as far as it is concerned with the explanation of the phenomena of Nature. Having considered these subjects I add, in Art. 398, that of the immutability of the primary elements of matter ; 398 according to my idea, these are quite simple in composition, of no extent, they are every- where unchangeable, & hence are splendidly adapted for explaining a continually recurring set of phenomena. From Art. 399 to Art. 406, 1 derive gravity from my Theory of forces, as if it were a 399 particular branch on a common trunk ; in this connection also I explain how it can happen that the fixed stars do not all coalesce into one mass, as would seem to be required under 406 universal gravitation. Then, as far as Art. 419, I deal with cohesion, which is also as it were another branch ; I show that this is not dependent upon quiescence, nor on motion that is the same for all parts, nor on the pressure of some fluid, nor on the idea that the attraction is greatest at actual contact, but on the limits between repulsion and attraction. I propose, & solve, a general problem relating to this, namely, why masses, once broken, do not again stick together, why the fibres are stretched or contracted before fracture takes place ; & I intimate which of my ideas relative to cohesion are the same as those held by other philosophers. In Art. 419, 1 pass on from cohesion to particles which are formed from a number of 4 J 9 cohering points ; & I consider these as far as Art. 426, & investigate the various distinctions 30 SYNOPSIS TOTIUS OPERIS ostendo nimirum, quo pacto varias induere possint figuras quascunque, quarum tenacissime sint ; possint autem data quavis figura discrepare plurimum in numero, & distributione punctorum, unde & oriantur admodum inter se diversae vires unius particulae in aliam, ac itidem diversae in diversis partibus ejusdem particulae respectu diversarum partium, vel etiam respectu ejusdem partis particulse alterius, cum a solo numero, & distributione punctorum pendeat illud, ut data particula datam aliam in datis earum distantiis, & superficierum locis, vel attrahat, vel repellat, vel respectu ipsius sit prorsus iners : turn illud addo, particulas eo dimcilius dissolubiles esse, quo minores sint ; debere autem in gravitate esse penitus uniformes, quaecunque punctorum dispositio habeatur, & in aliis proprietatibus plerisque debere esse admodum (uti observamus) diversas, quae diversitas multo major in majoribus massis esse debeat. 426 A num. 426 ad 446 de solidis, & fluidis, quod discrimen itidem pertinet ad varia cohaesionum genera ; & discrimen inter solida, & fluida diligenter expono, horum naturam potissimum repetens ex motu faciliori particularum in gyrum circa alias, atque id ipsum ex viribus circumquaque aequalibus ; illorum vero ex inaequalitate virium, & viribus quibusdam in latus, quibus certam positionem ad se invicem servare debeant. Varia autem distinguo fluidorum genera, & discrimen profero inter virgas rigidas, flexiles, elasticas, fragiles, ut & de viscositate, & humiditate ago, ac de organicis, & ad certas figuras determinatis corporibus, quorum efformatio nullam habet difficultatem, ubi una particula unam aliam possit in certis tantummodo superficiei partibus attrahere, & proinde cogere ad certam quandam positionem acquirendam respectu ipsius, & retinendam. Demonstro autem & illud, posse admodum facile ex certis particularum figuris, quarum ipsae tenacissimae sint, totum etiam Atomistarum, & Corpuscularium systema a mea Theoria repeti ita, ut id nihil sit aliud, nisi unicus itidem hujus veluti trunci foecundissimi ramus e diversa cohaesionis ratione prorumpens. Demum ostendo, cur non quaevis massa, utut constans ex homogeneis punctis, & circa se maxime in gyrum mobilibus, fluida sit ; & fluidorum resistentiam quoque attingo, in ejus leges inquirens. 446 A num. 446 ad 450 ago de iis, quae itidem ad diversa pertinent soliditatis genera, nimirum de elasticis, & mollibus, ilia repetens a magna inter limites proximos distantia, qua fiat, ut puncta longe dimota a locis suis, idem ubique genus virium sentiant, & proinde se ad priorem restituant locum ; hasc a limitum frequentia, atque ingenti vicinia, qua fiat, ut ex uno ad alium delata limitem puncta, ibi quiescant itidem respective, ut prius. Turn vero de ductilibus, & malleabilibus ago, ostendens, in quo a fragilibus discrepent : ostendo autem, haec omnia discrimina a densitate nullo modo pendere, ut nimirum corpus, quod multo sit altero densius, possit tarn multo majorem, quam multo minorem soliditatem, & cohaesionem habere, & quaevis ex proprietatibus expositis aeque possit cum quavis vel majore, vel minore densitate componi. 450 Num. 450 inquire in vulgaria quatuor elementa ; turn a num. 451 ad num. 467 persequor 452 chemicas operationes ; num. 452 explicans dissolutionem, 453 praecipitationem, 454, & 455 commixtionem plurium substantiarum in unam : turn num. 456, & 457 liquationem binis methodis, 458 volatilizationem, & effervescentiam, 461 emissionem efHuviorum, quae e massa constanti debeat esse ad sensum constans, 462 ebullitionem cum variis evaporationum generibus ; 463 deflagrationem, & generationem aeris ; 464 crystallizationem cum certis figuris ; ac demum ostendo illud num. 465, quo pacto possit fermentatio desinere ; & num. 466, quo pacto non omnia fermentescant cum omnibus. 467 A fermentatione num. 467 gradum facio ad ignem, qui mihi est fermentatio quaedam substantiae lucis cum sulphurea quadam substantia, ac plura inde consectaria deduce usque 471 ad num. 471 ; turn ab igne ad lumen ibidem transeo, cujus proprietates praecipuas, ex 472 quibus omnia lucis phaenomena oriuntur, propono num. 472, ac singulas a Theoria mea deduce, & fuse explico usque ad num. 503, nimirum emissionem num. 473, celeritatem 474, propagationem rectilineam per media homogenea, & apparentem tantummodo compene- trationem a num. 475 ad 483, pellucidatem, & opacitatem num. 483, reflexionem ad angulos aequales inde ad 484, refractionem ad 487, tenuitatem num. 487, calorem, & ingentes intestines motus allapsu tenuissimae lucis genitos, num. 488, actionem majorem corporum eleosorum, & sulphurosorum in lumen num. 489 : turn num. 490 ostendo, nullam resist- SYNOPSIS OF THE WHOLE WORK 31 between them. I show how it is possible for various shapes of all sorts to be assumed, which offer great resistance to rupture ; & how in a given shape they may differ very greatly in the number & disposition of the points forming them. Also that from this fact there arise very different forces for the action of one particle upon another, & also for the action of different parts of this particle upon other different parts of it, or on the same part of another particle. For that depends solely on the number & distribution of the points, so that one given particle either attracts, or repels, or is perfectly inert with regard to another given particle, the distances between them and the positions of their surfaces being also given. Then I state in addition that the smaller the particles, the greater is the diffi- culty in dissociating them ; moreover, that they ought to be quite uniform as regards gravitation, no matter what the disposition of the points may be ; but in most other properties they should be quite different from one another (which we observe to be the case) ; & that this difference ought to be much greater in larger masses. From Art. 426 to Art. 446, 1 consider solids & fluids, the difference between which is 426 also a matter of different kinds of cohesion. I explain with great care the difference between solids & fluids ; deriving the nature of the latter from the greater freedom of motion of the particles in the matter of rotation about one another, this being due to the forces being nearly equal ; & that of the former from the inequality of the forces, and from certain lateral forces which help them to keep a definite position with regard to one another. I distinguish between various kinds of fluids also, & I cite the distinction between rigid, flexible, elastic & fragile rods, when I deal with viscosity & humidity ; & also in dealing with organic bodies & those solids bounded by certain fixed figures, of which the formation presents no difficulty ; in these one particle can only attract another particle in certain parts of the surface, & thus urge it to take up some definite position with regard to itself, & keep it there. I also show that the whole system of the Atomists, & also of the Corpus- cularians, can be quite easily derived by my Theory, from the idea of particles of definite shape, offering a high resistance to deformation ; so that it comes to nothing else than another single branch of this so to speak most fertile trunk, breaking forth from it on account of a different manner of cohesion. Lastly, I show the reason why it is that not every mass, in spite of its being constantly made up of homogeneous points, & even these in a high degree capable of rotary motion about one another, is a fluid. I also touch upon the resistance of fluids, & investigate the laws that govern it. From Art. 446 to Art. 450, I deal with those things that relate to the different kinds 446 of solidity, that is to say, with elastic bodies, & those that are soft. I attribute the nature of the former to the existence of a large interval between the consecutive limits, on account of which it comes about that points that are far removed from their natural positions still feel the effects of the same kind of forces, & therefore return to their natural positions ; & that of the latter to the frequency & great closeness of the limits, on account of which it comes about that points that have been moved from one limit to another, remain there in relative rest as they were to start with. Then I deal with ductile and malleable solids, pointing out how they differ from fragile solids. Moreover I show that all these differ- ences are in no way dependent on density ; so that, for instance, a body that is much more dense than another body may have either a much greater or a much less solidity and cohesion than another ; in fact, any of the properties set forth may just as well be combined with any density either greater or less. In Art. 450 I consider what are commonly called the " four elements " ; then from 450 Art. 451 to Art. 467, I treat of chemical operations ; I explain solution in Art. 452, preci- 452 pitation in Art. 453, the mixture of several substances to form a single mass in Art. 454, 455, liquefaction by two methods in Art. 456, 457, volatilization & effervescence in Art. 458, emission of effluvia (which from a constant mass ought to be approximately constant) in. Art. 461, ebullition & various kinds of evaporation in Art. 462, deflagration & generation of gas in Art. 463, crystallization with definite forms of crystals in Art. 464 ; & lastly, I show, in Art. 465, how it is possible for fermentation to cease, & in Art. 466, how it is that any one thing does not ferment when mixed with any other thing. From fermentation I pass on, in Art. 467, to fire, which I look upon as a fermentation 467 of some substance in light with some sulphureal substance ; & from this I deduce several propositions, up to Art. 471. There I pass on from fire to light, the chief properties of 471 which, from which all the phenomena of light arise, I set forth in Art. 472 ; & I deduce 472 & fully explain each of them in turn as far as Art. 503. Thus, emission in Art. 473, velo- city in Art. 474, rectilinear propagation in homogeneous media, & a compenetration that is merely apparent, from Art. 475 on to Art. 483, pellucidity & opacity in Art. 483, reflec- tion at equal angles to Art. 484, & refraction to Art. 487, tenuity in Art. 487, heat & the great internal motions arising from the smooth passage of the extremely tenuous light in Art. 488, the greater action of oleose & sulphurous bodies on light in Art. 489. Then I 32 SYNOPSIS TOTIUS OPERIS entiam veram pati, ac num. 491 explico, unde sint phosphora, num. 492 cur lumen cum majo e obliquitate incidens reflectatur magis, num. 493 & 494 unde diversa refrangibilitas ortum ducat, ac num. 495, & 496 deduce duas diversas dispositiones ad asqualia redeuntes intervalla, unde num. 497 vices illas a Newtono detectas facilioris reflexionis, & facilioris transmissus eruo, & num. 498 illud, radios alios debere reflecti, alios transmitti in appulsu ad novum medium, & eo plures reflecti, quo obliquitas incidentise sit major, ac num. 499 & 500 expono, unde discrimen in intervallis vicium, ex quo uno omnis naturalium colorum pendet Newtoniana Theoria. Demum num. 501 miram attingo crystalli Islandicse proprietatem, & ejusdem causam, ac num. 502 diffractionem expono, quse est quaedam inchoata refractio, sive reflexio. 503 Post lucem ex igne derivatam, quse ad oculos pertinet, ago brevissime num. 503 de 504 sapore, & odore, ac sequentibus tribus numeris de sono : turn aliis quator de tactu, ubi 507 etiam de frigore, & calore : deinde vero usque ad num. 514 de electricitate, ubi totam 511 Franklinianam Theoriam ex meis principiis explico, eandem ad bina tantummodo reducens principia, quse ex mea generali virium Theoria eodem fere pacto deducuntur, quo prsecipi- 514 tationes, atque dissolutiones. Demum num. 514, ac 515 magnetismum persequor, tam directionem explicans, quam attractionem magneticam. 516 Hisce expositis, quas ad particulares .etiam proprietates pertinent, iterum a num. 516 ad finem usque generalem corporum complector naturam, & quid materia sit, quid forma, quse censeri debeant essentialia, quse accidentialia attributa, adeoque quid transformatio sit, quid alteratio, singillatim persequor, & partem hanc tertiam Theorise mesa absolve. De Appendice ad Metaphysicam pertinente innuam hie illud tantummodo, me ibi exponere de anima illud inprimis, quantum spiritus a materia differat, quern nexum anima habeat cum corpore, & quomodo in ipsum agat : turn de DEO, ipsius & existentiam me pluribus evincere, quae nexum habeant cum ipsa Theoria mea, & Sapientiam inprimis, ac Providentiam, ex qua gradum ad revelationem faciendum innuo tantummodo. Sed hsec in antecessum veluti delibasse sit satis. SYNOPSIS OF THE WHOLE WORK 33 show, in Art. 490, that it suffers no real resistance, & in Art. 491 I explain the origin of bodies emitting light, in Art. 492 the reason why light that falls with greater obliquity is reflected more strongly, in Art. 493, 494 the origin of different degrees of refrangibility, & in Art. 495, 496 I deduce that there are two different dispositions recurring at equal intervals ; hence, in Art. 497, I bring out those alternations, discovered by Newton, of easier reflection & easier transmission, & in Art. 498 I deduce that some rays should be reflected & others transmitted in the passage to a fresh medium, & that the greater the obli- quity of incidence, the greater the number of reflected rays. In Art. 499, 500 I state the origin of the difference between the lengths of the intervals of the alternations ; upon this alone depends the whole of the Newtonian theory of natural colours. Finally, in Art. 501, I touch upon the wonderful property of Iceland spar & its cause, & in Art. 502 I explain diffraction, which is a kind of imperfect refraction or reflection. After light derived from fire, which has to do with vision, I very briefly deal with taste & smell in Art. 503, of sound in the three articles that follow next. Then, in the S3 next four articles, I consider touch, & in connection with it, cold & heat also. After that, 54 as far as Art. 514, I deal with electricity ; here I explain the whole of the Franklin theory 57 by means of my principles ; I reduce this theory to two principles only, & these are 5 1 1 derived from my general Theory of forces in almost the same manner as I have already derived precipitations & solutions. Finally, in Art. 514, 515, I investigate magnetism, explaining 5H both magnetic direction attraction. These things being expounded, all of which relate to special properties, I once more consider, in the articles from 516 to the end, the general nature of bodies, what matter is, 516 its form, what things ought to be considered as essential, & what as accidental, attributes ; and also the nature of transformation and alteration are investigated, each in turn ; & thus I bring to a close the third part of my Theory. I will mention here but this one thing with regard to the appendix on Metaphysics ; namely, that I there expound more especially how greatly different is the soul from matter, the connection between the soul & the body, & the manner of its action upon it. Then with regard to GOD, I prove that He must exist by many arguments that have a close con- nection with this Theory of mine ; I especially mention, though but slightly, His Wisdom and Providence, from which there is but a step to be made towards revelation. But I think that I have, so to speak, given my preliminary foretaste quite sufficiently. [I] PHILOSOPHIC NATURALIS THEORIA In quo conveniat cum systemate Newtoniano, & Leibnitiano. Cujusmodi systema> Theoria exhibeat. PARS I Theorice expositio, analytica deductio^ & vindicatio. lRIUM mutuarum Theoria, in quam incidi jam ab Anno 1745, dum e notissimis principiis alia ex aliis consectaria eruerem, & ex qua ipsam simplicium materise elementorum constitutionem deduxi, systema exhibet medium inter Leibnitianum, & Newtonianum, quod nimirum & ex utroque habet plurimum, & ab utroque plurimum dissidet ; at utroque in immensum simplicius, proprietatibus corporum generalibus sane omnibus, & [2] peculiaribus quibusque praecipuis per accuratissimas demonstrationes deducendis est profecto mirum in modum idoneum. 2. Habet id quidem ex Leibnitii Theoria elementa prima simplicia, ac prorsus inex- tensa : habet ex Newtoniano systemate vires mutuas, quae pro aliis punctorum distantiis a se invicem aliae sint ; & quidem ex ipso itidem Newtono non ejusmodi vires tantummodo, quse ipsa puncta determinent ad accessum, quas vulgo attractiones nominant ; sed etiam ejusmodi, quae determinent ad recessum, & appellantur repulsiones : atque id ipsum ita, ut, ubi attractio desinat, ibi, mutata distantia, incipiat repulsio, & vice versa, quod nimirum Newtonus idem in postrema Opticse Quaestione proposuit, ac exemplo transitus a positivis ad negativa, qui habetur in algebraicis formulis, illustravit. Illud autem utrique systemati commune est cum hoc meo, quod quaevis particula materiae cum aliis quibusvis, utcunque remotis, ita connectitur, ut ad mutationem utcunque exiguam in positione unius cujusvis, determinationes ad motum in omnibus reliquis immutentur, & nisi forte elidantur omnes oppositas, qui casus est infinities improbabilis, motus in iis omnibus aliquis inde ortus habeatur. In quo differat a Leibnitiano & ipsi praestet. 3. Distat autem a Leibnitiana Theoria longissime, turn quia nullam extensionem continuam admittit, quae ex contiguis, & se contingentibus inextensis oriatur : in quo quidem dirficultas jam olim contra Zenonem proposita, & nunquam sane aut soluta satis, aut solvenda, de compenetratione omnimoda inextensorum contiguorum, eandem vim adhuc habet contra Leibnitianum systema : turn quia homogeneitatem admittit in elementis, omni massarum discrimine a sola dispositione, & diversa combinatione derivato, ad quam homogeneitatem in elementis, & discriminis rationem in massis, ipsa nos Naturae analogia ducit, ac chemicae resolutiones inprimis, in quibus cum ad adeo pauciora numero, & adeo minus inter se diversa principiorum genera, in compositorum corporum analysi deveniatur, id ipsum indicio est, quo ulterius promoveri possit analysis, eo ad majorem simplicitatem, & homogeneitatem devenire debere, adeoque in ultima demum resolutione ad homogenei- tatem, & simplicitatem summam, contra quam quidem indiscernibilium principium, & principium rationis sufficients usque adeo a Leibnitianis depraedicata, meo quidem judicio, nihil omnino possunt. in quo differat a A Distat itidem a Newtoniano systemate quamplunmum, turn in eo, quod ea, quae Newtoniano & ipsi XT . . r\ r\ r praestet. Newtonus in ipsa postremo (Juaestione (Jpticae conatus est expncare per tna pnncipia, gravitatis, cohsesionis, fermentationis, immo & reliqua quamplurima, quae ab iis tribus principiis omnino non pendent, per unicam explicat legem virium, expressam unica, & ex pluribus inter se commixtis non composita algebraica formula, vel unica continua geometrica curva : turn in eo, quod in mi-[3]-nimis distantiis vires admittat non positivas, sive attractivas, uti Newtonus, sed negativas, sive repulsivas, quamvis itidem eo majores in 34 A THEORY OF NATURAL PHILOSOPHY PART I Exposition ^ ^Analytical Derivation & Proof of the Theory I. ' ^i ^^ HE following Theory of mutual forces, which I lit upon as far back as the year Th e kind of sys- 1745, whilst I was studying various propositions arising from other very p^ents. 6 well-known principles, & from which I have derived the very constitu- tion of the simple elements of matter, presents a system that is midway between that of Leibniz & that of Newton ; it has very much in common with both, & differs very much from either ; &, as it is immensely more simple than either, it is undoubtedly suitable in a marvellous degree for deriving all the general properties of bodies, & certain of the special properties also, by means of the most rigorous demonstrations. 2. It indeed holds to those simple & perfectly non-extended primary elements upon what there is in which is founded the theory of Leibniz ; & also to the mutual forces, which vary as the * s ^" " { to $^ distances of the points from one another vary, the characteristic of the theory of Newton ; ton *& Leibniz. in addition, it deals not only with the kind of forces, employed by Newton, which oblige the points to approach one another, & are commonly called attractions ; but also it considers forces of a kind that engender recession, & are called repulsions. Further, the idea is introduced in such a manner that, where attraction ends, there, with a change of distance, repulsion begins ; this idea, as a matter of fact, was suggested by Newton in the last of his ' Questions on Optics ', & he illustrated it by the example of the passage from positive to negative, as used in algebraical formulas. Moreover there is this common point between either of the theories of Newton & Leibniz & my own ; namely, that any particle of matter is connected with every other particle, no matter how great is the distance between them, in such a way that, in accordance with a change in the position, no matter how slight, of any one of them, the factors that determine the motions of all the rest are altered ; &, unless it happens that they all cancel one another (& this is infinitely impro- bable), some motion, due to the change of position in question, will take place in every one of them. 3. But my Theory differs in a marked degree from that of Leibniz. For one thing, How it differs from, because it does not admit the continuous extension that arises from the idea of consecutive, non-extended points touching one another ; here, the difficulty raised in times gone by in opposition to Zeno, & never really or satisfactorily answered (nor can it be answered), with regard to compenetration of all kinds with non-extended consecutive points, still holds the same force against the system of Leibniz. For another thing, it admits homogeneity amongst the elements, all distinction between masses depending on relative position only, & different combinations of the elements ; for this homogeneity amongst the elements, & the reason for the difference amongst masses, Nature herself provides us with the analogy. Chemical operations especially do so ; for, since the result of the analysis of compound substances leads to classes of elementary substances that are so comparatively few in num- ber, & still less different from one another in nature ; it strongly suggests that, the further analysis can be pushed, the greater the simplicity, & homogeneity, that ought to be attained ; thus, at length, we should have, as the result of a final decomposition, homogeneity & simplicity of the highest degree. Against this homogeneity & simplicity, the principle of indiscernibles, & the doctrine of sufficient reason, so long & strongly advocated by the followers of Leibniz, can, in my opinion at least, avail in not the slightest degree. 4. My Theory also differs as widely as possible from that of Newton. For one thing, HOW it differs from, because it explains by means of a single law of forces all those things that Newton himself, * surpasses, the i i i i. . X f-\ , i i theory of Newton. in the last of his Questions on Uptics , endeavoured to explain by the three principles of gravity, cohesion & fermentation ; nay, & very many other things as well, which do not altogether follow from those three principles. Further, this law is expressed by a single algebraical formula, & not by one composed of several formulae compounded together ; or by a single continuous geometrical curve. For another thing, it admits forces that at very small distances are not positive or attractive, as Newton supposed, but negative or repul- 35 missum. 36 PHILOSOPHIC NATURALIS THEORIA infinitum, quo distantise in infinitum decrescant. Unde illud necessario consequitur, ut nee cohaesio a contactu immediate oriatur, quam ego quidem longe aliunde desumo ; nee ullus immediatus, &, ut ilium appellare soleo, mathematicus materiae contactus habeatur, quod simplicitatem, & inextensionem inducit elementorum, quae ipse variarum figurarum voluit, & partibus a se invicem distinctis composita, quamvis ita cohasrentia, ut nulla Naturae vi dissolvi possit compages, & adhaesio labefactari, quas adhaesio ipsi, respectu virium nobis cognitarum, est absolute infinita. Ubi de ipsa ctum 5. Quae ad ejusmodi Theoriam pertinentia hucusque sunt edita, continentur disserta- ante ; & quid pro- tionibus meis, De viribus vivis, edita Anno 1741;, De Lumine A. 1748, De Leee Continuitatis ml<mm " . T r^ ... . . rj . ... A. 1754, De Lege virium in natura existentium A. 1755, De divisibihtate materite, C5 principiis corporum A. 1757, ac in meis Supplementis Stayanae Philosophiae versibus traditae, cujus primus Tomus prodiit A. 1755 : eadem autem satis dilucide proposuit, & amplissimum ipsius per omnem Physicam demonstravit usum vir e nostra Societate doctissimus Carolus Benvenutus in sua Physics Generalis Synopsi edita Anno 1754. In ea Synopsi proposuit idem & meam deductionem aequilibrii binarum massarum, viribus parallelis animatarum, quas ex ipsa mea Theoria per notissimam legem compositionis virium, & aequalitatis inter actionem, & reac- tionem, fere sponte consequitur, cujus quidem in supplementis illis 4. ad lib. 3. mentionem feci, ubi & quae in dissertatione De centra Gravitatis edideram, paucis proposui ; & de centre oscillationis agens, protuli aliorum methodos praecipuas quasque, quae ipsius determinationem a subsidiariis tantummodo principiis quibusdam repetunt. Ibidem autem de sequilibrii centre agens illud affirmavi : In Natura nullce sunt rigidce virgce, infiexiles, & omni gravitate, ac inertia carentes, adeoque nee revera ullce leges pro Us conditcz ; & si ad genuina, & simpli- cissima natures principia, res exigatur, invenietur, omnia pendere a compositione virium, quibus in se invicem agunt particula materice ; a quibus nimirum viribus omnia Natures pb&nomena proficiscuntur. Ibidem autem exhibitis aliorum methodis ad centrum oscillationis perti- nentibus, promisi, me in quarto ejusdem Philosophiae tomo ex genuinis principiis investiga- turum, ut aequilibrii, sic itidem oscillationis centrum. Qua occasione hoc 6. Porro cum nuper occasio se mihi praebuisset inquirendi in ipsum oscillationis centrum turn 'opus." Cnp ex me i s principiis, urgente Scherffero nostro viro doctissimo, qui in eodem hoc Academico Societatis Collegio nostros Mathesim docet ; casu incidi in theorema simplicisimum sane, & admodum elegans, quo trium massarum in se mutuo agentium comparantur vires, [4] quod quidem ipsa fortasse tanta sua simplicitate effugit hucusque Mechanicorum oculos ; nisi forte ne effugerit quidem, sed alicubi jam ab alio quopiam inventum, & editum, me, quod admodum facile fieri potest, adhuc latuerit, ex quo theoremate & asquilibrium, ac omne vectium genus, & momentorum mensura pro machinis, & oscillationis centrum etiam pro casu, quo oscillatio fit in latus in piano ad axem oscillationis perpendiculari, & centrum percussionis sponte fluunt, & quod ad sublimiores alias perquisitiones viam aperit admodum patentem. Cogitaveram ego quidem initio brevi dissertatiuncula hoc theorema tantummodo edere cum consectariis, ac breve Theoriae meae specimen quoddam exponere ; sed paullatim excrevit opusculum, ut demum & Theoriam omnem exposuerim ordine suo, & vindicarim, & ad Mechanicam prius, turn ad Physicam fere universam applicaverim, ubi & quae maxima notatu digna erant, in memoratis dissertationibus ordine suo digessi omnia, & alia adjeci quamplurima, quae vel olim animo conceperam, vel modo sese obtulerunt scribenti, & omnem hanc rerum farraginem animo pervolventi. eiementa in- 7. Prima elementa materiae mihi sunt puncta prorsus indivisibilia, & inextensa, quae in i mr ftenso vacuo ita dispersa sunt, ut bina quaevis a se invicem distent per aliquod intervallum, quod quidem indefinite augeri potest, & minui, sed penitus evanescere non potest, sine conpenetratione ipsorum punctorum : eorum enim contiguitatem nullam admitto possi- bilem ; sed illud arbitror omnino certum, si distantia duorum materiae punctorum sit nulla, idem prorsus spatii vulgo concept! punctum indivisibile occupari ab utroque debere, & A THEORY OF NATURAL PHILOSOPHY 37 sive ; although these also become greater & greater indefinitely, as the distances decrease indefinitely. From this it follows of necessity that cohesion is not a consequence of imme- diate contact, as I indeed deduce from totally different considerations ; nor is it possible to get any immediate or, as I usually term it, mathematical contact between the parts of matter. This idea naturally leads to simplicity & non-extension of the elements, such as Newton himself postulated for various figures ; & to bodies composed of parts perfectly distinct from one another, although bound together so closely that the ties could not be broken or the adherence weakened by any force in Nature ; this adherence, as far as the forces known to us are concerned, is in his opinion unlimited. 5. What has already been published relating to this kind of Theory is contained in my when & where I dissertations, De Viribus vivis, issued in 1745, De Lumine, 1748, De Lege Continuitatis, * th^theory'* 1754, De Lege virium in natura existentium, 1755, De divisibilitate materia, y principiis & a promise that i corporum, 1757, & in my Supplements to the philosophy of Benedictus Stay, issued in verse, made> of which the first volume was published in 1755. The same theory was set forth with considerable lucidity, & its extremely wide utility in the matter of the whole of Physics was demonstrated, by a learned member of our Society, Carolus Benvenutus, in his Physics Generalis Synopsis published in 1754. In this synopsis he also at the same time gave my deduction of the equilibrium of a pair of masses actuated by parallel forces, which follows quite naturally from my Theory by the well-known law for the composition of forces, & the equality between action & reaction ; this I mentioned in those Supplements, section 4 of book 3, & there also I set forth briefly what I had published in my dissertation De centra Gravitatis. Further, dealing with the centre of oscillation, I stated the most note- worthy methods of others who sought to derive the determination of this centre from merely subsidiary principles. Here also, dealing with the centre of equilibrium, I asserted : " In Nature there are no rods that are rigid, inflexible, totally devoid of weight & inertia ; y so, neither are there really any laws founded on them. If the matter is worked back to the genuine W simplest natural principles, it will be found that everything depends on the com- position of the forces with which the particles of matter act upon one another ; y from these very forces, as a matter of fact, all phenomena of Nature take their origin." Moreover, here too, having stated the methods of others for the determination of the centre of oscillation, I promised that, in the fourth volume of the Philosophy, I would investigate by means of genuine principles, such as I had used for the centre of equilibrium, the centre of oscillation as well. 6. Now, lately I had occasion to investigate this centre of oscillation, deriving it from The occasion that my own principles, at the request of Father Scherffer, a man of much learning, who teaches |^ mathematics in this College of the Society. Whilst doing this, I happened to hit upon a matter. really most simple & truly elegant theorem, from which the forces with which three masses mutually act upon one another are easily to be found ; this theorem, perchance owing to its extreme simplicity, has escaped the notice of mechanicians up till now (unless indeed perhaps it has not escaped notice, but has at some time previously been discovered & published by some other person, though, as may very easily have happened, it may not have come to my notice). From this theorem there come, as the natural consequences, the equilibrium & all the different kinds of levers, the measurement of moments for machines, the centre of oscillation for the case in which the oscillation takes place sideways in a plane perpendicular to the axis of oscillation, & also the centre of percussion ; it opens up also a beautifully clear road to other and more sublime investigations. Initially, my idea was to publish in a short esssay merely this theorem & some deductions from it, & thus to give some sort of brief specimen of my Theory. But little by little the essay grew in length, until it ended in my setting forth in an orderly manner the whole of the theory, giving a demonstration of its truth, & showing its application to Mechanics in the first place, and then to almost the whole of Physics. To it I also added not only those matters that seemed to me to be more especially worth mention, which had all been already set forth in an orderly manner in the dissertations mentioned above, but also a large number of other things, some of which had entered my mind previously, whilst others in some sort pb truded themselves on my notice as I was writing & turning over in my mind all this conglomer- ation of material. 7. The primary elements of matter are in my opinion perfectly indivisible & non- The primary eie- extended points ; they are so scattered in an immense vacuum that every two of them are ^biVnon^xtended separated from one another by a definite interval ; this interval can be indefinitely & they are not increased or diminished, but can never vanish altogether without compenetration of the c points themselves ; for I do not admit as possible any immediate contact between them. On the contrary I consider that it is a certainty that, if the distance between two points of matter should become absolutely nothing, then the very same indivisible point of space, according to the usual idea of it, must be occupied by both together, & we have true 38 PHILOSOPHIC NATURALIS THEORIA haberi veram, ac omnimodam conpenetrationem. Quamobrem non vacuum ego quidem admitto disseminatum in materia, sed materiam in vacuo disseminatam, atque innatantem. Eorum inertias vis g j n n ; sce punctis admitto determinationem perseverandi in eodem statu quietis, vel cujusmodi. . r . r . ,. , . . , . J . . i * XT ' motus umiormis in directum l) m quo semel sint posita, si seorsum smgula in JNatura existant ; vel si alia alibi extant puncta, componendi per notam, & communem metho- dum compositionis virium, & motuum, parallelogrammorum ope, praecedentem motum cum mo-[5]-tu quern determinant vires mutuae, quas inter bina quaevis puncta agnosco a distantiis pendentes, & iis mutatis mutatas, juxta generalem quandam omnibus com- munem legem. In ea determinatione stat ilia, quam dicimus, inertiae vis, quae, an a libera pendeat Supremi Conditoris lege, an ab ipsa punctorum natura, an ab aliquo iis adjecto, quodcunque, istud sit, ego quidem non quaere ; nee vero, si velim quasrere, in- veniendi spem habeo ; quod idem sane censeo de ea virium lege, ad quam gradum jam facio. Eorundem vires g Censeo igitur bina quaecunque materiae puncta determinari asque in aliis distantiis mutuae in alus , y ,.. , - 1 . . . distantiis attrac- ad mutuum accessum, in alns ad recessum mutuum, quam ipsam determinationem appello tivae, in aliis re- v im, in priore casu attractivam, in posteriore repulsivam, eo nomine non agendi modum, sed pulsivae : v i n u m . , r . . . , '. . . ejusmodi exempia. ipsam determinationem expnmens, undecunque provemat, cujus vero magnitude mutatis distantiis mutetur & ipsa secundum certam legem quandam, quae per geometricam lineam curvam, vel algebraicam formulam exponi possit, & oculis ipsis, uti moris est apud Mechanicos repraesentari. Vis mutuae a distantia pendentis, & ea variata itidem variatae, atque ad omnes in immensum & magnas, & parvas distantias pertinentis, habemus exemplum in ipsa Newtoniana generali gravitate mutata in ratione reciproca duplicata distantiarum, qua; idcirco numquam e positiva in negativam migrare potest, adeoque ab attractiva ad repul- sivam, sive a determinatione ad accessum ad determinationem ad recessum nusquam migrat. Verum in elastris inflexis habemus etiam imaginem ejusmodi vis mutuae variatae secundum distantias, & a determinatione ad recessum migrantis in determinationem ad accessum, & vice versa. Ibi enim si duae cuspides, compresso elastro, ad se invicem accedant, acquirunt determinationem ad recessum, eo majorem, quo magis, compresso elastro, distantia decrescit ; aucta distantia cuspidum, vis ad recessum minuitur, donee in quadam distantia evanescat, & fiat prorsus nulla ; turn distantia adhuc aucta, incipit determinatio ad accessum, quae perpetuo eo magis crescit, quo magis cuspides a se invicem recedunt : ac si e contrario cuspidum distantia minuatur perpetuo ; determinatio ad accessum itidem minuetur, evanescet, & in determinationem ad recessum mutabitur. Ea determinatio oritur utique non ab immediata cuspidum actione in se invicem, sed a natura, & forma totius intermediae laminae plicatae ; sed hie physicam rei causam non merer, & solum persequor exemplum determinationis ad accessum, & recessum, quae determinatio in aliis distantiis alium habeat nisum, & migret etiam ab altera in alteram. virium earundero 10. Lex autem virium est ejusmodi, ut in minimis distantiis sint repulsivae, atque eo majores in infmitum, quo distantiae ipsae minuuntur in infinitum, ita, ut pares sint extinguen- [6]-dae cuivis velocitati utcunque magnae, cum qua punctum alterum ad alterum possit accedere, antequam eorum distantia evanescat ; distantiis vero auctis minuuntur ita, ut in quadam distantia perquam exigua evadat vis nulla : turn adhuc, aucta distantia, mutentur in attractivas, prime quidem crescentes, turn decrescentes, evanescentes, abeuntes in repulsivas, eodem pacto crescentes, deinde decrescentes, evanescentes, migrantes iterum in attractivas, atque id per vices in distantiis plurimis, sed adhuc perquam exiguis, donee, ubi ad aliquanto majores distantias ventum sit, incipiant esse perpetuo attractivae, & ad sensum reciproce (a) Id quidem respectu ejus spatii, in quo continemur nos, W omnia quis nostris observari sensibus possunt, corpora ; quod quiddam spatium si quiescat, nihil ego in ea re a reliquis differo ; si forte moveatur motu quopiam, quern motum ex hujusmodi determinatione sequi debeant ipsa materia puncta ; turn bcec mea erit quiedam non absoluta, sed respectiva inertia: vis, quam ego quidem exposui W in dissertatione De Maris aestu fcf in Supplementis Stayanis Lib. I. 13 ; ubi etiam illud occurrit, quam oh causam ejusmodi respectivam inertiam excogitarim, & quibus rationihus evinci putem, absolutam omnino demonstrari non posse ; sed ea hue non pertinent. A THEORY OF NATURAL PHILOSOPHY 39 compenetration in every way. Therefore indeed I do not admit the idea of vacuum interspersed amongst matter, but I consider that matter is interspersed in a vacuum & floats in it. 8. As an attribute of these points I admit an inherent propensity to remain in the The nat . ure ? f the same state of rest, or of uniform motion in a straight line, () in which they are initially the" possess. 1 * set, if each exists by itself in Nature. But if there are also other points anywhere, there is an inherent propensity to compound (according to the usual well-known composition of forces & motions by the parallelogram law), the preceding motion with the motion which is determined by the mutual forces that I admit to act between any two of them, depending on the distances & changing, as the distances change, according to a certain law common to them all. This propensity is the origin of what we call the ' force of inertia ' ; whether this is dependent upon an arbitrary law of the Supreme Architect, or on the nature of points itself, or on some attribute of them, whatever it may be, I do not seek to know ; even if I did wish to do so, I see no hope of finding the answer ; and I truly think that this also applies to the law of forces, to which I now pass on. 9. I therefore consider that any two points of matter are subject to a determination The mutual forces to approach one another at some distances, & in an equal degree recede from one another at Stw^*^!* other distances. This determination I call ' force ' ; in the first case ' attractive ', in the distances & repui- second case ' repulsive ' ; this term does not denote the mode of action, but the propen- ^mpies sity itself, whatever its origin, of which the magnitude changes as the distances change ; this kind, this is in accordance with a certain definite law, which can be represented by a geometrical curve or by an algebraical formula, & visualized in the manner customary with Mechanicians. We have an example of a force dependent on distance, & varying with varying distance, & pertaining to all distances either great or small, throughout the vastness of space, in the Newtonian idea of general gravitation that changes according to the inverse squares of the distances : this, on account of the law governing it, can never pass from positive to nega- tive ; & thus on no occasion does it pass from being attractive to being repulsive, i.e., from a propensity to approach to a propensity to recession. Further, in bent springs we have an illustration of that kind of mutual force that varies according as the distance varies, & passes from a propensity to recession to a propensity to approach, and vice versa. For here, if the two ends of the spring approach one another on compressing the spring, they acquire a propensity for recession that is the greater, the more the distance diminishes between them as the spring is compressed. But, if the distance between the ends is increased, the force of recession is diminished, until at a certain distance it vanishes and becomes absolutely nothing. Then, if the distance is still further increased, there begins a propensity to approach, which increases more & more as the ends recede further & further away from one another. If now, on the contrary, the distance between the ends is con- tinually diminished, the propensity to approach also diminishes, vanishes, & becomes changed into a propensity to recession. This propensity certainly does not arise from the imme- diate action of the ends upon one another, but from the nature & form of the whole of the folded plate of metal intervening. But I do not delay over the physical cause of the thing at this juncture ; I only describe it as an example of a propensity to approach & recession, this propensity being characterized by one endeavour at some distances & another at other distances, & changing from one propensity to another. 10. Now the law of forces is of this kind ; the forces are repulsive at very small dis- The Iaw . of forces tances, & become indefinitely greater & greater, as the distances are diminished indefinitely, for the pomts- in such a manner that they are capable of destroying any velocity, no matter how large it may be, with which one point may approach another, before ever the distance between them vanishes. When the distance between them is increased, they are diminished in such a way that at a certain distance, which is extremely small, the force becomes nothing. Then as the distance is still further increased, the forces are change-d to attractive forces ; these at first increase, then diminish, vanish, & become repulsive forces, which in the same way first increase, then diminish, vanish, & become once more attractive ; & so on, in turn, for a very great number of distances, which 1 are all still very^ minute : until, finally, when we get to comparatively great distances, they begin to be continually attractive & approxi- (a) This indeed holds true for that space in which we, and all bodies that can be observed by our senses, are contained. Now, if this space is at rest, I do not differ from other philosophers with regard to the matter in question ; but if perchance space itself moves in some way or other, what motion ought these points of matter to comply with owing to this kind of propensity ? In that case Ms force of inertia that I postulate is not absolute, but relative ; as indeed I explained both in the dissertation De Maris Aestu, and also in the Supplements to Stay's Philosophy, book I, section 13. Here also will be found the conclusions at which I arrived with regard to relative inertia of this sort, and the arguments by which I think it is proved that it is impossible to show that it is generally abxlute. But these things do not concern us at present. 4 PHILOSOPHI/E NATURALIS THEORIA proportionales quadratis distantiarum, atque id vel utcunque augeantur distantiae etiam in infinitum, vel saltern donee ad distantias deveniatur omnibus Planetarum, & Cometarum distantiis longe majores. Leg is simpiicitas ii. Hujusmodi lex primo aspectu videtur admodum complicata, & ex diversis legibus exprimibihs per temere j nter se coagmentatis coalescens ; at simplicissima, & prorsus incomposita esse potest, COIlLlIlUtllTl CUf VclIUi t i 1*1* A 1 1 " J" 1 expressa videlicet per unicam contmuam curvam, vel simphcem Algebraicam iormulam, uti innui superius. Hujusmodi curva linea est admodum apta ad sistendam oculis ipsis ejusmodi legem, nee requirit Geometram, ut id praestare possit : satis est, ut quis earn intueatur tantummodo, & in ipsa ut in imagine quadam solemus intueri depictas res qualescunque, virium illarum indolem contempletur. In ejusmodi curva eae, quas Geometrae abscissas dicunt, & sunt segmenta axis, ad quern ipsa refertur curva, exprimunt distantias binorum punctorum a se invicem : illae vero, quae dicuntur ordinatae, ac sunt perpendiculares lineee ab axe ad curvam ductae, referunt vires : quae quidem, ubi ad alteram jacent axis partem, exhibent vires attractivas ; ubi jacent ad alteram, rcpulsivas, & prout curva accedit ad axem, vel recedit, minuuntur ipsae etiam, vel augentur : ubi curva axem secat, & ab altera ejus parte transit ad alteram, mutantibus directionem ordinatis, abeunt ex positivis in negativas, vel vice versa : ubi autem arcus curvae aliquis ad rectam quampiam axi perpendicularem in infinitum productam semper magis accedit ita ultra quoscumque limites, ut nunquam in earn recidat, quern arcum asymptoticum appellant Geometrae, ibi vires ipsae in infinitum excrescunt. Forma curvae ips- ius. 12. Ejusmodi curvam exhibui, & exposui in dissertationibus De viribus vivis a Num. 51, De Lumine Num. 5, De Lege virium in Naturam existentium a Num. 68, & in sua Synopsi Physics Generalis P. Benvenutus eandem protulit a Num. 108. En brevem quandemejus ideam. In Fig. i, Axis C'AC habet in puncto A asymptotum curvae rectilineam AB indefinitam, circa quam habentur bini curvae rami hinc, & inde aequales, prorsus inter se, & similes, quorum alter DEFGHIKLMNOPQRSTV habet inprimis arcum ED [7] asympto- ticum, qui nimirum ad partes BD, si indefinite producatur ultra quoscunque limites, semper magis accedit ad rectam AB productam ultra quoscunque limites, quin unquam ad eandem deveniat ; hinc vero versus DE perpetuo recidit ab eadam recta, immo etiam perpetuo versus V ab eadem recedunt arcus reliqui omnes, quin uspiam recessus mutetur in accessum. Ad axem C'C perpetuo primum accedit, donee ad ipsum deveniat alicubi in E ; turn eodem ibi secto progreditur, & ab ipso perpetuo recedit usque ad quandam distantiam F, postquam recessum in accessum mutat, & iterum ipsum axem secat in G, ac flexibus continuis contor- quetur circa ipsum, quern pariter secat in punctis quamplurimis, sed paucas admodum ejusmodi sectiones figura exhibet, uti I, L, N, P, R. Demum is arcus desinit in alterum crus TpsV, jacens ex parte opposita axis respectu primi cruris, quod alterum crus ipsum habet axem pro asymptoto, & ad ipsum accedit ad sensum ita, ut distantiae ab ipso sint in ratione reciproca duplicata distantiarum a recta BA. Abscissae exprimen- d!nate Sta exprimen- tes vires. 13. Si ex quovis axis puncto a, b, d, erigatur usque ad curvam recta ipsi perpendicularis a S> ^ r ' ^h , segmentum axis Aa, Ab, Ad, dicitur abscissa, & refert distantiam duorum materiae punctorum quorumcunque a se invicem ; perpendicularis ag, br, db , dicitur ordinata, & exhibet vim repulsivam, vel attractivam, prout jacet respectu axis ad partes D, vel oppositas. Mutationes ordina- tarum, & virium iis expressarum. 14. Patet autem, in ea curvae forma ordinatam ag augeri ultra quoscunque limites, si abscissa Aa, minuatur pariter ultra quoscunque limites ; quae si augeatur, ut abeat in Ab, ordinata minuetur, & abibit in br, perpetuo imminutam in accessu b ad E, ubi evanescet : turn aucta abscissa in Ad, mutabit ordinata directionem in dh , ac ex parte opposita augebitur prius usque ad F, turn decrescet per il usque ad G, ubi evanescet, & iterum mutabit directionem regressa in mn ad illam priorem, donee post evanescentiam, & directionis mutationem factam in omnibus sectionibus I, L, N, P, R, fiant ordinatas op, vs, directionis constantis, & decrescentes ad sensum in ratione reciproca duplicata abscissarum Ao, Av. Quamobrem illud est manifestum, per ejusmodi curvam exprimi eas ipsas vires, initio A THEORY OF NATURAL PHILOSOPHY PHILOSOPHIC NATURALIS THEORIA o A THEORY OF NATURAL PHILOSOPHY 43 mately inversely proportional to the squares of the distances. This holds good as the distances are increased indefinitely to any extent, or at any rate until we get to distances that are far greater than all the distances of the planets & comets. 11. A law of this kind will seem at first sight to be very complicated, & to be the result The simplicity of of combining together several different laws in a haphazard sort of way ; but it can be of the law can ^ re ~ ^.t. i 1 i j o v j i v i 1 i r presented by means the simplest kind & not complicated in the slightest degree ; it can be represented for of a continuous instance by a single continuous curve, or by an algebraical formula, as I intimated above. curve - A curve of this sort is perfectly adapted to the .graphical representation of this sort of law, & it does not require a knowledge of geometry to set it forth. It is sufficient for anyone merely to glance at it, & in it, just as in a picture we are accustomed to view all manner of things depicted, so will he perceive the nature of these forces. In a curve of this kind, those lines, that geometricians call abscissae, namely, segments of the axis to which the curve is referred, represent the distances of two points from one another ; & those, which we called ordinates, namely, lines drawn perpendicular to the axis to meet the curve, repre- sent forces. These, when they lie on one side of the axis represent attractive forces, and, when they lie on the other side, repulsive forces ; & according as the curve approaches the axis or recedes from it, they too are diminished or increased. When the curve cuts the axis & passes from one side of it to the other, the direction of the ordinates being changed in consequence, the forces pass from positive to negative or vice versa. When any arc of the curve approaches ever more closely to some straight line perpendicular to the axis and indefinitely produced, in such a manner that, even if this goes on beyond all limits, yet the curve never quite reaches the line (such an arc is called asymptotic by geometricians), then the forces themselves will increase indefinitely. 12. I set forth and explained a curve of this sort in my dissertations De Firibus vivis The form of the (Art. 51), De Lumine (Art. 5), De lege virium in Natura existentium (Art. 68) ; and Father curve - Benvenutus published the same thing in his Synopsis Physicce Generalis (Art. 108). This will give you some idea of its nature in a few words. In Fig. i the axis C'AC has at the point A a straight line AB perpendicular to itself, which is an asymptote to the curve ; there are two branches of the curve, one on each side of AB, which are equal & similar to one another in every way. Of these, one, namely DEFGHIKLMNOPQRSTV, has first of all an asymptotic arc ED ; this indeed, if it is produced ever so far in the direction ED, will approach nearer & nearer to the straight line AB when it also is produced indefinitely, but will never reach it ; then, in the direction DE, it will continually recede from this straight line, & so indeed will all the rest of the arcs continually recede from this straight line towards V. The first arc continually approaches the axis C'C, until it meets it in some point E ; then it cuts it at this point & passes on, continually receding from the axis until it arrives at a certain distance given by the point F ; after that the recession changes to an approach, & it cuts the axis once more in G ; & so on, with successive changes of curvature, the curve winds about the axis, & at the same time cuts it in a number of points that is really large, although only a very few of the intersections of this kind, as I, L, N, P, R, are shown in the diagram. Finally the arc of the curve ends up with the other branch TpsV, lying on the opposite side of the axis with respect to the first branch ; and this second branch has the axis itself as its asymptote, & approaches it approximately in such a manner that the distances from the axis are in the inverse ratio of the squares of the distances from the straight line AB. 13. If from any point of the axis, such as a, b, or d, there is erected a straight line per- The abscissae re- pendicular to it to meet the curve, such as ag, br, or db then the segment of the axis, Aa, res ^Jg Ab, or Ad, is called the abscissa, & represents the distance of any two points of matter from forces, one another ; the perpendicular, ag, br, or dh, is called the ordinate, & this represents the force, which is repulsive or attractive, according as the ordinate lies with regard to the axis on the side towards D, or on the opposite side. 14. Now it is clear that, in a curve of this form, the ordinate ag will be increased Change in the or- beyond all bounds, if the abscissa Aa is in the same way diminished beyond all bounds ; & fbat tlfey reprSent! if the latter is increased and becomes Ab, the ordinate will be diminished, & it will become br, which will continually diminish as b approaches to E, at which point it will vanish. Then the abscissa being increased until it becomes Ad, the ordinate will change its direction as it becomes db, & will be increased in the opposite direction at first, until the point F is reached, when it will be decreased through the value il until the point G is attained, at which point it vanishes ; at the point G, the ordinate will once more change its direction as it returns to the position mn on the same side of the axis as at the start. Finally, after vanishing & changing direction at all points of intersection with the axis, such as I, L, N, P, R, the ordinates take the several positions indicated by op, vs : here the direction remains unchanged, & the ordinates decrease approximately in the inverse ratio of the squares of the abscissae Ao, Av. Hence it is perfectly evident that, by a curve of this kind, we can 44 PHILOSOPHIC NATURALIS THEORIA Discrimen hu us legis virium a gravitate N e w- toniana : ejus usus in Physica : ordo pertractandorum. Occasio inveniendae Theories ex consid- eraticine impulsus. V repulsivas, & imminutis in infinitum distantiis auctas in infinitum, auctis imminutas, turn evanescentes, abeuntes, mutata directione, in attractivas, ac iterum evenescentes, mutatasque per vices : donee demum in satis magna distantia evadant attractive ad sensum in ratione reciproca duplicata distantiarum. 15. Haec virium lex a Newtoniana gravitate differt in ductu, & progressu curvae earn exprimentis quse nimirum, ut in fig. 2, apud Newtonum est hyperbola DV gradus tertii, jacens tota citra axem, quern nuspiam secat, jacentibus omni-[8]-bus ordinatis vs, op, bt, ag ex parte attractiva, ut idcirco nulla habeatur mutatio e positivo ad negativum, ex attractione in repulsi- onem, vel vice versa ; caeterum utraque per ductum exponitur curvae continue habentis duo crura infinita asymptotica in ramis singulis utrinque in infinitum productis. Ex hujusmodi autem virium lege, & ex solis principiis Mechanicis notissimis, nimirum quod ex pluribus viribus, vel motibus componatur vis, vel motus quidam ope parallelogrammorum, quorum latera exprimant vires, vel mo- tus componentes, & quod vires ejusmodi in punctis singulis, tempusculis singulis aequalibus, inducant velocitates, vel motus proportion- ales sibi, omnes mihi profluunt generales, & praecipuae quacque particulars proprietates cor- porum,uti etiam superius innui, nee ad singulares proprietates derivandas in genere afHrmo, eas haberi per diversam combinationem, sed combinationes ipsas evolvo, & geometrice demon- stro, quae e quibus combinationibus phasnomena, & corporum species oriri debeant. Verum antequam ea evolvo in parte secunda, & tertia, ostendam in hac prima, qua via, & quibus positivis rationibus ad earn virium legem devenerim, & qua ratione illam elementorum materiae simplicitatem eruerim, turn quas difHcultatem aliquam videantur habere posse, dissolvam. 1 6. Cum anno 1745 De Viribus vivis dissertationem conscriberem, & omnia, quse a viribus vivis repetunt, qui Leibnitianam tuentur sententiam, & vero etiam plerique ex iis, qui per solam velocitatem vires vivas metiuntur, repeterem immediate a sola velocitate genita per potentiarum vires, quae juxta communem omnium Mechanicorum sententiam velocitates vel generant, vel utcunque inducunt proportionales sibi, & tempusculis, quibus agunt, uti est gravitas, elasticitas, atque aliae vires ejusmodi ; ccepi aliquant: o diligentius inquirere in earn productionem velocitatis, quae per impulsum censetur fieri, ubi tota velocitas momento temporis produci creditur ab iis, qui idcirco percussionis vim infinities majorem esse censent viribus omnibus, quae pressionem solam momentis singulis exercent. Statim illud mihi sese obtulit, alias pro percussionibus ejusmodi, quee nimirum momento temporis finitam velocitatem inducant, actionum leges haberi debere. FIG origo ejusdem ex 17. Verum re altius considerata, mihi illud incidit, si recta utamur ratiocinandi methodo, susTmrnedUatTalin eum agendi modum submovendum esse a Natura, quae nimirum eandem ubique virium lege Continuitatis. legem, ac eandem agendi rationem adhibeat : impulsum nimirum immediatum alterius corporis in alterum, & immediatam percussionem haberi non posse sine ilia productione finitse velocitatis facta momento temporis indivisibili, & hanc sine saltu quodam, & Isesione illius, quam legem Continuitatis appellant, quam quidem legem in Natura existere, & quidem satis [9] valida ratione evinci posse existimabam. En autem ratiocinationem ipsam, qua turn quidem primo sum usus, ac deinde novis aliis, atque aliis meditationibus illustravi, ac confirmavi. minus velox. Laesio legis Continu- 18. Concipiantur duo corpora aequalia, quae moveantur in directum versus eandem cOTpus^efocruTim- pl a g am > & id, quod praecedit, habeat gradus velocitatis 6, id vero, quod ipsum persequitur mediate incurrat in gradus 12. Si hoc posterius cum sua ilia velocitate illaesa deveniat ad immediatum contactum cum illo priore ; oportebit utique, ut ipso momento temporis, quo ad contactum devenerint, illud posterius minuat velocitatem suam, & illud primus suam augeat, utrumque per saltum, abeunte hoc a 12 ad 9, illo a 6 ad 9, sine ullo transitu per intermedios gradus n, & 7 ; 10, & 8 ; 9^, & 8i, &c. Neque enim fieri potest, ut per aliquam utcunque exiguam continui A THEORY OF NATURAL PHILOSOPHY 45 represent the forces in question, which are initially repulsive & increase indefinitely as the distances are diminished indefinitely, but which, as the distances increase, are first of all diminished, then vanish, then become changed in direction & so attractive, again vanish, & change their direction, & so on alternately ; until at length, at a distance comparatively great they finally become attractive & are sensibly proportional to the inverse squares of the distance. ic. This law of forces differs from the law of gravitation enunciated by Newton in Difference between . J -nii r i i i i this ' aw f forces the construction & development or the curve that represents it ; thus, the curve given in & Newton's law of Fie. 2, which is that according to Newton, is DV, a hyperbola of the third degree, lying gravitation ; i t s ii i r i i i i nil' 6 use ln Physics ; altogether on one side of the axis, which it does not cut at any point ; all the ordmates, the order in which such as vs, op, bt, ag lie on the side of the axis representing attractive forces, & there- ^ e t s ^ ects are to fore there is no change from positive to negative, i.e., from attraction to repulsion, or vice versa. On the other hand, each of the laws is represented by the construction of a continuous curve possessing two infinite asymptotic branches in each of its members, if produced to infinity on both sides. Now, from a law of forces of this kind, & with the help of well-known mechanical principles only, such as that a force or motion can be com- pounded from several forces or motions by the help of parallelograms whose sides represent the component forces or motions, or that the forces of this kind, acting on single points for single small equal intervals of time, produce in them velocities that are proportional to themselves ; from these alone, I say, there have burst forth on me in a regular flood all the general & some of the most important particular properties of bodies, as I intimated above. Nor, indeed, for the purpose of deriving special properties, do I assert that they ought to be obtained owing to some special combination of points ; on the contrary I consider the combinations themselves, & prove geometrically what phenomena, or what species of bodies, ought to arise from this or that combination. Of course, before I come to consider, both in the second part and in the third, all the matters mentioned above, I will show in this first part in what way, & by what direct reasoning, I have arrived at this law of forces, & by what argument I have made out the simplicity of the elements of matter ; then I will give an explanation of every point that may seem to present any possible difficulty. 16. In the year 1745, I was putting together my dissertation De Firibus vivis, & had The occasion that derived everything that they who adhere to the idea of Leibniz, & the greater number of o^my^L^Trom those who measure ' living forces ' by means of velocity only, derive from these ' living the consideration forces ' ; as, I say I had derived everything directly & solely from the velocity generated by of im P ulsiv e action, the forces of those influences, which, according to the generally accepted view taken by all Mechanicians, either generate, or in some way induce, velocities that are proportional to themselves & the intervals of time during which they act ; take, for instance, gravity, elasticity, & other forces of the same kind. I then began to investigate somewhat more carefully that production of velocity which is thought to arise through impulsive action, in which the whole of the velocity is credited with being produced in an instant of time by those, who think, because of that, that the force of percussion is infinitely greater than all forces which merely exercise pressure for single instants. It immediately forced itself upon me that, for percussions of this kind, which really induce a finite velocity in an instant of time, laws for their actions must be obtained different from the rest. 17. However, when I considered the matter more thoroughly, it struck me that, if The cause of we employ a straightforward method of argument, such a mode of action must be with- w ^ s the^pposftion drawn from Nature, which in every case adheres to one & the same law of forces, & the raised to the Law same mode of action. I came to the conclusion that really immediate impulsive action of he idea' one body on another, & immediate percussion, could not be obtained, without the pro- impulse, duction of a finite velocity taking place in an indivisible instant of time, & this would have to be accomplished without any sudden change or violation of what is called the Law of Continuity ; this law indeed I considered as existing in Nature, & that this could be shown to be so by a sufficiently valid argument. The following is the line of argument that I employed initially ; afterwards I made it clearer & confirmed it by further arguments & fresh reflection. 1 8. Suppose there are two equal bodies, moving in the same straight line & in the violation of the same direction ; & let the one that is in front have a degree of velocity represented by ^ tod^movrng 1 6, & the one behind a degree represented by 12. If the latter, i.e., the body that was be- more swiftly comes hind, should ever reach with its velocity undiminished, & come into absolute contact with, J"* with^another the former body which was in front, then in every case it would be necessary that, at the body moving more very instant of time at which this contact happened, the hindermost body should diminish slowlv - its velocity, & the foremost body increase its velocity, in each case by a sudden change : one of them would pass from 12 to 9, the other from 6 to 9, without any passage through the intermediate degrees, n & 7, 10 & 8, 9$ & 8f, & so on. For it cannot possibly happen 46 PHILOSOPHIC NATURALIS THEORIA temporis particulam ejusmodi mutatio fiat per intermedios gradus, durante contactu. Si enim aliquando alterum corpus jam habuit 7 gradus velocitatis, & alterum adhuc retinet 1 1 ; toto illo tempusculo, quod effluxit ab initio contactus, quando velocitates erant 12, & 6, ad id tempus, quo sunt n, & 7, corpus secundum debuit moveri cum velocitate majore, quam primum, adeoque plus percurrere spatii, quam illud, proinde anterior ejus superficies debuit transcurrere ultra illius posteriorem superficiem, & idcirco pars aliqua corporis sequentis cum aliqua antecedentis corporis parte compenetrari debuit, quod cum ob impenetrabilitatem, quam in materia agnoscunt passim omnes Physici, & quam ipsi tri- buendam omnino esse, facile evincitur, fieri omnino non possit ; oportuit sane, in ipso primo initio contactus, in ipso indivisibili momento temporis, quod inter tempus continuum praecedens contactum, & subsequens, est indivisibilis limes, ut punctum apud Geometras est limes indivisibilis inter duo continue lineae segmenta, mutatio velocitatum facta fuerit per saltum sine transitu per intermedias, laesa penitus ilia continuitatis lege, quae itum ab una magnitudine ad aliam sine transitu per intermedias omnino vetat. Quod autem in corporibus aequalibus diximus de transitu immediato utriusque ad 9 gradus velocitatis, recurrit utique in iisdem, vel in utcunque inaequalibus de quovis alio transitu ad numeros quosvis. Nimirum ille posterioris corporis excessus graduum 6 momento temporis auferri debet, sive imminuta velocitate in ipso, sive aucta in priore, vel in altero imminuta utcunque, & aucta in altero, quod utique sine saltu, qui omissis infinitis intermediis velocitatibus habeatur, obtineri omnino non poterit. Objectio petita a ig. Sunt, qui difficultatem omnem submoveri posse censeant, dicendo, id quidem ita se cofporum. dl ' habere debere, si corpora dura habeantur, quae nimirum nullam compressionem sentiant, nullam mutationem figurae ; & quoniam hsec a multis excluduntur penitus a Natura ; dum se duo globi contingunt, introcessione, [10] & compressione partium fieri posse, ut in ipsis corporibus velocitas immutetur per omnes intermedios gradus transitu facto, & omnis argumenti vis eludatur. Ea uti non posse, 2O fa mpr j m i s e a responsione uti non possunt, quicunque cum Newtono, & vero etiam qui admittunt ele- _, \ . . r . j o menta soiida, & cum plerisquc veterum Pnilosopnorum pnma elementa matenae omnino dura admittunt, & dura - soiida, cum adhaesione infinita, & impossibilitate absoluta mutationis figurae. Nam in primis elementis illis solidis, & duris, quae in anteriore adsunt sequentis corporis parte, & in praece- dentis posteriore, quae nimirum se mutuo immediate contingunt, redit omnis argumenti vis prorsus illaesa. Extensionem con- 2 i. Deinde vero illud omnino intelligi sane non potest, quo pacto corpora omnia partes primoT pores, 1 *! aliquas postremas circa superficiem non habeant penitus solidas, quae idcirco comprimi parietes soiidos, ac ornn i no non possint. In materia quidem, si continua sit, divisibilitas in infinitum haberi potest, & vero etiam debet ; at actualis divisio in infinitum difficultates secum trahit sane inextricablies ; qua tamen divisione in infinitum ii indigent, qui nullam in corporibus admittunt particulam utcunque exiguam compressionis omnis expertem penitus, atque incapacem. Ii enim debent admittere, particulam quamcunque actu interpositis poris distinctam, divisamque in plures pororum ipsorum velut parietes, poris tamen ipsis iterum distinctos. Illud sane intelligi non potest, qui fiat, ut, ubi e vacuo spatio transitur ad corpus, non aliquis continuus haberi debeat alicujus in se determinatae crassitudinis paries usque ad primum porum, poris utique carens ; vel quomodo, quod eodem recidit, nullus sit extimus, & superficiei externae omnium proximus porus, qui nimirum si sit aliquis, parietem habeat utique poris expertem, & compressionis incapacem, in quo omnis argumenti superioris vis redit prorsus illaesa. legis Con- 22. At ea etiam, utcunque penitus inintelligibili, sententia admissa, redit omnis eadem iprimis su^r argument! vis in ipsa prima, & ultima corporum se immediate contingentium superficie, vel debus, vel punctis. s { nullae continuae superficies congruant, in lineis, vel punctis. Quidquid enim sit id, in quo contactus fiat, debet utique esse aliquid, quod nimirum impenetrabilitati occasionem praestet, & cogat motum in sequente corpore minui, in prascedente augeri ; id, quidquid est, in quo exeritur impenetratibilitatis vis, quo fit immediatus contactus, id sane velocitatem mutare debet per saltum, sine transitu per intermedia, & in eo continuitatis lex abrumpi A THEORY OF NATURAL PHILOSOPHY 47 that this kind of change is made by intermediate stages in some finite part, however small, of continuous time, whilst the bodies remain in contact. For if at any time the one body then had 7 degrees of velocity, the other would still retain 1 1 degrees ; thus, during the whole time that has passed since the beginning of contact, when the velocities were respectively 12 Si 6, until the time at which they are 1 1 & 7, the second body must be moved with a greater velocity than the first ; hence it must traverse a greater distance in space than the other. It follows that the front surface of the second body must have passed beyond the back surface of the first body ; & therefore some part of the body that follows behind must be penetrated by some part of the body that goes in front. Now, on account of impenetrability, which all Physicists in all quarters recognize in matter, & which can be easily proved to be rightly attributed to it, this cannot possibly happen. There really must be, in the commencement of contact, in that indivisible instant of time which is an indivisible limit between the continuous time that preceded the contact & that subsequent to it (just in the same way as a point in geometry is an indivisible limit between two seg- ments of a continuous line), a change of velocity taking place suddenly, without any passage through intermediate stages ; & this violates the Law of Continuity, which absolutely denies the possibility of a passage from one magnitude to another without passing through intermediate stages. Now what has been said in the case of equal bodies concerning the direct passing of both to 9 degrees of velocity, in every case holds good for such equal bodies, or for bodies that are unequal in any way, concerning any other passage to any numbers. In fact, the excess of velocity in the hindmost body, amounting to 6 degrees, has to be got rid of in an instant of time, whether by diminishing the velocity of this body, or by increasing the velocity of the other, or by diminishing somehow the velocity of the one & increasing that of the other ; & this cannot possibly be done in any case, without the sudden change that is obtained by omitting the infinite number of intermediate velocities. 19. There are some people, who think that the whole difficulty can be removed by An objection de- saying that this is just as it should be, if hard bodies, such as indeed experience no com- ^ ed e x r ^nce nyil o1 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 re P'y cannot ton, & indeed many of the ancient philosophers as well, admit the primary elements of ^"admit^oiid * matter to be absolutely hard & solid, possessing infinite adhesion & a definite shape that it hard elements. is perfectly impossible to alter. For the whole force of my argument then applies quite unimpaired to those solid and hard primary elements that are in the anterior part of the body that is behind, & in the hindmost part of the body that is in front ; & certainly these parts touch one another immediately. 21. Next it is truly impossible to understand in the slightest degree how all bodies do Continuous exten- not have some of their last parts just near to the surface perfectly solid, & on that account mary ^resT* walls altogether incapable of being compressed. If matter is continuous, it may & must be sub- bounding them, ject to infinite divisibility ; but actual division carried on indefinitely brings in its train difficulties that are truly inextricable ; however, this infinite division is required by those who do not admit that there are any particles, no matter how small, in bodies that are perfectly free from, & incapable of, compression. For they must admit the idea that every particle is marked off & divided up, by the action of interspersed pores, into many boundary walls, so to speak, for these pores ; & these walls again are distinct from the pores them- selves. It is quite impossible to understand why it comes about that, in passing from empty vacuum to solid matter, we are not then bound to encounter some continuous wall of some definite inherent thickness from the surface to the first pore, this wall being everywhere devoid of pores ; nor why, which comes to the same thing in the end, there does not exist a pore that is the last & nearest to the external surface ; this pore at least, if there were one, certainly has a wall that is free from pores & incapable of compression ; & here then the whole force of the argument used above applies perfectly unimpaired. 22. Moreover, even if this idea is admitted, although it may be quite unintelligible, Violation of the then the whole force of the same argument applies to the first or last surface of the bodies ta^s'piace^any that are in immediate contact with one another ; or, if there are no continuous surfaces rate, in prime sur- congruent, then to the lines or points. For, whatever the manner may be in which contact takes place, there must be something in every case that certainly affords occasion for impenetrability, & causes the motion of the body that follows to be diminished, & that of the one in front to be increased. This, whatever it may be, from which the force of impene- trability is derived, at the instant at which immediate contact is obtained, must certainly change the velocity suddenly, & without any passage through intermediate stages ; & by 4 8 PHILOSOPHIC NATURALIS THEORIA debet, atque labefactari, si ad ipsum immediatum contactum illo velocitatum discrimine deveniatur. Id vero est sane aliquid in quacunque e sententiis omnibus continuam extensionem tribuentibus materise. Est nimirum realis affectio qusedam corporis, videlicet ejus limes ultimus realis, superficies, realis superficiei limes linea, realis lineae limes punctum, qua affectiones utcunque in iis sententiis sint prorsus inseparabiles [n] ab ipso corpore, sunt tamen non utique intellectu confictae, sed reales, quas nimirum reales dimensiones aliquas habent, ut superficies binas, linea unam, ac realem motum, & translationem cum ipso corpore, cujus idcirco in iis sententiis debent, esse affectiones quaedam, vel modi. Objectio petita a 2 7. Est, qui dicat, nullum in iis committi saltum idcirco, quod censendum sit, nullum vucemassa, &,. J r . .. , ,, i\/r x motns. quae super- habere motum, superficiem, Imeam, punctum, quae massam habeant nullam. Motus, mquit, ficiebus, & punctis a Mechanicis habet pro mensura massam in velocitatem ductam : massa autem est super- non convemant. _.. , . i j /-^ ficies baseos ducta in crassitudmem, sive altitudmem, ex. gr. m pnsmatis. Quo minor est ejusmodi crassitude, eo minor est massa, & motus, ac ipsa crassitudine evanescente, evanescat oportet & massa, & motus. Kesponsionis ini- 24. Verum qui sic ratiocinatur, inprimis ludit in ipsis vocibus. Massam vulgo appellant tacam.^punctmn! quantitatem materiae, & motum corporum metiuntur per massam ejusmodi, ac velocitatem. posita extensione At quemadmodum in ipsa geometrica quantitate tria genera sunt quantitatum, corpus, vel contmua, e - ^11^^ q UO( J trinam dimensionem habet, superficies quae binas, linae, quae unicam, quibus accedit linese limes punctum, omni dimensione, & extensione carens ; sic etiam in Physica habetur in communi corpus tribus extensionis speciebus praeditum ; superficies realis extimus corporis limes, praedita binis ; linea, limes realis superficiei, habens unicam; & ejusdem lineae indivisibilis limes punctum. Utrobique alterum alterius est limes, non pars, & quatuor diversa genera constituunt. Superficies est nihil corporeum, sed non & nihil superficial, quin immo partes habet, & augeri potest, & minui ; & eodem pacto linea in ratione quidem superficiei est nihil, sed aliquid in ratione linese ; ac ipsum demum punctum est aliquid in suo genere, licet in ratione lineae sit nihil. QUO pacto nomen 25. Hinc autem in iis ipsis massa quaedam considerari potest duarum dimensionum, vel motus 'debeat 8 con- unius, vel etiam nullius continuae dimensionis, sed numeri punctorum tantummodo, uti venire superficie- quantitas ejus genere designetur ; quod si pro iis etiam usurpetur nomen massae generaliter, bus, imeis, punctis. motus q uan titas definiri poterit per productum ex velocitate, & massa ; si vero massae nomen tribuendum sit soli corpori, turn motus quidem corporis mensura erit massa in velocitatem ducta ; superficiei, lineae, punctorum quotcunque motus pro mensura habebit quantitatem superficiei, vel lineae, vel numerum punctorum in velocitatem ducta ; sed motus utique iis omnibus speciebus tribuendus erit, eruntque quatuor motuum genera, ut quatuor sunt quantitatum, solidi, superficiei, lineae, punctorum ; ac ut altera harum erit nihil in alterius ratione, non in sua ; ita alterius motus erit nihil in ratione alterius sed erit sane aliquid in ratione sui, non purum nihil. Fore, ut ea laedatur saltern in velocitate punctorum. Motum passim r I2 i 2Q - g t q u id em jp S j Mechanici vulgo motum tribuunt & superficiebus & lineis, & tnbui punctis; ,'..* , . '. -m j fore, lit in eo ixda- punctis, ac centri gravitatis motum ubique nommant rhysici, quod centrum utique punctum i^ r Continuitatis est aliquod, non corpus trina praeditum dimensione, quam iste ad motus rationem, & appellationem requirit, ludendo, ut ajebam, in verbis. Porro in ejusmodi motibus exti- marum saltern superficierum, vel linearum, vel punctorum, saltus omnino committi debet, si ea ad contactum immediatum deveniant cum illo velocitatum discrimine, & continuitatis lex violari. 27. Verum hac omni disquisitione omissa de notione motus, & massae, si factum ex velocitate, & massa, evanescente una e tribus dimensionibus, evanescit ; remanet utique velocitas reliquarum dimensionum, quae remanet, si eae reapse remanent, uti quidem omnino remanent in superficie, & ejus velocitatis mutatio haberi deberet per saltum, ac in ea violari continuitatis lex jam toties memorata. -, t i exin ?P<! n e- 28. Haec quidem ita evidentia sunt, ut omnino dubitari non possit, quin continuitatis trabilitate admissa ,.,..,/ .-KT i i j- j in minimis parti- lex infnngi debeat, & saltus m Naturam induci, ubi cum velocitatis discrimine ad se invicem cuiis. & ejus confu- acce dant corpora, & ad immediatum contactum deveniant, si modo impenetrabilitas corporibus tribuenda sit, uti revera est. Earn quidem non in integris tantummodo corpori- bus, sed in minimis etiam quibusque corporum particulis, atque elementis agnoverunt Physici universi. Fuit sane, qui post meam editam Theoriam, ut ipsam vim mei argument} A THEORY OF NATURAL PHILOSOPHY 49 that the Law of Continuity must be broken & destroyed, if immediate contact is arrived at with such a difference of velocity. Moreover, there is in truth always something of this sort in every one of the ideas that attribute continuous extension to matter. There is some real condition of the body, namely, its last real boundary, or its surface, a real boundary of a surface, a line, & a real boundary of a line, a point ; & these conditions, however insepar- able they may be in these theories from the body itself, are nevertheless certainly not fictions of the brain, but real things, having indeed certain real dimensions (for instance, a surface has two dimensions, & a line one) ; they also have real motion & movement of trans- lation along with the body itself ; hence in these theories they must be certain conditions or modes of it. 23. Someone may say that there is no sudden change made, because it must be con- Objection derived sidered that a surface, a line or a point, having no mass, cannot have any motion. He may 1 mo/io^w^idi say that motion has, according to Mechanicians, as its measure, the mass multiplied by the do not accord with velocity ; also mass is the surface of the base multiplied by the thickness or the altitude, surfaces & P mis - as for instance in prisms. Hence the less the thickness, the less the mass & the motion ; thus, if the thickness vanishes, then both the mass & therefore the motion must vanish as well. 24. Now the man who reasons in this manner is first of all merely playing with words. Commencement of Mass is commonly called quantity of matter, & the motion of bodies is measured by mass the answer to tl ? ls : . i * i . , * - . * , . c -_* cl SUrlclCC, OF ii 11116, of this kind & the velocity. But, just as in a geometrical quantity there are three kinds of or a point, is some- quantities, namely, a body or a solid having three dimensions, a surface with two, & a line \ with one : to which is added the boundary of a line, a point, lacking dimensions altogether, is supposed to ex- & of no extension. So also in Physics, a body is considered to be endowed with three lst ' species of extension ; a surface, the last real boundary of a body, to be endowed with two ; a line, the real boundary of a surface, with one ; & the indivisible boundary of the line, to be a point. In both subjects, the one is a boundary of the other, & not a part of it ; & they form four different kinds. There is nothing solid about a surface ; but that does not mean that there is also nothing superficial about it ; nay, it certainly has parts & can be increased or diminished. In the same way a line is nothing indeed when compared with a surface, but a definite something when compared with a line ; & lastly a point is a definite something in its own class, although nothing in comparison with a line. 25. Hence also in these matters, a mass can be considered to be of two dimensions, or The manner in of one, or even of no continuous dimension, but only numbers of points, just as quantity of wn'^ma^and^the this kind is indicated. Now, if for these also, the term mass is employed in a generalized term motus is bound sense, we shall be able to define the quantity of motion by the product of the velocity & ! : ;^ p ^' to ? u j faces ' 1 Ii ' 1 1 1 1 1 * 1 1 1 i HOPS, <X pOintS. the mass. But if the term mass is only to be used in connection with a solid body, then indeed the motion of a solid body will be measured by the mass multiplied by the velocity ; but the motion of a surface, or a line, or any number of points will have as their measure the quantity of the surface, or line, or the number- of the points, multiplied by the velocity. Motion at any rate will be ascribed in all these cases, & there will be four kinds of motion, as there are four kinds of quantity, namely, for a solid, a surface, a line, or for points ; and, as each class of the latter will be as nothing compared with the class before it, but something in its own class, so the motion of the one will be as nothing compared with the motion of the other, but yet really something, & not entirely nothing, compared with those of its own class. 26. Indeed, Mechanicians themselves commonly ascribe motion to surfaces, lines & Motion is ascribed points, & Physicists universally speak of the motion of the centre of gravity ; this centre is minateiy 3 the'i^w undoubtedly some point, & not a body endowed with three dimensions, which the objector of Continuity is vio- demands for the idea & name of motion, by playing with words, as I said above. On the ' other hand, in this kind of motions of ultimate surfaces, or lines, or points, a sudden change must certainly be made, if they arrive at immediate contact with a difference of velocity as above, & the Law of Continuity must be violated. 27. But, omitting all debate about the notions of motion & mass, if the product of it is at least a fact the velocity & the mass vanishes when one of the three dimensions vanish, there will still fated^tf^the^idea remain the velocity of the remaining dimensions ; & this will persist so long as the dimen- of the velocity of sions persist, as they do persist undoubtedly in the case of a surface. Hence the change P mts - in its velocity must have been made suddenly, & thereby the Law of Continuity, which I have already mentioned so many times, is violated. 28. These things are so evident that it is absolutely impossible to doubt that the Law objection derived /./-! r - i i j j i . . j j . T iv from the admission of Continuity is infringed, & that a sudden change is introduced into Nature, when bodies O f impenetrability approach one another with a difference of velocity & come into immediate contact, if only in verv small P ar - we are to ascribe impenetrability to bodies, as we really should. And this property too, t ion. ' not in whole bodies only, but in any of the smallest particles of bodies, & in the elements as well, is recognized by Physicists universally. There was one, I must confess, who, after I 50 PHILOSOPHIC NATURALIS THEORIA infringeret, affirmavit, minimas corporum particulas post contactum superficierum com- penetrari non nihil, & post ipsam compenetrationem mutari velocitates per gradus. At id ipsum facile demonstrari potest contrarium illi inductioni, & analogiae, quam unam habemus in Physica investigandis generalibus naturae legibus idoneam, cujus inductionis vis quae sit, & quibus in locis usum habeat, quorum locorum unus est hie ipse impenetrabilitatis ad minimas quasque particulas extendendae, inferius exponam. Objectio a voce motus assumpta pro mutatione; confutatio ex reahtate motus 2 Q. Fuit itidem e Leibnitianorum familia, qui post evulgatam Theoriam meam cen- . ' ,./>- , j- j j j -i suerit, dimcultatem ejusmodi amoveri posse dicendo, duas monades sibi etiam mvicem occurrentes cum velocitatibus quibuscunque oppositis aequalibus, post ipsum contactum ..... . i, . . .' r ..... pergere moven sine locali progressione. Ham progressionem, ajebat, revera omnmo nihil esse, si a spatio percurso sestimetur, cum spatium sit nihil ; motum utique perseverare, & extingui per gradus, quia per gradus extinguatur energia ilia, qua in se mutuo agunt, sese premendo invicem. Is itidem ludit in voce motus, quam adhibet pro mutatione quacunque, & actione, vel actionis modo. Motus locaiis, & velocitas motus ipsius, sunt ea, quse ego quidem adhibeo, & quae ibi abrumpuntur per saltum. Ea, ut evidentissime constat, erant aliqua ante contactum, & post contactum mo-[i3]-mento temporis in eo casu abrumpuntur ; nee vero sunt nihil ; licet spatium pure imaginarium sit nihil. Sunt realis affectio rei mobilis fundata in ipsis modis localiter existendi, qui modi etiam relationes inducunt dis- tantiarum reales utique. Quod duo corpora magis a se ipsis invicem distent, vel minus ; quod localiter celerius moveantur, vel lentius ; est aliquid non imaginarie tantummodo, sed realiter diversum ; in eo vero per immediatum contactum saltus utique induceretur in eo casu, quo ego superius sum usus. Qui Continuitatu, 30. Et sane summus nostri aevi Geometra, & Philosophus Mac-Laurinus, cum etiam ipse jegem summover- co ni s j onem corporum contemplatus vidisset, nihil esse, quod continuitatis legem in collisione corporum facta per immediatum contactum conservare, ac tueri posset, ipsam continuitatis legem deferendam censuit, quam in eo casu omnino violari affirmavit in eo opere, quod de Newtoni Compertis inscripsit, lib. I, cap. 4. Et sane sunt alii nonnulli, qui ipsam con- tinuitatis legem nequaquam admiserint, quos inter Maupertuisius, vir celeberrimus, ac de Republica Litteraria optime meritus, absurdam etiam censuit, & quodammodo inexplica- bilem. Eodem nimirum in nostris de corporum collisione contemplationibus devenimus Mac-Laurinus, & ego, ut viderimus in ipsa immediatum contactum, atque impulsionem cum continuitatis lege conciliari non posse. At quoniam de impulsione, & immediate corporum contactu ille ne dubitari quidem posse arbitrabatur, (nee vero scio, an alius quisquam omnem omnium corporum immediatum contactum subducere sit ausus antea, utcunque aliqui aeris velum, corporis nimirum alterius, in collisione intermedium retinuerint) continuitatis legem deseruit, atque infregit. Theorise exortus, t^'t U f fien 31. Ast ego cum ipsam continuitatis legem aliquanto diligentius considerarim, & , quibus ea innititur, perpenderim, arbitratus sum, ipsam omnino e Natura submoveri non posse, qua proinde retenta contactum ipsum immediatum submovendum censui in collisionibus corporum, ac ea consectaria persecutus, quae ex ipsa continuitate servata sponte profluebant, directa ratiocinatione delatus sum ad earn, quam superius exposui, virium mutuarum legem, quae consectaria suo quaeque ordine proferam, ubi ipsa, quae ad continuitatis legem retinendam argumenta me movent, attigero. Lex Continuitatis 32. Continuitatis lex, de qua hie agimus, in eo sita est, uti superius innui, ut quaevis quid sit : discn- j i i- T i- r men inter status, quantitas, dum ab una magmtudme ad aliam migrat, debeat transire per omnes intermedias & incrementa. ejusdem generis magnitudines. Solet etiam idem exprimi nominandi transitum per gradus intermedios, quos quidem gradus Maupertuisius ita accepit, quasi vero quaedam exiguae accessiones fierent momento temporis, in quo quidem is censuit violari jam necessario legem ipsam, quae utcunque exiguo saltu utique violatur nihilo minus, quam maximo ; cum nimi-[l4]-rum magnum, & parvum sint tantummodo respectiva ; & jure quidem id censuit ; si nomine graduum incrementa magnitudinis cujuscunque momentanea intelligerentur. A THEORY OF NATURAL PHILOSOPHY 51 had published my Theory, endeavoured to overcome the force of the argument I had used by asserting that the minute particles of the bodies after contact of the surfaces were subject to compenetration in some measure, & that after compenetration the velocities were changed gradually. But it can be easily proved that this is contrary to that induction & analogy, such as we have in Physics, one peculiarly adapted for the investigation of the general laws of Nature. What the power of this induction is, & where it can be used (one of the cases is this very matter of extending impenetrability to the minute particles of a body), I will set forth later. 29. There was also one of the followers of Leibniz who, after I had published my Objection to the Theory, expressed his opinion that this kind of difficulty could be removed by saying that used fora"change^ two monads colliding with one another with any velocities that were equal & opposite refutation from the ,,,., .. . .-I , , TT reality of local mo- would, alter they came into contact, go on moving without any local progression, rle tion. added that that progression would indeed be absolutely nothing, if it were estimated by the space passed over, since the space was nothing ; but the motion would go on & be destroyed by degrees, because the energy with which they act upon one another, by mutual pressure, would be gradually destroyed. He also is playing with the meaning of the term motus, which he uses both for any change, & for action & mode of action. Local motion, & the velocity of that motion are what I am dealing with, & these are here broken off suddenly. These, it is perfectly evident, were something definite before contact, & after contact in an instant of time in this case they are broken off. Not that they are nothing ; although purely imaginary space is nothing. They are real conditions of the movable thing depending on its modes of extension as regards position ; & these modes induce relations between the distances that are certainly real. To account for the fact that two bodies stand at a greater distance from one another, or at a less ; or for the fact that they are moved in position more quickly, or more slowly ; to account for this there must be some- thing that is not altogether imaginary, but real & diverse. In this something there would be induced, in the question under consideration, a sudden change through immediate contact. 30. Indeed the finest geometrician & philosopher of our times, Maclaurin, after he too There are some who had considered the collision of solid bodies & observed that there is nothing which could i^ d oi continuity 5 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, & Th e origin of my pondering over the fundamental ideas on which it depends, came to the conclusion that thisLaw, as'shouid it certainly could not be withdrawn altogether out of Nature. Hence, since it had to be be done, retained, I came to the conclusion that immediate contact in the collision of solid bodies must be got rid of ; &, investigating the deductions that naturally sprang from the conservation of continuity, I was led by straightforward reasoning to the law that I have set forth above, namely, the law of mutual forces. These deductions, each set out in order, I will bring forward when I come to touch upon those arguments that persuade me to retain the Law of Continuity. 32. The Law of Continuity, as we here deal with it, consists in the idea that, as I J he nature of the . j , ..''... . , , Law of Continuity ; intimated above, any quantity, in passing from one magnitude to another, must pass through distinction between all intermediate magnitudes of the same class. The same notion is also commonly expressed stat< ~ s & incre - , , .,,. ,. -11 ments. by saying that the passage is made by intermediate stages or steps ; these steps indeed Maupertuis accepted, but considered that they were very small additions made in an instant of time. In this he thought that the Law of Continuity was already of necessity violated, the law being indeed violated by any sudden change, no matter how small, in no less a degree than by a very great one. For, of a truth, large & small are only relative terms ; & he rightly thought as he did, if by the name of steps we are to understand momentaneous PHILOSOPHIC NATURALIS THEORIA Geometriae usus ad earn exponendam : momenta punctis, tempera continua lineis expressa. Fluxus ordinatae transeuntis per m agnit u d i nes omnes intermedias. Idem in quantitate variabili expressa : aequivocatio in voce gradus. FKMH K' M' D' FIG. 3. Verum id ita intelligendum est ; ut singulis momentis singuli status respondeant ; incre- menta, vel decrementa non nisi continuis tempusculis. 33. Id sane admodum facile concipitur ope Geometriae. Sit recta quaedam AB in fig. 3, ad quam referatur quaedam alia linea CDE. Exprimat prior ex iis tempus, uti solet utique in ipsis horologiis circularis peripheria ab indicis cuspide denotata tempus definire. Quemadmodum in Geometria in lineis puncta sunt indivisibiles limites continuarum lineas partium, non vero partes linese ipsius ; ita in tempore distinguenda; erunt partes continui temporis respondentes ipsis lines partibus, continue itidem & ipsas, a mo- mentis, quae sunt indivisibiles earum partium limites, & punctis respondent ; nee inpos- terum alio sensu agens de tempore momenti nomen adhibebo, quam eo indivisibilis limitis ; particulam vero temporis utcunque exiguam, & habitam etiam pro infinitesima, tempusculum appellabo. 34. Si jam a quovis puncto rectae AB, ut F, H, erigatur ordinata perpendicularis FG, HI, usque ad lineam CD ; ea poterit repraesentare quantitatem quampiam continuo variabilem. Cuicunque momento temporis F, H, respondebit sua ejus quantitatis magnitudo FG, HI ; momentis autem intermediis aliis K, M, aliae magnitudines, KL, MN, respondebunt ; ac si a puncto G ad I continua, & finita abeat pars linese CDE, facile patet & accurate de- monstrari potest, utcunque eadem contorqueatur, nullum fore punctum K intermedium, cui aliqua ordinata KL non respondeat ; & e converse nullam fore ordinatam magnitu- dinis intermediae inter FG, HI, quae alicui puncto inter F, H intermedio non respondeat. 35. Quantitas ilia variabilis per hanc variabilem ordinatam expressa mutatur juxta continuitatis legem, quia a magnitudine FG, quam habet momento temporis F, ad magni- tudinem HI, quae respondet momento temporis H, transit per omnes intermedias magnitu- dines KL, MN, respondentes intermediis momentis K, M, & momento cuivis respondet determinata magnitudo. Quod si assumatur tempusculum quoddam continuum KM utcunque exiguum ita, ut inter puncta L, N arcus ipse LN non mutet recessum a recta AB in accessum ; ducta LO ipsi parallela, habebitur quantitas NO, quas in schemate exhibito est incrementum magnitudinis ejus quantitatis continuo variatae. Quo minor est ibi temporis particula KM, eo minus est id incrementum NO, & ilia evanescente, ubi congruant momenta K, M, hoc etiam evanescit. Potest quaevis magnitudo KL, MN appellari status quidam variabilis illius quantitatis, & gradus nomine deberet potius in-[i5]-telligi illud incrementum NO, quanquam aliquando etiam ille status, ilia magnitudo KL nomine gradus intelligi solet, ubi illud dicitur, quod ab una magnitudine ad aliam per omnes intermedios gradus transeatur ; quod quidem aequivocationibus omnibus occasionem exhibuit. status singuios 36. Sed omissis aequivocationibus ipsis, illud, quod ad rem facit, est accessio incremen- menta^vero'utcun" torum facta non momento temporis, sed tempusculo continuo, quod est particula continui que parva tem- temporis. Utcunque exiguum sit incrementum ON, ipsi semper respondet tempusculum respondere C ntinuis q. u ddam KM continuum. Nullum est in linea punctum M ita proximum puncto K, ut sit primum post ipsum ; sed vel congruunt, vel intercipiunt lineolam continua bisectione per alia intermedia puncta perpetuo divisibilem in infinitum. Eodem pacto nullum est in tempore momentum ita proximum alteri praecedenti momento, ut sit primum post ipsum, sed vel idem momentum sunt, vel inter jacet inter ipsa tempusculum continuum per alia intermedia momenta divisibile in infinitum ; ac nullus itidem est quantitatis continuo variabilis status ita proximus praecedenti statui, ut sit primus post ipsum accessu aliquo momentaneo facto : sed differentia, quae inter ejusmodi status est, debetur intermedio continuo tempusculo ; ac data lege variationis, sive natura lineae ipsam exprimentis, & quacunque utcunque exigua accessione, inveniri potest tempusculum continuum, quo ea accessio advenerit. Transitus sine sal- 37- Atque sic quidem intelligitur, quo pacto fieri possit transitus per intermedias tu, etiamapositivis magnitudines omnes, per intermedios status, per gradus intermedios, quin ullus habeatur ad negativa perm- , . r -, . ... , ' " hiium, quod tamen saltus utcunque exiguus momento temporis factus. Notari mud potest tantummodo, m" eS s t ed Vere u 'ida 1 m mutat i nem ner i alicubi per incrementa, ut ubi KL abit, in MN per NO ; alicubi per reaiis status, 1 " ' decrementa, ut ubi K'L' abeat in N'M' per O'N' ; quin immo si linea CDE, quse legem A THEORY OF NATURAL PHILOSOPHY 53 increments of any magnitude whatever. But the idea should be interpreted as follows : single states correspond to single instants of time, but increments or decrements only to small intervals of continuous time. 33. The idea can be very easily assimilated by the help of geometry. Explanation by the Let AB be any straight line (Fig. 3), to which as axis let any other line CDE be referred. " n s s e ta t f s ^eTes^ Let the first of them represent the time, in the same manner as it is customary to specify ted by points, con- the time in the case of circular clocks by marking off the periphery with the end of a pointer. 1^ "^s*** f Now, just as in geometry, points are the indivisible boundaries of the continuous parts of a line, so, in time, distinction must be made between parts of continuous time, which cor- respond to these parts of a line, themselves also continuous, & instants of time, which are the indivisible boundaries of those parts of time, & correspond to points. In future I shall not use the term instant in any other sense, when dealing with time, than that of the indivisible boundary ; & a small part of time, no matter how small, even though it is considered to be infinitesimal, I shall term a tempuscule, or small interval of time. 34. If now from any points F,H on the straight line AB there are erected at right angles T . he flux . f the or ~ to it ordinates FG, HI, to meet the line CD ; any of these ordinates can be taken to repre- through^ ail *inter S sent a quantity that is continuously varying. To any instant of time F, or H, there will mediate values, correspond its own magnitude of the quantity FG, or HI ; & to other intermediate instants K, M, other magnitudes KL, MN will correspond. Now, if from the point G, there pro- ceeds a continuous & finite part of the line CDE, it is very evident, & it can be rigorously proved, that, no matter how the curve twists & turns, there is no intermediate point K, to which some ordinate KL does not correspond ; &, conversely, there is no ordinate of magnitude intermediate between FG & HI, to which there does not correspond a point intermediate between F & H. 35. The variable quantity that is represented by this variable ordinate is altered in The same holds accordance with the Law of Continuity ; for, from the magnitude FG, which it has at able 1 quantity w the instant of time F, to the magnitude HI, which corresponds to the instant H, it passes represented ; equi- through all intermediate magnitudes KL, MN, which correspond to the intermediate oUhe 1( term Itep^ instants K, M ; & to every instant there corresponds a definite magnitude. But if we take a definite small interval of continuous time KM, no matter how small, so that between the points L & N the arc LN does not alter from recession from the line AB to approach, & draw LO parallel to AB, we shall obtain the quantity NO that in the figure as drawn is the increment of the magnitude of the continuously varying quantity. Now the smaller the interval of time KM, the smaller is this increment NO ; & as that vanishes when the instants of time K, M coincide, the increment NO also vanishes. Any magnitude KL, MN can be called a state of the variable quantity, & by the name step we ought rather to under- stand the increment NO ; although sometimes also the state, or the magnitude KL is accustomed to be called by the name step. For instance, when it is said that from one magnitude to another there is a passage through all intermediate stages or steps ; but this indeed affords opportunity for equivocations of all sorts. 36. But, omitting all equivocation of this kind, the point is this : that addition of single states cor- .' 1-11 < i 11 . respond to instants, increments is accomplished, not m an instant 01 time, but in a small interval of con- but increments tinuous time, which is a part of continuous time. However small the increment ON may however sma11 to i i i Tru if mi > intervals of con- DC, there always corresponds to it some continuous interval KM. 1 here is no point M tinuous time. in the straight line AB so very close to the point K, that it is the next after it ; but either the points coincide, or they intercept between them a short length of line that is divisible again & again indefinitely by repeated bisection at other points that are in between M & K. In the same way, there is no instant of time that is so near to another instant that has gone before it, that it is the next after it ; but either they are the same instant, or there lies between them a continuous interval that can be divided indefinitely at other inter- mediate instants. Similarly, there is no state of a continuously varying quantity so very near to a preceding state that it is the next state to it, some momentary addition having been made ; any difference that exists between two states of the same kind is due to a continuous interval of time that has passed in the meanwhile. Hence, being given the law of variation, or the nature of the line that represents it, & any increment, no matter how small, it is possible to find a small interval of continuous time in which the increment took place. 37. In this manner we can understand how it is possible for a passage to take place Passages without through all intermediate magnitudes, through intermediate states, or through intermediate from^positive 1 8 to stages, without any sudden change being made, no matter how small, in an instant of time, negative through T' 11 1111 i 111- /i zero : zero how- It can merely be remarked that change in some places takes place by increments (as when ever ; s not real i y KL becomes MN by the addition of NO), in other places by decrements (as when K'L' nothing, but acer- ' tain real state. 54 PHILOSOPHIC NATURALIS THEORIA variationis exhibit, alicubi secet rectam, temporis AB, potest ibidem evanescere magnitude, ut ordinata M'N', puncto M' allapso ad D evanesceret, & deinde mutari in negativam PQ, RS, habentem videlicet directionem contrariam, quae, quo magis ex oppositae parte crescit, eo minor censetur in ratione priore, quemadmodum in ratione possessionis, vel divitiarum, pergit perpetuo se habere pejus, qui iis omnibus, quae habebat, absumptis, aes alienum contrahit perpetuo majus. Et in Geometria quidem habetur a positivo ad negativa transitus, uti etiam in Algebraicis formulis, tarn transeundo per nihilum, quam per innnitum, quos ego transitus persecutus sum partim in dissertatione adjecta meis Sectionibus Conicis, partim in Algebra 14, & utrumque simul in dissertatione De Lege Continuitatis ; sed in Physica, ubi nulla quantitas in innnitum excrescit, is casus locum non habet, & non, nisi transeundo per nihilum, transitus fit a positi-[i6]-vis ad negativa, ac vice versa ; quanquam, uti inferius innuam, id ipsum sit non nihilum revera in se ipso, sed realis quidem status, & habeatur pro nihilo in consideration quadam tantummodo, in qua negativa etiam, qui sunt veri status, in se positivi, ut ut ad priorem seriem pertinentes negative quodam modo, negativa appellentur. Proponitur pro- ,_ Exposita hoc pacto, & vindicata continuitatis lege, earn in Natura existere plerique banda existentia _, ., J . . r . . .... ... P . . ,-, r . legis Continuitat.s. Philosophi arbitrantur, contradicentibus nonnullis, uti supra mnui. Ego, cum in earn primo inquirerem, censui, eandem omitti omnino non posse ; si earn, quam habemus unicam, Naturae analogiam, & inductionis vim consulamus, ope cujus inductionis earn demonstrare conatus sum in pluribus e memoratis dissertationibus, ac eandem probationem adhibet Benvenutus in sua Synopsi Num. 119; in quibus etiam locis, prout diversis occasionibus conscripta sunt, repetuntur non nulla. Ejus probatio ab ,g Longum hie esset singula inde excerpere in ordinem redacta : satis erit exscribere mductione satis ,. Jy . _ , ~ . P . r ,-, -n i j ampia. dissertatioms De lege Continuitatis numerum 138. Post mductionem petitam praecedente numero a Geometria, quae nullum uspiam habet saltum, atque a motu locali, in quo nunquam ab uno loco ad alium devenitur, nisi ductu continue aliquo, unde consequitur illud, dis- tantiam a dato loco nunquam mutari in aliam, neque densitatem, quae utique a distantiis pendet particularum in aliam, nisi transeundo per intermedias ; fit gradus in eo numero ad motuum velocitates, & ductus, quas magis hie ad rem faciunt, nimirum ubi de velocitate agimus non mutanda per saltum in corporum collisionibus. Sic autem habetur : " Quin immo in motibus ipsis continuitas servatur etiam in eo, quod motus omnes in lineis continuis fiunt nusquam abruptis. Plurimos ejusmodi motus videmus. Planetae, & cometse in lineis continuis cursum peragunt suum, & omnes retrogradationes fiunt paullatim, ac in stationibus semper exiguus quidem motus, sed tamen habetur semper, atque hinc etiam dies paullatim per auroram venit, per vespertinum crepusculum abit, Solis diameter non per saltum, sed continuo motu supra horizontem ascendit, vel descendit. Gravia itidem oblique projecta in lineis itidem pariter continuis motus exercent suos, nimirum in parabolis, seclusa ^aeris resistentia, vel, ea considerata, in orbibus ad hyperbolas potius accedentibus, & quidem semper cum aliqua exigua obliquitate projiciuntur, cum infinities infinitam improbabilitatem habeat motus accurate verticalis inter infinities infinitas inclinationes, licet exiguas, & sub sensum non cadentes, fortuito obvenienfe, qui quidem motus in hypothesi Telluris^motae a parabolicis plurimum distant, & curvam continuam exhibent etiam pro casu projectionis accurate verticalis, quo, quiescente penitus Tellure, & nulla ventorum vi deflectente motum, haberetur [17] ascensus rectilineus, vel descensus. Immo omnes alii motus a gravitate pendentes, omnes ab elasticitate, a vi magnetica, continuitatem itidem servant ; cum earn servent vires illse ipsae, quibus gignuntur. Nam gravitas, cum decrescat in ratione reciproca duplicata distantiarum, & distantise per saltum mutari non possint, mutatur per omnes intermedias magnitudines. Videmus pariter, vim magneticam a distantiis pendere lege continua ; vim elasticam ab inflexione, uti in laminis, vel a distantia, ut in particulis aeris compressi. In iis, & omnibus ejusmodi viribus, & motibus, quos gignunt, continuitas habetur semper, tarn in lineis quae describuntur, quam in velocitatibus, quae pariter per omnes intermedias magnitudines mutantur, ut videre est in pendulis, in ascensu corporum gravium, A THEORY OF NATURAL PHILOSOPHY 55 becomes N'M' by the subtraction of O'N') ; moreover, if the line CDE, which represents the law of variation, cuts the straight AB, which is the axis of time, in any point, then the magnitude can vanish at that point (just as the ordinate M'N' would vanish when the point M' coincided with D), & be changed into a negative magnitude PQ, or RS, that is to say one having an opposite direction ; & this, the more it increases in the opposite sense, the less it is to be considered in the former sense (just as in the idea of property or riches, a man goes on continuously getting worse off, when, after everything he had has been taken away from him, he continues to get deeper & deeper into debt). In Geometry too we have this passage from positive to negative, & also in algebraical formulae, the passage being made not only through nothing, but also through infinity ; such I have discussed, the one in a dissertation added to my Conic Sections, the other in my Algebra ( 14), & both of them together in my essay De Lege Continuitatis ; but in Physics, where no quantity ever increases to an infinite extent, the second case has no place ; hence, unless the passage is made through the value nothing, there is no passage from positive to negative, or vice versa. Although, as I point out below, this nothing is not really nothing in itself, but a certain real state ; & it may be considered as nothing only in a certain sense. In the same sense, too, negatives, which are true states, are positive in themselves, although, as they belong to the first set in a certain negative way, they are called negative. 38. Thus explained & defended, the Law of Continuity is considered by most philoso- I propose to prove phers to exist in Nature, though there are some who deny it, as I mentioned above. I, LaVof^Continuity 6 when first I investigated the matter, considered that it was absolutely impossible that it should be left out of account, if we have regard to the unparalleled analogy that there is with Nature & to the power of induction ; & by the help of this induction I endeavoured to prove the law in several of the dissertations that I have mentioned, & Benvenutus also used the same form of proof in his Synopsis (Art. 119). In these too, as they were written on several different occasions, there are some repetitions. 39. It would take too long to extract & arrange in order here each of the passages in Proof by induction these essays ; it will be sufficient if I give Art. 138 of the dissertation De Lege Continuitatis. s ~^^ for the After induction derived in the preceding article from geometry, in which there is no sudden change anywhere, & from local motion, in which passage from one position to another never takes place unless by some continuous progress (the consequence of which is that a distance from any given position can never be changed into another distance, nor the density, which depends altogether on the distances between the particles, into another density, except by passing through intermediate stages), the step is made in that article to the velocities of motions, & deductions, which have more to do with the matter now in hand, namely, where we are dealing with the idea that the velocity is not changed suddenly in the collision of solid bodies. These are the words : " Moreover in motions themselves continuity is preserved also in the fact that all motions take place in continuous lines that are not broken anywhere. We see a great number of motions of this kind. The planets & the comets pursue their courses, each in its own continuous line, & all retrogradations are gradual ; & in stationary positions the motion is always slight indeed, but yet there is always some ; hence also daylight comes gradually through the dawn, & goes through the evening twilight, as the diameter of the sun ascends above the horizon, not suddenly, but by a continuous motion, & in the same manner descends. Again heavy bodies projected obliquely follow their courses in lines also that are just as continuous ; namely, in para- bolae, if we neglect the resistance of the air, but if that is taken into account, then in orbits that are more nearly hyperbolae. Now, they are always projected with some slight obli- quity, since there is an infinitely infinite probability against accurate vertical motion, from out of the infinitely infinite number of inclinations (although slight & not capable of being observed), happening fortuitously. These motions are indeed very far from being para- bolae, if the hypothesis that the Earth is in motion is adopted. They give a continuous curve also for the case of accurate vertical projection, in which, if the Earth were at rest, & no wind-force deflected the motion, rectilinear ascent & descent would be obtained. All other motions that depend on gravity, all that depend upon elasticity, or magnetic force, also preserve continuity ; for the forces themselves, from which the motions arise, preserve it. For gravity, since it diminishes in the inverse ratio of the squares of the dis- tances, & the distances cannot be changed suddenly, is itself changed through every inter- mediate stage. Similarly we see that magnetic force depends on the distances according to a continuous law ; that elastic force depends on the amount of bending as in plates, or according to distance as in particles of compressed air. In these, & all other forces of the sort, & in the motions that arise from them, we always get continuity, both as regards the lines which they describe & also in the velocities which are changed in similar manner through all intermediate magnitudes ; as is seen in pendulums, in the ascent of heavy 56 PHILOSOPHISE NATURALIS THEORIA & in aliis mille ejusmodi, in quibus mutationes velocitatis fiunt gradatim, nee retro cursus reflectitur, nisi imminuta velocitate per omnes gradus. Ea diligentissime continuitatem servat omnia. Hinc nee ulli in naturalibus motibus habentur anguli, sed semper mutatio directionis fit paullatim, nee vero anguli exacti habentur in corporibus ipsis, in quibus utcunque videatur tennis acies, vel cuspis, microscopii saltern ope videri solet curvatura, quam etiam habent alvei fluviorum semper, habent arborum folia, & frondes, ac rami, habent lapides quicunque, nisi forte alicubi cuspides continuae occurrant, vel primi generis, quas Natura videtur affectare in spinis, vel secundi generis, quas videtur affectare in avium unguibus, & rostro, in quibus tamen manente in ipsa cuspide unica tangente continuitatem servari videbimus infra. Infinitum esset singula persequi, in quibus continuitas in Natura observatur. Satius est generaliter provocare ad exhibendum casum in Natura, in quo eontinuitas non servetur, qui omnino exhiberi non poterit." Duplex inductionis 40. Inductio amplissima turn ex hisce motibus, ac velocitatibus, turn ex aliis pluribus vimhabeatittductio exem P n ' s > <l U3e habemus in Natura, in quibus ea ubique, quantum observando licet depre- incompieta. hendere, continuitatem vel observat accurate, vel affcctat, debet omnino id efficere, ut ab ea ne in ipsa quidem corporum collisione recedamus. Sed de inductionis natura, & vi, ac ejusdem usu in Physica, libet itidem hie inserere partem numeri 134, & totum 135, disserta- tionis De Lege Continuitatis. Sic autem habent ibidem : " Inprimis ubi generales Naturae leges investigantur, inductio vim habet maximam, & ad earum inventionem vix alia ulla superest via. Ejus ope extensionem, figurabilitem, mobilitatem, impenetrabilitatem corporibus omnibus tribuerunt semper Philosophi etiam veteres, quibus eodem argumento inertiam, & generalem gravitatem plerique e recentioribus addunt. Inductio, ut demon- strationis vim habeat, debet omnes singulares casus, quicunque haberi possunt percurrere. Ea in Natu-[i8]-rae legibus stabiliendis locum habere non potest. Habet locum laxior qusedam inductio, quae, ut adhiberi possit, debet esse ejusmodi, ut inprimis in omnibus iis casibus, qui ad trutinam ita revocari possunt, ut deprehendi debeat, an ea lex observetur, eadem in iis omnibus inveniatur, & ii non exiguo numero sint ; in reliquis vero, si quse prima fronte contraria videantur, re accuratius perspecta, cum ilia lege possint omnia conciliari ; licet, an eo potissimum pacto concilientur, immediate innotescere, nequaquam possit. Si eae conditiones habeantur ; inductio ad legem stabiliendam censeri debet idonea. Sic quia videmus corpora tarn multa, quae habemus prae manibus, aliis corporibus resistere, ne in eorum locum adveniant, & loco cedere, si resistendo sint imparia, potius, quam eodem perstare simul ; impenetrabilitatem corporum admittimus ; nee obest, quod qusedam corpora videamus intra alia, licet durissima, insinuari, ut oleum in marmora, lumen in crystalla, & gemmas. Videmus enim hoc phsenomenum facile conciliari cum ipsa impene- trabilitate, dicendo, per vacuos corporum poros ea corpora permeare. (Num. 135). Praeterea, qusecunque proprietates absolutae, nimirum quae relationem non habent ad nostros sensus, deteguntur generaliter in massis sensibilibus corporum, easdem ad quascunque utcunque exiguas particulas debemus transferre ; nisi positiva aliqua ratio obstet, & nisi sint ejusmodi, quae pendeant a ratione totius, seu multitudinis, contradistincta a ratione partis. Primum evincitur ex eo, quod magna, & parva sunt respectiva, ac insensibilia dicuntur ea, quse respectu nostrae molis, & nostrorum sensuum sunt exigua. Quare ubi agitur de proprietatibus absolutis non respectivis, quaecunque communia videmus in iis, quse intra limites continentur nobis sensibiles, ea debemus censere communia etiam infra eos limites : nam ii limites respectu rerum, ut sunt in se, sunt accidentales, adeoque siqua fuisset analogise Isesio, poterat ilia multo facilius cadere intra limites nobis sensibiles, qui tanto laxiores sunt, quam infra eos, adeo nimirum propinquos nihilo. Quod nulla ceciderit, indicio est, nullam esse. Id indicium non est evidens, sed ad investigationis principia pertinet, quae si juxta A THEORY OF NATURAL PHILOSOPHY 57 bodies, & in a thousand other things of the same kind, where the changes of velocity occur gradually, & the path is not retraced before the velocity has been diminished through all degrees. All these things most strictly preserve continuity. Hence it follows that no sharp angles are met with in natural motions, but in every case a change of direction occurs gradually ; neither do perfect angles occur in bodies themselves, for, however fine an edge or point in them may seem, one can usually detect curvature by the help of the microscope if nothing else. We have this gradual change of direction also in the beds of rivers, in the leaves, boughs & branches of trees, & stones of all kinds ; unless, in some cases perchance, there may be continuous pointed ends, either of the first kind, which Nature is seen to affect in thorns, or of the second kind, which she is seen to do in the claws & the beak of birds ; in these, however, we shall see below that continuity is still preserved, since we are left with a single tangent at the extreme end. It would take far too long to mention every single thing in which Nature preserves the Law of Continuity ; it is more than sufficient to make a general statement challenging the production of a single case in Nature, in which continuity is not preserved ; for it is absolutely impossible for any such case to be brought forward." 40. The effect of the very complete induction from such motions as these & velocities, induction of a two- as well as from a large number of other examples, such as we have in Nature, where Nature * old , kil \ d ' when e c ,...-..& why incomplete in every case, as far as can be gathered from direct observation, maintains continuity or induction has vaii- tries to do so, should certainly be that of keeping us from neglecting it even in the case of collision of bodies. As regards the nature & validity of induction, & its use in Physics, I may here quote part of Art. 134 & the wjiole of Art. 135 from my dissertation De Lege Continuitatis, The passage runs thus : " Especially when we investigate the general laws of Nature, induction has very great power ; & there is scarcely any other method beside it for the discovery of these laws. By its assistance, even the ancient philosophers attributed to all bodies extension, figurability, mobility, & impenetrability ; & to these properties, by the use of the same method of reasoning, most of the later philosophers add inertia & universal gravitation. Now, induction should take account of every single case that can possibly happen, before it can have the force of demonstration ; such induction as this has no place in establishing the laws of Nature. But use is made of an induction of a less rigorous type ; in order that this kind of induction may be employed, it must be of such a nature that in all those cases particularly, which can be examined in a manner that is bound to lead to a definite conclusion as to whether or no the law in question is followed, in all of them the same result is arrived at ; & that these cases are not merely a few. Moreover, in the other cases, if those which at first sight appeared to be contradictory, on further & more accurate investigation, can all of them be made to agree with the law ; although, whether they can be made to agree in this way better than in any other whatever, it is impossible to know directly anyhow. If such conditions obtain, then it must be considered that the induction is adapted to establishing the law. Thus, as we see that so many of the bodies around us try to prevent other bodies from occupying the position which they themselves occupy, or give way to them if they are not capable of resisting them, rather than that both should occupy the same place at the same time, therefore we admit the impenetrability of bodies. Nor is there anything against the idea in the fact that we see certain bodies penetrating into the innermost parts of others, although the latter are very hard bodies ; such as oil into marble, & light into crystals & gems. For we see that this phenomenon can very easily be reconciled with the idea of impenetrability, by supposing that the former bodies enter and pass through empty pores in the latter bodies (Art. 135). In addition, whatever absolute properties, for instance those that bear no relation to our senses, are generally found to exist in sensible masses of bodies, we are bound to attribute these same properties also to all small parts whatsoever, no matter how small they may be. That is to say, unless some positive reason prevents this ; such as that they are of such a nature that they depend on argument having to do with a body as a whole, or with a group of particles, in contradistinction to an argument dealing with a part only. The proof comes in the first place from the fact that great & small are relative terms, & those things are called insensible which are very small with respect to our own size & with regard to our senses. Therefore, when we consider absolute, & not relative, properties, whatever we perceive to be common to those contained within the limits that are sensible to us, we should consider these things to be still common to those beyond those limits. For these limits, with regard to such matters as are self-contained, are accidental ; & thus, if there should be any violation of the analogy, this would be far more likely to happen between the limits sensible to us, which are more open, than beyond them, where indeed they are so nearly nothing. Because then none did happen thus, it is a sign that there is none. This sign is not evident, but belongs to the principles of investigation, which generally proves successful if it is carried out in accordance with certain definite wisely 5 8 PHILOSOPHIC NATURALIS THEORIA quasdam prudentes regulas fiat, successum habere solet. Cum id indicium fallere possit ; fieri potest, ut committatur error, sed contra ipsum errorem habebitur praesumptio, ut etiam in jure appellant, donee positiva ratione evincatur oppositum. Hinc addendum fuit, nisi ratio positiva obstet. Sic contra hasce regulas peccaret, qui diceret, corpora quidem magna compenetrari, ac replicari, & inertia carere non posse, compenetrari tamen posse, vel replicari, vel sine inertia esse exiguas eorum partes. At si proprietas sit respectiva, respectu nostrorum sensuum, ex [19] eo, quod habeatur in majoribus massis, non debemus inferre, earn haberi in particulis minoribus, ut est hoc ipsum, esse sensibile, ut est, esse coloratas, quod ipsis majoribus massis competit, minoribus non competit ; cum ejusmodi magnitudinis discrimen, accidentale respectu materiae, non sit accidentale respectu ejus denominationis sensibile, coloratum. Sic etiam siqua proprietas ita pendet a ratione aggregati, vel totius, ut ab ea separari non possit ; nee ea, ob rationem nimirum eandem, a toto, vel aggregate debet transferri ad partes. Est de ratione totius, ut partes habeat, nee totum sine partibus haberi potest. Est de ratione figurabilis, & extensi, ut habeat aliquid, quod ab alio distet, adeoque, ut habeat partes ; hinc eae proprietates, licet in quovis aggregate particularum materiae, sive in quavis sensibili massa, inveniantur, non debent inductionis vi transferri ad particulas quascunque." Et impenetrabili- 41. Ex his patet, & impenetrabilitatem, & continuitatis legem per ejusmodi inductionis ultatem tvtad""pCT genus abunde probari, atque evinci, & illam quidem ad quascunque utcunque exiguas inductionem : "* ad particulas corporum, hanc ad gradus utcunque exiguos momento temporis adjectos debere ipsam quid requu-a- exten( jj < Requiritur autem ad hujusmodi inductionem primo, ut ilia proprietas, ad quam probandam ea adhibetur, in plurimis casibus observetur, aliter enim probabilitas esset exigua ; & ut nullus sit casus observatus, in quo evinci possit, earn violari. Non est necessarium illud, ut in iis casibus, in quibus primo aspectu timeri possit defectus proprietatis ipsius, positive demonstretur, earn non deficere ; satis est, si pro iis casibus haberi possit ratio aliqua conciliandi observationem cum ipsa proprietate, & id multo magis, si in aliis casibus habeatur ejus conciliationis exemplum, & positive ostendi possit, eo ipso modo fieri aliquando conciliationem. Ejus appiicatio ad 42. Id ipsum fit, ubi per inductionem impenetrabilitas corporum accipitur pro generali impenetrab;htatem. j e g e ]sj aturaEi Nam impenetrabilitatem ipsam magnorum corporum observamus in exemplis sane innumeris tot corporum, quae pertractamus. Habentur quidem & casus, in quibus earn violari quis credent, ut ubi oleum per ligna, & marmora penetrat, atque insinuatur, & ubi lux per vitra, & gemmas traducitur. At praesto est conciliatio phasnomeni cum impenetra- bilitate, petita ab eo, quod ilia corpora, in quse se ejusmodi substantiae insinuant, poros habeant, quos 633 permeent. Et quidem haec conciliatio exemplum habet manifestissimum in spongia, quae per poros ingentes aqua immissa imbuitur. Poros marmorum illorum, & multo magis vitrorum, non videmus, ac multo minus videre possumus illud, non insinuari eas substantias nisi per poros. Hoc satis est reliquae inductionis vi, ut dicere debeamus, eo potissimum pacto se rem habere, & ne ibi quidem violari generalem utique impenetrabilitatis legem. simiiisad continu- [ 2O ] 43- Eodem igitur pacto in lege ipsa continuitatis agendum est. Ilia tarn ampla itatem : duo cas- inductio, quam habemus, debet nos movere ad illam generaliter admittendam etiam pro iis quibus ea n< videatur casibus, in quibus determinare immediate per observations non possumus, an eadem lacdi - habeatur, uti est collisio corporum ; ac si sunt casus nonnulli, in quibus eadem prima fronte violari videatur ; ineunda est ratio aliqua, qua ipsum phsenomenum cum ea lege conciliari possit, uti revera potest. Nonnullos ejusmodi casus protuli in memoratis dissertationibus, quorum alii ad geometricam continuitatem pertinent, alii ad physicam. In illis prioribus non immorabor ; neque enim geometrica continuitas necessaria est ad hanc physicam propugnandam, sed earn ut exemplum quoddam ad confirmationem quandam inductionis majoris adhibui. Posterior, ut saepe & ilia prior, ad duas classes reducitur ; altera est eorum casuum, in quibus saltus videtur committi idcirco, quia nos per saltum omittimus intermedias quantitates : rem exemplo geometrico illustro, cui physicum adjicio. A THEORY OF NATURAL PHILOSOPHY 59 chosen rules. Now, since the indication may possibly be fallacious, it may happen that an error may be made ; but there is presumption against such an error, as they call it in law, until direct evidence to the contrary can be brought forward. Hence we should add : unless some positive argument is against it. Thus, it would be offending against these rules to say that large bodies indeed could not suffer compenetration, or enfolding, or be deficient in inertia, but yet very small parts of them could suffer penetration, or enfolding, or be without inertia. On the other hand, if a property is relative with respect to our senses, then, from a result obtained for the larger masses we cannot infer that the same is to be obtained in its smaller particles ; for instance, that it is the same thing to be sensible, as it is to be coloured, which is true in the case of large masses, but not in the case of small particles ; since a distinction of this kind, accidental with respect to matter, is not accidental with respect to the term sensible or coloured. So also if any property depends on an argu- ment referring to an aggregate, or a whole, in such a way that it cannot be considered apart from the whole, or the aggregate ; then, neither must it (that is to say, by that same argument), be transferred from the whole, or the aggregate, to parts of it. It is on account of its being a whole that it has parts ; nor can there be a whole without parts. It is on account of its being figurable & extended that it has some thing that is apart from some other thing, & therefore that it has parts. Hence those properties, although they are found in any aggregate of particles of matter, or in any sensible mass, must not however be transferred by the power of induction to each & every particle." 41. From what has been said it is quite evident that both impenetrability & the Law Both impenetra- of Continuity can be proved by a kind of induction of this type ; & the former must be c^bf dTm""^ extended to all particles of bodies, no matter how small, & the latter to all additional steps, strated by indue- however small, made in an instant of time. Now, in the first place, to use this kind of quired for this S pur- induction, it is required that the property, for the proof of which it is to be used, must be pose. observed in a very large number of cases ; for otherwise the probability would be very small. Also it is required that no case should be observed, in which it can be proved that it is violated. It is not necessary that, in those cases in which at first sight it is feared that there may be a failure of the property, that it should be directly proved that there is no failure. It is sufficient if in those cases some reason can be obtained which will make the observation agree with the property ; & all the more so, if in other cases an example of reconciliation can be obtained, & it can be positively proved that sometimes reconciliation can be obtained in that way. 42. This is just what does happen, when the impenetrability of solid bodies is accepted Application of in- as a law of Nature through inductive reasoning. For we observe this impenetrability of tr'abiuty.* lmpene " large bodies in innumerable examples of the many bodies that we consider. There are indeed also cases, in which one would think that it was violated, such as when oil penetrates wood and marble, & works its way through them, or when light passes through glasses & gems. But we have ready a means of making these phenomena agree with impenetrability, derived from the fact that those bodies, into which substances of this kind work their way, possess pores which they can permeate. There is a very evident example of this recon- ciliation in a sponge, which is saturated with water introduced into it by means of huge pores. We do not see the pores of the marble, still less those of glass ; & far less can we see that these substances do not penetrate except by pores. It satisfies the general force of induction if we can say that the matter can be explained in this way better than in any other, & that in this case there is absolutely no contradiction of the general law of impene- trability. 43. In the same way, then, we must deal with the Law of Continuity. The full Similar application " induction that we possess should lead us to admit in general this law even in those cases in ^sisses "oT cases which it is impossible for us to determine directly by observation whether the same law which there seems holds good, as for instance in the collision of bodies. Also, if there are some cases in which * the law at first sight seems to be violated, some method must be followed, through which each phenomenon can be reconciled with the law, as is in every case possible. I brought forward several cases of this kind in the dissertations I have mentioned, some of which pertained to geometrical continuity, & others to physical continuity. I will not delay over the first of these : for geometrical continuity is not necessary for the defence of the physical variety ; I used it as an example in confirmation of a wider induction. The latter, as well as very frequently the former, reduces to two classes ; & the first of these classes is that class in which a sudden change seems to have been made on account of our having omitted the intermediate quantities with a jump. I give a geometrical illustration, and then add one in physics. 6o PHILOSOPHISE NATURALIS THEORIA Exemplum geome- tricum primi gene- ris, ubi nos inter- mcdias magnitu- dines omittimus. Quando id accidat exempla physica dierum, & oscilla- tionum consequen- tium. 44. In axe curvae cujusdam in fig. 4. sumantur segmenta AC, CE, EG aequalia, & erigantur ordinatae AB, CD, EF, GH. Area; BACD, DCEF, FEGH videntur continue cujusdam seriei termini ita, ut ab ilia BACD acl DCEF, & inde ad FEGH immediate transeatur, & tamen secunda a prima, ut & tertia a secunda, differunt per quanti- tates finitas : si enim capiantur CI, EK sequales BA, DC, & arcus BD transferatur in IK ; area DIKF erit incrementum se- cundae supra primam, quod videtur imme- diate advenire totum absque eo, quod unquam habitum sit ejus dimidium, vel quaevis alia pars incrementi ipsius ; ut idcirco a prima ad secundam magnitudinem areae itum sit sine transitu per intermedias. At ibi omittuntur a nobis termini intermedii, qui continuitatem servant ; si enim ac aequalis FIG. 4. AC motu continue feratur ita, ut incipiendo ab AC desinat in CE ; magnitude areae BACD per omnes intermedias bacd abit in magnitu- dinem DCEF sine ullo saltu, & sine ulla violatione continuitatis. 45. Id sane ubique accidit, ubi initium secundae magnitudinis aliquo intervallo distal ab initio primas ; sive statim veniat post ejus finem, sive qua vis alia lege ab ea disjungatur. Sic in pliysicis, si diem concipiamus intervallum temporis ab occasu ad occasum, vel etiam ab ortu ad occasum, dies praecedens a sequent! quibusdam anni temporibus differt per plura secunda, ubi videtur fieri saltus sine ullo intermedio die, qui minus differat. At seriem quidem continuam ii dies nequaquam constituunt. Concipiatur parallelus integer Telluris, in quo sunt continuo ductu disposita loca omnia, quae eandem latitudinem geographicam habent ; ea singula loca suam habent durationem diei, & omnium ejusmodi dierum initia, ac fines continenter fluunt ; donee ad eundem redeatur locum, cujus pre-[2i]-cedens dies est in continua ilia serie primus, & sequens postremus. Illorum omnium dierum magni- tudines continenter fluunt sine ullo saltu : nos, intermediis omissis, saltum committimus non Natura. Atque huic similis responsio est ad omnes reliquos casus ejusmodi, in quibus initia, & fines continenter non fluunt, sed a nobis per saltum accipiuntur. Sic ubi pendulum oscillat in acre ; sequens oscillatio per finitam magnitudinem distat a praecedente ; sed & initium & finis ejus finite intervallo temporis distat a prascedentis initio, & fine, ac intermedii termini continua serie fluente a prima oscillatione ad secundam essent ii, qui haberentur, si primae, & secundae oscillationis arcu in aequalem partium numerum diviso, assumeretur via confecta, vel tempus in ea impensum, inter jacens inter fines partium omnium proportion- alium, ut inter trientem, vel quadrantem prioris arcus, & trientem,vel quadrantem posterioris, quod ad omnes ejus generis casus facile transferri potest, in quibus semper immediate etiam demonstrari potest illud, continuitatem nequaquam violari. Exempla secundi 46. Secunda classis casuum est ea, in qua videtur aliquid momento temporis peragi, at ne iOTime UtS sed & tamen peragitur tempore successive, sed perbrevi. Sunt, qui objiciant pro violatione non momento' tem- continuitatis casum, quo quisquam manu lapidem tenens, ipsi statim det velocitatem quandam finitam : alius objicit aquae e vase effluentis, foramine constitute aliquanto infra superficiem ipsius aquae, velocitatem oriri momento temporis finitam. At in priore casu admodum evidens est, momento temporis velocitatem finitam nequaquam produci. Tempore opus est, utcunque brevissimo, ad excursum spirituum per nervos, & musculos, ad fibrarum tensionem, & alia ejusmodi : ac idcirco ut velocitatem aliquam sensibilem demus lapidi, manum retrahimus, & ipsum aliquandiu, perpetuo accelerantes, retinemus. Sic etiam, ubi tormentum bellicum exploditur, videtur momento temporis emitti globus, ac totam celeritatem acquirere ; at id successive fieri, patet vel inde, quod debeat inflammari tota massa pulveris pyrii, & dilatari aer, ut elasticitate sua globum acceleret, quod quidem fit omnino per omnes gradus. Successionem multo etiam melius videmus in globe, qui ab elastro sibi relicto propellatur : quo elasticitas est major, eo citius, sed nunquam momento temporis velocitas in globum inducitur. AppUcatio ipsorum 47. Hsec exempla illud praestant, quod aqua per pores spongiae ingressa respectu ad emuxum 1I aquK impenetrabilitatis, ut ea responsione uti possimus in aliis casibus omnibus, in quibus accessio e vase. aliqua magnitudinis videtur fieri tota momento temporis ; ut nimirum dicamus fieri tempore A THEORY OF NATURAL PHILOSOPHY 61 44. In the axis of any curve (Fig. 4) let there be taken the segments AC, CE, EG equal Geometrical ex- to one another ; & let the ordinates AB, CD, EF, GH be erected. The areas BACD, DCEF, kind where 6 Ivl FEGH seem to be terms of some continuous series such that we can pass directly from BACD omit . intermediate to DCEF and then on to FEGH, & yet the second differs from the first, & also the third from the second, by a finite quantity. For if CI, EK are taken equal to BA, DC, & the arc BD is transferred to the position IK ; then the area DIKF will be the increment of the second area beyond the first ; & this seems to be directly arrived at as a whole without that which at any one time is considered to be the half of it, or indeed any other part of the increment itself : so that, in consequence, we go from the first to the second magnitude of area without passing through intermediate magnitudes. But in this case we omit intermediate terms which maintain the continuity ; for if ac is equal to AC, & this is carried by a continuous motion in such a way that, starting from the position AC it ends up at the position CE, then the magnitude of the area BACD will pass through all intermediate values such as bacd until it reaches the magnitude of the area DCEF without any sudden change, & hence without any breach of continuity. 45. Indeed this always happens when the beginning of the second magnitude is distant when this will by a definite interval from the beginning of the first ; whether it comes immediately after ha ppen = physical the end of the first or is disconnected from it by some other law. Thus in physics, if we casTof Consecutive look upon the day as the interval of time between sunset & sunset, or even between sunrise da y^ OI . consecutive & sunset, the preceding day differs from that which follows it at certain times of the year by several seconds ; in which case we see that there is a sudden change made, without there being any intermediate day for which the change is less. But the fact is that these days do not constitute a continuous series. Let us consider a complete parallel of latitude on the Earth, along which in a continuous sequence are situated all those places that have the same geographical latitude. Each of these places has its own duration of the day, & the begin- nings & ends of days of this kind change uninterruptedly ; until we get back again to the same place, where the preceding day is the first of that continuous series, & the day that fol- lows is the last of the series. The magnitudes of all these days continuously alter without there being any sudden change : it was we who, by omitting the intermediates, made the sudden change, & not Nature. Similar to this is the answer to all the rest of the cases of the same kind, in which the beginnings & the ends do not change uninterruptedly, but are observed by us discontinuously. Similarly, when a pendulum oscillates in air, the oscillation that follows differs from the oscillation that has gone before by a finite magnitude. But both the begin- ning & the end of the second differs from the beginning & the end of the first by a finite inter- val of time ; & the intermediate terms in a continuously varying series from the first oscillation to the second would be those that would be obtained, if the arcs of the first & second oscilla- tions were each divided into the same number of equal parts, & the path traversed (or the time spent in traversing the path) is taken between the ends of all these proportional paths ; such as that between the third or fourth part of the first arc & the third or fourth part of the second arc. This argument can be easily transferred so as to apply to all cases of this kind ; & in such cases it can always be directly proved that there is no breach of continuity. 46. The second class of cases is that in which something seems to have been done in an Examples of the instant of time, but still it is really done in a continuous, but very short, interval of time. ^iS? the^chan'e There are some who bring forward, as an objection in favour of a breach of continuity, the is very rapid, but case in which a man, holding a stone in his hand, gives to it a definite velocity all at once ; f^an^nstant^of another raises an objection that favours a breach of continuity, in the case of water flowing time. from a vessel, where, if an opening is made below the level of the surface of the water, a finite velocity is produced in an instant of time. But in the first case it is perfectly clear that a finite velocity is in no wise produced in an instant of time. For there is need of time, although this is exceedingly short, for the passage of cerebral impulses through the nerves and muscles, for the tension of the fibres, and other things of that sort ; and therefore, in order to give a definite sensible velocity to the stone, we draw back the hand, and then retain the stone in it for some time as we continually increase its velocity forwards. So too when an engine of war is exploded, the ball seems to be driven forth and to acquire the whole of its speed in an instant of time. But that it is done continuously is clear, if only from the fact that the whole mass of the gunpowder has to be inflamed and the gas has to be expanded in order that it may accelerate the ball by its elasticity ; and this latter certainly takes place by degrees. The continuous nature of this is far better seen in the case of a ball propelled by releasing a spring ; here the stronger the elasticity, the greater the speed ; but in no case is the speed imparted to the ball in an instant of time. 47. These examples are superior to that of water entering through the pores of a sponge, Application of which we employed in the matter of impenetrability ; so that we can make use of this reply *.g s s . e particularly in all other cases in which some addition to a magnitude seems to have taken place entirely in to the flow of water an instant of time. Thus, without doubt we may say that it takes place in an exceedingly from a vesse1 ' 62 PHILOSOPHIC NATURALIS THEORIA brevissimo, utique per omnes intermedias magnitudines, ac illsesa penitus lege continuitatis. Hinc & in aquae effluentis exemplo res eodem redit, ut non unico momento, sed successive aliquo tempore, & per [22] omnes intermedias magnitudines progignatur velocitas, quod quidem ita se habere optimi quique Physici affirmant. Et ibi quidem, qui momento temporis omnem illam velocitatem progigni, contra me affirmet, principium utique, ut ajunt, petat, necesse est. Neque enim aqua, nisi foramen aperiatur, operculo dimoto, effluet ; remotio vero operculi, sive manu fiat, sive percussione aliqua, non potest fieri momento temporis, sed debet velocitatem suam acquirere per omnes gradus ; nisi illud ipsum, quod quaerimus, supponatur jam definitum, nimirum an in collisione corporum communicatio motus fiat momento temporis, an per omnes intermedios gradus, & magni- tudines. Verum eo omisso, si etiam concipiamus momento temporis impedimentum auferri, non idcirco momento itidem temporis omnis ilia velocitas produceretur ; ilia enim non a percussione aliqua, sed a pressione superincumbentis aquae orta, oriri utique non potest, nisi per accessiones continuas tempusculo admodum parvo, sed non omnino nullo : nam pressio tempore indiget, ut velocitatem progignat, in communi omnium sententia. Transitus ad meta- continuis ut in Geometria. 48. Illaesa igitur esse debet continuitatis lex, nee ad earn evertendam contra inductionem, tam uberem quidquam poterunt casus allati hucusque, vel iis similes. At ejusdem con- umcus, tinuitatis aliam metaphysicam rationem adinveni, & proposui in dissertatione De Lege Continuitatis, petitam ab ipsa continuitatis natura, in qua quod Aristoteles ipse olim notaverat, communis esse debet limes, qui praecedentia cum consequentibus conjungit, qui idcirco etiam indivisibilis est in ea ratione, in qua est limes. Sic superficies duo solida dirimens & crassitudine caret, & est unica, in qua immediatus ab una parte fit transitus ad aliam ; linea dirimens binas superficiei continuae partes latitudine caret ; punctum continuae lineae segmenta discriminans, dimensione omni : nee duo sunt puncta contigua, quorum alterum sit finis prioris segmenti, alterum initium sequentis, cum duo contigua indivisibilia, & inextensa haberi non possint sine compenetratione, & coalescentia quadam in unum. idem in tempore 49. Eodem autem pacto idem debet accidere etiam in tempore, ut nimirum inter tempus > ti ua^'evide" 6 contmuum praecedens, & continuo subsequens unicum habeatur momentum, quod sit tius in quibusdam. indivisibilis terminus utriusque ; nee duo momenta, uti supra innuimus, contigua esse possint, sed inter quodvis momentum, & aliud momentum debeat intercedere semper continuum aliquod tempus divisibile in infinitum. Et eodem pacto in quavis quantitate, quae continuo tempore duret, haberi debet series quasdam magnitudinum ejusmodi, ut momento temporis cuivis respondeat sua, quae praecedentem cum consequente conjungat, & ab ilia per aliquam determinatam magnitudinem differat. Quin immo in illo quantitatum genere, in quo [23] binae magnitudines simul haberi non possunt, id ipsum multo evidentius conficitur, nempe nullum haberi posse saltum immediatum ab una ad alteram. Nam illo momento temporis, quo deberet saltus fieri, & abrumpi series accessu aliquo momentaneo, deberent haberi duae magnitudines, postrema seriei praecedentis, & prima seriei sequentis. Id ipsum vero adhuc multo evidentius habetur in illis rerum statibus, in quibus ex una parte quovis momento haberi debet aliquis status ita, ut nunquam sine aliquo ejus generis statu res esse possit ; & ex alia duos simul ejusmodi status habere non potest. inde cur motus ip- r o> \& quidem satis patebit in ipso locali motu, in quo habetur phsenomenum omnibus calls non fiat, nisi > . . * , . r r . ...... \ ,. . , . . per Hneam contin- sane notissimum, sed cujus ratio non ita facile ahunde redditur, inde autem patentissima est, Corpus a quovis loco ad alium quemvis devenire utique potest motu continuo per lineas quascunque utcunque contortas, & in immensum productas quaquaversum, quae numero infinities infinitae sunt : sed omnino debet per continuam aliquam abire, & nullibi inter- ruptam. En inde rationem ejus rei admodum manifestam. Si alicubi linea motus abrum- peretur ; vel momentum temporis, quo esset in primo puncto posterioris lineae, esset posterius eo momento, quo esset in puncto postremo anterioris, vel esset idem, vel anterius ? In primo, & tertio casu inter ea momenta intercederet tempus aliquod continuum divisibile in infinitum per alia momenta intermedia, cum bina momenta temporis, in eo sensu accepta, in quo ego hie ea accipio, contigua esse non possint, uti superiusexposui. Quamobrem in A THEORY OF NATURAL PHILOSOPHY 63 short interval of time, and certainly passes through every intermediate magnitude, and that the Law of Continuity is not violated. Hence also in the case of water flowing from a vessel it reduces to the same example : so that the velocity is generated, not in a single instant, but in some continuous interval of time, and passes through all intermediate magni- tudes ; and indeed all the most noted physicists assert that this is what really happens. Also in this matter, should anyone assert in opposition to me that the whole of the speed is produced in an instant of time, then he must use a petitio principii, as they call it. For the water can-not flow out, unless the hole is opened, & the lid removed ; & the removal of the lid, whether done by hand or by a blow, cannot be effected in an instant of time, but must acquire its own velocity by degrees ; unless we suppose that the matter under investi- gation is already decided, that is to say, whether in collision of bodies communication of motion takes place in an instant of time or through all intermediate degrees and magnitudes. But even if that is left out of account, & if also we assume that the barrier is removed in an instant of time, none the more on that account would the whole of the velocity also be produced in an instant of time ; for it is impossible that such velocity can arise, not from some blow, but from a pressure arising from the superincumbent water, except by continuous additions in a very short interval of time, which is however not absolutely nothing ; for pressure requires time to produce velocity, according to the general opinion of everybody. 48. The Law of Continuity ought then to be subject to no breach, nor will the cases Passing to a meta- hitherto brought forward, nor others like them, have any power at all to controvert this haveT'smrie'iinUt law in opposition to induction so copious. Moreover I discovered another argument, a in the case of con- metaphysical one, in favour of this continuity, & published it in my dissertation De Lege g'^n^iy 1 " 11 ^' &S "* Continuitatis, having derived it from the very nature of continuity ; as Aristotle himself long ago remarked, there must be a common boundary which joins the things that precede to those that follow ; & this must therefore be indivisible for the very reason that it is a boundary. In the same way, a surface of separation of two solids is also without thickness & is single, & in it there is immediate passage from one side to the other ; the line of separation of two parts of a continuous surface lacks any breadth ; a point determining segments of a continuous line has no dimension at all ; nor are there two contiguous points, one of which is the end of the first segment, & the other the beginning of the next ; for two contiguous indivisibles, of no extent, cannot possibly be considered to exist, unless there is compenetration & a coalescence into one. 49. In the same way, this should also happen with regard to time, namely, that between similarly for time a preceding continuous time & the next following there should be a single instant, which ^ y . mor^evi- is the indivisible boundary of either. There cannot be two instants, as we intimated above, dent in some than contiguous to one another ; but between one instant & another there must always intervene m others - some interval of continuous time divisible indefinitely. In the same way, in any quantity which lasts for a continuous interval of time, there must be obtained a series of magnitudes of such a kind that to each instant of time there is its corresponding magnitude ; & this magnitude connects the one that precedes with the one that follows it, & differs from the former by some definite magnitude. Nay even in that class of quantities, in which we cannot have two magnitudes at the same time, this very point can be deduced far more clearly, namely, that there cannot be any sudden change from one to another. For at that instant, when the sudden change should take place, & the series be broken by some momen- tary definite addition, two magnitudes would necessarily be obtained, namely, the last of the first series & the first of the next. Now this very point is still more clearly seen in those states of things, in which on the one hand there must be at any instant some state so that at no time can the thing be without some state of the kind, whilst on the other hand it can never have two states of the kind simultaneously. 50. The above will be sufficiently clear in the case of local motion, in regard to which Hence the reason the phenomenon is perfectly well known to all ; the reason for it, however, is not so easily ^Jj^ Recurs;:! 10 " derived from any other source, whilst it follows most clearly from this idea. A body can continuous line, get from any one position to any other position in any case by a continuous motion along any line whatever, no matter how contorted, or produced ever so far in any direction ; these lines being infinitely infinite in number. But it is bound to travel by some continuous line, with no break in it at any point. Here then is the reason of this phenomenon quite clearly explained. If the motion in the line should be broken at any point, either the instant of time, at which it was at the first point of the second part of the line, would be after the instant, at which it was at the last point of the first part of the line, or it would be the same instant, or before it. In the first & third cases, there would intervene between the two instants some definite interval of continuous time divisible indefinitely at other intermediate instants ; for two instants of time, considered in the sense in which I have PHILOSOPHIC NATURALIS THEORIA primo casu in omnibus iis infinitis intermediis momentis nullibi esset id corpus, in secundo casu idem esset eodem illo memento in binis locis, adeoque replicaretur ; in terio haberetur replicatio non tantum respectu eorum binorum momentorum, sed omnium etiam inter- mediorum, in quibus nimirum omnibus id corpus esset in binis locis. Cum igitur corpus existens nee nullibi esse possit, nee simul in locis pluribus ; ilia vias mutatio, & ille saltus haberi omnino non possunt. 51. Idem ope Geometric magis adhuc oculis ipsis subjicitur. Exponantur per rectam AB tempora, ac per ordinatas ad lineas CD, EF, abruptas alicubi, diversi status rei cujuspiam. e metaphysica, Ductis ordinatis DG, EH, vel punctum H iaceret post G, ut in Fie. c : vel cum ipso ibus exemphs . / i i j . r T . o J r congrueret, ut in 6 ; vel ipsum prsccederet, ut in 7. In pnmo casu nulla responderet ordinata omnibus punctis rectae GH ; in secundo binae responderent GD, & HE eidem puncto G ; in tertio vero binae HI, & HE puncto H, binas GD, GK puncto G, & binae LM, LN Illustratio ejus i ex Geo- ratiocina- tione pluribus exempl D E. D G H FIG. 5. B A GH FIG. 6. H L G FIG. 7. puncto cuivis intermedio L ; nam ordinata est relatio quaedam distantly, quam habet punctum curvae cum puncto axis sibi respondente, adeoque ubi jacent in recta eadem perpendiculari axi bina curvarum puncta, habentur binae ordinatae respondentes eidem puncto axis. Quamobrem si nee o-[24]-mni statu carere res possit, nee haberi possint status simul bini ; necessario consequitur, saltum ilium committi non posse. Saltus ipse, si deberet accidere, uti vulgo fieri concipitur, accideret binis momentis G, & H, quae sibi in fig. 6 immediate succederent sine ullo immediato hiatu, quod utique fieri non potest ex ipsa limitis ratione, qui in continuis debet esse idem, & antecedentibus, & consequentibus communis, uti diximus. Atque idem in quavis reali serie accidit ; ut hie linea finita sine puncto primo, & postremo, quod sit ejus limes, & superficies sine linea esse non potest ; unde fit, ut in casu figurae 6 binae ordinatae necessario respondere debeant eidem puncto : ita in quavis finita reali serie statuum primus terminus, & postremus haberi necessario debent ; adeoque si saltus fit, uti supra de loco diximus ; debet eo momento, quo saltus confici dicitur, haberi simul status duplex ; qui cum haberi non possit : saltus itidem ille haberi omnino non potest. Sic, ut aliis utamur exemplis, distantia unius corporis ab alio mutari per saltum non potest, nee densitas, quia dux simul haberentur distantiae, vel duae densitates, quod utique sine replicatione haberi non potest ; caloris itidem, & frigoris mutatio in thermometris, ponderis atmosphaerae mutatio in barometris, non fit per saltum, quia binae simul altitudines mercurii in instrumento haberi deberent eodem momento temporis, quod fieri utique non potest ; cum quovis momento determinate unica altitude haberi debeat, ac unicus determinatus caloris gradus, vel frigoris ; quae quidem theoria innumeris casibus pariter aptari potest. 52. Contra hoc argumentum videtur primo aspectu adesse aliquid, quod ipsum pforsus non esse conjun- evertat, & tamen ipsi illustrando idoneum est maxime. Videtur nimirum inde erui, gend s in creatione M o J & annihiiatione, ac impossibilem esse & creationem rei cujuspiam, Scintentum. 01 enim conjungendus est ejus soiutio. postremus terminus praecedentis seriei cum primo sequentis ;" in ipso transitu a non esse ad esse, vel vice versa, debebit utrumque conjungi, ac idem simul erit, & non erit, quod est absurdum. Responsio in promptu est. Seriei finita; realis, & existentis, reales itidem, & existentes termini esse debent ; non vero nihili, quod nullas proprietates habet, quas exigat, Hinc si realium statuum seriei altera series realium itidem statuum succedat, quae non sit communi termino conjuncta ; bini eodem momento debebuntur status, qui nimirum sint bini limites earundem. At quoniam non esse est merum nihilum ; ejusmodi series limitem nullum extremum requirit, sed per ipsum esse immediate, & directe excluditur. Quamobrem primo, & postremo momento temporis ejus continui, quo res est, erit utique, nee cum hoc esse suum non esse conjunget simul ; at si densitas certa per horam duret, turn momento temporis in aliam mutetur duplam, duraturam itidem per alteram sequentem horam ; momento temporis, [25] quod horas dirimit, binae debebunt esse densitates simul, nimirum & simplex, & dupla, quae sunt reales binarum realium serierum termini. Objectio ab esse, & A THEORY OF NATURAL PHILOSOPHY 65 considered them, cannot be contiguous, as I explained above. Wherefore in the first case, at all those infinite intermediate instants the body would be nowhere at all ; in the second case, it would be at the same instant in two different places & so there would be replication. In the third case, there would not only occur replication in respect of these two instants but for all those intermediate to them as well, in all of which the body would forsooth be in two places at the same time. Since then a body that exists can never be nowhere, nor in several places at one & the same time, there can certainly be no alteration of path & no sudden change. 51. The same thing can be visualized better with the aid of Geometry. illustration of this Let times be represented by the straight line AB, & diverse states of any thing by SSyT^STS ordinates drawn to meet the lines CD, EF, which are discontinuous at some point. If the reasoning being ordmates DG, EH are drawn, either the point H will fall after the point G, as in Fig. 5 ; or it will coincide with it, as in Fig. 6 ; or it will fall before it, as in Fig. 7. In the first case, no ordinate will correspond to any one of the points of the straight line GH ; in the second case, GD and HE would correspond to the same point G ; in the third case, two ordinates, HI, HE, would correspond to the same point H, two, GD, GK, to the same point G, and two, LM, LN, to any intermediate point L. Now the ordinate is some relation as regards distance, which a point on the curve bears to the point on the axis that corresponds with it ; & thus, when two points of the curve lie in the same straight line perpendicular to the axis, we have two ordinates corresponding to the same point of the axis. Wherefore, if the thing in question can neither be without some state at each instant, nor is it possible that there should be two states at the same time, then it necessarily follows that the sudden change cannot be made. For this sudden change, if it is bound to happen, would take place at the two instants G & H, which immediately succeed the one the other without any direct gap between them ; this is quite impossible, from the very nature of a limit, which should be the same for,& common to, both the antecedents & the consequents in a continuous set, as has been said. The same thing happens in any series of real things ; as in this case there cannot be a finite line without a first & last point, each to be a boundary to it, neither can there be a surface without a line. Hence it comes about that in the case of Fig. 6 two ordinates must necessarily correspond to the same point. Thus, in any finite real series of states, there must of necessity be a first term & a last ; & so if a sudden change is made, as we said above with regard to position, there must be at the instant, at which the sudden change is said to be accomplished, a twofold state at one & the same time. Now since this can never happen, it follows that this sudden change is also quite impossible. Similarly, to make use of other illustrations, the distance of one body from another can never be altered suddenly, no more can its density ; for there would be at one & the same time two distances, or two densities, a thing which is quite impossible without replication. Again, the change of heat, or cold, in thermometers, the change in the weight of the air in barometers, does not happen suddenly ; for then there would necessarily be at one & the same time two different heights for the mercury in the instrument ; & this could not possibly be the case. For at any given instant there must be but one height, & but one definite degree of heat, & but one definite degree of cold ; & this argument can be applied just as well to innu- merable other cases. 52. Against this argument it would seem at first sight that there is something ready to hand which overthrows it altogether ; whilst as a matter of fact it is peculiarly fitted to together of existence exemplify it. It seems that from this argument it follows that both the creation of any * non-existence a.t > i -11 rf >r T i < -i i the time of creation thing, & its destruction, are impossible, r or, it the last term of a series that precedes is to O r annihilation ; & be connected with the first term of the series that follows, then in the passage from a state its solution. of existence to one of non-existence, or vice versa, it will be necessary that the two are connected together ; & then at one & the same time the same thing will both exist & not exist, which is absurd. The answer to this is immediate. For the ends of a finite series that is real & existent must themselves be real & existent, not such as end up in absolute nothing, which has no properties. Hence, if to one series of real states there succeeds another series of real states also, which is not connected with it by a common term, then indeed there must be two states at the same instant, namely those which are their two limits. But since non-existence is mere nothing, a series of this kind requires no last limiting term, but is immediately & directly cut off by fact of existence. Wherefore, at the first & at the last instant of that continuous interval of time, during which the matter exists, it will certainly exist ; & its non-existence will not be connected with its existence simultaneously. On the other hand if a given density persists for an hour, & then is changed in an instant of time into another twice as great, which will last for another hour ; then in that instant of time which separates the two hours, there would have to be two densities at one & the same time, the simple & the double, & these are real terms of two real series. 66 PHILOSOPHIC NATURALIS THEORIA Unde hue transfer- enda solutio ipsa. Solutio petita ex geometrico exem- plo. Solutio physica atione. ex meta- consider- Illustratio ulterior geometrica. Applicatio ad crea- tionem, & annihi- lationem. D F i \ F D f m m* \ G G' P L 5 \ MJVI, ' A B C E H H'E'C 7 FIG. 8. 53. Id ipsum in dissertatione De lege virium in Natura existentium satis, ni fallor, luculenter exposui, ac geometricis figuris illustravi, adjectis nonnullis, quae eodem recidunt, & quae in applicatione ad rem, de qua agimus, & in cujus gratiam haec omnia ad legem con- tinuitatis pertinentia allata sunt, proderunt infra ; libet autem novem ejus dissertationis numeros hue transferre integros, incipiendo ab octavo, sed numeros ipsos, ut & schematum numeros mutabo hie, ut cum superioribus consentiant. 54. " Sit in fig. 8 circulus GMM'wz, qui referatur ad datam rectam AB per ordinatas HM ipsi rectae perpendiculares ; uti itidem perpendiculares sint binae tangentes EGF, E'G'F'. Concipiantur igitur recta quaedam indefinita ipsi rectse AB perpendicularis, motu quodam continuo delata ab A ad B. Ubi ea habuerit, positionem quamcumque GD, quae praecedat tangentem EF, vel C'D', quae consequatur tangentem E'F' ; ordinata ad circulum nulla erit, sive erit impossibilis, & ut Geometrae loquuntur, imaginaria. Ubicunque autem ea sit inter binas tangentes EGF, E'G'F', in HI, HT, occurret circulo in binis punctis M, m, vel M', m', & habebitur valor ordinate HM, HOT, vel H'M', H'm'. Ordinata quidem ipsa respondet soli inter- vallo EE' : & si ipsa linea AB referat tempus ; momentum E est limes inter tempus praecedens continuum AE, quo ordinata non est, & tempus continuum EE' subsequens, quo ordinata est ; punc- tum E' est limes inter tempus praecedens EE', quo ordinata est, & subsequens E'B, quo non est. Vita igitur quaedam ordinatae est tempus EE' ; ortus habetur in E, interitus in E'. Quid autem in ipso ortu, & interitu ? Habetur-ne quoddam esse ordinatas, an non esse ? Habetur utique esse, nimi- rum EG, vel E'G', non autem non esse. Oritur tota finitae magnitudinis ordinata EG, interit tota finite magnitudinis E'G', nee tamen ibi conjungit esse, & non esse, nee ullum absurdum secum trahit. Habetur momento E primus terminus seriei sequentis sine ultimo seriei praecedentis, & habetur momento E' ultimus terminus seriei praecedentis sine primo termino seriei sequentis." 55. " Quare autem id ipsum accidat, si metaphysica consideratione rem perpendimus, statim patebit. Nimirum veri nihili nullae sunt verae proprietates : entis realis verae, & reales proprietates sunt. Quaevis realis series initium reale debet, & finem, sive primum, & ultimum terminum. Id, quod non est, nullam habet veram proprietatem, nee proinde sui generis ultimum terminum, aut primum exigit. Series praecedens ordinatae nullius, ultimum terminum non [26] habet, series consequens non habet primum : series realis contenta intervallo EE', & primum habere debet, & ultimum. Hujus reales termini terminum ilium nihili per se se excludunt, cum ipsum esse per se excludat non esse." 56. " Atque id quidem manifestum fit magis : si consideremus seriem aliquam praecedentem realem, quam exprimant ordinatae ad lineam continuam PLg, quae respondeat toti tempori AE ita, ut cuivis momento C ejus temporis respondeat ordinata CL. Turn vero si momento E debeat fieri saltus ab ordinata Eg ad ordinatam EG : necessario ipsi momento E debent respondere binae ordinatae EG, Eg. Nam in tota linea PLg non potest deesse solum ultimum punctum g ; cum ipso sublato debeat adhuc ilia linea terminum habere suum, qui terminus esset itidem punctum : id vero punctum idcirco fuisset ante contiguum puncto g, quod est absurdum, ut in eadem dissertatione De Lege Continuitatis demonstravimus. Nam inter quodvis punctum, & aliud punctum linea aliqua interjacere debet ; quae si non inter jaceat ; jam ilia puncta in unicum coalescunt. Quare non potest deesse nisi lineola aliqua gL ita, ut terminus seriei praecedentis sit in aliquo momento C praecedente momentum E, & disjuncto ab eo per tempus quoddam continuum, in cujus temporis momentis omnibus ordi'nata sit nulla." 57. " Patet igitur discrimen inter transitum a vero nihilo, nimirum a quantitate imaginaria, ad esse, & transitum ab una magnitudine ad aliam. In primo casu terminus nihili non habetur ; habetur terminus uterque seriei veram habentis existentiam, & potest quantitas, cujus ea est series, oriri, vel occidere quantitate finita, ac per se excludere non esse. In secundo casu necessario haberi debet utriusque seriei terminus, alterius nimirum postre- mus, alterius primus. Quamobrem etiam in creatione, & in annihilatione potest quantitas oriri, vel interire magnitudine finita, & primum, ac ultimum esse erit quoddam esse, quod secum non conjunget una non esse. Contra vero ubi magnitude realis ab una quantitate ad A THEORY OF NATURAL PHILOSOPHY 67 c*. I explained this very point clearly enough, if I mistake not, in my dissertation The s " 166 from n i JIT- . ' . a T -11 j v i ... i A 'IT ^ which the solution D,? lege vmum in Natura existentium, & 1 illustrated it by geometrical figures ; also I made u to be borrowed. some additions that reduced to the same thing. These will appear below, as an application to the matter in question ; for the sake of which all these things relating to the Law of Continuity have been adduced. It is allowable for me to quote in this connection the whole of nine articles from that dissertation, beginning with Art. 8 ; but I will here change the numbering of the articles, & of the diagrams as well, so that they may agree with those already given. 54. " In Fig. 8, let GMM'm be a circle, referred to a given straight line AB as axis, by Sotoion derived means of ordinates HM drawn perpendicular to that straight line ; also let the two tan- exampief" gents EGF, E'G'F' be perpendiculars to the axis. Now suppose that an unlimited straight line perpendicular to the axis AB is carried with a continuous motion from A to B. When it reaches some such position as CD preceding the tangent EF, or as C'D' subsequent to the tangent E'F', there will be no ordinate to the circle, or it will be impossible &, as the geometricians call it, imaginary. Also, wherever it falls between the two tangents EGF, E'G'F', as at HI or HT, it will meet the circle in two points, M, m or M', m' ; & for the value of the ordinate there will be obtained HM & Hm, or H'M' & H'm'. Such an ordinate will correspond to the interval EE' only ; & if the line AB represents time, the instant E is the boundary between the preceding continuous time AE, in which the ordinate does not exist, the subsequent continuous time EE', in which the ordinate does exist. The point E' is the boundary between the preceding time EE', in which the ordinate does exist, & the subsequent time E'B, in which it does not ; the lifetime, as it were, of the ordinate, is EE' ; its production is at E & its destruction at E'. But what happens at this production & destruction ? Is it an existence of the ordinate, or a non-existence I Of a truth there is an existence, represented by EG & E'G', & not a non-existence. The whole ordinate EG of finite magnitude is produced, & the whole ordinate E'G' of finite magnitude is destroyed; & yet there is no connecting together of the states of existence & non-existence, nor does it bring in anything absurd in its train. At the instant E we get the first term of the sub- sequent series without the last term of the preceding series ; & at the instant E' we have the last term of the preceding series without the first term of the subsequent series." 55. " The reason why this should happen is immediately evident, if we consider the Sol tion from a matter metaphysically. Thus, to absolute nothing there belong no real properties ; but Sderatwn!* the properties of a real absolute entity are also real. Any real series must have a real beginning & end, or a first term & a last. That which does not exist can have no true property ; & on that account does not require a last term of its kind, or a first. The preceding series, in which there is no ordinate, does not have a last term ; & the subsequent series has likewise no first term ; whilst the real series contained within the interval EE' must have both a first term & a last term. The real terms of this series of themselves exclude the term of no value, since the fact of existence of itself excludes non-existence" 56. " This indeed will be still more evident, if we consider some preceding series of Further illustration i 11 i i i i T.T r i i by geometry. real quantities, expressed by the ordinates to the curved line PLg ; & let this curve correspond to the whole time AE in such a way that to every instant C of the time there corresponds an ordinate CL. Then, if at the instant E there is bound to be a sudden change from the ordinate Eg to the ordinate EG, to that instant E there must of necessity correspond both the ordinates EG, Eg. For it is impossible that in the whole line PLg the last point alone should be missing ; because, if that point is taken away, yet the line is Bound to have an end to it, & that end must also be a point ; hence that point would be before & contiguous to the point g ; & this is absurd, as we have shown in the same dissertation De Lege Continuitatis. For between any one point & any other point there must lie some line ; & if such a line does not intervene, then those points must coalesce into one. Hence nothing can be absent, except it be a short length of line gL, so that the end of the series that precedes occurs at some instant, C, preceding the instant E, & separated from it by an interval of continuous time, at all instants of which there is no ordinate." 157. "Evidently, then, there is a distinction between passing from absolute nothing, Application to crea- f' . '' . ... . , . Y tion& annihilation. i.e., from an imaginary quantity, to a state of existence, & passing from one magnitude to another. In the first case the term which is naught is not reckoned in ; the term at either end of a series which has real existence is given, & the quantity, of which it is the series, can be produced or destroyed, finite in amount ; & of itself it will exclude non- existence. In the second case, there must of necessity be an end to either series, namely the last of the one series & the first of the other. Hence, in creation & annihilation, a quantity can be produced or destroyed, finite in magnitude ; & the first & last state of existence will be a state of existence of some kind ; & this will not associate with itself a state of non-existence. But, on the other hand, where a real magnitude is bound 68 PHILOSOPHIC NATURALIS THEORIA Aliquando videri nihtium id, quod est aliquid. Ordinatam nullam, ut & distantiam nullam existentium esse compenetra- tionem. Ad idem pertinere seriei realis genus earn distan t i a m nullam, & aliquam. Alia, quje videntur nihil, & sunt ali- quid : discrimen inter radicem ima- ginariam, & zero. aliam transire debet per saltum ; momento temporis, quo saltus committitur, uterque terminus haberi deberet. Manet igitur illaesum argumentum nostrum metaphysicum pro exclusione saltus a creatione & annihilatione, sive ortu, & interitu." 58. "At hie illud etiam notandum est ; quoniam ad ortum, & interitum considerandum geometricas contemplationes assumpsimus, videri quidem prima fronte, aliquando etiam realis seriei terminum postremum esse nihilum ; sed re altius considerata, non erit vere nihilum ; sed status quidam itidem realis, & ejusdem generis cum prsecedentibus, licet alio nomine insignitus." [27] 59. " Sit in Fig. 9. Linea AB, ut prius, ad quam linea qusedam PL deveniat in G (pertinet punctum G ad lineam PL, E ad AB continuatas, & sibi occurrentes ibidem), & sive pergat ultra ipsam in GM', sive retro resiliat per GM'. Recta CD habebit ordinatam CL, quas evanescet, ubi puncto C abeunte in E, ipsa CD abibit in EF, turn in positione ulteriori rectse perpendicularis HI, vel abibit in nega- tivam HM, vel retro positiva regredietur in HM'. Ubi linea altera cum altera coit, & punctum E alterius cum alterius puncto G congreditur, ordinata CL videtur abire in nihilum ita, ut nihilum, quemadmodum & supra innuimus, sit limes quidam inter seriem ordinatarum positivarum CL, & negativarum HM ; vel positivarum CL, & iterum posi- tivarum HM'. Sed, si res altius considere- tur ad metaphysicum conceptum reducta, in situ EF non habetur verum nihilum. In situ CD, HI habetur distantia quaedam punctorum C, L ; H, M : in situ EF habetur eorundem punctorum compene- tratio. Distantia est relatio quaedam FJG binorum modorum, quibus bina puncta existunt ; compenetratio itidem est relatio binorum modorum, quibus ea existunt, quae compenetratio est aliquid reale ejusdem prorsus generis, cujus est distantia, constituta nimirum per binos reales existendi modos." 60. " Totum discrimen est in vocabulis, quae nos imposuimus. Bini locales existendi modi infinitas numero relationes possunt constituere, alii alias. Hae omnes inter se & differunt, & tamen simul etiam plurimum conveniunt ; nam reales sunt, & in quodam genere congruunt, quod nimirum sint relationes ortae a binis localibus existendi modis. Diversa vero habent nomina ad arbitrarium instituta, cum alise ex ejusmodi relationibus, ut CL, dicantur distantiae positivae, relatio EG dicatur compenetratio, relationes HM dicantur distantiae negativse. Sed quoniam, ut a decem palmis distantiae demptis 5, relinquuntur 5, ita demptis aliis 5, habetur nihil (non quidem verum nihil, sed nihil in ratione distantiae a nobis ita appellatae, cum remaneat compenetratio) ; ablatis autem aliis quinque, remanent quinque palmi distantiae negativae ; ista omnia realia sunt, & ad idem genus pertinent ; cum eodem prorsus modo inter se differant distantia palmorum 10 a distantia palmorum 5, haec a distantia nulla, sed reali, quas compenetrationem importat, & haec a distantia negativa palmorum 5. Nam ex prima ilia quantitate eodem modo devenitur ad hasce posteriores per continuam ablationem palmorum 5. Eodem autem pacto infinitas ellipses, ab infinitis hyperbolis unica interjecta parabola discriminat, quae quidem unica nomen peculiare sortita est, cum illas numero infinitas, & a se invicem admodum discrepantes unico vocabulo com- plectamur ; licet altera magis oblonga ab altera minus oblonga plurimum itidem diversa sit." [28] 61. " Et quidem eodem pacto status quidam realis est quies, sive perseverantia in eodem modo locali existendi ; status quidam realis est velocitas nulla puncti existentis. nimirum determinatio perseverandi in eodem loco ; status quidam realis puncti existentis est vis nulla, nimirum determinatio retinendi praecedentem velocitatem, & ita porro ; plurimum haec discrepant a vero non esse. Casus ordinatae respondentis lineae EF in fig. 9, differt plurimum a casu ordinatae circuli respondentis lineae CD figurae 8 : in prima existunt puncta, sed compenetrata, in secunda alterum punctum impossible est. Ubi in solutione problematum devenitur ad quantitatem primi generis, problema determinationem peculiarem accipit ; ubi devenitur ad quantitatem secundi generis, problema evadit impossible ; usque adeo in hoc secundo casu habetur verum nihilum, omni reali proprietate carens ; in illo primo habetur aliquid realibus proprietatibus praeditum, quod ipsis etiam solutionibus problematum, & constructionibus veras sufficit, & reales determinationes ; cum realis, non imaginaria sit radix equationis cujuspiam, quae sit = o, sive nihilo aequalis." A THEORY OF NATURAL PHILOSOPHY 69 to pass suddenly from one quantity to another, then at the instant in which the sudden change is accomplished, both terms must be obtained. Hence, our argument on metaphysical grounds in favour of the exclusion of a sudden change from creation or annihilation, or production & destruction, remains quite unimpaired." 58. " In this connection the following point must be noted. As we have used geometrical Sometimes what is ideas for the consideration of production & destruction, it seems also that sometimes reall y some thingap- the last term of a real series is nothing. But if we go deeper into the matter, we find that it is not in reality nothing, but some state that is also real and of the same kind as those that precede it, though designated by another name." 59. " In Fig. 9, let AB be a line, as before, which some line PL reaches at G (where the When the ordinate point G belongs to the line PL, & E to the line AB, both being produced to meet one whe^thT'dlst^n' 13 another at this point) ; & suppose that PL either goes on beyond the point as GM, or between two exis- recoils along GM'. Then the straight line CD will contain the ordinate CL, which will ^ t J 1 gs . u no " _ & , . . . ' . thing, there is com- vanish when, as the point L, gets to H, L-D attains the position r,r ; & after that, in the penetration. further position of the perpendicular straight line HI, will either pass on to the negative ordinate HM or return, once more positive, to HM'. Now when the one line meets the other, & the point E of the one coincides with the point G of the other, the ordinate CL seems to run off into nothing in such a manner that nothing, as we remarked above, is a certain boundary between the series of positive ordinates CL & the negative ordinates HM, or between the positive ordinates CL & the ordinates HM' which are also positive. But if the matter is more deeply considered & reduced to a metaphysical concept, there is not an absolute nothing in the position EF. In the position CD, or HI, we have given a certain distance between the points C,L, or H,M ; in the position EF, there is compenetration of these points. Now distance is a relation between the modes of existence of two points ; also compenetration is a relation between two modes of existence ; & this compenetration is something real of the very same nature as distance, founded as it is on two real modes of existence." 60. " The whole difference lies in the words that we have given to the things in question. s ' no ' distance Two local modes of existence can constitute an infinite number of relations, some of one kmdT^f series "of sort & some of another. All of these differ from one another, & yet agree with one real quantities as i i i j r ia j i j j ' some ' distance. another in a high degree ; ior they are real & to a certain extent identical, since indeed they are all relations arising from a pair of local modes of existence. But they have different names assigned to them arbitrarily, so that some of the relations of this kind, as CL, are called positive distances, the relation EG is called compenetration, & relations like HM are called negative distances. But, just as when five palms of distance are taken away from ten palms, there are left five palms, so when five more are taken away, there is nothing left (& yet not really nothing, but nothing in comparison with what we usually call distance ; for compenetration is left). Again, if we take away another five, there remain five palms of negative distance. All of these are real & belong to the same class ; for they differ amongst themselves in exactly the same way, namely, the distance of ten palms from the distance of five palms, the latter from ' no ' distance (which however is something real that denotes compenetration), & this again from a negative distance of five palms. For starting with the first quantity, the others that follow are obtained in the same manner, by a continual subtraction of five palms. In a similar manner a single intermediate parabola discriminates between an infinite number of ellipses & an infinite number of hyperbolas ; & this single curve receives a special name, whilst under the one term we include an infinite number of them that to a certain extent are all different from one another, although one that is considerably elongated may be very different from another that is less elongated." 61. "In the same way, rest, i.e., a perseverance in the same mode of local existence, other things that is some real state ; so is ' no ' velocity a real state of an existent point, namely, a propensity ^ndVet^re^eaJi^ to remain in the same place ; so also is ' no ' force a real state of an existent point, namely, something ; d i s- a propensity to retain the velocity that it has already; & so on. All these differ from a '~" a state of non-existence in the highest degree. The case of the ordinate corresponding & zero/ to the line EF in Fig. 9 differs altogether from the case of the ordinate of the circle corresponding to the line CD in Fig. 8. In the first there exist two points, but there is compenetration of these points ; in the other case, the second point cannot possibly exist. When, in the solution of problems, we arrive at a quantity of the first kind, the problem receives a special sort of solution ; but when the result is a quantity of the second kind, the problem turns out to be incapable of solution. So much indeed that, in this second case, there is obtained a true nothing that lacks every real property ; in the first case, we get something endowed with real properties, which also supplies true & real values to the solutions & constructions of the problems. For the root of any equation that = o, or is equal to nothing, is something that is real, & is not an imaginary thing." 70 PHILOSOPHIC NATURALIS THEORIA Conciusip prosolu- fa. " Firmum igitur manebit semper. & stabile, seriem realem quamcunque. quas tione ejus objec- . ~ . , , , v ... a i r contmuo tempore finito duret, debere habere primum prmcipium, & ultimum nnem realem, sine ullo absurdo, & sine conjunctione sui esse cum non esse, si forte duret eo solo tempore : dum si prascedenti etiam exstitit tempore, habere debet & ultimum terminum seriei praecedentis, & primum sequentis, qui debent esse unicus indivisibilis communis limes, ut momentum est unicus indivisibilis limes inter tempus continuum praecedens, & subsequens. Sed haec de ortu, & interitu jam satis." Apphcatio leg is ft- ij t igitur contrahamus iam vela, continuitatis lex & inductione, & metaphysico contmuitatis ad J , , . . i . .. . . r ' . coiiisionem corpo- argumento abunde nititur, quas idcirco etiam in velocitatis commumcatione retmeri omnmo rum - debet, ut nimirum ab una velocitate ad aliam numquam transeatur, nisi per intermedias velocitates omnes sine saltu. Et quidem in ipsis motibus, & velocitatibus inductionem habuimus num. 39, ac difficultates solvimus num. 46, & 47 pertinentes ad velocitates, quae videri possent mutatse per saltum. Quod autem pertinet ad metaphysicum argumentum, si toto tempore ante contactum subsequentis corporis superficies antecedens habuit 12 gradus velocitatis, & sequenti 9, saltu facto momentaneo ipso initio contactus ; in ipso momento ea tempora dirimente debuisset habere & 12, & 9 simul, quod est absurdum. Duas enim velocitates simul habere corpus non potest, quod ipsum aliquanto diligentius demonstrabo. DUO velocitatum g, Velocitatis nomen, uti passim usurpatur a Mechanicis, asquivocum est; potest genera, potentials, T r r . T. . r & actuaiis. enim sigmncare velocitatem actuaiem, quas nimirum est relatio quaedam in motu asquabm spatii percursi divisi per tempus, quo percurritur ; & potest significare [29] quandam, quam apto Scholiasticorum vocabulo potentialem appello, quae nimirum est determinatio, ad actuaiem, sive determinatio, quam habet mobile, si nulla vis mutationem inducat, percur- rendi motu asquabili determinatum quoddam spatium quovis determinato tempore, quas quidem duo & in dissertatione De Viribus Fivis, & in Stayanis Supplements distinxi, distinctione utique .necessaria ad aequivocationes evitandas. Prima haberi non potest momento temporis, sed requirit tempus continuum, quo motus fiat, & quidem etiam motum aequabilem requirit ad accuratam sui mensuram ; secunda habetur etiam momento quovis determinata ; & hanc alteram intelligunt utique Mechanici, cum scalas geometricas effor- mant pro motibus quibuscunque difformibus, sive abscissa exprimente tempus, & ordinata velocitatem, utcunque etiam variatam, area exprimat spatium : sive abscissa exprimente itidem tempus, & ordinata vim, area exprimat velocitatem jam genitam, quod itidem in aliis ejusmodi scalis, & formulis algebraicis fit passim, hac potentiali velocitate usurpata, quas sit tantummodo determinatio ad actuaiem, quam quidem ipsam intelligo, ubi in collisione corporum earn nego mutari posse per saltum ex hoc posteriore argumento. ^5' J am vero velocitates actuales non posse simul esse duas in eodem mobili, satis patet ; potentials 'simul quia oporteret, id mobile, quod initio dati cujusdam temporis fuerit in dato spatii puncto, ne^etur n< vei exf<*a- ^ n omn ibus sequentibus occupare duo puncta ejusdem spatii, ut nimirum spatium percursum tur compenetratfo. sit duplex, alterum pro altera velocitate determinanda, adeoque requireretur actuaiis replicatio, quam non haberi uspiam, ex principio inductionis colligere sane possumus admodum facile. Cum nimirum nunquam videamus idem mobile simul ex eodem loco discedere in partes duas, & esse simul in duobis locis ita, ut constet nobis, utrobique esse illud idem. At nee potentiales velocitates duas simul esse posse, facile demonstratur. Nam velocitas potentialis est determinatio ad existendum post datum tempus continuum quodvis in dato quodam puncto spatii habente datam distantiam a puncto spatii, in quo mobile est eo temporis momento, quo dicitur habere illam potentialem velocitatem determinatam. Quamobrem habere simul illas duas potentiales velocitates est esse determinatum ad occu- panda eodem momento temporis duo puncta spatii, quorum singula habeant suam diversam distantiam ab eo puncto spatii, in quo turn est mobile, quod est esse determinatum ad replicationem habendam momentis omnibus sequentis temporis. Dicitur utique idem mobile a diversis causis acquirere simul diversas velocitates, sed eae componuntur in unicam ita, ut singulas constituant statum mobilis, qui status respectu dispositionum, quas eo momento, in quo turn est, habet ipsum mobile, complectentium omnes circumstantias praeteritas, & praesentes, est tantummodo conditionatus, non absolutus ; nimirum ut con- tineant determi-[3o]-nationem, quam ex omnibus praeteritis, & praesentibus circumstantiis haberet ad occupandum illud determinatum spatii punctum determinato illo momento A THEORY OF NATURAL PHILOSOPHY 71 62. "Hence in all cases it must remain a firm &stable conclusion that any real series, Conclusion in ,.,, , c . . . .1 i, c i r- i favour of a solution which lasts for some finite continuous time, is bound to have a first beginning & a final O f this difficulty. end, without any absurdity coming in, & without any linking up of its existence with a state of non-existence, if perchance it lasts for that interval of time only. But if it existed at a previous time as well, it must have both a last term of the preceding series & a first term of the subsequent series ; just as an instant is a single indivisible boundary between the continuous time that precedes & that which follows. But what I have said about production & destruction is already quite enough." 63. But, to come back at last to our point, the Law of Continuity is solidly founded Application of the both on induction & on metaphysical reasoning ; & on that account it should be retained ^The* co5ision"af in every case of communication of velocity. So that indeed there can never be any passing solid bodies. from one velocity to another except through all intermediate velocities, & then without any sudden change. We have employed induction for actual motions & velocities in Art. 39 & solved difficulties with regard to velocities in Art. 46, 47, in cases in which they might seem to be subject to sudden changes. As regards metaphysical argument, if in the whole time before contact the anterior surface of the body that follows had 12 degrees of velocity & in the subsequent time had 9, a sudden change being made at the instant of first contact ; then at the instant that separates the two times, the body would be bound to have 12 degrees of velocity, & 9, at one & the same time. This is absurd ; for a body cannot at the same time have two velocities, as I will now demonstrate somewhat more carefully. 64. The term velocity, as it is used in general by Mechanicians is equivocal. For it Two kinds of veio- may mean actual velocity, that is to say, a certain relation in uniform motion given by Clty< P tentlal & the space passed over divided by the time taken to traverse it. It may mean also something which, adopting a term used by the Scholastics, I call potential velocity. The latter is a propensity for actual velocity, or a propensity possessed by the movable body (should no force cause an alteration) for traversing with uniform motion some definite space in any definite time. I made the distinction between these two meanings, both in the dissertation De Firibus Fivis & in the Supplements to Stay's Philosophy ; the distinction being very necessary to avoid equivocations. The former cannot be obtained in an instant of time, but requires continuous time for the motion to take place ; it also requires uniform motion in order to measure it accurately. The latter can be determined at any given instant ; & it is this kind that is everywhere intended by Mechanicians, when they make geometrical measured diagrams for any non-uniform velocities whatever. In which, if the abscissa represents time & the ordinate velocity, no matter how it is varied, then the area will express the distance passed over ; or again, if the abscissa represents time & the ordinate force, then the area will represent the velocity already produced. This is always the case, for other scales of the same kind, whenever algebraical formulae & this potential velocity are employed ; the latter being taken to be but the propensity for actual velocity, such indeed as I understand it to be, when in collision of bodies I deny from the foregoing argument that there can be any sudden change. 65. Now it is quite clear that there cannot be two actual velocities at one & the same I4 is impossible time in the same moving body. For, then it would be necessary that the moving body, have two velocities" which at the beginning of a certain time occupied a certain given point of space, should at either actual or all times afterwards occupy two points of that space ; so that the space traversed would be ^given) or we are twofold, the one space being determined by the one velocity & the other by the other, forced to admit, Thus an actual replication would be required ; & this we can clearly prove in a perfectly penetration ' S simple way from the principle of induction. Because, for instance, we never see the same movable body departing from the same place in two directions, nor being in two places at the same time in such a way that it is clear to us that it is in both. Again, it can be easily proved that it is also impossible that there should be two potential velocities at the same time. For potential velocity is the propensity that the body has, at the end of any given continuous time, for existing at a certain given point of space that has a given distance from that point of space, which the moving body occupied at the instant of time in which it is said to have the prescribed potential velocity. Wherefore to have at one & the same time two potential velocities is the same thing as being prescribed to occupy at the same instant of time two points of space ; each of which has its own distinct distance from that point of space that the body occupied at the start ; & this is the same thing as prescribing that there should be replication at all subsequent instants of time. It is commonly said that a movable body acquires from different causes several velocities simultaneously ; but these velocities are compounded into one in such a way that each produces a state of the moving body ; & this state, with regard to the dispositions that it has at that instant (these include all circumstances both past & present), is only conditional, not absolute. That is to say, each involves the propensity which the body, on account of all past & present circumstances, would have for occupying that prescribed point of space at that particular 72 PHILOSOPHISE NATURALIS THEORIA temporis ; nisi aliunde ejusmodi determinatio per conjunctionem alterius causae, quae turn agat, vel jam egerit, mutaretur, & loco ipsius alia, quae composita dicitur, succederet. Sed status absolutus resultans ex omnibus eo momento praasentibus, & prseteritis circumstantiis ipsius mobilis, est unica determinatio ad existendum pro quovis determinato momento temporis sequentis in quodam determinato puncto spatii, qui quidem status pro circum- stantiis omnibus praeteritis, & prsesentibus est absolutus, licet sit itidem conditionatus pro futuris : si nimirum esedem, vel alias causa; agentes sequentibus momentis non mutent determinationem, & punctum illud loci, ad quod revera deveniri deinde debet dato illo momento temporis, & actu devenitur ; si ipsae nihil aliud agant. Porro patet ejusmodi status ex omnibus prseteritis, & praesentibus circumstantiis absolutes non posse eodem momento temporis esse duos sine determinatione ad replicationem, quam ille conditionatus status resultans e singulis componentibus velocitatibus non inducit ob id ipsum, quod conditionatus est. Jam vero si haberetur saltus a velocitate ex omnibus prsteritis, & praesentibus circumstantiis exigente, ex. gr. post unum minutum, punctum spatii distans per palmos 6 ad exigentem punctum distans per palmos 9 ; deberet eo momento temporis, quo fieret saltus, haberi simul utraque determinatio absoluta respectu circumstantiarum omnium ejus momenti, & omnium praeteritarum ; nam toto prsecedenti tempore habita fuisset realis series statuum cum ilia priore, & toto sequenti deberet haberi cum ilia posteriore, adeoque eo momento, simul utraque, cum neutra series realis sine reali suo termino stare possit. Quovis momento 66. Praeterea corporis, vel puncti existentis potest utique nulla esse velocitas actualis, de n be U re hTbeTe saltern accurate talis ; si nimirum difformem habeat motum, quod ipsum etiam semper in statum reaiem ex Natura accidit, ut demonstrari posse arbitror, sed hue non pertinet ; at semper utique potentialis' 6 li!itlS haberi debet aliqua velocitas potentialis, vel saltern aliquis status, qui licet alio vocabulo appellari soleat, & dici velocitas nulla, est tamen non nihilum quoddam, sed realis status, nimirum determinatio ad quietem, quanquam hanc ipsam, ut & quietem, ego quidem arbitrer in Natura reapse haberi nullam, argumentis, quae in Stayanis Supplementis exposui in binis paragraphis de spatio, ac tempore, quos hie addam in fine inter nonnulla, quae hie etiam supplementa appellabo, & occurrent primo, ac secundo loco. Sed id ipsum itidem nequaquam hue pertinet. lis etiam penitus praetermissis, eruitur e reliquis, quae diximus, admisso etiam ut existente, vel possibili in Natura motu uniformi, & quiete, utramque velocitatem habere conditiones necessarias ad [31] hoc, ut secundum argumentum pro continuitatis lege superius allatum vim habeat suam, nee ab una velocitate ad alteram abiri possit sine transitu per intermedias. ento te^oris'trari" ^7' P atet auteni j nmc illud evinci, nee interire momento temporis posse, nee oriri sin ab una veioci- velocitatem totam corporis, vel puncti non simul intereuntis, vel orientis, nee hue transferri demonstrat liai & P osse quod de creatione, & morte diximus ; cum nimirum ipsa velocitas nulla corporis, vel vindicatur. puncti existentis, sit non purum nihil, ut monui, sed realis quidam status, qui simul cum alio reali statu determinatae illius intereuntis, vel orientis velocitatis deberet conjungi ; unde etiam fit, ut nullum effugium haberi possit contra superiora argumenta, dicendo, quando a 12 gradibus velocitatis transitur ad 9, durare utique priores 9, & interire reliquos tres, in quo nullum absurdum sit, cum nee in illorum duratione habeatur saltus, nee in saltu per interitum habeatur absurdi quidpiam, ejus exemplo, quod superius dictum fuit, ubi ostensum est, non conjungi non esse simul, & esse. Nam in primis 12 gradus velocitatis non sunt quid compositum e duodecim rebus inter se distinctis, atque disjunctis, quarum 9 manere possint, 3 interire, sed sunt unica determinatio ad existendum in punctis spatii distantibus certo intervallo, ut palmorumi2, elapsis datis quibusdam temporibus aequalibus quibusvis. Sic etiam in ordinatis GD, HE, quae exprimunt velocitates in fig. 6, revera, in mea potissimuim Theoria, ordinata GD non est quaedam pars ordinatae HE communis ipsi usque ad D, sed sunt duae ordinatae, quarum prima constitit in relatione distantiaa, puncti curvae D a puncto axis G, secunda in relatione puncti curvae E a puncto axis H, quod estibi idem, ac punctum G. A THEORY OF NATURAL PHILOSOPHY 73 instant of time ; were it not for the fact that that particular propensity is for other reasons altered by the conjunction of another cause, which acts at the time, or has already done so ; & then another propensity, which is termed compound, will take the place of the former. But the absolute propensity, which arises from the combination of all the past & present circumstances of the moving body for that instant, is but a single propensity for existing at any prescribed instant of subsequent time in a certain prescribed point of space ; & this state is absolute for all past & present circumstances, although it may be conditional for future circumstances. That is to say, if the same or other causes, acting during subsequent instants, do not change that propensity, & the point of space to which it ought to get thereafter at the given instant of time, & which it actually does reach if these causes have no other effect. Further, it is clear that we cannot have two such absolute states, arising from all past & present circumstances, at the same time without prescribing replication ; & this the conditional state arising from each of the component velocities does not induce because of the very fact that it is conditional. If now there should be a jump from the velocity, arising out of all the past & present circumstances, which, after one minute for example, compels a point of space to move through 6 palms, to a velocity that compels the point to move through 9 palms ; then, at the instant of time, in which the sudden change takes place, there would be each of two absolute propensities in respect of all the circum- stances of that instant & all that had gone before, existing simultaneously. For in the whole of the preceding time there would have been a real series of states having the former velocity as a term, & in the whole of the subsequent time there must be one having the latter velocity as a term ; hence at that particular instant each of them must occur at one & the same time, since neither real series can stand good without each having its own real end term. 66. Again, it is at least possible that the actual velocity of a body, or of an existing At any instant an point, may be nothing ; that is to say, if the motion is non-uniform. Now, this always ^^l *? ""** is the case in Nature ; as I think can be proved, but it does not concern us at present. But, arising from a kind at any rate, it is bound to have some potential velocity, or at least some state, which, y P tentlal vel - although usually referred to by another name, & the velocity stated to be nothing, yet is not definitely nothing, but is a real state, namely, a propensity for rest. I have come to the conclusion, however, that in Nature there is not really such a thing as this state, or absolute rest, from arguments that I gave in the Supplements to Stay's Philosophy in two paragraphs concerning space & time ; & these I will add at the end of the work, amongst some matters, that I will call by the name of supplements in this work as well ; they will be placed first & second amongst them. But that idea also does not concern us at present. Now, putting on one side these considerations altogether, it follows from the rest of what I have said that, if we admit both uniform motion & rest as existing in Nature, or even possible, then each velocity must have conditions that necessarily lead to the conclusion that according to the argument given above in support of the Law of Continuity it has its own corresponding force, & that no passage from one velocity to another can be made except through intermediate stages. 67. Further, it is quite clear that from this it can be rigorously proved that the whole Rigorous proof that e i . i . . . , 9 J r , . it is impossible to velocity of a body cannot perish or arise in an instant of time, nor for a point that does pas s from one veio- not perish or arise along with it ; nor can our arguments with regard to production & cit y to a* 11 ? 1 in 1-1 i r i T-i i.** an instant of time. destruction be made to refer to this. For, since that no velocity of a body, or of an existing point, is not absolutely nothing, as I remarked, but is some real state ; & this real state is bound to be connected with that other real state, namely, that of the prescribed velocity that is being created or destroyed. Hence it comes about that there can be no escape from the arguments I have given above, by saying that when the change from twelve degrees of velocity is made to nine degrees, the first nine at least endure, whilst the remaining three are destroyed ; & then by asserting that there is nothing absurd in this, since neither in the duration of the former has there been any sudden change, nor is there anything absurd in the jump caused by the destruction of the latter, according to the instance of it given above, where it was shown that non-existence & existence must be disconnected. For in the first place those twelve degrees of velocity are not something compounded of twelve things distinct from, & unconnected with, one another, of which nine can endure & three can be destroyed ; but are a single propensity for existing, after the lapse of any given number of equal times of any given length, in points of space at a certain interval, say twelve palms, away from the original position. So also, with regard to the ordinates GD, HE, which in Fig. 6. express velocities, it is the fact that (most especially in my Theory) the ordinate GD is not some part of the ordinate HE, common with it as far as the point D ; but there are two ordinates, of which the first depends upon the relation of the distance of the point D of the curve from the point G on the axis, & the second upon the relation of the distance of point E on the curve from the point H on the axis, which is here the 74 PHILOSOPHIC NATURALIS THEORIA Relationem distantiae punctorum D, & G constituunt duo reales modi existendi ipsorum, relationem distantias punctorum D. & E duo reales modi existendi ipsorum, & relationem distantiae punctorum H, & E duo reales modi existendi ipsorum. Haec ultima relatio constat duobus modis realibus tantummodo pertinentibus ad puncta E, & H, vel G, & summa priorum constat modis realibus omnium trium, E, D, G. Sed nos indefinite con- cipimus possibilitatem omnium modorum realium intermediorum, ut infra dicemus, in qua praecisiva, & indefinita idea stat mini idea spatii continui ; & intermedii modi possibles inter G, & D sunt pars intermediorum inter E, & H. Praeterea omissis etiam hisce omnibus ipse ille saltus a velocitate finita ad nullam, vel a nulla ad finitam, haberi non potest. Cur adhibita col- 68. Atque hinc ego quidem potuissem etiam adhibere duos globos asquales, qui sibi eaiuicm^aKanTpro mv * cem occurrant cum velocitatibus sequalibus, quae nimirum in ipso contactu deberent Thcoria deducenda. momento temporis intcrirc ; sed ut hasce ipsas considerationes evitarem de transitu a statu reali ad statum itidem realem, ubi a velocitate aliqua transitur ad velocitatem nullam ; adhibui potius [32] in omnibus dissertationibus meis globum, qui cum 12 velocitatis gradibus assequatur alterum praecedentem cum 6 ; ut nimirum abeundo ad velocitatem aliam quamcunque haberetur saltus ab una velocitate ad aliam, in quo evidentius esset absurdum. Quo pacto mutata velocitate poten- tial! per saltum, non mutetur per saltum actualis. 69. Jam vero in hisce casibus utique haberi deberet saltus quidam, & violatio legis continuitatis, non quidem in velocitate actuali, sed in potentiali, si ad contactum deveniretur cum velocitatum discrimine aliquo determinato quocunque. In velocitate actuali, si earn metiamur spatio, quod conficitur, diviso per tempus, transitus utique fieret per omnes intermedias, quod sic facile ostenditur ope Geometriae. In fig. 10 designent AB, BC bina tempora ante & post contactum, & momento quolibet H sit velocitas potentialis ilia major HI, quae aequetur velocitati primae AD : quovis autem momento Q posterioris temporis sit velocitas potentialis minor QR, quae aequetur velocitati cuidam data: CG. Assumpto quovis tempore HK determinatae magnitudinis, area IHKL divisa per tempus HK, sive recta HI, exhibebit velocitatem actualem. Moveatur tempus HK versus B, & donee K adveniat ad B, semper eadem habebitur velocitatis men- sura ; eo autem progressoin O ultra B, sed adhuc H existente in M citra B, spatium illi tem- pori respondens componetur ex binis MNEB, BFPO, quorum summa si dividatur per MO ; jam nee erit MN aequalis priori AD, nee BF, ipsa minor per datam quantitatem FE ; sed facile demonstrari potest (&), capta VE asquali D! ~ L V N E Y Irrcgularitas alia in cxpressione act- ualis velocitatis. \ "1 1 X \ p; R T G 1 1 1 1 AH K M B OQ S C FIG. 10. IL, vel HK, sive MO, & ducta recta VF, quae secet MN in X, quotum ex illo divisione prodeuntem fore MX, donee, abeunte toto illo tempore ultra B in QS, jam area QRTS divisa per tempus QS exhibeat velocitatem constantem QR. 70. Patet igitur in ea consideratione a velocitate actuali praecedente HI ad sequentem QR transiri per omnes intermedias MX, quas continua recta VF definiet ; quanquam ibi etiam irregulare quid oritur inde, quod velocitas actualis XM diversa obvenire debeat pro diversa magnitudine temporis assumpti HK, quo nimirum assumpto majore, vel minore removetur magis, vel minus V ab E, & decrescit, vel crescit XM. Id tamen accidit in motibus omnibus, in quibus velocitas non manet eadem toto tempore, ut nimirum turn etiam, si velocitas aliqua actualis debeat agnosci, & determinari spatio diviso per tempus ; pro aliis, atque aliis temporibus assumptis pro mensura alias, atque alias velocitatis actualis mensuras ob-[33]-veniant, secus ac accidit in motu semper aequabili, quam ipsam ob causam, velocitatis actualis in motu difformi nulla est revera mensura accurata, quod supra innui sed ejus idea praecisa, ac distincta aequabilitatem motus requirit, & idcirco Mechanic! in difformibus motibus ad actualem velocitatem determinandam adhibere solent spatiolum infinitesimo tempusculo percursum, in quo ipso motum habent pro aequabili. (b) Si enim producatur OP usque ad NE in T, erit ET = VN, ob VE = MO =NT. Est autem VE : VN : : EF : NX ; quart VN X EF = VE X NX, sive posito ET pro VN, W MO pro VE, erit ET XEF =MO X NX. Totum MNTO est MO X MN, pars FETP est = EY X EF. Quafe residuus gnomon NMOPFE est MOx(MN-NX), sive est MO X MX, quo diviso per MO babetur MX. A THEORY OF NATURAL PHILOSOPHY 75 same as the point G. The relation of the distance between the points D & G is determined by the two real modes of existence peculiar to them, the relation of the distance between the points D & E by the two real modes of existence peculiar to them, & the relation of the distance between the points H & E by the two real modes of existence peculiar to them. The last of these relations depends upon the two real modes of existence that pertain to the points E & H (or G), & upon these alone ; the sum of the first & second depends upon all three of the modes of the points E, D, & G. But we have some sort of ill-defined conception of the possibility of all intermediate real modes of existence, as I will remark later ; & on this disconnected & ill-defined idea is founded my conception of continuous space ; also the possible intermediate modes between G & D form part of those intermediate between E & H. Besides, omitting all considerations of this sort, -that sudden change from a finite velocity to none at all, or from none to a finite, cannot happen. 68. Hence I might just as well have employed two equal balls, colliding with one wh y the collision another with equal velocities, which in truth at the moment of contact would have to be the b sameTirecfion destroyed in an instant of time. But, in order to avoid the very considerations just stated is employed for the with regard to the passage from a real state to another real state (when we pass from a In " definite velocity to none), I have preferred to employ in all my dissertations a ball having 1 2 degrees of velocity, which follows another ball going in front of it with 6 degrees ; so that, by passing to some other velocity, there would be a sudden change from one velocity to another ; & by this means the absurdity of the idea would be made more evident. 69. Now, at least in such cases as these, there is bound to be some sudden change & a breach of the Law of Continuity, not indeed in the actual velocity, but in the potential sudden change in velocity, if the collision occurs with any given difference of velocities whatever. In the ^ ^^T^ might actual velocity, measured by the space traversed divided by the time, the change will at any not 'be a sudden rate be through all intermediate stages ; & this can easily be shown to be 50 by the aid of ^^veioclty 16 ***" Geometry. In Fig. 10 let AB, BC represent two intervals of time, respectively before & after contact ; & at any instant let the potential velocity be the greater velocity HI, equal to the . first velocity AD ; & at any instant Q of the time subsequent to contact let the potential velocity be the less velocity QR, equal to some given velocity CG. If any prescribed interval of time HK be taken, the area IHKL divided by the time HK, i.e., the straight line HI, will represent the actual velocity. Let the time HK be moved towards B ; then until K comes to B, the measure of the velocity will always be the same. If then, K goes on beyond B to O, whilst H still remains on the other side of B at M ; then the space corre- sponding to that time will be composed of the two spaces MNEB, BFPO. Now, if the sum of these is divided by MO, the result will not be equal to either MN (which is equal to the first AD), or BF (which is less than MN by the given quantity FE). But it can easily be proved ( ) that, if VE is taken equal to IL, or HK, or MO, & the straight line VF is drawn to cut MN in X ; then the quotient obtained by the division will be MX. This holds until, when the whole of the interval of time has passed beyond B into the position QS, the area QRTS divided by the time QS now represents a constant velocity equal to QR. 70. From the foregoing reasoning it is therefore clear that the change from the A further irregu- preceding actual velocity HI to the subsequent velocity QR is made through all intermediate lari ty m the repre- r , . . TV/TTT- i i MI i i i i i >> sentation of actual velocities such as MX, which will be determined by the continuous straight line VF. There velocity, is, however, some irregularity arising from the fact that the actual velocity XM must turn out to be different for different magnitudes of the assumed interval of time HK. For, according as this is taken to be greater or less, so the point V is removed to a greater or less distance from E ; & thereby XM will be decreased or increased correspondingly. This is the case, however, for all motions in which the velocity does not remain the same during the whole interval ; as for instance in the case where, if any actual velocity has to be found & determined by the quotient of the space traversed divided by the time taken, far other & different measures of the actual velocities will arise to correspond with the different intervals of time assumed for their measurement ; which is not the case for motions that are always uniform. For this reason there is no really accurate measure of the actual velocity in non-uniform motion, as I remarked above ; but a precise & distinct idea of it requires uniformity of motion. Therefore Mechanicians in non-uniform motions, as a means to the determination of actual velocity, usually employ the small space traversed in an infinitesimal interval of time, & for this interval they consider that the motion is uniform. (b) For if OP be produced to meet NE in T, then EY = VN ; for VE = MO = NT. Moreover VE : VN=EF : NX ; and therefore VN.EF=VE.NX. Hence, replacing VN hy EY, and. VE hy MO, we have EYEF=MO.NX. Now, the whole MNYO = MO.MN, and the part FEYP= ET.EF. Hence the remainder (the gnomon NMOPFE) = MO.(MN NX) = MO.MX .- and this, on division by MO, will give MX. 76 PHILOSOPHIC NATURALIS THEORIA " \mmc yi- At velocitas potcntialis, quas singulis momentis temporis respondet sua, mutaretur citatum non posse utique per saltum ipso momento B, quo deberet haberi & ultima velocitatum praecedentium entia ni vciodtatum r " ^' ^ P" ma sequentium BF, quod cum haberi nequeat, uti demonstratum est, fieri non potest per secundum ex argumentis, quae adhibuimus pro lege continuitatis, ut cum ilia velocitatum inasqualitate deveniatur ad immediatum contactum ; atque id ipsum excludit etiam inductio, quam pro lege continuitatis in ipsis quoque velocitatibus, atque motibus primo loco proposui. Prpmovenda ana- 72. Atque hoc demum pacto illud constitit evidenter, non licere continuitatis legem deserere in collisione corporum, & illud admittere, ut ad contactum immediatum deveniatur cum illaesis binorum corporum velocitatibus integris. Videndum igitur, quid necessario consequi debeat, ubi id non admittatur, & haec analysis ulterius promovenda. ifaberimu- 73' Q uon i am a ^ immediatum contactum devenire ea corpora non possunt cum praece- tationem veiocita- dentibus velocitatibus ; oportet, ante contactum ipsum immediatum incipiant mutari auk mutat Ue Vlm> velocitates ipsae, & vel ea consequentis corporis minui, vel ea antecedentis augeri, vel utrumque simul. Quidquid accidat, habebitur ibi aliqua mutatio status, vel in altero corpore, vel in utroque, in ordine ad motum, vel quietem, adeoque habebitur aliqua mutationis causa, quaecunque ilia sit. Causa vero mutans statum corporis in ordine ad motum, vel quietem, dicitur vis ; habebitur igitur vis aliqua, quae effectum gignat, etiam ubi ilia duo corpora nondum ad contactum devenerint. Earn vim debere 74. Ad impediendam violationem continuitatis satis esset, si ejusmodi vis ageret in . iSf-SSi & alterum tantummodo e binis corporibus, reducendo praecedentis velocitatem ad gradus 12, agere m panes op- . r. . ' . positas. vel sequentis ad 6. Videndum igitur aliunde, an agere debeat in alterum tantummodo, an in utrumque simul, & quomodo. Id determinabitur per aliam Naturae legem, quam nobis inductio satis ampla ostendit, qua nimirum evincitur, omnes vires nobis cognitas agere utrinque & aequaliter, & in partes oppositas, unde provenit principium, quod appellant actionis, & reactionis aequalium ; est autem fortasse quaedam actio duplex semper aequaliter agens in partes oppositas. Ferrum, & magnes aeque se mutuo trahunt ; elastrum binis globis asqualibus interjectum aeque utrumque urget, & aequalibus velocitatibus propellit ; gravitatem ipsam generalem mutuam esse osten-[34]-dunt errores Jovis, ac Saturni potissi- mum, ubi ad se invicem accedunt, uti & curvatura orbitae lunaris orta ex ejus gravitate in terram comparata cum aestu maris orto ex inaequali partium globi terraquei gravitate in Lunam. Ipsas nostrae vires, quas nervorum ope exerimus, semper in partes oppositas agunt, nee satis valide aliquid propellimus, nisi pede humum, vel etiam, ut efficacius agamus, oppositum parietem simul repellamus. En igitur inductionem, quam utique ampliorem etiam habere possumus, ex qua illud pro eo quoque casu debemus inferre, earn ibi vim in utrumque corpus agere, quae actio ad aequalitatem non reducet inaequales illas velocitates, nisi augeat praecedentis, minuat consequentis corporis velocitatem ; nimirum nisi in iis producat velocitates quasdam contrarias, quibus, si solae essent, deberent a se invicem recedere : sed quia eae componuntur cum praecedentibus ; hasc utique non recedunt, sed tantummodo minus ad se invicem accedunt, quam accederent. Hinc dicendam 75. Invenimus igitur vim ibi debere esse mutuam, quae ad partes oppositas agat, & quae esse sua na t ura determinet per sese ilia corpora ad recessum mutuum a se invicem. Hujusmodi quaerendam ejus . . . . . ./ . . 11 i /~ j i legem. igitur vis ex nomims denmtione appellari potest vis repulsiva. Uuaerendum jam ulterius, qua lege progredi debeat, an imminutis in immensum distantiis ad datam quandam mensuram deveniat, an in infinitum excrescat ? Ea vi debere totum 76. Ut in illo casu evitetur saltus ; satis est in allato exemplo ; si vis repulsiva, ad quam crimen at eHdi ante delati sumus, extinguat velocitatum differentiam illam 6 graduum, antequam ad contactum contactum. immediatum corpora devenirent : quamobrem possent utique devenire ad eum contactum eodem illo momento, quo ad aequalitatem velocitatum deveniunt. At si in alio quopiam casu corpus sequens impellatur cum velocitatis gradibus 20, corpore praecedente cum suis 6 ; A THEORY OF NATURAL PHILOSOPHY 77 71. The potential velocity, each corresponding to its own separate instant of time, The conclusion is ij f i j jj i ^.t i. t n a i t nat immediate would certainly be changed suddenly at that instant ot time r> ; & at this point we are contact with a dif- bound to have both the last of the preceding velocities, BE, & the first of the subsequent ference of velocities velocities, BF. Now, since (as has been already proved) this is impossible, it follows from the second of the arguments that I used to prove the Law of Continuity, that it cannot come about that the bodies come into immediate contact with the inequality of velocities in question. This is also excluded by induction, such as I gave in the first place for the Law of Continuity, in the case also of these velocities & motions. 72. In this manner it is at length clearly established that it is not right to neglect the immediate contact Law of Continuity in the collision of bodies, & admit the idea that they can come into ^Sysis^tobe ca^ immediate contact with the whole velocities of both bodies unaltered. Hence, we must ried further, now investigate the consequences that necessarily follow when this idea is not admitted ; & the analysis must be carried further. 73. Since the bodies cannot come into immediate contact with the velocities they had There must be then, at first, it is necessary that those velocities should commence to change before that immediate change in the v'eio a contact ; & either that of the body that follows should be diminished, or that of the one cit y ' & therefore going in front should be increased, or that both these changes should take place together, causes the change! 1 Whatever happens, there will be some change of state at the time, in one or other of the bodies, or in both, with regard to motion or rest ; & so there must be some cause for this change, whatever it is. But a cause that changes the state of a body as regards motion or rest is called force. Hence there must be some force, which gives the effect, & that too whilst the two bodies have not as yet come into contact. 74. It would be enough, to avoid a breach of the Law of Continuity, if a force of The f rce o must V 6 i i.ii 11 r -I IT i i i i < i i i mutual, & act m this kind should act on one of the two bodies only, altering the velocity of the body in opposite directions, front to 12 degrees, or that of the one behind to 6 degrees. Hence we must find out, from other considerations, whether it should act on one of the two bodies only, or on both of them at the same time, & how. This point will be settled by another law of Nature, which sufficiently copious induction brings before us ; that is, the law in which it is estab- lished that all forces that are known to us act on both bodies, equally, and in opposite directions. From this comes the principle that is called ' the principle of equal action & reaction ' ; perchance this may be a sort of twofold action that always produces its effect equally in opposite directions. Iron & a loadstone attract one another with the same strength ; a spring introduced between two balls exerts an equal action on either ball, & generates equal velocities in them. That universal gravity itself is mutual is proved by the aberrations of Jupiter & of Saturn especially (not to mention anything else) ; that is to say, the way in which they err from their orbits & approach one another mutually. So also, when the curvature of the lunar orbit arising from its gravitation towards the Earth is compared with the flow of the tides caused by the unequal gravitation towards the Moon of different parts of the land & water that make up the Earth. Our own bodily forces, which produce their effect by the help of our muscles, always act in opposite direc- tions ; nor have we any power to set anything in motion, unless at the same time we press upon the earth with our feet or, in order to get a better purchase, upon something that will resist them, such as a wall opposite. Here then we have an induction, that can be made indeed more ample still ; & from it we are bound in this case also to infer that the force acts on each of the two bodies. This action will not reduce to equality those two unequal velocities, unless it increases that of the body which is in front & diminishes that of the one which follows. That is to say, unless it produces in them velocities that are opposite in direction ; & with these velocities, if they alone existed, the bodies would move away from one another. But, as they are compounded with those they had to start with, the bodies do not indeed recede from one another, but only approach one another less quickly than they otherwise would have done. 75. We have then found that the force must be a mutual force which acts in opposite Hence the force directions ; one which from its very nature imparts to those bodies a natural propensity p u * sive *r ^"1^ for mutual recession from one another. Hence a force of this kind, from the very meaning governing it is now of the term, may be called a repulsive force. We have now to go further & find the law to ^ found - that it follows, & whether, when the distances are indefinitely diminished, it attains any given measure, or whether it increases indefinitely. 76. In this case, in order that any sudden change may be avoided, it is sufficient, in The whole differ- the example under consideration, if the repulsive force, to which our arguments have led veiocities W must *be us, should destroy that difference of 6 degrees in the velocities before the bodies should destroyed by the have come into immediate contact. Hence they might possibly at least come into contact t ^ e at the instant in which they attained equality between the velocities. But if in another case, say, the body that was behind were moving with 20 degrees of velocity, whilst the 78 PHILOSOPHL/E NATURALIS THEORIA turn vero ad contactum deveniretur cum differentia velocitatum majore, quam graduum 8. Nam illud itidem amplissima inductione evincitur, vires omnes nobis cognitas, quas aliquo tempore agunt, ut velocitatem producant, agere in ratione temporis, quo agunt, & sui ipsius. Rem in gravibus oblique descendentibus experimenta confirmant ; eadem & in elastris institui facile possunt, ut rem comprobent ; ac id ipsum est fundamentum totius Mechanicae, quae inde motuum leges eruit, quas experimenta in pendulis, in projectis gravibus, in aliis pluribus comprobant, & Astronomia confirmat in caelestibus motibus. Quamobrem ilia vis repulsiva, quae in priore casu extinxit 6 tantummodo gradus discriminis, si agat breviore tempore in secundo casu, non poterit extinguere nisi pauciores, minore nimirum velocitate producta utrinque ad partes contrarias. At breviore utique tempore aget : nam cum majore velocitatum discrimine velocitas respectiva est major, ac proinde accessus celerior. [35] Extingueret igitur in secundo casu ilia vis minus, quam 6 discriminis gradus, si in primo usque ad contactum extinxit tantummodo 6. Superessent igitur plures, quam 8 ; nam inter 20 & 6 erant 14, ubi ad ipsum deveniretur contactum, & ibi per saltum deberent velocitates mutari, ne compenetratio haberetur, ac proinde lex continuitatis violari. Cum igitur id accidere non possit ; oportet, Natura incommodo caverit per ejusmodi vim, quae in priore casu aliquanto ante contactum extinxerit velocitatis discrimen, ut nimirum imminutis in secundo casu adhuc magis distantiis, vis ulterior illud omne discrimen auferat, elisis omnibus illis 14 gradibus discriminis, qui habebantur. Earn vim debere augeri in infinitum, imminutis, & qui- dem in infinitum, distantiis : habente virium curva ali- quam asymptotum in origine abscissa- rum. 77. Quando autem hue jam delati sumus, facile est ulterius progredi, & illud con- siderare, quod in secundo casu accidit respectu primi, idem accidere aucta semper velocitate consequentis corporis in tertio aliquo respectu secundi, & ita porro. Debebit igitur ad omnem pro omni casu evitandum saltum Natura cavisse per ejusmodi vim, quae imminutis distantiis crescat in infinitum, atque ita crescat, ut par sit extinguendas cuicunque velocitati, utcunque magnae. Devenimus igitur ad vires repulsivas imminutis distantiis crescentes in infinitum, nimirum ad arcum ilium asymptoticum ED curae virium in fig. i propositum. Illud quidem ratiocinatione hactenus instituta immediate non deducitur, hujusmodi incrementa virium auctarum in infinitum respondere distantiis in infinitum imminutis. Posset pro hisce corporibus, quae habemus prae manibus, quasdam data distantia quascunque esse ultimus limes virium in infinitum excrescentium, quo casu asymptotus AB non transiret per initium distantiae binorum corporum, sed tanto intervallo post ipsum, quantus esset ille omnium distantiarum, quas remotiores particulse possint acquirere a se invicem, limes minimus ; sed aliquem demum esse debere extremum etiam asymptoticum arcum curvas habentem pro asymptote rectam transeuntem per ipsum initium distantiae, sic evincitur ; si nullus ejusmodi haberetur arcus ; particulae materiae minores, & primo collocatae in distantia minore, quam esset ille ultimus limes, sive ilia distantia asymptoti ab initio distantias binorum punctorum materiae, in mutuis incursibus velocitatem deberent posse mutare per saltum, quod cum fieri nequeat, debet utique aliquis esse ultimus asymptoticus arcus, qui asymptotum habeat transeuntem per distantiarum initium, & vires inducat imminutis in infinitum distantiis crescentes in infinitum ita, ut sint pares velocitati extin- guendae cuivis, utcunque magnae. Ad summum in curva virium haberi possent plures asymptotici arcus, alii post alios, habentes ad exigua intervalla asymptotes inter se parallelas, qui casus itidem uberrimum aperit contemplationibus fcecundissimis campum, de quo aliquid inferius ; sed aliquis arcus asympto-[36]-ticus postremus, cujusmodi est is, quern in figura i proposui, haberi omnino debet. Verum ea perquisitione hie omissa, pergendum est in consideratione legis virium, & curvae earn exprimentis, quae habentur auctis distantiis. vim in majoribus tractfvam, ^ 78. In primis gravitas omnium corporum in Terram, quam quotidie experimur, satis , evmc i t > repulsionem illam, quam pro minimis distantiis invenimus, non extendi ad distantias secante axem in quascunque, sed in magnis jam distantiis haberi determinationem ad accessum, quam vim aliquo hmite. attractivam nominavimus. Quin immo Keplerianae leges in Astronomia tarn feliciter a Newtono adhibitae ad legem gravitatis generalis deducendam, & ad cometas etiam traductas, A THEORY OF NATURAL PHILOSOPHY 79 I? 3 O 8o PHILOSOPHIC NATURALIS THEORIA o A THEORY OF NATURAL PHILOSOPHY 81 body in front still had its' original 6 degrees ; then they would come into contact with a difference of velocity greater than 8 degrees. For, it can also be proved by the fullest possible induction that all forces known to us, which act for any intervals of time so as to produce velocity, give effects that are proportional to the times for which they act, & also to the magnitudes of the forces themselves. This is confirmed by experiments with heavy bodies descending obliquely ; the same things can be easily established in the case of springs so as to afford corroboration. Moreover it is the fundamental theorem of the whole of Mechanics, & from it are derived the laws of motion ; these are confirmed by experiments with pendulums, projected weights, & many other things ; they are corroborated also by astronomy in the matter of the motions of the heavenly bodies. Hence the repulsive force, which in the first case destroyed only 6 degrees difference of velocity, if it acts for a shorter time in the second case, will not be able to destroy aught but a less number of degrees, as the velocity produced in the two bodies in opposite directions is less. Now it certainly will act for a shorter time ; for, owing to the greater difference of velocities, the relative velocity is greater & therefore the approach is faster. Hence, in the second case the force would destroy less than 6 degrees of the difference, if in the first case it had, just at contact, destroyed 6 degrees only. There would therefore be more than 8 degrees left over (for, between 20 & 6 there are 14) when contact happened, & then the velocities would have to be changed suddenly unless there was compenetration ; & thereby the Law of Continuity would be violated. Since, then, this cannot be the case, Nature would be sure to guard against this trouble by a force of such a kind as that which, in the former case, extinguished the difference of velocity some time before contact ; that is to say, so that, when the distances are still further diminished in the second case, a further force eliminates all that difference, all of the 14 degrees of difference that there were originally being destroyed. 77. Now, after that we have been led so far, it is easy to go on further still & to consider ' nie fon : e mus * "* that, what happens in the second case when compared with the first, will happen also in SThe distances Ire a third case, in which the velocity of the body that follows is once more increased, when diminished, also compared with the second case ; & so on, & so on. Hence, in order to guard against any Sn-ve"^* 6 forces has sudden change at all in every case whatever, Nature will necessarily have taken measures an asymptote at the for this purpose by means of a force of such a kind that, as the distances are diminished the ongm force increases indefinitely, & in such a manner that it is capable of destroying any velocity, however great it may be. We have arrived therefore at repulsive forces that increase as the distances diminish, & increase indefinitely ; that is to say, to the asymptotic arc, ED, of the curve of forces exhibited in Fig. i . It is indeed true that by the reasoning given so far it is not immediately deduced that increments of the forces when increased to infinity correspond with the distances diminished to infinity. There may be for these bodies, such as we have in consideration, some fixed distance that acts as a boundary limit to forces that increase indefinitely ; in this case the asymptote AB will not pass through the beginning of the distance between the two bodies, but at an interval after it as great as the least limit of all distances that particles, originally more remote, might acquire from one another. But, that there is some final asymptotic arc of the curve having for its asymptote the straight line passing through the very beginning of the distance, is proved as follows. If there were no arc of this kind, then the smaller particles of matter, originally collected at a distance less than this final limit would be, i.e., less than the distance of the asymptote from the beginning of the distance between the two points of matter, must be capable of having- their velocities, on collision with one another, suddenly changed. Now, as this is impossible, then at any rate there must be some asymptotic arc, which has an asymptote passing through the very beginning of the distances ; & this leads us to forces that, as the distances are indefinitely diminished, increase indefinitely in such a way that they are capable of destroying any velocity, no matter how large it may be. In general, in a curve of forces there may be several asymptotic arcs, one after the other, having at short intervals asymptotes parallel to one another ; & this case also opens up a very rich field for fruitful investigations, about which I will say something later. But there must certainly be some one final asymptotic arc of the kind that I have given in Fig. i. However, putting this investigation on one side, we must get on with the consideration of the law of forces, & the curve that represents them, which are obtained when the distances are increased. 78. First of all, the gravitation of all bodies towards the Earth, which is an everyday The force at greater experience, proves sufficiently that the repulsion that we found for very small distances fv^he^curve^cut- does not extend to all distances ; but that at distances that are now great there is a ting the axis at propensity for approach, which we have called an attractive force. Moreover the Keplerian s Laws in astronomy, so skilfully employed by Newton to deduce the law of universal gravitation, & applied even to the comets, show perfectly well that gravitation extends, 82 PHILOSOPHIC NATURALIS THEORIA satis ostendunt, gravitatem vel in infinitum, vel saltern per totum planetarium, & come- tarium systema extendi in ratione reciproca duplicata distantiarum. Quamobrem virium curva arcum habet aliquem jacentem ad partes axis oppositas, qui accedat, quantum sensu percipi possit, ad earn tertii gradus hyperbolam, cujus ordinatae sunt in ratione reciproca duplicata distantiarum, qui nimirum est ille arcus STV figuras I. Ac illud etiam hinc patet, esse aliquem locum E, in quo curva ejusmodi axem secet, qui sit limes attractionum, & repulsionum, in quo ab una ad alteram ex iis viribus transitus fiat. Plures esse debere, 79. Duos alios nobis indicat limites ejusmodi, sive alias duas intersectiones, ut G & I, linStes 3 P n3enomen um vaporum, qui oriuntur ex aqua, & aeris, qui a fixis corporibus gignitur ; cum in iis ante nulla particularum repulsio fuerit, quin immo fuerit attractio, ob cohaerentiam, qua, una parte retracta, altera ipsam consequebatur, & in ilia tanta expansione, & elasticitatis vi satis se manifesto prodat repulsio, ut idcirco a repulsione in minimis distantiis ad attractionem alicubi sit itum, turn inde iterum ad repulsionem, & iterum inde ad generalis gravitatis attractiones. Effervescentiae, & fermentationes adeo diversae, in quibus cum adeo diversis velocitatibus eunt, ac redeunt, & jam ad se invicem accedunt, jam recedunt a se invicem particulae, indicant utique ejusmodi limites, atque transitus multo plures ; sed illos prorsus evincunt substantise molles, ut cera, in quibus compressiones plurimse acquiruntur cum distantiis admodum adversis, in quibus, tamen omnibus limites haberi debent ; nam, anteriore parte ad se attracta, posteriores earn sequuntur, eadem propulsa, illae recedunt, distantiis ad sensum non mutatis, quod ob illas repulsiones in minimis distantiis, quae contiguitatem impediunt, fieri alio modo non potest, nisi si limites ibidem habeantur in iis omnibus distantiis inter attractiones, & repulsiones, quae nimirum requi- runtur ad hoc, ut pars altera alteram consequatur retractam, vel prsecedat propulsam. Hinc tota curvae 80. Habentur igitur plurimi limites, & plurimi flexus curvse hinc, & inde ab axe prseter ayroptotL m & b Tu- ^ uos arcus > quorum prior ED in infinitum protenditur, & asymptoticus est, alter STV, ribus flexibus, ac [37] si gravitas generalis in infinitum protenditur, est asymptoticus itidem, & ita accedit ad crus illud hyperbolae gradus tertii, ut discrimen sensu percipi nequeat : nam cum ipso penitus congruere omnino non potest ; non enim posset ab eodem deinde discedere, cum duarum curvarum, quarum diversa natura est, nulli arcus continui, utcunque exigui, possint penitus congruere, sed se tantummodo secare, contingere, osculari possint in punctis quotcunque, & ad se invicem accedere utcumque. Hinc habetur jam tota forma curvae virium, qualem initio proposui, directa ratiocinatione a Naturae phsenomenis, & genuinis principiis deducta. Remanet jam determinanda constitutio primorum elementorum materiae ab iis viribus deducta, quo facto omnis ilia Theoria, quam initio proposui, patebit, nee erit arbitraria quaedam hypothesis, ac licebit progredi ad amovendas apparentes quasdam difHcultates, & ad uberrimam applicationem ad omnem late Physicam qua exponendam, qua tantummodo, ne hoc opus plus aequo excrescat, indicandam. Hinc elementorum 81. Quoniam, imminutis in infinitum distantiis, vis repulsiva augetur in infinitum ; m facile patet, nullam partem materias posse esse contiguam alteri parti : vis enim ilia repulsiva carens partibus. protinus alteram ab altera removeret. Quamobrem necessario inde consequitur, prima materiae elementa esse omnino simplicia, & a nullis contiguis partibus composita. Id quidem immediate, & necessario fluit ex ilia constitutione virium, quae in minimis distantiis sunt repulsivae, & in infinitum excrescunt. Soiutio objectionis 82. Objicit hie fortasse quispiam illud, fieri posse, ut particulae primigenias materias petitaeex eo quod s j nt com p O sitae quidem, sed nulla Naturae vi divisibiles a se invicem, quarum altera tota vires repulsivas r . . ^ . .... . . . ,. .. i i habere possent non respectu altenus totius habeat vires illas in minimis distantiis repulsivas, vel quarum pars puncta smguia, se q u3ev i s respectu reliquarum partium eiusdem particulae non solum nullam habeat repulsivam particulae primi- T-. 1,1 '-.,, J . r ,. ,.,. r . . geniae. vim, sed habeat maximam illam attractivam, qua; ad ejusmodi cohaesionem requintur : eo pacto evitari debere quemvis immediatum impulsum, adeoque omnem saltum, & con- tinuitatis laesionem. At in primis id esset contra homogeneitatem materiae, de qua agemus infra : nam eadem materiae pars in iisdem distantiis respectu quarundam paucissimarum partium, cum quibus particulam suam componit, haberet vim repulsivam, respectu autem A THEORY OF NATURAL PHILOSOPHY 83 either to infinity or at least to the limits of the system including all the planets & comets, in the inverse ratio of the squares of the distances. Hence the curve will have an arc lying on the opposite side of the axis, which, as far as can be perceived by our senses, approximates to that hyperbola of the third degree, of which the ordinates are in the inverse ratio of the squares of the distances ; & this indeed is the arc STV in Fig. i. Now from this it is evident that there is some point E, in which a curve of this kind cuts the axis ; and this is a limit-point for attractions and repulsions, at which the passage from one to the other of these forces is made. 79. The phenomenon of vapour arising from water, & that of gas produced from There are bound to fixed bodies lead us to admit two more of these limit-points, i.e., two other intersections, ^Syof'tiSep^ say, at G & I. Since in these there would be initially no repulsion, nay rather there sages, with corre- would be an attraction due to cohesion, by which, when one part is retracted, another 1 " 6 hmit generally followed it : & since in the former, repulsion is clearly evidenced by the greatness of the expansion, & by the force of its elasticity ; it therefore follows that there is, somewhere or other, a passage from repulsion at very small distances to attraction, then back again to repulsion, & from that back once more to the attractions of universal gravitation. Effervescences & fermentations of many different kinds, in which the particles go & return with as many different velocities, & now approach towards & now recede from one another, certainly indicate many more of these limit-points & transitions. But the existence of these limit-points is perfectly proved by the case of soft substances like wax ; for in these substances a large number of compressions are acquired with very different distances, yet in all of these there must be limit-points. For, if the front part is drawn out, the part behind will follow ; or if the former is pushed inwards, the latter will recede from it, the distances remaining approximately unchanged. This, on account of the repulsions existing at very small distances, which prevent contiguity, can- not take place in any way, unless there are limit-points there in all those distances between attractions & repulsions ; namely, those that are requisite to account for the fact that one part will follow the other when the latter is drawn out, & will recede in front of the latter when that is pushed in. 80. Therefore there are a large number of limit-points, & a large number of flexures Hence we get the on the curve, first on one side & then on the other side of the axis, in addition to two whole for h of t ^ e arcs, one of which, ED, is continued to infinity & is asymptotic, & the other, STV, is asymptotes, many asymptotic also, provided that universal gravitation extends to infinity. It approximates flexures & many J i r j- r i i r i i i i -11 111 i intersections with to the form of the hyperbola of the third degree mentioned above so closely that the the axis, difference from it is imperceptible ; but it cannot altogether coincide with it, because, in that case it would never depart from it. For, of two curves of different nature, there cannot be any continuous arcs, no matter how short, that absolutely coincide ; they can only cut, or touch, or osculate one another in an indefinitely great number of points, & approximate to one another indefinitely closely. Thus we now have the whole form of the curve of forces, of the nature that I gave at the commencement, derived by a straight- forward chain of reasoning from natural phenomena, & sound principles. It only remains for us now to determine the constitution of the primary elements of matter, derived from these forces ; : in this manner the whole of the Theory that I enunciated at the start will become quite clear, & it will not appear to be a mere arbitrary hypothesis. We can proceed to remove certain apparent difficulties, & to apply it with great profit to the whole of Physics in general, explaining some things fully &, to prevent the work from growing to an unreasonable size, merely mentioning others. 81. Now, because the repulsive force is indefinitely increased when the distances are The simplicity of indefinitely diminished, it is quite easy to see clearly that no part of matter can be contiguous ments^oT^att^r " to any other part ; for the repulsive force would at once separate one from the other, they are altogether Therefore it necessarily follows that the primary elements of matter are perfectly simple, w^ 110 "* P ar ts. & that they are not composed of any parts contiguous to one another. This is an immediate & necessary deduction from the constitution of the forces, which are repulsive at very small distances & increase indefinitely. 82. Perhaps someone will here raise the objection that it may be that the primary Solution of the ob- particles of matter are composite, but that they cannot be disintegrated by any force in j n e tlo ^ SS ertion d that Nature; that one whole with regard to another whole may possibly have those forces single points can- that are repulsive at very small distances, whilst any one part with regard to any other part ?OTces, a u7 P t h'at of the same particle may not only have no repulsive force, but indeed may have a very primary particles great attractive force such as is required for cohesion of this sort ; that, in this way, we can have them - are bound to avoid all immediate impulse, & so any sudden change or breach of continuity. But, in the first place, this would be in opposition to the homogeneity of matter, which we will consider later ; for the same part of matter, at the same distances with regard to those very few parts, along with which it makes up the particle, would have a repulsive 8 4 PHILOSOPHISE NATURALIS THEORIA aliarum omnium attractivam in iisdem distantiis, quod analogic adversatur. Deinde si a Deo agente supra vires Naturae sejungerentur illas partes a se invicem, turn ipsius Naturae vi in se invicem incurrerent ; haberetur in earum collisione saltus naturalis, utut praesup- ponens aliquid factum vi agente supra Naturam. Demum duo turn cohaesionum genera deberent haberi in Natura admodum diversa, alterum per attractionem in minimis distantiis, alterum vero longe alio pacto in elementarium particularum massis, nimirum per limites cohaesionis ; adeoque multo minus simplex, & minus uniformis evaderet Theoria. An elementa sint [38] extensa : argumen- ta pro virtual! eor- um extensione. 83. Simplicitate & incompositione elementorum defmita, dubitari potest, an ea sint etiam inextensa, an aliquam, utut simplicia, extensionem habeant ejus generis, quam virtualem extensionem appellant Scholastici. Fuerunt enim potissimum inter Peripateticos, qui admiserint elementa simplicia, & carentia partibus, atque ex ipsa natura sua prorsus indivisibilia, sed tamen extensa per spatium divisibile ita, ut alia aliis ma jus etiam occupent spatium, ac eo loco, quo unum stet, possint, eo remote, stare simul duo, vel etiam plura ; ac sunt etiamnum, qui ita sentiant. Sic etiam animam rationalem hominis utique prorsus indivisibilem censuerunt alii per totum corpus diffusam : alii minori quidem corporis parti, sed utique parti divisibili cuipiam, & extensae, praesentem toti etiamnum arbitrantur. Deum autem ipsum praesentem ubique credimus per totum utique divisibile spatium, quod omnia corpora occupant, licet ipse simplicissimus sit, nee ullam prorsus compositionem admittat. Videtur autem sententia eadem inniti cuidam etiam analogiae loci, ac temporis. Ut enim quies est conjunctio ejusdem puncti loci cum serie continua omnium moment- orum ejus temporis, quo quies durat : sic etiam ilia virtualis extensio est conjunctio unius momenti temporis cum serie continua omnium punctorum spatii, per quod simplex illud ens virtualiter extenditur ; ut idcirco sicut ilia quies haberi creditur in Natura, ita & haec virtualis extensio debeat admitti, qua admissa poterunt utique ilia primse materiae elementa esse simplicia, & tamen non penitus inextensa. Exciuditur virtu- rite appiicato. 84. At ego quidem arbitror, hanc itidem sententiam everti penitus eodem inductionis principio, ex quo alia tarn multa hucusque, quibus usi sumus, deduximus. Videmus enim in his corporibus omnibus, quae observare possumus, quidquid distinctum occupat locum, distinctum esse itidem ita, ut etiam satis magnis viribus adhibitis separari possint, quae diversas occupant spatii partes, nee ullum casum deprehendimus, in quo magna haec corpora partem aliquam habeant, quae eodem tempore diversas spatii partes occupet, & eadem sit. Porro haec proprietas ex natura sua ejus generis est, ut aeque cadere possit in magnitudines, quas per sensum deprehendimus, ac in magnitudines, quae infra sensuum nostrorum limites sunt ; res nimirum pendet tantummodo a magnitudine spatii, per quod haberetur virtualis extensio, quae magnitudo si esset satis ampla, sub sensus caderet. Cum igitur nunquam id comperiamus in magnitudinibus sub sensum cadentibus, immo in casibus innumeris deprehendamus oppositum : debet utique res transferri ex inductionis principio supra exposito ad minimas etiam quasque materiae particulas, ut ne illae quidem ejusmodi habeant virtualem extensionem. Responsioadexem- [39] 85. Exempla, quae adduntur, petita ab anima rational}, & ab omnipraesentia plum anima & Dei. j) e j } n j^ positive evincunt, cum ex alio entium genere petita sint ; praeterquam quod nee illud demonstrari posse censeo, animam rationalem non esse unico tantummodo, simplici, & inextenso corporis puncto ita praesentem, ut eundem locum obtineat, exerendo inde vires quasdam in reliqua corporis puncta rite disposita, in quibus viribus partim necessariis, & partim liberis, stet ipsum animae commercium cum corpore. Dei autem praesentia cujusmodi sit, ignoramus omnino ; quem sane extensum per spatium divisibile nequaquam dicimus, nee ab iis modis omnem excedentibus humanum captum, quibus ille existit, cogitat, vult, agit, ad humanos, ad materiales existendi, agendique modos, ulla esse potest analogia, & deductio. itidem ad analo- 86. Quod autem pertinet ad analogiam cum quiete, sunt sane satis valida argumenta, giam cum quiete. q u ibus, ut supra innui, ego censeam, in Natura quietem nullam existere. Ipsam nee posse A THEORY OF NATURAL PHILOSOPHY 85 force ; but it would have an attractive force with regard to all others, at the very same distances ; & this is in opposition to analogy. Secondly, if, due to the action of GOD surpassing the forces of Nature, those parts are separated from one another, then urged by the forces of Nature they would rush towards one another ; & we should have, from their collision, a sudden change appertaining to Nature, although conveying a presumption that something was done by the action of a supernatural force. Lastly, with this idea, there would have to be two kinds of cohesion in Nature that were altogether different in constitution ; one due to attraction at very small distances, & the other coming about in a far different way in the case of masses of elementary particles, that is to say, due to the limit-points of cohesion. Thus a theory would result that is far less simple & less uniform than mine. 83. Taking it for granted, then, that the elements are simple & non-composite, whether the ele- there can be no doubt as to whether they are also non-extended or whether, although ments are extended; , , , , .' , 1-1 i V certain arguments simple, they have an extension of the kind that is termed virtual extension by the m favour of virtual Scholastics. For there were some, especially among the Peripatetics, who admitted elements extension. that were simple, lacking in all parts, & from their very nature perfectly indivisible ; but, for all that, so extended through divisible space that some occupied more room than others ; & such that in the position once occupied by one of them, if that one were removed, two or even more others might be placed at the same time ; & even now there are some who are of the same opinion. So also some thought that the rational soul in man, which certainly is altogether indivisible, was diffused throughout the whole of the body ; whilst others still consider that it is present throughout the whole of, indeed, a smaller part of the body, but yet a part that is at any rate divisible & extended. Further we believe that GOD Himself is present everywhere throughout the whole of the undoubtedly divisible space that all bodies occupy ; & yet He is onefold in the highest degree & admits not of any composite nature whatever. Moreover, the same idea seems to depend on an analogy between space & time. For, just as rest is a conjunction with a continuous series of all the instants in the interval of time during which the rest endures ; so also this virtual extension is a conjunction of one instant of time with a continuous series of all the points of space throughout which this one-fold entity extends virtually. Hence, just as rest is believed to exist in Nature, so also are we bound to admit virtual extension ; & if this is admitted, then it will be possible for the primary elements of matter to be simple, & yet not absolutely non-extended. 84. But I have come to the conclusion that this idea is quite overthrown by that same virtual extension principle of induction, by which we have hitherto deduced so many results which we have is r .excluded^ by the employed. For we see, in all those bodies that we can bring under observation, that auction 6 correctly whatever occupies a distinct position is itself also a distinct thing ; so that those that occupy a PP lied - different parts of space can be separated by using a sufficiently large force ; nor can we detect a case in which these larger bodies have any part that occupies different parts of space at one & the same time, & yet is the same part. Further, this property by its very nature is of the sort for which it is equally probable that it happens in magnitudes that we can detect by the senses & in magnitudes which are below the limits of our senses. In fact, the matter depends only upon the size of the space, throughout which the virtual extension is supposed to exist ; & this size, if it were sufficiently ample, would become sensible to us. Since then we never find this virtual extension in magnitudes that fall within the range of our senses, nay rather, in innumerable cases we perceive the contrary ; the matter certainly ought to be transferred by the principle of induction, as explained above, to any of the smallest particles of matter as well ; so that not even they are admitted to have such virtual extension. 85. The illustrations that are added, derived from a consideration of the rational Reply to the soul & the omnipresence of GOD, prove nothing positively ; for they are derived from s^uf&'cot) 6 ' another class of entities, except that, I do not think that it can even be proved that the rational soul does not exist in merely a single, simple, & non-extended point of the body ; so that it maintains the same position, & thence it puts forth some sort of force into the remaining points of the body duly disposed about it ; & the intercommunication between the soul & the body consists of these forces, some of which are involuntary whilst others are voluntary. Further, we are absolutely ignorant of the nature of the presence of GOD ; & in no wise do we say that He is really extended throughout divisible space ; nor from those modes, surpassing all human intelligence, by which HE exists, thinks, wills & acts, can any analogy or deduction be made which will apply to human or material modes of existence & action. 86. Again, as regards the analogy with rest, we have arguments that are sufficiently Again with regard IT T i j i i i_ t ^v vr .. ' to the analogy with strong to lead us to believe, as I remarked above, that there is no such thing m Nature rest . as absolute rest. Indeed, I proved that such a thing could not be, by a direct argument 86 PHILOSOPHISE NATURALIS THEORIA existere, argumento quodam positive ex numero combinationum possibilium infinite contra alium finitum, demonstravi in Stayanis Supplementis, ubi de spatio, & tempore quae juxta num. 66 occurrent infra Supplementorum i, & 2 ; numquam vero earn existere in Natura, patet sane in ipsa Newtoniana sententia de gravitate generali, in qua in planetario systemate ex mutuis actionibus quiescit tantummodo centrum commune gravi- tatis, punctum utique imaginarium, circa quod omnia planetarum, cometarumque corpora moventur, ut & ipse Sol ; ac idem accidit fixis omnibus circa suorum systematum gravitatis centra ; quin immo ex actione unius systematis in aliud utcunque distans, in ipsa gravitatis centra motus aliquis inducetur ; & generalius, dum movetur quaecunque materiae particula, uti luminis particula qusecunque ; reliquae omnes utcunque remotae, quas inde positionem ab ilia mutant, mutant & gravitatem, ac proinde moventur motu aliquo exiguo, sed sane motu. In ipsa Telluris quiescentis sententia, quiescit quidem Tellus ad sensum, nee tota ab uno in alium transfertur locum ; at ad quamcunque crispationem maris, rivuli decursum, muscae volatum, asquilibrio dempto, trepidatio oritur, perquam exigua ilia quidem, sed ejusmodi, ut veram quietem omnino impediat. Quamobrem analogia inde petita evertit potius virtualem ejusmodi simplicium elementorum extensionem positam in conjunctione ejusdem momenti temporis cum serie continua punctorum loci, quam comprobet. in quo deficiat ana- 87. Sed nee ea ipsa analogia, si adesset, rem satis evinceret ; cum analogiam inter tempus, logia loci, & tem- l ocum videamus in aliis etiam violari : nam in iis itidem paragraphis Supplementorum demonstravi, nullum materiae punctum unquam redire ad punctum spatii quodcunque, in quo semel fuerit aliud materiae punctum, ut idcirco duo puncta materiae nunquam conjungant idem [40] punctum spatii ne cum binis quidem punctis temporis, dum quam- plurima binaria punctorum materiae conjungunt idem punctum temporis cum duobus punctis loci ; nam utique coexistunt : ac praeterea tempus quidem unicam dimensionem habet diuturnitatis, spatium vero habet triplicem, in longum, latum, atque profundum. inextensio utilis 88. Quamobrem illud jam tuto inferri potest, haec primigenia materiae elementa, non ad exciudendum so i um esse s i m pli c ia, ac indivisibilia, sed etiam inextensa. Et quidem haec ipsa simplicitas, transitum momen- , t ; . i i_ j ji_ taneum a densitate & inextensio elementorum praestabit commoda sane plunma, quibus eadem adnuc magis nuiia ad summam. f u i c itur, ac comprobatur. Si enim prima elementa materiae sint quaedam partes solidse, ex partibus compositae, vel etiam tantummodo extensae virtualiter, dum a vacuo spatio motu continue pergitur per unam ejusmodi particulam, fit saltus quidam momentaneus a densitate nulla, quae habetur in vacuo, ad densitatem summam, quae habetur, ubi ea particula spatium occupat totum. Is vero saltus non habetur, si elementa simplicia sint, & inextensa, ac a se invicem distantia. Turn enim omne continuum est vacuum tantum- modo, & in motu continue per punctum simplex fit transitus a vacuo continue ad vacuum continuum. Punctum illud materiae occupat unicum spatii punctum, quod punctum spatii est indivisibilis limes inter spatium praecedens, & consequens. Per ipsum non immoratur mobile continue motu delatum, nee ad ipsum transit ab ullo ipsi immediate proximo spatii puncto, cum punctum puncto proximum, uti supra diximus, nullum sit ; sed a vacuo continue ad vacuum continuum transitur per ipsum spatii punctum a materiae puncto occupatum. itidem ad hoc, ut go,. Accedit, quod in sententia solidorum, extensorumque elementorum habetur illud, possit, ut p"test densitatem corporis minui posse in infinitum, augeri autem non posse, nisi ad certum limitem minui in infinitum. i n q uo increment! lex necessario abrumpi debeat. Primum constat ex eo, quod eadem particula continua dividi possit in particulas minores quotcunque, quae idcirco per spatium utcunque magnum diffundi potest ita, ut nulla earum sit, quae aliquam aliam non habeat utcunque libuerit parum a se distantem. Atque eo pacto aucta mole, per quam eadem ilia massa diffusa sit, eaque aucta in ratione quacunque minuetur utique densitas in ratione itidem utcunque magna. Patet & alterum : ubi enim omnes particulae ad contactum devenerint ; densitas ultra augeri non poterit. Quoniam autem determinata quaedam erit utique ratio spatii vacui ad plenum, nonnisi in ea ratione augeri poterit densitas, cujus augmentum, ubi ad contactum deventum fuerit, adrumpetur. At si elementa sint puncta penitus indivisibilia, & inextensa ; uti augeri eorum distantia poterit in infinitum, ita utique poterit etiam minui pariter in ratione quacunque ; cum A THEORY OF NATURAL PHILOSOPHY 87 founded upon the infiniteness of a number of possible combinations as against the finiteness of another number, in the Supplements to Stay's Philosophy, in connection with space & time ; these will be found later immediately after Art. 14 of the Supplements, I and II. That it never does exist in Nature is really clear in the Newtonian theory of universal gravitation ; according to this theory, in the planetary system the common centre of gravity alone is at rest under the action of the mutual forces ; & this is an altogether imaginary point, about which all the bodies of the planets & comets move, as also does the sun itself. Moreover the same thing happens in the case of all the fixed stars with regard to the centres of gravity of their systems ; & from the action of one system on another at any distance whatever from it, some motion will be imparted to these very centres of gravity. More generally, so long as any particle of matter, so long as any particle of light, is in motion, all other particles, no matter how distant, which on account of this motion have their distance from the first particle altered, must also have their gravitation altered, & consequently must move with some very slight motion, but yet a true motion. In the idea of a quiescent Earth, the Earth is at rest approximately, nor is it as a whole translated from place to place ; but, due to any tremulous motion of the sea, the downward course of rivers, even to the fly's flight, equilibrium is destroyed & some agitation is produced, although in truth it is very slight ; yet it is quite enough to prevent true rest altogether. Hence an analogy deduced from rest contradicts rather than corroborates virtual extension of the simple elements of Nature, on the hypothesis of a conjunction of the same instant of time with a continuous series of points of space. 87. But even if the foregoing analogy held good, it would not prove the matter Where the analogy satisfactorily ; since we see that in other ways the analogy between space & time is impaired. 2^ pace and tlme For I proved, also in those paragraphs of the Supplements that I have mentioned, that no point of matter ever returned to any point of space, in which there had once been any other point of matter ; so that two points of matter never connected the same point of space with two instants of time, let alone with more ; whereas a huge number of pairs of points connect the same instant of time with two points of space, since they certainly coexist. Besides, time has but one dimension, duration ; whilst space has three, length, breadth & depth. 88. Therefore it can now be safely accepted that these primary elements of matter Non-extension use- are not only simple & indivisible, but also that they are non-extended. Indeed this a u n \nstanTaneous very simplicity & non-extension of the elements will prove useful in a really large number passage from no of cases for still further strengthening & corroborating the results already obtained. J^-one. a Very For if the primary elements were certain solid parts, themselves composed of parts or even virtually extended only, then, whilst we pass by a continuous motion from empty space through one particle of this kind, there would be a sudden change from a density that is nothing when the space is empty, to a density that is very great when the particle occupies the whole of the space. But there is not this sudden change if we assume that the elements are simple, non-extended & non-adjacent. For then the whole of space is merely a continuous vacuum, &, in the continuous motion by a simple point, the passage is made from continuous vacuum to continuous vacuum. The one point of matter occupies but one point of space ; & this point of space is the indivisible boundary between the space that precedes & the space that follows. There is nothing to prevent the moving point from being carried through it by a continuous motion, nor from passing to it from any point of space that is in immediate proximity to it : for, as I remarked above, there is no point that is the next point to a given point. But from continuous vacuum to continuous vacuum the passage is made through that point of space which is occupied by the point of matter. 89. There is also the point, that arises in the theory of solid extended elements, namely Also for the idea that the density of a body can be diminished indefinitely, but cannot be increased except j^^a'^ ^can up to a certain fixed limit, at which the law of increase must be discontinuous. The first be decreased, comes from the fact that this same continuous particle can be divided into any number mdefinltel y- of smaller particles ; these can be diffused through space of any size in such a way that there is not one of them that does not have some other one at some little (as little as you will) distance from itself. In this way the volume through which the same mass is diffused is increased ; & when that is increased in any ratio whatever, then indeed the density will be diminished in the same ratio, no matter how great the ratio may be. The second thing is also evident ; for when the particles have come into contact, the density cannot be increased any further. Moreover, since there will undoubtedly be a certain determinate ratio for the amount of space that is empty compared with the amount of space that is full, the density can only be increased in that ratio ; & the regular increase of density will be arrested when contact is attained. But if the elements are points that are perfectly indivisible & non-extended, then, just as their distances can be increased indefinitely, 88 PHILOSOPHIC NATURALIS THEORIA in [41] ratione quacunque lineola quaecunque secari sane possit : adeoque uli nullus est limes raritatis auctae, ita etiam nullus erit auctae densitatis. Et ad excludendum 9- Sed & illud commodum accidet, quod ita omne continuum coexistens eliminabitur continuum extcn- e Natura, in quo explicando usque adeo dcsudarunt, & fere incassum, Philosophi, ncc idcirco sum, & in infinitum j ** r j i i in existentibus. divisio ulla realis entis in innmtum produci potent, nee naerebitur, ubi quaeratur, an numerus partium actu distinctarum, & separabilium, sit finitus, an infinitus ; nee alia ejusmodi sane innumera, quae in continui compositione usque adeo negotium facessunt Philosophis, jam habebuntur. Si enim prima materiae elementa sint puncta penitus inextensa, & indivisibilia, a se invicem aliquo intervallo disjuncta ; jam erit finitus punctorum numerus in quavis massa : nam distantiae omnes finitae erunt ; infinitesimas enim quantitates in se determinatas nullas esse, satis ego quidem, ut arbitror, luculcnter demonstravi & in disser- tatione De Natura, t$ Usu infinitorum, ac infinite parvorum, & in dissertatione DC Lege Continuitatis, & alibi. Intervallum quodcunque finitum erit, & divisibile utique in infinitum per interpositionem aliorum, atque aliorum punctorum, quae tamen singula, ubi fuerint posita, finita itidem erunt, & aliis pluribus, finitis tamen itidem, ubi extiterint, locum reliquent, ut infinitum sit tantummodo in possibilibus, non autem in existentibus, in quibus possibilibus ipsis omnem possibilium seriem idcirco ego appellare soleo constantem terminis finitis in infinitum, quod quaecunque, quae existant, finita esse debeant, sed nullus sit existentium finitus numerus ita ingens, ut alii, & alii majores, sed itidem finiti, haberi non possint, atque id sine ullo limite, qui nequeat praeteriri. Hoc autcm pacto, sublato ex existentibus omni actuali infinite, innumerae sane difficultates auferentur. inextensionem 91. Cum igitur & positive argumento. a lege virium positive demonstrata desumpto, qua'rend^m^e simplicitas, & inextensio primorum materiae elementorum deducatur, tam multis aliis homogeneitate. vel indiciis fulciatur, vel emolumentis inde derivatis confirmetur ; ipsa itidem admitti jam debet, ac supererit quaerendum illud tantummodo, utrum haec elementa homogenca censeri debeant, & inter se prorsus similia, ut ea initio assumpsimus, an vero heterogenea, ac dissimilia. Homogeneitatem 92. Pro homogeneitate primorum materiae elementorum illud est quoddani veluti genefta 1 te a primi! n( & P rm cipium, quod in simplicitate, & inextensione conveniant, ac etiam vires quasdam habeant uitimi asymptotici utique omnia. Deinde curvam ipsam virium eandem esse omnino in omnibus illud indicat, omnibus' P "' S ve ^ e tiani evincit, quod primum crus repulsivum impenetrabilitatem secum trahens, & postremum attractivum gravitatem definiens, omnino communia in omnibus sint : nam corpora omnia aeque impenetrabilia sunt, & vero etiam aeque gravia pro quantitate materiae suae, uti satis [42] evincit aequalis velocitas auri, & plumse cadentis in Boyliano recipiente Si reliquus curvae arcus intermedius esset difformis in diversis materiae punctis ; infinities probabilius esset, difformitatem extendi etiam ad crus primum, & ultimum, cum infinities plures sint curvae, quae, cum in reliquis differant partibus, differant plurimum etiam in hisce extremis, quam quae in hisce extremis tantum modo tam arete consentiant. Et hoc quidem argumento illud etiam colligitur, curvam virium in quavis directione ab eodem primo materiae elemento, nimirum ab eodem materiae puncto eandem esse, cum & primum impenetrabilitatis, & postremum gravitatis crus pro omnibus directionibus sit ad sensum idem. Cum primum in dissertatione De Firibus Vivis hanc Theoriam protuli, suspicabar diversitatem legis ' virium respondentis diversis directionibus ; sed hoc argumento adi majorem simplicitatem, & uniformitatem deinde adductus sum. Diversitas autem legum virium pro diversis particulis, & pro diversis respectu ejusdem particulae directionibus, habetur utique ex diverso numero, & positione punctorum earn componentium, qua de re inferius aliquid. i contra deduci 93- Nee vero huic homogeneitati opponitur inductionis principium, quo ipsam ex principio indis- Leibnitiani oppugnare solent, nee principium rationis sufficients, atque indiscernibilium, cermbUium, & rati- . . T TV- /" j- -j onis sufficients. quod supenus innui numero 3. Innmtam Divini v_onditons mentem, ego quidem omnino. arbitror, quod & tam multi Philosophi censuerunt, ejusmodi perspicacitatem habere, atque intuitionem quandam, ut ipsam etiam, quam individuationem appellant, omnino similium individuorum cognoscat, atque ilia inter se omnino discernat. Rationis autem sufficientis A THEORY OF NATURAL PHILOSOPHY 89 so also can they just as well be diminished in any ratio whatever. For it is certainly possible that a short line can be divided into parts in any ratio whatever ; & thus, just as there is no limit to increase of rarity, so also there is none to increase of density. qo. The theory of non-extension is also convenient for eliminating from Nature all ^ lso -/ or excludm s 7 / 1 1 1 1 ! 1 1 Ml 11 1 6 *" ea a C011 " idea of a coexistent continuum to explain which philosophers have up till now laboured tinuum in existing so very hard & generally in vain. Assuming non-extension, no division of a real entity thm R s - that can be carried on indefinitely ; we shall not be brought to a standstill when we seek to find out whether the number of parts that are actually distinct & separable is finite or infinite ; nor with it will there come in any of those other truly innumerable difficulties that, with the idea of continuous composition, have given so much trouble to philosophers. For if the primary elements of matter are perfectly non-extended & indivisible points separated from one another by some definite interval, then the number of points in any given mass must be finite ; because all the distances are finite. I proved clearly enough, I think, in the dissertation De Natura, & Usu infinitorum ac infinite parvorum, & in the dissertation De Lege Continuitatis, & in other places, that there are no infinitesimal quantities determinate in themselves. Any interval whatever will be finite, & at least divisible indefinitely by the interpolation of other points, & still others ; each such set however, when they have been interpolated, will be also finite in number, & leave room for still more ; & these too, when they existed, will also be finite in number. So that there is only an infinity of possible points, but not of existing points ; & with regard to these possible points, I usually term the whole series of possibles a series that ends at finite limits at infinity. This for the reason that any of them that exist must be finite in number ; but there is no finite number of things that exist so great that other numbers, greater & greater still, but yet all finite, cannot be obtained ; & that too without any limit, which cannot be surpassed. Further, in this way, by doing away with all idea of an actual infinity in existing things, truly countless difficulties are got rid of. 91. Since therefore, by a direct argument derived from a law of forces that has been Non-extension directly proved, we have both deduced the simplicity & non-extension of the primary w" 5 have a now e to elements of matter, & also we have strengthened the theory by evidence pointing towards investigate homo- it, or corroborated it by referring to the advantages to be derived from it ; this theory gen ought now to be accepted as true. There only remains the investigation as to whether these elements ought to be considered to be homogeneous & perfectly similar to one another, as we assumed at the start, or whether they are really heterogeneous & dissimilar. 92. In favour of the homogeneity of the primary elements of matter we have so to Homogeneity for speak some foundation derived from the fact that all of them agree in simplicity & non- V oca!ted St f rom a a extension, & also that they are all endowed with forces of some sort. Now, that this consideration of curve of forces is exactly the same for all of them is indicated or even proved by the fact O f 6 the fir?t 86 last that the first repulsive branch necessitating impenetrability, & the last attractive branch a s y"m p t o t i c determining gravitation, are exactly the same in all respects. For all bodies are equally c ^l S forces* impenetrable ; & also all are equally heavy in proportion to the amount of matter contained in them, as is sufficiently proved by the equal velocity of the piece of gold & the feather when falling in Boyle's experiment. If the remaining intermediate arc of the curve were non-uniform for different points of matter, it would be infinitely more probable that the non-uniformity would extend also to the first & last branches also ; for there are infinitely more curves which, when they differ in the remaining parts, also differ to the greatest extent in the extremes, than there are curves, which agree so closely only in these extremes. Also from this argument we can deduce that the curve of forces is indeed exactly the same from the same point of matter, in any direction whatever from the same primary element of matter ; for both the first branch of impenetrability & the last branch of gravitation are the same, so far as we can perceive, for all directions. When I first published this Theory in my dissertation De Firibus Fivis, I was inclined to believe that there was a diversity in the law of forces corresponding to diversity of direction ; but I was led by the argument given above to the greater simplicity & the greater uniformity derived therefrom. Further, diversity of the laws of forces for diverse particles, & for different directions with the same particle, is certainly to be obtained from the diverse number & position of the points composing it ; about which I shall have something to say later. 93. Nor indeed is there anything opposed to this idea of homogeneity to be derived Notl ?i n s t? b r i i r i J i o rr t o /_ . brought against from the principle of induction, by means of which the followers of Leibniz usually raise this from the doc- an objection to it ; nor from the principle of sufficient reason, & of indiscernibles, that fc es f . ind j. scern : J . , . . . -r r i i -TO i_ ibles & sufficient 1 mentioned above in Art. 3. I am indeed quite convinced, & a great many other reason. 1 philosophers too have thought, that the Infinite Will of the Divine Founder has a perspicacity & an intuition of such a nature that it takes cognizance of that which is called individuation amongst individuals that are perfectly similar, & absolutely 90 PHILOSOPHIC NATURALIS THEORIA principium falsum omnino esse censeo, ac ejusmodi, ut omnem verse libertatis ideam omnino tollat ; nisi pro ratione, ubi agitur de voluntatis determinatione, ipsum liberum arbitrium, ipsa libera determinatio assumatur, quod nisi fiat in voluntate divina, quaccunque existunt, necessario existunt, & qusecunque non existunt, ne possibilia quidem erunt, vera aliqua possibilitate, uti facile admodum demonstratur ; quod tamen si semel admittatur, mirum sane, quam prona demum ad fatalem necessitatem patebit via. Quamobrem potest divina voluntas determinari ex toto solo arbitrio suo ad creandum hoc individuum potius, quam illud ex omnibus omnino similibus, & ad ponendum quodlibet ex iis potius eo loco, quo ponit, quam loco alterius. Sed de rationis sufficientis principio haec ipsa fusius pertractavi turn in aliis locis pluribus, turn in Stayanis Supplementis, ubi etiam illud ostendi, id prin- cipium nullum habere usum posse in iis ipsis casibus, in quibus adhibetur, & praedicari solet tantopere, atque id idcirco, quod nobis non innotescant rationes omnes, quas tamen oporteret utique omnes nosse ad hoc, ut eo principio uti possemus, amrmando, nullam esse rationem sufncientem pro hoc potius, quam pro illo [43] alio : sane in exemplo illo ipso, quod adhiberi solet, Archimedis hoc principio aequilibrium determinantis, ibidem ostendi, ex ignoratione causarum, sive rationum, quse postea detectae sunt, ipsum in suae investigationis progressu errasse plurimum, deducendo per abusum ejus principii sphsericam figuram marium, ac Telluris. combinatiombus. Posse etiam puncta 94. Accedit & illud, quod ilia puncta materiae, licet essent prorsus similia in simplicitate, dlfierrTin aiiis 11S> & extensione, ac mensura virium, pendentium a distantia, possent alias habere proprietates metaphysicas diversas inter se, nobis ignotas, quae ipsa etiam apud ipsos Leibnitianos discriminarent. Non vaierehicprin- 95. Quod autem attinet ad inductionem, quam Leibnitiani desumunt a dissimilitudine, a^ma^sis^eas^de! quam observamus in rebus omnibus, cum nimirum nusquam ex. gr. in amplissima silva reperire ferre ex diversis sit duo folia prorsus similia ; ea sane me nihil movet ; cum nimirum illud discrimen sit p rO pri e tas relativa ad rationem aggregati, & nostros sensus, quos singula materiae elementa non afficiunt vi sufficiente ad excitandam in animo ideam, nisi multa sint simul, & in molem majorem excrescant. Porro scimus utique combinationes ejusdem numeri terminorum in immensum excrescere, si ille ipse numerus sit aliquanto major. Solis 24 litterulis Alphabet! diversimodo combinatis formantur voces omnes, quibus hue usque usa sunt omnia idiomata, quae extiterunt, & quibus omnia ilia, quae possunt existere, uti possunt. Quid si numerus earum existeret tanto major, quanto major est numerus puuctorum materiae in quavis massa sensibili ? Quod ibi diversus est litterarum diversarum ordo, id in punctis etiam prorsus homogeneis sunt positiones, & distantia, quibus variatis, variatur utique forma, & vis, qua sensus afficitur in aggregatis. Quanto major est numerus combinationum diversarum possibilium in massis sensibilibus, quam earum massarum, quas possumus observare, & inter se conferre (qui quidem ob distantias, & directiones in infinitum variabiles praescindendo ab aequilibrio virium, est infinitus, cum ipso aequilibrio est immen- sus) ; tanto major est improbabilitas duarum massarum omnino similium, quam omnium aliquantisper saltern inter se dissimilium. Physica ratio dis- 96. Et quidem accedit illud etiam, quod alicujus dissimilitudinis in aggregatis physicam massU 1 ut 1 in 1 fo r iiu US I 1100 ! 116 rationem cernimus in iis etiam casibus, in quibus maxime inter se similia esse deberent. Cum enim mutuae vires ad distantias quascunque pertineant ; status uniuscu- jusque puncti pendebit saltern aliquantisper a statu omnium aliorum punctorum, quae sunt in Mundo. Porro utcunque puncta quaedam sint parum a se invicem remota, uti sunt duo folia in eadem silva, & multo magis in eodem ramo ; adhuc tamen non eandem prorsus relationem distantiae, & virium habent ad reliqua omnia materiae puncta, quae [44] sunt in Mundo, cum non eundem prorsus locum obtineant ; & inde jam in aggregate discrimen aliquod oriri debet, quod perfectam similitudinem omnino impediat. Sed illud earn inducit magis, quod quae maxime conferunt ad ejusmodi dispositionem, necessario respectu diversarum frondium diversa non nihil esse debeant. Omissa ipsa earum forma in semine, solares radii, humoris ad nutritionem necessarii quantitas, distantia, a qua debet is progredi, ut ad locum suum deveniat, aura ipsa, & agitatio inde orta, non sunt omnino similia, sed diversitatem aliquam habent, ex qua diversitas in massas inde efformatas redundat. A THEORY OF NATURAL PHILOSOPHY 91 distinguishes them one from the other. Moreover, I consider that the principle of sufficient reason is altogether false, & one that is calculated to take away all idea of true freewill. Unless free choice or free determination is assumed as the basis of argument, in discussing the determination of will, unless this is the case with the Divine Will, then, whatever things exist, exist because they must do so, & whatever things do not exist will not even be possible, i.e., with any real possibility, as is very easily proved. Nevertheless, once this idea is accepted, it is truly wonderful how it tends to point the way finally to fatalistic necessity. Hence the Divine Will is able, of its own pleasure alone, to be determined to the creation of one individual rather than another out of a whole set of exactly similar things, & to the setting of any one of these in the place in which it puts it rather than in the place of another. But I have discussed these very matters more at length, besides several other places, in the Supplements to Stay's Philosophy ; where I have shown that the principle cannot be employed in those instances in which it is used & generally so strongly asserted. The reason being that all possible reasons are not known to us ; & yet they should certainly be known, to enable us to employ the principle by stating that there is no sufficient reason in favour of this rather than that other. In truth, in that very example of the principle generally given, namely, that of Archimedes' determination of equilibrium by means of it, I showed also that Archimedes himself had made a very big mistake in following out his investigation because of his lack of knowledge of causes or reasons that were discovered in later days, when he deduced a spherical figure for the seas & the Earth by an abuse of this principle. 94. There is also this, that these points of matter, although they might be perfectly it is possible for similar as regards simplicity & extension, & in having the measure of their forces depen- ^"^ese^ro erties dent on their distances, might still have other metaphysical properties different from one but to disagree in another, & unknown to us ; & these distinctions also are made by the followers of others - Leibniz. 95. As regards the induction which the followers of Leibniz make from the lack of The principle does similitude that we see in all things, (for instance such as that there never can be found in n t . hold g d here T_ i j i i vi \ i i . , * induction from the largest wood two leaves exactly alike), their argument does not impress me in the masses; they differ slightest degree. For that distinction is a property that is concerned with reasoning for .n account of an aggregate, & also with our senses ; & these senses single elements of matter cannot tionsof their parta. influence with sufficient force to excite an idea in the mind, except when there are many of them together at a time, & they develop into a mass of considerable size. Further it is well known that combinations of the same number of terms increase enormously, if that number itself increase a little. From the 24 letters of the alphabet alone, grouped together in different ways, are formed all the words that have hitherto been used in all expressions that have existed, or can possibly come into existence. What then if their number were increased to equal the number of points of matter in any sensible mass ? Corresponding to the different order of the several letters in the one, we have in perfectly homogeneous points also different positions & distances ; & if these are altered at least the form & the force, which affect our senses in the groups, are altered as well. How much greater is the number of different combinations that are possible in sensible masses than the number of those masses that we can observe & compare with one another (& this number, on account of the infinitely variable distances & directions of the forces, when equilibrium is precluded, is infinite, since including equilibrium it is very great) ; just so much greater is the improbability of two masses being exactly similar than of their being all at least slightly different from one another. 96. There is also this point in addition ; we discern a physical reason as well for some Physical reason for dissimilarity in groups for those cases too, in which they ought to be especially similar to the difference in .1 -n i f 11 -11 T r i ' several masses, as one another, ror since mutual forces pertain to all possible distances, the state of any in leaves. one point will depend upon, at least in some slight degree, the state of all other points that are in the universe. Further, however short the distance between certain points may be, as of two leaves in the same wood, much more so on the same branch, still for all that they do not have quite the same relation as regards distance & forces as all the rest of the points of matter that are in the universe, because they do not occupy quite the same place. Hence in a group some distinction is bound to arise which will entirely prevent perfect similarity. Moreover this tendency is all the stronger, because those things which especially conduce to this sort of disposition must necessarily be somewhat different with regard to different leaves. For the form itself being absent in the seed, the rays of the sun, the quantity of moisture necessary for nutrition, the distance from which it has to proceed to arrive at the place it occupies, the air itself & the continual motion derived from this, these are not exactly similar, but have some diversity ; & from this diversity there proceeds a diversity in the masses thus formed. 92 PHILOSOPHIC NATURALIS THEORIA simiiitudine quaii- 97. Patet igitur, varietatem illam a numero pendere combinationum possibilium in ^ numero punctorum necessario ad sensationem, & circumstantiarum, quae ad formationem geneitatem, quam massze sunt neccssariae, adeoque ejusmodi inductionem extend! ad elementa non posse. * ' Q u i n immo ilia tanta similitude, quae cum exigua dissimilitudine commixta invenitur in tarn multis corporibus, indicat potius similitudinem ingentem in elementis. Nam ob tantum possibilium combinationum numerum, massae elementorum etiam penitus homo- geneorum debent a se invicem differre plurimum, adeoque si elementa heterogenea sint, in immensum majorem debent habere dissimilitudinem, quam ipsa prima elementa, ex quibus idcirco nullae massas, ne tantillum quidem, similes provenire deberent. Cum elementa multo minus dissimilia esse debeant, quam aggregata elementorum, multo magis ad elementorum homogeneitatem valere debet ilia quaecunque similitudo, quam in corporibus observamus, potissimum in tarn multis, quae ad eandem pertinent speciem, quam ad homogeneitatem eorundem tarn exiguum illud discrimen, quod in aliis tarn multis observatur. Rem autem penitus conficit ilia tanta similitudo, qua superius usi sumus, in primo crure exhibente impenetrabilitatem, & in postremo exhibente gravitatem generalem, quae crura cum ob hasce proprietates corporibus omnibus adeo generales, adeo inter se in omnibus similia sint, etiam reliqui arcus curvae exprimentis vires omnimodam similitudinem indicant pro corporibus itidem omnibus. Homogeneitatem 98. Superest, quod ad hanc rem pertinet, illud unum iterum hie monendum, quod insinuarr' ^xem* ip sum etiam initio hujus Operis innui, ipsam Naturam, & ipsum analyseos ordinem nos plum a libris, lit- ducere ad simplicitatem & homogeneitatem elementorum, cum nimirum, quo analysis ns> pul promovetur magis, eo ad pauciora, & inter se minus discrepantia principia deveniatur, uti patet in resolutionibus Chemicis. Quam quidem rem ipsum litterarum, & vocum exemplum multo melius animo sistet. Fieri utique possent nigricantes litteras, non ductu atramenti continue, sed punctulis rotundis nigricantibus, & ita parum a se invicem remotis, ut inter- valla non nisi ope microscopii discerni possent, & quidem ipsae litterarum formae pro typis fieri pos-[45]-sent ex ejusmodi rotundis sibi proximis cuspidibus constantes. Concipiatur ingens quaedam bibliotheca, cujus omnes libri constent litteris impressis, ac sit incredibilis in ea multitude librorum conscriptorum linguis variis, in quibus omnibus forma charac- terum sit eadem. Si quis scripturae ejusmodi, & linguarum ignarus circa ejusmodi libros, quos omnes a se invicem discrepantes intueretur, observationem institueret cum diligenti contemplatione ; primo quidem inveniret vocum farraginem quandam, quae voces in quibusdam libris occurrerent saepe, cum eaedem in aliis nusquam apparent, & inde lexica posset quaedam componere totidem numero, quot idiomata sunt, in quibus singulis omnes ejusdem idiomatis voces reperirentur, quae quidem numero admodum pauca essent, discri- mine illo ingenti tot, tarn variorum librorum redacto ad illud usque adeo minus discrimen, quod contineretur lexicis illis, & haberetur in vocibus ipsa lexica constituentibus. At inquisitione promota, facile adverteret, omnes illas tarn varias voces constare ex 24 tantummodo diversis litteris, discrimen aliquod inter se habentibus in ductu linearum, quibus formantur, quarum combinatio diversa pareret omnes illas voces tarn varias, ut earum combinatio libros efformaret usque adeo magis a se invicem discrepantes. Et ille quidem si aliud quodcunque sine microscopic examen institueret, nullum aliud inveniret magis adhuc simile elementorum genus, ex quibus diversa ratione combinatis orirentur ipsae litterse ; at microscopic arrepto, intueretur utique illam ipsam litterarum composi- tionem e punctis illis rotundis prorsus homogeneis, quorum sola diversa positio, ac distributio litteras exhiberet. Appiicatio exempli 99. Haec mihi quaedam imago videtur esse eorum, quae cernimus in Natura. Tarn a<^ Naturae analy- mu \ t { } tam var ;j fift ijb r j corpora sunt, & quae ad diversa pertinent regna, sunt tanquam diversis conscripta linguis. Horum omnium Chemica analysis principia quaedam invenit minus inter se difrormia, quam sint libri, nimirum voces. Hae tamen ipsae inter se habent discrimen aliquod, ut tam multas oleorum, terrarum, salium species eruit Chemica analysis e diversis corporibus. Ulterior analysis harum, veluti vocum, litteras minus adhuc inter se difformes inveniret, & ultima juxta Theoriam meam deveniret ad homogenea punctula, quae ut illi circuli nigri litteras, ita ipsa diversas diversorum corporum particulas per solam dispositionem diversam efformarent : usque adeo analogia ex ipsa Naturae consideratione A THEORY OF NATURAL PHILOSOPHY 93 97. It is clear then that this variety depends on the number of possible combinations Homogeneity is to to be found for the number of points that are necessary to make the mass sensible, & ^ m d< ^ ort * ot of the circumstances that arenecessary for the formation of the mass ; & so it is not similitude in some possible that the induction should be extended to the elements. Nay rather, the great heterogeneity from" similarity that is found accompanied by some very slight dissimilarity in so many bodies dissimilarity. points more strongly to the greatest possible similarity of the elements. For on account of the great number of the possible combinations, even masses of elements that are perfectly homogeneous must be greatly different from one another ; & thus if the elements are heterogeneous, the masses must have an immensely greater dissimilarity than the primary elements themselves ; & therefore no masses formed from these ought to come out similar, not even in the very slightest degree. Since the elements are bound to be much less dissimilar than aggregates formed from these elements, homogeneity of the elements must be indicated by that certain similarity that we observe in bodies, especially in so many of those that belong to the same species, far more strongly than heterogeneity of the elements is indicated by the slight differences that are observed in so many others. The whole discussion is made perfectly complete by that great similarity, which we made use of above, that exists in the first branch representing impenetrability, & in the last branch representing universal gravitation ; for since these branches, on account of properties that are so general to all bodies, are so similar to one another in all cases, they indicate complete similarity of the remaining arc of the curve expressing the forces for all bodies as well. 98. Naught that concerns this subject remains but for me to once more mention in Homogeneity is this connection that one thing, which I have already remarked at the beginning of this anftysis of Nature" work, namely, that Nature itself & the method of analysis lead us towards simplicity & example taken homogeneity of the elements ; since in truth the farther the analysis is pushed, the fewer anc j dots ' the fundamental substances we arrive at & the less they differ from one another ; as is to be seen in chemical experiments. This will be presented to the mind far more clearly by an illustration derived from letters & words. Suppose we have made black letters, not by drawing a continuous line with ink, but by means of little black dots which are at such small distances from one another that the intervals cannot be perceived except with the aid of a microscope & indeed such forms of letters may be made as types from round points of this sort set close to one another. Now imagine that we have a huge library, all the books in it consisting of printed letters, & let there be an incredible multitude of books printed in various languages, in all which the form of the characters is the same. If anyone, who was ignorant of such compositions or languages, started on a careful study of books of this kind, all of which he would perceive differed from one another ; then first of all he would find a medley of words, some of which occurred frequently in certain books whilst they never appeared at all in others. Hence he could compose lexicons, as many in number as there are languages ; in each of these all words of the same language would be found, & these would indeed be very few in number ; for the immense multiplicity of words in this numerous collection of books of so many kinds is now reduced to what is still a multiplicity, but smaller, than is contained in the lexicons & the words forming these lexicons. Now if he continued his investigation, he would easily perceive that the whole of these words of so many different kinds were formed from 24 letters only ; that these differed in some sort from one another in the manner in which the lines forming them were drawn ; that the different combinations of these would produce the whole of that great variety of words, & that combinations of these words would form books differing from one another still more widely. Now if he made yet another examination without the aid of a microscope, he would not find any other kind of elements that were more similar to one another than these letters, from a combination of which in different ways the letters themselves could be produced. But if he took a microscope, then indeed would he see the mode of formation of the letters from the perfectly homogeneous round points, by the different position & distribution of which the letters were depicted. 99. This seems to me to be a sort of picture of what we perceive in Nature. Those Application of the i,- 7 - 7 . ,..,. r ., i T n i 1-111 illustration to the books, so many m number & so different in character are bodies, & those which belong analysis of Nature. to the different kingdoms are written as it were in different tongues. Of all of these, chemical analysis finds out certain fundamental constituents that are less unlike one another than the books ; these are the words. Yet these constituent substances have some sort of difference amongst themselves, & thus chemical analysis produces a large number of species of oils, earths & salts from different bodies. Further analysis of these, like that of the words, would disclose the letters that are still less unlike one another ; & finally, according to my Theory, the little homogeneous points would be obtained. These, just as the little black circles formed the letters, would form the diverse particles of diverse bodies through diverse arrangement alone. So far then the analogy derived from such a 94 PHILOSOPHIC NATURALIS THEORIA derivata non ad difformitatem, sed ad conformitatem elementorum nos ducit. Transitus a pro- ioo. Atque hoc demum pacto ex principiis certis & vulgo receptis, per legitimam, ad consectariorum seriem devenimus ad omnem illam, quam initio proposui, Theoriam, nimirum ad legem virium mutuarum, & ad constitutionem primorum materiae elementorum ex ilia ipsa virium lege derivatorum. [46] Videndum jam superest, quam uberes inde fructus per universam late Physicam colligantur, explicatis per earn unam praecipuis cor- porum proprietatibus, & Naturae phaenomenis. Sed antequam id aggredior, praecipuas quasdam e difficultatibus, quae contra Theoriam ipsam vel objectae jam sunt, vel in oculos etiam sponte incurrunt, dissolvam, uti promisi. Legem virium non ioi. Contra vires mutuas illud sclent objicere, illas esse occultas quasdam qualitates, in distans, a nec esse ve ^ etiam actionem in distans inducere. His satisfit notione virium exhibita numero 8, occuitam quaiita- & 9. Illud unum praeterea hie addo, admodum manifestas eas esse, quarum idea admodum facile efformatur, quarum existentia positive argumento evincitur, quarum effectus multi- plices continue oculis observantur. Sunt autem ejusmodi hae vires. Determinationis ad accessum, vel recessum idea efformatur admodum facile. Constat omnibus, quid sit accedere, quid recedere ; constat, quid sit esse indifferens, quid determinatum ; adeoque & determinationis ad accessum, vel recessum habetur idea admodum sane distincta. Argumenta itidem positiva, quae ipsius ejusmodi determinationis existentiam probant, superius prolata sunt. Demum etiam motus varii, qui ab ejusmodi viribus oriuntur, ut ubi corpus quoddam incurrit in aliud corpus, ubi partem solidi arreptam pars alia sequitur, ubi vaporum, vel elastrorum particulae se invicem repellunt, ubi gravia descendunt, hi motus, inquam, quotidie incurrant in oculos. Patet itidem saltern in genere forma curvae ejusmodi vires exprimentis. Haec omnia non occuitam, sed patentem reddunt ejusmodi virium legem. Quid adhuc lateat : IO2. Sunt quidem adhuc quaedam, quae ad earn pertinent, prorsus incognita, uti est admittendam om- numerus, & distantia intersectionum curvae cum axe, forma arcuum intermediorum, atque nino : quo pacto .. . ,. -11 i -11 i i i evitetur hie actio alia ejusmodi, quae quidem longe superant humanum captum, & quas me solus habuit in distans. omnia simul prae oculis, qui Mundum condidit ; sed id omnino nil officit. Nee sane id ipsum in causa esse debet, ut non admittatur illud, cujus existentiam novimus, & cujus proprietates plures, & effectus deprehendimus ; licet alia multa nobis incognita eodem pertinentia supersint. Sic aurum incognitam, occultamque substantiam nemo appellant, & multo minus ejusdem existentiam negabit idcirco, quod admodum probabile sit, plures alias latere ipsius proprietates, olim forte detegendas, uti'aliae tarn multae subinde detectae sunt, & quia non patet oculis, qui sit particularum ipsum componentium textus, quid, & qua ratione Natura ad ejus compositionem adhibeat. Quod autem pertinet ad actionem in distans, id abunde ibidem praevenimus, cum inde pateat fieri posse, ut punctum quodvis in se ipsum agat, & ad actionis directionem, ac energiam determinetur ab altero puncto, vel ut Deus juxta liberam sibi legem a se in Natura condenda stabilitam motum progignat in utroque pun-[47]-cto. Illud sane mihi est evidens, nihilo magis occuitam esse, vel explicatu, & captu difficilem productionem. motus per hasce vires pendentes a certis distantiis, quam sit productio motus vulgo concepta per immediatum impulsum, ubi ad motum determinat impenetrabilitas, quae itidem vel a corporum natura, vel a libera conditoris lege repeti debet. sine impuisione 103. Et quidem hoc potius pacto, quam per impulsionem, in motuum causas, & leges Mst'hucus^'u^N inquirendum esse, illud etiam satis indicat, quod ubi hue usque, impuisione omissa, vires turam, & menus ex- adhibitae sunt a distantiis pendentes, ibi sane tantummodo accurate definita sunt omnia, phcajidam. impost- at q ue determinata, & ad calculum redacta cum phaenomenis congruunt ultra, quam sperare liceret, accuratissime. Ego quidem ejusmodi in explicando, ac determinando felicitatem nusquam alibi video in universa Physica, nisi tantummodo in Astronomia mechanica, quae abjectis vorticibus, atque omni impuisione submota, per gravitatem generalem absolvit omnia, ac in Theoria luminis, & colorum, in quibus per vires in aliqua distantia agentes, & reflexionem, & refractionem, & diffractionem Newtonus exposuit, ac priorum duarum, potissimum leges omnes per calculum, & Geometriam determinavit, & ubi ilia etiam, quae ad diversas vices facilioris transmissus, & facilioris reflexionis, quas Physici passim relinquunt A THEORY OF NATURAL PHILOSOPHY 95 consideration of Nature leads us not to non-uniformity but to uniformity of the elements. 100. Thus at length, from known principles that are commonly accepted, by a Pa^g ,, fro the ... , , , .' . . r r , i i % i n-<i i T i proof of the Theory legitimate series of deductions, we have arrived at the whole of the I heory that I enunciated to the considera- at the start ; that is to say. at a law of mutual forces & the constitution of the primary tion f . objections , ' i i ,- i re XT i r i ' against it. elements of matter derived from that law of forces. Now it remains to be seen what a bountiful harvest is to be gathered throughout the wide field of general physics ; for from this one theory we obtain explanations of all the chief properties of bodies, & of the phenomena of Nature. But before I go on to that, I will give solutions of a few of the principal difficulties that have been raised against the Theory itself, as well as some that naturally meet the eye, according to the promise I made. 10 1. The objection is frequently brought forward against mutual forces that they The law o{ forces , J . * . . , . . j. mi does not necessi- are some sort of mysterious qualities or that they necessitate action at a distance. This tate action at a is answered by the idea of forces outlined in Art. 8, & 9. In addition, I will make just distance, nor is it one remark, namely, that it is quite evident that these forces exist, that an idea of them quTuty. " can be easily formed, that their existence is demonstrated by direct reasoning, & that the manifold results that arise from them are a matter of continual ocular observation. Moreover these forces are of the following nature. The idea of a propensity to approach or of a propensity to recede is easily formed. For everybody knows what approach means, and what recession is ; everybody knows what it means to be indifferent, & what having a propensity means ; & thus the idea of a propensity to approach, or to recede, is perfectly distinctly obtained. Direct arguments, that prove the existence of this kind of propensity, have been given above. Lastly also, the various motions that arise from forces of this kind, such as when one body collides with another body, when one part of a solid is seized & another part follows it, when the particles of gases, & of springs, repel one another, when heavy bodies descend, these motions, I say, are of everyday occurrence before our eyes. It is evident also, at least in a general way, that the form of the curve represents forces of this kind. In all of these there is nothing mysterious ; on the contrary they all tend to make the law of forces of this kind perfectly plain. 102. There are indeed certain things that relate to the law of forces of which we are What is so far un altogether ignorant, such as the number & distances of the intersections of the curve J^ ^ idmittedm with the axis, the shape of the intervening arcs, & other things of that sort ; these indeed ail detail ; the way far surpass human understanding, & He alone, Who founded the universe, had the whole ^ action 1 at h a to* before His eyes. But truly there is no reason on that account, why a thing, whose existence tance is eliminated, we fully recognize, & many of the properties & results of which are readily understood, should not be accepted ; although certainly there do remain many other things pertaining to it that are unknown to us. For instance, nobody would call gold an unknown & mysterious substance, & still less would deny its existence, simply because it is quite probable that many of its properties are unknown to us, to be discovered perhaps in the future, as so many others have been already discovered from time to time, or because it is not visually apparent what is the texture of the particles composing it, or why & in what way Nature adopts that particular composition. Again, as regards action at a distance, we amply guard against this by the same means ; for, if this is admitted, then it would be possible for any point to act upon itself, & to be determined as to its direction of action & energy apart from another point, or that God should produce in either point a motion according to some arbitrary law fixed by Him when founding the universe. To my mind indeed it is clear that motions produced by these forces depending on the distances are not a whit more mysterious, involved or difficult of understanding than the production of motion by immediate impulse as it is usually accepted ; in which impenetrability determines the motion, & the latter has to be derived just the same either from the nature of solid bodies, or from an arbitrary law of the founder of the universe. 103. Now, that the investigation of the causes & laws of motion are better made As far as we have by my method, than through the idea of impulse, is sufficiently indicated by the fact that, f^' m0 re UI ciearJy where hitherto we have omitted impulse & employed forces depending on the distances, explained without only in this way has everything been accurately defined & determined, & when reduced g^ 1 what to calculation everything agrees with the phenomena with far more accuracy than we will be so too. could possibly have expected. Indeed I do not see anywhere such felicity in explaining & determining the matters of general physics, except only in celestial mechanics ; in which indeed, rejecting the idea of vortices, & doing away with that of impulse entirely, Newton gave a solution of everything by means of universal gravitation ; & in the theory of light & colours, where by means of forces acting at some distance he explained reflection, refraction & diffraction ; &, especially in the two first mentioned, he determined all the laws by calculus & Geometry. Here also those things depending on alternate fits of easier transmission & easier reflection, which physicists everywhere leave almost 9 6 PHILOSOPHISE NATURALIS THEORIA fere intactas, ac alia multa admodum feliciter determinantur, explicanturque, quod & ego praestiti in dissertatione De Lumine, & praestabo hie in tertia parte ; cum in ceteris Physicae partibus plerumque explicationes habeantur subsidariis quibusdam principiis innixae & vagas admodum. Unde jam illud conjectare licet, si ab impulsione immediata penitus recedatur, & sibi constans ubique adhibeatur in Natura agendi ratio a distantiis pendens, multo sane facilius, & certius explicatum iri cetera ; quod quidem mihi omnino successit, ut patebit inferius, ubi Theoriam ipsam applicavero ad Naturam. Non fieri saltum in tracfiva ad repui- sivam. 104. Solent & illud objicere, in hac potissimo Theoria virium committi saltum ilium, a( ^ quern evitandum ea inprimis admittitur ; fieri enim transitum ab attractionibus ad repulsiones per saltum, ubi nimirum a minima ultima repulsione ad minimam primam attractionem transitur. At isti continuitatis naturam, quam supra exposuimus, nequaquam intelligunt. Saltus, cui evitando Theoria inducitur, in eo consistit, quod ab una magnitudine ad aliam eatur sine transitu per intermedias. Id quidem non accidit in casu exposito. Assumatur quaecunque vis repulsiva utcunque parva ; turn quaecunque vis attractiva. Inter eas intercedunt omnes vires repulsivae minores usque ad zero, in quo habetur deter- minatio ad conservandum praecedentum statum quietis, vel motus uniformis in directum : turn omnes vires attractivae a z^-[48]-ro usque ad earn determinatam vim, & omnino nullus erit ex hisce omnibus intermediis statibus, quern aliquando non sint habitura puncta, quae a repulsione abeunt ad attractionem. Id ipsum facile erit contemplari in fig. i, in qua a vi repulsiva br ad attractionem dh itur utique continue motu puncti b ad d transeundo per omnes intermedias, & per ipsum zero in E sine ullo saltu ; cum ordinata in eo motu habitura sit omnes magnitudines minores priore br usque ad zero in E ; turn omnes oppositas majores usque ad posteriorem dh. Qui in ea veluti imagine mentis oculos defigat, is omnem apparentem difficultatem videbit plane sibi penitus evanescere. Nuiium esse post- JO P Q uo d autem additur de postremo repulsionis gradu, & primo attractionis nihil remum attractions, - 1 . ... r ,.... r , ., A: primum repuisio- sane probaret, quando etiam essent aliqui n gradus postrerm, & primi ; nam ab altero ms gradum, qm si eorum transiretur ad alterum per intermedium illud zero, & ex eo ipso, quod illi essent essent, adhuc tran- .....*. . ,. . . sire per omnes in- postremus, ac primus, mhil omitteretur mtermeaium, quae tamen sola intermedn omissio termedios. continuitatis legem evertit, & saltum inducit. Sed nee habetur ullus gradus postremus, aut primus, sicut nulla ibi est ordinata postrema, aut prima, nulla lineola omnium minima. Data quacunque lineola utcunque exigua, aliae ilia breviores habentur minores, ac minores ad infinitum sine ulla ultima, in quo ipso stat, uti supra etiam monuimus, continuitatis natura. Quamobrem qui primum, aut ultimum sibi confingit in lineola, in vi, in celeritatis gradu, in tempusculo, is naturam continuitatis ignorat, quam supra hie innui, & quam ego idcirco initio meae dissertationis De Lege Continuitatis abunde exposui. potest cuipiam saltern illud, ejusmodi legem virium, & curvam, quam in curvae, & duobus fig. I protuli, esse nimium complicatam, compositam, & irregularem, quae nimirum coalescat virium genenbus. ex } n g en ti numero arcuum jam attractivorum, jam repulsivorum, qui inter se nullo pacto cohaereant ; rem eo redire, ubi erat olim, cum apud Peripateticos pro singulis proprietatibus corporum singulae qualitates distinctae, & pro diversis speciebus diversae formae substantiales confingebaritur ad arbitrium. Sunt autem, qui & illud addant, repulsionem, & attractionem esse virium genera inter se diversa ; satius esse, alteram tantummodo adhibere, & repulsionem explicare tantummodo per attractionem minorem. repuisivam positive I0 7- Inprimis quod ad hoc postremum pertinet, satis patet, per positivam meae demonstrari prater Theoriae probationem immediate evinci repulsionem ita, ut a minore attractione repeti omnino non possit ; nam duae materiae particulae si etiam solae in Mundo essent, & ad se invicem cum aliqua velocitatum inaequalitate accederent, deberent utique ante contactum ad sequalitatem devenire vi, quse a nulla attractione pendere posset. A THEORY OF NATURAL PHILOSOPHY 97 untouched, & many other matters were most felicitously determined & explained by him ; & also that which I enunciated in the dissertation De Lumine, & will repeat in the third part of this work. For in other parts of physics most of the explanations are independent of, & disconnected from, one another, being based on several subsidiary principles. Hence we may now conclude that if, relinquishing all idea of immediate impulses, we employ a reason for the action of Nature that is everywhere the same & depends on the distances, the remainder will be explained with far greater ease & certainty ; & indeed it is altogether successful in my hands, as will be evident later, when I come to apply the Theory to Nature. 104. It is very frequently objected that, in this Theory more especially, a sudden change There is no sudden is made in the forces, whilst the theory is to be accepted for the very purpose of avoiding sitfwPfrom aiTat- such a thing. For it is said that the transition from attractions to repulsions is made tractive to a repui- suddenly, namely, when we pass from the last extremely minute repulsive force to the first extremely minute attractive force. But those who raise these objections in no wise understand the nature of continuity, as it has been explained above. The sudden change, to avoid which the Theory has been brought forward, consists in the fact that a passage is made from one magnitude to another without going through the intermediate stages. Now this kind of thing does not take place in the case under consideration. Take any repulsive force, however small, & then any attractive force. Between these two there lie all the repulsive forces that are less than the former right down to zero, in which there is the propensity for preserving the original state of rest or of uniform motion in a straight line ; & also all the attractive forces from zero up to the prescribed attractive force, & there will be absolutely no one of all these intermediate states, which will not be possessed at some time or other by the points as they pass from repulsion to attraction. This can be readily understood from a study of Fig. I, where indeed the passage is made from the repulsive force br to the attractive force dh by the continuous motion of a point from b to d ; the passage is made through every intermediate stage, & through zero at E, without any sudden change. For in this motion there will be obtained as ordinates all magnitudes, less than the first one br, down to zero at E, & after that all magnitudes of opposite sign greater than zero as far as the last ordinate dh. Anyone, who will fix his intellectual vision on this as on a sort of pictorial illustration cannot fail to perceive for himself that all the apparent difficulty vanishes completely. i OS. Further, as regards what is said in addition about the last stage of repulsion & T^ 1 " 6 & no ! ast , r . . 11 11 TI 1111 stage of attraction, the first stage of attraction, it would really not matter, even if there were these so called an d no first for re- last & first stages ; for, from one of them to the other the passage would be made through puisjon ; and even 6 ,.' . . , i c if there were, the the one intermediate stage, namely zero ; since it passes zero, & because they are the nrst passage would be & last, therefore no intermediate stage is omitted. Nevertheless the omission of this p ade through ail intermediate alone would upset the law of continuity, & introduce a sudden change. But, as a matter of fact, there cannot possibly be a last stage or a first ; just as there cannot be a last ordinate or a first in the curve, that is to say, a short line that is the least of them all. Given any short line, no matter how short, there will be others shorter than it, less & less in infinite succession without any limit whatever ; & in this, as we remarked also above, there lies the nature of continuity. Hence anyone who brings forward the idea of a first or a last in the case of a line, or a force, or a degree of velocity, or an interval of time, must be ignorant of continuity ; this I have mentioned before in this work, & also for this very reason I explained it very fully at the beginning of my dissertation De Lege Continuitatis. 1 06. It may seem to some that at least a law of forces of this nature, & the curve g b ^ c s t t 10 ? he r appar d expressing it, which I gave in Fig. I, is very complicated, composite & irregular, being e nt composite cha- indeed made up of an immense number of arcs that are alternately attractive & repulsive, ^ te [ h f t & that these are joined together according to no definite plan ; & that it reduces to O f forces, the same thing as obtained amongst the ancients, since with the Peripatetics separate distinct qualities were invented for the several properties of bodies, & different substantial forms for different species. Moreover there are some who add that repulsion & attraction are kinds of forces that differ from one another ; & that it would be quite enough to use only the latter, & to explain repulsion merely as a smaller attraction. 107. First of all, as regards the last objection, it is clear enough from what has been p^ssibie^o prove directly proved in my Theory that the existence of repulsion has been rigorously demonstrated directly the exist- in such a way that it cannot possibly be derived from the idea of a smaller attraction. For f^ce f apart PU from two particles of matter, if they were also the only particles in the universe, & approached attraction. one another with some difference of velocity, would be bound to attain to an equality of velocity on account of a force which could not possibly be derived from an attraction of any kind, H PHILOSOPHIC NATURALIS THEORIA tiva, & negativa. Hinc nihu pbstare, Io g i Deinde vef o quod pertinet ad duas diversas species attractionis, & repulsionis ; si diversi suit gene- ., . , ,. . ,^. r .r-ii-i i ris; sed esse ejus- id quidem licet ita se haberet, m-[49j-hil sane obesset, cum positive argumento evmcatur dem uti sunt posi- & re pulsio. & attractio, uti vidimus; at id ipsum est omnino falsum. Utraque vis ad . f . . . . ^ . . . eandem pertinet speciem, cum altera respectu alterms negativa sit, & negativa a positivis specie non differant. Alteram negativam esse respectu alterius, patet inde, quod tantum- modo differant in directione, quae in altera est prorsus opposita direction! alterius ; in altera enim habetur determinatio ad accessum, in altera ad recessum, & uti recessus, & accessus sunt positivum, ac negativum ; ita sunt pariter & determinationes ad ipsos. Quod autem negativum, & positivum ad eandem pertineant speciem, id sane patet vel ex eo principio : magis, W minus non differunt specie. Nam a positive per continuam subtrac- tionem, nimirum diminutionem, habentur prius minora positiva, turn zero, ac demum negativa, continuando subtractionem eandem. Probatio hujus a progressu, & re- gressu, in fluvio. 109. Id facile patet exemplis solitis. Eat aliquis contra fluvii directionem versus locum aliquem superiori alveo proximum, & singulis minutis perficiat remis, vel vento too hexapedas, dum a cursu fluvii retroagitur per hexapedas 40 ; is habet progressum hexapedarum 60 singulis minutis. Crescat autem continue impetus fluvii ita, ut retroagatur per 50, turn per 60, 70, 80, 90, 100, no, 120, &c. Is progredietur per 50, 40, 30, 20, 10, nihil ; turn regredietur per 10, 20, quae erunt negativa priorum ; nam erat prius 100 50, 100 60, 10070,100 80,100 90, turn 100 100=0,100 no, = 10, 100 120 = 20, et ita porro. Continua imminutione, sive subtractione itum est a positivis in negativa, a progressu ad regressum, in quibus idcirco eadem species mansit, non duae diversae. Probatio ex Alge- bra, & Geometria : applicatio ad omnes quantitates varia- biles. An habeatur trans- itus e positivis in negativa ; investi- gatio ex sola curv- arum natura. B FHN MAC FIG. ii. i to. Idem autem & algebraicis formulis, & geometricis lineis satis manifeste ostenditur. Sit formula 10 x, & pro x ponantur valores 6, 7, 8, 9, 10, n, 12, &c. ; valor formulae exhibebit 4, 3, 2, I, o, I, 2, &c., quod eodem redit, ubi erat superius in progressu, & regressu, qui exprimerentur simulper formulam 10 x. Eadem ilia formula per continuam mutationem valoris x migrat e valore positive in negativum, qui aeque ad eandem formulam pertinent. Eodem pacto in Geometria in fig. u,siduae lineae MN, OP referantur invicem per ordinatas AB, CD, &c. parallelas inter se, secent autem se in E ; continue motu ipsius ordinatae a positive abitur in negativum, mutata directione AB, CD, quae hie habentur pro positivis, in FG, HI, post evanescentiam in E. Ad eandem lineam continuam OEP aeque pertinet omnis ea ordinatarum series, nee est altera linea, alter locus geometricus OE, ubi ordinatae sunt positivae, ac EP, ubi sunt nega- tivae. Jam vero variabilis quantitatis cujusvis natura, & lex plerumque per formulam aliquam analyticam, semper per ordinatas ad lineam aliquam exprimi potest ; si [50] enim singulis ejus statibus ducatur perpendicularis respondens ; vertices omnium ejusmodi perpendicularium erunt utique ad lineam quandam continuam. Si ea linea nusquam ad alteram abeat axis partem, si ea formula nullum valorem negativum habeat ; ilia etiam quantitas semper positiva manebit. Sed si mutet latus linea, vel formula valoris signum ; ipsa ilia quantitatis debebit itidem ejusmodi mutationem habere. Ut autem a formulae, vel lineae exprimentis natura, & positione respectu axis mutatio pendet ; ita mutatio eadem a natura quantitatis illius pendebit ; & ut nee duas formulae, nee duae lineae speciei diversae sunt, quae positiva exhibent, & negativa ; ita nee in ea quantitate duae erunt naturae, duae species, quarum altera exhibeat positiva, altera negativa, ut altera progressus, altera regressus ; altera accessus, altera recessus ; & hie altera attractiones, altera repulsiones exhibeat ; sed eadem erit, unica, & ad eandem pertinens quantitatis speciem tota. in. Quin immo hie locum habet argumentum quoddam, quo usus sum in dissertatione De Lege Continuitatis, quo nimirum Theoria virium attractivarum, & repulsivarum pro diversis distantiis, multo magis rationi consentanea evincitur, quam Theoria ^ virium tantummodo attractivarum, vel tantummodo repulsivarum. Fingamus illud, nos ignorare penitus, quodnam virium genus in Natura existat, an tantummodo attractivarum, vel repulsivarum tantummodo, an utrumque simul : hac sane ratiocinatione ad earn perquisi- tionem uti liceret. Erit utique aliqua linea continua, quae per suas ordinatas ad axem exprimentem distantias, vires ipsas determinabit, & prout ipsa axem secuerit, vel non A THEORY OF NATURAL PHILOSOPHY 99 108. Next, as regards attraction & repulsion being of different species, even if it Hence it does not were a fact that they were so, it would not matter in the slightest degree, since by rigorous S at different h kmds! argument the existence of both attraction & repulsion is proved, as we have seen ; but but as a matter of really the supposition is untrue. Both kinds of force belong to the same species ; for one same^kmdnusVas is negative with regard to the other, & a negative does not differ in species from positives. a positive and a That the one is negative with regard to the other is evident from the fact that they only negatlve are so - differ in direction, the direction of one being exactly the opposite of the direction of the other ; for in the one there is a propensity to approach, in the other a propensity to recede ; & just as approach & recession are positive & negative, so also are the propensities for these equally so. Further, that such a negative & a positive belong to the same species, is quite evident from the principle the greater & the less are not different in kind. For from a positive by continual subtraction, or diminution, we first obtain less positives, then zero, & finally negatives, the same subtraction being continued throughout. 109. The matter is easily made clear by the usual illustrations. Suppose a man Demonstration by to go against the current of a river to some place on the bank up-stream; & suppose "veTndretrogS that he succeeds in doing, either by rowing or sailing, 100 fathoms a minute, whilst he motion on a river. is carried back by the current of the river through 40 fathoms ; then he will get forward a distance of 60 fathoms a minute. Now suppose that the strength of the current continually increases in such a way that he is carried back first 50, then 60, 70, 80, 90, ipo, no, 120, &c. fathoms per minute. His forward motion will be successively 50, 40, 30, 20, 10 fathoms per minute, then nothing, & then he will be carried backward through 10, 20, &c. fathoms a minute ; & these latter motions are the negatives of the former. For first of all we had 100 50, 100 60, 100 70, 100 80, 100 90, then 100 100 (which = o), then 100 no (which = 10), 100 120 (which = 20), and so on. By a continual diminution or subtraction we have passed from positives to negatives, from a progressive to a retrograde motion ; & therefore in these there was a continuance of the same species, and there were not two different species. no. Further, the same thing is shown plainly enough by algebraical formulae, & Proof from algebra by lines in geometry. Consider the formula 10 x, & for x substitute the values, 6, pucatfon^ O y : a n 7, 8, 9, 10, n, 12, &c. ; then the value of the formula will give in succession 4, 3, 2, variable quantities. I, o, i, 2, &c. ; & this comes to the same thing as we had above in the case of the progressive & retrograde motion, which may be expressed by the formula 10 x, all together. This same formula passes, by a continuous change in the value of x, from a positive value to a negative, which equally belong to the same formula. In the same manner in geometry, in Fig. 1 1, if two lines MN, OP are referred to one another by ordinates AB, CD, & also cut one another in E ; then by a continuous motion of the ordinate itself it passes from positive to negative, the direction of AB, CD, which are here taken to be positive, being changed to that of FG, HI, after evanescence at E. To the same continuous line OEP belongs equally the whole of this series of ordinates ; & OE, where the ordinates are positive, is not a different line, or geometrical locus from EP, where the ordinates are negative. Now the nature of any variable quantity, & very frequently also the law, can be expressed by an algebraical formula, & can always be expressed by some line ; for if a perpendicular be drawn to correspond to each separate state of the quantity, the vertices of all perpendiculars so drawn will undoubtedly form some continuous line. If the line never passes over to the other side of the axis, if the formula has no negative value, then also the quantity will always remain positive. But if the line changes side, or the formula the sign of its value, then the quantity itself must also have a change of the same kind. Further, as the change depends on the nature of the formula & the line expressing it, & its position with respect to the axis ; so also the same change will depend on the nature of the quantity ; & just as there are not two formulas, or two lines of different species to represent the positives & the negatives, so also there will not be in the quantity two natures, or two species, of which the one yields positives & the other negatives, as the one a progressive & the other a retrograde motion, the one approach & the other recession, & in the matter under consideration the one will give attractions & the other repulsions. But it will be one & the same nature & wholly belonging to the same spec es of quantity. in. Lastly, this is the proper place for me to bring forward an argument that I used whether there can i i . T\ T /-. -T. r . ,..,!.. , .be a transition in the dissertation De Lege Continmtatis ; by it indeed it is proved that a theory of attractive {rom positive to & repulsive forces for different distances is far more reasonable than one of attractive negative ; mves- forces only, or of repulsive forces only. Let us imagine that we are quite ignorant of the of 8 thenature of the kind of forces that exist in Nature, whether they are only attractive or only repulsive, or curv e only, both ; it would be allowable to use the following reasoning to help us to investigate the matter. Without doubt there will be some continuous line which, by means of ordinates drawn from it to an axis representing distances, will determine the forces ; & according ioo PHILOSOPHIC NATURALIS THEORIA cent. qi secuerit ; vires erunt alibi attractive, alibi repulsivae ; vel ubique attractive tantum, aut repulsive tantum. Videndum igitur, an sit ration! consentaneum magis, lineam ejus naturae, & positionis censere, ut axem alicubi secet, an ut non secet. Transitum deduci U2. Inter rectas axem rectilineum unica parallela ducta per quod vis datum punctum sint 0> curvse, P quas non secat j omnes alie numero infinitae secant alicubi. Curvarum nulla est, quam infinitae recte secent, quam numero rectae secare non possint ; & licet aliquae curvae ejus naturae sint, ut eas aliquae rectae non secent ; tamen & eas ipsas aliae infinite numero recte secant, & infinite numero curve, quod Geometrie sublimioris peritis est notissimum, sunt ejus nature, ut nulla prorsus sit recta linea, a qua possint non secari. Hujusmodi ex. gr. est parabola ilia, cujus ordinate sunt in ratione triplicata abscissarum. Quare infinite numero curve sunt, & infinite numero rectae, que sectionem necessario habeant, pro quavis recta, que non habeat, & nulla est curva, que sectionem cum axe habere non possit. Ergo inter casus possibles multo plures sunt ii, qui sectionem admittunt, quam qui ea careant ; adeoque seclusis rationibus aliis omnibus, & sola casuum probabilitate, & rei [51] natura abstracte considerata, multo magis rationi consentaneum est, censere lineam illam, que vires exprimat, esse unam ex iis, que axem secant, quam ex iis, que non secant, adeoque & ejusmodi esse virium legem, ut attractiones, & repulsiones exhibeat simul pro diversis distantiis, quam ut alteras tantummodo referat ; usque adeo rei natura considerata non solam attractionem, vel solam repulsionem, sed utramque nobis objicit simul. punctis a recta. secabiles Ulterior perqui- u* ged eodem argumento licet ultenus quoque progredi, & primum etiam difficultatis sitio: curvarum J , o -j i genera : quo aiti- caput amovere, quod a sectionum, & idcirco etiam arcuum jam attractivorum, jam repulsi- ores, eo in piuribus vorum multiplicitate desumitur. Curvas lineas Geometre in quasdam classes dividunt uni , ......... P e anaiyseos, que earum naturam expnmit per mas, quas Analyste appellant, equationes, & que ad varies gradus ascendunt. Aequationes primi gradus exprimunt rectas ; equati- ones secundi gradus curvas primi generis ; equationes tertii gradus curvas secundi generis, atque ita porro ; & sunt curve, que omnes gradus transcendunt finite Algebre, & que idcirco dicuntur transcendentes. Porro illud demonstrant Geometre in Analysi ad Geometriam applicata, lineas, que exprimuntur per equationem primi gradus, posse secari a recta in unico puncto ; que equationem habent gradus secundi, tertii, & ita porro, secari posse a recta in punctis duobus, tribus, & ita porro : unde fit, ut curva noni, vel nonagesimi noni generis secari possit a recta in punctis decem, vel centum. itidem sum plures in eo- J am vero curvae primi generis sunt tantummodo tres conice sectiones, ellipis, parabola, hyperbola, adnumerato ellipsibus etiam circulo, que quidem veteribus quoque Geometris innotuerunt. Curvas secundi generis enumeravit Newtonus omnium primus, & sunt circiter octoginta ; curvarum generis tertii nemo adhuc numerum exhibuit accura- tum, & mirum sane, quantus sit is ipse illarum numerus. Sed quo altius assurgit curve genus, eo plures in eo genere sunt curve, progressione ita in immensum crescente, ut ubi aliquanto altius ascenderit genus ipsum, numerus curvarum omnem superet humane imaginationis vim. Idem nimirum ibi accidit, quod in combinationibus terminorum, de quibus supra mentionem fecimus, ubi diximus a 24 litterulis omnes exhiberi voces linguarum omnium, & que fuerunt, aut sunt, & que esse possunt. Deductio inde piu- jjr I n de iam pronum est argumentationem hujusmodi instituere. Numerus rimarum mtersec- .. J .. . ,.. J .. tionum, axis, & linearum, que axem secare possint in punctis quamplunmis, est in immensum major earum curvae exprimentis numero, quae non possint, nisi in paucis, vel unico : igitur ubi agitur de linea exprimente legem virium, ei, qui nihil aliunde sciat, in immensum probabilius erit, ejusmodi lineam esse ex prio-[52]-rum genere unam, quam ex genere posteriorum, adeoque ipsam virium naturam plurimos requirere transitus ab attractionibus ad repulsiones, & vice versa, quam paucos, vel nullum. - Sed omissa ista conjecturali argumentatione quadam, formam curve exprimentis simpiicem: in quo vires positive argumento a phenomenis Nature deducto nos supra determinavimus cum plurimis intersectionibus, que transitus ejusmodi quamplurimos exhibeant. Nee ejusmodi curva debet esse e piuribus arcubus temere compaginata, & compacta : diximus enim, 11 * A THEORY OF NATURAL PHILOSOPHY 101 as it will cut the axis, or will not, the forces will be either partly attractive & partly repulsive, or everywhere only attractive or only repulsive. Accordingly it is to be seen if it is more reasonable to suppose that a line of this nature & position cuts the axis anywhere, or does not. 112. Amongst straight lines there is only one, drawn parallel to the rectilinear axis, intersection is to through any given point that does not cut the axis; all the rest (infinite in number) will the factThat t f here cut it somewhere. There is no curve that an infinite number of straight lines cannot cut ; are more lines that & although there are some curves of such a nature that some straight lines do not cut them, thL^es^hat^o yet there are an infinite number of other straight lines that do cut these curves ; & there not. are an infinite number of curves, as is well-known to those versed in higher geometry, of such a nature that there is absolutely not a single straight line by which they cannot be cut. An example of this kind of curve is that parabola, in which the ordinates are in the triplicate ratio of the abscissae. Hence there are an infinite number of curves & an infinite number of straight lines which necessarily have intersection, corresponding to any straight line that has not ; & there is no curve that cannot have intersection with an axis. Therefore amongst the cases that are possible, there are far more curves that admit intersection than those that are free from it ; hence, putting all other reasons on one side, & considering only the probability of the cases & the nature of the matter on its own merits, it is far more reasonable to suppose that the line representing the forces is one of those, which cut the axis, than one of those that do not cut it. Thus the law of forces is such that it yields both attractions & repulsions (for different distances), rather than such that it deals with either alone. Thus far the nature of the matter has been considered, with the result that it presents to us, not attraction alone, nor repulsion alone, but both of these together. 113. But we can also proceed still further adopting the same line of argument, & Further investiga- first of all remove the chief point of the difficulty, that is derived from the multiplicity S^L.^JILhi ri* */i i i p i i i curves , nit, iijgiicr of the intersections, & consequently also of the arcs alternately attractive & repulsive, their order, the Geometricians divide curves into certain classes by the help of analysis, which expresses wWcV^a ^teaight their nature by what the analysts call equations ; these equations rise to various degrees, line can cut them. Equations of the first degree represent straight lines, equations of the second degree represent curves of the first class, equations of the third degree curves of the second class, & so on. There are also curves which transcend all degrees of finite algebra, & on that account these are called transcendental curves. Further, geometricians prove, in analysis applied to geometry, that lines that are expressed by equations of the first degree can be cut by a straight line in one point only ; those that have equations of the second, third, & higher degrees can be cut by a straight line in two, three, & more points respectively. Hence it comes about that a curve of the ninth, or the ninety-ninth class can be cut by a straight line in ten, or in a hundred, points. 114. Now there are only three curves of the first class, namely the conic sections, the As the class gets parabola, the ellipse & the hyperbola; the circle is included under the name of ellipse; gh " of that & these three curves were known to the ancient geometricians also. Newton was the class becomes im- first of all persons to enumerate the curves of the second class, & there are about eighty mensel y greater. of them. Nobody hitherto has stated an exact number for the curves of the third class ; & it is really wonderful how great is the number of these curves. Moreover, the higher the class of the curve becomes, the more curves there are in that class, according to a progression that increases in such immensity that, when the class has risen but a little higher, the number of curves will altogether surpass the fullest power of the human imagination. Indeed the same thing happens in this case as in combinations of terms ; we mentioned the latter above, when we said that by means of 24 little letters there can be expressed all the words of all languages that ever have been, or are, or can be in the future. 115. From what has been said above we are led to set up the following line of argument. Hence we deduce The number of lines that can cut the axis in very many points is immensely greater than that there . are ^ , , , .... ' , ' r . . . ' f> many intersections the number of those that can cut it in a few points only, or in a single point. Hence, when O f the axis and the the line representing the law of forces is in question, it will appear to one. who otherwise ? urve representing i i i i i r i 111 , forces. knows nothing about its nature, that it is immensely more probable that the curve is of the first kind than that it is of the second kind ; & therefore that the nature of the forces must be such as requires a very large number of transitions from attractions to repulsions & back again, rather than a small number or none at all. 116. But, omitting this somewhat conjectural line of reasoning, we have already it may be that the determined, by what has been said above, the form of the curve representing forces by a j|? I^SSlJ 8 . ' rxr /iii simple , tnecuarac- ngorous argument derived trom the phenomena of Nature, & that there are very many teristic of simplicity intersections which represent just as many of these transitions. Further, a curve of this mcurves - 102 PHILOSOPHIC NATURALIS THEORIA notum esse Geometris, infinita esse curvarum genera, quae ex ipsa natura sua debeant axem in plurimis secare punctis, adeoque & circa ipsum sinuari ; sed praeter hanc generalem responsionem desumptam a generali curvarum natura, in dissertatione De Lege Firium in Natura existentium ego quidem directe demonstravi, curvam illius ipsius formae, cujusmodi ea est, quam in fig. i exhibui, simplicem esse posse, non ex arcubus diversarum curvarum compositam. Simplicem autem ejusmodi curvam affirmavi esse posse : earn enim simplicem appello, quae tota est uniformis naturae, quae in Analysi exponi possit per aequationem non resolubilem in plures, e quarum multiplicatione eadem componatur cujuscunque demum ea curva sit generis, quotcunque habeat flexus, & contorsiones. Nobis quidem altiorum generum curvae videntur minus simplices ; quh nimirum nostrae humanae menti, uti pluribus ostendi in dissertatione De Maris Aestu, & in Stayanis Supplementis, recta linea videtur omnium simplicissima, cujus congruentiam in superpositione intuemur mentis oculis evidentissime, & ex qua una omnem nos homines nostram derivamus Geometriam ; ac idcirco, quae lineae a recta recedunt magis, & discrepant, illas habemus pro compositis, & magis ab ea simplicitate, quam nobis confinximus, recedentibus. At vero lineae continuae, & uniformis naturae omnes in se ipsis sunt aeque simplices ; & aliud mentium genus, quod cujuspiam ex ipsis proprietatem aliquam aeque evidenter intueretur, ac nos intuemur congruentiam rectarum, illas maxime simplices esse crederet curvas lineas, ex ilia earum proprietate longe alterius Geometrise sibi elementa conficeret, & ad illam ceteras referret lineas, ut nos ad rectam referimus ; quas quidem mentes si aliquam ex. gr. parabolae pro- prietatem intime perspicerent, atque intuerentur, non illud quaarerent, quod nostri Geometrae quaerunt, ut parabolam rectificarent, sed, si ita loqui fas est, ut rectam parabolarent. Problema continens 1 1 7. Et quidem analyseos ipsius profundiorem cognitionem requirit ipsa investigatio naturam curvaeana- aequationis, qua possit exprimi curva ems formae, quae meam exhibet virium legem. lytice expnmendam. /! j- 11 ji -i Quamobrem hie tantummodo exponam conditiones, quas ipsa curva habere debet, & quibus aequatio ibi inventa satis facere [53] debeat. (c) Continetur autem id ipsum num. 75, illius dissertationis, ubi habetur hujusmodi Problema : Invenire naturam curvce, cujus abscissis exprimentibus distantias, ordinal exprimant vires, mutatis distantiis utcunque mutatas, y in datis quotcunque limitibus transeuntes e repulsivis in attractivas, ac ex attractivis in repulsivas, in minimis autem distantiis repulsivas, W ita crescentes, ut sint pares extinguendce cuicunque velocitati utcunque magnce. Proposito problemate illud addo : quoniam posuimus mutatis distantiis utcunque mutatas, complectitur propositio etiam rationem quee ad rationem reciprocam duplicatam distantiarum accedat, quantum libuerit, in quibusdam satis magnis distantiis. Conditiones ejus problematis. 1 18. His propositis numero illo 75, sequenti numero propono sequentes sex conditiones, quae requirantur, & sufficiant ad habendam curvam, quse quaeritur. Primo : ut sit regularis, ac simplex, & non composita ex aggregate arcuum diversarum curvarum. Secundo : ut secet axem C'AC figures i. tantum in punctis quibusdam datis ad binas distantias AE', AE ; AG', AG ; y ita porro cequales (d) bine, y inde. Tertio : ut singulis abscissis respondeant singulcs ordinatcf. ( e ) Quarto : ut sumptis abscissis cequalibus hinc, y inde ab A, respondeant ordinal* (c) Qui velit ipsam rei determinationem videre, poterit hie in fine, ubi Supphmentorum, 3. exhibebitur solutio problematis, qua in memorata dissertatione continetur a num. 77. ad no. Sed W numerorum ordo, & figurarum mutabitur, ut cum reliquis hujusce operis cohtereat. Addetur prieterea eidem . postremum scholium pertinens ad qu<sstionem agitatam ante has aliquot annos Parisiis ; an vis mutua inter materite particulas debeat omnino exprimi per solam aliquam distantiee potenttam, an possit per aliquam ejus functionem ; W constabit, posse utique per junctionem, ut hie ego presto, qute uti superiore numero de curvts est dictum, est in se eeque simplex etiam, ubi nobis potentias ad ejus expressionem adhibentibus videatur admodum composita. (d) Id, ut y quarta conditio, requiritur, ut curva utrinque sit sui similis, quod ipsam magis uniformem reddit ; quanquam de illo crure, quod est citra asymptotum AB, nihil est, quod soliciti simus ; cum ob vim repulsivam imminutis distantiis ita in infinitum excrescentem, non possit abscissa distantiam exprimens unquam evadere zero, W abire in negativam. (e) Nam singulis distantiis singulte vires respondent. A THEORY -OF NATURAL PHILOSOPHY 103 kind is not bound to be built up by connecting together a number of independent arcs. For, as I said, it is well known to Geometricians that there are an infinite number of classes of curves that, from their very nature, must cut the axis in a very large number of points, & therefore also wind themselves about it. Moreover, in addition to this general answer to the objector, derived from the general nature of curves, in my dissertation De Lege Firium in Natura existentium, I indeed proved in a straightforward manner that a curve, of the form that I have given in Fig. i, might be simple & not built up of arcs of several different curves. Further, I asserted that a simple curve of this kind was perfectly feasible ; for I call a curve simple, when the whole of it is of one uniform nature. In analysis, this can be expressed by an equation that is not capable of being resolved into several other equations, such that the former is formed from the latter by multiplication ; & that too, no matter of what class the curve may be, or how many flexures or windings it may have. It is true that the curves of higher classes seem to us to be less simple ; this is so because, as I have shown in several places in the dissertation De Marts Aestu, & the supplements to Stay's Philosophy, a straight line seems to our human mind to be the simplest of all lines ; for we get a real clear mental perception of the congruence on superposition in the case of a straight line, & from this we human beings form the whole of our geometry. On this account, the more that lines depart from straightness & the more they differ, the more we consider them to be composite & to depart from that simplicity that we have set up as our standard. But really all lines that are continuous & of uniform nature are just as simple as one another. Another kind of mind, which might form an equally clear mental perception of some property of any one of these curves, as we do the congruence of straight lines, might believe these curves to be the simplest of all & from that property of these curves build up the elements of a far different geometry, referring all other curves to that one, just as we compare them with a straight line. Indeed, these minds, if they noticed & formed an extremely clear perception of some property of, say, the parabola, would not seek, as our geometricians do, to rectify the parabola ; they would endeavour, if one may use the words, to parabolify a straight line. 1 17. The investigation of the equation, by which a curve of the form that will represent p T bl !J n . ' i ~ j' i i j f i ir 1T71. r "* the analytical my law of forces can be expressed, requires a deeper knowledge 01 analysis itselt. Wnereiore expression of the I will here do no more than set out the necessary requirements that the curve must fulfil nature of the curve. & those that the equation thereby discovered must satisfy. (c) It is the subject of Art. 75 of the dissertation De Lege Firium, where the following problem is proposed. Required to find the nature of the curve, whose abscissa represent distances & whose ordinates represent forces that are changed as the distances are changed in any manner, y pass from attractive forces to repulsive, & from repulsive to attractive, at any given number of limit-points ; further, the forces are repulsive at extremely small distances and increase in such a manner that they are capable of destroying any velocity, however great it may be. To the problem as there proposed I now add the following : As we have used the words are changed as the distances are changed in any manner, the proposition includes also the ratio that approaches as nearly as you please to the reciprocal ratio of the squares of the distances, whenever the distances are sufficiently great. 1 1 8. In addition to what is proposed in this Art. 75, I set forth in the article that The * of follows it the following six conditions ; these are the necessary and sufficient conditions for determining the curve that is required. (i) The curve is regular & simple, & not compounded of a number of arcs of different curves. (ii) It shall cut the axis C'AC of Fig. I, only in certain given points, whose distances, AE',AE, AG', AG, and so on, are equal (<t) in pairs on each side of A [see p. 80]. (iii) To each abscissa there shall correspond one ordinate y one only, (f) (iv) To equal abscisses, taken one on each side of A, there shall correspond equal ordinates. (c) Anyone who desires to see the solution of the -problem will be able to do seat the end of this work; it will be found in 3 of the Supplements ; it is the solution of the problem, as it was given in the dissertation mentioned above, from Art. 77 to no. But here both the numbering of the articles W of the diagrams have been changed, so as to agree with the rest of the work. In addition, at the end of this section, there will be found a final note dealing with a question that was discussed some years ago in Paris. Namely, whether the mutual force between particles of mat- ter is bound to be expressible by some one power of the distance only, or by some function of the distance. It will be evident that at any rate it may be expressible by a function as I here assert ; y that function, as has been stated in the article above, is perfectly simple in itself also ; whereas, if we adhere to an expression by means of powers, the curve will seem to be altogether complex. (d) This, y the fourth condition too, is required to make the curve symmetrical, thus giving it greater uniformity ; although we are not concerned with the branch on the other side of the asymptote AB at all. For, on account of the repulsive force at very small distances increasing indefinitely in such a manner as postulated, it is impossible that the abscissa that represents the distance should ever become zero y then become negative. (e) For to each distance one force, & and only one, corresponds. 104 PHILOSOPHIC NATURALIS THEORIA czquales. Quinto : ut babeant rectam AB pro asymptoto, area asymptotica BAED existente ( ) infinita. Sexto : ut arcus binis quibuscunque intersectionibus terminati possint variari, ut libuerit, fcf? ad quascunque distantias recedere ab axe C'AC, ac accedcre ad quoscunque quarum- cunque curvarum arcus, quantum libuerit, eos secanda, vel tangendo, vel osculando ubicunque, ff quomodocunque libuerit. soiutio IrT U attrac IS4] IT 9- Verum quod ad multiplicitatem virium pertinet, quas diversis jam Physici tionem gravitatis nominibus appellant, illud hie etiam notari potest, si quis singulas seorsim considerare velit, licere illud etiam, hanc curvam in se unicam per resolutionem virium cogitatione nostra, atque fictione quadam, dividere in plures. Si ex. gr. quis velit considerare in materia gravitatem generalem accurate reciprocam distantiarum quadratis ; poterit sane is describere ex parte attractiva hyperbolam illam, quae habeat accurate ordinatas in ratione reciproca duplicata distantiarum, quse quidem erit quaedam velut continuatio cruris VTS, turn singulis ordinatis ag, dh curvae virium expressae in fig. I. adjungere ordinatas hujus novae hyperbolae ad partes AB incipiendo a punctis curvae g, b, & eo pacto orietur nova quaedam curva, quae versus partes pV coincidet ad sensum cum axe oC, in reliquis locis ab eo distabit, & contor- quebitur etiam circa ipsum, si vertices F, K, O distiterint ab axe magis, quam distet ibidem hyperbola ilia. Turn poterit dici, puncta omnia materiae habere gravitatem decrescentem accurate in ratione reciproca duplicata distantiarum, & simul habere vim aliam expressam ab ilia nova curva : nam idem erit, concipere simul hasce binas leges virium, ac illam praecedentem unicam, & iidem effectus orientur. Hujus posterioris vis resolutio in alias plures. 1 20. Eodem pacto haec nova curva potest dividi in alias duas, vel plures, concipiendo aliam quamcunque vim, ut ut accurate servantem quasdam determinatas leges, sed simul mutando curvam jam genitam, translatis ejus punctis per intervalla aequalia ordinatis respondentibus novae legi ass.umptae. Hoc pacto habebuntur plures etiam vires diversae, quod aliquando, ut in resolutione virium accidere diximus, inserviet ad faciliorem deter- minationem effectuum, & ea erit itidem vera virium resolutio quaedam ; sed id omne erit nostrae mentis partus quidam ; nam reipsa unica lex virium habebitur, quam in fig. I . exposui, & quae ex omnibus ejusmodi legibus componetur. Non obesse theo- r i a m gravitatis ; distantiis locum non habet. 121. Quoniam autem hie mentio injecta est gravitatis decrescentis accurate in ratione cujusiex1n a minimis reciproca duplicata distantiarum ; cavendum, ne cui difficultatem aliquam pariat illud, '""" m quod apud Physicos, & potissimum apud Astronomiae mechanicae cultores, habetur pro comperto, gravitatem decrescere in ratione reciproca duplicata distantiarum accurate, cum in hac mea Theoria lex virium discedat plurimum ab ipsa ratione reciproca duplicata distantiarum. Inprimis in minoribus distantiis vis integra, quam in se mutuo exercent particulae, omnino plurimum discrepat a gravitate, quae sit in ratione reciproca duplicata distantiarum. Nam & vapores, qui tantam exercent vim ad se expandendos, repulsionem habent utique in illis minimis distantiis a se invicem, non attractionem ; & ipsa attractio, quae in cohaesione se prodit, est ilia quidem in immensum major, quam quae ex generali gravitate consequitur ; cum ex ipsis Newtoni compertis attractio gravitati respondens [55] in globes homogeneos diversarum diametrorum sit in eadem ratione, in qua sunt globorum diametri, adeoque vis ejusmodi in exiguam particulam est eo minor gravitate corporum in Terram, quo minor est diameter particulae diametro totius Terrae, adeoque penitus insen- sibilis. Et idcirco Newtonus aliam admisit vim pro cohaesione, quae decrescat in ratione majore, quam sit reciproca duplicata distantiarum ; & multi ex Newtonianis admiserunt vim respondentem huic formulae - 3 + - v cujus prior pars respectu posterioris sit in immensum minor, ubi x sit in immensum major unitate assumpta ; sit vero major, ubi x sit in immensum minor, ut idcirco in satis magnis distantiis evanescente ad sensum prima parte, vis remaneat quam proxime in ratione reciproca duplicata distantiarum x, in minimis vero distantiis sit quam proxime in ratione reciproca triplicata : usque adeo ne apud Newtonianos quidem servatur omnino accurate ratio duplicata distantiarum. EX pianetarum I2 2. Demonstravit quidem Newtonus, in ellipsibus planetariis, earn, quam Astronomi q^ampro lineam apsidum nominant, & est axis ellipseos, habituram ingentem motum, si ratio virium ime, non accurate. a re ciproca duplicata distantiarum aliquanto magis aberret, cumque ad sensum quiescant (f) Id requiritur, quia in Mecbanica demonstrator, aream curves, cujus abscissa fxprimant distantias, 13 ordinatx vires, exprimere incrementum, vel decrementum quadrati velocitatis : quare ut illte vires sint pares extinguendte veloci- tati cuivis utcunque magna, debet ilia area esse omni finita major. A THEORY OF NATURAL PHILOSOPHY 105 (v) The straight line AB shall be an asymptote, and the asymptotic area BAED shall be infinite. (f) (vi) The arcs lying between any two intersections may vary to any extent, may recede to any distances whatever from the axis C AC, and approximate to any arcs of any curves to any degree of closeness, cutting them, or touching them, or osculating them, at any points and in any manner. 119. Now, as regards the multiplicity of forces which at the present time physicists call Resolution of the by different names, it can also here be observed that, if anyone wants to consider one of these U j^f of N^wtonUn separately, the curve though it is of itself quite one-fold can yet be divided into several attraction of parts by a sort of mental & fictitious resolution of the forces. Thus, for instance, if f^fother 1 force* 1 d anyone wishes to consider universal gravitation of matter exactly reciprocal to the squares of the distances ; he can indeed describe on the attractive side the hyperbola which has its ordinates accurately in the inverse ratio of the squares of the distances, & this will be as it were a continuation of the branch VTS. Then he can add on to every ordinate, such as ag, dh, the ordinates of this new hyperbola, in the direction of AB, starting in each case from points on the curve, as g,h ; & in this way there will be obtained a fresh curve, which for the part pV will approximately coincide with the axis 0C, & for the remainder will recede from it & wind itself about it, if the vertices F,K,O are more distant from the axis than the corresponding point on the hyperbola. Then it can be stated that all points of matter have gravitation accurately decreasing in the inverse square of the distance, together with another force represented by this new curve. For it comes to the same thing to think of these two laws of forces acting together as of the single law already given ; & the results that arise will be the same also. 1 20. In the same manner this new curve can be divided into two others, or several The resolution of others, by considering some other force, in some way or other accurately obeying certain ^o several other fixed laws, & at the same time altering the curve just obtained by translating the points of it forces. through intervals equal to the ordinates corresponding to the new law that has been taken. In this manner several different forces will be obtained ; & this will be sometimes useful, as we mentioned that it would be in resolution of forces, for determining their effects more readily ; & will be a sort of true resolution of forces. But all this will be as it were only a conception of our mind ; for, in reality, there is a single law of forces, & that is the one which I gave in Fig. i, & it will be the compounded resultant of all such forces as the above. 121. Moreover, since I here make mention of gravitation decreasing accurately in the The .. t. heor .y * , r11 . .. 111 1111 gravitation is not inverse ratio ot the squares ot the distances, it is to be remarked that no one should make in opposition ; this any difficulty over the fact that, amongst physicists & more especially those who deal with l! J^ 'H d t es v not hol n celestial mechanics, it is considered as an established fact that gravitation decreases accurately distances. in the inverse ratio of the squares of the distances, whilst in my Theory the law of forces is very different from this ratio. Especially, in the case of extremely small distances, the whole force, which the particles exert upon one another, will differ very much in every case from the force of gravity, if that is supposed to be inversely proportional to the squares of these distances. For, in the case of gases, which exercise such a mighty force of self-expansion, there is certainly repulsion at those very small distances from one another, & not attraction ; again, the attraction that arises in cohesion is immensely greater than it ought to be according to the law of universal gravitation. Now, from the results obtained by Newton, the attraction corresponding to gravitation in homogeneous spheres of different diameters varies as the diameters of the spheres ; & therefore this kind of force for the case of a tiny particle is as small in proportion to the gravitation of bodies to the Earth as the diameter of the particle is small in proportion to the diameter of the whole Earth ; & is thus insensible altogether. Hence Newton admitted another force in the case of cohesion, decreasing in a greater ratio than the inverse square of the distances ; also many of the followers of Newton have admitted a force corresponding to the formula, a'x 3 + b'x 2 ; in this the first term is immensely less than the second, when x is immensely greater than some distance assumed as unit distance ; & immensely greater, when x is immensely less. By this means, at sufficiently great distances the first part practically vanishes & the force remains very approximately in the inverse ratio of the squares of the distances x ; whilst, at very small distances, it is very nearly in the inverse ratio of the cubes of the distances. Thus indeed, not even amongst the followers of Newton has the inverse ratio of the squares of the distances been altogether rigidly adhered to. 122. Now Newton proved, in the case of planetary elliptic orbits, that that which The law follows Astronomers call the apsidal line, i.e., the axis of the ellipse, would have a very great motion, not 7 , if the ratio of the forces varied to any great extent from the inverse ratio of the squares from the apheiia of of the distances ; & since as far as could be observed the lines of apses were stationary (f) This is required because in Mechanics it is shown that the area of a curve, whose abscissa r'present distances y ordinates forces, represents the increase or decrease of the square of the velocity. Hence in order that the forces should be capable of destroying any velocity however great, this area must be greater than any finite area. io6 PHILOSOPHIC NATURALIS THEORIA in earum orbitis apsidum linese, intulit, earn rationem observari omnino in gravitate. At id nequaquam evincit, accurate servari illam legem, sed solum proxime, neque inde ullum efficax argumentum contra meam Theoriam deduci potest. Nam inprimis nee omnino quiescunt illae apsidum lineae, sive, quod idem est, aphelia planetarum, sed motu exiguo quidem, at non insensibili prorsus, moventur etiam respectu fixarum, adeoque motu non tantummodo apparente, sed vero. Tribuitur is motus perturbationi virium ortae ex mutua planetarum actione in se invicem ; at illud utique hue usque nondum demonstratum est, ilium motum accurate respondere actionibus reliquorum planetarum agentium in ratione reciproca duplicata distantiarum ; neque enim adhuc sine contemptibus pluribus, & approximationibus a perfectione, & exactitudine admodum remotis solutum est problema, quod appellant, trium corporum, quo quasratur motus trium corporum in se mutuo agentium in ratione reciproca duplicata distantiarum, & utcunque projectorum, ac illae ipsae adhuc admodum imperfectae solutiones, quae prolatae hue usque sunt, inserviunt tantummodo particularibus quibusdam casibus, ut ubi unum corpus sit maximum, & remotissimum, quemadmodum Sol, reliqua duo admodum minora & inter se proxima, ut est Luna, ac Terra, vel remota admodum a majore, & inter se, ut est Jupiter, & Saturnus. Hinc nemo hucusque accuratum instituit, aut etiam instituere potuit calculum pro actione perturbativa omnium planetarum, quibus si accedat actio perturbativa cometarum, qui, nee scitur, quam multi sint, nee quam longe abeant ; multo adhuc magis evidenter patebit, nullum inde confici posse argumentum [56] pro ipsa penitus accurata ratione reciproca duplicata distantiarum. I2 3- Clairautius quidem in schediasmate ante aliquot annos impresso, crediderat, ex autem hanc legem ipsis motibus Kneje apsidum Lunae colligi sensibilem recessum a ratione reciproca duplicata a uantum iftmerit[ m distantiae, & Eulerus in dissertatione De Aberrationibus Jovis, W Saturni, quas premium retulit ab Academia Parisiensi an. 1748, censuit, in ipso Jove, & Saturno haberi recessum admodum sensibilem ab ilia ratione ; sed id quidem ex calculi defectu non satis product! sibi accidisse Clairautius ipse agnovit, ac edidit ; & Eulero aliquid simile fortasse accidit : nee ullum habetur positivum argumentum pro ingenti recessu gravitatis generalis a ratione duplicata distantiarum in distantia Lunae, & multo magis in distantia planetarum. Vero nee ullum habetur argumentum positivum pro ratione ita penitus accurata, ut discrimen sensum omnem prorsus effugiat. At & si id haberetur ; nihil tamen pati posset inde Theoria mea ; cum arcus ille meae curvae postremus VT possit accedere, quantum libuerit, ad arcum illius hyperbolae, quae exhibet legem gravitatis reciprocam quadratorum dis- tantiae, ipsam tangendo, vel osculando in punctis quotcunque, & quibuscunque ; adeoque ita possit accedere, ut discrimen in iis majoribus distantiis sensum omnem effugiat, & effectus nullum habeat sensibile discrimen ab effectu, qui responderet ipsi legi gravitatis ; si ea accurate servaret proportionem cum quadratis distantiarum reciproce sumptis. Difficuitas a Mau- 124. Nee vero quidquam ipsi meae virium Theorias obsunt meditationes Maupertuisii, tionemaxfma^Nlw- ingeniosae illae quidem, sed meo judicio nequaquam satis conformes Natune legibus circa tonianae legis. legem virium decrescentium in ratione reciproca duplicata distantiarum, cujus ille perfec- tiones quasdam persequitur, ut illam, quod in hac una integri globi habeant eandem virium legem, quam singulae particulae. Demonstravit enim Newtonus, globos, quorum singuli paribus a centre distantiis homogenei sint, & quorum particulae minimae se attrahant in ratione reciproca duplicata distantiarum, se itidem attrahere in eadem ratione distantiarum reciproca duplicata. Ob hasce perfectiones hujus Theoriae virium ipse censuit hanc legem reciprocam duplicatam distantiarum ab Auctore Naturae selectam fuisse, quam in Natura esse vellet. Prima responsio : 125. At mihi quidem inprimis nee unquam placuit, nee placebit sane unquam in n !^Jf 8 T s ^rwt 8 investieatione Naturae causarum fmalium usus, quas tantummodo ad meditationem quandam, onmcs, *x jjcricui~ o f i i t XT 1*1* * "VT iones, ac seiigi et- contemplationemque, usui esse posse abitror, ubi leges JNaturse aliunde innotuennt. JNam ^J$L*5* nee perfectiones omnes innotescere nobis possunt, qui intimas rerum naturas nequaquam III grH.ilclIU pcrlcC~ * f . * a C tionum. inspicimus, sed externas tantummodo propnetates quasdam agnoscimus, & lines omnes, quos Naturae Auctor sibi potuit [57] proponere, ac proposuit, dum Mundum conderet, A THEORY OF NATURAL PHILOSOPHY 107 in the orbits of each, he deduced that the ratio of the inverse square of the distances was exactly followed in the case of gravitation. But he only really proved that that law was very approximately followed, & not that it was accurately so ; nor from this can any valid argument against my Theory be brought forward. For, in the first place these lines of apses, or what comes to the same thing, the aphelia of the planets are not quite stationary ; but they have some motion, slight indeed but not quite insensible, with respect to the fixed stars, & therefore move not only apparently but really. This motion is attributed to the perturbation of forces which arises from the mutual action of the planets upon one another. But the fact remains that it has never up till now been proved that this motion exactly corresponds with the actions of the rest of the planets, where this is in accordance with the inverse ratio of the squares of the distances. For as yet the problem of three bodies, as they call it, has not been solved except by much omission of small quantities & by adopting approximations that are very far from truth and accuracy ; in this problem is investigated the motion of three bodies acting mutually upon one another in the inverse ratio of the squares of the distances, & projected in any manner. Moreover, even these still only imperfect solutions, such as up till now have been published, hold good only in certain particular cases ; such as the case in which one of the bodies is very large & at a very great distance, the Sun for instance, whilst the other two are quite small in comparison & very near one another, as are the Earth and the Moon, or at a large distance from the greater & from one another as well, as Jupiter & Saturn. Hence nobody has hitherto made, nor indeed could anybody make, an accurate calculation of the disturbing influence of all the other planets combined. If to this is added the disturbing influence of the comets, of which we neither know the number, nor how far off they are ; it will be still more evident that from this no argument can be built up in favour of a perfectly exact observance of the inverse ratio of the squares of the distances. 123. Clairaut indeed, in a pamphlet printed several years ago, asserted his belief that The same thing is he had obtained from the motions of the line of apses for the Moon a sensible discrepancy J? ^ ^duced from , , . r i i AIT--I i i r>^r T the res t of astro- from the inverse square of the distance. Also Euler, in his dissertation De Aberratiombus nomy ; moreover Jovis, y Saturni, which carried off the prize given by the Paris Academy, considered that thls Iaw of . mi 1 e /, ,. T . <-, , r . ' .. , ,. *', can approximate in the case of Jupiter & Saturn there was quite a sensible discrepancy from that ratio, to the other as But Clairaut found out, & proclaimed the fact, that his result was indeed due to a defect nearl y as is desired. in his calculation which had not been carried far enough ; & perhaps something similar happened in Euler's case. Moreover, there is no positive argument in favour of a large discrepancy from the inverse ratio of the squares of the distances for universal gravitation in the case of the distance of the Moon, & still more in the case of the distances of the planets. Neither is there any rigorous argument in favour of the ratio being so accurately observed that the difference altogether eludes all observation. But even if this were the case, my Theory would not suffer in the least because of it. For the last arc VT of my curve can be made to approximate as nearly as is desired to the arc of the hyperbola that represents the law of gravitation according to the inverse squares of the distances, touching the latter, or osculating it in any number of points in any positions whatever ; & thus the approximation can be made so close that at these relatively great distances the difference will be altogether unnoticeable, & the effect will not be sensibly different from the effect that would correspond to the law of gravitation, even if that exactly conformed to the inverse ratio of the squares of the distances. 124. Further, there is nothing really to be objected to my Theory on account of the Objection arising meditations of Maupertuis ; these are certainly most ingenious, but in my opinion in no p r ct ion fccord* way sufficiently in agreement with the laws of Nature. Those meditations of his, I mean, ing to Maupertuis, with regard to the law of forces decreasing in the inverse ratio of the squares of the distances ; j^ f w the Newtoman for which law he strives to adduce certain perfections as this, that in this one law alone complete spheres have the same law of forces as the separate particles of which they are formed. For Newton proved that spheres, each of which have equal densities at equal distances from the centre, & of which the smallest particles attract one another in the inverse ratio of the squares of the distances, themselves also attract one another in the same ratio of the inverse squares of the distances. On account of such perfections as these in this Theory of forces, Maupertuis thought that this law of the inverse squares of the distances had been selected by the Author of Nature as the one He willed should exist in Nature. 125. Now, in the first place I was never satisfied, nor really shall I ever be satisfied, First reply to this ; with the use of final causes in the investigation of Nature ; these I think can only be employed perfections^ 'not for a kind of study & contemplation, in such cases as those in which the laws of Nature known ; and even have already been ascertained from other methods. For we cannot possibly be acquainted ^sdlcted^fo^'fhe with all perfections ; for in no wise do we observe the inmost nature of things, but all we sake of greater per- know are certain external properties. Nor is it at all possible for us to see & know all fl the intentions which the Author of Nature could and did set before Himself when He founded io8 PHILOSOPHIC NATURALIS THEORIA videre, & nosse omnino non possumus. Quin immo cum juxta ipsos Leibnitianos inprimis, aliosque omnes defensores acerrimos principii rationis sufficients, & Mundi perfectissimi, qui inde consequitur, multa quidem in ipso Mundo sint mala, sed Mundus ipse idcirco sit optimus, quod ratio boni ad malum in hoc, qui electus est, omnium est maxima ; fieri utique poterit, ut in ea ipsius Mundi parte, quam hie, & nunc contemplamur, id, quod electum fuit, debuerit esse non illud bonum, in cujus gratiam tolerantur alia mala, sed illud malum, quod in aliorum bonorum gratiam toleratur. Quamobrem si ratio reciproca duplicata distantiarum esset omnium perfectissima pro viribus mutuis particularum, non inde utique sequeretur, earn pro Natura fuisse electam, & constitutam. Eandem legem nee I2 6. At nee revera perfectissima est, quin immo meo quidem judicio est omnino pcrfcctam esse, nee r v 1 i ... in corporibus, non imperfecta, & tarn ipsa, quam aliae plunmse leges, quas requirunt attractionem immmutis utique accurate distantiis crcscentcm in ratione reciproca duplicata distantiarum, ad absurda deducunt ' plurima, vel saltern ad inextricabiles difficultates, quod ego quidem turn alibi etiam, turn inprimis demonstravi in dissertatione De Lege Firium in Natura existentium a num. 59. (g) Accedit autem illud, quod ilia, qua; videtur ipsi esse perfectio maxima, quod nimirum eandem sequantur legem globi integri, quam particulae minimae, nulli fere usui est in Natura ; si res accurate ad exactitudinem absolutam exigatur ; cum nulli in Natura sint accurate perfecti globi paribus a centre distantiis homogenei, nam praeter non exiguam inaequalitatem interioris textus, & irregularitatem, quam ego quidem in Tellure nostra demonstravi in Opere, quod de Litteraria Expeditione per Pontificiam ditionem inscripsi, in reliquis autem planetis, & cometis suspicari possumus ex ipsa saltern analogia, prater scabritiem superficiei, quaj utique est aliqua, satis patet, ipsa rotatione circa proprium axem induci in omnibus compressionem aliquam, quae ut ut exigua, exactam globositatem impedit, adeoque illam assumptam perfectionem maximam corrumpit. Accedit autem & illud, quod Newtoniana determinatio rationis reciprocal duplicatae distantiarum locum habet tantummodo in globis materia continua constantibus sine ullis vacuolis, qui globi in Natura non existunt, & multo minus a me admitti possunt, qui non vacuum tantummodo disseminatum in materia, ut Philosophi jam sane passim, sed materiam in immenso vacuo innatantem, & punctula a se invicem remota, ex quibus, qui apparentes globi fiant, illam habere proprietatem non possunt rationis reciprocal duplicatae distantiarum, adeoque nee illius perfectionis creditas maxime perfectam, absolutamque applicationem. o ex prae- \<:$\ \2j. Demum & illud nonnullis difficultatem parit summam in hac Theoria juuiv-.w pro impul- ~ * ' . . . . . . f i. i i .. sione, & ex testi- Virium, quod censeant, phaenomena omnia per impulsionem explicari debere, & immedi- monio sensuum : a tum contactum, quern ipsum credant evidenti sensuum testimonio evinci ; hinc huiusmodi responsio ad hanc . r n XT i posteriorem. nostras vires immechamcas appellant, & eas, ut & Newtomanorum generalem gravitatem, vel idcirco rejiciunt, quod mechanicae non sint, & mechanismum, quem Newtoniana labefactare coeperat, penitus evertant. Addunt autem etiam per jocum ex serio argumento petito a sensibus, baculo utendum esse ad persuadendum neganti contactum. Quod ad sensuum testimonium pertinet, exponam uberius infra, ubi de extensione agam, quae eo in genere habeamus praejudicia, & unde : cum nimirum ipsis sensibus tribuamus id, quod nostrae ratiocinationis, atque illationis vitio est tribuendum. Satis erit hie monere illud, ubi corpus ad nostra organa satis accedat, vim repulsivam, saltern illam ultimam, debere in organorum ipsorum fibris excitare motus illos ipsos, qui excitantur in communi sententia ab impenetrabilitate, & contactu, adeoque eundem tremorem ad cerebrum propagari, & eandem excitari debere in anima perceptionem, quae in communi sententia excitaretur ; quam ob rem ab iis sensationibus, quae in hac ipsa Theoria Virium haberentur, nullum utique argumentum desumi potest contra ipsam, quod ullam vim habeant utcunque tenuem. Felicius explicari 128. Quod pertinet ad explicationem phaenomenorum per impulsionem immediatam, sione*- "eam^nus- rnonui sane superius, quanto felicius, ea prorsus omissa, Newtonus explicarit Astronomiam, quam positive pro- & Opticam ; & patebit inferius, quanto felicius phaenomena quaeque praecipua sine ulla immediata impulsione explicentur. Cum iis exemplis, turn aliis, commendatur abunde ea ratio explicandi phsenomena, quae adhibet vires agentes in aliqua distantia. Ostendant (g) Qute hue pertinent, (J continentur novem numeris ejus Dissertations incipiendo a 59, habentur in fine Supplem. 4- A THEORY OF NATURAL PHILOSOPHY 109 the Universe. Nay indeed, since in the doctrine of the followers of Leibniz more especially, and of all the rest of the keenest defenders of the principle of sufficient reason, and a most perfect Universe which is a direct consequence of that idea, there may be many evils in the Universe, and yet the Universe may be the best possible, just because the ratio of good to evil, in this that has been chosen, is the greatest possible. It might certainly happen that in this part of the Universe, which here & now we are considering, that which was chosen would necessarily be not that goodness in virtue of which other things that are evil are tolerated, but that evil which is tolerated because of the other things that are good. Hence, even if the inverse ratio of the squares of the distances were the most perfect of all for the mutual forces between particles, it certainly would not follow from that fact that it was chosen and established for Nature. 126. But this law as a matter of fact is not the most perfect of all; nay rather, in This law is neither my opinion, it is altogether imperfect. Both it, & several other laws, that require ^ ec ^Tor i0 Dod t - attraction at very small distances increasing in the inverse ratio of the squares of the distances ies that are not lead to very many absurdities ; or at least, to insuperable difficulties, as I showed in the exactl y spherical, dissertation De Lege Virium in Natura existentium in particular, as well as in other places. (g) In addition there is the point that the thing, which to him seems to be the greatest perfection, namely, the fact that complete spheres obey the same law as the smallest particles composing them, is of no use at all in Nature ; for there are in Nature no exactly perfect spheres having equal densities at equal distances from the centre. Besides the not insignificant inequality & irregularity of internal composition, of which I proved the existence in the Earth, in a work which I wrote under the title of De Litteraria Ex-peditione per Pontificiam ditionem, we can assume also in the remaining planets & the comets (at least by analogy), in addition to roughness of surface (of which it is sufficiently evident that at any rate there is some), that there is some compression induced in all of them by the rotation about their axes. This compression, although it is indeed but slight, prevents true sphericity, & therefore nullifies that idea of the greatest perfection. There is too the further point that the Newtonian determination of the inverse ratio of the squares of the distances holds good only in spheres made up of continuous matter that is free from small empty spaces ; & such spheres do not exist in Nature. Much less can I admit such spheres ; for I do not so much as admit a vacuum disseminated throughout matter, as philosophers of all lands do at the present time, but I consider that matter as it were swims in an immense vacuum, & consists of little points separated from one another. These apparent spheres, being composed of these points, cannot have the property of the inverse ratio of the squares of the distances ; & thus also they cannot bear the true & absolute application of that perfection that is credited so highly. 127. Finally, some persons raise the greatest objections to this Theory of mine, because Objection founded they consider that all the phenomena must be explained by impulse and immediate contact ; "mpui^and'on the this they believe to be proved by the clear testimony of the senses. So they call forces testimony of the like those I propose non-mechanical, and reject them, just as they also reject the universal th?s latter. rep ' y t0 gravitation of Newton, for the alleged reason that they are not mechanical, and overthrow altogether the idea of mechanism which the Newtonian theory had already begun to undermine. Moreover, they also add, by way of a joke in the midst of a serious argument derived from the senses, that a stick would be useful for persuading anyone who denies contact. Now as far as the evidence of the senses is concerned, I will set forth below, when I discuss extension, the prejudices that we may form in. such cases, and the origin of these prejudices. Thus, for instance, we may attribute to the senses what really ought to be attributed to the imperfection of our reasoning and inference. It will be enough just for the present to mention that, when a body approaches close enough to our organs, my repulsive force (at any rate it is that finally), is bound to excite in the nerves of those organs the motions which, according to the usual idea, are excited by impenetrability and contact ; & that thus the same vibrations are sent to the brain, and these are bound to excite the same perception in the mind as would be excited in accordance with the usual idea. Hence, from these sensations, which are also obtained in my Theory of Forces, no argument can be adduced against the theory, which will have even the slightest validity. 128. As regards the explanation of phenomena by means of immediate contact I, h s a ver ^ thin e g is ^^ indeed, mentioned above how much more happily Newton had explained Astronomy and without the idea of Optics by omitting it altogether ; and it will be evident, in what follows, how much more |^^ lse ^ nowhere happily every one of the important phenomena is explained without any idea of immediate rigorously proved contact. - Both by these instances, and by many others, this method of explaining phenomena, to exist - by employing forces acting at a distance, is strongly recommended. Let objectors bring (g) That which refers to this point, & which is contained in nine articles of the dissertation commencing with Art. 59, is to bf found at the end of this work as Supplement IV, no PHILOSOPHISE NATURALIS THEORIA isti vel unicum exemplum, in quo positive probare possint, per immediatam impulsionem communicari motum in Natura. Id sane ii praestabunt nunquam ; cum oculorum testi- monium ad excludendas distantias illas minimas, ad quas primum crus repulsivum pertinet, & contorsiones curvae circa axem, quae oculos necessario fugiunt, adhibere non possint ; cum e contrario ego positive argumento superius excluserim immediatum contactum omnem, & positive probaverim, ipsum, quern ii ubique volunt, haberi nusquam. Vires hujus Theo- I2 g j) e no minibus quidem non esset, cur solicitudinem haberem ullam ; sed ut & rise pertineread ve- ...,*,.., . ~t . . , ' . . ,. ... rum, nee occuitum in nsdem aliquid prasjudicio cmdam, quod ex communi loquendi usu provenit, mud mechanismum. notandum duco, Mechanicam non utique ad solam impulsionem immediatam fuisse restrictam unquam ab iis, qui de ipsa tractarunt, sed ad liberos inprimis adhibitam contem- plandos motus, qui independenter ab omni impulsione habeantur. Quae Archimedes de aequilibrio tradidit, quse Galilaeus de li-[59]-bero gravium descensu, ac de projectis, quae de centralibus in circulo viribus, & oscillationis centre Hugenius, quae Newtonus generaliter de motibus in trajectoriis quibuscunque, utique ad Mechanicam pertinent, & Wolfiana & Euleriana, & aliorum Scriptorum Mechanica passim utique ejusmodi vires, & motus inde ortos contemplatur, qui fiant impulsione vel exclusa penitus, vel saltern mente seclusa. Ubicunque vires agant, quae motum materiae gignant, vel immutent, & leges expandantur, secundum quas velocitas oriatur, mutetur motus, ac motus ipse determinetur ; id omne inprimis ad Mechanicam pertinet in admodum propria significatione acceptam. Quam- obrem ii maxime ea ipsa propria vocum significatione abutuntur, qui impulsionem unicam ad Mechanismum pertinere arbitrantur, ad quern haec virium genera pertinent multo magis, qu33 idcirco appellari jure possunt vires Mechanic*?, & quidquid per illas fit, jure affirmari potest fieri per Mechanismum, nee vero incognitum, & occuitum, sed uti supra demonstra- vimus, admodum patentem, a manifestum. Discrimen inter j -m E o dem etiam pacto in omnino propria significatione usurpare licebit vocem con- contactum mathe- J . . .. * T i i maticum, & physi- tactus ; licet intervallum semper remaneat aliquod ; quanquam ego ad aequivocationes evi- cum : hunc did tandas soleo distinguere inter contactum Mathematicum, in quo distantia sit prorsus nulla, proprie contactum. ni j- a. o 1 & contactum Physicum, in quo distantia sensus effugit omnes, & vis repulsiva satis magna ulteriorem accessum per nostras vires inducendum impedit. Voces ab hominibus institutae sunt ad significandas res corporeas, & corporum proprietates, prout nostris sensibus subsunt, iis, quae continentur infra ipsos, nihil omnino curatis. Sic planum, sic laeve proprie dicitur id, in quo nihil, quod sensu percipi possit, sinuetur, nihil promineat ; quanquam in communi etiam sententia nihil sit in Natura mathematice planum, vel laeve. Eodem pacto & nomen contactus ab hominibus institutum est, ad exprimendum physicum ilium contactum tantum- modo, sine ulla cura contactus mathematics, de quo nostri sensus sententiam ferre non possunt. Atque hoc quidem pacto si adhibeantur voces in propria significatione ilia, quae ipsarum institutioni respondeat ; ne a vocibus quidem ipsis huic Theoriae virium invidiam creare poterunt ii, quibus ipsa non placet. extensionis sit orta. Transitus ab ob- j^j. Atque haec de iis, quae contra ipsam virium legem a me propositam vel objecta Theoriam virium sunt hactenus, vel objici possent, sint satis, ne res in infinitum excrescat. Nunc ad ilia ad objections con- transibimus, quae contra constitutionem elementorum materiae inde deductam se menti tra puncta. .*.... i . j. oiferunt, in quibus itidem, quae maxime notatu digna sunt, persequar. Objectio ab idea 132. Inprimis quod pertinet ad hanc constitutionem elementorum materise, sunt puncti inextensi, multi, qui nullo pacto in animum sibi possint inducere, ut admittant puncta prorsus qua caremus : re- , r i n T 11 j A - J sponsio : unde idea mdi-[6o]-visibiha, & mextensa, quod nullam se dicant habere posse eorum ideam. At id a - hominum genus praejudiciis quibusdam tribuit multo plus aequo. Ideas omnes, saltern eas, quae ad materiam pertinent, per sensus hausimus. Porro sensus nostri nunquam potuerunt percipere singula elementa, quae nimirum vires exerunt nimis tenues ad movendas fibras, & propagandum motum ad cerebrum : massis indiguerunt, sive elementorum aggregatis, quae ipsas impellerent collata vi. Haec omnia aggregata constabant partibus, quarum partium extremae sumptae hinc, & inde, debebant a se invicem distare per aliquod intervallum, nee ita exiguum. Hinc factum est, ut nullam unquam per sensus acquirere potuerimus ideam pertinentem ad materiam, quae simul & extensionem, & partes, ac divisibilitatem non involverit. Atque idcirco quotiescunque punctum nobis animo sistimus, nisi reflexione utamur, habemus ideam globuli cujusdam perquam exigui, sed tamen globuli rotundi, habentis binas superficies oppositas distinctis. A THEORY OF NATURAL PHILOSOPHY in forward but a single instance in which they can positively prove that motion in Nature is communicated by immediate impulse. Of a truth they will never produce one ; for they cannot use the testimony of the eyes to exclude those very small distances to which the first repulsive branch of my curve refers & the windings about the axis ; for these necessarily evade ocular observation. Whilst I, on the other hand, by the rigorous argument given above, have excluded all idea of immediate contact ; & I have positively proved that the thing, which they wish to exist everywhere, as a matter of fact exists nowhere. 129. There is no reason why I should trouble myself about nomenclature ; but, as The forces in this in that too there is something that, from the customary manner of speaking, gives rise to ^j^/^ot to an a kind of prejudice, I think it should be observed that Mechanics was certainly never occult mechanism, restricted to immediate impulse alone by those who have dealt with it ; but that in the first place it was employed for the consideration of free motions, such as exist quite independently of any impulse. The work of Archimedes on equilibrium, that of Galileo on the free descent of heavy bodies & on projectiles, that of Huygens on central forces in a circular orbit & on the centre of oscillation, what Newton proved in general for motion on all sorts of trajectories ; all these certainly belong to the science of Mechanics. The Mechanics of Wolf, Euler & other writers in different lands certainly treats of such forces as these & the motions that arise from them, & these matters have been accomplished with the idea of impulse excluded altogether, or at least put out of mind. Whenever forces act, & there is an investigation of the laws in accordance with which velocity is produced, motion is changed, or the motion itself is determined ; the whole of this belongs especially to Mechanics in a truly proper signification of the term. Hence, they greatly abuse the proper signification of terms, who think that impulse alone belongs to the science of Mechanics ; to which these kinds of forces belong to a far greater extent. Therefore these forces may justly be called Mechanical ; & whatever comes about through their action can be justly asserted to have come about through a mechanism ; & one too that is not unknown or mysterious, but, as we proved above, perfectly plain & evident. 130. Also in the same way we may employ the term contact in an altogether special Distinction be- sense ; the interval may always remain something definite. Although, in order to avoid ticai n and m ph^ e skai ambiguity, I usually distinguish between mathematical contact, in which the distance is contact ; the latter absolutely nothing, & -physical contact, in which the distance is too small to affect our C anedorftact > . per y senses, and the repulsive force is great enough to prevent closer approach being induced by the forces we are considering. Words are formed by men to signify corporeal things & the properties of such, as far as they come within the scope of the senses ; & those that fall beneath this scope are absolutely not heeded at all. Thus, we properly call a thing plane or smooth, which has no bend or projection in it that can be perceived by the senses ; although, in the general opinion, there is nothing in Nature that is mathematically plane or smooth. In the same way also, the term contact was invented by men to express physical contact only, without any thought of mathematical contact, of which our senses can form no idea. In this way, indeed, if words are used in their correct sense, namely, that which corresponds to their original formation, those who do not care for my Theory of forces cannot from those words derive any objection against it. 131. I have now said sufficient about those objections that either up till now have Passing on from been raised, or might be raised, against the law of forces that I have proposed ; otherwise ^y^Theorf ""of the matter would grow beyond all bounds. Now we will pass on to objections against forces to objections the constitution of the elements of matter derived from it, which present themselves to the a g amst P mts - mind ; & in these also I will investigate those that more especially seem worthy of remark. 132. First of all, as regards the constitution of the elements of matter, there are indeed Potion to^the many persons who cannot in any way bring themselves into that frame of mind to admit tended points, the existence of points that are perfectly indivisible and non-extended ; for they say that which we postu- * < ' T* i r 1_ 13. tc , reply , tiic they cannot form any idea of such points. But that type of men pays more heed than origin of the idea is right to certain prejudices. We derive all our ideas, at any rate those that relate to of extension. matter, from the evidences of our senses. Further, our senses never could perceive single elements, which indeed give forth forces that are too slight to affect the nerves & thus propagate motion to the brain. The senses would need masses, or aggregates of the elements, which would affect them as a result of their combined force. Now all these aggregates are made up of parts ; & of these parts the two extremes on the one side and on the^ other must be separated from one another by a certain interval, & that not an insignificant one. Hence it comes about that we could never obtain through the senses any idea relating to matter, which did not involve at the same time extension, parts & divisibility. So, as often as we thought of a point, unless we used our reflective powers, we should get the idea of a sort of ball, exceedingly small indeed, but still a round ball, having two distinct and opposite faces. "2 PHILOSOPHISE NATURALIS THEORIA idea m puncti 133. Quamobrem ad concipiendum punctum indivisibile, & inextensum ; non debemus refl^xionemT'quo- consulere ideas > q uas immediate per sensus hausimus ; sed earn nobis debemus efformare modo ejus idea per reflexionem. Reflexione adhibita non ita difficulter efformabimus nobis ideam ejusmodi. negativa acqmra- N am i n p r imi s u bi & extensionem, & partium compositionem conceperimus ; si utranque negemus ; jam inextensi, & indivisibilis ideam quandam nobis comparabimus per negati- onem illam ipsam eorum, quorum habemus ideam ; uti foraminis ideam habemus utique negando existentiam illius materias, quas deest in loco foraminis. Quomodo ejus idea 134. Verum & positivam quandam indivisibilis, & inextensi puncti ideam poterimus posfit^per itmlte" com P arare n bis ope Geometrias, & ope illius ipsius ideas extensi continui, quam per sensus & limitum inter- hausimus, & quam inferius ostendemus, fallacem esse, ac fontem ipsum fallacies ejusmodi aperiemus, quas tamen ipsa ad indivisibilium, & inextensorum ideam nos ducet admodum claram. Concipiamus planum quoddam prorsus continuum, ut mensam, longum ex. gr. pedes duos ; atque id ipsum planum concipiamus secari transversum secundum longitudinem ita, ut tamen iterum post sectionem conjungantur partes, & se contingant. Sectio ilia erit utique limes inter partem dexteram & sinistram, longus quidem pedes duos, quanta erat plani longitude, at latitudinis omnino expers : nam ab altera parte immediate motu continue transitur ad alteram, quse, si ilia sectio crassitudinem haberet aliquam, non esset priori contigua. Ilia sectio est limes secundum crassitudinem inextensus, & indivisibilis, cui si occurrat altera sectio transversa eodem pacto indivisibilis, & inextensa ; oportebit utique, intersectio utriusque in superficie plani concepti nullam omnino habeat extensionem in partem quamcumque. Id erit punctum peni-[6i]-tus indivisibile, & inextensum, quod quidem punctum, translate piano, movebitur, & motu suo lineam describet, longam quidem, sed latitudinis expertem. Natura inextensi, j^c. Quo autem melius ipsius indivisibilis natura concipi possit ; quasrat a nobis quod non potest . /" r , . , . ^. . . ' ". esse inextenso con- quispiam, ut aliam faciamus ejus planae massas sectionem, quas priori ita sit proxima, ut tiguum in Uneis. n ihil prorsus inter utramque intersit. Respondebimus sane, id fieri non posse : vel enim inter novam sectionem, & veteram intercedet aliquid ejus materias, ex qua planum con- tinuum constare concipimus, vel nova sectio congruet penitus cum praecedente. En quomodo ideam acquiremus etiam ejus naturas indivisibilis illius, & inextensi, ut aliud indivisibile, & inextensum ipsi proximum sine medio intervallo non admittat, sed vel cum eo congruat, vel aliquod intervallum relinquat inter se, & ipsum. Atque hinc patebit etiam illud, non posse promoveri planum ipsum ita, ut ilia sectio promoveatur tantummodo per spatium latitudinis sibi asqualis. Utcunque exiguus fuerit motus, jam ille novus sectionis locus distabit a praecedente per aliquod intervallum, cum sectio sectioni contigua esse non possit. Eademin punctis : 136. Hasc si ad concursum sectionum transferamus, habebimus utique non solum ideam idea puncti eeo- ..,...,.,.. . , . ,. ... v j -i metricf transiata puncti indivisibilis, & inextensi, sed ejusmodi naturae puncti ipsius, ut aliud punctum sibi ad physicum, & contiguum habere non possit, sed vel congruant, vel aliquo a se invicem intervallo distent. Et hoc pacto sibi & Geometrae ideam sui puncti indivisibilis, & inextensi, facile efformare possunt, quam quidem etiam efformant sibi ita, ut prima Euclidis definitio jam inde incipiat : punctum est, cujus nulla pars est. Post hujusmodi ideam acquisitam illud unum intererit inter geometricum punctum, & punctum physicum materiae, quod hoc secundum habebit proprietates reales vis inertias, & virium illarum activarum, quas cogent duo puncta ad se invicem accedere, vel a se invicem recedere, unde net, ut ubi satis accesserint ad organa nostrorum sensuum, possint in iis excitare motus, qui propagati ad cerebrum, perceptiones ibi eliciant in anima, quo pacto sensibilia erunt, adeoque materialia, & realia, non pure imaginaria. Punctorum exist- j-- g n jgjtur per reffexionem acquisitam ideam punctorum realium, materialium, entiam aliunde . .. , J .( . . . , . . r r . demonstrari : per indivisibilium, inextensorum, quam inter ideas ab infantia acquisitas per sensus mcassum ideam acquisitam q uaer i m us. Idea ejusmodi non evincit eorum existentiam. Ipsam quam nobis exhibent ea tantum concipi. ^ . . J . , , . . ..*-."... positiva argumenta superms facta, quod mmirum, ne admittatur in colhsione corporum saltus, quern & inductio, & impossibilitas binarum velocitatum diversarum habendarum omnino ipso momento, quo saltus fieret, excludunt, oportet admittere in materia vires, quas repulsivae sint in minimis distantiis, & iis in infinitum imminutis augeantur in infinitum ; A THEORY OF NATURAL PHILOSOPHY 113 133. Hence for the purpose of forming an idea of a point that is indivisible & non- The idea of a point extended, we cannot consider the ideas that we derive directly from the senses ; but we ^"refleTti obtai h rxed must form our own idea of it by reflection. If we reflect upon it, we shall form an idea a negative >n ideaof of this sort for ourselves without much difficulty. For, in the first place, when we have con- rt may ^ ac( i uired - ceived the idea of extension and composition by parts, if we deny the existence of both, then we shall get a sort of idea of non-extension & indivisibility by that very negation of the existence of those things of which we already have formed an idea. For instance, we have the idea of a hole by denying the existence of matter, namely, that which is absent from the position in which the hole lies. 134. But we can also get an idea of a point that is indivisible & non-extended, by HOW a positive idea the aid of geometry, and by the help of that idea of an extended continuum that we derive ^^ ^of^bourf from the senses ; this we will show below to be a fallacy, & also we will open up the very daries, and inter- source of this kind of fallacy, which nevertheless will lead us to a perfectly clear idea of ^g ns of boun " indivisible & non-extended points. Imagine some thing that is perfectly plane and continuous, like a table-top, two feet in length ; & suppose that this plane is cut across along its length ; & let the parts after section be once more joined together, so that they touch one another. The section will be the boundary between the left part and the right part ; it will be two feet in length (that being the length of the plane before section), & altogether devoid of breadth. For we can pass straightaway by a continuous motion from one part to the other part, which would not be contiguous to the first part if the section had any thickness. The section is a boundary which, as regards breadth, is non-extended & indivisible ; if another transverse section which in the same way is also indivisible & non-extended fell across the first, then it must come about that the intersection of the two in the surface of the assumed plane has no extension at all in any direction. It will be a point that is altogether indivisible and non-extended ; & this point, if the plane be moved, will also move and by its motion will describe a line, which has length indeed but is devoid of breadth. 135. The nature of an indivisible itself can be better conceived in the following way. The nature of a Suppose someone should ask us to make another section of the plane mass, which shall lie hin~ g ext w I hich so near to the former section that there is absolutely no distance between them. We cannot he next to should indeed reply that it could not be done. For either between the new section & ^ S the old there would intervene some part of the matter of which the continuous plane was concerned, composed ; or the new section would completely coincide with the first. Now see how we acquire an idea also of the nature of that indivisible and non-extended thing, which is such that it does not allow another indivisible and non-extended thing to lie next to it without some intervening interval ; but either coincides with it or leaves some definite interval between itself & the other. Hence also it will be clear that it is not possible so to move the plane, that the section will be moved only through a space equal to its own breadth. However slight the motion is supposed to be, the new position of the section would be at a distance from the former position by some definite interval ; for a section cannot be contiguous to another section. 136. If now we transfer these arguments to the intersection of sections, we shall truly Th . e same thing for have not only the idea of an indivisible & non-extended point, but also an idea of the ^geometrical point nature of a point of this sort ; which is such that it cannot have another point contiguous transferred to a to it, but the two either coincide or else they are separated from one another by some interval. riafpoLt^ In this way also geometricians can easily form an idea of their own kind of indivisible & non-extended points ; & indeed they do so form their idea of them, for the first defi- nition of Euclid begins : A -point is that which has no parts. After an idea of this sort has been acquired, there is but one difference between a geometrical point & a physical point of matter ; this lies in the fact that the latter possesses the real properties of a force of inertia and of the active forces that urge the two points to approach towards, or recede from, one another ; whereby it comes about that when they have approached sufficiently near to the organs of our senses, they can excite motions in them which, when propagated to the brain, induce sensations in the mind, and in this way become sensible, & thus material and real, & not imaginary. 137. See then how by reflection the idea of real, material, indivisible, non-extended The existence of points can be acquired ; whilst we seek for it in vain amongst those ideas that we have o^herwise^demon- acquired since infancy by means of the senses. But an idea of this sort about things does strated ; they can not prove that these things exist. That is just what the rigorous arguments given above through ^cquir- point out to us ; that is to say, because, in order that in the collision of solids a sudden ing an idea of them, change should not be admitted (which change both induction & the impossibility of there being two different velocities at the same instant in which the change should take place), it had to be admitted that in matter there were forces which are repulsive ^at very small distances, & that these increased indefinitely as the distances were diminished. I ii4 PHILOSOPHIC NATURALIS THEORIA unde fit, ut duse particulae materiae sibi [62] invicem contiguae esse non possint : nam illico vi ilia repulsiva resilient a se invicem, ac particula iis constans statim disrumpetur, adeoque prima materiae elementa non constant contiguis partibus, sed indivisibilia sunt prorsus, atque simplicia, & vero etiam ob inductionem separabilitatis, ac distinctionis eorum, quae occupant spatii divisibilis partes diversas, etiam penitus inextensa. Ilia idea acquisita per reflexionem illud praestat tantummodo, ut distincte concipiamus id, quod ejusmodi rationes ostendunt existere in Natura, & quod sine reflexione, & ope illius supellectilis tantummodo, quam per sensus nobis comparavimus ab ipsa infantia, concipere omnino non liceret. Ceterum simplicium, & inextensorum notionem non ego primus in Physicam aiiis quoque ad- induco. Eorum ideam habuerunt veteres post Zenonem, & Leibnitiani monades suas & "rastare "hanc s i m P^ ces utique volunt, & inextensas ; ego cum ipsorum punctorum contiguitatem auferam, eorum theoriam. & distantias velim inter duo quaelibet materiae puncta, maximum evito scopulum, in quern utrique incurrunt, dum ex ejusmodi indivisibilibus, & inextensis continuum extensum componunt. Atque ibi quidem in eo videntur mini peccare utrique, quod cum simplicitate, & inextensione, quam iis elementis tribuunt, commiscent ideam illam imperfectam, quam sibi compararunt per sensus, globuli cujusdam rotundi, qui binas habeat superficies a se distinctas, utcumque interrogati, an id ipsum faciant, omnino sint negaturi. Neque enim aliter possent ejusmodi simplicibus inextensis implere spatium, nisi concipiendo unum elementum in medio duorum ab altero contactum ad dexteram, ab altero ad laevam, quin ea extrema se contingant; in quo, praeter contiguitatem indivisibilium, & inextensorum impossibilem, uti supra demonstravimus, quam tamen coguntur admittere, si rem altius perpenderint ; videbunt sane, se ibi illam ipsam globuli inter duos globules inter jacentis ideam admiscere. impugnatur con- 139. Nee ad indivisibilitatem, & inextensionem elementorum conjungendas cum formats ^b^inex- con t inua extensione massarum ab iis compositarum prosunt ea, quae nonnulli ex Leibniti- tensis petita ab anorum familia proferunt, de quibus egi in una adnotatiuncula adjecta num. 13. dissertationis impenetrabiiitate. j) g Mater its Divisibilitate, ? Principiis Corporum, ex qua, quae eo pertinent, hue libet transferre. Sic autem habet : Qui dicunt, monades non compenetrari, quia natura sua impenetrabiles sunt, ii difficultatem nequaquam amovenf ; nam si e? natura sua impenetrable s sunt, y continuum debent componere, adeoque contigua esse ; compenetrabuntur simul, W non compenetrabuntur, quod ad absurdum deducit, W ejusmodi entium impossibilitatem evincit. Ex omnimodfs inextensionis, & contiguitatis notione evincitur, compenetrari debere argumento contra Zenonistas institute per tot stecula, if cui nunquam satis responsum est. Ex natura, qua in [63] iis supponitur, ipsa compenetratio excluditur, adeoque habetur contradictio, & absurdum. inductionem a 140'. Sunt alii, quibus videri poterit, contra haec ipsa puncta indivisibilia, & inextensa sensibihbus com- ,1 ., T . . , ^ . . . . r . . r . K . positis, & extensis adniberi posse mductionis prmcipmm, a quo contmuitatis legem, & alias propnetates haud vaiere contra derivavimus supra, quae nos ad haec indivisibilia, & inextensa puncta deduxerunt. Videmus puncta simplicia, & t* *. . . ... ... ,. . .. ... inextensa. enim in matena omni, quae se uspiam nostns objiciat sensibus, extensionem, divisibihtatem, partes ; quamobrem hanc ipsam proprietatem debemus transferre ad elementa etiam per inductionis principium. Ita ii : at hanc difficultatem jam superius praeoccupavimus, ubi egimus de inductionis principio. Pendet ea proprietas a ratione sensibilis, & aggregati, cum nimirum sub sensus nostros ne composita quidem, quorum moles nimis exigua sit, cadere possint. Hinc divisibilitatis, & extensionis proprietas ejusmodi est ; ut ejus defectus, si habeatur alicubi is casus, ex ipsa earum natura, & sensuum nostrorum constitutione non possit cadere sub sensus ipsos, atque idcirco ad ejusmodi proprietates argumentum desumptum ab inductione nequaquam pertingit, ut nee ad sensibilitatem extenditur. Per ipsam etiam 141. Sed etiam si extenderetur, esset adhuc nostrae Theoriae causa multo melior in eo, tensT^Hn^uctioms q 110 ^ circa, extensionem, & compositionem partium negativa sit. Nam eo ipso, quod habitam ipsum ex- continuitate admissa, continuitas elementorum legitima ratiocinatione excludatur, excludi omnino debet absolute ; ubi quidem illud accidit, quod a Metaphysicis, & Geometris nonnullis animadversum est jam diu, licere aliquando demonstrare propositionem ex A THEORY OF NATURAL PHILOSOPHY 115 From this it comes about that two particles of matter cannot be contiguous ; for thereupon they would recoil from one another owing to that repulsive force, & a particle composed of them would at once be broken up. Thus, the primary elements of matter cannot be composed of contiguous parts, but must be perfectly indivisible & simple ; and also on account of the induction from separability & the distinction between those that occupy different divisible parts of space, they must be perfectly non-extended as well. The idea acquired by reflection only yields the one result, namely, that through it we may form a clear conception of that which reasoning of this kind proves to be existent in Nature ; of which, without reflection, using only the equipment that we have got together for ourselves by means of the senses from our infancy, we could not have formed any conception. 138. Besides, I was not the first to introduce the notion of simple non-extended points Simple and into physics. The ancients from the time of Zeno had an idea of them, & the followers are^admitt of Leibniz indeed suppose that their monads are simple & non-extended. I, since I do others as well ; but not admit the contiguity of the points themselves, but suppose that any two points of ^m is "the 7 best." matter are separated from one another, avoid a mighty rock, upon which both these others come to grief, whilst they build up an extended continuum from indivisible & non-extended things of this sort. Both seem to me to have erred in doing so, because they have mixed up with the simplicity & non-extension that they attribute to the elements that imperfect idea of a sort of round globule having two surfaces distinct from one another, an idea they have acquired through the senses ; although, if they were asked if they had made this supposition, they would deny that they had done so. For in no other way can they fill up space with indivisible and non-extended things of this sort, unless by imagining that one element between two others is touched by one of them on the right & by the other on the left. If such is their idea, in addition to contiguity of indivisible & non-extended things (which is impossible, as I proved above, but which they are forced to admit if they consider the matter more carefully) ; in addition to this, I say, they will surely see that they have introduced into their reasoning that very idea of the two little spheres lying between two others. I3Q. Those arguments that some of the Leibnitian circle put forward are of no use The deduction from , i ~ r T -i ! o r i i -i. impenetrability of for the purpose of connecting indivisibility & non-extension of the elements with continuous a conciliation of extension of the masses formed from them. I discussed the arguments in question in extension ^j 1 ^ a short note appended to Art. 13 of the dissertation De Materies Divisibilitate and extendeTthings. Principiis Corporum ; & I may here quote from that dissertation those things that concern us now. These are the words : Those, who say that monads cannot be corn-penetrated, because they are by nature impenetrable, by no means remove the difficulty. For, if they are both by nature impenetrable, & also at the same time have to make up a continuum, i.e., have to be contiguous, then at one & the same time they are compenetrated & they are not compenetrated ; y this leads to an absurdity \3 proves the impossibility of entities of this sort. For, from the idea of non-extension of any sort, & of contiguity, it is proved by an argument instituted against the Zenonists many centuries ago that there is bound to be compenetration ; & -this argument has never been satisfactorily answered. From the nature that is ascribed to them, this compenetration is excluded. Thus there is a contradiction 13 an absurdity. 140. There are others, who will think that it is possible to employ, for the purpose induction derived of opposing the idea of these indivisible & non-extended points, the principle of induction, ^T'senslSf 3 <m*- by which we derived the Law of Continuity & other properties, which have led us to pound, and ex- these indivisible & non-extended points. For we perceive (so they say) in all matter, a vau ed for r the f pur that falls under our notice in any way, extension, divisibility & parts. Hence we must pose of opposing transfer this property to the elements also by the principle of induction. Such is their argument. But we have already discussed this difficulty, when we dealt with the principle of induction. The property in question depends on a reasoning concerned with a sensible body, & one that is an aggregate ; for, in fact, not even a. composite body can come within the scope of our senses, if its mass is over-small. Hence the property of divisibility & extension is such that the absence of this property (if this case ever comes about), from the very nature of divisibility & extension, & from the constitution of our senses, cannot fall within the scope of those senses. Therefore an argument derived from induction will not apply to properties of this kind in any way, inasmuch as the extension does not reach the point necessary for sensibility. 141. But even if this point is reached, there would only be all the more reason for our Extension Theory from the fact that it denies extension and composition by parts. For, from the very exclusion of fact that, if continuity be admitted, continuity of the elements is excluded by legitimate exte ^ s e io argument, it follows that continuity ought to be absolutely excluded in all cases. For in d u Ctio n. that case we get an instance of the argument that has been observed by metaphysicists and some geometers for a very long time, namely, that a proposition may sometimes be n6 PHILOSOPHIC NATURALIS THEORIA assumpta veritate contradictoriae propositionis ; cum enim ambae simul verae esse non possint, si ab altera inferatur altera, hanc posteriorem veram esse necesse est. Sic nimirum, quoniam a continuitate generaliter assumpta defectus continuitatis consequitur in materiae elementis, & in extensione, defectum hunc haberi vel inde eruitur : nee oberit quidquam principium inductionis physicae, quod utique non est demonstrativum, nee vim habet, nisi ubi aliunde non demonstretur, casum ilium, quern inde colligere possumus, improbabilem esse tantummodo, adhuc tamen haberi, uti aliquando sunt & falsa veris probabiliora. Cujusmodi con- 142. Atque hie quidem, ubi de continuitate seipsam excludente mentio injecta est, TheoiSadrnittatur n tandum & illud, continuitatis legem a me admitti, & probari pro quantitatibus, quae quid sit spatium, magnitudinem mutent, quas nimirum ab una magnitudine ad aliam censeo abire non posse, & tempus. n j g - transean t per intermedias, quod elementorum materiae, quse magnitudinem nee mutant, nee ullam habent variabilem, continuitatem non inducit, sed argumento superius facto penitus summovet. Quin etiam ego quidem continuum nullum agnosco coexistens, uti & supra monui ; nam nee spatium reale mihi est ullum continuum, sed [64] imaginarium tantummodo, de quo, uti & de tempore, quae in hac mea Theoria sentiam, satis luculenter exposui in Supplementis ad librum i. Stayanae Philosophise (*). Censeo nimirum quodvis materiae punctum, habere binos reales existendi modos, alterum localem, alterum tem- porarium, qui num appellari debeant res, an tantummodo modi rei, ejusmodi litem, quam arbitror esse tantum de nomine, nihil omnino euro. Illos modos debere admitti, ibi ego quidem positive demonstro : eos natura sua immobiles esse, censeo ita, ut idcirco ejusmodi existendi modi per se inducant relationes prioris, & posterioris in tempore, ulterioris, vel citerioris in loco, ac distantiae cujusdam deter minatae, & in spatio determinatae positionis etiam, qui modi, vel eorum alter, necessario mutari debeant, si distantia, vel etiam in spatio sola mutetur positio. Pro quovis autem modo pertinente ad quodvis punctum, penes omnes infinites modos possibiles pertinentes ad quodvis aliud, mihi est unus, qui cum eo inducat in tempore relationem coexistentiae ita, ut existentiam habere uterque non possit, quin simul habeant, & coexistant ; in spatio vero, si existunt simul, inducant relationem compenetrationis, reliquis omnibus inducentibus relationem distantiae temporarise, vel localis, ut & positionis cujusdam localis determinatae. Quoniam autem puncta materiae existentia habent semper aliquam a se invicem distantiam, & numero finita sunt ; finitus est semper etiam localium modorum coexistentium numerus, nee ullum reale continuum efformat. Spatium vero imaginarium est mihi possibilitas omnium modorum localium confuse cognita, quos simul per cognitionem praecisivam concipimus, licet simul omnes existere non possint, ubi cum nulli sint modi ita sibi proximi, vel remoti, ut alii viciniores, vel remotiores haberi non possint, nulla distantia inter possibiles habetur, sive minima omnium, sive maxima. Dum animum abstrahimus ab actuali existentia, & in possibilium serie finitis in infinitum constante terminis mente secludimus tarn minimae, quam maximae distantiae limitem, ideam nobis efformamus continuitatis, & infinitatis in spatio, in quo idem spatii punctum appello possibilitatem omnium modorum localium, sive, quod idem est, realium localium punctorum pertinentium ad omnia materiae puncta, quae si existerent, compenetrationis relationem inducerent, ut eodem pacto idem nomino momentum tem- poris temporarios modos omnes, qui relationem inducunt coexistentiae. Sed de utroque plura in illis dissertatiunculis, in quibus & analogiam persequor spatii, ac temporis multiplicem. Ubi habeat con- [65] 143. Continuitatem igitur agnosco in motu tantummodo, quod est successivum u i bi lit aff e e 1 ctet Na t Ura ^ u ^' non coexistens, & in eo itidem solo, vel ex eo solo in corporeis saltern entibus legem continuitatis admitto. Atque hinc patebit clarius illud etiam, quod superius innui, Naturam ubique continuitatis legem vel accurate observare, vel affectare saltern. ^ Servat in motibus, & distantiis, affectat in aliis casibus multis, quibus continuity, uti etiam supra definivimus, nequaquam convenit, & in aliis quibusdam, in quibus haberi omnino non pptest continuitas, quae primo aspectu sese nobis objicit res non aliquanto intimius inspectantibus, ac perpendentibus : ex. gr. quando Sol oritur supra horizontem, si concipiamus Solis discum (h) Binte dissertatiunculis, qua hue pertinent, inde excerptte habentur hie Supplementorum I, 13 2, quarum mentio facta est etiam superius num. 66, W 86. con- A THEORY OF NATURAL PHILOSOPHY 117 proved by assuming the truth of the contradictory proposition. For since both propositions cannot be true at the same time, if from one of them the other can be inferred, then the latter of necessity must be the true one. Thus, for instance, because it follows, from the assumption of continuity in general, that there is an absence of continuity in the elements of matter, & also in the case of extension, we come to the conclusion that there is this absence. Nor will any principle of physical induction be prejudicial to the argument, where the induction is not one that can be proved in every case ; neither will it have any validity, except in the case where it cannot be proved in other ways that the conclusion that we can come to from the argument is highly improbable but yet is to be held as true ; for indeed sometimes things that are false are more plausible than the true facts. 142. Now, in this connection, whilst incidental mention has been made of the exclusion xhe sort of of continuity, it should be observed that the Law of Continuity is admitted by me, & tinuum that is proved for those quantities that change their magnitude, but which indeed I consider Th^r^fthe^ature cannot pass from one magnitude to another without going through intermediate stages ; of s P a ce and time, but that this does not lead to continuity in the case of the elements of matter, which neither change their magnitude nor have anything variable about them ; on the contrary it proves quite the opposite, as the argument given above shows. Moreover, I recognize no co- existing continuum, as I have already mentioned ; for, in my opinion, space is not any real continuum, but only an imaginary one ; & what I think about this, and about time as well, as far as this Theory is concerned, has been expounded clearly enough in the supplements to the first book of Stay's Philosophy. (A) For instance, I consider that any point of matter has two modes of existence, the one local and the other temporal ; I do not take the trouble to argue the point as to whether these ought to be called things, or merely modes pertaining to a thing, as I consider that this is merely a question of terminology. That it is necessary that these modes be admitted, I prove rigorously in the supplements mentioned above. I consider also that they are by their very nature incapable of being displaced ; so that, of themselves, such modes of existence lead to the relations of before & after as regards time, far & near as regards space, & also of a given distance & a given position in space. These modes, or one of them, must of necessity be changed, if the distance, or even if only the position in space is altered. Moreover, for any one mode belonging to any point, taken in conjunction with all the infinite number of possible modes pertaining to any other point, there is in my opinion one which, taken in conjunction with the first mode, leads as far as time is concerned to a relation of coexistence ; so that both cannot have existence unless they have it simultaneously, i.e., they coexist. But, as far as space is concerned, if they exist simultaneously, the conjunction leads to a relation of compenetration. All the others lead to a relation of temporal or of local distance, as also of a given local position. Now since existent points of matter always have some distance between them, & are finite in number, the number of local modes of existence is also always finite ; & from this finite number we cannot form any sort of real continuum. But I have an ill-defined idea of an imaginary space as a possibility of all local modes, which are precisely conceived as existing simultaneously, although they cannot all exist simul- taneously. In this space, since there are not modes so near to one another that there cannot be others nearer, or so far separated that there cannot be others more so, there cannot therefore be a distance that is either the greatest or the least of all, amongst those that are possible. So long as we keep the mind free from the idea of actual existence &, in a series of possibles consisting of an indefinite number of finite terms, we mentally exclude the limit both of least & greatest distance, we form for ourselves a conception of continuity & infinity in space. In this, I define the same point of space to be the possibility of all local modes, or what comes to the same thing, of real local points pertaining to all points of matter, which, if they existed, would lead to a relation of compenetration ; just as I define the same instant of time as all temporal modes, which lead to a relation of coexistence. But there is a fuller treatment of both these subjects in the notes referred to ; & in them I investigate further the manifold analogy between space & time. 143. Hence I acknowledge continuity in motion only, which is something successive where there is con- i TJ . . . . , ' , f . V . . tmuity in Nature ; and not co-existent ; & also in it alone, or because or it alone, in corporeal entities at any W here Nature does rate, lies my reason for admitting the Law of Continuity. From this it will be all the no more than at- more clear that, as I remarked above, Nature accurately observes the Law of Continuity, j t eml or at least tries to do so. Nature observes it in motions & in distance, & tries to in many other cases, with which continuity, as we have defined it above, is in no wise in agree- ment ; also in certain other cases, in which continuity cannot be completely obtained. This continuity does not present itself to us at first sight, unless we consider the subjects somewhat more deeply & study them closely. For instance, when the sun rises above the horizon, (h) The two notes, which refer to this matter, have been quoted in this work as supplements IS- II : these have been already referred to in Arts. 66 & 86 above. n8 PHILOSOPHISE NATURALIS THEORIA ut continuum, & horizontem ut planum quoddam ; ascensus Solis fit per omnes magnitudines ita, ut a primo ad postremum punctum & segmenta Solaris disci, & chordae segmentorum crescant transeundo per omnes intermedias magnitudines. At Sol quidem in mea Theoria non est aliquid continuum, sed est aggregatum punctorum a se invicem distantium, quorum alia supra illud imaginarium planum ascendunt post alia, intervallo aliquo temporis inter- posito semper. Hinc accurata ilia continuitas huic casui non convenit, & habetur tantummodo in distantiis punctorum singulorum componentium earn massam ab illo imaginario piano. Natura tamen etiam hie continuitatem quandam affectat, cum nimirum ilia punctula ita sibi sint invicem proxima, & ita ubique dispersa, ac disposita, ut apparens quaedam ibi etiam continuitas habeatur, ac in ipsa distributione, a qua densitas pendet, ingentes repentini saltus non riant. Exempla continu- 144. Innumera ejus rei exempla liceret proferre, in quibus eodem pacto res pergit. it at is apparent gj c j n fl uv i orum alveis, in frondium flexibus, in ipsis salium, & crystallorum, ac aliorum tantum : unde ea ..... . ,., . *., ortum ducat. corporum angulis, in ipsis cuspidibus unguium, quae acutissimae in quibusdam ammalibus apparent nudo oculo ; si microscopio adhibito inspiciantur ; nusquam cuspis abrupta prorsus, nusquam omnino cuspidatus apparet angulus, sed ubique flexus quidam, qui curvaturam habeat aliquam, & ad continuitatem videatur accedere. In omnibus tamen iis casibus vera continuitas in mea Theoria habetur nusquam ; cum omnia ejusmodi corpora constent indivisibilibus, & a se distantibus punctis, quse continuam superficiem non efformant, & in quibus, si quaevis tria puncta per rectas lineas conjuncta intelligantur ; triangulum habebitur utique cum angulis cuspidatis. Sed a motuum, & virium continuitate accurata etiam ejusmodi proximam continuitatem massarum oriri censeo, & a casuum possibilium multitudine inter se collata, quod ipsum innuisse sit satis. Motuum omnium 145- Atque hinc fiet manifestum, quid respondendum ad casus quosdam, qui eo continuitas in -pertinent, & in quibus violari quis crederet F661 continuitatis legem. Quando piano aliquo line is continuis r .*. f r . n n nusquam inter- speculo lux excipitur, pars relrmgitur, pars renectitur : in renexione, & retractione, uti earn ruptis, aut mutatis. o li m creditum est fieri, & etiamnum a nonnullis creditur, per impulsionem nimirum, & incursum immediatum, fieret violatio quaedam continui motus mutata linea recta in aliam ; sed jam hoc Newtonus advertit, & ejusmodi saltum abstulit, explicando ea phenomena per vires in aliqua distantia agentes, quibus fit, ut quaevis particula luminis motum incurvet paullatim in accessu ad superficiem re flectentem, vel refringentem ; unde accessuum, & recessuum lex, velocitas, directionum flexus, omnia juxta continuitatis legem mutantur. Quin in mea Theoria non in aliqua vicinia tantum incipit flexus ille, sed quodvis materiae punctum a Mundi initio unicam quandam continuam descripsit orbitam, pendentem a continua ilia virium lege, quam exprimit figura I , quae ad distantias quascunque protenditur ; quam quidem lineae continuitatem nee liberae turbant animarum vires, quas itidem non nisi juxta continuitatis legem exerceri a nobis arbitror ; unde fit, ut quemadmodum omnem accuratam quietem, ita omnem accurate rectilineum motum, omnem accurate circularem, ellipticum, parabolicum excludam ; quod tamen aliis quoque sententiis omnibus commune esse debet ; cum admodum facile sit demonstrare, ubique esse perturbationem quandam, & mutationum causas, quae non permittant ejusmodi linearum nobis ita simplicium accuratas orbitas in motibus. Apparens saltus in 146. Et quidem ut in iis omnibus, & aliis ejusmodi Natura semper in mea Theoria diffusione reflexi, accuratissimam continuitatem observat, ita & hie in reflexionibus, ac refractionibus luminis. ac refracti luminis. . ,. , . ..'..,. , , , '. . , At est ahud ea in re, in quo continuitatis violatio quaedam haben videatur, quam, qui rem altius perpendat, credet primo quidem, servari itidem accurate a Natura, turn ulterius progressus, inveniet affectari tantummodo, non servari. Id autem est ipsa luminis diffusio, atque densitas. Videtur prima fronte discindi radius in duos, qui hiatu quodam intermedio a se invicem divellantur velut per saltum, alia parte reflexa, ali refracta, sine ullo intermedio flexu cujuspiam. Alius itidem videtur admitti ibidem saltus quidam : si enim radius integer excipiatur prismate ita, ut una pars reflectatur, alia transmittatur, & prodeat etiam e secunda superficie, turn ipsum prisma sensim convertatur ; ubi ad certum devenitur in conversione angulum, lux, quae datam habet refrangibilitatem, jam non egreditur, sed reflectitur in totum ; ubi itidem videtur fieri transitus a prioribus angulis cum superficie semper minoribus, sed jacentibus ultra ipsam, ad angulum reflexionis aequalem angulo A THEORY OF NATURAL PHILOSOPHY 119 if we think of the Sun's disk as being continuous, & the horizon as a certain plane ; then the rising of the Sun is made through all magnitudes in such a way that, from the first to the last point, both the segments of the solar disk & the chords of the segments increase by passing through all intermediate magnitudes. But, in my Theory, the Sun is not something continuous, but is an aggregate of points separate from one another, which rise, one after the other, above that imaginary plane, with some interval of time between them in all cases. Hence accurate continuity does not fit this case, & it is only observed in the case of the distances from the imaginary plane of the single points that compose the mass of the Sun. Yet Nature, even here, tries to maintain a sort of continuity ; for instance, the little points are so very near to one another, & so evenly spread & placed that, even in this case, we have a certain apparent continuity, and even in this distribution, on which the density depends, there do not occur any very great sudden changes. 144. Innumerable examples of this apparent continuity could be brought forward, in Examples of con- which the matter comes about in the same manner. Thus, in the channels of rivers, the ^"reiy apparent' 3 bends in foliage, the angles in salts, crystals and other bodies, in the tips of the claws that its origin, appear to the naked eye to be very sharp in the case of certain animals ; if a microscope were used to examine them, in no case would the point appear to be quite abrupt, or the angle altogether sharp, but in every case somewhat rounded, & so possessing a definite curvature & apparently approximating to continuity. Nevertheless in all these cases there is nowhere true continuity according to my Theory ; for all bodies of this kind are composed of points that are indivisible & separated from one another ; & these cannot form a continuous surface ; & with them, if any three points are supposed to be joined by straight lines, then a triangle will result that in every case has three sharp angles. But I consider that from the accurate continuity of motions & forces a very close approximation of this kind arises also in the case of masses ; &, if the great number of possible cases are compared with one another, it is sufficient for me to have just pointed it out. 145. Hence it becomes evident how we are to refute certain cases, relating to this The . continuity of matter, in which it might be considered that the Law of Continuity was violated. When uous lines ""is light falls upon a plane mirror, part is refracted & part is reflected. In reflection & nowhere inter- refraction, according to the idea held in olden times, & even now credited by some people, rup e namely, that it took place by means of impulse & immediate collision, there would be a breach of continuous motion through one straight line being suddenly changed for another. But already Newton has discussed this point, & has removed any sudden change of this sort, by explaining the phenomena by means of forces acting at a distance ; with these it comes about that any particle of light will have its path bent little by little as it approaches a reflecting or refracting surface. Hence, the law of approach and recession, the velocity, the alteration of direction, all change in accordance with the Law of Continuity. Nay indeed, in my Theory, this alteration of direction does not only begin in the immediate neighbourhood, but any point of matter from the time that the world began has described a single continuous orbit, depending on the continuous law of forces, represented in Fig. i, a law that extends to all distances whatever. I also consider that this continuity of path is undisturbed by any voluntary mental forces, which also cannot be exerted by us except in accordance with the Law of Continuity. Hence it comes about that, just as I exclude all idea of absolute rest, so I exclude all accurately rectilinear, circular, elliptic, or parabolic motions. This too ought to be the general opinion of all others ; for it is quite easy to show that there is everywhere some perturbation, & reasons for alteration, which do not allow us to have accurate paths along such simple lines for our motions. 146. Just as in all the cases I have mentioned, & in others like them, Nature always Apparent discon- '.-,,, J .. i i i i i tinuity in diffusion in my Theory observes the most accurate continuity, so also is this done here in the case O f ren ected and re- of the reflection and refraction of light. But there is another thing in this connection, fracted light. in which there seems to be a breach of continuity ; & anyone who considers the matter fairly deeply, will think at first that Nature has observed accurate continuity, but on further consideration will find that Nature has only endeavoured to do so, & has not actually observed it ; that is to say, in the diffusion of light, & its density. At first sight the ray seems to be divided into two parts, which leave a gap between them & diverge from one another as it were suddenly, the one part being reflected & the other part refracted without any intermediate bending of the path. It also seems that another sudden change must be admitted ; for suppose that a beam of light falls upon a prism, & part of it is reflected & the rest is transmitted & issues from the second surface, and that then the prism is gradually rotated ; when a certain angle of rotation is reached, light, having a given refrangibility, is no longer transmitted, but is totally reflected. Here also it seems that there is a sudden transition from the first case in which the angles made^with the surface by the issuing rays are always less than the angle of incidence, & lie on the far side of the surface, to the latter case in which the angles of reflection are equal to 120 PHILOSOPHIC NATURALIS THEORIA incidentiae, & jacentem citra, sine ulla reflexione in angulis intermediis minoribus ab ipsa superficie ad ejusmodi finitum angulum. Apparens concili- 14.7. Huic cuidam velut laesioni continuitatis videtur responderi posse per illam lucem Unuitafe pel radios q ua3 reflectitur, vel refrin-[67]-gitur irregulariter in quibusvis angulis. Jam olim enim irregulariter disper- observatum est illud, ubi lucis radius reflectitur, non reflecti totum ita, ut angulus reflexionis aequetur angulo incidentiae, sed partem dispergi quaquaversus ; quam ob causam si Solis radius in partem quandam speculi incurrat, quicunque est in conclavi, videt, qui sit ille locus, in quern incurrit radius, quod utique non fieret, nisi e solaribus illis directis radiis etiam ad oculum ipsius radii devenirent, egressi in omnibus iis directionibus, quae ad omnes oculi positiones tendunt ; licet ibi quidem satis intensum lumen non appareat, nisi in directione faciente angulum reflexionis aequalem incidentiae, in qua resilit maxima luminis pars. Et quidem hisce radiis redeuntibus in angulis hisce inaequalibus egregie utitur Newtonus in fine Opticae ad explicandos colores laminarum crassarum : & eadem irregularis dispersio in omnes plagas ad sensum habetur in tenui parte, sed tamen in aliqua, radii refracti. Hinc inter vividum ilium reflexum radium, & refractum, habetur intermedia omnis ejusmodi radiorum series in omnibus iis intermediis angulis prodeuntium, & sic etiam ubi transitur a refractione ad reflexionem in totum, videtur per hosce intermedios angulos res posse fieri citissimo transitu per ipsos, atque idcirco illaesa perseverare continuitas. Cur ea apparens 148. Verum si adhuc altius perpendatur res ; patebit in ilia intermedia serie non haberi dSitio 1 pe^contiii- accuratam continuitatem, sed apparentem quandam, quam Natura affectat, non accurate ujtatem yiae cujus- servat illaesam. Nam lumen in mea Theoria non est corpus quoddam continuum, quod vis puncti diffundatur continue per illud omne spatium, sed est aggregatum punctorum a se invicem disjunctorum, atque distantium, quorum quodlibet suam percurrit viam disjunctam a proximi via per aliquod intervallum. Continuitas servatur accuratissime in singulorum punctorum viis, non in diffusione substantiae non continuae, & quo pacto ea in omnibus iis motibus servetur, & mutetur, mutata inclinatione incidentiae, via a singulis punctis descripta sine saltu, satis luculenter exposui in secunda parte meae dissertationis De Lumine a num. 98. Sed haec ad applicationem jam pertinent Theoriae ad Physicam. QUO pacto servetur 149. Haud multum absimiles sunt alii quidam casus, in quibus singula continuitatem bu^dam^casibusTui observant, non aggregatum utique non continuum, sed partibus disjunctis constans. quibus videtur tedi. Hujusmodi est ex. gr. altitude cujusdam domus, quae aedificatur de novo, cui cum series nova adjungitur lapidum determinatae cujusdam altitudinis, per illam additionem repente videtur crescere altitude domus, sine transitu per altitudines intermedias : & si dicatur id non esse Naturae opus, sed artis ; potest difficultas transferri facile ad Naturae opera, ut ubi diversa inducuntur glaciei strata, vel in aliis incrustationibus, ac in iis omnibus casibus, in quibus incrementum fit per externam applicationem partium, ubi accessiones finitae videntur acquiri simul totae sine [68] transitu per intermedias magnitudines. In iis casibus continuitas servatur in motu singularum partium, quae accedunt. Illae per lineam quandam continuam, & continua velocitatis mutatione accedunt ad locum sibi deditum : quin immo etiam posteaquam eo advenerunt, pergunt adhuc moveri, & nunquam habent quietem nee absolutam, nee respectivam respectu aliarum partium, licet jam in respectiva positione sensibilem mutationem non subeant : parent nimirum adhuc viribus omnibus, quae respondent omnibus materiae punctis utcunque distantibus, & actio proximarum partium, quae novam adhaesionem parit, est continuatio actionis, quam multo minorem exercebant, cum essent procul. Hoc autem, quod pertineant ad illam domum, vel massam, est aliquid non in se determinatum, quod momento quodam determinato fiat, in quo saltus habeatur, sed ab aestimatione quadam pendet nostrorum sensuum satis crassa ; ut licet perpetuo accedant illae partes, & pergant perpetuo mutare positionem respectu ipsius massae ; turn incipiant censeri ut pertinentes ad illam domum, vel massam : cum desinit respectiva mutatio esse sensibilis, quae sensibilitatis cessatio fit ipsa etiam quodammodo per gradus omnes, & continue aliquo tempore, non vero per saltum. Generate responsio ISO- Hinc distinctius ibi licebit difHcultatem omnem amovere dicendo, non servari de emta. 3 similes m " mutationem continuam in magnitudinibus earum rerum, quae continuae non sunt, & magnitudinem non habent continuam, sed sunt aggregata rerum disjunctarum ; vel in iis rebus, quae a nobis ita censentur aliquod totum constituere, ut magnitudinem aggregati non A THEORY OF NATURAL PHILOSOPHY 121 the angles of incidence & lie on the near side of the surface, without any reflection for rays at intermediate angles with the surface less than a certain definite angle. 147. It seems that an explanation of this apparent breach of continuity can be given Apparent recontiii- by means of light that is reflected or refracted irregularly at all sorts of angles. For long ago of Continuity 6 effect it was observed that, when a ray of light is reflected, it is not reflected entirely in such a * ed fa y means of manner that the angle of reflection is equal to the angle of incidence, but that a part of it is dispersed in all directions. For this reason, if a ray of light from the Sun falls upon some part of a mirror, anybody who is in the room sees where the ray strikes the mirror ; & this certainly would not be the case, unless some of the solar rays reached his eye directly issuing from the mirror in all those directions that reach to all positions that the eye might be in. Nevertheless, in this case the light does not appear to be of much intensity, unless the eye is in the position facing the angle of reflection equal to the angle of incidence, along which the greatest part of the light rebounds. Newton indeed employed in a brilliant way these rays that issue at irregular angles at the end of his Optics to explain the colours of solid laminae. The same irregular dispersion in all directions takes place as far as can be observed in a small part, but yet in some part, of the refracted ray. Hence, in between the intense reflected & refracted rays, we have a whole series of intermediate rays of this sort issuing at all intermediate angles. Thus, when the transition is effected from refraction to total reflection, it seems that it can be done through these intermediate angles by an extremely rapid transition through them, & therefore continuity remains unimpaired. 148. But if we inquire into the matter yet more carefully, it will be evident that in Why this is only an that intermediate series there is no accurate continuity, but only an apparent continuity ; atimT* the^true & this Nature tries to maintain, but does not accurately observe it unimpaired. For, reconciliation is in my Theory, light is not some continuous body, which is continuously diffused through t!nurty h of ^ath^or all the space it occupies ; but it is an aggregate of points unconnected with & separated any point of light, from one another ; & of these points, any one pursues its own path, & this path is separated from the path of the next point to it by a definite interval. Continuity is observed perfectly accurately for the paths of the several points, not in the diffusion of a substance that is not continuous ; & the manner in which continuity is preserved in all these motions, & the path described by the several points is altered without sudden change, when the angle of incidence is altered, I have set forth fairly clearly in the second part of my dissertation De Lumine, Art. 98. But in this work these matters belong to the application of the Theory to physics. 140. There are certain cases, not greatly unlike those already given, in which each The manner in / i _j i_ ' V j r which continuity part preserves continuity, but not so the whole, which is not continuous but composed ot is m aintained m separate parts. For an instance of this kind, take the height of a new house which is being certain cases in built ; as a fresh layer of stones of a given height is added to it, the height of the house ^ on account of that addition seems to increase suddenly without passing through intermediate heights. If it is said that that is not a work of Nature, but of art ; then the same difficulty can easily be transferred to works of Nature, as when different strata of ice are formed, or in other incrustations, and in all cases in which an increment is caused by the external application of parts, where finite additions seem to be acquired all at once without any passage through intermediate magnitudes. In these cases the continuity is preserved in the motions of the separate parts that are added. These reach the place allotted to them along some continuous line & with a continuous change of velocity. Further, after they have reached it, they still continue to move, & never have absolute rest ; no, nor even relative rest with respect to the other parts, although they do not now suffer a sensible change in their relative positions. Thus, they still submit to the action of all the forces that correspond to all points of matter at any distances whatever ; and the action of the parts nearest to them, which produces a new adhesion, is the continuation of the action that they exert to a far smaller extent when they are some distance away. Moreover, in the fact that they belong to that house or mass, there is something that is not determinate in itself, because it happens at a determinate instant in which the sudden change takes place ; but it depends on a somewhat rough assessment by our senses. So that, although these parts are continually being added, & continually go on changing their position with respect to the mass, they both begin to be thought of as belonging to that house or mass, & the relative change ceases to be sensible ; also this cessation of sensibility itself also takes place to some extent through all stages, and in some continuous interval of time, & not by a sudden jump. KO. From this consideration we may here in a clearer manner remove all difficulty -' * . , . - jrt,'U* simuar cd.b by saying that a continuous change is not maintained in the magnitudes ot those tmngs, derived from this, which are not themselves continuous, & do not possess continuous magnitude, but are aggregates of separate things. That is to say, in those things that are thus considered as forming a certain whole, in such a way that the magnitude of the aggregate is not determined 122 PHILOSOPHIC NATURALIS THEORIA determinent distantias inter eadem extrema, sed a nobis extrema ipsa assumantur jam alia, jam alia, quae censeantur incipere ad aggregatum pertinere, ubi ad quasdam distantias devenerint, quas ut ut in se juxta legem continuitatis mutatas, nos a reliquis divellimus per saltum, ut dicamus pertinere eas partes ad id aggregatum. Id accidit, ubi in objectis casibus accessiones partium novae fiunt, atque ibi nos in usu vocabuli saltum facimus ; ars, & Natura saltum utique habet nullum. Alii casus in quibus 151. Non idem contingit etiam, ubi plantas, vel animantia crescunt, succo se insinuante 'uibus Ur ' hab'etur P er tubulos fibrarum, & procurrente, ubi & magnitude computata per distantias punctorum soium proxima, non maxime distantium transit per omnes intermedias ; cum nimirum ipse procursus fiat accurata contmm- p ef omnes intermedias distantias. At quoniam & ibi mutantur termini illi, qui distantias determinant, & nomen suscipiunt altitudinis ipsius plantas ; vera & accurata continuitas ne ibi quidem observatur, nisi tantummodo in motibus, & velocitatibus, ac distantiis singularum partium : quanquam ibi minus recedatur a continuitate accurata, quam in superioribus. In his autem, & in illis habetur ubique ilia alia continuitas quasdam apparens, & affectata tantummodo a Natura, quam intuemur etiam in progressu substantiarum, ut incipiendo ab inanima-[69]-tis corporibus progressu facto per vegetabilia, turn per quasdam fere semianimalia torpentia, ac demum animalia perfectiora magis, & perfectiora usque ad simios homini tarn similes. Quoniam & harum specierum, ac existentium individuorum in quavis specie numerus est finitus, vera continuitas haberi non potest, sed ordinatis omnibus in seriem quandam, inter binas quasque intermedias species hiatus debet esse aliquis necessario, qui continuitatem abrumpat. In omnibus iis casibus habentur discretas quasdam quantitates, non continues ; ut & in Arithmetica series ex. gr. naturalium numerorum non est continua, sed discreta ; & ut ibi series ad continuam reducitur tantummodo, si generaliter omnes intermedias fractiones concipiantur ; sic & in superiore exemplo quasdam velut continua series habebitur tantummodo ; si concipiantur omnes intermedias species possibiles. uitatem. Conciusio pertinens 152. Hoc pacto excurrendo per plurimos justmodi casus, in quibus accipiuntur ad ea, quse veram, a g gre g ata rerum a se invicem certis intervallis distantium, & unum aliquid continuum non (X CcL, CJ1.13E cLttCCtcl" OO O iill** tam habent contin- constituentium, nusquam accurata occurret continuitatis lex, sed per quandam dispersionem quodammodo affectata, & vera continuitas habebitur tantummodo in motibus, & in iis, quas a motibus pendent, uti sunt distantiae, & vires determinatas a distantiis, & velocitates a viribus ortae ; quam ipsam ob causam ubi supra num. 39 inductionem pro lege continuitatis assumpsimus, exempla accepimus a motu potissimum, & ab iis, quae cum ipsis motibus connectuntur, ac ab iis pendent. 153. Sed jam ad aliam difficultatem gradum faciam, quae non nullis negotium ingens 3ito facessit, & obvia est etiam, contra hanc indivisibilium, & inextensorum punctorum Theoriam ; ' & quod nimirum ea nullum habitura sint discrimen a spiritibus. Ajunt enim, si spiritus ejusmodi vires habeant, praestituros eadem phaenomena, tolli nimirum corpus, & omnem corporeae substantiae notionem sublata extensione continua, quae sit prascipua materias proprietas ita pertinens ad naturam ipsius ; ut vel nihil aliud materia sit, nisi substantia praedita extensione continua ; vel saltern idea corporis, & materiae haberi non ppssit ; nisi in ea includatur idea extensionis continuae. Multa hie quidem congeruntur simul, quae nexum aliquem inter se habent, quae hie seorsum evolvam singula. Difficultates petitae a discrimine debito inter materiam spiritum. DifferrehKcpuncta 154. Inprimis falsum omnino est, nullum esse horum punctorum discrimen a spiritibus. fm^netrabUttlte^ Di scrimen potissimum materiae a spiritu situm est in hisce duobus, quod _ materia_ est nSitatem, a e - sensibilis, & incapax cogitationis, ac voluntatis, spiritus nostros sensus non afficit, & cogitare capadtatem cogit- p Otest) ac ve lle. Sensibilitas autem non ab extensione continua oritur, sed ab impene- trabilitate, qua fit, ut nostrorum organorum fibrae tendantur a corporibus, quae ipsis sistuntur, & motus ad cerebrum pro-[7o]-pagetur. Nam si extensa quidem essent corpora, sed impenetrabilitate carerent ; manu contrectata fibras non sisterent, nee motum ullum in iis progignerent, ac eadem radios non reflecterent, sed liberum intra se aditum luci prasberent. Porro hoc discrimen utrumque manere potest integrum, & manet inter mea indivisibilia hasc puncta, & spiritus. Ipsa impenetrabilitatem habent, & sensus nostros afficiunt, ob illud primum crus asymptoticum exhibens vim illam repulsivam primam ; spiritus autem, quos impenetrabilitate carere credimus, ejusmodi viribus itidem carent, & sensus nostros idcirco nequaquam afficiunt, nee oculis inspectantur, nee ^manibus palpari possunt. Deinde in meis hisce punctis ego nihil admitto aliud, nisi illam virium legem cum inertias vi conjunctam, adeoque ilia volo prorsus incapacia cogitationis, & voluntatis. A THEORY OF NATURAL PHILOSOPHY I2 3 by the distances between the same extremes all the time, but the extremes we take are different, one after another ; & these are considered to begin to belong to the aggregate when they attain to certain distances from it ; &, although in themselves changed in accordance with the Law of Continuity, we separate them from the rest in a discontinuous manner, by saying that these parts belong to the aggregate. This comes about, whenever in the cases under consideration fresh additions of parts take place ; & then we make a discontinuity in the use of a term ; art, as well as Nature, has no discontinuity. 151. It is not the same thing however in the case of the growth of plants or animals, which is due to a life-principle insinuating itself into, & passing along the fine tubes of the fibres ; here the magnitude, calculated by means of the distance between the points furthest from one another, passes through all intermediate distances ; for the flow of the life-principle takes place indeed through all intermediate distances. But, since here also the extremes are changed, which determine the distances, & denominate the altitude of the plant ; not even in this case is really accurate continuity observed, except only in the motions & velocities and distances of the separate parts ; however there is here less departure from accurate continuity, than there was in the examples given above. In both there is indeed that kind of apparent continuity, which Nature does no more than try to maintain ; such as we also see in the series of substantial things, which starting from inanimate bodies, continues through vegetables, then through certain sluggish semianimals, & lastly, through animals more & more perfect, up to apes that are so like to man. Also, since the number of these species, & the number of existent individuals of any species, is finite, it is impossible to have true continuity ; but if they are all ordered in a series, between any two intermediate species there must necessarily be a gap ; & this will break the continuity. In all these cases we have certain discrete, & not continuous, quantities ; just as, for instance, the arithmetical series of the natural numbers is not continuous, but discrete. Also, just as the series is reduced to continuity only by mentally introducing in general all the intermediate fractions ; so also, in the example given above a sort of continuous series is obtained, if & only if all intermediate possible species are so included. 152. In the same way, if we examine a large number of cases of the same kind, in which aggregates of things are taken, separated from one another by certain definite intervals, & not composing a single continuous whole, an accurate continuity law will never be met with, but only a sort of counterfeit depending on dispersion. True continuity will only be obtained in motions, & in those things that depend on motions, such as distances & forces determined by distances, & velocities derived from such forces. It was for this very reason that, when we adopted induction for the proof of the Law of Continuity in Art. 39 above, we took our examples mostly from motion, & from those things which are connected with motions & depend upon them. 153. Now I will pass on to another objection, which some people have made a great to-do about, and which has also been raised in opposition to this Theory of indivisible & non-extended points ; namely, that there will be no difference between my points & spirits. For, they say that, if spirits were endowed with such forces, they would show the same phenomena as bodies, & that bodies & all idea of corporeal substance would be done away with by denying continuous extension ; for this is one of the chief properties of matter, so pertaining to Nature itself ; so that either matter is nothing else but substance endowed with continuous extension, or the idea of a body and of matter cannot be obtained without the inclusion of the idea of continuous extension. Here indeed there are many matters all jumbled together, which have no connection with one another ; these I will now separate & discuss individually. 154. First of all it is altogether false that there is no difference between my points & spirits. The most important difference between matter & spirit lies in the two facts, that matter is sensible & incapable of thought, whilst spirit does not affect the senses, but can think or will. Moreover, sensibility does not arise from continuous extension, but from impenetrability, through which it comes about that the fibres of our organs are subjected to stress by bodies that are set against them & motions are thereby propagated to the brain. For if indeed bodies were extended, but lacked impenetrability, they would not resist the fibres of the hand when touched, nor produce in them any motion ; nor would they reflect light, but allow it an uninterrupted passage through themselves. Further, it is possible that each of these distinctions should hold good independently ; & they do so between these indivisible points of mine & spirits. My points have impenetrability & affect our senses, because of that first asymptotic branch representing that first repulsive force ; but spirits, which we suppose to lack impenetrability, lack also forces of this kind, and therefore can in no wise affect our senses, nor be examined by the eyes, nor be felt by the hands. Then, in these points of mine, I admit nothing else but the law of forces conjoined with the force of inertia ; & hence I intend them to be incapable Cases in which there is a breach of continuity ; others in which the con- tinuity is only very nearly, but not accurately, ob- served. Conclusion as re- gards those things that possess true continuity, and those that have a counterfeit continu- ity. Objections derived from the distinc- tion that has to be made between matter & spirit. These points differ from spirits on account o f their impenet rability, their being sen- sible, & their inca- pacity for thought. I2 4 PHILOSOPHIC NATURALIS THEORIA Si possibilis sub- earn nee materiam spiritum. Quamobrem discrimen essentiae illud utrumque, quod inter corpus, & spiritum agnoscunt omnes, id & ego agnosco, nee vero id ab extensione, & compositione continua desumitur, sed ab iis, quae cum simplicitate, & inextensione aeque conjungi possunt, & cohaerere cum ipsis. 155. At si substantiae capaces cogitationis & voluntatis haberent ejusmodi virium legem, an non eosdem praestarent effectus respectu nostrorum sensuum, quos ejusmodi puncta ? capax cogitationis ; Respondebo sane, me hie non quaerere, utrum impenetrabilitas, & sensibilitas, quae ab iis f nec v i r ibus pendent, conjungi possint cum facultate cogitandi, & volendi, quae quidem quaestio eodem redit, ac in communi sententia de impenetrabilitate extensorum, ac compositorum relata ad vim cogitandi, & volendi. Illud ajo, notionem, quam habemus partim ex observationibus tarn sensuum respectu corporurh, quam intimae conscientiae respectu spiritus, una cum reflexione, partim, & vero etiam circa spiritus potissimum, ex principiis immediate revelatis, vel connexis cum principiis revelatis, continere pro materia impenetrabilitatem, & sensibilitatem, una cum incapacitate cogitationis, & pro spiritu incapacitatem afHcicndi per impenetrabilitatem nostros sensus, & potentiam cogitandi, ac volendi, quorum priores illas ego etiam in meis punctis admitto, posteriores hasce in spiritibus ; unde fit, ut mea ipsa puncta materialia sint, & eorum massae constituant corpora a spiritibus longissime discrepantia. Si possibile sit illud substantiae genus, quod & hujusmodi vires activas habeat cum inertia conjunctas, & simul cogitare possit, ac velle ; id quidem nee corpus erit, nee spiritus, sed tertium quid, a corpore discrepans per capacitatem cogitationis, & voluntatis, discrepans autem a spiritu per inertiam, & vires hasce nostras, quae impenetrabilitatem inducunt. Sed, ut ajebam, ea quaestio hue non pertinet, & aliunde resolvi debet ; ut aliunde utique debet resolvi quaestio, qua quaeratur, an substantia extensa, & impenetrabilis [71] hasce proprietates conjungere possit cum facultate cogitandi, volendique. Nihil amitti, 156. Nee vero illud reponi potest, argumentum potissimum ad evincendum, materiam amisso argumento cogitare non posse, deduci ab extensione, & partium compositione, quibus sublatis, omne id eorum, qui a com- r to , . r , . VT i positione partium lundamentum prorsus corruere, & ad materialismum sterm viam. JNam ego sane non video, deducunt incapaci- quid argument! peti possit ab extensione, & partium compositione pro incapacitate cogitandi, & volendi. Sensibilitas, praecipua corporum, & materiae proprietas, quae ipsam adeo a spiritibus discriminat, non ab extensione continua, & compositione partium pendet, uti vidimus, sed ab impenetrabilitate, quae ipsa proprietas ab extensione continua, & compositione non pendet. Sunt qui adhibent hoc argumentum ad excludendam capacitatem cogitandi a materia, desumptum a compositione partium : si materia cogitaret ; singulae ejus partes deberent singulas cogitationis partes habere, adeoque nulla pars objectum perciperet ; cum nulla haberet earn perceptionis partem, quam habet altera. Id argumentum in mea Theoria amittitur ; at id ipsum, meo quidem judicio, vim nullam habet. Nam posset aliquis respondere, cogitationem totam indivisibilem existere in tota massa materiae, quae certa partium dispositione sit praedita, uti anima rationalis per tarn multos Philosophos, ut ut indivisibilis, in omni corpore, vel saltern in parte corporis aliqua divisibili existit, & ad ejusmodi praesentiam praestandam certa indiget dispositione partium ipsius corporis, qua semel laesa per vulnus, ipsa non potest ultra ibi esse ; atque ut viventis corporei, sive animalis rationalis natura, & determinatio habetur per materiam divisibilem, & certo modo constructam, una cum anima indivisibili ; ita ibi per indivisibilem cogitationem inhaerentem divisibili materise natura, & determinatio cogitantis haberetur. Unde aperte constat eo argumento amisso, nihil omnino amitti, quod jure dolendum sit. Etiam si quidpiam 157. Sed quidquid de eo argumento censeri debeat, nihil refert, nee ad infirmandam iam^poStive ^prob- Theoriam positivis, & validis argumentis comprobatam, ac e solidissimis principiis directa ari, & in ea manere ratiocinatione deductani, quidquam potest unum, vel alterum argumentum amissum, quod intemter1ain me & a ^ probandam aliquam veritatem aliunde notam, & a revelatis principiis aut directe, aut spiritum. indirecte confirmatam, ab aliquibus adhibeatur, quando etiam vim habeat aliquam, quam, uti ostendi, superius allatum argumentum omnino non habet. Satis est, si ilia Theoria cum ejusmodi veritate conjungi possit, uti haec nostra cum immaterialitate spirituum con- jungitur optime, cum retineat pro materia inertiam, impenetrabilitatem, sensibilitatem, incapacitatem cogitandi, & pro spiritibus retineat incapacitatem afHciendi sensus nostros per impenetrabilitatem, & facultatem cogitandi, ac volendi. [72] Ego quidem in ipsius A THEORY OF NATURAL PHILOSOPHY 125 of thought or will. Wherefore I also acknowledge each of those essential differences between matter and spirit, which are acknowledged by everyone ; but by me it is not deduced from extension and continuous composition, but, just as correctly, from things that can be conjoined with simplicity & non-extension, & can combine with them. 155. Now if there were substances capable of thought & will that also had a law of if it were possible forces of this kind, is it possible that they would produce the same effects with respect to substanc<Pthatwas our senses, as points of this sort ? Truly, I will answer that I do not seek to know in this both endowed with connection, whether impenetrability & sensibility, which depend on these forces, can capsTbieofthoughT be conjoined with the faculty of thinking & willing ; indeed this question comes to the it would be neither same thing as the general idea of the relations of impenetrability of extended & composite matter nor s P' nt - things to the power of thinking & willing. I will say but this, that we form our ideas, partly from observations, of the senses in the case of bodies, & of the inner consciousness in the case of spirits, together with reflections upon them, partly, & indeed more especially in the case of spirits, from directly revealed principles, or matters closely connected with revealed principles ; & these ideas involve for matter impenetrability, sensibility, combined with incapacity for thought, & for spirit an incapacity for affecting our senses by means of impenetrability, together with the capacity for thinking and willing. I admit the former of these in the case of my points, & the latter for spirits ; so that these points of mine are material points, & masses of them compose bodies that are far different from spirits. Now if it were possible that there should be some kind of substance, which has both active forces of this kind together with a force of inertia & also at the same time is able to think and will ; then indeed it will neither be body nor spirit, but some third thing, differing from a body in its capacity for thought & will, & also from spirit by possessing inertia and these forces of mine, which lead to compenetration. But as I was saying, that question does not concern me now, & the answer must be found by other means. So by other means also must the answer be found to the question, in which we seek to know whether a substance that is extended & impenetrable can conjoin these two properties with the faculty of thinking and willing. 156. Now it cannot be ignored that an argument of great importance in proving that Nothing is lost matter is incapable of thought is deduced from extension & composition by parts ; & ^ n ^g^^ 1 " 1 ^ if these are denied, the whole foundation breaks down, & the way is laid open to materialism, those who deduce But really I do not see what in the way of argument can be derived from extension & i? t&%~* y j; r ... , *, i i i *ii* n *i "v v i_ f lutJU 5 i *^ irom com- composition by parts, to support incapacity for thinking and willing, bensibmty, the cruel position by parts. property of bodies & of matter, which is so much different from spirits, does not depend on continuous extension & composition by parts, as we have seen, but on impenetrability ; & this latter property does not depend on continuous extension & composition. There are some, who use the following argument, derived from composition by parts, to exclude from matter the capacity for thought : If matter were to think, then each of its parts would have a separate part of the thought, & thus no part would have perception of the object of thought ; for no part can have that part of the perception that another part has. This argument is neglected in my Theory ; but the argument itself, at least so I think, is unsound. For one can reply that the complete thought exists as an indivisible thing in the whole mass of matter, which is endowed with a certain arrangement of parts, in^the same way as the rational soul in the opinion of so many philosophers exists, although it is indivisible, in the whole of the body, or at any rate in a certain divisible part of the body ; & to maintain a presence of this kind there is need for a definite arrangement of the parts of the body, which if at any time impaired by a wound would no longer exist there. Thus, just as from the nature of a living body, or of a rational animal, determination arises from matter that is divisible & constructed on a definite plan, in conjunction with an indivisible mind ; so also in this case by means of indivisible thought inherent in the nature of divisible matter, there is a propensity for thought. From this it is very plain that, if this argument is dismissed, there will be nothing neglected that we have any reason to regret. 157. But whatever opinion we are to form about this argument, it makes no difference, Even a something nor can it weaken a Theory that has been corroborated by direct & valid arguments, & iheVheorrcln C be deduced from the soundest principles by a straightforward chain of reasoning, if we leave P^J^. in & a d t ^t out one or other of the arguments, which have been used by some for the purpose of ^f^fi rem ain in testing some truth that is otherwise known & confirmed by revealed principles either j^^fj*^ directly or indirectly ; even when the argument has some validity, which, as I have shown, matter & spir i t . that adduced above has not in any way. It is sufficient if that theory can be conjoined with such a truth ; just as this Theory of mine can be conjoined in an excellent manner with the immateriality of spirits. For it retains for matter inertia, impenetrability, sensibility, & incapacity for thinking, & for spirits it retains the incapacity for affecting our senses by impenetrability, & the faculty of thinking or willing. Indeed I assume the 126 PHILOSOPHIC NATURALIS THEORIA materiae, & corporeae substantias definitione ipsa assumo incapacitatem cogitandi, & volendi, & dico corpus massam compositam e punctis habentibus vim inertiae conjunctam cum viribus activis expressis in fig. i,& cum incapacitate cogitandi, ac volendi, qua definitione admissa, evidens est, materiam cogitare non posse ; quae erit metaphysica quaedam conclusio, ea definitione admissa, certissima : turn ubi solae rationes physicae adhibeantur, dicam, haec corpora, quae meos afficiunt sensus, esse materiam, quod & sensus afficiant per illas utique vires, & non cogitent. Id autem deducam inde, quod nullum cogitationis indicium praestent ; quae erit conclusio tantum physica, circa existentiam illius materiae ita definitae, aeque physice certa, ac est conclusio, quae dicat lapides non habere levitatem, quod nunquam earn prodiderint ascendendo sponte, sed semper e contrario sibi relict! descenderint. Sensus omnino fain 158. Quod autem pertmet ad ipsam corporum, & materiae ideam, quae videtur exten- ^nultat^in^xten- si nem continuam, & contactum partium involvere, in eo videntur mihi quidem Cartesian! sionis, quam nobis inprimis, qui tantopere contra prasjudicia pugnare sunt visi, praejudiciis ipsis ante omnes alios indulsisse. Ideam corporum habemus per sensus ; sensus autem de continuitate accurata judicare omnino non possunt, cum minima intervalla sub sensus non cadant. Et quidem omnino certo deprehendimus illam continuitatem, quam in plerisque corporibus nobis objiciunt sensus nostri, nequaquam haberi. In metallis, in marmoribus, in vitris, & crystallis continuitas nostris sensibus apparet ejusmodi, ut nulla percipiamus in iis vacua spatiola, nullos poros, in quo tamen hallucinari sensus nostros manifesto patet, turn ex diversa gravitate specifica, quae a diversa multitudine vacuitatum oritur utique, turn ex eo, quod per ilia insinuentur substantiae plures, ut per priora oleum diffundatur, per posteriora liberrime lux transeat, quod quidem indicat, in posterioribus hisce potissi- mum ingentem pororum numerum, qui nostris sensibus delitescunt. Fons prajudici- 159- Quamobrem jam ejusmodi nostrorum sensuum testimonium, vel potius noster orum : haberi pro eor um ratiociniorum usus, in hoc ipso genere suspecta esse debent, in quo constat nos nulhs in se, quas , .... . 11- v -i_ sunt nuiia in nostris decipi. Suspican igitur licet, exactam continuitatem sine urns spatiolis, ut in majonbus sensibus : eorum corporibus ubique deest, licet sensus nostri illam videantur denotare, ita & in minimis quibusvis particulis nusquam haberi, sed esse illusionem quandam sensuum tantummodo, & quoddam figmentum mentis, reflexione vel non utentis, vel abutentis. Est enim solemne illud hominibus, atque usitatum, quod quidem est maximorum praejudiciorum fons, & origo praecipua, ut quidquid in nostris sensibus est nihil, habeamus pro nihilo absolute. Sic utique per tot saecula a multis est creditum, & nunc etiam a vulgo creditur, [73] quietem Telluris, & diurnum Solis, ac fixarum motum sensuum testimonio evinci, cum apud Philosophos jam constet, ejusmodi qusestionem longe aliunde resolvendam esse, quam per sensus, in quibus debent eaedem prorsus impressiones fieri, sive stemus & nos, & Terra, ac moveantur astra, sive moveamur communi motu & nos, & Terra, ac astra consistant. Motum cognoscimus per mutationem positionis, quam objecti imago habet in oculo, & quietem per ejusdem positionis permanentiam. Tarn mutatio, quam permanentia fieri possunt duplici modo : mutatio, primo si nobis immotis objectum movea- tur ; & permanentia, si id ipsum stet : secundo, ilia, si objecto stante moveamur nos ; haec, si moveamur simul motu communi. Motum nostrum non sentimus, nisi ubi nos ipsi motum inducimus, ut ubi caput circumagimus, vel ubi curru delati succutimur. Idcirco habemus turn quidem motum ipsum pro nullo, nisi aliunde admoneamur de eodem motu per causas, quae nobis sint cognitae, ut ubi provehimur portu, quo casu vector, qui jam diu assuevit idese littoris stantis, & navis promotae per remos, vel vela, corrigit apparentiam illius, terrceque urbesque recedunt, & sibi, non illis, motum adjudicat. Eorum correctio 160. Hinc Philosophus, ne fallatur, non debet primis hisce ideis acquirere, quas e ubi deprehendatur, se nsationibus haurimus, & ex illis deducere consectaria sine diligent! perquisitione, ac in ea modo al cum tl s a en n quae ab infantia deduxit, debet diligenter inquirere. Si inveniat, easdem illas sensuum suum apparentia perceptiones duplici modo aeque fieri posse ; peccabit utique contra Logicae etiam naturalis leges, si alterum modum prze altero pergat eligere, unice, quia alterum antea non viderat, & pro nullo habuerat, & idcirco alteri tantum assueverat. Id vero accidit in casu nostro : A THEORY OF NATURAL PHILOSOPHY 127 incapacity for thinking & willing in the very definition of matter itself & corporeal substance ; & I say that a body is a mass composed of points endowed with a force of inertia together with such active forces as are represented in Fig. i, & an incapacity for thinking & willing. If this definition is taken, it is clear that matter cannot think ; & this will be a sort of metaphysical conclusion, which will follow with absolute certainty from the acceptation of the definition. Again, where physical arguments are alone employed, I say that such bodies as affect our senses are matter, because they affect the senses by means of the forces under consideration, & do not think. I also deduce the same conclusion from the fact that they afford no evidence of thought. This will be a conclusion that is solely physical with regard to the existence of matter so defined ; & it will be just as physically true as the conclusion that says that stones do not possess levity, deduced from the fact that they never display such a thing by an act of spontaneous ascent, but on the contrary always descend if left to themselves. 158. With regard to the idea of bodies & matter, which seems to involve continuous The senses are extension, it seems to me indeed that in this matter the Cartesians in particular, who have altogether at fault . r i i_ m the greatness of appeared to impugn pre judgments with so much vigour, have given themselves up to these the continuity of prejudgments more than anyone else. We obtain the idea of bodies through the senses ; f^^'^beiieve 5 '' and the senses cannot in any way judge on a matter of accurate continuity ; for very small intervals do not fall within the scope of the senses. Indeed we quite take it for granted that the continuity, which our senses meet with in a large number of bodies, does not really exist. In metals, marble, glass & crystals there appears to our senses to be continuity, of such sort that we do not perceive in them any little empty spaces, or pores ; but in this respect the senses have manifestly been deceived. This is clear, both from their different specific gravities, which certainly arises from the differences in the numbers of the empty spaces ; & also from the fact that several substances will insinuate themselves through their substance. For instance, oil will diffuse itself through the former, & light will pass quite freely through the latter ; & this indeed indicates, especially in the case of the latter, an immense number of pores ; & these are concealed from our senses. 159. Hence such evidence of our senses, or rather our employment of such arguments, The origin of pre- must now lie open to suspicion in that class, in which it is known that we have been deceived, j^fjdered : as^o- We may then suspect that accurate continuity without the presence of any little empty thing, which are spaces such as is certainly absent from bodies of considerable size, although our senses SSe 1 srases ar^con- seem to remark its presence is also nowhere existent in any of their smallest particles ; cemed ; examples but that it is merely an illusion of the senses, & a sort of figment of the brain through its not using, or through misusing, reflection. For it is a customary thing for men (& a thing that is frequently done) to consider as absolutely nothing something that is nothing as far as the senses are concerned ; & this indeed is the source & principal origin of the greatest prejudices. Thus for many centuries it was credited by many, & still is believed by the unenlightened, that the Earth is at rest, & that the daily motions of the Sun & the fixed stars is proved by the evidence of the senses ; whilst among philosophers it is now universally accepted that such a question has to be answered in a far different manner from that by means of the senses. Exactly the same impressions are bound to be obtained, whether we & the Earth stand still & the stars are moved, or we & the Earth are moved with a common motion & the stars are at rest. We recognize motion by the change of position, which the image of an object has in the eye ; and rest by the permanence of that position. Now both the change & the permanence can come about in two ways. Firstly, if we remain at rest, there is a change of position if the object is moved, & permanence if it too is at rest ; secondly, if we move, there is a change if the object is at rest, & permanence if we & it move with a motion common to both. We do not feel ourselves moving, unless we ourselves induce the motion, as when we turn the head, or when we are jolted as we are borne in a vehicle. Hence we consider that the motion is nothing, unless we are made to notice in other ways that there is motion by causes that are known to us. Thus, when " we leave the harbour" a passenger who has for some time been accustomed to the idea of a shore remaining still, & of a ship being propelled by oars or sails, corrects the apparent motion of the shore ; &, as " the land & buildings recede" he attributes the motion to himself and not to them. 160. Hence, the philosopher, to avoid being led astray, must not seek to obtain from ^ctionjrf ^ these primary ideas that we derive from the senses, or deduce from them, consequential known that the theorems, without careful investigation; & he must carefully study those things that matter^ ^annot^ be he has deduced from infancy. If he find that these very perceptions by the senses can agreement with come about in two ways, one of which is as probable as the other ; then he will certainly hat the is JV e commit an offence against the laws of natural logic, if he should proceed to choose one some other way . method in preference to the other, solely for the reason that previously he had not seen the one & took no account of it, & thus had become accustomed to the other. Now 128 PHILOSOPHIC NATURALIS THEORIA sensationes habebuntur eaedem, sive materia constet punctis prorsus inextensis, & distantibus inter se per intervalla minima, quae sensum fugiant, ac vires ad ilia intervalla pertinentes organorum nostrorum fibras sine ulla sensibili interruptione afficiant, sive continua sit, & per immediatum contactum agat. Patebit autem in tertia hujusce operis parte, quo pacto proprietates omnes sensibiles corporum generales, immo etiam ipsorum prsecipua discrimina, cum punctis hisce indivisibilibus conveniant, & quidem multo sane melius, quam in communi sententia de continua extensione materiae. Quamobrem errabit contra rectae ratiocinationis usum, qui ex praejudicio ab hujusce conciliationis, & alterius hujusce sensationum nostrarum causae ignoratione inducto, continuam extensionem ut proprietatem necessariam corporum omnino credat, & multo magis, qui censeat, materialis substantive ideam in ea ipsa continua extensione debere consistere. Ordo idearum, quas esse per tactum. 161. Verum quo magis evidenter constet horum prsejudiciorum origo, afferam hie dissertationis De Materia Divisibilita-\j4\-te, & Principiis Corporum, numeros tres inci- piendo a 14, ubi sic : " utcunque demus, quod ego omnino non censeo, aliquas esse innatas ideas, & non per sensus acquisitas ; illud procul dubio arbitror omnino certum, ideam corporis, materiae, rei corporeae, rei materialis, nos hausisse ex sensibus. Porro ideas prims omnium, quas circa corpora acquisivimus per sensus, fuerunt omnino eae, quas in nobis tactus excitavit, & easdem omnium frequentissimas hausimus. Multa profecto in ipso materno utero se tactui perpetuo offerebant, antequam ullam fortasse saporum, aut odorum, aut sonorum, aut colorum ideam habere possemus per alios sensus, quarum ipsarum, ubi eas primum habere ccepimus, multo minor sub initium frequentia fuit. Idese autem, quas per tactum habuimus, ortae sunt ex phsenomenis hujusmodi. Experiebamur palpando, vel temere impingendo resistentiam vel a nostris, vel a maternis membris ortam, quae cum nullam interruptionem per aliquod sensibile intervallum sensui objiceret, obtulit nobis ideam impenetrabilitatis, & extensionis continuae : cumque deinde cessaret in eadem directione, alicubi resistentia, & secundum aliam directionem exerceretur ; terminos ejusdem quanti- tatis concepimus, & figurse ideam hausimus." Quae fuerint turn consideranda : in- fantia ad eas re- flexiones, inepta : in quo ea sita sit. 162. " Porro oriebantur haec phsenomena a corporibus e materia jam efformatis, non a singulis materiae particulis, e quibus ipsa corpora componebantur. Considerandum diligenter erat, num extensio ejusmodi esset ipsius corporis, non spatii cujusdam, per quod particulae corpus efformantes diffunderentur : num ea particulse ipsae iisdem proprietatibus essent praeditae : num resistentia exerceretur in ipso contactu, an in minimis distantiis sub sensus non cadentibus vis aliqua impedimento esset, quae id ageret, & resistentia ante ipsum etiam contactum sentiretur : num ejusmodi proprietates essent intrinsecae ipsi materiae, ex qua corpora componuntur, & necessariae : an casu tantum aliquo haberentur, & ab extrinseco aliquo determinante. Haec, & alia sane multa considerate diligentius oportuisset : sed erat id quidem tempus maxime caliginosum, & obscurum, ac reflexionibus minus obviis minime aptum. Praster organorum debilitatem, occupabat animum rerum novitas, phaenomenorum paucitas, & nullus, aut certe satis tenuis usus in phaenomenis ipsis inter se comparandis, & ad certas classes revocandis, ex quibus in eorum leges, & causas liceret inquirere & systema quoddam efformare, quo de rebus extra nos positis possemus ferre judicium. Nam in hac ipsa phaenomenorum inopia, in hac efformandi systematis difficultate, in hoc exiguo reflexionum usu, magis etiam, quam in organorum imbecillitate, arbitror, sitam esse infantiam." inde [75] 163. "In hac tanta rerum caligine ea prima sese obtulerunt animo, quae^ minus orta extensionis j ta jndagine, minus intentis reflexionibus indigebant, eaque ipsa ideistoties repetitis altius continuae ut essen- . . .- 1 tiaiis, odorum, &c., impressa sunt, & tenacius adhaeserunt, & quendam veluti campum nacta prorsus vacuum, ut accidentaiium. & ac jhuc immunem, suo quodammodo jure quandam veluti possessionem inierunt. Inter- valla, quae sub sensum nequaquam cadebant, pro nullis habita : ea, quorum ideae^ semper simul conjunctae excitabantur, habita sunt pro iisdem, vel arctissimo, & necessario^ nexu inter se conjunctis. Hinc illud effectum est, ut ideam extensionis continuae, ideam A THEORY OF NATURAL PHILOSOPHY 129 that is just what happens in the case under consideration. The same sensations will be experienced, whether matter consists of points that are perfectly non-extended & distant from one another by very small intervals that escape the senses, & forces pertaining to those intervals affect the nerves of our organs without any sensible interruption ; or whether it is continuous and acts by immediate contact. Moreover it will be clearly shown, in the third part of this work, how all the general sensible properties of bodies, nay even the principal distinctions between them as well, will fit in with these indivisible points ; & that too, in a much better way than is the case with the common idea of continuous extension of matter. Wherefore he will commit an offence against the use of true reasoning, who, from a prejudgment derived from this agreement & from ignorance of this alter- native cause for our sensations, persists in believing that continuous extension is an absolutely necessary property of bodies ; and much more so, one who thinks that the very idea of material substance must depend upon this very same continuous extension. 161. Now in order that the source of these prejudices may be the more clearly known, Order of the ideas I will here quote, from the dissertation De Materice Divisibilitate & Princi-pii Corporum, ^ b^ies^tte three articles, commencing with Art. 14, where we have : " Even if we allow (a thing quite first ideas come opposed to my way of thinking) that some ideas are innate & are not acquired through o^ifch the SenSe the senses, there is no doubt in my mind that it is quite certain that we derive the idea of a body, of matter, of a corporeal thing, or a material thing, through the senses. Further, the very first ideas, of all those which we have acquired about bodies through the senses, would be in every circumstance those which have excited our sense of touch, & these also are the ideas that we have derived on more occasions than any other ideas. Many things continually present themselves to the sense of touch actually in the very womb of our mothers, before ever perchance we could have any idea of taste, smell, sound, or colour, through the other senses ; & of these latter, when first we commenced to have them, there were to start with far fewer occasions for experiencing them. Moreover the ideas which we have obtained through the sense of touch have arisen from phenomena of the following kind. We experienced a resistance on feeling, or on accidental contact with, an object ; & this resistance arose from our own limbs, or from those of our mothers. Now, since this resistance offered no opposition through any interval that was perceptible to the senses, it gave us the idea of impenetrability & continuous extension ; & then when it ceased in the original direction at any place & was exerted in some other direction, we conceived the boundaries of this quantity, & derived the idea of figure." 162. " Furthermore, these phenomena will have arisen from bodies already formed from Such things de- matter, not from the single particles of matter of which the bodies themselves were composed. S^time ^tae tf It would have to be considered carefully whether such extension was a property of the tude of inf'ancy^for body itself, & not of some space through which the particles forming the body were su .f h . r eflection ; on j-rc JIT -11 i r i i i wnat the y ma Y be diffused ; whether the particles themselves were endowed with the same properties ; founded, whether the resistance was exerted only on actual contact, or whether, at very small distances such as did not fall within the scope of the senses, some force would act as a hindrance & produce the same effect, and resistance would be felt even before actual contact ; whether properties of this kind would be intrinsic in the matter of which the bodies are composed, & necessary to its existence ; or only possessed in certain cases, being due to some external influence. These, & very many other things, should have been investigated most carefully ; but the period was indeed veiled in mist & obscurity to a great degree, & very little fitted for aught but the most easy thought. In addition to the weakness of the organs, the mind was occupied with the novelty of things & the rareness of the phenomena ; & there was no, or certainly very little, use made of comparisons of these phenomena with one another, to reduce them to definite classes, from which it would be permissible to investigate their laws & causes & thus form some sort of system, through which we could bring the judgment to bear on matters situated outside our own selves. Now, in this very paucity of phenomena, in this difficulty in the matter of forming a system, in this slight use of the powers of reflection, to a greater extent even than in the lack of development of the organs, I consider that infancy consists." 163. " In this dense haze of things, the first that impressed themselves on the mind Th ence P r J' u <^- i 1*1 -i i 11 i n i mcnis> di c uci i vcu. were those which required a less deep study & less intent investigation ; & these, since that continuity of the ideas were the more often renewed, made the greater impression & became fixed J^^J 1 ^ S the more firmly in the mind, & as it were took possession of, so to speak, a land that they continuity of odours found quite empty & hitherto immune, by a sort of right of discovery. Intervals, which &c - * accidental. in no wise came within the scope of the senses, were considered to be nothing ; those things, the ideas of which were always excited simultaneously & conjointly, were considered as identical, or bound up with one another by an extremely close & necessary bond. Hence the result is that we have formed the idea of continuous extension, the idea of 130 PHILOSOPHISE NATURALIS THEORIA impenetrabilitatis prohibentis ulteriorem motum in ipso tantum contactu corporibus affinxerimus, & ad omnia, quae ad corpus pertinent, ac ad materiam, ex qua ipsum constat, temere transtulerimus : quse ipsa cum primum insedissent animo, cum frequcntissimis, immo perpetuis phaenomenis, & experimentis confirmarentur ; ita tenaciter sibi invicem adhseserunt, ita firmiter ideae corporum immixta sunt, & cum ea copulata ; ut ea ipsa pro primis corporibus, & omnium corporearum rerum, nimirum etiam materiae corpora compo- nentis, ejusque partium proprietatibus maxime intrinsecis, & ad naturam, atque essentiam earundem pertinentibus, & turn habuerimus, & nunc etiam habeamus, nisi nos praejudiciis ejusmodi liberemus. Extensionem nimirum continuam, impenetrabilitatem ex contactu, compositionem ex partibus, & figuram, non solum naturae corporum, sed etiam corporeae materiae, & singulis ejusdem partibus, tribuimus tanquam proprietates essentiales : csetera, quae serius, & post aliquem reflectendi usum deprehendimus, colorem, saporem, odorem sonum, tanquam accidentales quasdam, & adventitias proprietates consideravimus." propositiones 164. Ita ego ibi, ubi Theoriam virium deinde refero, quam supra hie exposui, ac ad Theoriamcontinen? P r3 ecipuas corporum proprietates applico, quas ex ilia deduco, quod hie praestabo in parte tis. tertia. Ibi autem ea adduxeram ad probandam primam e sequentibus propositionibus, quibus probatis & evincitur Theoria mea, & vindicatur : sunt autem hujusmodi : i. Nullo prorsus argumento evincitur materiam habere extensionem continuam, W non potius constare e punctis prorsus indivisibilibus a se per aliquod intervallum distantibus ; nee ulla ratio seclusis pr&judiciis suadet extensionem ipsam continuam potius, quam compositionem e punctis prorsus indivisibilibus, inextensis, y nullum continuum extensum constituentibus. 2. Sunt argumenta, y satis valida ilia quidem, qua hanc compositionem e punctis indivisibilibus evincant extensioni ipsi continues pr&ferri oportere. Quo pacto con- 165. At quodnam extensionis genus erit istud, quod e punctis inextensis, & spatio coaiescan^lnmassas imaginario, sive puro nihilo [76] constat ? Quo pacto Geometria locum habere poterit, tenaces: transitus ubi nihil habetur reale continue extensum? An non punctorum ejusmodi in vacuo dam Partem secun " innatantium congeries erit, ut quaedam nebula unico oris flatu dissolubilis prorsus sine ulla consistent! figura, solidate, resistentia ? Haec quidem pertinent ad illud extensionis ,& cohaesionis genus, de quo agam in tertia parte, in qua Theoriam applicabo ad Physicam, ubi istis ipsis difficultatibus faciam satis. Interea hie illud tantummodo innuo in antecessum, me cohaesionem desumere a limitibus illis, in quibus curva virium ita secat axem, ut a repulsione in minoribus distantiis transitus fiat ad attractionem in majoribus. Si enim duo puncta sint in distantia alicujus limitis ejus generis, & vires, quae immutatis distantiis oriuntur, sint satis magnae, curva secante axem ad angulum fere rectum, & longissime abeunte ab ipso ; ejusmodi distantiam ea puncta tuebuntur vi maxima ita, ut etiam insensibiliter compressa resistant ulteriori compressioni, ac distracta resistant ulteriori distractioni ; quo pacto si multa etiam puncta cohaereant inter se, tuebuntur utique positionem suam, & massam constituent formae tenacissimam, ac eadem prorsus phsenomena exhibentem, quae exhiberent solidae massulae in communi sententia. Sed de hac re uberius, uti monui, in parte tertia : nunc autem ad secundam faciendus est gradus. A THEORY OF NATURAL PHILOSOPHY 131 impenetrability preventing further motion only on the absolute contact of bodies ; & then we have heedlessly transferred these ideas to all things that pertain to a solid body, and to the matter from which it is formed. Further, these ideas, from the time when they first entered the mind, would be confirmed by very frequent, not to say continual, phenomena & experiences. So firmly are they mutually bound up with one another, so closely are they intermingled with the idea of solid bodies & coupled with it, that we at the time considered these two things as being just the same as primary bodies, & as peculiarly intrinsic properties of all corporeal things, nay further, of the very matter from which bodies are composed, & of its parts ; indeed we shall still thus consider them, unless we free ourselves from prejudgments of this nature. To sum up, we have attributed continuous extension, impenetrability due to actual contact, composition by parts, & shape, as if they were essential properties, not only to the nature of bodies, but also to corporeal matter & every separate part of it ; whilst others, which we comprehend more deeply & as a consequence of some considerable use of thought, such as colour, taste, smell & sound, we have considered as accidental or adventitious properties." 164. Such are the words I used ; & then I stated the Theory of forces which I have A pair of proposi- expounded in the previous articles of this work, and I applied the theory to the principal tation 0f containing properties of bodies, deducing them from it ; & this I will set forth in the third part the whole of y of the present work. In the dissertation I had brought forward the arguments quoted *" in order to demonstrate the truth of the first of the following theorems. If these theorems are established, then my Theory is proved & verified; they are as follows : i. There is absolutely no argument that can be brought forward to prove that matter has continuous extension, y that it is not rather made up of perfectly indivisible points separated from one another by a definite interval ; nor is there any reason apart from prejudice in favour of continuous extension in preference to composition from points that are perfectly indivisible, non-extended, & forming no extended continuum of any sort. 2. There are arguments, W fairly strong ones too, which will prove that this composition from indivisible points is preferable to continuous extension. 165. Now what kind of extension can that be which is formed out of non-extended The manner in o t i 5 TT /-i 1111 which groups of points & imaginary space, i.e., out of pure nothing ? How can Geometry be upheld points coalesce into if no thing is considered to be actually continuously extended ? Will not groups of points, tenacious masses : n t i 11-1 i i T i i i i n & then we pass on floating in an empty space of this sort be like a cloud, dissolving at a single breath, & to the second part. absolutely without a consistent figure, or solidity, or resistance ? These matters pertain to that kind of extension & cohesion, which I will discuss in the third part, where I apply my Theory to physics & deal fully with these very difficulties. Meanwhile I will here merely remark in anticipation that I derive cohesion from those limit-points, in which the curve of forces cuts the axis, in such a way that a transition is made from repulsion at smaller distances to attraction at greater distances. For if two points are at the distance that corresponds to that of any of the limit-points of this kind, & the forces that arise when the distances are changed are great enough (the curve cutting the axis almost at right angles & passing to a considerable distance from it), then the points will maintain this distance apart with a very great force ; so that when they are insensibly compressed they will resist further compression, & when pulled apart they resist further separation. In this way also, if a large number of points cohere together, they will in every case maintain their several positions, & thus form a mass that is most tenacious as regards its form ; & this mass will exhibit exactly the same phenomena as little solid masses, as commonly understood, exhibit. But I will discuss this more fully, as I have remarked, in the third part ; for now we must pass on to the second part. [77] PARS II Theories *Applicato ad Mechanicam Ante appHcatipnem 166. Considerabo in hac secunda parte potissimum generates quasdam leges aequilibrii consideratio'curvs! & motus tam punctorum, quam massarum, quae ad Mechanicam utique pertinent, & ad plurima ex iis, quae in elementis Mechanics passim traduntur, ex unico principio, & adhibito constant! ubique agendi modo, demonstranda viam sternunt pronissimam. Sed prius praemittam nonnulla quae pertinent ad ipsam virium curvam, a qua utique motuum, phaenomena pendent omnia. Quid in ea con- 167. In ea curva consideranda sunt potissimum tria, arcus curvae, area comprehensa siderandum. i nter axemj & arcum, quam general ordinata continue fluxu, ac puncta ilia, in quibus curva secat axem. Diversa arcnum 1 68. Quod ad arcus pcrtinet, alii dici possunt repulsivi, & alii attractivi, prout nimirum asymptotic! "tiam J acent ac * partes cruris asymptotici ED, vel ad contrarias, ac terminant ordinatas exhibentes numero infiniti. vires repulsivas, vel attractivas. Primus arcus ED debet omnino esse asymptoticus ex parte repulsiva, & in infinitum productus : ultimus TV, si gravitas cum lege virium reciproca duplicata distantiarum protenditur in infinitum, debet itidem esse asymptoticus ex parte attractiva, & itidem natura sua in infinitum productus. Reliquos figura I exprimit omnes finitos. Verum curva Geometrica etiam ejus naturae, quam exposuimus, posset habere alia itidem asymptotica crura, quot libuerit, ut si ordinata mn in H abeat in infinitum. Sunt nimirum curvae continuae, & uniformis naturae, quae asymptotes habent plurimas, & habere possunt etiam numero infinitas. (') Arcus intermedii. [78] 169. Arcus intermedii, qui se contorquent circa axem, possunt etiam alicubi, ubi ad ipsum devenerint, retro redire, tangendo ipsum, atque id ex utralibet parte, & possent itidem ante ipsum contactum inflecti, & redire retro, mutando accessum in recessum, ut in fig. i. videre est in arcu P^R. Arcus prostremus 170. Si gravitas gencralis legem vis proportionalis inverse quadrate distantiae, quam 36 non accurate servat, sed quamproxime, uti diximus in priore parte, retinet ad sensum non mutatam solum per totum planetarium, & cometarium systema, fieri utique poterit, ut curva virium non habeat illud postremum crus asymptoticum TV, habens pro asymptoto ipsam rectam AC, sed iterum secet axem, & se contorqueat circa ipsum.(*) Turn vero inter (i) S* ex. gr. in fig. 12. cyclois continua CDEFGH (3e., quam generet punctum peripheries circuli continue revoluti supra rectam AB, qute natura sua protenditur utrinque in infinitum, adeoque in infinitis punctis C, E, G, I, &c. occurrit basi AB. Si ubicunque ducatur qutevis ordinata PQ, productaturque in R ita, ut sit PR tertia post PQ, y datam quampiam rectam ; punctum R frit ad curvam continuum constantem totidem ramis MNO, VXY, yr., quot erunt arcus Cycloidales CDE, EFG, i3c,, quorum ramorum singuli habebunt bina crura asymptotica, cum ordinata PQ in accessu ad omnia puncta, C, E, G, &c. decrescat ultra quoscunque Unites, adeoque ordinata PR crescat ultra limites quoscunque. Erunt hie quidem omnes asymptoti CK, EL, GS &c. parallels inter se, & perpendiculares basi AB, quod in aliis curvis non est necessarium, cum etiam divergentes utcunque possint esse. Erunt autem y totidem numero, quot puncta. ilia C, E, G &c., nimirum infinite. Eodem autem pacto curvarum quarumlibet singuli occursus cum axe in curvis per eas hac eadem lege genitis bina crura asymptotica generant, cruribus ipsis jacentibus, vel, ut hie, ad eandem axis partem, ubi curva genetrix ab eo regreditur retro post appulsum, vel etiam ad partes oppositas, ubi curva genetrix ipsum secet, ac transiliat : cumque possit eadem curva altiorum generum secari in punctis plurimis a recta, vel contingi ; poterunt utique haberi y rami asymptotici in curva eadem continua, quo libuerit data numero. (k)Nam ex ipsa Geometrica continuitate, quam persecutus sum in dissertatione De Lege Continuitatis, y in dissertatione De Transformatione Locorum Geometricorum adjecta Sectionibus Conicis, exhibui necessitatem generalem secundi illius cruris asymptotici redeuntis ex infinite. Quotiescunque enim curva aliqua saltern algebraica habet asymptoticum crus aliquod, debet necessario habere y alterum ipsi respondens, y habens pro asymptoto eandem rectam : sed id habere 132 A THEORY OF NATURAL PHILOSOPHY 133 o o m O O \ '34 PHILOSOPHIC NATURALIS THEORIA PART II ^Application of the Theory to Mechanics 1 66. I will consider in this second part more especially certain general laws of Consideration of equilibrium, & motions both of points & masses ; these certainly belong to the science of proceeding w^t'h Mechanics, & they smooth the path that is most favourable for proving very many of those tne application to theorems, that are everywhere expounded in the elements of Mechanics, from a single principle, & in every case by the constant employment of a single method of dealing with them. But, before I do that, I will call attention to a few points that pertain to the curve of forces itself, upon which indeed all the phenomena of motions depend. 167. With regard to the curve, there are three points that are especially to be considered ; The points we have namely, the arcs of the curve, the area included between the axis & the curve swept out regard'tolt. 1 by the ordinate by its continuous motion, & those points in which the curve cuts the axis. 1 68. As regards the arcs, some may be called repulsive, & others attractive, according The different kinds indeed as they lie on the same side of the axis as the asymptotic branch ED or on the opposite totkfarc's may even side, & terminate ordinates that represent repulsive or attractive forces. The first arc be infinite in num- ED must certainly be asymptotic on the repulsive side of the axis, & continued indefinitely. r ' The last arc TV, if gravity extends to indefinite distances according to a law of forces in the inverse ratio of the squares of the distances, must also be asymptotic on the attractive side of the axis, & by its nature also continued indefinitely. All the remaining arcs are represented in Fig. I as finite. But a geometrical curve, of the kind that we have expounded, may also have other asymptotic branches, as many in number as one can wish ; for instance, suppose the ordinate mn at H to go away to infinity. There are indeed curves, that are continuous & uniform, which have very many asymptotes, & such curves may even have an infinite number of asymptotes. (') 169. The intermediate arcs, which wind about the axis, can also, at any point where intermediate arcs, they reach it, return backwards & touch it ; and they can do this on either side of it ; they may also be reflected and recede before actual contact, the approach being altered into a recession, as is to be seen in Fig. i with regard to the arc P^/yR. 170. If universal gravity obeys the law of a force inversely proportional to the square of The ultimate arc the distance (which, as I remarked in the first part, it only obeys as nearly as possible, but [ t P y eS poss not exactly), sensibly unchanged only throughout the planetary & cometary system, it will asymptotic, certainly be the case that the curve of forces will not have the last arm PV asymptotic with the straight line AC as the asymptote, but will again cut the axis & wind about it. (*) Then (i) Let, for example, in Fig. 12, CDEFGH &c. be a. continuous cycloid, generated by a point on the circumference of a circle rolling continuously along the straight line AB ; this by its nature extends on either side to infinity, W thus meets the base AB in an infinite number of points such as C, E, G, I, &c. // at every point there is drawn an ordinate such as PQ, and this is produced to R, so that PR is a third proportional to PQ W some given straight line ; then the point R will trace out a continuous curve consisting of as many branches, MNO, VXY, &c., as there are cycloidal arcs, CDE, EFG, &c. ; each of these branches will have a pair of asymptotic arms, since the ordinate PQ on approaching any one of the points C,E,G, &c., will decrease beyond all limits, (3 thus the ordinate PR will increase beyond all limits. In this curve then there will be CK, EL, GS, &c., all asymptotes parallel to one another & perpendicular to the base AB ; this is not necessarily the case in other curves, since they may be also inclined to one another in any manner. Further they will be as many in number as there are points such as C, E, G, &c., that is to say, infinite. Again, in a similar way, the several intersections of any curves you please with the axis give rise to a pair of asymptotic arms in curves derived from them according to the same law ; and these arms lie, either on the same side of the axis, as in this case, where the original curve leaves the axis once more after approaching it, or indeed on opposite sides of the axis, where the original curve cuts W crosses it. Also, since it is possible for the same curve of higher orders to be cut in a large number of points, or to be touched, there will possibly be also asymptotic arms in this same continuous curve equal to any given number you please. (k) For, from the principle of geometrical continuity itself, which I discussed in my dissertation De Lege Continuitatis and in the dissertation De Transformatione Locorum Geometricorum appended to my Sectionum Conicarum Elementa, / showed the necessity for the second asymptotic arm returning from infinity. For as often as an algebraical curve has at least one asymptotic arm, it must also have another that corresponds to it y has the same straight line 135 136 PHILOSOPHIC NATURALIS THEORIA alios casus innumeros, qui haberi possent, unum censeo speciminis gratia hie non omitten- dum ; incredibile enim est, quam ferax casuum, quorum singuli sunt notatu dignissimi, unica etiam hujusmodi curva esse possit. shnufum curTserte I 7 I> Si in % H * n axe C'C sint segmenta AA', A'A" numero quocunque, quorum Mundoru'm mag- posteriora sint in immensum majora respectu praecedentium, & per singula transeant, donaikfm propor " asympto-[79]-ti AB, A'B', A"B" perpendiculares axi ; possent inter binas quasque asymptotes esse curvae ejus formae, quam in fig. I habuimus, & quae exhibetur hie in DEFI &c., D'E'F'F, &c., in quibus primum crus ED esset asymptoticum repulsivum, postremum SV attractivum, in singulis vero intervallum EN, quo arcus curvae contorquetur, sit perquam exiguum respectu intervalli circa S, ubi arcus diutissime perstet proximus hyperbolae habenti ordinatas in ratione reciproca duplicata distantiarum, turn vero vel immediate abiret in arcum asymptoticum attractivum, vel iterum contorqueretur utcunque usque ad ejusmodi asymptoticum attractivum arcum, habente utroque asymptotico arcu aream infinitam ; in eo casu collocate quocunque punctorum numero inter binas quascunque asymptotes, vel inter binaria quotlibet, & rite ordinato, posset exurgere quivis, ut ita dicam, Mundorum numerus, quorum singuli essent inter se simillimi, vel dissimillimi, prout arcus EF&cN, E'F'&cN' essent inter se similes, vel dissimiles, atque id ita, ut quivis ex iis nullum haberet commercium cum quovis alio ; cum nimirum nullum punctum posset egredi ex spatio incluso iis binis arcubus, hinc repulsive, & inde attractive ; & ut omnes Mundi minorum dimensionum simul sumpti vices agerent unius puncti respectu proxime majoris, qui constaret ex ejusmodi massulis respectu sui tanquam punctualibus, dimensione nimirum omni singulorum, respectu ipsius, & respectu distantiarum, ad quas in illo devenire possint, fere nulla ; unde & illud consequi posset, ut quivis ex ejusmodi tanquam Mundis nihil ad sensum perturbaretur a motibus, & viribus Mundi illius majoris, sed dato quovis utcunque magno tempore totus Mundus inferior vires sentiret a quovis puncto materiae extra ipsum posito accedentes, quantum libuerit, ad aequales, & parallelas quae idcirco nihil turbarent respectivum ipsius statum internum. Omissis subiimiori- 172. Sed ea jam pertinent ad applicationem ad Physicam, quae quidem hie innui areas pr0greSSUS ad tantumm do, ut pateret, quam multa notatu dignissima considerari ibi possent, & quanta sit hujusce campi fcecunditas, in quo combinationes possibiles, & possibiles formae sunt sane infinities infinitae, quarum, quae ab humana mente perspici utcunque possunt, ita sunt paucae respectu totius, ut haberi possint pro mero nihilo, quas tamen omnes unico intuitu prsesentes vidit, qui Mundum condidit, DEUS. Nos in iis, quae consequentur, simpliciora tantummodo qusedam plerumque consectabimur, quae nos ducant ad phaeno- mena iis conformia, quae in Natura nobis pervia intuemur, & interea progrediemur ad areas arcubus respondentes. Cuicunque axis 173. Aream curvae propositae cuicunque, utcunque exiguo, axis segmento respondentem aream e "respondere P osse ess e utcunque magnam, & aream respondentem cuicunque, utcunque magno, [80] utcunque magnam posse esse utcunque parvam, facile patet. Sit in fig. 15, MQ segmentum axis utcunque secundjT" 1 de^non- parvum, vel magnum ; ac detur area utcunque magna, vel parva. Ea applicata ad MQ stratio. exhibebit quandam altitudinem MN ita, ut, ducta NR parallela MQ, sit MNRQ aequalis areae datae, adeoque assumpta QS dupla QR, area trianguli MSQ erit itidem aequalis areae datae. Jam vero pro secundo casu satis patet, posse curvam transire infra rectam NR, uti transit XZ, cujus area idcirco esset minor, quam area MNRQ ; nam esset ejus pars. potest vel ex eadem parte, vel ex opposita ; W crus ipsum jacere potest vel ad easdem plagas partis utriuslibet cum priore crure, vel ad oppositas, adeoque cruris redeuntis ex infinite poshiones quatuor esse possunt. Si in fig. 13 crus ED abeat in infinitum, existente asymptoto ACA', potest regredi ex parte A vel ut HI, quod crus facet ad eandem plagam, velut KL, quod, facet ad oppositam ; y ex parte A', vel ut MN, ex eadem plaga, vel ut OP, ex opposita. In posteriore ex iis duabus dissertationibus profero exempla omnium ejusmodi regressuum ; ac secundi, ($ quarti casus exempla exhibet etiam superior genesis, si curva generans contingat axem, vel secet, ulterius progressa respectu ipsius. Inde autem fit, ut crura asymptotica rectilineam babentia asymptotum esse non possint, nisi numero part, ut & radices imaginarite in eequationibus algebraicis. Verum hie in curva virium, in qua arcus semper debet progredi, ut singulis distantiis, sive abscissis, singula vires, sive ordinatts respondeant, casus primus, & tertius haberi non possunt. Nam ordinata RQ cruris DE occurreret alicubi in S, S' cruribus etiam HI, MN , adeoque relinquentur soli quartus, & secundus, quorum usus erit infra. A THEORY OF NATURAL PHILOSOPHY 137 o CD en, PHILOSOPHIC NATURALIS THEORIA LJ O \ (0 o A THEORY OF NATURAL PHILOSOPHY 139 there is one, out of an innumerable number of other cases that may possibly happen, which I think for the sake of an example should not be omitted here ; for it is incredible how prolific in cases, each of which is well worth mentioning, a single curve of this kind can be. 171. If, in Fig. 14, there are any number of segments AA', A'A", of which each that A series of similar follows is immensely great with regard to the one that precedes it ; & if through each c " rve . s - wlth a s ^ ies point there passes an asymptote, such as AB, A'B', A"B", perpendicular to the axis ; then tionai in magnitude, between any two of these asymptotes there may be curves of the form given in Fig. i. These are represented in Fig. 14 by DEFI &c., D'E'F'I' &c. ; & in these the first arm E would be asymptotic & repulsive, & the last SV attractive. In each the interval EN, where the arc of the curve is winding, is exceedingly small compared with the interval near S, where the arc for a very long time continues closely approximating to the form of the hyperbola having its ordinates in the inverse ratio of the squares of the distances ; & then, either goes off straightway into an asymptotic & attractive arm, or once more winds about the axis until it becomes an asymptotic attractive arc of this kind, the area corresponding to either asymptotic arc being infinite. In such a case, if a number of points are assembled between any pair of asymptotes, or between any number of pairs you please, correctly arranged, there can, so to speak, arise from them any number of universes, each of them being similar to the other, or dissimilar, according as the arcs EF . . . . N, E'F' .... N' are similar to one another, or dissimilar ; & this too in such a way that no one of them has any communication with any other, since indeed no point can possibly move out of the space included between these two arcs, one repulsive & the other attractive ; & such that all the universes of smaller dimensions taken together would act merely as a single point compared with the next greater universe, which would consist of little point-masses, so to speak, of the same kind compared with itself, that is to say, every dimension of each of them, compared with that universe & with respect to the distances to which each can attain within it, would be practically nothing. From this it would also follow that any one of these universes would not be appreciably influenced in any way by the motions & forces of that greater universe ; but in any given time, however great, the whole inferior universe would experience forces, from any point of matter placed without itself, that approach as near as possible to equal & parallel forces ; these therefore would have no influence on its relative internal state. 172. Now these matters really belong to the application of the Theory to physics ; & Leaving out more indeed I only mentioned them here to show how many things there may be well worth abstruse matters, j. . -i a i .,..,. ' - 9 . n i i r we pass on to areas. considering in that section, & how great is the fertility of this field of investigation, m which possible combinations & possible forms are truly infinitely infinite ; of these, those that can be in any way comprehended by the human intelligence are so few compared with the whole, that they can be considered as a mere nothing. Yet all of them were seen in clear view at one gaze by GOD, the Founder of the World. We, in what follows, will for the most part investigate only certain of the more simple matters which will lead us to phenomena in conformity with those things that we contemplate in Nature as far as our intelligence will carry us ; meanwhile we will proceed to the areas corresponding to the arcs. 173. It is easily shown that the area corresponding to any segment of the axis, however To any segment of small, can be anything, no matter how great ; & the area corresponding to any segment, corrt'spo^a'rfy however great, can be anything, no matter how small. In Fig. 1 5 , let MQ be a segment of the area, however axis, no matter how small, or great; & let an area be given, no matter how great, or SSi ; proof^the small. If this area is applied to MQ a certain altitude MN will be given, such that, if NR second part of this is drawn parallel to MQ, then MNRQ will be equal to the given area ; & thus, if QS is a taken equal to twice QR, the area of the triangle MSQ will also be equal to the given area. Now, for the second case it is sufficiently evident that a curve can be drawn below the straight line NR, in the way XZ is shown, the area under which is less than the area MNRQ ; as its asymptote ; & this can take place with either the same part of the line or with the other part ; also the arm itself can lie either on the same side of either of the two parts, or on the opposite side. Thus there may be four positions of the arm that returns from infinity. If, in Fig. 13, the arm ED goes off to infinity, the asymptote being ACA, it may return from the direction of A, either like HI, wheie the arm lies on the same side of the asymptote or as KL which lies 'on the opposite side of it ; or from the direction of A', either as MN, on the same side, or as, DP, on the opposite side. In the second of these two dissertations, I have given examples of all regressions of this sort ; y the method of generation given above will yield examples of the second W fourth cases, if the generating curve touches the axis, or cuts it & passes over beyond it. Further, it thus comes about that asymptotic arms having a rectilinear asymptote cannot exist except in pairs, just like imaginary roots in algebraical equations. But here in the curve of forces, in which the arc must always proceed in such a manner that to each distance or abscissa there corresponds a single force or ordinate, the first tf third cases cannot occur. For the ordinate RQ of the arm DE would meet somewhere, in S, S', the branches HI, MN as well. Hence only the fourth & second cases are left ; W these we will make use of later. 140 PHILOSOPHIC NATURALIS THEORIA Demonst ratio primse. Aream asympto- ticam posse esse infinitam, vel fini- tam magnitudinis cujuscunque. Areas exprimere incrementa, vel decrementa quad- ati velocitatis. Quin immo licet ordinata QV sit utcunque magna ; facile patet, posse arcum MaV ita accedere ad rectas MQ, QV ; ut area inclusa iis rectis, & ipsa curva, minuatur infra quoscunque determinatos limites. Potest enim jacere totus arcus intra duo triangula QaM, QaV, quorum altitudines cum minui possint, quantum libuerit, stantibus basibus MQ, QV, potest utique area ultra quoscunque limites imminui. Pos- set autem ea area esse minor quacunque data ; etiamsi QV esset asymptotus, qua de re paullo inferius. 174. Pro primo autem casu vel curva secet axem extra MQ, ut in T, vel in altero extremo, ut in M ; fieri poterit, ut ejus arcus TV, vel MV transeat per aliquod punctum V jacens ultra S, vel etiam per ipsum S ita, ut curvatura ilium ferat, quemad- modum figura exhibet, extra triangulum MSQ, quo casu patet, aream curvae respondentem intervallo MQ fore majorem, quam sit area trianguli MSQ, adeoque quam sit area data ; erit enim ejus trianguli area pars areae pertinentis ad curvam. Quod si curva etiam secaret alicubi axem, ut in H inter M, & Q, turn vero fieri posset, ut area respondens alteri e segmentis MH, QH esset major, quam area data . simul, & area alia assumpta, qua area assumpta esset minor area respondens segmento, alteri adeoque excessus prioris supra posteriorem remaneret major, quam area data. 175. Area asymptotica clausa inter asymptotum, & ordinatam quamvis, ut in fig. I BA#g, potest esse vel infinita, vel finita magnitudinis cujusvis ingentis, vel exiguae. Id quidem etiam geometrice demonstrari potest, sed multo facilius demonstratur calculo integrali admodum elementari ; & in Geometriae sublimioris elementis habentur theoremata, ex quibus id admodum facile deducitur 0. Generaliter nimi-[8l]-rum area ejusmodi est infinita ; si ordinata crescit in ratione reciproca abscissarum simplici, aut majore : & est finita ; si crescit in ratione multiplicata minus, quam per unitatem. 176. Hoc, quod de areis dictum est, necessarium fuit ad applicationem ad Mechanicam, ut nimirum habeatur scala quaedam velocitatum, quae in accessu puncti cujusvis ad aliud punctum, vel recessu generantur, vel eliduntur ; prout ejus motus conspiret directione vis, vel sit ipsi contrarius. Nam, quod innuimus & supra in adnot. (/) ad num. 118., ubi vires exprimuntur per ordinatas, & spatia per abscissas, area, quam texit ordinata, exprimit incrementum, vel decrementum quadrati velocitatis, quod itidem ope Geometrise demon- stratur facile, & demonstravi tam in dissertatione De Firibus Vivis, quam in Stayanis Supplements ; sed multo facilius res conficitur ope calculi integralis. () M H FIG. 15. (1) Sit Aa in Fig. I =x, ag=y ; ac sit #"y = I ; erit y = *-" >/ ", y dx elementum areee=x~ m/ *dx, cujus integrate *fn + A, addita constanti A, sive ob x~* > " > =y, habebitur ?xy + A. Quoniam incipit area in A, in n~m " n-m origine abscissarum ; si nm fuerit numerus positivus, adeoque n major, quam m ; area erit finita, ac valor A =o; area vero erit ad rectangulum AaXag, ut in ad n m, quod rectangulum, cum ag possit esse magna, & parva, ut libuerit, potest esse magnitudinis cujusvis. Is valor fit infinitus, si facto m =n, divisor evaaatQ; adeoque multo magis fit infinitus valor area, si m sit major, quam n. Unde constat, aream fore infiniiam, quotiescunque ordinatte crescent in ratione reciproca simplici, y majore ; secus fore finitam. (m) Sit u vis, c celeritas, t tempus, s spatium : erit u at = dc, cum celeritatis incrementum sit proportionale vi, W tempusculo ; ac erit c dt = ds, cum spatiolum confectwm respondeat velocitati, & tempusculo. Hinc eruitur dt = , W pariter dt =, adeoque- = W c dc = u ds. Porro 2c dc est incrementum quadrati vekcitatis cc, i3 u ds c u c in bypotbesi, quod ordinata sit w, & spatium s sit abscissa, est areola respondens spatiolo ds confecto. Igitur incrementum quadrati velocitatis conspirante vi, adeoque decrementum vi contraria, respondet arete respondent spatiolo percurso quovis infinitesimo tempusculo ; & proinde tempore etiam quovis finito incrementum, vel decrementum quadrati velocitatis respondet arece pertinenti ad partem axis referentem spatium percursum. Hinc autem illud sponte consequitur : si per aliquod spatium vires in singulis punctis eeedem permaneant, mobile autem adveniat cum velocitate quavis ad ejus initium ; diferentiam quadrati velocitatis finalis a quadrate velocitatis initialis fore semper eandem, quts idcirco erit tola velocitas finalis in casu, in quo mobile initio illius spatii haberet velocitatem nullam. Quare, quod nobis erit inferius usui, quadratum velocitatis finalis, conspirante vi cum directione motus, tzquabitur binis quadratis binarum velocitatum, ejus, quam babuit initio, W ejus,.quam acquisivisset in fine, si initio ingressum fuisset sine ulla velocitate. A THEORY OF NATURAL PHILOSOPHY 141 for it is part of it. Again, although the ordinate QV may be of any size, however great, it is easily shown that an arc MoV can approach so closely to the straight lines MQ, QV that the area included between these lines & the curve shall be diminished beyond any limits whatever. For it is possible for the curve to lie within the two triangles QaM, QaV ; & since the altitudes of these can be diminished as much as you please, whilst the bases MQ, QV remain the same, therefore the area can indeed be diminished beyond all limits whatever. Moreover it is possible for this area to be less than any given area, even although QV should be an asymptote ; we will consider this a little further on. 174. Again, for the first case, either the curve will cut the axis beyond MQ, as at T, or at either end, as at M. Then it is possible for it to happen that an arc of it, TV or MV, will pass through some point V lying beyond S, or even through S itself, in such a way that its curvature will carry it, as shown in the diagram, outside the triangle MSQ ; in this case it is clear that the area of the curve corresponding to the interval MQ will be greater than the area of the triangle MSQ, & therefore greater than the given area, for the area of this triangle is part of the area belonging to the curve. But if the curve should even cut the axis anywhere, as at H, between M & Q, then it would be possible for it to come about that the area corresponding to one of the two segments MH, QH would be greater than the given area together with some other assumed area ; & that the area corresponding to the other segment should be less than this assumed area ; and thus the excess of the former over the latter would remain greater than the given area. 175. An asymptotic area, bounded by an asymptote & any ordinate, like BAag in Fig. i, can be either infinite, or finite of any magnitude either very great or very small. This can indeed be also proved geometrically, but it can be demonstrated much more easily by an application of the integral calculus that is quite elementary ; & in the elements of higher geometry theorems are obtained from which it is derived quite easily. In general, it is true, an area of this kind is infinite ; namely when the ordinate increases in the simple inverse ratio of the abscissse, or in a greater ratio ; and it is finite, if it increases in this ratio multiplied by something less than unity. 176. What has been said with regard to areas was a necessary preliminary to the application of the Theory to Mechanics ; that is to say, in order that we might obtain a diagrammatic representation of the velocities, which, on the approach of any point to another point, or on recession from it, are produced or destroyed, according as its motion is in the same direction as the direction of the force, or in the opposite direction. For, as we also remarked above, in note (/) to Art. 118, when the forces are represented by ordinates & the distances by abscissae, the area that the ordinate sweeps out represents the increment or decrement of the square of the velocity. This can also be easily proved by the help of geometry ; & I gave the proof both in the dissertation De Firibus Fivis & in the Supplements to Stay's Philosophy ; but the matter is much more easily made out by the aid of the integral calculus. () Proof part. of the first An a s y m p totic area may be either infinite or equal to any finite area whatever. The areas represent the increments or decrements of the square of the velo- city. (1) In Fig. iletAa = x,ag = y; y let x m y" = I. Then will y x~ the element of area y dx = x~ m/ * dx : the integral of this is - x <-"/"+ A, where a constant A is added ; or, since x~ m/ *=y, we shall have-^ X v + A n-m n-m ' Now, since the area is initially A, at the origin of the abscissa, if n-m happened to be a positive number, y thus n greater than m, then the area will be finite, y the value of A will be = o. Also the area will be to the rectangle Aa.ag as n is to n-m ; y this rectangle, since ag can be either great or small, as you please, may be of any magnitude whatever. The value is infinite, if by making m equal to n the divisor becomes equal to zero ; & thus the value of the area becomes all the more infinite, if m is greater than n. Hence it follows that the area will be infinite, whenever the ordinates increase in a simple inverse ratio, or in a greater ratio ; otherwise it will be finite. (m) Let u be the force, c the velocity, t the time, y s the distance. Then will u dt dc, since the increment of the velocity is proportional to the force, y to the small interval of time. Also c dt = ds, since the distance traversed corresponds with the velocity W the small interval of time. Hence it follows that dt = dc/u, y similarly dt ds/c, y therefore dc/u = ds/c, & c dc u ds. Further, ^c dc is the increment of the square of the velocity c', y u ds, on the hypothesis that the ordinate represents u, y the abscissa the distance s, is the small area corresponding to the small distance traversed. Hence the increment of the square of the velocity, when in the direction of the force, y the decrement when opposite in direction to the force, is represented by the area corresponding to ds, the small distance traversed in any infinitely short time. Hence also, in any finite interval of time, the increment or decrement of the square of the velocity will be represented by the area corresponding to that part of the axis which represents the distance traversed. Hence also it follows immediately that, if through any distance the force on each of the points remains as before, but the moving body arrives at the beginning of it with any velocity, then the difference between the square of the final velocity y the square of the initial velocity will always be the same ; y this therefore will be the total final velocity, in the case where the moving body had no velocity at the beginning of the distance. Hence, the square of the final velocity, when the motion is in the same direction as the force, will be equal to the sum of the squares of the velocity which it had at the beginning y of the velocity it would have acquired at the end, if it had at the beginning started without any velocity ; a theorem that we shall make use of later. I 4 2 PHILOSOPHIC NATURALIS THEORIA Atque id ips u m, licet segmenta axis sint dimidia spatio- rum percursorum a singulis punctis. Si arese sint partim attractivae, partim repulsivae, assumen- dam esse differen- tiam earundem. 177. Duo tamen hie tantummodo notanda sunt ; primo quidem illud : si duo puncta ad se invicem accedant, vel a se invicem recedant in ea recta, quae ipsa conjungit, segmenta illius [82] axis, qui exprimit distantias, non expriment spatium confectum ; nam moveri debebit punctum utrumque : adhuc tamen ilia segmenta erunt proportionalia ipsi spatio confecto, eorum nimirum dimidio ; quod quidem satis est ad hoc, ut illae areae adhuc sint proportionales incrementis, vel decrementis quadrati velocitatum, adeoque ipsa exprimant. 178. Secundo loco notandum illud, ubi areae respondentes dato cuipiam spatio sint partim attractive, partim repulsivae, earum differentiam, quae oritur subtrahendo summam omnium repulsivarum a summa attractivarum, vel vice versa, exhibituram incrementum illud, vel decrementum quadrati velocitatis ; prout directio motus respectivi conspiret cum vi, vel oppositam habeat directionem. Quamobrem si interea, dum per aliquod majus intervallum a se invicem recesserunt puncta, habuerint vires directionis utriusque ; ut innotescat, an celeritas creverit, an decreverit & quantum ; erit investigandum, an areas omnes attractivae simul, omnes repulsivas simul superent, an deficiant, & quantum ; inde enim, & a velocitate, quae habebatur initio, erui poterit quod quaeritur. ^ e arcubus, & areis ; nunc aliquanto diligentius considerabimus tangentis: sectio- ilia axis puncta, ad quae curva appellit. Ea puncta vel sunt ejusmodi, ut in iis curva axem ducT enera UmltUm secet > cu jusmodi in fig. I sunt E,G,I, &c., vel ejusmodi, ut in iis ipsa curva axem contingat tantummodo. Primi generis puncta sunt ea, in quibus fit transitus a repulsionibus ad attractiones, vel vice versa, & hsec ego appello limites, quod nimirum sint inter eas opposi- tarum directionum vires. Sunt autem hi limites duplicis generis : in aliis, aucta distantia, transitur a repulsione ad attractionem : in aliis contra ab attractione ad repulsionem. Prioris generis sunt E,I,N,R ; posterioris G,L,P : & quoniam, posteaquam ex parte repulsiva in una sectione curva transiit ad partem attractivam ; in proxime sequent! sectione debet necessario ex parte attractiva transire ad repulsivam, ac vice versa ; patet, limites fore alternatim prioris illius, & hujus posterioris generis. t P rro linrites prioris generis, a limitibus posterioris ingens habent inter se dis- differant': limites crimen. Habent illi quidem hoc commune, ut duo puncta collocata in distantia unius cohaesionls' & n n h' 111 ^ 8 cujuscunque nullam habeant mutuam vim, adeoque si respective quiescebant, pergant itidem respective quiescere. At si ab ilia respectiva quiete dimoveantur ; turn vero in limite primi generis ulteriori dimotioni resistent, & conabuntur priorem distantiam recu- perare, ac sibi relicta ad illam ibunt ; in limite vero secundi generis, utcunque parum dimota, sponte magis fugient, ac a priore distantia statim recedent adhuc magis. Nam si distantia minuatur ; habebunt in limite prioris generis vim repulsivam, quae obstabit uteriori accessui, & urgebit puncta ad mutuum recessum, quern sibi relicta acquirent, [83] adeoque tendent ad illam priorem distantiam : at in limite secundi generis habebunt attractionem, qua adhuc magis ad se accedent, adeoque ab ilia priore distantia, quae erat major, adhuc magis sponte fugient. Pariter si distantia augeatur, in primo limitum genere a vi attractiva, quse habetur statim in distantia majore ; habebitur resistentia ad ulteriorem recessum, & conatus ad minuendam distantiam, ad quam recuperandam sibi relicta tendent per accessum ; at in limitibus secundi generis orietur repulsio, qua sponte se magis adhuc fugient, adeoque a minore ilia priore distantia sponte magis recedent. Hinc illos prioris generis limites, qui mutuse positionis tenaces sunt, ego quidem appellavi limites coh&sionis, & secundi generis limites appellavi limites non cobasionis. Duo genera tactuum. 181. Ilia puncta, in quibus curva axem tangit, sunt quidem terminus quidam virium, quae ex utraque parte, dum ad ea acceditur, decrescunt ultra quoscunque limites, ac demum ibidem evanescunt ; sed in iis non transitur ab una virium directione ad aliam. Si con- tactus fiat ab arcu repulsive ; repulsiones evanescunt, sed post contactum remanent itidem repulsiones ; ac si ab arcu attractive, attractionibus evanescentibus attractiones iterum immediate succedunt. Duo puncta collocata in ejusmodi distantia respective quiescunt ; A THEORY OF NATURAL PHILOSOPHY '43 177. However, there are here two things that want noting only. The first of them The same result is this, that if two points approach one another or recede from one another in the straight holds good even ,...., , r i ,. , ,. i when the segments line joining them, the segments of the axis, which expresses distances, do not represent of the axis are the the distances traversed ; for both points will have to move. Nevertheless the segments ' ialves of the d is- 11 -11 i i i T i i i if f i . . i , tances traversed by will still be proportional to the distance traversed, namely, the half of it ; & this indeed is single points, sufficient for the areas to be still proportional to the increments or decrements of the squares of the velocities, & thus to represent them. 178. In the second place it is to be noted that, where the areas corresponding to any if the areas are given interval are partly attractive & partly repulsive, their difference, obtained by p^ti* tt 2SS2 & subtracting the sum of all those that are repulsive from the sum of those that are attractive, their difference or vice versa, will represent the increment, or the decrement, of the square of the velocity, must be taken - according as the direction of relative motion is in the same direction as the force, or in the opposite direction. Hence, if, during the time that the points have receded from one another by some considerable interval, they had forces in each direction ; then in order to ascertain whether the velocity had been increased or decreased, & by how much, it will have to be considered whether all the attractive areas taken together are greater or less than all the repulsive areas taken together, & by how much. For from this, & from the velocity which initially existed, it will be possible to deduce what is required. 179. So much for the arcs & the areas; now we must consider in a rather more careful Approach of the manner those points of the axis to which the curve approaches. These points are either ^en it cSa^or such that the curve cuts the axis in them, for instance, the points E, G, I, &c. in Fig. I : touches it; two or such that the curve only touches the axis at the points. Points of the first kind are u^ns^/'ihnit- those in which there is a transition from repulsions to attractions, or vice-versa ; & these points. I call limit-points or boundaries, since indeed they are boundaries between the forces acting in opposite directions. Moreover these limit-points are twofold in kind ; in some, when the distance is increased, there is a transition from repulsion to attraction ; in others, on the contrary, there is a transition from attraction to repulsion. The points E, I, N, R are of the first kind, and G, L, P are of the second kind. Now, since at one intersection, the curve passes from the repulsive part to the attractive part, at the next following intersection it is bound to pass from the attractive to the repulsive part, & vice versa. It is clear then that the limit-points will be alternately of the first & second kinds. 1 80. Further, there is a distinction between limit-points of the first & those of the in what they agree second kind. The former kind have this property in common ; namely that, if two points * iff j . w ^ u *? t y are situated at a distance from one another equal to the distance of any one of these limit- points of cohesion points from the origin, they will have no mutual force ; & thus, if they are relatively & of non - cohesic i. at rest with regard to one another, they will continue to be relatively at rest. Also, if they are moved apart from this position of relative rest, then, for a limit-point of the first kind, they will resist further separation & will strive to recover the original distance, & will attain to it if left to themselves ; but, in a limit-point of the second kind, however small the separation, they will of themselves seek to get away from one another & will immediately depart from the original distance still more. For, if the distance is diminished, they will have, in a limit-point of the first kind, a repulsive force, which will impede further approach & impel the points to mutual recession, & this they will acquire if left to themselves ; thus they will endeavour to maintain the original distance apart. But in a limit-point of the second kind they will have an attraction, on account of which they will approach one another still more ; & thus they will seek to depart still further from the original distance, which was a greater one. Similarly, if the distance is increased, in limit-points of the first kind, due to the attractive force which is immediately obtained at this greater distance, there will be a resistance to further recession, & an endeavour to diminish the distance ; & they will seek to recover the original distance if left to themselves by approaching one another. But, in limit-points of the second class, a repulsion is produced, owing to which they try to get away from one another still further ; & thus of themselves they will depart still more from the original distance, which was less. On this account indeed I have called those limit-points of the first kind, which are tenacious of mutual position, limit-points of cohesion, & I have termed limit-points of the second kind limit-points of non-cohesion. 181. Those points in which the curve touches the axis are indeed end-terms of series Two kinds of con- of forces, which decrease on both sides, as approach to these points takes place, beyond tactt all limits, & at length vanish there ; but with such points there is no transition from one direction of the forces to the other. If contact takes place with a repulsive arc, the repulsion vanishes, but after contact remains still a repulsion. If it takes place with an attractive arc, attraction follows on immediately after a vanishing attraction. Two points situated such a distance remain in a state of relative rest ; but in the first case they will 144 PHILOSOPHIC NATURALIS THEORIA pro forma curvae prope sectionem. sed in prime casu resistunt soli compressioni, non etiam distractioni, in secundo resistunt huic soli, non illi. l ^ 2 ' Limites cohsesionis possunt esse validissimi, & languidissimi. Si curva ibi quasi ad pcrpendiculum secat axem, & ab eo longissime recedit ; sunt validissimi : si autem ip Sum secet in angulo perquam exiguo, & parum ab ipso recedat ; erunt languidissimi. Primum genus limitum cohsesionis exhibet in fig. I arcus tNy, secundum cNx. In illo assumptis in axe Nz, NM utcunque exiguis, possunt vires zt, uy, & areae Nzt, Nwy esse utcumque magnas, adeoque, mutatis utcunque parum distantiis, possunt haberi vires ab ordinatis expressae utcunque magnae, quae vi comprimenti, vel distrahenti, quantum libuerit, valide resistant, vel areae utcunque magnae, quae velocitates quantumlibet magnas respectivas elidant, adeoque sensibilis mutatio positionis mutuae impediri potest contra utcunque magnam vel vim prementem, vel celeritatem ab aliorum punctorum actionibus impressam. In hoc secundo genere limitum cohaesionis, assumptis etiam majoribus segmentis Nz, Nw, possunt & vires zc,ux, & areae Nzf , Ntf, esse quantum libuerit exiguae, & idcirco exigua itidem, quantum libuerit, resistentia, quae mutationem vetet. P ssunt autem hi Hmites esse quocunque, utcunque magno numero ; cum ro, utcunque proxi- demonstratum sit, posse curvam in quotcunque, & quibuscunque punctis axem secare. mos, vel remotes p oss unt idcirco etiam esse utcunque inter se proximi, vel remoti, ut [84] alicubi intervallum originis' abscissa- inter duos proximos limites sit etiam in quacunque ratione majus, quam sit distantia ordme praecedentis ab origine abscissarum A ; alibi in intervallo vel exiguo, vel ingenti sint quam- plurimi inter se ita proximi, ut a se invicem distent minus, quam pro quovis assumpto, aut dato intervallo. Id evidenter fluit ex eo ipso, quod possint sectiones curvae cum axe haberi quotcunque, & ubicunque. Sed ex eo, quod arcus curvae ubicunque possint habere positiones quascunque, cum ad datas curvas accedere possint, quantum libuerit, sequitur, quod limites ipsi cohaesionis possint alii aliis esse utcunque validiores, vel languidiores, atque id quocunque ordine, vel sine ordine ullo ; ut nimirum etiam sint in minoribus distantiis alicubi limites validissimi, turn in majoribus languidiores, deinde itidem in majoribus multo validiores, & ita porro ; cum nimirum nullus sit nexus necessarius inter distantiam limitis ab origine abscissarum, & ejus validitatem pendentem ab inclinatione, & recessu arcus secantis respectu axis, quod probe notandum est, futurum nimirum usui ad ostendendum, tenacitatem, sive cohaesionem, a densitate non pendere. similes. Quse positio rectae jg^.. I n utroque limitum genere fieri potest, ut curva in ipso occursu cum axe pro infinite 3 rarissima! tangente habeat axem ipsum, ut habeat ordinatam, ut aliam rectam aliquam inclinatam. quae frequentissima. J n primo casu maxime ad axem accedit, & initio saltern languidissimus est limes ; in secundo maxime recedit, & initio saltern est validissimus ; sed hi casus debent esse rarissimi, si uspiam sunt : nam cum ibi debeat & axem secare curva, & progredi, adeoque secari in puncto eodem ab ordinata producta, debebit habere flexum contrarium, sive mutare directionem flexus, quod utique fit, ubi curva & rectam tangit simul, & secat. Rarissimos tamen debere esse ibi hos flexus, vel potius nullos, constat ex eo, quod flexus contrarii puncta in quovis finito arcu datae curvae cujusvis numero finite esse debent, ut in Theoria curvarum demonstrari potest, & alia puncta sunt infinita numero, adeoque ilia cadere in intersectiones est infinities improbabilius. Possunt tamen saepe cadere prope limites : nam in singulis contorsionibus curvae saltern singuli flexus contrarii esse debent. Porro quamcunque directionem habuerit tangens, si accipiatur exiguus arcus hinc, & inde a limite, vel maxime accedet ad rectam, vel habebit curvaturam ad sensum aequalem, & ad sensum aequali lege progredientem utrinque, adeoque vires in aequali distantia exigua a limite erunt ad sensum hinc, & inde aequales ; sed distantiis auctis poterunt & diu aequalitatem retinere, & cito etiam ab ea recedere. Transitus per infi- 185. Hi quidem sunt limites per intersectionem curvae cum axe, viribus evanescentibus as tl m"toticis ribUS m *P SO li m i te - At possunt [85] esse alii limites, ac transitus ab una directione virium ad aliam non per evanescentiam, sed per vires auctas in infinitum, nimirum per asymptoticos A THEORY OF NATURAL PHILOSOPHY 145 resist compression only, & not separation ; and in the second case the latter only, but not the former. 182. Limit-points may be either very strong or very weak. If the curve cuts the axis The limit-points of at the point almost at right angles, & goes off to a considerable distance from it, they o h ^eak ?ccordf are very strong. But if it cuts the axis at a very small angle & recedes from it but little, to the form of the then they will be very weak. The arc *Ny in Fig. i represents the first kind of limit- ^Hint * iVater- points of cohesion, and the arc cNx the second kind. At the point N, if Nz, N are section. taken along the axis, no matter how small, the forces zt, uy, & the areas Nzt, Ny may be of any size whatever ; & thus, if the distances are changed ever so little, it is possible that there will be forces represented by ordinates ever so great ; & these will strongly resist the compressing or separating force, be it as great as you please ; also that we shall have areas, ever so large, that will destroy the relative velocities, no matter how great they may be. Thus, a sensible change of relative position will be hindered in opposition to any impressed force, however great, or against a velocity generated by the actions upon them of other points. In the second kind of limit-points of cohesion, if also segments Nz, Nw are taken of considerable size even, then it is possible for both the forces zc, ux, & the areas Nzc, Nux to be as small as you please ; & therefore also the resistance that opposes the change will be as small as you please. 183. Moreover, there can be any number of these limit-points, no matter how great ; The limit-points for it has been proved that the curve can cut the axis in any number of points, & anywhere. are m <J efimte as rrM F i i r i i i i f i i 6 g cl i Cl S HUTU DCr, I herefore it is possible for them to be either close to or remote from one another, without their proximity to any restriction whatever, so that the interval between any two consecutive limit-points one^another^&^th^ at any place shall even bear to the distance of the first of the two from A, the origin of order of their occur- abscissae, a ratio that is greater than unity. In other words, in any interval, either very ^toe <SgVof P ab- small or very large, there may be an exceedingly large number of them so close to one scissae. another, that they are less distant from one another than they are from any chosen or given interval. This evidently follows from the fact that the intersections of the curve with the axis can happen any number of times & anywhere. Again, from the fact that arcs of the curve can anywhere, owing to their being capable of approximating as closely as you please to given curves, have any positions whatever, it follows that these limit-points of cohesion can be some of them stronger than others, or weaker, in any manner ; & that too, in any order, or without order. So that, for instance, we may have at small distances anywhere very strong limit-points, then at greater distances weaker ones, & then again at still greater distances much stronger ones, & so on. That is to say, since there is no necessary connection between the distance of a limit-point from the origin of abscissae and its strength, which depends on the inclination of the intersecting arc & the distance it recedes from the axis. It is well that this should be made a note of ; for indeed it will be used later to prove that tenacity or cohesion does not depend on density. 184. In each of these kinds of limit-points it may happen that the curve, where it What position of meets the axis, may have the axis itself as its tangent, or the ordinate, or any other straight touchkig^he* curve line inclined to the axis. In the first case it approximates very closely to the axis, & at a limit-point is close to the point at any rate it is a very weak limit-point ; in the second case, it departs ^at magt^fr* from the axis very sharply, & close to the point at any rate it is a very strong limit-point, quent ; small arcs But these two cases must be of very rare occurrence, if indeed they ever occur. For, since fTm^tVo'in t* are at the point the curve is bound to cut the axis & go on, & thus be cut in the same point equal & similar, by the ordinate produced, it is bound to have contrary-flexure ; that is to say, a change in the direction of its curvature, such as always takes place at a point where the curve both touches a straight line & cuts it at the same time. Yet, that these flexures must occur very rarely at such points, or rather never occur at all, is evident from the fact that in any finite arc of any given curve the number of points of contrary-flexure must be finite, as can be proved in the theory of curves ; & other points are infinite in number ; hence that the former should happen at the points of intersection with the axis is infinitely improbable. On the other hand they may often fall close to the limit-points ; for in each winding of the curve about the axis there must be at least one point of contrary-flexure. Further, whatever the direction of the tangent, if a very small arc of the curve is taken on each side of the limit-point, this arc will either approximate very closely to the straight line, or will have its curvature the same very nearly, & will proceed very nearly according to the same law on each side ; & thus the forces, at equal small distances on each side of the limit- point will be very nearly equal to one another ; but when the distances are increased, they can either maintain this equality, for some considerable time, or indeed very soon depart from it. 185. The limit-points so far discussed are those obtained through the intersection Passage through of the curve with the axis, where the forces vanish at the limit-point. But there may be other limit-points ; the transition from one direction of the forces to another L 146 PHILOSOPHIC NATURALIS THEORIA curvse arcus. Diximus supra num. 168. adnot. (i), quando crus asymptoticum abit in infinitum, debere ex infinite regredi crus aliud habens pro asymptote eandem rectam, & posse regredi cum quatuor diversis positionibus pendentibus a binis partibus ipsius rectae, & binis plagis pro singulis rectae partibus ; sed cum nostra curva debeat semper progredi, diximus, relinqui pro ea binas ex ejusmodi quatuor positionibus pro quovis crure abeunte in infinitum, in quibus nimirum regressus fiat ex plaga opposita. Quoniam vero, progre- diente curva, abire potest in infinitum tarn crus repulsivum, quam crus attractivum , jam iterum fiunt casus quatuor possibiles, quos exprimunt figurae 16, 17, 18, & 19, in quibus omnibus est axis ACS, asymptotus DCD', crus recedens in infinitum EKF, regrediens ex infinite GMH. D A I C B FIG. 1 6. I C D B D FIG. 17. Quatuor eorum 186. In fig. 16. cruri repulsivo EKF succedit itidem repulsivum GMH ; in fig. 17 f^nXntesTontac- repulsivo attractivum ; in 1 8 attractive attractivum; in 19 attractive repulsivum. Primus tibus, bini Hmiti- & tertius casus respondent contactibus. Ut enim in illis evanescebat vis ; sed directionem ionU, al aHer COh non non mutabat ; ita & hie abit quidem in infinitum, sed directionem non mutat. Repulsioni cohaesionis. IK in fig. 1 6 succedit repulsio LM ; & attractioni in fig. 18 attractio. Quare ii casus non habent limites quosdam. Secundus, & quartus habent utique limites ; nam in fig. 17 repulsion! IK succedit attractio LM ; & in fig. 19 attractioni repulsio ; atque idcirco secundus continet limitem cobasionis, quartus limitem non cohcesionis. Nuiium in Natura vero eum utcunque. 187. Ex istis casibus a nostra curva censeo removendos esse omnes praeter solum quartum ; & in hoc ipso removenda omnia crura, in quibus ordinata crescit in ratione ipsum minus, quam simplici reciproca distantiarum a limite. Ratio excludendi est, ne haberi aliquando vis infinita possit, quam & per se se absurdam censeo, & idcirco praeterea, quod infinita vis natura sua velocitatem infinitam requirit a se generandam finito tempore. Nam in primo, & secundo casu punctum collocatum in ea distantia ab alio puncto, quam habet I, ab origine abscissarum, abiret ad C per omnes gradus virium auctarum in infinitum, & in C deberet habere vim infinitam ; in tertio vero idem accideret puncto collocate in distantia, quam habet L. At in quarto casu accessum ad C prohibet ex parte I attractio IK, & ex parte L repulsio LM. Sed quoniam, si eae crescant in ratione reciproca minus, quam simplici distantiarum CI,CL ; area FKICD, vel GMLCD erit finita, adeoque punctum impulsum versus C velocitate majore, quam quae respondeat illi areae, debet transire per omnes virium magnitudines usque ad vim absolute infinitam in C, quae ibi [86] praeterea & attractiva esse deberet, & repulsiva, limes videlicet omnium & attracti- varum, & repulsivarum ; idcirco ne hie quidem casus admitti debet, nisi cum hac conditione, ut ordinata crescat in ratione reciproca simplici distantiarum a C, vel etiam majore, ut nimirum area infinita evadat, & accessum a puncto C prohibeat. A THEORY OF NATURAL PHILOSOPHY may occur, not with evanescence of the forces, but through the forces increasing indefinitely, that is to say through asymptotic arcs of the curve. We said above, in Note () to Art. 1 68, when an asymptotic arm goes off to infinity, there must be another asymptotic arm returning from infinity having the same straight line for an asymptote ; & it may return in four different positions, which depend on the two parts of the straight line & the two sides of each part of the straight line. But, since our curve must always go forward, we said that for it there remained only two out of these four positions, for any arm going off to infinity ; that is to say, those in which the return is made on the opposite side of the straight line. However, since, whilst the curve goes forward, either a repulsive or an attractive arm can go off to infinity, here again we must have four possible cases, represented in Figs. 16, 17, 18, 19, in all of which ACB is the axis, DCD' the asymptote, EKF the arm going off to infinity, & GMH the arm returning from infinity. I C B n cr FIG 19. 1 86. In Fig. 1 6, to a repulsive arm EKF there succeeds an arm that is also repulsive; in Fig. 17, to a repulsive succeeds an attractive ; in Fig. 18, to an attractive succeeds an attractive ; and in Fig. 19, to an attractive succeeds a repulsive. The first & third cases correspond to contacts. For, just as in contact, the force vanished, but did not change its direction, so here also the force indeed becomes infinite but does not change its direction. In Fig. 1 6, to the repulsion IK there succeeds the repulsion LM, & in Fig. 18 to an attraction an attraction ; & thus these two cases cannot have any limit-points. But the second & fourth cases certainly have limit-points ; for, in Fig. 17, to the repulsion IK there succeeds the attraction LM, & in Fig. 19 to an attraction a repulsion ; & thus the second case contains a limit-point of cohesion, & the fourth a limit-point of non-cohesion. 187. Out of these cases I think that all except the last must be barred from our curve ; & even with that all arms must be rejected for which the ordinates increase in a ratio less than the simple reciprocal of the distances from the limit-point. My reasons for excluding these are to avoid the possibility of there being at any time an infinite force (which of itself I consider to be impossible), & because, in addition to that, an infinite force, by its very nature necessitates the creation by it of an infinite velocity in a finite time. For, in the first & second cases, a point, situated at the distance from another point equal to that which I has from the origin of abscissae, would go off to C through all stages of forces increased indefinitely, & at C would be bound to have an infinite force. In the third case, too, the same thing would happen to a point situated at a distance equal to that of L. Now, in the fourth case, the approach to C is restrained, from the side of I by the attraction IK, & from the side of L by the repulsion LM. However, since, if these forces increase in a ratio that is less than the simple reciprocal ratio of the distances CI, CL, then the area FKICD or the area GMLCD will be finite ; thus the point, being impelled towards C with a velocity that is greater than that corresponding to the area, must pass through all magnitudes of the forces up to a force that is absolutely infinite at C ; and this force must besides be both attractive & repulsive, the limit so to speak of all attractive & repulsive forces. Hence not even this case is admissible, unless with the condition that the ordinate increases in the simple reciprocal ratio of the distances from C, or in a greater ; that is to say, the area must turn out to be infinite and so restrain the approach towards the point C. Four kinds of them ; two corre- sponding to contact, & two to limit- points, of which the one is a limit-point of cohesion & the other of non-cohe- sion. None of these ex- cept the last admis- sible in Nature ; & not even that in general. 148 PHILOSOPHIC NATURAL! S THEORIA Transitus per eum 188. Quando habeatur hie quartus casus in nostra curva cum ea conditione ; turn bills: teqaibai quidem nullum punctum collocatum ex alters parte puncti C poterit ad alteram transilire, distantiis constet, quacunque velocitate ad accessum impellatur versus alterum punctum, vel ad recessum eum non haben. u j i i TJ ab ipso, impediente transitum area repulsiva mnnita, vel innnita attractiva. Inde vero facile colligitur, eum casum non haberi saltern in ea distantia, quae a diametris minimarum particularum conspicuarum per microscopia ad maxima protenditur fixarum intervalla nobis conspicuarum per telescopia : lux enim liberrime permeat intervallum id omne. Quamobrem si ejusmodi limites asymptotici sunt uspiam, debent esse extra nostrae sensibi- litatis sphaeram, vel ultra omnes telescopicas fixas, vel citra microscopicas moleculas. Transitus ad puncta 189. Expositas hisce, quae ad curva virium pertinebant, aggrediar simpliciora quaadam, iae, & massas. maxime notatu digna sunt, ac pertinent ad combinationem punctorum primo quidem duorum, turn trium, ac deinde plurium in massa etiam coalescentium, ubi & vires mutuas, & motus quosdam, & vires, quas in alia exercent puncta, considerabimus. in limitibus ; icp. Duo puncta posita in distantia aequali distantiae limitis cujuscunque ab origine ? 1 ^ ' abscissarum, ut in fig. i. AE, AG, AI, &c, (immo etiam si curva alicubi axem tangat, aequali distantiae contactus ab eodem), ac ibi posita sine ulla velocitate, quiescent, ut patet, quia nullam habebunt ibi vim mutuam : posita vero extra ejusmodi limites, incipient statim ad se invicem accedere, vel a se invicem recedere per intervalla aequalia, prout fuerint sub arcu attractivo, vel repulsive. Quoniam autem vis manebit semper usque ad proximum limitem directionis ejusdem ; pergent progredi in ea recta, quae ipsa urgebat prius, usque ad distantiam limitis proximi, motu semper accelerate, juxta legem expositam num. 176, ut nimirum quadrata velocitatum integrarum, quae acquisitae jam sunt usque ad quodvis momentum (nam velocitas initio ponitur nulla) respondeant areis clausis inter ordinatam respondentem puncto axis terminanti abscissam, quae exprimebat distantiam initio motus, & ordinatam respondentem puncto axis terminanti abscissam, quae exprimit distantiam pro eo sequent! momento. Atque id quidem, licet interea occurrat contactus aliquis ; quamvis enim in eo vis sit nulla, tamen superata distantia per velocitatem jam acquisitam, statim habentur iterum [87] vires ejusdem directionis, quae habebatur prius, adeoque perget acceleratio prioris motus. Motus post proxi- l q I p rO ximus limes erit ems generis, cuius generis diximus limites cohaesioms, in quo mum limitem super- . . 7 . ,. . , J . i j ^ i atum, & osciiiatio. nimirum si distantia per repulsionem augebatur, succedet attractio ; si vero minuebatur per attractionem, succedet e contrario repulsio, adeoque in utroque casu limes erit ejusmodi, ut in distantiis minoribus repulsionem, in majoribus attractionem secum ferat. In eo limite in utroque casu recessus mutui, vel accessus ex praecedentibus viribus, incipiet, velocitas motus minui vi contraria priori, sed motus in eadem directione perget ; donee sub sequent! arcu obtineatur area curvae aequalis illi, quam habebat prior arcus ab initio motus usque ad limitem ipsum. Si ejusmodi aequalitas obtineatur alicubi sub arcu sequente ; ibi, extincta omni praecedenti velocitate, utrumque punctum retro reflectet cursum ; & si prius accedebant, incipient a se invicem recedere ; si recedebant, incipient accedere, atque id recuperando per eosdem gradus velocitates, quas amiserant, usque ad limitem, quern fuerant prsetergressa ; turn amittendo, quas acquisiverant usque ad dis- tantiam, quam habuerant initio ; viribus nimirum iisdem occurrentibus in ingressu, & areolis curvae iisdem per singula tempuscula exhibentibus quadratorum velocitatis incre- menta, vel decrementa eadem, quae fuerant antea decrementa, vel incrementa. Ibi autem iterum retro cursum reflectent, & oscillabunt circa ilium cohaesionis limitem, quern fuerant praetergressa, quod facient hinc, & inde perpetuo, nisi aliorum externorum punctorum viribus perturbentur, habentia velocitatem maximam in plagam utramlibet in distantia ipsius illius limitis cohaesionis. Casus osdiiationis jo.2. Quod si ubi primum transgressa sunt proximum limitem cohaesionis, offendant ' S arcum ita minus validum praecedente, qui arcus nimirum ita minorem concludat aream, quam praecedens, ut tota ejus area sit aequalis, vel etiam minor, quam ilia praecedentis arcus area, quae habetur ab ordinata respondente distantiae habitse initio motus, usque ad A THEORY OF NATURAL PHILOSOPHY 149 1 88. When, if ever, this fourth case occurs in our curve, then indeed no point situated Passage through a on either side of the point C will be able to pass through it to the other side, no matter L^dTslmpo'ssibiel what the velocity with which it is impelled to approach towards, or recede from, the other distances at which point ; for the infinite repulsive area, or the infinite attractive area, will prevent such $, e re are^iio such passage. Now, it can easily be derived from this, that this case cannot happen at any rate limit-points. in the distance lying between the diameters of the smallest particles visible under the microscope & the greatest distances of the stars visible to us through the telescope ; for light passes with the greatest freedom through the whole of this interval. Therefore, if there are ever any such asymptotic limit-points, they must be beyond the scope of our senses, either superior to all telescopic stars, or inferior to microscopic molecules. 1 89. Having thus set forth these matters relating to the curve of forces, I will now w ? n w pass on to discuss some of the simpler things that are more especially worth mentioning with regard ^^ s . of matter> & to combination of points ; & first of all I will consider a combination of two points, then of three, & then of many, coalescing into masses ; & with them we will discuss their mutual forces, & certain motions, and forces, which they exercise on other points. 190. Two points situated at a distance apart equal to the distance of any limit-point Rest at Hmit- from the origin of abscissae, like AE, AG, AI, &c. in Fig. I (or indeed also where the r curve touches the axis anywhere, equal to the distance of the point of contact from the without them. origin), & placed in that position without any velocity, will be relatively at rest ; this is evident from the fact that they have then no mutual force ; but if they are placed at any other distance, they will immediately commence to move towards one another or away from one another through equal intervals, according as they lie below an attractive or a repulsive arc. Moreover, as the force always remains the same in direction as far as the next following limit-point, they continue to move in the same straight line which contained them initially as far as the distance apart equal to the distance of the next limit-point from the origin, with a motion that is continually accelerated according to the law given in Art. 176 ; that is to say, in such a manner that the squares of the whole velocities which have been already acquired up to any instant (for the velocity at the commencement is supposed to be nothing) will correspond to the areas included between the ordinate corresponding to the point of the axis terminating the abscissa which the distance traversed since motion began and the ordinate corresponding to the point on the axis terminating the abscissa which expresses the distance for the next instant after it. This is still the case, even if a contact should occur in the meantime. For, although at a point where contact occurs the force is nothing, yet, this distance being passed by the velocity already acquired, immediately afterwards there will be forces having the same direction as before ; and thus the acceleration of the former motion will proceed. 191. The next limit-point will be one of the kind we have called limit-points of cohesion, Motion after the namely, one in which, if the distance is increased by repulsion, then attraction follows ; passed^osc^Sion 5 but if the distance is diminished by attraction, then on the contrary repulsion will follow ; & thus, in either case, the limit-point will be of such a kind, that it gives a repulsion at smaller distances & an attraction at larger. In this limit-point, in either case, the separation or approach, due to the forces that have preceded, will be changed, & the velocity of motion will begin to be diminished by a force opposite to the original force, but the motion will continue in the same direction ; until an area of the curve under the arc that follows the limit-point becomes equal to the area under the former arc from the commencement of the motion as far as the limit-point. If equality of this kind is obtained somewhere under the subsequent arc, then, the whole of the preceding velocity being destroyed, both the points will return along their paths ; & if at the start they approached one another, they will now begin to recede from one another, or if they originally receded from one another, they will now commence to approach ; and as they do this, they will regain by the same stages the velocities which they lost, as far as the limit-point which they passed through ; then they will lose those which they had acquired, until they reach the distance apart which they had at the commencement. That is to say, the same forces occur on the return path, & the same little areas of the curve for the several short intervals of time represent increments or decrements of the squares of the velocities which are the same as were formerly decrements or increments. Then again they will once more retrace their paths, & they will oscillate about the limit-point of cohesion which they had passed through ; & this they will do, first on this side & then on that, over & over again, unless they are disturbed by forces due to other points outside them ; & their greatest velocity in either direction will occur at a distance apart equal to that of the limit-point of cohesion from the origin. 192. But if, when they first passed through the nearest limit-point of cohesion, they The case of a larger happened to come to an arc representing forces so much weaker than those of the preceding ti-ii f , 11 i ft T arc that the whole area of it was equal to, or even less than, the area of the preceding arc, reckoning from the ordinate corresponding to the distance apart at the commencement 150 PHILOSOPHIC NATURALIS THEORIA limitem ipsum ; turn vero devenient ad distantiam alterius limitis proximi priori, qui idcirco erit limes non cohaesionis. Atque ibi quidem in casu sequalitatis illarum arearum consistent, velocitatibus prioribus elisis, & nulla vi gignente novas. At in casu, quo tota ilia area sequentis arcus fuerit minor, quam ilia pars areae praecedentis, appellent ad dis- tantiam ejus limitis motu quidem retardate, sed cum aliqua velocitate residua, quam distantiam idcirco praetergressa, & nacta vires directionis mutatse jam conspirantes cum directione sui motus, non, ut ante, oppositas, accelerabunt motum usque ad distantiam limitis proxime sequentis, quam praetergressa precedent, sed motu retardato, ut in priore ; & si area sequentis arcus non sit par extinguendae ante suum finem toti [88] velocitati, quae fuerat residua in appulsu ad distantiam limitis praecedentis non cohaesionis, & quae acquisita est in arcu sequent! usque ad limitem cohsesionis proximum ; turn puncta appellent ad distantiam limitis non cohaesionis sequentis, ac vel ibi sistent, vel progredientur itidem, eritque semper reciprocatio quaedam motus perpetuo accelerati, turn retardati ; donee deveniatur ad arcum ita validum, nimirum qui concludat ejusmodi aream, ut tota velocitas acquisita extinguatur : quod si accidat alicubi, & non accidat in distantia alicujus limitis ; cursum reflectent retro ipsa puncta, & oscillabunt perpetuo. Velocitatis muta- 193. Porro in hujusmodi motu patet illud, dum itur a distantia limitis cohaesionis ^"^abeat^maxU a ^ distantiam limitis non cohaesionis, velocitatem semper debere augeri ; turn post mum, & minimum transitum per ipsam debere minui, usque ad appulsum ad distantiam limitis non cohaesionis, extmgui possit a d eO q ue habebitur semper in ipsa velocitate aliquod maximum in appulsu ad distantiam limitis cohaesionis, & minimum in appulsu ad distantiam limitis non cohaesionis. Quamo- brem poterit quidem sisti motus in distantia limitis hujus secundi generis ; si sola existant ilia duo puncta, nee ullum externum punctum turbet illorum motum : sed non poterit sisti in distantia limitis illius primi generis ; cum ad ejusmodi distantias deveniatur semper motu accelerate. Praeterea patet & illud, si ex quocunque loco impellantur velocitatibus aequalibus vel alterum versus alterum, vel ad partes oppositas, debere haberi reciprocationes easdem auctis semper aeque velocitatibus utriusque, dum itur versus distantiam limitis primi generis, & imminutis, dum itur versus distantiam limitis secundi generis. oscMatlo S ma/**? X 94' P atet & illud, si a distantia limitis primi generis dimoveantur vi aliqua, vel non esse debeat, & unde ita uigenti velocitate impressa, oscillationem fore perquam exiguam, saltern si quidam " CJUS mag va h"dus fuerit limes ; nam velocitas incipiet statim minui, & ei vi statim vis contraria invenietur, ac puncta parum dimota a loco suo, turn sibi relicta statim retro cursum reflect- ent. At si dimoveantur a distantia limitis secundi generis vi utcunque exigua ; oscillatio erit multo major, quia necessario debebunt progredi ultra distantiam sequentis limitis primi generis, post quern motus primo retardari incipiet. Quin immo si arcus proximus hinc, & inde ab ejusmodi limite secundi generis concluserit aream ingentem, ac majorem pluribus sequentibus contrariae directionis, vel majorem excessu eorundem supra areas interjacentes directionis suae ; turn vero oscillatio poterit esse ingens : nam fieri poterit, ut transcurrantur hinc, & inde limites plurimi, antequam deveniatur ad arcum ita validum, ut velocitatem omnem elidat, & motum retro reflectat. Ingens itidem oscillatio esse poterit, si cum ingenti vi dimoveantur puncta a distantia limitum generis utriuslibet ; ac res tota pendet a velocitate initiali, & ab areis, quae post oc-[8Q]-currunt, & quadratum velocitatis vel augent, vel minuunt quantitate sibi proportionali. Accessum debere 195. Utcunque magna sit velocitas, qua dimoveantur a distantia limitum ilia duo swt^ saltern a^pmno p uncta> utcunque validos inveniant arcus conspirantes cum velocitatis directione, si ad recess um posse se invicem accedunt, debebunt utique alicubi motum retro reflectere, vel saltern sistere, cas^'^o^bms 1 saltern advenient ad distantias illas minimas, quae respondent arcui asymptotico, exiguae differentiae cujus area est capax cxtinguendse cujuscunque velocitatis utcunque magnae. At si velocitatis ingentis. rece( j ant a se mv i ce m, fieri potest, ut deveniant ad arcum aliquem repulsivum validissimum, cujus area sit major, quam omnis excessus sequentium arearum attractivarum supra repul- A THEORY OF NATURAL PHILOSOPHY 151 of the motion up to the limit-point ; then indeed they will arrive at a distance apart equal to that of the limit-point next following the first one, which will therefore be a limit-point of non-cohesion. Here they will stop, in the case of equality between the areas in question ; for the preceding velocities have been destroyed & no fresh ones will be generated. But in the case when the whole of the area under the second arc is less than the said part of the first area, they will reach a distance apart equal to that of the limit-point with a motion that is certainly diminished ; but some velocity will be left, & this distance will therefore be passed, & the points, coming under the influence of forces changed in direction so that they now act in the same sense as their own motion, will accelerate their motion as far as the next following limit-point ; & having passed through this they will go on, but with retarded motion as in the first case. Then, if the area of the subsequent arc is not capable before it ends of destroying the whole of the velocity which remained on attaining the distance of the preceding limit-point of non-cohesion, & that which was acquired in the arc that followed it up to the next limit-point of cohesion, then the points will move to a distance apart equal to that of the next following limit-point from the origin, & will either stop there or proceed ; & there will always be a repetition of the motion, continually accelerated & retarded. Until at length it comes to an arc so strong, that is to say, one under which the area is such, that the whole velocity acquired is destroyed ; & when this happens anywhere, & does not happen at a distance equal to that of any limit-point, then the points will retrace their paths & oscillate continuously. 193. Further in this kind of motion it is clear that along the path from the distance Alternate changes of a limit-point of cohesion to a limit-point of non-cohesion the velocity is bound to be of velocity ; where always increasing ; then after passing through the latter it must decrease up to its arrival at the value! & ""a^mln? distance of a limit-point of non-cohesion. Thus, there will always be in the velocity a P um valu e ; where maximum on arrival at a distance equal to that of a limit-point of cohesion, & a minimum ' maybe estr yed. on arrival at a distance of a limit-point of non-cohesion. Hence indeed the motion may possibly cease at a limit-point of this second kind, if the two points exist by themselves, & no other point influences their motion from without. But it cannot cease at a distance of a limit-point of the first kind ; for it will always arrive at distances of this kind with an accelerated motion. Moreover it is also clear that, if they are urged from any given position with equal velocities, either towards one another or in opposite directions, the same alternations must be had as before, the velocities being increased equally for each point whilst they are moving up to a distance of a limit-point of the first kind, & diminished whilst they are moving up to a distance of a limit-point of the second kind. 194. It is evident also that, if the points are moved from a distance apart equal to that of The limit-points a limit-point of the first kind by some force (especially when the velocity thus impressed oscfflation mus^be is not extremely great), then the oscillation will be exceedingly small, at least so long as the krger; & the thing limit-point is a fairly strong one. For the velocity will commence to be diminished tude" immediately, & to the force another force will be obtained at once, acting in opposition to it ; & the points, being moved but little from their original position, will immediately afterwards retrace their paths if left to themselves. But if they are moved from a distance apart equal to that of a limit-point of the second kind by any force, no matter how small, then the oscillation will be much greater ; for, of necessity, they are bound to go on beyond the distance equal to that of the next following limit-point of the first kind ; & not until this has been done, will the motion begin to be retarded. Nay, if the next arc on each side of such a limit-point of the second kind should include a very large area, and one that is greater than several of those subsequent to them, which are opposite in direction, or greater than the excess of these over the intervening areas that are in the same direction, then indeed the oscillation may be exceedingly large. For it may be that very many limit-points on either side are traversed before an arc is arrived at, which is sufficiently strong to destroy the whole of the velocity & reverse the direction of motion. A very large oscillation will also be possible, if the points are moved from a distance apart equal to that of a limit-point of either kind by an exceedingly large force. The whole thing depends on the initial velocity & the areas which occur subsequently, & either increase or decrease the square of the velocity by a quantity that is proportional to the areas themselves. 195. However great the velocity may be, with which the two points are moved from Approach is bound a distance equal to that of any limit-point, no matter how strong are the arcs they come *^g S \* t t a h n e y % upon, which are in the same direction as that of the velocity ; yet, if they approach one repulsive arc, but another, they are bound somewhere to have their motion reversed, or at least to come on pa Sdennitety ; S a to rest ; for, at all events, they must finally attain to those very small distances that correspond noteworthy case to an asymptotic arm, the area of which is capable of destroying any velocity whatever, arSerencTfar a S very no matter how great. But, if they recede from one another, it may happen that they come great velocity. to some very strong repulsive arc, the area of which is greater than the whole of the excess of the subsequent attractive arcs above those that are repulsive, as far as the very weak 152 PHILOSOPHIC NATURALIS THEORIA sivas, usque ad languidissimum ilium arcum postremi cruris gravitatem exhibentis. Turn vero motus acquisitus ab illo arcu nunquam poterit a sequentibus sisti, & puncta ilia recedent a se invicem in immensum : quin immo si ille arcus repulsivus cum sequentibus repulsivis ingentem habeat areae excessum supra arcus sequentes attractivos ; cum ingenti velocitate pergent puncta in immensum recedere a se invicem ; & licet ad initium ejus tarn validi arcus repulsivi deveniant puncta cum velocitatibus non parum diversis ; tamen velocitates recessuum post novum ingens illud augmentum erunt parum admodum discrepantes a se invicem : nam si ingentis radicis quadrate addatur quadratum radicis multo minoris, quamvis non exiguae ; radix extracta ex summa parum admodum differet a radice priore. Demonstratio ad- modum simplex. A BD FIG. 20. Quid accidat binis punctis, cum sunt sola, quid possit accidere actionibus aliorum externis. Si limites sint a se invicem r e m o t i, m u t a t a multum distantia r e d i ri retro : secus, si sint proximi. 196. Id quidem ex Euclidea etiam Geometria manifestum fit. Sit in fig. 20 AB linea longior, cui addatur ad perpendiculum BC, multo minor, quam fit ipsa ; turn centre A, intervallo AC, fiat semicirculus occurrens AB hinc, & inde in E, D. Quadrate AB addendo quadratum BC habetur quadratum AC, sive AD ; & tamen haec excedit prsece- dentem radicem AB per solam BD, quae semper est minor, quam BC, & est ad ipsam, ut est ipsa ad totam BE. Exprimat AB velocitatem, quam in punctis quiescentibus gigneret arcus ille repulsivus per suam aream, una cum differentia omnium sequentium arcuum repulsivorum supra omnes sequentes attractivos : exprimat autem BC velocitatem, cum qua advenitur ad distantiam respondentem initio ejus arcus : exprimet AC velocitatem, qu33 habebitur, ubi jam distantia evasit major, & vis insensibilis, ac ejus excessus supra priorem AB erit BD, exiguus sane etiam re- spectu BC, si BC fuerit exigua respectu AB, adeoque multo magis respectu AB ; & ob eandem rationem perquam exigua area sequentis cruris attractivi ingentem illam jam acqui- sitam velocitatem nihil ad sensum mutabit, quae permanebit ad sensum eadem post recessum in immensum. 197. Haec accident binis punctis sibi relictis, vel impulsis [90] in recta, qua junguntur, cum oppositis velocitatibus aequalibus, quo casu etiam facile demonstratur, punctum, quod illorum distantiam bifariam secat, debere quiescere ; nunquam in hisce casibus poterit motus extingui in adventu ad distantiam limitis cohaesionis, & multo minus poterunt ea bina puncta consistere extra distantiam limitis cujuspiam, ubi adhuc habeatur vis aliqua vel attractiva, vel repulsiva. Verum si alia externa puncta agant in ilia, poterit res multo aliter se habere. Ubi ex. gr. a se recedunt, & velocitates recessus augeri deberent in accessu ad distantiam limitis cohaesionis ; potest externa compressio illam velocitatem minuere, & extinguere in ipso appulsu ad ejusmodi distantiam. Potest externa compressio cogere ilia puncta manere immota etiam in ea distantia, in qua se validissime repellunt, uti duae cuspides elastri manu compressae detinentur in ea distantia, a qua sibi relictas statim recederent : & simile quid accidere potest vi attractivae per vires externas distrahentes. 198. Turn vero diligenter notandum discrimen inter casus varies, quos inducit varia arcuum curvae natura. Si puncta sint in distantia alicujus limitis cohassionis, circa quern sint arcus amplissimi, ita, ut proximi limites plurimum inde distent, & multo magis etiam, quam sit tota distantia proximi citerioris limitis ab origine abscissarum ; turn poterunt externa vi comprimente, vel distrahente redigi ad distantiam multis vicibus minorem, vel majorem priore ita, ut semper adhuc conentur se restituere ad priorem positionem recedendo, vel accedendo, quod nimirum semper adhuc sub arcu repulsive permaneat, vel attractive. At si ibi frequentissimi limites, curva saepissime secante axem ; turn quidem post compressionem, vel distractionem ab externa vi factam, poterunt sisti in multo minore, vel majore distantia, & adhuc esse in distantia alterius limitis cohaesionis sine ullo conatu ad recuperandum priorem locum. Superiorum usus in 199. Haec omnia aliquanto fusius considerare libuit, quia in applicatione ad Physicam magno usui erunt infra haec ipsa, & multo magis hisce similia, quae massis respondent habentibus utique multo uberiores casus, quam bina tantummodo habeant puncta. Ilia ingens agitatio cum oscillationibus variis, & motibus jam acceleratis, jam retardatis, jam retro reflexis, fermentationes, & conflagrationes exhibebit : ille egressus ex ingenti arcu A THEORY OF NATURAL PHILOSOPHY 153 arc of the last branch which represents gravity. Then indeed the velocity acquired through that arc can never be stopped by the subsequent arcs, & the points will recede from one another to an immense distance. Nay further, if that repulsive arc taken together with the subsequent repulsive arcs has a very great excess of area over the subsequent attractive arcs, then the points will continue to recede to an immense distance from one another with a very great velocity ; &, although points arrive at this repulsive arc, which is so strong, with considerably different velocities, yet the velocities after this fresh & exceedingly great increase will be very little different from one another. For, if to the square of a very great number there is added the square of a number that is much less, although not in itself very small, the square root of the sum differs very little from the first number. 196. This indeed is very evident from Euclidean geometry even. In Fig. 20, let The demonstration AB be a fairly long line, to which is added, perpendicular to it, BC, which is much less ls P erfectl y sun P Ie - than AB. Then, with centre A, & radius AC, describe a semicircle meeting AB on either side in E & D. On adding the square on BC to the square on AB, we get the square on AC or AD ; & yet this exceeds the former root AB by BD only, which is always less than BC, bearing the same ratio to it as BC bears to the whole length BE. Suppose that AB represents the velocity which the repulsive arc, owing to the area under it, would generate in points initially at rest, together with the difference for all the subsequent repulsive arcs over all the subsequent attractive arcs ; also let BC represent the velocity with which the distance corresponding to the beginning of this arc is reached ; then AC will represent the velocity which is obtained when the distance has already become of considerable amount, & the force insensible. Now the excess of this above the former velocity AB will be represented by BD ; & this is really very small compared with BC, if BC were very small compared with AB ; & therefore much more so with regard to AB. For the same reason, the very small area under the subsequent attractive branch will not sensibly change the very great velocity acquired so far ; this will remain sensibly the same after recession to a huge distance. 197. These things will take place in the case of two points left to themselves, or impelled What may happen along the straight line joining them with velocities that are equal & opposite ; in such the^areT^then? a case it can be easily proved that the middle point of the distance between them is bound selves ; what may to remain at rest. The motion in the cases we have discussed can never be destroyed ^teif 11 under th the altogether on arrival at a distance equal to that of a limit-point of cohesion, & much less actions of other will the two points be able to stop at a distance apart that is not equal to that of some them 3 extemal to limit-point, as far as which there is some force acting, either attractive or repulsive. But if other external points act upon them, we may have altogether different results. For instance, in a case where they recede from one another, & the velocities would therefore be bound to be increased as they approached a distance equal to that of a limit-point of cohesion, an external compression may diminish that velocity, & completely destroy it as it approaches the distance of that limit-point. An external compression may even force the points to remain motionless at a distance for which they repel one another very strongly ; just as the two ends of a spring compressed by the hands are kept at a distance from which if left to themselves they will immediately depart. A similar thing may come about in the case of an attractive force when there are external tensile forces. 198. Now, a careful note must be made of the distinctions between the various cases, if the limit-points which arise from the various natures of the arcs of the curve. If our points are at a distance fj e far a part, there .,_. i i 111 is a tenaency to of any limit-point of cohesion, on each side of which the arcs are very wide, so that the return if the dis- nearest limit-points are very far distant from it, & also much more so than the nearest j^sfderabSH-han^e; limit-point to the left is distant from the origin of abscissae ; they may, under the action but this is not the' of an external force causing either compression or tension, be reduced after many alternations l^tsare very'dose to a distance, either less, or greater, than the original distance, in such a way that they together, will always strive however to revert to their old position by receding from or approaching towards one another ; for indeed they will still always remain under a repulsive, or an attractive arc. But if, near the limit-point in question, the limit-points on either side occur at very frequent intervals ; then indeed, after compression, or separation, caused by an external force, they may stop at a much less, or a much greater, distance apart, & still be at a distance equal to that of another limit-point of cohesion, without there being any endeavour to revert to their original position. 199. All these considerations I have thought it a good thing to investigate somewhat The use of the at length ; for they will be of great service later in the application of the Theory to physics, ^ ve facts "* phy " both these considerations, & others like them to an even greater degree ; namely those that correspond to masses, for which indeed there are far more cases than for a system of only two points. The great agitation, with its various oscillations & motions that are sometimes accelerated, sometimes retarded, & sometimes reversed, will represent fermentations 154 PHILOSOPHIC NATURALIS THEORIA repulsive cum velocitatibus ingentibus, quas ubi jam ad ingentes deventum est distantias, parum admodum a se invicem differant, nee ad sensum mutentur quidquam per immensa intervalla, luminis emissionem, & propagationem uniformem., ac ferme eandem celeritatem in quovis ejusdem speciei radio fixarum, Solis, flammse, cum exiguo discrimine inter diversos coloratos radios ; ilia vis permanens post compressionem ingentem, vel diffractionem elasticitati explicandae in-[9i]-serviet ; quies ob frequentiam limitum, sine conatu ad priorem recuperandam figuram, mollium corporum ideam suggeret ; quae quidem hie innuo in antecessum, ut magis haereant animo, prospicienti jam nine insignes eorum usus. Motus binorum 2OO. Quod si ilia duo puncta proiiciantur oblique motibus contrarns, & aequalibus punctorum oblique j- -11 j i rr projectorum. P er directiones, quae cum recta jungente ipsa ilia duo puncta angulos aequales efficiant ; turn vero punctum, in quo recta ilia conjungens secatur bifariam, manebit immotum ; ipsa autem duo puncta circa id punctum gyrabunt in curvis lineis aaqualibus, & contrariis, quae data lege virium per distantias ab ipso puncto illo immoto (uti daretur, data nostra curva virium figurae i, cujus nimirum abscissae exprimunt distantias punctorum a se invicem, adeoque eorum dimidiae distantias a puncto illo medio immoto) invenitur solutione pro- blematis a Newtono jam olim soluti, quod vocant inversum problema virium centralium, cujus problematis generalem solutionem & ego exhibui syntheticam eodem cum Newtoniana recidentem, sed non nihil expolitam, in Stayanis Supplementis ad lib. 19. Casus, in quo duo 201. Hie illud notabo tantummodo, inter infinita curvarum genera, quae describi sc 1 Hbere debe &pira < ies P ossunt 5 cum nulla sit curva, quas assumpto quovis puncto pro centre virium describi circa medium im- non possit cum quadam virium lege, quae definitur per Problema directum virium centralium, esse innumeras, quas in se redeant, vel in spiras contorqueantur. Hinc fieri potest, ut duo puncta delata sibi obviam e remotissimis regionibus, sed non accurate in ipsa recta, quae ilia jungit (qui quidem casus accurati occursus in ea recta est infinities improbabilior casu deflexionis cujuspiam, cum sit unicus possibilis contra infinites), non recedant retro, sed circa punctum spatii medium immotum gyrent perpetuo sibideinceps semper proxima, intervallo etiam sub sensus non cadente ; qui quidem casus itidem diligenter notandi sunt, cum sint futuri usui, ubi de cohaesione, & mollibus corporibus agendum erit. Theorema de statu 202. Si utcunque alio modo projiciantur bina puncta velocitatibus quibuscunque ; enSfter m 1n U 'maSs P test facile ostendi illud : punctum, quod est medium in recta jungente ipsa, debere centri gravitatis quiescere, vel progredi uniformiter in directum, & circa ipsum vel quietum, vel uniformiter progrediens, debere haberi vel illas oscillationes, vel illarum curvarum descriptiones. Verum id generalius pertinet ad massas quotcunque, & quascunque, quarum commune gravitatis centrum vel quiescit, vel progreditur uniformiter in directum a viribus mutuis nihil turbatum. Id theorema Newtonus proposuit, sed non satis demonstravit. Demon- strationem accuratissimam, ac generalem simul,& non per casuum inductionem tantummodo, inveni, ac in dissertation e De Centra Gravitatis proposui, quam ipsam demonstrationem hie etiam inferius exhibebo. Accessum aiterius [92] 203. Interea hie illud postremo loco adnotabo, quod pertinet ad duorum quodvis ad aiterius punctorum motum ibi usui futurum : si duo puncta moveantur viribus mutuis tantummodo, aequari recessui ex & ultra ipsa assumatur planum quodcunque ; accessus aiterius ad illud planum secundum directionem quamcunque, aequabitur recessui aiterius. Id sponte consequitur ex eo, quod eorum absoluti motus sint aequales, & contrarii ; cum inde fiat, ut ad directionem aliam quamcunque redacti aequales itidem maneant, & contrarii, ut erant ante. Sed de aequilibrio, & motibus duorum punctorum jam satis. Transitus ad syste- 204. Deveniendo ad systema trium punctorum, uti etiam pro punctis quotcunque, trium "binagene res > si generaliter pertractari deberet, reduceretur ad haac duo problemata, quorum alterum alia probiemata. pertinet ad vires, & alterum ad motus : I. Data positions, 5? distantia mutua eorum punc- torum, invenire magnitudinem, & directionem vis, qua urgetur quodvis ex ipsis, composites a viribus, quibus urgetur a reliquis, quarum singularum virium lex communis datur per curvam figure primce. 2. Data ilia lege virium figures -primes invenire motus eorum punctorum, quorum singula cum datis velocitatibus projiciantur ex datis locis cum datis directionibus. Primum facile solvi potest, & potest etiam ope curva; figurae i determinari lex virium A THEORY OF NATURAL PHILOSOPHY 155 & conflagrations. The starting forth from a very large repulsive arc with very great velocities, which, as soon as very great distances have been reached, are very little different from one another ; nor are they sensibly changed in the slightest degree for very great intervals ; this will represent the emission & uniform propagation of light, & the approximately equal velocities in any ray of the same kind from the stars, the sun, and a flame, with a very slight difference between rays of different colours. The force persisting after compression, or separation, will serve to explain elasticity. The lack of motion due to the frequent occurrence of limit-points, without any endeavour towards recovering the original configuration, will suggest the idea of soft bodies. I mention these matters here in anticipation, in order that they may the more readily be assimilated by a mind that already sees from what has been said that there is an important use for them. 200. But if the two points are projected obliquely with velocities that are equal and The motion of two opposite to one another, in directions making equal angles with the straight line joining obikmei pro ^ ected the two points ; then, the point in which the straight line joining them is bisected will remain motionless ; the two points will gyrate about this middle point in equal curved paths in opposite directions. Moreover, if the law of forces is given in terms of the distances from that motionless point (as it will be given when our curve of forces in Fig. i is given, where the abscissae represent the distances of the points from one another, & therefore the halves of these abscissae represent -the distances from the motionless middle point), then we arrive at a solution of the problem already solved by Newton some time ago, which is called the inverse -problem of central forces. Of this problem I also gave a general synthetic solution that was practically the same thing as that of Newton, not altogether devoid of neatness, in the Supplements to Stay's Philosophy, Book 3, Art. 19. 201. At present I will only remark that, amongst the infinite number of different The case in which curves that can be described, there are an innumerable number which will either re-enter boumT^o^teswibe their paths, or wind in spirals ; for there is no curve that, having taken any point whatever spirals about the for the centre of forces, cannot be described with some law of forces, which is determined j^^ nless mlddle by the direct problem of central forces. Hence it may happen that two points approaching one another from a long way off, but not exactly in the straight line joining them and the case of accurate approach along the straight line joining them is infinitely more improbable than the case in which there is some deviation, since the former is only one possible case against an infinite number of others then the points will not reverse their motion and recede, but will gyrate about a motionless middle point of space for evermore, always remaining very near to one another, the distance between them not being appreciable by the senses. These cases must be specially noted ; for they will be of use when we come to consider cohesion & soft bodies. 202. If two points are projected in any manner whatever with any velocities whatever, Theorem on the it can readily be proved that the middle point of the line joining them must remain at steady state of the 1 . , r , . , , . r , , , , J . P , , . . central point &, rest or move uniformly in a straight line ; and that about this point, whether it is at rest more generally, or is moving uniformly, the oscillations or descriptions of the curved paths, referred to of the . cen , tre of i ' T> i 11 i r c gravity in the case above, must take place. But this, more generally, is a property relating to masses, of any of masses. number or kind, for which the common centre of gravity is either at rest or moves uniformly in a straight line, in no wise disturbed by the mutual forces. This theorem was enunciated by Newton, but he did not give a satisfactory proof of it. I have discovered a most rigorous demonstration, & one that is at the same time general, & I gave it in the dissertation De Centra Gravitatis ; this demonstration I will also give here in the articles that follow. 203. Lastly, I will here mention in passing something that refers to the motion of The approach of two points, which will be of use later, in connection with that subject. If two points poLts towards^y move subject to their mutual forces only, & any plane is taken beyond them both, then plane is equal to the approach of one of them to that plane, measured in any direction, will be equal to the other^rom 1 it 'on recession of the other. This follows immediately from the fact that their absolute motions account of 'the are equal & opposite ; for, on that account, it comes about that the resolved parts in any mutual force - other direction also remain equal & opposite, as they were to start with. However, I have said enough for the present about the equilibrium & motions of two points. 204. When we come to consider systems of three points, as also systems of any number Extension to a of points, the whole matter in general will reduce to these two problems, of which the system of three tt 11 . T, . . . , points ; two general one refers to forces and the other to motions, i. Being given the position and the mutual problems. distance of the points, it is required to find the magnitude and direction of the force, to which any one of them is subject ; this force being the resultant of the forces due to the remaining points, and each of these latter being found by a general law which is given by the curve of Fig. i. 2. Being given the law of forces represented by Fig. I, it is required to find the motions of the points, when each of them is projected with known velocities from given initial positions in given directions. The first of these problems is easily solved ; and also, by the aid of 156 PHILOSOPHIC NATURALIS THEORIA generaliter pro omnibus distantiis assumptis in quavis recta positionis datae,' a que id tarn geometrice determinando per puncta curvas, quae ejusmodi legem exhibeant, ac determinent sive magnitudinem vis absolutae, sive magnitudines binarum virium, in quas ea concipiatur resoluta, & quarum altera sit perpendicularis data? illi rectse, altera secundum illam agat ; quam exhibendo tres formulas analyticas, quas id praestent. Secundum omnino generaliter acceptum, & ita, ut ipsas curvas describendas liceat definire in quovis casu vel constructione, vel caculo, superat (licet puncta sint tantummodo tria) vires methodorum adhuc cognit- arum : & si pro tribus punctis substituantur tres massae punctorum, est illud ipsum celeberrimum problema quod appellant trium corporum, usque adeo qusesitum per haec nostra tempora, & non nisi pro peculiaribus quibusdam casibus, & cum ingentibus limita- tionibus, nee adhuc satis promoto ad accurationem calculo, solutum a paucissimis nostri asvi Geometris primi ordinis, uti diximus num. 122. Theorema de motu 205. Pro hoc secundo casu illud est notissimum, si tria puncta sint in fig. 21 A, C, B, puncti habentis ac- ,. J . . T. i > IT- T-> i tionem cum aiiis & distantia AB duorum divisa semper bifanam in D, ac ducta CD, & assumpto ejus binis - triente DE, utcunque moveantur eadem puncta motibus compositis a projectionibus quibus- cunque, & mutatis viribus ; punctum E debere vel quiescere semper, vel progredi in directum motu uniformi. Pendet id a general! theore- mate de centre gravitatis, cujus & superius injecta est mentio, & de quo age-[93]-mus infra pro massis quibuscunque. Hinc si sibi re- linquantur, accedet C ad E, & rectae AB punctum medium D ibit ipsi obviam versus ipsum cum velocitate dimidia ejus, quam ipsum habebit, vel contra recedent, vel hinc, aut inde movebuntur in latus, per lineas tamen similes, atque ita, ut C, & D semper respectu puncti E immoti ex adverse sint, in quo motu tam directio . _ rectae AB, quam directio rectae CD, & ejus incli- " _ *^ natio ad AB, plerumque mutabitur. Determmatio vis 2 o6. Quod pertinet ad inveniendam vim pro quacunque positione puncti C respectu ejusdem composite A T> r -i T r e binis viribus. punctorum A, & B, ea facile sic mvemetur. In fig. i assumendae essent abscissae in axe asquales rectis AC, BC figurae 21, & erigendae ordinatas ipsis respondentes, quae vel ambae essent ex parte attractiva, vel ambae ex parte repulsiva ; vel prima attractiva, & secunda repulsiva ; vel prima repulsiva & secunda attractiva. In primo casu sumendae essent CL, CK ipsis aequales (figura 21 exhibet minores, nenimis excrescat) versus A, &B ; in secundo CN, CM ad partes oppositas A,B : in tertio CL versus A, & CM ad partes oppositas B ; in quarto CN ad partes oppositas A, & CK versus B. Tam complete parallelogrammo LCKF, vel MCNH, vel LCMI, vel KCNG, diameter CF, vel CH, vel CI, vel CG exprimeret directionem, & magnitudinem vis compositae, qua urgetur C a reliquis binis punctis. Methodus constru- 207. Hinc si assumantur ad arbitrium duo loca qusecunque punctorum A, & B, ad expi? <l u3e referendum sit tertium C ; ducta quavis recta DEC indefinita, ex quovis ejus puncto mat vim ejusmodi. posset erigi recta ipsi perpendicularis, & asqualis illi diametro, ut CF in primo casu, ac haberetur curva exprimens vim absolutam puncti in eo siti, & solicitati a viribus, quas habet cum ipsis A, & B. Sed satis esset binas curvas construere, alteram, quae exprimeret vim redactam ad directionem DC per perpendiculum FO, ut CO ; alteram, quae exprimeret vim perpendicularem OF : nam eo pacto haberentur etiam directiones vis absolutae ab iis compositae per ejusmodi binas ordinatas. Oporteret autem ipsam ordinatam curvas utriuslibet assumere ex altera plaga ipsius CD, vel ex altera opposita ; prout CO jaceret versus D, vel ad plagam oppositam pro prima curva; & prout OF jaceret ad alteram partem rectae DC, vel ad oppositam, pro secunda. Expressio magis 208. Hoc pacto datis locis A, B pro singulis rectis egressis e puncto medio D duas * 115 Per super " haberentur diversae curvae, quae diversas admodum exhiberent virium leges ; ac si quasre- retur locus geometricus continuus, qui exprimeret simul omnes ejusmodi leges pertinentes ad omnes ejusmodi curvas, sive indefinite exhiberet omnes vires pertinentes ad omnia A THEORY OF NATURAL PHILOSOPHY 157 the curve given in Fig. i, the law of forces can be determined in general for any assumed distances along any straight line given in position. Moreover, this can be effected either by constructing geometrically curves through sets of points, which represent a law of this sort & give either the magnitude of the absolute force, or the magnitudes of the pair of forces into which it may be considered to be resolved, the one acting perpendicularly to the given straight line & the other in its direction ; or else by writing down three analytical formulae, which will represent its value. The second, if treated perfectly generally, & in such a manner that the curves to be described can be assigned in any case whatever, either by construction or by calculation, is (even when there are only three points in question) beyond the power of all methods known hitherto. Further, if instead of three points we have three masses of points, then we have the well-known problem that is called " the problem of three bodies." The solution of this problem is still sought after in our own times ; & has only been solved in certain special cases, with great limitations by a very few of the geometricians of our age belonging to the highest rank, & even then with insufficient accuracy of calculation ; as was pointed out in Art. 122. 205. As for this second case, it is very well known that, if in Fig. 21, A,C,B, are three Theorem with re- points, & the distance between two of them, A & B, is always bisected at D, & CD is joined, g'Vyjj m ^ r & DE is taken equal to one third of DC, then, however these points move under the influence the action of two of the forces compounded from the forces of any projection whatever & the mutual forces, other P mts - the point E must always remain at rest or proceed in a straight line with uniform motion. This depends on a general theorem with regard to the centre of gravity, about which passing mention has already been made, & with which we shall deal in what follows for the case of any masses whatever. From this it follows that, if they are left to themselves, the point C will approach the point E, & D, the middle point of the straight line AB, will move in the opposite direction towards E with half the velocity of C ; or, on the contrary, both C & D will recede from E ; or they will move, one in one direction & the other in the opposite direction : nevertheless they will follow similar paths, in such a manner that C & D will always be on opposite sides of the stable point E ; & in this motion, the direc- tion of the straight line AB, that of the straight line DE, & the inclination of the latter to AB will usually be altered. 206. As regards the determination of the force for any position of the point C with Determination o f regard to the points A & B, that is easily effected in the following manner. Take, in Fig. I, compound^ abscissa measured along the axis equal to the straight lines AC & BC of Fig. 21 ; draw two forces. the ordinates corresponding to them, which may be either both on the attractive side of the axis, or both on the repulsive side ; or the first on the attractive & the second on the repulsive ; or the first on the repulsive & the second on the attractive side. In the first case, take CL, CK, equal to these ordinates (in Fig. 21 they are reduced so as to prevent the figure from being too large) ; let them be taken in the direction of A & B ; similarly, in the second case, take CN & CM in the opposite directions to those of A & B ; and, in the third case, take CL in the direction of A, & CM in the direction opposite to that of B ; whilst, in the fourth case, take CN in the direction opposite to that of A, & CK in the direction of B. Then, completing the parallelogram LCKF, or MCNH, or LCMI, or KCNG, the diagonal CF, or CH, or CI, or CG, will represent the direction & the magnitude of the resultant force, which is exerted upon the point C by the remaining two points. 207. Hence, if any two positions are taken at random as those of the points A & B, The method of & to these the third point C is referred ; & if any straight line DEC is drawn of indefinite curve' wWch^m in length ; then from any point of it a straight line can be erected perpendicular to it, & general express a equal to the diagonal of the parallelogram, for instance CF in the first case. From these force of this sort ' perpendiculars a curve will be obtained, which will represent the absolute force on a point situated in the straight line DEC, & under the action of the forces exerted upon it by the points A & B. However, it would be more satisfactory if two curves were constructed ; one of which would represent the force resolved along the direction DC by means of a perpendicular FO, such as CO ; & the other to represent the perpendicular force OF. For, in this way, we should also obtain the directions of the absolute forces compounded from these resolved parts, by means of the two ordinates of this kind. Moreover, we ought to take these ordinates of either of the curves on the one side or the other of the straight line CD, according as CO would be towards D, or away from it, in the first curve, & according as OF would be away from the straight line CD, on the one side or on the other, in the second curve. 208. In this way, given the positions of A & B, for each straight line drawn through A m re g eneral the point D, we should obtain distinct curves ; & these would represent altogether different of ^ surface. 71 laws of forces. If then a continuous geometrical locus is required, which would simultaneously represent all the laws of this kind relating to every curve of this sort, or express in general all the forces pertaining to all points such as C, wherever they might I 5 8 PHILOSOPHIC NATURALIS THEORIA puncta C, ubicunque collocata ; oporteret erigere in omnibus punctis C rectas normales piano ACB, alteram aequalem CO, [94] alteram OF, & vertices ejusmodi normalium determinarent binas superficies quasdam continuas, quarum altera exhiberet vires in directione CD attractivas ad D, vel repulsivas respectu ipsius, prout, cadente O citra, vel ultra C, normalis ilia fuisset erecta supra, vel infra planum ; & altera pariter vires perpen- diculares. Ejusmodi locus geometricus, si algebraice tractari deberet, esset ex iis, quos Geometrse tractant tribus indeterminatis per unicam aequationem inter se connexis ; ac data aequatione ad illam primam curvam figurse I, posset utique inveniri tam sequatio ad utramlibet curvam respondentem singulis rectis DC, constans binis tantum indeterminatis, quam sequatio determinans utramlibet superficiem simul indefinite per tres indetermin- atas. () Methpdus determi- [gel 20Q. Si pro duobus punctis tantummodo agentibus in tertium daretur numerus nandi vim composi- L7 T J . r . , . . . . ., tam ex viribus re- quicunque punctorum positorum in datis locis, ac agentium in idem punctum, posset utique spicientibus puncta constructione simili inveniri vis, qua sineula agunt in ipsum collocatum in quovis assumpto quotcunque. , . . ' ~*. . .. . r , ., . .? loci puncto, ac vis ex ejusmodi viribus composita denniretur tam directione, quam magnitudine, per notam virium compositionem. Posset etiam analysis adhiberi ad expri- mendas curvas per asquationes duarum indeterminatarum pro rectis quibuscunque, & (") si omnia puncta jaceant in eodem piano, superficies per asquationem trium. [96] Mirum autem, quanta inde diversarum legum combinatio oriretur. Sed & ubi duo tantummodo puncta agant in tertium, incredibile dictu est, quanta diversitas legum, & curvarum inde erumpat. Manente etiam distantia AB, leges pertinentes ad diversas inclinationes rectae DC ad AB, admodum diversse obveniunt inter se : mutata vero punctorum A, B distantia (n) Stantibus in fig. 22 punctis ADBCKFLO, ut in fig. 21, ducantur perpendicula BP, AQ in CD, qute dabuntur data inclinations DC, y punctis B, A, ac pariter dabuntur y DP, DQ. Dicatur prtsterea DC = x, y dabuntur analytice CQ, CP. Quare ob angulos rectos P, Q, dabuntur etiam analytice CB, CA. Denominentur CK=, CL =z, CF =y. Quoniam datur AB, y dantur analytice AC, CB ; dabitur analytice ex applicatione Algebrte ad Trigonometriam sinus anguli ACB per x, y datas quantitates, qui est idem, ac sinus anguli CKF complements ad duos rectos. Datur autem idem ex datis analytice valoribus CK = , KF = CL =z, CF =y ; quare habetur ibi una tequatio per x, y, z, u, y constantes. Si pr<eterea valor CB ponatur pro valore abscissae in tequatione curvte figurte I ; acquiritur altera tequatio per valores CK, CB, sive per x,u, y constantes. Eodem facto invenietur ope tequationis curvte figure I tertia tequatio per AC, & CL, adeoque per x, z, y constantes. Quare jam babebuntur tequationes tres per x,u,z,y, y con- stantes, qute, eliminates u, y z, reducentur ad unicam per x,y, & constantes, ac ea primam illam curvam definiet. Quod si queeratur' tequatio ad secundam curvam, cujus ordinata est CO, vel tertiam, cujus ordinata OF, T>p inveniri itidem poterit. Nam datur analytice sinus anguli DCB = , W * trianguk FCK datur analytice \sD pTT sinus FCK ===- X sin CKF. Quare datur analytice etiam sinus Cr differentite OCF, adeoque & ejus cosinus, & inde, ac ex CF datur analytice OF, vel CO. Sz igitur altera ex illis dicatur p, acquiri- tur nova tequatio, cujus ope una cum superioribus eliminari poterit pristerea una alia indeterminata ; adeoque eliminata CF =y, habebitur unica tequatio per x,p, y constantes, qua exhibebit utramlibet e reliquis curvis determinantibus legem virium CO, vel OF. Pro tequatione cum binis indeterminatis, quts exhibebit locum- ad superficiem, ducatur CR perpendicularis ad AB, y dicatur DR x, RC = q, denominatis, ut prius, CK =, CL = z, CF = v ; y quoniam dantur AD, DB ; dabuntur analytice per x, y constantes AR, RB, adeoque per x, q, & constantes AC, CB, W factis omnibus reliquis, ut prius, kabebuntur quatuor tequationes per x,q,u,z,y,p, y constantes, qute eliminatis valoribus u,z,y, reducentur ad unicam datam per constantes, y tres indeterminatas x,p,q, sive DR, RC, y CO, vel OF, qute exhibebit qutesitum locum ad superficiem. Calculus quidem esset immensus, sed patet methodus, qua deveniri possit ad tequationem qutesitam. Mirum autem, quanta curvarum, y superficierum, adeoque y legum virium varietas obvenerit, mutata tantummodo distantia AB binorum punctorum agentium in tertium, qua mutata, mutatur tola lex, y tequatio. (o) Htec conditio punctorum jacentium in eodem piano necessaria fuit pro loco ad superficiem, y pro tequatione, qute legem virium exhibeat per tequationem indeterminatarum tantummodo trium : at si puncta sint plura, y in eodem piano non jaceant, quod punctis tantummodo tribus accidere omnino non potest ; turn vero locus ad superficiem, y tequatio trium indeterminatarum non sufficit, sed ad earn generaliter exprimendam legem Geometria omnis est incapax, y analysis indiget tequatione indeterminatarum quatuor. Primum patet ex eo, quod si manentibus punctis A, B, exeat punctum C ex data quodam piano, pro quo constructus sit locus ad superficiem ; liceret converters circa rectam AB planum illud cum superficie curva legem virium determinate, donee ad punctum C deveniret planum ipsum : turn enim erecto perpendiculo usque ad superficiem illam curvam, definiretur per ipsum vis agens secundum rectam CD, vel ipsi perpendicularis, prout locus ille ad curvam superficiem constructus fuerit pro altera ex iis. A THEORY OF NATURAL PHILOSOPHY 159 be situated ; we should have to erect at every point C normals to the plane ACB, one of them equal to CO & the other to OF. The ends of these normals would determine two continuous surfaces ; & of these, the one would represent the forces in the direction CD, attractive or repulsive with respect to the point D, according as the normal was erected above or below this plane, whether C fell on the near side or on the far side of D ; & similarly the other would represent the perpendicular forces. A geometrical locus of this kind, if it has to be treated algebraically, is such as geometricians deal with by means of three unknowns connected together by a single equation ; &, if the equation to the primary curve of Fig. i is given, it would in all cases be possible to find, not only the equations to the two curves corresponding to each & every straight line DC, involving only two unknowns, but also the equations for both the surfaces corresponding to the general determination, by means of three unknowns. () 209. If instead of only two points acting upon a third we are given any number of The method of points situated in given positions, & acting on the same point, it would be possible, by a force ""compounded similar construction in each case, to find the force, with which each acts on the point from the forces due situated in any chosen position ; & the force compounded from forces of this kind would points 7 The 1 great be determined, both in position & magnitude, by the well-known method for composition "]^ s r & variet y of forces. Also analysis could be employed to represent the curves by equations involving two unknowns for any straight lines ; & () provided that all the points were in the same plane, the surface could be represented by an equation involving three unknowns. But it is marvellous what a huge number of different laws arise. But, indeed, it is incredible, even when there are only two points acting on a third, how great a number of different laws & curves are produced in this way. Even if the distance AB remains the same, the laws with respect to different inclinations of the straight line CD to the straight line AB, come out quite different to one another. But when the distance of the points A & B from (n) In Fig. 22, let the -points A,D,B,C,K,F,L,O be in the same positions as in Fig. 21, y let BP, AQ be drawn perpendicular to CD ; then these will be known, if the inclination of CD y the positions of A y B are known: y so also will DP W DQ be known. Further, suppose DC = x, then CQ y CP will be given analytically. Hence on account of the right angles at P y Q, CB y CA will also be given analytically. Suppose CK = w, CL = z, CF = y. Since AB is known, y AC, CB are given analytically, by an application of algebra to trigonometry, the sine of the angle ACB is also known analytically in terms of x y known quantities ; y this is the same thing as the sine of the supplementary angle CKF. Moreover the same thing will be given in terms of the known analytical values of 3K = u, KF = CL = z, CF = y. Hence there is obtained in this case an equation involving x,y,z,u, y constants. If, in addition, the value CB is substituted for the value of the abscissa in the equation of the curve in Fig. I, another equation will be obtained in terms of the values of CK, CB, i.e. in terms of x, u, y constants. In a similar way by the help of the equation of the curve of Fig. I, there can be found a third equation in terms of AC y CL, i.e., in terms of #,z, y constants. Now, snce there will be' thus obtained three equations in terms of x,y,z,u, y constants, these, on eliminating u,z, will reduce to a single equation involving x,y, y constants ; y this will be the equation defining the first curve. Again, / the equation to the second curve is required, of which the ordinate is CO, or of a third curve for which the ordinate is CF, it will be possible to find either of these as well. For the sine of the angle DCB is analytically given, being equal to BP/CB ; y from the triangle FCK, the sine of the angle FCK is given, being equal to 'CKF.(FK/CF). There fore the sine of the difference OOP is also given analytically, y therefore also its cosine; y from this y the value of CF, the value of OF or CO will be given analytically. If then one or the other of them is denoted by p, a new equation will be obtained: by the help of this y one of the equations given above, another of the unknowns can be eliminated. If then, we eliminate CF = y, a single equation will be obtained in terms of x,p, y constants, which will be that of one or other of the remaining curves determining the law of forces for CO or OF. For an equation in three unknowns, which will represent the surface, draw CR perpendicular to AB, y let DR=# RC = q ; y, as before, let CK = it, CL = z, CF = y. Then, since AD, DB are given, AR y RB are also given analytically in terms of x y constants : y therefore AC y CB are given in terms of x,q, y constants : y if all the rest of the work is done as before, four equations will be obtained in terms of x,q,u,z,y,p, y constants. These, on eliminating the values u,z,y, will reduce to a single equation in terms of constants y the three unknowns x,p,q, or DR, RC, y CO or OF ; this equation will represent the surface required. The calculation would indeed be enormous ; but the method, by which the required equation might be obtained is perfectly clear. But it is wonderful what a great number of curves y surfaces, y therefore of laws of force, would be met with, if merely the distance between A y B, the two points which act upon the third, is changed ; for if this alone is changed, the whole law is altered y so too is the equation. (o) This condition, that the points should all lie in the same plane, is necessary for the determination of the surface, y for the equation, which will express the law of the forces by an equation involving only three unknowns. If the points are numerous, y they do not all lie in the same plane (which is quite impossible in the case of only three points), then indeed a surface locus, y an equation in three unknowns, will not be sufficient; indeed, to express the law generally, the whole of geometry is powerless, y analysis requires an equation in four unknowns. The first point is clear from the fact that if, whilst the points A y B remain where they were, the point C moves out of the given plane, with regard to which the construction for the surface locus was made, it would be right to rotate about the straight line AB that plane together with its curved surface, which determines the law of forces, until the plane passes through the point C. For then, if a perpendicular is drawn to meet the curved surface, this would define the force acting along the straight line CD, or perpendicular to it, according as the locus to the curved surface had been constructed for the one or for the other of them. 160 PHILOSOPHIC NATURALIS THEORIA a se invicem, leges etiam pertinentes ad eandem inclinationem DC differunt inter se plurimum ; & infinitum esset singula persequi ; quanquam earum variationum cognitio, si obtineri utcunque posset, mirum in modum vires imaginationis extenderet, & objiceret discrimina quamplurima scitu dignissima, & maximo futura usui, atque incredibilem Theoriae foecunditatem ostenderet. distantiis 1 2IO> ^8 ^ c s i m pliciora quaedam, ac faciliora, & usum habitura in sequentibus, ac in ac ejus usus pro applicatione ad Physicam inprimis attingam tantummodo ; sed interea quod ad generalem nu'iiaVinUs s^mma P ert i net determinationem expositam, duo adnotanda proponam. Primo quidem in ipsa virium simpiicium. trium punctorum combinatione occurrit jam hie nobis praeter vim determinantem ad accessum, & recessum, vis urgens in latus, ut in fig. 21, praeter vim CF, vel CH, vis CI, vel CG. Id erit infra magno usui ad explicanda solidorum phaenomena, in quibus, inclinato fundo virgse solidae, tola virga, & ejus vertex moventur in latus, ut certam ad basim positionem acquirant. Deinde vero illud : haec omnia curvarum, & legum discrimina tam quae [97] pertinent ad diversas directiones rectarum DC, data distantia punctorum A, B, quam quae pertinent ad diversas distantias ipsorum punctorum A, B, data etiam directione DC, ac hasce vires in latus haberi debere in exiguis illis distantiis, in quibus curva figurae I circa axem contorquetur, ubi nimirum mutata parum admodum distantia, vires singulorem punctorum mutantur plurimum, & e repulsivis etiam abeunt in attractivas, ac vice versa, & ubi respectu alterius puncti haberi possit attractio, respectu alterius repulsio, quod utique requiritur, ut vis dirigatur extra angulum ACB, & extra ipsi ad verticem oppositum. At in majoribus distantiis, in quibus jam habetur illud postremum crus figurae I exprimens arcum attractivum ad sensum in ratione reciproca duplicata distantiarum, vis in punctum C a punctis A, B inter se proximis, utcunque ejusmodi distantia mutetur, & quaecunque fuerit inclinatio CD ad AB, erit semper ad sensum eadem, directa ad sensum ad punctum D, ad sensum proportionalis reciproce quadrato distantiae DC ab ipso puncto D, & ad sensum dupla ejus, quam in curva figurae i requireret distantia DC. At secundum sit manifestum ex eo, quod si puncta agenda sint etiam omnia in eodem piano, y punctum, cufus vis composita quteritur, in quavis recta posita extra ipsum planum, relationes omnes distantiarum a reliquis punctis, ac directionum, a quibus pendent vires singulorum, y compositio ipsarum virium, longe alia essent, ac in quavis recta in eodem piano posita, uti facile videre est. Hinc pro quovis puncto loci ubicunque assumpto sua responderet vis composita, y quarta aliqua plaga, seu dimensio, prater longum, latum, & profundum, requireretur ad ducendas ex omnibus punctis spatii rectas Us viribus proportionales, quarum rectarum vertices locum continuum aliqucm exhiberent determinantem virium legem. Sed quod Geometria non assequitur, assequeretur quarta alia dimensio mente concepta, ut si conciperetur spatium totum plenum materia continua, quod in mea sententia cogitatione tantummodo effingi potest, W ea esset in omnibus spatii punctis densitatis diverse, vel diversi pretii ; turn ilia diversa densitas, vel illud pretium, vel quidpiam ejusmodi, exhibere posset legem virium ipsi respondentium, ques nimirum ipsi essent proportionales. Sed ibi iterum ad determmandam directtonem vis composite non esset satis resolutio in duas vires, alteram secundum rectam transcuntem per datum punctum ; altcram ipsi perpendicularem ; ed requirerentur tres, nimirum vel omnes secundum tres datas directiones, vel tendentes per rectas, qua per data tria puncta transeant, vel quavis alia certa lege definitas : adeoque tria loca ejusmodi ad spatium, quarta aliqua dimensione, vel qualitate affectum requirerentur, qu<e tribus ejusmodi plusquam Geometricis legibus vis composite legem definirent, turn quod pertinet ad ejus magnitudinem, turn quod ad directionem. quod non assequitur Geometria, assequeretur Analysis ope aquationis quatuor indeterminatarum : si enim planum, quod libuerit, ut ACB, y in eo quavis recta AB, ac in ipsa recta quodvis punitum D ; turn quovis Ferum conciperetur . M hujus segmento DR appellate x, quavis recta RC ipsi perpendiculari y, quavis tertia perpendicular! ad totum planum z, per hasce tres indeterminatas involveretur positio puncti spatii cujuscumque, in quo collocatum esset punctum materiel, cufus vis quteritur. Punctorum agentium utcunque collocatorum ubicunque vel intra id planum, vel extra, possent definiri positiones per ejusmodi tres rectas, datas utique pro singulis, si eorum positiones dentur. Per eas, y per illas x,y,z, posset utique haberi distantia cujuscumque ex Us punctis agentibus, y positione datis, a puncto indefinite accepto ; adeoque ope aquationis figurtz I posset haberi analytice per aquationes quasdam, ut supra, vis ad singula agentia puncta pertinens, y per easdem rectas ejus etiam directio resoluta in tres parallelas illis x,y,z. Hinc haberetur analytice omnium summa pro singulis ejusmodi directionibus per aliam aquationem derivatam ab ejus summa denominatione, ea nimirum facia = u, ac expunctis omnibus subsidiariis valoribus, methodo non absimili ei, quam adhibuimus superius pro loco ad superficiem, deveniretur ad unam aquationem constitutam illis quatuor indeterminatis x,y,z,u, y constantibus ; ac tres ejusmodi aquationes pro tribus directionibus vim omnem compositam definirent. Sed hac innuisse sit satis, qua nimirum y altiora sunt, y ob ingentem complicationcm casuum, ac nostra humantf mentis imbecillitatem nulli nobis inferius futura sunt usui. A THEORY OF NATURAL PHILOSOPHY 161 one another is also changed, the laws corresponding to the same inclination of DC are altogether different to one another ; & it would be an interminable task to consider them all, case by case. However, a comprehensive insight into their variations, if it could be obtained, would enlarge the powers of imagination to a marvellous extent ; it would bring to the notice a very large number of characteristics that would be well worth knowing & most useful for further work ; & it would give a demonstration of the marvellous fertility of my Theory. 210. First of all, therefore, I will here only deal slightly with certain of the more simple The lateral force at cases, such as will be of use in what follows, & later when considering the application to tances, Tits use \n Physics. But meanwhile, I will enunciate two theorems, applying to the general deter- t] } e consideration mination set forth above, which should be noted. Firstly, in the case of the combination absence^ this of three points, we have here already met with, in addition to a force inducing approach f ? rce at great . . ' . T-,. . IT. ,. X-,T-< ,"ITT ^-.T 2Z^~. rr distances, the sum & recession, i.e., in rig. 21, in addition to a force CF or CH, a force CI or CG, urging of the simple forces the point C to one side. This will be of great service to us in explaining certain phenomena in the latter case - of solids ; for instance, the fact that, if the bottom of a solid rod is inclined, the whole rod, including its top, is moved to one side & takes up a definite position with respect to the base. Secondly, there is the fact that we are bound to have all these differences of curves & laws, not only those corresponding to different directions of the straight lines DC when the distance between the points A & B is given, but also those corresponding to different distances of the points A & B when the direction of DC is given ; & that we are bound to have these lateral forces for very small distances, for which the curve in Fig. I twists about the axis ; for then indeed, if the change in distance is very slight, the change in the forces corresponding to the several points is very great, & even passes from repulsion to attraction & vice versa ; & also there may be attraction for one point & repulsion for another ; & this must be the case if the direction of the force has to be without the angle ACB, or the angle vertically opposite to it. But, at distances that are fairly large, for which we have already seen that there is a final branch of the curve of Fig. i that represents attraction approximately in the ratio of the inverse square of the distance, the force on the point C, due to two points A & B very near to one another, will be approximately the same, no matter how this distance may be altered, or what the inclination of CD to AB may be ; its direction is approximately towards D ; & its magnitude will be approximately in inverse proportion to the square of DC, its distance from the point D ; that is to say, it will be approximately double of that to which in Fig. I the distance DC would correspond. The second point is evident from the fact that, if all the points acting are all in the same plane, 5? the point for which the resultant farce is required, lies in any straight line situated without that plane, even then all the relations between the distances from the remaining points as well as between their directions, will be altogether different from those for any straight line situated in the same plane, as can be easily seen. Hence, for any point of space chosen at random there would be a corresponding force ; W a fourth region, or dimension, in addition to length, breadth, & depth, would be required, in order to draw through each point of space straight lines proportional to these forces, the ends of which straight lines would give a continuous locus determining the law for the forces. But 'what can not be attained by the use of geometry, could be attained, by imagining another, a fourth, dimension (just as if the whole of space were imagined to be full of eontinuous matter, which in my opinion can only be a mental fiction) ; W this would be of different density, or different value, at all points of space. Then the different density, or value, or something of that kind, might represent the law of forces corresponding to it, these indeed being proportional to it. But here again, in order to find the direction of the resultant force, resolution into two forces, the one along the straight line passing through the given -point, y the other perpendicular to it, would, not be sufficient. Three resolved parts would be required, either all in three given directions, or along straight lines passing through three given points, or defined by some other fixed law. Thus, three regions of this kind in space possessed of some fourth dimension or quality would be required ; y these would define, by three ultra-geometrical laws of this sort, the law of the resultant force both as regards magnitude & direction. But what cannot be obtained with the help of geometry could be obtained by the aid of analysis by employing an equation with four unknowns. For, if we take any arbitrary plane, as ACB, y in it any straight line AB, y in this straight line any point D / then, calling any segment of it x, any straight line perpendicular to it y, y any third straight line perpendicular to the whole plane z, there would be contained in these three unknowns the position of any point in space, at which is situated a point of matter, for which the force is required. The positions of the acting points, however W wherever they may be situated, either within the plane or without it, could be defined by three straight lines of this sort ; y these would in all cases be known for each point, if the positions of the points are given. By means of these, y the former straight lines denoted by x,y,z, there could he obtained in all cases the distance of each of the acting points, that are given in position, from any point assumed indefinitely. Thus by the help of the equation to the curve of Fig. I, there could be obtained analytically, by means of certain equations similar to those above, the force corresponding to each of the acting points ; also from the same straight lines, its direction as well, by resolving along three parallels to x, y, y z. Hence there could be obtained analytically the sum of all of them for each of these directions, by means of another equation derived from the symbol used for the sum (for instance, let this be called u] ; y, eliminating all the subsidiary values, by a method not unlike that which was used above for the surface locus, we should arrive at a single equation in terms of the four unknowns, x, y, z, u, y constants. Three equations of this sort, one for each of the three directions, would determine the resultant force completely. But let it suffice merely to have mentioned these things ; for indeed they are too abstruse, y, on account of the enormous Complexity of cares, y the disability of the human intelligence, will not be of any use to us later. 1 62 PHILOSOPHIC NATURALIS THEORIA Demonstratio post- remi theorematis. 2 u. Id quidem facile demonstratur. Si enim AB respectu DC sit perquam exigua, , . ,-, ^ . . /-.T^ i i r i A n anguius AL.D erit perquam exiguus, & a recta CL) ad sensum bitanam sectus : distantias AC, CB erunt ad se invicem ad sensum in ratione sequalitatis, adeoque & vires CL, CK ambae attractive debebunt ad sensum aequales esse inter se, & proinde LCKF ad sensum rhombus, diametro CF ad sensum secante angulum LCK bifariam, quae rhombi proprietas est, & ipsa CF congruente cum CO, ac (ob angulum FCK insensibilem, & CKF ad sensum aequalem duobus rectis) aequali ad sensum binis CK, KF, sive CK, CL, simul sumptis ; quae singulae cum sint quam proxime in ratione reciproca duplicata distantiarum CB, BA ; erunt & eadem, & earum summa ad sensum in ratione reciproca duplicata distantiae CD. suia summa tarum massae, ingens 212. Porro id quidem commune est etiam massulis constantibus quocunque punctorum qu;is mas- _ _ -^ _ ... *** exercet in numero. Mutata alarum combmatione, vis composita a vinbus singulorum agens in oM- p unc tum distans a massula ipsa per intervallum perquam exiguum, nimirum ejusmodi, in remo- in quo curva figurae I circa axem contorquetur, debet mutare plurimum tarn intensitatem ^us^ quse suanij q uam directionem, & fieri utique potest, quod infra etiam in aliquo simpliciore & reciproce, casu trium punctorum videbimus, ut in alia combinatione punctorum massulae pro eadem dlS distantia a medio repulsiones praevaleant, in alia attractiones, in alia oriatur vis in latus ad perpendiculum, ac in eadem constitutione massulae pro diversis directionibus admodum diversae sint vires pro eadem etiam distantia a medio. At in magnis illis distantiis, in quibus singulorum punctorum vires jam attractive sunt omnes, & directiones, ob molem massulae tarn exiguam respectu ingentis distantias, ad sensum conspirant, vis com-[98] -posita ex omnibus dirigetur necessario ad punctum aliquod intra massulam situm, adeoque ad sensum ejus directio erit eadem, ac directio rectae tendentis ad mediam massulam, & aequabitur vis ipsa ad sensum summae virium omnium punctorum constituentium ipsam massulam, adeoque erit attractiva semper, & ad sensum proportionalis in diversis etiam massulis numero punctorum directe, & quadrate distantias a medio massulae ipsius reciproce ; sive generaliter erit in ratione composita ex directa simplici massarum, & reciproca duplicata distantiarum. Multo autem majus erit discrimen in exiguis illis distantiis, si non unicum punctum a massula ilia solicitetur, sed massula alia, cujus vis componatur e singulis viribus singulorum suorum punctorum, quod tamen in massula etiam respectu massulae admodum remotae evanescet, singulis ejus punctis vires habentibus ad sensum aequales, & agentes in eadem ad sensum directione ; unde net, ut vis motrix ejus massulae solicitatae, orta ab actionibus illius alterius remotae massulae, sit ad sensum proportionalis numero punctorum, quas habet ipsa, numero eorum, quae habet altera, & quadrate distantiae, quaecunque sit diversa dispositio punctorum in utralibet, quicunque numerus. Unde necessaria 213. Mirum sane, quantum in applicatione ad Physicam haec animadversio habitura unTf^mitTsTn s ^ usum '> nam inde constabit, cur omnia corporum genera gravitatem acceleratricem gravitate, differ- habeant proportionalem massae, in quam tendunt, & quadrato distantiae, adeoque in a- superficie Terrae aurum, & pluma cum aequali celeritate descendant seclusa resistentia, vim autem totam, quam etiam pondus appellamus, proportionalem praeterea massae suae, adeoque in ordine ad gravitatem nullum sit discrimen, quascunque differentia habeatur inter corpora, quae gravitant, & in quae gravitant, sed ad solam demum massam, & distantiam res omnis deveniat ; at in iis proprietatibus, quae pendent a minimis distantiis, in quibus nimirum fiunt reflexionis lucis, & refractiones cum separatione colorum pro visu, vellicationes fibrarum palati pro gustu, incursus odoriferarum particularum pro odoratu, tremor communicatus particulis aeris proximis, & propagatus usque ad tympanum auriculare pro auditu, asperitas, ac aliae sensibiles ejusmodi qualitates pro tactu, tot cohaesionum tarn diversa genera, secretiones, nutritionesque, fermentationes, conflagrationes, displosiones, dissolutiones. prascipitationes, ac alii effectus Chemici omnes, & mille alia ejusmodi, quae diversa corpora a se invicem discernunt, in iis, inquam, tantum sit discrimen, & vires tarn variae, ac tarn A THEORY OF NATURAL PHILOSOPHY 163 211. The latter theorem can be easily demonstrated. For, if AB is very small compared S[ oof of thelattcr with DC, the angle ACB will be very small, & will be very nearly bisected by the straight line CD. The distances AC, CB will be approximately equal to one another ; & thus the forces CL, CK, which are both attractive, must be approximately equal to one another. Hence, LCKF is approximately a rhombus, & the diagonal CF very nearly bisects the angle LCK, that being a property of a rhombus ; CF will fall along CO, &, because the angle FCK is exceedingly small & CKF very nearly two right angles, CF will be very nearly equal to CK & KF, or CK & CL, taken together. Now each of these are as nearly as possible in the inverse ratio of the square of the distances CB, CA ; & these will be the same, & their sum therefore approximately inversely proportional to the square of the distance DC. 212. Further this theorem is also true in general for little masses consisting of points, J.*i ere is . a hu f e ,. T-,, , jjf it- difference in the whatever their number may be. Ine force compounded from the several forces acting forces which a small on a point, whose distance from the mass is very small, i.e., such a distance as that for which, mass , exerts on , a r .-11 7 11 i T i i small mass that in Fig. i, the curve is twisted about the axis, must be altered very greatly if the combination is very near to it; of the points is altered ; & this is so, both as regards its intensity, & as regards its direction. possibie^nrformH* It may even happen, as will be seen later in the more simple case of three points, that in in the forces due one combination of the points forming the little mass, & for one & the same distance from these'var 6 dlrectf' the mean point, repulsions will preponderate, in another case attractions, & in another as the masses, & case there will arise a perpendicular lateral force. Also for the same constitution of the squaref'of the dls 6 mass, for the same distance from the mean point, there may be altogether different forces tances. for different directions. But, for considerable distances, where the forces due to the several points are now attractive, & their directions practically coincide owing to the dimensions of the little mass being so small compared with the greatness of the distance, the force compounded from all of them will necessarily be directed towards some point within the mass itself ; & thus its direction will be approximately the same as the straight line drawn through the mean centre of the mass ; & the force itself will be equal approximately to the sum of all the forces due to the points composing the little mass. Hence, it will always be an attractive force ; & in different masses, it will be approximately proportional to the number of points directly, & to the square of the distance from the mean centre of the mass inversely. That is, in general, it will be in the ratio compounded of the simple direct ratio of the masses & the inverse duplicate ratio of the distances. Further, the differences will be far greater, in the case of very small distances, if not a single point alone, but another mass, is under the action of the little mass under consideration ; for in this case, the force is compounded from the several forces on each of the points that constitute it ; & yet these differences will also disappear in the case of a mass acted on by a mass considerably remote from it, since each of the points composing it is under the influence of forces that are approximately equal & act in practically the same direction. Hence it comes about that the motive force of the mass acted upon, which is produced by the action of the other mass remote from it, is approximately proportional to the number of points in itself, to the number of points in the other mass, & to the square of the distance between them, whatever the difference in the disposition of the points, or their number, may be for either mass. 213. It is indeed wonderful what great use can be made of this consideration in the He ice we have application of my Theory to Physics ; for, from it it will be clear why all classes of bodies bodies? uniformity have an accelerating gravity, proportional to the mass on which they act, & to the square m ^ e c * se ? f of the distance [inversely] ; & hence that, on the surface of the Earth, a piece of gold & a uniformity in the feather will descend with equal velocity, when the resistance of the air is eliminated. It ca f e of numerous will be clear also that the whole force, which we call the weight, is in addition proportional to the mass itself ; & thus, without exception, there is no difference as regards gravity, no matter what difference there may be between the bodies which gravitate, or towards which they gravitate ; the whole matter reducing finally to a consideration of mass & distance alone. However, for those properties that depend on very small distances, for instance, where we have reflection of light, & refraction with separation of colours, with regard to sight, the titillation of the nerves of the palate, with regard to taste, the inrush of odoriferous particles where smell is concerned, the quivering motion communicated to the nearest particles of the air & propagated onwards till it reaches the drum of the ear for sound, roughness & other such qualities as may be felt in the case of touch, the large number of kinds of cohesion that are so different from one another, secretion, nutrition, fermentation, conflagration, explosion, solution, precipitation, & all the rest of the effects met with in Chemistry, & a thousand other things of the same sort, which distinguish different bodies from one another ; for these, I say, the differences become as great, the forces and the motions become as different, as the differences in the phenomena, 1 64 PHILOSOPHISE NATURALIS THEORIA vis in duo puncta puacti positi in recta jungente ipsa. vei in recta secante hanc bi- fariam, & ad angu- los rectos directa secundum eandem rectam. varii motus, qui tarn varia phaenomena, & omnes specificas tot corporum differentias inducunt, consensu Theoriae hujus cum omni Natura sane admirabili. Sed hsec, quas hue usque dicta sunt ad massas pertinent, & ad amplicationem ad Physicam : interea peculiaria quaedam persequar ex innumeris iis, quas per-[99]-tinent ad diversas leges binorum punctorum agentium in tertium. 214. Si libeat considerare illas leges, quas oriuntur in recta perpendiculari ad AB , T _ . . , _. . . . T- . . ...... r r . , ... , ,. ducta per D, vel m ipsa AB hmc, & inde producta, mprimis facile est videre mud, direc- tionem vis compositas utrobique fore eandem cum ipsa recta sine ulla vi in latus, & sine ulla . . . " , . . . _ , *, . . _ declinatione a recta, quas tendit ad ipsum D, vel ab ipso. Pro recta AB res constat per sese . nam v j res Qjgg q U33 a d bina ea puncta pertinent, vel habebunt directionem eandem, , t * . . r ,. -1 j vel oppositas, jacente ipso tertio puncto in directum cum utroque e pnonbus : unde fit, ut vis composita asquetur summae, vel differentias virium singularum componentium, quae in eadem recta remaneat. Pro recta perpendiculari facile admodum demonstratur. Si enim in fig. 23 recta DC fuerit perpendicularis ad AB sectam bifariam in D, erunt AC, BC aequales inter se. Quare vires, quibus C agitatur ab A, & B, sequales erunt, & proinde vel ambae attractivae, ut CL, CK, vel ambae repulsivae, ut CN, CM. Quare vis composita CF, vel CH, erit diameter rhombi, adeoque secabit bifariam angulum LCK, vel NCM ; quos angulos cum bifariam secet etiam recta DC, ob asqualitatem triangulorum DCA, DCB, patet, ipsas CF, CH debere cum eadem congruere. Quamobrem in hisce casibus evane- scit vis ilia perpendicularis FO, quae in prsecedentibus binis figuris habebatur, ac in iis per unicam aequationem res omnis absolvitur (f), quarum ea, quae ad posteriorem casum pertinet, admodum facile invenitur. exhfbentis casus posterioris. 2I 5' ^egem pro recta perpendiculari rectae jungenti duo puncta, & asque distanti ab utroque exhibet fig. 24, quse vitandae confusionis causa exhibetur, ubi sub numero 24 habetur littera B, sed quod ad ejus constructionem pertinet, habetur separatim, ubi sub num. 24 habetur littera A ; ex quibus binis figuris fit unica ; si puncta XYEAE' censeantur utrobique eadem. In ea X, Y sunt duo materiae puncta, & ipsam XY recta CC* secat bifariam in A. Curva, quae vires compositas ibi exhibet per ordinatas, constructa est ex fig. I, quod fieri potest, inveniendo vires singulas singulorum punctorum, turn vim com- positam ex iis more consueto juxta [100] generalem constructionem numeri 205 ; sed etiam sic facilius idem praestatur ; centro Y intervallo cujusvis abscissae Ad figurae I in- veniatur in figura 24 sub littera A in recta CC' punctum d, sumaturque de versus Y aequalis ordinatae dh figuras i , ductoque ea perpendiculo in CA, erigatur eidem CA itidem perpendicularis dh dupla da versus plagam electam ad arbitrium pro attractionibus, vel versus oppositam, prout ilia ordinata in fig. I attractionem, vel repulsionem expresserit, & erit punctum h ad curvam exprimentem legem virium, qua punctum ubicunque collocatum in recta C'C solicitatur a binis X, Y. de proprietates. Demonstratio facilis est : si enim ducatur dX, & in ea sumatur dc aequalis de, ac compleatur rhombus debc ; patet fore ejus verticem b in recta dA secante angulum XdY bifariam, cujus diameter db exprimet vim compositam a binis de, dc, quae bifariam secaretur a diametro altera ec, & ad angulos rectos, adeoque in ipso illo puncto a ; & dh, dupla da, aequabitur db exprimenti vim, quae respectu A erit attractiva, vel repulsiva, prout ilia dh figurae I fuerit itidem attractiva, vel repulsiva. 2I 7- Porro ex ipsa constructione patet, si centro Y, intervallis AE, AG, AI figuras i inveniantur in recta CAC' hujus figurae positae sub littera B puncta E, G, I, &c, ea fore limites respectu novae curvas ; & eodem pacto reperiri posse limites E', G', Y, &c. ex parte opposita A ; in iis enim punctis evanescente de figuras ejusdem positae sub A, evadit nulla da, & db. Notandum tamen, ibi in figura posita sub B mutari plagam attractivam in (p) Ducta enim LK in Fig. 23. ipsam FC secabit alicubi in I bifariam, W ad angulos rectos ex rhombi natura. Dicatur CD = x, CF = y, DB = a, W erit CB = Vaa + xx, fcf CD = *.CB = Vaa + xx : : CI = Jy.CK = \/aa + xx, quo valore posito in tequatione curvie figura I pro valore ordinata, y vaa + xx ffo valore abscissa, habebitur immediate cfquatio nova per x, y, W constants, qua ejusmodi curvam deUrminabit, A THEORY OF NATURAL PHILOSOPHY 165 O i66 PHILOSOPHIC NATURALIS THEORIA o ^> A THEORY OF NATURAL PHILOSOPHY 167 & all the specific differences between the large number of bodies which they yield ; the agreement between the Theory & the whole of Nature is truly remarkable. But what has so far been said refers to masses, & to the application of the Theory to Physics. Before we come to this, however, I will discuss certain particular cases, out of an innumerable number of those which refer to the different laws concerning the action of two points on a third. 214. If we wish to consider the laws that arise in the case of a straight line drawn The force exerted through D perpendicular to AB, or in the case of AB itself produced on either side, first by. two points on a of all it is easily seen that the direction of the resultant force in either case will coincide the?* straight* 1 line with the line itself without any lateral force or any declination from the straight line which Joining them, or in is drawn towards or away from D. In the case of AB itself the matter is self-evident ; whicifblsects it'at for the forces which pertain to the two points either have the same direction as one another, "8 ht angles. or are opposite in direction, since the third point lies in the same straight line as each of the two former points. Whence it comes about that the resultant force is equal to the sum, or the difference, of the two component forces ; & it will be in the same straight line as they. In the case of the line at right angles, the matter can be quite easily demonstrated. For, if in Fig. 23 the straight line DC were perpendicular to AB, passing through its middle point, then will AC, BC be equal to one another. Hence, the forces, by which C is influenced by A & B, will also be equal ; secondly, they will either be both attractive, as CL, CK, or they will be both repulsive, as CN, CM. Hence the resultant force, CF, or CH, will be the diagonal of a rhombus, & thus it will bisect the angle LCK, or NCM. Now since these angles are also bisected by the straight line DC, on account of the equality of the triangles DCA, DCB, it is evident that CF, CH must coincide with DC. Therefore, in these cases the perpendicular force FO, which was obtained in the two previous figures, will vanish. Also in these cases, the whole matter can be represented by a single equation (?) ; & the one, which refers to the latter case, can be found quite easily. 215. The law in the case of the straight line perpendicular to the straight line joining Construction for the two points, & equally distant from each, is graphically given in Fig. 24 ; to avoid * he curve . e iv ^s . . . ^ 1r .' . -r- i -i i r r .the law in the confusion the curve itself is given in rig. 243, whilst the construction for it is given separately second case. in Fig. 24A. These two figures are but one & the same, if the points X,Y,E,A,E' are supposed to be the same in both. Then, in the figure, X,Y are two points of matter, & the straight line CC' bisects XY at A. The curve, which here gives the resultant forces by means of the ordinates drawn to it, is constructed from that of Fig. i : & this can be done, by finding the forces for the points, each for each, then the force compounded from them in the usual manner according to the general construction given in Art. 205. But the same thing can be more easily obtained thus :, With centre Y, & radius equal to any abscissa Ad in Fig. i, construct a point d in the straight line CC', of Fig. 24A, & mark off de towards Y equal to the ordinate db in Fig. i ; draw ea perpendicular to CA, & erect a perpendicular, dh, to the same line CA also, so that dh = 2ae ; this perpendicular should be drawn towards the side of CA which is chosen at will to represent attractions, or towards the opposite side, according as the ordinate in Fig. i represents an attraction or a repulsion ; then the point h will be a point on the curve expressing the law of forces, with which a point situated anywhere on the line CC' will be influenced by the two points X & Y. . 216. The demonstration is easy. For, if dX is drawn, & in it dc is taken equal to de, Proof of the fore- & the rhombus debc is completed, then it is clear that the point b will fall on the straight gng construction, line dA. bisecting the angle X/Y ; & the diagonal of this rhombus represents the resultant of the two forces de, dc. Now, this diagonal is bisected at right angles by the other diagonal ec, & thus, at the point a in it. Also dh, being double of da, will be equal to db, which expresses the resultant force ; this will be attractive with respect to A, or repulsive, according as the ordinate dh in Fig. I is also attractive or repulsive. 217. Further, from the construction, it is evident that, if with centre Y & radii Further properties respectively equal to AE, AG, AI in Fig. i, there are found in the straight line CAC' of sort. ' Fig. 248 the points E, G, I, &c, then these will be limit-points for the new curve ; & that in the same way limit-points E', G', I', &c. may be found on the opposite side of A. For, since at these points, in Fig. 24A, de vanishes, it follows that da & db become nothing also. Yet it must be noted that, in this case, in Fig. 248, there is a change from the attractive (p) For, if in Fig. 23, LK is drawn, it will cut FC somewhere, in I say ; & it will be at right angles to it on account of the nature of a rhombus. Sup-pose CD = x, CF = y, DB = a ; then CB = \/(a z + x 2 ), W toe have CD (or x) : CB (or ^(a* + x*) = CI (or Jy) : CK, /. CK = y.yV + x*)/2x ; y if this value is substituted in the equation of the curve in Fig. I instead of the ordinate, W ^/ (a z + x z ) for the abscissa, we shall get straightaway a new equation in x, y, \ constants ; & Ms will determine a curve of the kind under consideration. 1 68 PHILOSOPHIC NATURALIS THEORIA repulsivam, & vice versa ; nam in toto tractu CA vis attractiva ad A habet directionem CC', & in tractu AC' vis itidem attractiva ad A habet directionem oppositam C'C. Deinde facile patebit, vim in A fore nullam, ubi nimirum oppositae vires se destruent, adeoque ibi debere curvam axem secare ; ac licet distantiae AX, AY fuerint perquam exiguse, ut idcirco repulsiones singulorum punctorum evadant maximse ; tamen prope A vires erunt perquam exiguae ob inclinationes duarum virium ad XY ingentes, & contrarias ; & si ipsae AY, AX fuerint non majores, quam sit AE figurae I ; postremus arcus EDA erit repulsivus ; secus si fuerint majores, quam AE, & non majores, quam AG, atque ita porro ; cum vires in exigua distantia ab A debeant esse ejus directionis, quam in fig. I requirunt abscissas paullo majores, quam sit haec YA. Postrema crura T/>V,T"/>'V, patet, fore attractiva ; & si in figura I fuerint asymptotica, fore asymptotica etiam hie ; sed in A nullum erit asymptoticum crus. 2l8> At curva C l uae ex hibet in fig. 25 legem virium pro recta CC' transeunte per duo casus prioris. puncta X, Y, est admodum diversa a priore. Ea facile construitur : satis est pro quovis ejus puncto d assumere in fig. I duas abscissae aequales, alteram Yd hujus figurae, alteram Xd ejusdem, & sumere hie db aequalem [101] summae, vel differentiae binarum ordinatarum pertinentium ad eas abscissas, prout fuerint ejusdem directionis, vel contrariae, & earn ducere ex parte attractiva, vel repulsiva, prout ambae ordinatae figurae I, vel earum major, attractiva fuerit, vel repulsiva. Habebitur autem asymptotus bYc, & ultra ipsam crus asymptoticum DE, citra ipsam autem crus itidem asymptoticum dg attractivum respectu A, cui attractivum, sed directionis mutatas respectu CC', ut in fig. superiore diximus, ad partes oppositas A debet esse aliud g'd', habens asymptotum c'V transeuntem per X ; ac utrumque crus debet continuari usque ad A, ubi curva secabit axem. Hoc postremum patet ex eo, quod vires oppositae in A debeant elidi ; illud autem prius ex eo, quod si a sit prope Y, & ad ipsum in infinitum accedat, repulsio ab Y crescat in infinitum, vi, quae provenit ab X, manente finita ; adeoque tam summa, quam differentia debet esse vis repulsiva respectu Y, & proinde attractiva respectu A, quae imminutis in infinitum distantiis ab Y augebitur in infinitum. Quare ordinata ag in accessu ad bYc crescet in infinitum ; unde consequitur, arcum gd fore asymptoticum respectu Yc ; & eadem erit ratio pro a'g', & arcu g'd' respectu b'Xc'. Ejus curvae pro- 219. Poterit autem etiam arcus curvae interceptus asymptotis bYc, b'Xc' sive cruribus at mutata puncto- * mutata ^S> ^'g' secare alicubi axem, ut exhibet figura 26 ; quin immo & in locis pluribus, si nimirum distantia puncto- AY sit satis major, quam AE figurae i, ut ab Y habeatur alicubi citra A attractio, & ab X curva casus*aiterius! repulsio, vel ab X repulsio major, quam repulsio ab Y. Ceterum sola inspectione postremarum duarum figurarum patebit, quantum discrimen inducat in legem virium, vel curvam, sola distantia punctorum X, Y. Utraque enim figura derivata est a figura I, & in fig. 25 assumpta est XY sequalis AE figurae I, in fig. 26 aequalis AI, ejusdem quae variatio usque adeo mutavit figurse genitae ductum ; & assumptis aliis, atque aliis distantiis punc- torum X, Y, aliae, atque aliae curvae novae provenirent, quae inter se collatae, & cum illis, quae habentur in recta CAC' perpendiculari ad XAY, uti est in fig. 24 ; ac multo magis cum iis, quae pertinentes ad alias rectas mente concipi possunt, satis confirmant id, quod supra innui de tanta multitudine, & varietate legum provenientium a sola etiam duo- rum punctorum agentium in tertium dispositione diversa ; ut & illud itidem patet ex sola etiam harum trium curvarum delineatione, quanta sit ubique conformitas in arcu illo attractive TpV, ubique conjuncta cum tanto discrimine in arcu se circa axem contorquente. genera hujus 220. Verum ex tanto discriminum numero unum seligam maxime notatu dignum, Usima! ' g & maximo nobis usui futurum inferius. Sit in fig. 2jC 'AC axis idem, ac in fig. i, & quin- que arcus consequenter accept! alicubi GHI, IKL, LMN, NOP, PQR sint aequales prorsus inter se, ac similes. Ponantur autem bina puncta B', B hinc, & inde ab A in fig. 28 [102] ad intervallum aequale dimidiae amplitudini unius e quinque iis arcubus, uti uni GI, vel IL ; in fig. 29 ad intervallum aequale integrae ipsi amplitudini ; in fig. 30 ad intervallum aequale duplae ; sint autem puncta L, N in omnibus hisce figuris eadem, & quaeratur, quae futura sit vis in quovis puncto g in intervallo LN in hisce tribus posi- tionibus punctorum B', B. A THEORY OF NATURAL PHILOSOPHY 169 1 7 o PHILOSOPHISE NATURALIS THEORIA A THEORY OF NATURAL PHILOSOPHY 171 side to the repulsive side, & vice versa. For along the whole portion CA, the force of attraction towards A has the direction CC', whilst for the portion AC', the force of attraction also towards A has the direction C'C. Secondly, it will be clear,/ seen that the force at A will be nothing ; for there indeed the forces, being equal & opposite, cancel one another, & so the curve cuts the axis there ; & although the distances AX, AY would be very small, & thus the repulsions due to each of the two points would be Immensely great, nevertheless, close to A, the resultants would be very small, on account of the inclinations of the two forces to XY being extremely great & oppositely inclined. Also if AY, AX were not greater than AE in Fig. i, the last arc would be repulsive ; & attractive, if they were greater than AE, but not greater than AG, & so on ; for the forces at very small distances from A must have their directions the same as that required in Fig. I for abscissae that are slightly greater than YA. The final branches TpV, T'p'V will plainly be attractive ; &, if in Fig. i they were asymptotic, they would also be asymptotic in this case ; but there will not be an asymptotic branch at A. 218. But the curve, in Fig. 25, which expresses the law of forces for the straight line Construction fo- CC', when it passes through the points X,Y, is quite different from the one just considered. j| the^aw^'tte It is easily constructed ; it is sufficient, for any point d upon it, to take, in Fig. i, two first case, abscissae, one equal to Yd, & the other equal to Xd ; & then, for Fig. 25, to take dh equal to the sum or the difference of the two ordinates corresponding to these abscissas, according as they are in the same direction or in opposite directions ; &, according as each ordinate, or the greater of the two, in Fig. i, is attractive or repulsive, to draw dh on the attractive or repulsive side of CC'. Moreover there will be obtained an atymptote bYc ; on the far side of this there will be an asymptotic branch DE, & on the near side of it there will also be an asymptotic branch dg, which will be attractive with respect to A ; & with respect to this part, there must be another branch g'd', which is attractive but, since the direction with regard to CC' is altered, as we mentioned in the case of the preceding figure, falling on the opposite side of CC' ; this has an asymptote c'b' passing through X. Also each branch must be continuous up to the point A, where it cuts thVaxis. This last fact is evident from the consideration that the equal & opposite forces at A must cancel one another ; & the former is clear from the fact that, if a is very near to Y, & approaches indefinitely near to it, the repulsion due to Y increases indefinitely, whilst the force due to X remains finite. Thus, both the sum & the difference must be repulsive with respect to Y, & therefore attractive with respect to A ; & this, as the distance from Y is diminished indefinitely, will increase indefinitely. Hence the ordinate ag, when approaching bYc, increases indefinitely : & it thus follows that the arc gd will be asymptotic with respect to Yc ; & the reasoning will be the same for a'g', & the arc g'd', with respect to b'Xc', 219. Again, it is even possible that the arc intercepted between the asymptotes bYc, The properties of b'Xc', i.e., between the branches dg, d'g', to cut the axis somewhere, as is shown in Fig. 26 ; encescor^espo'n'dSg nay rather, it may cut it in more places than one, for instance, if AY is sufficiently greater to changed dis- than AE in Fig. i ; so that, at some place on the near side of A, there is obtained an attraction p^s ^"clrnpari 6 from the point Y & a repulsion from the point X, or a repulsion from X greater than the son with the curve repulsion fiom Y. Besides, by a mere inspection of the last two figures, it will be evident other case. in the how great a difference in the law of forces, & the curve, may be derived from the mere distance apart of the points X & Y. For both figures are derived from Fig. I, &, in Fig. 25, XY is taken equal to AE in Fig. i , whilst, in Fig. 26, it is taken equal to AI of Fig. i ; & this variation alone has changed the derived figure to such a degree as is shown. If other distances, one after another, are taken for the points X & Y, fresh curves, one after the other, will be produced. If these are compared with one another, & with those that are obtained for a straight line CAC' perpendicular to XAY, like the one in Fig 24, nay, far more, if they are compared with those, referring to other straight lines, that can be imagined, will sufficiently confirm what has been said above with regard to the immense number & variety of the laws arising from a mere difference of disposition of the two points that act on the third. Also, from the drawing of merely these three curves, it is plainly seen what great uniformity there is in all cases for the attractive arc TpV, combined always with a great dissimilarity for the arc that is twisted about the axis. _ 220. But I will select, from this great number of different cases, one which is worth T ^ Tee classes of notice in a high degree, which also will be of the greatest service to us later. In Fig. 27, we u let CAC' be the same axis as in Fig. i, & let the five arcs, GHI, IKL, LMN, NOP, PQR taken consecutively anywhere along it, be exactly equal & like one another. Moreover, in Fig. 28, let the two points B & B', one on each side of A, be taken at a distance equal to half the width of one of these five arcs, i.e., half of the one GL, or LI ; in Fig. 29, at 3. distance equal to the whole of this width ; &, in Fig. 30, at a distance equal to double the width ; also let the points L,N be the same in all these figures. It is required to find the force at any point g in the interval LN, for these three positions of the points B & B'. I 7 2 PHILOSOPHISE NATURALIS THEORIA Determinatip vis compositaa in iis- dem. 221. Si in Fig. 27 capiantur hinc, & inde ab ipso g intervalla sequalia intervallis AB', AB reliquarum trium figurarum ita, ut ge, gi respondeant figurae 28 ; gc, gm figures 29 ; ga, go figurae 30 ; patet, intervallum ei fore aequale amplitudini LN, adeoque Le, Ni aequales fore dempto communi Lz, sed puncta e, i debere cadere sub arcus proximos directionum contrariarum ; ob arcuum vero aequalitatem fore aequalem vim ef vi contrariae il, adeoque in fig. 28 vim ab utraque compositam, respondentem puncto g, fore nullam. At quoniam gc, gm integrae amplitudini aequantur ; cadent puncta c, m sub arcus IKL, NOP, conformes etiam directione inter se, sed directionis contraries respectu arcus LMN, eruntque asquales wzN, cl ipsi gL, adeoque attractiones mn, cd, & repulsioni gh aequales, & inter se ; ac idcirco in figura 29 habebitur vis attractiva gh composita ex iis binis dupla repulsivae figurae 27. Demum cum ga, go sint sequales duplae amplitudini, cadent puncta a, o sub arcus GHI, PQR conformis directionis inter se, & cum arcu LMN, eruntque pariter binae repulsiones ab, op aequales repulsioni gh, & inter se. Quare vis ex iis com- positae pro fig. 30 erit repulsio gh dupla repulsionis gh figurae 27, & aequalis attraction! figurae 29. vhn'in tractu*' 0116 222< ^^ igi tur j am patet, loci geometric! exprimentis vim compositam, qua bina tinuo nuiiam, in puncta B', B agunt in tertium, partem, quae respondet intervallo eidem LN, fore in prima aha attractionem, e tribus eorum positionibus propositis ipsum axem LN, in secunda arcum attractivum in aha repulsionem, T , ,,., . . , . i i , r manente distantia ; LMN, in tertia repulsivum, utroque reccdente ab axe ubique duplo plus, quam in fig. Ph y sica 27 ; ac pro quovis situ puncti g in toto intervallo LN in primo e tribus casibus fore prorsus nullam, in secundo fore attractionem, in tertio repulsionem aequalem ei, quam bina puncta B', B exercerent in tertium punctum situm in g, si collocarentur simul in A, licet in omnibus hisce casibus distantia puncti ejusdem g a medio systematis eorundem duorum punctorum, sive a centre particulae constantis iis duobus punctis sit omnino eadem. Possunt autem in omnibus hisce casibus puncta B', B esse simul in arctissimis limitibus cohaesionis inter se,- adeoque particulam quandam constantis positionis constituere. Aequalitas ejusmodi accurata inter arcus, & amplitudines, ac limitum distantias in figura I non dabitur uspiam ; cum nullus arcus curvae derivatae utique continuae, deductae nimirum certa lege a curva continua, possit congruere accurate cum recta ; at poterunt ea omnia ad sequalitatem accedere, quantum [103] libuerit ; poterunt haec ipsa discrimina haberi ad sensum per tractus continues aliis modis multo adhuc pluribus, immo etiam pluribus in immensum, ubi non duo tantummodo puncta, sed immensus eorum numerus constituat massulas, quae in se agant, & ut in hoc simplicissimo exemplo deprompto e solo trium punctorum systemate, multo magis in systematis magis compositas, & plures idcirco variationes admit- tentibus, in eadem centrorum distantia, pro sola varia positione punctorum componentium massulas ipsas vel a se mutuo repelli, vel se mutuo attrahere, vel nihil ad sensum agere in se invicem. Quod si ita res habet, nihil jam mirum accidet, quod quaedam substantial inter se commixtse ingentem acquirant intestinarum partium motum per effervescentiam, & fermentationem, quas deinde cesset, particulis post novam commixtionem respective quiescentibus ; quod ex eodem cibo alia per secretionem repellantur, alia in succum nutrititium convertantur, ex quo ad eandem prseterfluente distantiam alia aliis partibus solidis adhaereant, & per alias valvulas transmittantur, aliis libere progredientibus. Sed adhuc multa supersunt notatu dignissima, quae pertinent ad ipsum etiam adeo simplex trium punctorum systema. Alius casus vis nul- lius trium puncto- rum positorum in directum e x dis- tantiis limitum : tres alii in quorum binis vis nulla ex elisione contrari- arum. 223. Jaceant in figura 31 tria puncta A,D,B, in directum : ea poterunt respective quiescere, si omnibus mutuis viribus careant, quod fieret, si tres distantiae AD, DB, AB omnes essent distantiae limitum ; sed potest haberi etiam quies respectiva per elisionem contrariarum virium. Porro virium mutuarum casus diversi tres esse poterunt : vel enim punctum medium D ab utroque extremorum A, B attrahitur, vel ab utroque repellitur, vel ab altero attrahitur, ab altero repellitur. In hoc postremo casu, patet, non haberi quietem respectivam ; cum debeat punctum medium moveri versus extremum attrahens recedendo simul ab altero extremo repellente. In reliquis binis casibus poterit utique A THEORY OF NATURAL PHILOSOPHY 173 c' C B'A B FIG. 28. C' B' A B FIG. 29. cV 1 I g C B' T B L N FIG. 30. '74 PHILOSOPHIC NATURALIS THEORIA H R C C' FIG. 27. L n. ," 5 B'A B FIG. 28. B' A B B 7 FIG .29. B L N FIG. 30. A THEORY OF NATURAL PHILOSOPHY 175 221. If, in Fig. 27, we take, on either side of this point g, intervals that are equal to Determination of the intervals AB', AB of the other three figures ; so that ge, gi correspond to Fig. 28 ; gc, gm to Fig. 29 ; & ga, go to Fig. 30 ; then it is plain that the interval ei will be equal to the width LN, & thus, taking away the common part Lz, we have L & Ni equal to one another, but the points e & i must fall under successive arcs of opposite directions. Now, on account of the equality of the arcs, the force ef will be equal to the opposite force il ; thus, in Fig. 28, the force compounded from the two, corresponding to the point g, will be nothing. Again, in Fig. 29, since gc, gm are each equal to the whole width of an arc, the points c & m fall under arcs IKL, NOP, which lie in the same direction as one another, but in the opposite direction to the arc LMN. Hence, mN, c\ will be equal to gL ; & thus the attractions mn, cd will be equal to the repulsion gb, & to one another. Therefore, in Fig. 29, we shall have an attractive force, compounded of these two, which is double of the repulsive force in Fig. 27. Lastly, in Fig. 30, since ga, go are equal to double the width of an arc, the points a & o will fall beneath arcs GHI, PQR, lying in the same direction as one another, & as that of the arc LMN as well. As before, the two repulsions, ab, op will be equal to the repulsion gb, & to one another. Hence, in Fig. 30, the force compounded from the two of them will be a repulsion gh which is double of the repulsion gh in Fig. 27, & equal to the attraction in Fig. 29. 222. Therefore, from the preceding article, it is now evident that the part of the in one arrange- geometrical locus representing the resultant force, with which two points B', B act " '* region upon a third, corresponding to the same interval LN, will be the axis LN itself in the first no force at ail, in of the three stated positions of the points ; in the second position it will be an attractive attraction" 5 * ^a arc LMN, & in the third a repulsive arc ; each of these will recede from the axis at all third a repulsion points along it to twice the corresponding distance shown in Fig. 27. So, for any position maining^constant of the point g in the whole interval LN, the force will be nothing at all in the first of the this result is of the three cases, an attraction in the second, & a repulsion in the third. This latter will be -physics. l equal to that which the two points B', B would exert on the third point, if they were both situated at the same time at the point A. And yet, in all these three cases, the distance of the point g under consideration remains absolutely the same, measured from the centre of the system of the same two points, or from the mean centre of a particle formed from them. Moreover, in all three cases, the points B',B may be in the positions defining the strongest limits of cohesion with regard to one another, & so constitute a particle fixed in position. Now we never can have such accurate equality as this between the arcs, the widths, & the distances of the limit-points ; for no arc of the derived curve, which is every- where continuous because it is obtained by a given law from a continuous curve, can possibly coincide accurately with a straight line ; but there could be an approximation to equality for all of them, to any degree desired. The same distinctions could be obtained, approximately for continuous regions in very many more different ways, nay the number of ways is immeasurable ; in which the number of points constituting the little masses is not two only, but a very large number, acting upon one another ; &, as in this very simple case derived from a consideration of a single system of three points, so, much more in systems that are more complex & on that account admitting of more variations, corresponding to a single variation of the points composing the masses, whilst the distance between the masses themselves remains the same, there may be either mutual repulsion, mutual attraction, or no mutual action to any appreciable extent. But, that being the case, there is nothing wonderful in the fact that certain substances, when mixed together, acquire a huge motion of their inmost parts, as in effervescence & fermentation ; this motion ceasing & the particles attaining relative rest after rearrangement. There is nothing wonderful in the fact that from the same food some things are repelled by secretion, whilst others are converted into nutritious juices ; & that from these juices, though flowing past at exactly the same distances, some things adhere to some solid parts & some to others ; that some are transmitted through certain little passages, some through others, whilst some pass along uninterruptedly. However, there yet remain many things with regard to this ever so simple system of three points ; & these are well worth our attention. 223. In Fig. 31, let A,D,B be three points in a straight line. These will be at rest Another instance i 11 i n i ., , of no force in the with regard to one another if they lack all mutual forces ; & this would be the case, it the case of three points three distances AD, DB, AB were all distances corresponding to limit-points. In addition, sit V?;, t f. d i " a i 111 i i ,...*- in f ft. straight line at the relative rest could be obtained owing to elimination of equal & opposite iorces. .further, distances corre- there will be three different cases with regard to the mutual forces. For, either the middle spending to Hmit- . -i r i i > T> 11 ii i r V. points. Ihree point D is attracted by each of the outside points A & B, or it is repelled by each of them, others, in two of or it is attracted by one of them & repelled by the other. In the last case, it is evident ^^res^tenl'farce that relative rest could not obtain ; for the middle point must then be moved towards the arises from an eii- outside point that is attracting it, & recede from the other outside point which is repelling it at the same time. But in the other two cases, it is at least possible that there may be PHILOSOPHIC NATURALIS THEORIA In eorum altero nisus ad recuper- andam positionem, in altero ad magis ab ea recedendum, si incipiant inde removeri. res haberi : nam vires attractive, vel repulsive, quas habet medium punctum, possunt esse aequales ; turn autem extrema puncta debebunt itidem attrahi a medio in primo casu, repelli in secundo ; quae si se invicem e contrario aeque repellant in casu primo, attrahant in secundo ; poterunt mutuse vires elidi omnes. 224. Adhuc autem ingens est discrimen inter hosce binos casus. Si nimirum puncta ilia a directione rectae lineae quidquam removeantur, ut nimirum medium punctum D distet jam non nihil a recta AB, delatam in C, in secundo casu adhuc magis sponte recedet inde, & in primo accedet iterum ; vel si vi aliqua externa urgeatur, conabitur recuperare positionem priorem, & ipsi urgenti vi resi- stet. Nam binae repulsiones CM, CN adhuc habebuntur in secundo casu in ipso primo recessu a D (licet ese mutatis jam satis distan- tiis BD, AD inBC, AC, evadere possint at- tractiones) & vim com-[i opponent direc- tam per CH contrariam directioni tendenti ad rectam AB. At in primo casu habebuntur attractiones CL, CK, quae component vim CF directam versus AB, quo casu attractio AP cum repulsione AR, et attractio BV, cum repulsione BS component vires AQ, BT, quibus puncta A, B ibunt obviam puncto C redeunti ad rectam transituram per illud T"V /"* FIG. 31. Theoria generalior indicata : t r i u m punctorum jacen- tium in directum : vis maxima ad conservandam dis- tantiam. M R A B S punctum E, quod est in triente rectae DC, & de quo supra mentionem fecimus num. 205. 225. Haec Theoria generaliter etiam non rectilineae tantum, sed & cuivis position! trium massarum applicari potest, ac applicabitur infra, ubi etiam generale simplicissimum, ac fcecundissimum theorema eruetur pro comparatione virium inter se : sed hie interea evolvemus nonnulla, quae pertinent ad simpliciorem hunc casum trium punctorum. In- primis fieri utique potest, ut ejusmodi tria puncta positionem ad sensum rectilineam retineant cum prioribus distantiis, utcunque magna fuerit vis, quae ilia dimovere tentet, vel utcunque magna velocitas impressa fuerit ad ea e suo respectivo statu deturbanda. Nam vires ejusmodi esse possunt, ut tarn in eadem directione ipsius rectas, quam in directione ad earn perpendicular!, adeoque in quavis obliqua etiam, quae in eas duas resolvi cogitatione potest, validissimus exurgat conatus ad redeundum ad priorem locum, ubi inde discesserint puncta. Contra vim impressam in directione ejusdem rectae satis est, si pro puncto medio attractio plurimum crescat, aucta distantia ab utrolibet extreme, & plurimum decrescat eadem imminuta ; ac pro utrovis puncto extreme satis est, si repulsio decrescat plurimum aucta distantia ab extreme, & attractio plurimum crescat, aucta distantia a medio, quod secundum utique fiet, cum, ut dictum est, debeat attractio medii in ipsum crescere, aucta distantia. Si haec ita se habuerint, ac vice versa ; differentia virium vi extrinsecae resistet, sive ea tenet contrahere, sive distrahere puncta, & si aliquod ex iis velocitatem in ea directione acquisiverit utcunque magnam, poterit differentia virium esse tanta, ut extinguat ejusmodi respectivam velocitatem tempusculo, quantum libuerit, parvo, & post percursum spatiolum, quantum libuerit, exiguum. Quid ubi vis exter- & virgae flexiiis. 226. Quod si vis urgeat perpendiculariter, ut ex.gr. punctum medium D moveatur per rectam DC perpendicularem ad AB ; turn vires CK, CL possunt utique esse ita validae, ut vis composita CF sit post recessum, quantum libuerit, exiguum satis magna ad ejusmodi vim elidendam, vel ad extinguendam velocitatem impressam. In casu vis, quas constanter urgeat, & punctum D versus C, & puncta A, B ad partes oppositas, habebitur inflexio ; ac in casu vis, quae agat in eadem directione rectae jungentis puncta, habebitur contractio, seu distractio ; sed vires resistentes ipsis poterunt esse ita validae, ut & inflexio, & contractio, vel distractio, sint prorsus insensibiles ; [105] ac si actione externa velocitas imprimatur punctis ejusmodi, quae flexionem, vel contractionem, aut distractionem inducat, turn ipsa puncta permittantur sibi libera ; habebitur oscillatio quasdam, angulo jam in alteram plagam obverso, jam in alteram oppositam, ac longitudine ejus veluti virgae constantis iis tribus punctis jam aucta, jam imminuta, fieri poterit ; ut oscillatio ipsa sensum omnem effugiat, quod quidem exhibebit nobis ideam virgae, quam vocamus rigidam, & solidam, contractionis nimirum, & dilatationis incapacem, quas proprietates nulla virga in Natura [The reader should draw a more general figure for Art. 224 & 227, taking AD, DB unequal and CD not at right angles to AB.] A THEORY OF NATURAL PHILOSOPHY 177 relative rest ; for the attractive, or repulsive, forces which are acting on the middle point may be equal. But then, in these cases, the outside points must be respectively attracted, or repelled by the middle point ; & if they are equally & oppositely repelled by one another in the first case, & attracted by one another in the second case, then it will be possible for all the mutual forces to cancel one another. 224. Further, there is also a very great difference between these two cases. For in one of these instance, if the points are moved a small distance out of the direct straight line, so that endeavour 6 towards the middle point D, say, is now slightly off the straight line AB, being transferred to C, a recovery of posi- then, if left to itself, it will recede still further from it in the first case, & will approach t^warts^iurther it once more in the second case. Or, if it is acted on by some external force, it will endeavour recession from it, to recover its position & will resist the force acting on it. For two repulsions, CM, CN, moved o will at first be obtained in the second case, at the first instant of motion from the position position. D ; although indeed these may become attractions when the distances BD, AD are sufficiently altered into the distances BC, AC. These will give a resultant force acting along CH in a direction away from the straight line AB. But in the first case we shall have two attractions CL, CK ; & these will give a force directed towards AB. In this case, the attraction AP combined with the repulsion AR, & the attraction BV combined with the repulsion BY, will give resultant forces, AQ, BT, under the action of which the points A,B will move in the opposite direction to that of the point C, as it returns to the straight line passing through that point E, which is a third of the way along the straight line DC, of which mention was made above in Art. 205. 225. This Theory can also be applied more generally, to include not only a position Enunciation of a of the three points in a straight line but also any position whatever. This application more general theory will be made in what follows, where also a general theorem, of a most simple & fertile nature ly^g ^ a straight will be deduced for comparison of forces with one another. But for the present we will line ; possibility of consider certain points that have to do with this more simple case of three points. First tendSg^conser^ of all, it may come about that three points of this kind may maintain a position practically vation of distance, in a straight line, no matter how great the force tending to drive them from it may be, or no matter how great a velocity may be impressed upon them for the purpose of disturbing them from their relative positions. For there may be forces of such a kind that both in the direction of the straight line, & perpendicular to it, & hence in any oblique direction which may be mentally resolved into the former, there may be produced an extremely strong endeavour towards a return to the initial position as soon as the points had departed from it. To counterbalance the force impressed in the direction of the same straight line itself, it is sufficient if the attraction for the middle point should increase by a large amount when the distance from either of the outside points is increased, & should be decreased by a large amount if this distance is decreased. For either of the outside points it is sufficient if the repulsion should greatly decrease, as the distance is increased, from the outside point, and the attraction should greatly increase, as the distance is increased, from the middle point ; & this second requirement will be met in every case, since, as has been said, and attraction on it of the middle point will necessarily increase when the distance is increased. If matters should turn out to be as stated, or vice versa, then the difference of the forces will resist the external force, whether it tries to bring the points together or to drive them apart ; & if any one of them should have acquired a velocity in the direction of the straight line, no matter how great, there will be a possibility that the difference of the forces may be so great that it will destroy any relative velocity of this kind, in any interval of time, no matter how short the time assigned may be ; & this, after passing over any very small assigned space, no matter how small. 226. But if the force acts perpendicularly, so that, for instance, the point D moves what happens if along the line DC perpendicular to AB, then the forces CK, CL, can in any case be so ^es n^tTct ah^g strong that the resultant force CF may become, after a recession of any desired degree the straight line ; of smallness, large enough to eliminate any force of this kind, or to destroy any impressed velocity. In the case of a force continually urging the point D towards C, & the points A & B in the opposite direction, there will be some bending ; & in the case of a force acting in the same direction as the straight line joining the two points, there will be some contraction or distraction. But the forces resisting them may be so strong that the bending, the contraction, or the distraction will be altogether inappreciable. If by external action a velocity is impressed on points of this kind, & if this induces bending, contraction or distraction, & if the points are then left to themselves, there will be produced an oscillation, in which the angle will jut out first on one side & then on the other side ; & the length of, so to speak, the rod consisting of the three points will be at one time increased & at another decreased ; & it may possibly be the case that the oscillation will be totally unappreciable ; & this indeed will give us the idea of a rod, such as we call rigid & solid, incapable of being contracted or dilated ; these properties are possessed by no rod in Nature perfectly N I 7 8 PHILOSOPHLE NATURALIS THEORIA habet accurata tales, sed tantummodo ad sensum. Quod si vires sint aliquanto debiliores, turn vero & inflexio ex vi externa mediocri, & oscillatio, ac tremor erunt majores, & jam hinc ex simplicissimo trium punctorum systemate habebitur species quaedam satis idonea ad sistendum animo discrimen, quod in Natura observatur quotidie oculis. inter virgas rigidas, ac eas, quae sunt flexiles, & ex elasticitate trementes. Systemate inflexo 227. Ibidem si binse vires, ut AQ, BT fuerint perpendiculares ad AB, vel etiam vls r ^ncti^medii utcunque parallels inter se, tertia quoque erit parallela illis, & aequalis earum summae, contraria extremis, sed directionis contrariae. Ducta enim CD parallela iis, turn ad illam KI parallela BA, " ' erit ob CK > y B sequales, triangulum CIK aequale simili BTV, sive TBS, adeoque CI squalls BT, IK aequalis BS, sive AR, vel QP. Quare si sumpta IF aequali AQ ducatur KF ; erit triangulum FIK aequale AQP, ac proinde FK aequalis, & parallela AP, sive LC, & CLFK parallelogrammum, ac CF, diameter ipsius, exprimet vim puncti C utique parallelam viribus AQ, BT, & asqualem earum summae, sed directionis contrariae. Quoniam vero est SB ad BT, ut BD ad DC ; ac AQ ad AR, ut DC ad DA ; erit ex aequalitate perturbata AQ ad BT, ut BD ad DA, nimirum vires in A, & B in ratione reciproca distantiarum AD, DB a recta CD ducta per C secundum directionem virium. & summae. Postremum theo- rema generate, ubi etiam tria puncta non jaceant in di- rectum. Equilibrium trium punctorum non in directum jacentium impossible sine vi externa, nisi sint in distantiis limi- tum : cum iis qui nisus ad retinen- dam formam syste- matis. 228. Ea, quas hoc postremo numero demonstravimus, aeque pertinent ad actiones mutuas trium punctorum habentium positionem mutuam quamcunque, etiam si a rectilinea recedat quantumlibet ; nam demonstratio generalis est : sed ad massas utcunque inaequales, & in se agentes viribus etiam divergentibus, multo generalius traduci possunt, ac traducentur inferius, & ad aequilibrii leges, & vectem, & centra oscillationis ac percussionis nos deducent. Sed interea pergemus alia nonnulla persequi pertinentia itidem ad puncta tria, quae in directum non jaceant. 229. Si tria puncta non jaceant in directum, turn vero sine externis viribus non poterunt esse in aequilibrio ; nisi omnes tres distantiae, quae latera trianguli constituunt, sint dis- tantiae limitum figurae i. Cum enim vires illae mutuae non habeant [106] directiones oppositas ; sive unica vis ab altero e reliquis binis punctis agat in tertium punctum, sive ambae ; haberi debebit in illo tertio puncto motus, vel in recta, quae jungit ipsum cum puncto agente, vel in diagonali parallelogrammi, cujus latera binas illas exprimant vires. Quamobrem si assumantur in figura I tres distantiae limitum ejusmodi, ut nulla ex iis sit major reliquis binis simul sumptis, & ex ipsis constituatur triangulum, ac in singulis angu- lorum cuspidibus singula materiae puncta collocentur ; habebitur systema trium punctorum quiescens, cujus punctis singulis si imprimantur velocitates aequales, & parallelae ; habebitur systema progrediens quidem, sed respective quiescens ; adeoque istud etiam systema habebit ibi suum quemdam limitem, sed horum quoque limitum duo genera erunt : ii, qui orientur ab omnibus tribus limitibus cohaesionis, erunt ejusmodi, ut mutata positione, conentur ipsam recuperare, cum debeant conari recuperare distantias : ii vero, in quibus etiam una e tribus distantiis fuerit distantia limitis non cohaesionis, erunt ejusmodi, ut mutata positione : ab ipsa etiam sponte magis discedat systema punctorum eorundem. Sed consideremus jam casus quosdam peculiares, & elegantes, & utiles, qui hue pertinent. Eiegans theoria 230. Sint in fig. 32 tria puncta A,E,B ita collocata, ut tres distantise AB, AE, BE sint ratto eiupsTs binis distantiae limitum cohaesionis, & postremae duae aiiis occupantibus sint aequales. Focis A, B concipiatur ellipsis transiens per E, cujus axis transversus sit FO, conjugatus EH, centrum D : sit in fig. I AN aequalis semiaxi transverse hujus DO, sive BE, vel AE, ac sit DB hie minor, quam in fig. I amplitude proximorum arcuum LN, NP, & sint in eadem fig. i arcus ipsi NM, NO similes, & aequales ita, ut ordinatae uy, zt, aeque distantes ab N, sint inter se aequales. Inprimis si punctum materiae sit hie in E ; nullum ibi habebit vim, cum AE, BE sint aequales distantiae AN limitis N figurae I ; ac eadem est ratio pro puncto collocate in H. Quod si fuerit in O, itidem erit in aequilibrio. Si enim assumantur in fig. I Az, AM aequales hisce BO, AO ; erunt Nz, foco : vis nulla in verticibus axium. illius aequales DB, DA hujus, adeoque & inter se. Quare & vires illius zt, uy erunt aequales inter se, quae cum pariter oppositae directionis sint, se mutuo elident ; ac eadem ratio est pro collocatione in F. Attrahetur hie utique A, & repelletur B ab O ; sed si limes, qui respondet distantiae AB, sit satis validus ; ipsa puncta nihil ad sensum discedent a focis A THEORY OF NATURAL PHILOSOPHY 179 accurately, but only approximately. But if the forces are somewhat more feeble, then indeed, under the action of a moderate external force, the bending, the oscillation & the vibration will all be greater ; & from this extremely simple system of three points we now obtain several kinds of cases that are adapted to giving us a mental conception of the differences, that meet our eyes every day in Nature, between rigid rods & those that are flexible & elastically tremulous. 227. At the same time, if the two forces, represented by AQ, BT, were perpendicular In * s y stem dis - A T> n i i_ ^i. ^L- j f u f i n i torted by parallel to AB, or parallel to one another in any manner, then the third force would also be parallel forces the force on to them, equal to their sum, but in the opposition direction. For, if CD is drawn parallel the middle point is to the forces, & KI parallel to BA to meet CD in I, then, since CK & VB are equal to direction t thaTof one another, the triangle CIK will be equal to the similar triangle BTV, or to the triangle the outside forces, TBS ; & therefore CI will be equal to BT, IK to BS or AR or QP. Hence if IF is taken sumT* equal to AQ & KF is drawn, then the triangle FIK will be equal to AQP, & thus FK will be equal parallel to AP or LC, CLFK will be a- parallelogram, & its diagonal CF will represent the force for the point C, in every case parallel to the forces AQ, BT, & equal to their sum, but opposite in direction. But, because SB : BT : : BD : DC, & AQ : AR : : DC : DA ; hence, ex cequali we have AQ : BT : : BD : DA, that is to say, the forces on A & B are in the inverse ratio of the distances AD & DB, drawn from the straight line CD in the direction of the forces. 228. What has been proved in the last article applies equally to the mutual actions The last theorem in of three points having any relative positions whatever, even if it departs from a rectilinear fhe tiTree^point^do position to any extent you may please. For the demonstration is general ; &, further, the not He in a straight results can be deduced much more generally for masses that are in every manner unequal, line- & that act upon one another even with diverging forces ; & they will be thus deduced later ; & these will lead us to the laws of equilibrium, the lever, & the centres of oscillation & percussion. But meanwhile we will go straight on with our consideration of some matters relating in the same manner to three points, which do not lie in a straight line. 229. If the three points do not lie in a straight line, then indeed without the presence Equilibrium of of an external force they cannot be in equilibrium ; unless all three distances, which form ^o^no* ^ in^a the sides of the triangle, are those corresponding to the limit-points in Fig. I. For, since straight line isim- the mutual forces do not have opposite directions, either a single force from one of the F^^^^^ 1 ^ ^ . i i i i TT i iri presence 01 an remaining two points acts on the third, or two such forces. Hence there must be for that external force, third point some motion, either in the direction of the straight line joining it to the acting "? B less 3l . th H is l^ ' o j o o are at distances point, or along the diagonal of the parallelogram whose sides represent those two forces, corresponding to Therefore, if in Fig. i we take three limit-distances of such a kind, that no one of them is 8 ' in his greater than the other two taken together, & if from them a triangle is formed & at each case, to 'conserve vertical angle a material point is situated, then we shall have a system of three points at rest, sygte^" 11 of the If to each point of the system there is given a velocity, and these are all equal & parallel to one another, we shall have a system which moves indeed, but which is relatively at rest. Thus also that system will have a certain limit of its own ; moreover, of such limits there are also two kinds. Namely, those that arise from all three limit-points being those of cohesion which will be such that, if the relative position is altered, they will strive to recover it ; for they are bound to try to restore the distances. Secondly, those in which one of the three distances corresponds to a limit-point of non-cohesion, which will be such that, if the relative position is altered, the system will of its own accord depart still more from it. However, let us now consider certain special cases, that are both elegant & useful, for which this is the appropriate place. 230. In Fig. 32, let the three points A,E,B be so placed that the three distances AB, An elegant theory AE, BE correspond to limit-points of cohesion, & let the two last be equal to one another. i^the'periineter of Suppose that an ellipse, whose foci are A & B, passes through E ; let the transverse axis of an ellipse, each of this be FO, & the conjugate axis EH, & the centre D. In Fig. i, let AN be equal to behVpiaced irTa the transverse semiaxis DO of Fig. 32, that is equal to BE or AE ; also in the latter figure focus ; no force at let DB be less than the width of the successive arcs LN, NP of Fig. i ; also, in Fig. i, let en the arcs NM, NO be similar & equal, so that the ordinates uy, zt, which are equidistant from N, are equal to one another. Then, first of all, if in Fig. 32, the point of matter is situated at E, there will be no force upon it ; for AE, BE are equal to the distance AN of the limit-point N in Fig. i ; & the argument is the same for a point situated at H. Further, if it is at O, it will in like manner be in equilibrium. For, if in Fig. i we take Az, Au equal to AO, BO of Fig. 32, then Nz, NM of the former figure will be equal to DB, DA of the latter ; & thus equal also to one another. Hence also the forces in that figure, zt & uy, will be equal to one another ; & since they are likewise opposite in direction, they will cancel one another ; & the argument is the same for a point situated at F. Here in every case A is attracted & B is repelled from O ; but if the limit-point, which corresponds to the distance AB is strong enough, the points will not depart to any appreciable extent i8o PHILOSOPHISE NATURALIS THEORIA In reliquis puncti perimetri vis direc- ta per ipsam peri- metrum versus ver- tices axis conju- gati. Analogia verticum binorum axium cum limitibus cur- vae virium. Quando limites contrario m o d o positi : casus ele- gantissimi alterna- tionis p 1 u r i u m limitum in peri- metro ellipseos. N ellipseos, in quibus fuerant collocata, vel si debeant discedere ob limitem minus validum, considerari poterunt per externam vim ibidem immota, ut contemplari liceat solam relationem tertii puncti ad ilia duo. 231. Manet igitur immotum, ac sine vi, punctum collocatum tarn in verticibus axis con- jugati ejus ellipseos, quam in verticibus axis transversi ; & si ponatur in quovis puncto C [107] perimetri ejus ellipseos, turn ob AC, CB simul aequales in ellipsi axi transverse, sive duplo semiaxi DO ; erit AC tanto longior, quam ipsa DO, quanto BC brevior ; adeoque si jam in fig. I sint AM, Az aequales hisce AC, BC ; habe- buntur ibi utique uy, zt itidem aequales inter se. Quare hie attractio CL sequabitur repulsioni CM, & LIMC erit rhombus, in quo inclinatio 1C secabit bifariam angulum LCM ; ac proinde si ea utrinque producatur in P, & Q ; angulus ACP, qui est idem, ac LCI, erit aequalis angulo BCQ, qui est ad verticem oppositus angulo ICM. Quse cum in ellipsi sit notissima proprietas tangentis relatae ad focos ; erit ipsa PQ tangens. Quamobrem dirigetur vis puncti C in latus secundum tangentem, sive secundum directionem arcus elliptici, atque id, ubicunque fuerit punctum in perimetro ipsa, versus verticem propiorem axis conjugati, & sibi relictum ibit per ipsam perimetrum versus eum verticem, nisi quatenus ob vim centrifugam motum non nihil adhuc magis incurvabit. 232. Quamobrem hie jam licebit contemplari in hac curva perimetro vicissitudinem limitum prorsus analogorum limitibus cohaesionis, & non cohaesionis, qui habentur in axe rectilineo curvae primigeniae figures I. Erunt limites quidam in E, in F, in H, in O, in quibus nimirum vis erit nulla, cum in omnibus punctis C intermediis sit aliqua. Sed in E, & H erunt ejusmodi, ut si utravis ex parte punctum dimoveatur, per ipsam perimetrum, debeat redire versus ipsos ejusmodi limites, sicut ibi accidit in limitibus cohaesionis ; at in F, & O erit ejusmodi, ut in utramvis partem, quantum libuerit, parvum inde punctum dimotum fuerit, sponte debeat inde magis usque recedere, prorsus ut ibi accidit in limitibus non cohaesionis. 233. Contrarium accideret, si DO aequaretur distantiae limitis non cohaesionis : turn enim distantia BC minor haberet attractionem CK, distantia major AC repulsionem CN, & vis composita per diagonalem CG rhombi CNGK haberet itidem directionem tangentis ellipseos ; & in verticibus quidem axis utriusque haberetur limes quidam, sed punctum in perimetro collocatum tenderet versus vertices axis transversi, non versus vertices axis conjugati, & hi referrent limites cohaesionis, illi e contrario limites non cohaesionis. Sed adhuc major analogia in perimetro harum ellipsium habebitur cum axe curvae primigeniae figurae I ; si fuerit DO asqualis distantiae limitis cohaesionis AN illius, & DB in hac major, quam in fig. i amplitude NL, NP ; multo vero magis, si ipsa hujus DB superet plures ejusmodi amplitudines, ac arcuum aequalitas maneat hinc, & inde per totum ejusmodi spatium. Ubi enim AC hujus figurae fiet aequalis abscissae AP illius, etiam BC hujus fiet pariter aequalis AL illius. Quare in ejusmodi loco habebitur limes, & ante ejusmodi locum versus A distantia [108] longior AC habebit repulsionem, & BC brevior attractionem, ac rhombus erit KGNC, & vis dirigetur versus O. Quod si alicubi ante in loco adhuc propriore O distantiae AC, BC aequarentur abscissis AR, AI figurae i ; ibi iterum esset limes ; sed ante eum locum rediret iterum repulsio pro minore distantia, attractio pro majore, & iterum rhombi diameter jaceret versus verticem axis conjugati E. Generaliter autem ubi semiaxis transversus aequatur distantiae cujuspiam limitis cohaesionis, & distantia punctorum a centre ellipseos, sive ejus eccentricitas est major, quam intervallum dicti limitis a pluribus sibi proximis hinc, & inde, ac maneat aequalitas arcuum, habebuntur in singulis quadrantibus perimetri ellipeos tot limites, quot limites transibit eccentricitas hinc translata in axem figurae I, a limite illo nominato, qui terminet in fig. i semiaxem transversum hujus ellipseos ; ac praetererea habebuntur limites in verticibus amborum ellipseos axium ; eritque incipiendo ab utrovis vertice axis conjugati in gyrum per ipsam perimetrum is limes primus cohaesionis, turn illi proximus esset non cohaesionis, deinde A THEORY OF NATURAL PHILOSOPHY 181 from the foci of the ellipse, in which they were originally situated ; or, if they are forced to depart therefrom owing to the insufficient strength of the limit-point, they may be considered to be kept immovable in the same place by means of an external force, so that we may consider the relation of the third point to those two alone. 2i>i. A point, then, which is situated at one of the vertices of the conjugate axis of At remaining points , ,. J . , ' , r i_ i o j -L of tne perimeter the ellipse or at one of the vertices of the transverse axis remains motionless & under the the force directed action of no force. If it is placed at any point C in the perimeter of the ellipse, then, since alon s the perimeter A ^i /~.T> i i IT i i Till .is towards the ver- AC, CB taken together are m the ellipse equal to the transverse axis, or double the semi- tices of the conju- axis DO, AC will be as much longer than DO as BC is shorter. Hence, if in Fig. i AM, 8 ate axis - Az are equal to these lines AC, BC, we shall have in every case, in Fig. I, uy, zt also equal to one another. Therefore, in Fig. 32, the attraction CL will be equal to the repulsion CM, & LIMC will be a rhombus, in which the inclination 1C will bisect the angle LCM. Hence if it is produced on either side to P & Q, the angle ACP, which is the same as the angle LCI will be equal to the angle BCQ, which is vertically opposite to the angle ICM. Now this is a well-known property with respect to the tangent referred to the foci in the case of an ellipse ; & therefore PQ is the tangent. Hence the force on the point C is directed laterally along the tangent, i.e., in the direction of the arc of the ellipse ; & this is true, no matter where the point is situated on the perimeter, & the force is towards the nearest vertex of the conjugate axis ; if left to itself, the point will travel along the perimeter towards that vertex, except in so far as its motion is disturbed somewhat in addition, owing to centrifugal force. 232. Hence we can consider in this curved perimeter the alternation of limit-points Analogy between as being perfectly analogous to those of cohesion & non-cohesion, which were obtained in two ^xes 63 * the the rectilinear axis of the primary curve of Fig. I. There will be certain limit-points at limit-points of the E, F, H, O, in which there is no force, whilst in all intermediate points such as C there c will be some force. But at E & H they will be such that, if the point is moved towards either side along the perimeter, it must return towards such limit-points, just as it has to do in the case of limit-points of cohesion in Fig. I. But at F & O, the limit-point would be such that, if the point is moved therefrom to either side by any amount, no matter how small, it must of its own accord depart still further from it ; exactly as it fell out in Fig. i for the limit-points of non-cohesion. 233. Just the contrary would happen, if DO were equal to the distance corresponding when the limit to a limit-point of non-cohesion. For then the smaller distance BC would have an P omi f are disposed ATT- i T A ^i i /^XT i i r -i , in the opposite attraction CK, & the greater distance AC a repulsion CJN ; the resultant force along the way ; most elegant diagonal CG of the rhombus CNGK would in the same way have its direction along the instances of aiter- ,,,. . . r -i i 111 .,..9 nation of several tangent to the ellipse, & at the vertices of either axis there would be certain limit-points ; limit-points in the but a point situated in the perimeter would tend towards the vertices of the transverse g^ 6 *" of the axis, & not towards the vertices of the conjugate axis ; & the latter are of the nature of limit-points of cohesion & the former of non-cohesion. However, a still greater analogy in the case of the perimeter of these ellipses with the axis of the primary curve of Fig. i would be obtained, if DO were taken equal to the distance corresponding to the limit-point of cohesion AN in that figure, & in the present figure DB were taken greater than the width of NL, NP in Fig. i ; much more so, if DB were greater than several of these widths, & the equality between the areas on one side & the other held good throughout the whole of the space taken. For where AC in the present figure becomes equal to the abscissa AP of the former, BC in the present figure will likewise become equal to AL in the former. Hence at a position of this kind there will be a limit-point ; & before a position of this kind, towards O, the longer distance AC will have a repulsion & the shorter distance BC an attraction, KGNC will be a rhombus, & the force will be directed towards O. But if at some position, on the side of O, & still nearer to O, the distances AC, BC were equal to the abscissae AR, AI of Fig. I, then again there would be a limit-point ; but before that position there would return once more a repulsion for the smaller distance & an attraction for the greater, & once more the diagonal of the rhombus would lie in the direction of E, the vertex of the conjugate axis. Moreover, in general, whenever the transverse semiaxis is equal to the distance corresponding to any limit-point of cohesion, & the distance of the points from the centre of the ellipse, i.e., its eccentricity, is greater than the interval between the said limit-point & several successive limit-points on either side of it, & the equality of the arcs holds good, then for each quadrant of the perimeter of the ellipse there will be as many limit-points as the number of limit-points in the axis of Fig. I that the eccentricity will cover when transferred to it from the present figure, measured from that limit-point mentioned as terminating in Fig. I the transverse semiaxis of the ellipse of the present figure ; in addition there will be limit-points at the vertices of both axes of the ellipse. Beginning at either vertex -of the conjugate axis, & going round the perimeter, the first limit-point will be one of cohesion, then the next to it one of non-cohesion, then PHILOSOPHISE NATURALIS THEORIA alter cohaesionis, & ita porro, donee redeatur ad primum, ex quo incceptus fuerit gyrus, vi in transitu per quemvis ex ejusmodi limitibus mutante directionem in oppositam. Quod si semiaxis hujus ellipseos aequetur distantiae limitis non cohsesionis figurae i ; res ecdem ordine pergit cum hoc solo discrimine, quod primus limes, qui habetur in vertice semiaxis conjugati sit limes non cohaesionis, turn eundo in gyrum ipsi proximus sit cohsesionis limes, deinde iterum non cohaesionis, & ita porro. Perimetn piunum 2 , . Verum est adhuc alia quaedam analogia cum iis limitibus ; si considerentur elhpsium aequiva- JT .. . .... , . .. '..... lentes limitibus. plures ellipses nsdem illis iocis, quarum semiaxes ordine suo aequentur distantns, in altera cujuspiam e limitibus cohaesionis figuras I, in altera limitis non cohaesionis ipsi proximi, & ita porro alternatim, communis autem ilia eccentricitas sit adhuc etiam minor quavis amplitudine arcuum interceptorum limitibus illis figurse I, ut nimirum singulae ellipsium perimetri habeant quaternos tantummodo limites in quatuor verticibus axium. Ipsae ejusmodi perimetri totae erunt quidam veluti limites relate ad accessum, & recessum a centro. Punctum collocatum in quavis perimetro habebit determinationem ad motum secundum directionem perimetri ejusdem ; at collocatum inter binas perimetros diriget semper viam suam ita, ut tendat versus perimetrum definitam per limitem cohaesionis figurae I, & recedat a perimetro definita per limitem non cohaesionis ; ac proinde punctum a perimetro primi generis dimotum conabitur ad illam redire ; & dimotum a perimetro secundi generis, sponte illam adhuc magis fugiet, ac recedet. Demonstrate. 235. Sint enim in fig. 33. ellipsium FEOH, F'E'O'H', F"E"O"H" semiaxes DO, D'O', D"O" aequales primus di-[iO9]-stantiae AL limitis non cohaesionis figurae i ; secundus distantiae AN limitis cohaesionis ; tertius distantiae AP limitis iterum non cohaesionis, & primo quidem collocetur C aliquanto ultra perimetrum mediam F'E'O'H' : erunt AC, BC majores, quam si essent in perimetro, adeoque in fig. I factis AM, Az majoribus, quam essent prius, decrescet repulsio zt, crescet attractio uy ; ac proinde hie in parallelogram mo LCMI erit attractio CL major, quam repulsio CM, & idcirco accedet directio diagonalis CI magis ad CL, quam ad CM, & inflectetur introrsum versus perimetrum mediam. Contra vero si C' sit intra perimetrum mediam, factis BC', AC' minoribus, quam si essent in perimetro media ; crescet repulsio C'M', & decrescet attractio C'L', adeoque directio C'l' accedet magis ad priorem C'M', quam ad posteriorem C'L', & vis dirigetur extrorsum versus eandem mediam perimetrum. Contrarium autem accideret ob rationem omnino similem in vicinia primae vel tertias perimetri : atque inde patet, quod fuerat propositum. blematum s e g e s, sed minus utilis : immensa combina- tionum varietas. Alias curvas eiiip- 236. Quoniam arcus hinc, & inde a quovis limite non sunt prorsus aequales ; quanquam, das 13 ; ampfa^pro- ut su P ra observavimus num. 184, exigui arcus ordinatas ad sensum aequales hinc, & inde habere debeant ; curva, per cujus tangentem perpetuo dirigatur vis, licet in exigua eccen- tricitate debeat esse ad sensum ellipsis, tamen nee in iis erit ellipsis accurate, nee in eccentricitatibus majoribus ad ellipses multum accedet. Erunt tamen semper aliquae curvae, quae determinent continuam directionem virium, & curvse etiam, quae trajectoriam describendam definiant, habita quoque ratione vis centifugae : atque hie quidem uberrima seges succrescit problematum Geometrise, & Analysi exercendae aptissimorum ; sed omnem ego quidem ejusmodi perquisitionem omittam, cujus nimirum ad Theoriae applicationem usus mihi idoneus occurrit nullus ; & quae hue usque vidimus, abunde sunt ad ostendendam elegantem sane analogiam alternationis in directione virium agentium in latus, cum virium primigeniis simplicibus, ac harum limitum cum illarum limitibus, & ad ingerendam animo semper magis casuum, & combinationum diversarum ubertatem tantam in solo etiam trium punctorum systemate simplicissimo ; unde conjectare liceat, quid futurum sit, ubi immensus quidam punctorum numerus coalescat in massulas constituentes omnem hanc usque adeo inter se diversorum corporum multitudinem sane immensam. Conversio t o t i u s systematis illaesi : impulsu per peri- metrum ellip sees oscil latio: idea liquationis, & con- glaciationis. 237. At praeterea est & alius insignis, ac magis determinatus fructus, quern ex ejusmodi contemplationibus capere possumus, usui futurus etiam in applicatione Theoriae ad Physicam. Si nimirum duo puncta A, & B sint in distantia limitis cohaesionis satis validi, & punctum tertium collocatum in vertice axis conjugati in E distantiam a reliquis habeat, quam habet limes itidem cohaesionis satis validus ; poterit sane [no] vis, qua ipsum retinetur in eo vertice, esse admodum ingens pro utcunque exigua dimotione ab eo loco, A THEORY OF NATURAL PHILOSOPHY 183 FIG. 33. i8 4 PHILOSOPHISE NATURALIS THEORIA FIG. 33. A THEORY OF NATURAL PHILOSOPHY 185 another of cohesion, & so on, until we arrive at the first of them, from which the circuit was commenced ; & the force changes direction as we pass through each of the limit-points of this kind to the exactly opposite direction. But if the semiaxis of this ellipse is equal to the distance corresponding to a limit-point of non-cohesion in Fig. i, the whole matter goes on as before, with this difference only, namely, that the first limit-point at the vertex of the conjugate semiaxis becomes one of non-cohesion ; then, as we go round, the next to it is one of cohesion, then again one of non-cohesion, & so on. 234. Now there is yet another analogy with these limit-points. Let us consider a The perimeters of number of ellipses having the same foci, of which the semiaxes are in order equal to the sev< r ral ellipses ,. ,....,-,. , - . . equivalent to limit- distances corresponding to limit-points in .tig. I, namely to one of cohesion for one, to points. that of non-cohesion next to it for the second, & so on alternately ; also suppose that the eccentricity is still smaller than any width of the arcs between the limit-points of Fig. I, so that each of the elliptic perimeters has only four limit-points, one at each of the four vertices of the axes. The whole set of such perimeters will be somewhat of the nature of limit-points as regards approach to, or recession from the centre. A point situated in any one of the perimeters will have a propensity for motion along that perimeter. If it is situated between two perimeters, it will always direct its force in such a way that it will tend towards a perimeter corresponding to a limit-point of cohesion in Fig. i, & will recede from a perimeter corresponding to a limit-point of non-cohesion. Hence, if a point is disturbed out of a position on a perimeter of the first kind, it will endeavour to return to it ; but if disturbed from a position on a perimeter of the second kind, it will of its own accord try to get away from it still further, & will recede from it. 235. In Fig. 33, of the semiaxes DO, DO', DO" of the ellipses FEOH, F'E'O'H', Demonstration. F"E"O"H". let the first be equal to the distance corresponding to AL, a limit-point of non-cohesion in Fig. I, the second to AN, one of cohesion, the third to AP, one of non- cohesion. In the first place, let the point C be situated somewhere outside the middle perimeter F'E'O'H' ; then AC, BC will be greater than if they were drawn to the perimeter. Hence, in Fig. I, since AM, Az would be made greater than they were formerly, the repulsion zt would decrease, & the attraction uy would increase. Therefore, in Fig. 33, in the parallelogram LCMI, the attraction CL will be greater than the repulsion CM, & so the direction of the diagonal CI will approach more nearly to CL than to CM, & will be turned inwards towards the middle perimeter. On the other hand, if C' is within the middle perimeter, BC', AC' are made smaller than if they were drawn to the middle perimeter ; the repulsion C'M' will increase, & the attraction C'L' will decrease, & thus the direction of CT will approach more nearly to the former, C'M', than to the latter, C'L' ; & the force will be directed outwards towards the middle perimeter. Exactly the opposite would happen in the neighbourhood of the first or third perimeter, & the reasoning would be similar. Hence, the theorem enunciated is evidently true. 236. Now, since the arcs on either side of any chosen limit-point are not exactly equal, other curves to & yet, as has been mentioned above in Art. 184, very small arcs on either side are bound b ,f. substltuted for i IT i 11 i ellipses ; an ample to nave approximately equal ordmates ; the curve, along the tangent to which the force crop of theorems, is continually directed, although for small eccentricity it must be practically an ellipse, b "g a " ot v^|et h US oi yet will neither be an ellipse accurately in this case, nor approach very much to the form combinations. of an ellipse for larger eccentricity. Nevertheless, there will always be certain curves determining the continuous direction of the force, & also curves determining the path described when account is taken of the centrifugal force. Here indeed there will spring up a most bountiful crop of problems well-adapted for the employment of geometry & analysis. But I am going to omit all discussion of that kind ; for I can find no fit use for them in the application of my Theory. Also those which we have already seen are quite suitable enough to exhibit the truly elegant analogy between the alternation in direction of forces acting in a lateral direction & the simple primary forces, between the limit-points of the former & those of the latter ; also for impressing on the mind more & more the great wealth of cases & different combinations to be met with even in the single very simple system of three points. From this it may be conjectured what will happen when an immeasurable number of points coalesce into small masses, from which are formed all that truly immense multitude of bodies so far differing from one another. 237. In addition to the above, there is another noteworthy & more determinate result Rotation of the to be derived from considerations of this kind, & one that will be of service in the application ** ' e sy ., s , t m t 1 -r>i T.I i-i ! -i A n T> T T mtact ) oscillation oi the 1 neory to rnysics. r or instance, it the two points A & is are at a distance corresponding along the perimeter to a limit-point of cohesion that is sufficiently strong, & the third point situated at the of the ellipse due to T-, , . . . , . ' 1-1 an impulse ; the vertex r, oi the conjugate axis is at a distance from the other two which corresponds to idea of liquefaction a limit-point of cohesion that is also sufficiently strong, then the force retaining the point & congelation, at that vertex might be great enough, for any slight disturbance from that position, to prevent it from being moved any further, unless through the action of a huge external 1 86 PHILOSOPHIC NATURALIS THEORIA ut sine ingenti externa vi inde magis dimoveri non possit. Turn quidem si quis impediat motum puncti B, & circa ipsum circumducat punctum A, ut in fig. 34 abeat in A' ; abibit utique & E versus E', ut servetur forma trianguli AEB, quam necessario requirit conver- satio distantiarum, sive laterum inducta a limitum validitate, & in qua sola poterit respective quiescere systema, ac habebitur idea quaedam soliditatis cujus & supra injecta est mentio. At si stantibus in fig. 32 punctis A, B per quaspiam vires externas, quae eorum motum impediant, vis aliqua exerceatur in E ad ipsum a sua positione deturbandum ; donee ea fuerit medio- cris, dimovebit illud non nihil ; turn, ilia cessante, ipsum se resti- tuet, & oscillabit hinc, & inde ab illo vertice per perimetrum curvae cujusdam proximse arcui elliptico. Quo major fuerit vis externa dimovens, eo major oscillatio net ; sed si non fuerit tanta, ut punctum a vertice axis conjugati recedens deveniat ad verticem axis transversi ; semper retro cursus reflectetur, & de- scribetur minus, quam semiellipsis. Verum si vis externa coegerit percurrere totum quadrantem, & transilire ultra verticem axis transversi ; turn verogyrabit punctum circumquaque per totam FIG. 34. perimetrum motu continue, quern a vertice axis conjugati ad verticem transversi retardabit, turn ab hoc ad verticem conjugati accelerabit, & ita porro, nee sistetur periodicus conversionis motus, nisi exteriorum punctorum impedimentis occurrentibus, quae sensim celeritatem imminuant, & post ipsos ejusmodi motus periodicos per totum ambitum reducant meras oscillationes, quas contrahant, & pristinam debitam positionem restituant, in qua una haberi potest quies respectiva. An non ejusmodi aliquid accidit, ubi solida corpora, quorum partes certam positionem servant ad se invicem, ingenti agitatione accepta ab igneis particulis liquescunt, turn iterum refrigescentes, agitatione sensim cessante per vires, quibus igneae particulae emittuntur, & evolant, positionem prio- rem recuperant, ac tenacissime iterum servant, & tuentur ? Sed haec de trium punctorum systemate hucusque dicta sint satis. Systema punctorum 238. Quatuor, & multo magis plurium, punctorum systemata multo plures nobis quatuor, in eodem . . J , .. . . ,',..., . r -p, piano cum distan- vanationes objicerent ; si rite ad examen vocarentur ; sed de us id unum innuam. H,a tiis hmitum, suao q u id e m in piano eodem possunt positionem mutuam tueri tenacissime ; si singulorum forma; tenax. f. . r ,. . ....... . ,. , ,. distantiae a reliquis sequentur distantns hmitum satis validorum tigurae I : neque emm in eodem piano positionem respectivam mutare possunt, aut aliquod ex iis exire e piano ducto per reliqua tria, nisi mutet distantiam ab aliquo e reliquis, cum datis trium punctorum distantiis mutuis detur triangulum, quod constituere debent, turn datis distantiis quarti a duobus detur itidem ejus positio respectu eorum in eodem piano, & detur distantia ab eorum tertio, quae, si id punctum exeat e [in] priore piano, sed retineat ab iis duobus distantiam priorem, mutari utique debet, ut facili negotio demonstrari potest. Alia ratio system- 239. Quin immo in ipsa ellipsi considerari possunt puncta quatuor, duo in focis, & quatuor ^^eodem a ^ a ^ uo nmc > & ' m< ^ e a vertice axis conjugati in ea distantia a se invicem, ut vi mutua piano cum idea repulsiva sibi invicem elidant vim, qua juxta praecedentem Theoriam urgentur in ipsum flexiiis "Systema verticem ; quo quidem pacto rectangulum quoddam terminabunt, ut exhibet fig. 35, in eorundem forms punctis A, B, C, D. Atque inde si supra angulos quadratae basis assurgant series ejusmodi nes^rifmrticufa- punctorum exhibentium series continuas rectangulorum, habebitur quaedam adhuc magis rum pyramidaiium. praecisa idea virgae solidae, in qua si basis ima inclinetur ; statim omnia superiora puncta movebuntur in latus, ut rectangulorum illorum positionem retineant,& celeritas conversionis erit major, vel minor, prout major fuerit, vel minor vis ilia in latus, quae ubi fuerit aliquanto languidior, multo serius progredietur vertex, quam fundum, . & inflectetur virga, quae inflexio in omni virgarum genere apparet adhuc multo magis manifesta, si celeritas conversionis C O fuerit ingens. Sed extra idem planum possunt quatuor puncta collocata ita, ut positionem suam validissime tueantur, etiam ope unicae distantiae limitis unici satis validi. Potest enim fieri pyramis regularis, cujus latera singula triangularia habeant ejusmodi distantiam. Turn ea pyramis constituet particulam quandam suae figurae tenacissimam, quae in puncta, vel pyra- /\ 3 mides ejusmodi aliquanto remotiores ita poterit agere, ut ejus FlG 35 puncta respectivum situm nihil ad sensum mutent. Ex quatuor ejusmodi particulis in aliam majorem pyramidem dispositis fieri poterit particula secundi ordinis aliquanto minus tenax ob majorem distantiam particularum primi earn componen- A THEORY OF NATURAL PHILOSOPHY 187 force. In that case, if the motion of the point B were prevented, & the point A were set in motion round B, so that in Fig. 34 it moved to A', then the point E would move off to E' as well, so as to conserve the form of the triangle AEB, as is required by the conservation of the sides or distances which is induced by the strength of the limits ; & the system can be relatively at rest in this form only ; thus we get an idea of a certain solidity, of which casual mention has" already been made above. But if, in Fig. 2, whilst the points A,B, are kept stationary by means of an external force preventing their motion, some force is exerted on the point at E to disturb it from its position, then, as long as the force is only moderate, it will move the point a little ; & afterwards, when the force ceases, the point will recover its position, & will then oscillate on each side of the vertex along a perimeter of the curve that closely approximates to an elliptic arc. The greater the external force producing the motion, the greater the oscillation will be ; but if it is not so great as to make the point recede from the vertex of the conjugate axis until it reaches the vertex of the transverse axis, its path will always be retraced, & the arc described will be less than a semi-ellipse. But if the external force should compel the point to traverse a whole quadrant & pass through the vertex of the transverse axis, then indeed the point will make a complete circuit of the whole perimeter with a continuous motion ; this will be retarded from the vertex of the conjugate axis to that of the transverse axis, then accelerated from there onwards to the vertex of the conjugate axis, & so on ; there will not be any periodic reversal of motion, unless there are impediments met with from external points that appreciably diminish the speed ; in which case, following on such periodic motions round the whole circuit, there will be a return to mere oscillations ; & these will be shortened, & the original position restored, the only one in which there can possibly be relative rest. Probably something of this sort takes place, when solid bodies whose parts maintain a definite position with regard to one another, if subjected to the enormous agitation produced by fiery particles, liquefy ; & once more freezing, as the agitation practically ceases on account of forces due to the action of which the fiery particles are driven out & fly off, recover their initial position & again keep & preserve it most tenaciously. But let us be content with what has been said above with regard to a system of three points for the present. 238. Systems of four, & much more so for more, points would yield us many more varia- A system of four tions, if they were examined carefully one after the other ; but I will only mention one thing jj^^ces^orre* about such systems. It is possible that such systems, in one plane, may conserve their rela- spending to Hmit- tive positions very tenaciously, if the distances of each from the rest are equal to the dis- ^ves^tslform. 0011 tances in Fig. i corresponding to limit-points of sufficient strength. For neither can they change their relative position in the plane, nor can any one of them leave the plane drawn through the other three ; since, if the distances of three points from one another is given, then we are given the triangle which they must form ; & then being given the distances of the fourth point from two of these, we are also given the position of this fourth point from them, & therefore also the distance from the third of them. If the point should depart from the plane mentioned, & yet preserve its former distances from the two points the distance from the third point must be changed in any case, as can be easily proved. 239. Again, we may consider the case of four points in the ellipse, two being at the A further consider- foci, & the other two on either side of a vertex of the conjugate axis at such a distance from a { io " ou f ^i^f 6 , one another, that the mutual repulsive force between them will cancel the force with which connection with they are urged towards that vertex, according to the preceding theorem. Thus, they are the idea of rigid & f i . T->- i A T, V-i T-V flexible rods; a. at the vertices of a rectangle, as is shown in rig. 35, where they occupy the points A,B,U,D. system of four Hence, if we have a series of points of this kind to stand above the four angles of the quadratic P mts m the fo . rm r i 11 i r i ^ of a pyramid; base, so as to represent continuous series of rectangles, we shall obtain from this supposition different arrange- a more precise idea than hitherto has been possible of a solid rod, in which, if the lowest ments f particular r r v i n T i 1-1 pyramids. set or points is inclined, all the points above are immediately moved sideways, but so that they retain the positions in their rectangles ; & the speed of rotation will be greater or less according as the force sideways was greater or less ; even where this force is somewhat feeble, the top will move considerably later than the base & the rod will be bent ; & the amount of bending in every kind of rod will be still more apparent if the speed of rotation is very great. Again, four points not in the same plane can be so situated that they preserve their relative position very tenaciously ; & that too, when we make use of but a single distance corresponding to a limit-point of sufficient strength. For they can form a regular pyramid, of which each of the sides of the triangles is of a length equal to this distance. Then this pyramid will constitute a particle that is most tenacious as regards its form ; & this will be able to act upon points, or pyramids of the same kind, that are more remote, in such a manner that its points do not alter their relative position in the slightest degree for all practical purposes. From four particles of this kind, arranged to form a larger pyramid, we can obtain a particle of the second order, somewhat less tenacious of form on account of the greater distance between the particles of the first order that compose it ; 1 88 PHILOSOPHIC NATURALIS THEORIA tium, qua fit, ut vires in easdem ab externis punctis impressae multo magis inaequales inter se sint,^quam fuerint in punctis constituentibus particulas ordinis primi ; ac eodem pacto ex his secundi ordinis particulis fieri possunt particulse ordinis tertii adhuc minus tenaces figurae suae, atque ita porro, donee ad eas deventum sit multo majores, sed adhuc multo magis mobiles, atque variabiles, ex quibus pendent chemica; operationes, & ex quibus haec ipsa crassiora corpora componuntur, ubi id ipsum accideret, quod Newtonus in postrema Optics questione proposuit de particulis suis primigeneis, & elementaribus, alias diversorum ordi- num particulas efformantibus. Sed de particularibus hisce systematis determinati punc- torum numeri jam satis, ac ad massas potius generaliter considerandas faciemus gradum. Transitus ad 240. In massis primum nobis se offerunt considerandas elegantissimse sane, ac foccund- massas : quid cen- m **t* ' j T->I trum gravitatis : issimae, & utilissimae propnetates centn gravitatis, quse quidem e nostra 1 heona sponte theoremata hk de propemodum fluunt, aut saltern eius ope evidentissime demonstrantur. Porro centrum eo demonstrando. L *. . .., . . . . . , . gravitatis a gravium aequihbno nomen accepit suum, a quo etiam ejus consideratio ortum duxit ; sed id quidem a gravi-[ii2]-tate non pendet, sed ad massam potius pertinet. Quamobrem ejus definitionem proferam ab ipsa gravitate nihil omnino pendentem, quan- quam & nomen retinebo, & innuam, unde originem duxerit ; turn demonstrabo accuratissime, in quavis massa haberi aliquod gravitatis centrum, idque unicum, quod quidem passim omittere solent, & perperam ; deinde ad ejus proprietatem praecipuam exponendam gradum faciam, demonstrando celeberrimum theorema a Newtono propositum, centrum gravitatis commune massarum, sive mihi punctorum quotcunque, & utcunque disposi- torum, quorum singula moveantur sola inertiae vi motibus quibuscunque, qui in singulis punctis uniformes sint, in diversis utcunque diversi, vel quiescere, vel moveri uniformiter in directum : turn vero mutuas actiones quascunque inter puncta quaelibet, vel omnia simul, nihil omnino turbare centri communis gravitatis statum quiescendi vel movendi uniformiter in directum, unde nobis & actionis, ac reactionis aequalitas in massis quibusque, & principia collisiones corporum definientia, & alia plurima sponte provenient. Sed aggrediamur ad rem ipsam. Definitio centri 241. Centrum igitur commune gravitatis punctorum quotcunque. & utcunque gravitatis non j- n f -j j j i pendens ab idea dispositorum, appellabo id punctum, per quod si ducatur planum quodcunque ; summa gravitatis : ejus distantiarum perpendicularium ab eo piano punctorum omnium jacentium ex altera idea 8 communi. C * ejusdem parte, sequatur summa distantiarum ex altera. Id quidem extenditur ad quas- cunque, & quotcunque massas ; nam eorum singulae punctis utique constant, & omnes simul sunt quaedam punctorum diversorum congeries. Nomen traxit ab aequilibrio gravium, & natura vectis, de quibus agemus infra : ex iis habetur illud, singula pondera ita connexa per virgas inflexiles, ut moveri non possint, nisi motu circa aliquem horizontalem axem, exerere ad conversionem vim proportionalem sibi, & distantiae perpen- diculari a piano verticali ducto per axem ipsum ; unde fit, ut ubi ejusmodi vires, vel, ut ea vocant, momenta virium hinc, & inde asqualia fuerint, habeatur aequilibrium. Porro ipsa pondera in nostris gravibus, in quibus gravitatem concipimus, ac etiam ad sensum experimur, proportionalem in singulis quantitati materiae, & agentem directionibus inter se parallelis, proportionalia sunt massis, adeoque punctorum eas constituentium numero ; quam ob rem idem est, ea pondera in distantias dncere, ac assumere summam omnium distantiarum omnium punctorum ab eodem piano. Quod si igitur respectu aggregati cujuscunque punctorum materiae quotcunque, & quomodocunque dispositorum sit aliquod punctum spatii ejusmodi, ut, ducto per ipsum quovis piano, summa distantiarum ab illo punctorum jacentium ex parte altera aequetur summse distantiarum jacentium ex altera ; concipiantur autem singula ea puncta animata viribus aequalibus, & parallelis, cujusmodi sunt vires, quas in nostris gravibus concipimus ; illud utique consequitur, [113] suspense utcunque ex ejusmodi puncto, quale definivimus gravitatis centrum, omni eo systemate, cujus systematis puncta viribus quibuscunque, vel conceptis virgis inflexibilus, & gravitate carentibus, positionem mutuam, & respectivum statum, ac distantias omnino servent, id systema fore in aequilibrio ; atque illud ipsum requiri, ut in aequilibrio sit. Si enim vel unicum planum ductum per id punctum sit ejusmodi, ut summae illae distantiarum non sint aequales hinc, & inde ; converse systemate omni ita, ut illud planum evadat verticale, jam non essent aequales inter se summae momentorum hinc, & inde, & altera pars alteri prseponderaret. Verum haec quidem, uti supra monui, fuit occasio quaedam nominis imponendi ; at ipsum punctum ea lege determinatum longe ulterius extenditur, quam A THEORY OF NATURAL PHILOSOPHY 189 for from this fact it comes about that the forces impressed upon these from external points are much more unequal to one another. than they would be for the points constituting particles of the first order. In the same manner, from these particles of the second order we might obtain particles of the third order, still less tenacious of form, & so on ; until at last we reach those which are much greater, still more mobile, & variable particles, which are concerned in chemical operations ; & to those from which are formed the denser bodies, with regard to which we get the very thing set forth by Newton, in his last question in Optics, with respect to his primary elemental particles, that form other particles of different orders. We have now, however, said enough concerning these systems of a definite number of points, & we will proceed to consider masses rather more generally. 240. In dealing with masses, the first matters that present themselves for our considera- Passing on to tion are certain really very elegant, as well as most fertile & useful properties of the centre of "ntr? of "gravity^ gravity. These indeed come forth almost spontaneously from my Theory, or at least are Theorems to be demonstrated most clearly by means of it. Further, the centre of gravity derived its name from the equilibrium of heavy (gravis) bodies, & the first results in connection with the former were developed by means of the latter ; but in reality it does not depend on gravity, but rather is related to masses. On this account, I give a definition of it, which in no way depends on gravity, although I will retain the name, & will mention whence it derived its origin. Then I will prove with the utmost rigour that in every body there is a centre of gravity, & one only (a thing which is usually omitted by everybody, quite unjustifiably). Then I will proceed to expound its chief property, by proving the well-known theorem enunciated by Newton ; that the centre of gravity of masses, or, in my view, of any number of points in any positions, each of which is moved in any manner by the force of inertia alone, this force being uniform for the separate points but maybe non-uniform to any extent for different points, will be either at rest or will move uniformly in a straight line. Finally, I will show that any mutual action whatever between the points, or all of them taken together, will in no way disturb the state of rest or of uniform motion in a straight line of the centre of gravity. From which the equality of action & reaction in all bodies, & the principles governing the collision of solids, & very many other things will arise of them- selves. However let us set to work on the matter itself. 241. Accordingly, I will call the common centre of gravity of any number of points, Definition of the situated in any positions whatever, that point which is such that, if through it any plane in^tndent^fTn 7 is drawn, the sum of the perpendicular distances from the plane of all the points lying on idea of gravitation ; one side of it is equal to the sum of the distances of all the points on the other side of it. ^ dtfaiSorT The definition applies also to masses, of any sort or number whatever ; for each of the the usual idea, latter is made up of points, & all of them taken together are certain groups of different points. The name is taken from the equilibrium of weights (gravis), & from the principle of the lever, with which we shall deal later. Hence we obtain the principle that each of the weights, connected together by rigid rods in such a manner that the only motion possible to them is one round a horizontal axis, will exert a turning force proportional to itself & to its perpendicular distance from a vertical plane drawn through this axis. From which it comes about that, when the forces of this sort (or, as they are called, the moments of the forces) are equal to one another on this side & on that, then there is equilibrium. Further, the weights in our heavy bodies, in which we conceive the existence of gravity (& indeed find by experience that there is such a thing) proportional in each to the quantity of matter, & acting in directions parallel to one another, are proportional to the masses, & thus to the number of points that go to form them. Therefore, the product of the weights into the distances comes to the same thing as the sum of all the distances of all the points from the plane. If then, for an aggregate of points of matter, of any sort & number whatever, situated in any way, there is a point of space of such a nature that, for any plane drawn through it, the sum of the distances from it of all points lying on one side of it is equal to the sum of the distances of all the points lying on the other side of it ; if moreover each of the points is supposed to be endowed with a force, & these forces are all equal & parallel to one another, & of such a kind as we conceive the forces in our weights to be ; then it follows directly that, if the whole of this system is suspended in any way from a point of the sort we have defined the centre of gravity to be, the points of the system, on account of certain assumed forces or rigid weightless rods, preserving their mutual position, their relative state & their distances absolutely unchanged, then the system will be in equilibrium. Such a point is to be found, in order that the system may be in equilibrium. For, if any one plane can be drawn through the point, such that the sum of the distances on the one side are not equal to those on the other side, & thewhole system is turned so that this plane becomes vertical, then the sums of the moments will not be equal to one another on each side, but one part will outweigh the other part. This indeed, as I said above, was the idea that gave rise to the term centre of gravity ; but the point determined by this rule has 190 PHILOSOPHIC NATURALIS THEORIA Corollarium g e it- erate pertinens ad summas distanti- arum omnium punctorum massse a piano transeunte per centrum gravi- tatis xquales utrin- que. Bi.n a theoremata per tinentia ad planum parallel urn piano distantiarum aequalium cum eorum demonstra- tiouibus. Com pie me n turn demonstrationis ut e x t e n d[a t u r ad omnes casus. ad solas massas animatas viribus asqualibus, & parallelis, cujusmodi concipiuntur a nobis in nostris gravibus, licet ne in ipsis quidem accurate sint tales. Quamobrem assumpta superiore definitione, quae a gravitatis, & sequilibrii natura non pendet, progrediar ad deducenda inde corollaria quaaedam, quae nos ad ejus proprietates demonstrandas deducant. 242. Primo quidem si aliquod fuerit ejusmodi planum, ut binae summae distantiarum perpendicularium punctorum omnium hinc & inde acceptorum aequenter inter se : aequabuntur & summae distantiarum acceptarum secundum quancunque aliam directionem datam, & communem pro omnibus. Erit enim quaevis distantia perpendicularis ad quanvis in dato angulo inclinatam semper in eadem ratione, ut patet. Quare & sunimae illarum ad harum summas erunt in eadem ratione, ac asqualitas summarum alterius binarii utriuslibet secum trahet aequalitatem alterius. Quare in sequentibus, ubi distantias nominavero, nisi exprimam perpendiculares, intelligam generaliter distantias acceptas in quavis directione data. 243. Quod si assumatur planum aliud quodcunque parallelum piano habenti aequales hinc, & inde distantiarum summas ; summa distantiarum omnium punctorum jacentium ex parte altera superabit summam jacentium ex altera, excessu aequali distantiae planorum acceptae secundum directionem eandem ductae in nwmerum punctorum : & vice versa si duo plana parallela sint, ac is excessus alterius summas supra summam alterius in altero ex iis aequetur eorum distantiae ductae in numerum punctorum ; planum alterum habebit oppositarum distantiarum summas aequales. Id quidem facile concipitur ; si concipiatur, planum distantiarum aequalium moveri versus illud alterum planum motu parallelo secundum earn directionem, secundum quam sumuntur distantiae. In eo motu distantiae singulse ex altera parte crescunt, ex altera decrescunt continue tantum, quantum promo- vetur planum, & si aliqua distantia evanescit interea ; jam eadem deinde incipit tantundem ex parte contraria crescere. Quare patet excessum omnium citeriorum [114] distantiarum supra omnes ulteriores aequari progressui plani toties sumpto, quot puncta habentur, & in regressu destruitur e contrario, quidquid in ejusmodi progressu est factum, atque idcirco ad aequalitatem reditur. Verum ut demonstratio quam accuratissima evadat, exprimat in fig. 36 recta AB planum distantiarum aequalium, & CD planum ipsi parallelum, ac omnia puncti distribui poterunt in classes tres, in quorum prima sint omnia jacentia citra utrumque planum, ut punctum E ; in secunda omnia puncta jacentia inter utrumque, ut F, in tertia omnia puncta adhuc jacentia ultra utrumque, ut G. Rectae autem per ipsa ductae in directione data quacunque, occurrant rectae AB in M, 'H, K, & rectae CD in N, I, L ; ac sit quaedam reacta direc- tionis ejusdem ipsis AB, CD occurrens in O, P. Patet, ipsam OP fore aequalem ipsis MN, HI, KL. Dicatur jam summa omnium punctorum E primae classis E, & distantiarum omnium EM summa e ; punctorum F secundae classis F, & distantiarum / ; punctorum G tertiae classis summa G, & distantiarum earundem g ; distantia vero OP dicatur O. Patet, sum- mam omnium MN fore E X O ; summam omnium HI fore F X O ; summam omnium KL fore G X O ; erit autem quaevis EN EM +MN ; quaevis FI = HI FH ; quaevis GL = KG KL. Quare summa omnium EN erit e + E xO ; summa omnium FI = F x O /, & summa omnium GL = g G X O ; adeoque summa omnium distantiarum punctorum jacentium citra planum CD, primae nimi- rum, ac secundae classis, erit e + E xO+F X O /, & summa omnium jacentium ultra, nimirum classis tertiae, erit g G X O. Quare excessus prioris summae supra secundam erit e + E X O -f F xO / g+ G xO; adeoque si prius fuerit e = f + g ; delete e f g, totus excessus erit E x O + F X O + G X O, sive (E + F + G) X O, summa omnium punctorum ducta in distantiam planorum ; & vice versa si is excessus respectu secundi plani CD fuerit aequalis huic summas ductae in distantiam O, oportebit e f /aequetur nihilo, adeoque sit e = f -\- g, nimirum respectu primi plani AB summas distantiarum hinc, & inde aequales esse. 244. Si aliqua puncta sint in altero ex iis planis, ea superioribus formulis contineri possunt, concepta zero singulorum distantia a piano, in quo jacent ; sed & ii casus involvi facile possent, concipiendo alias binas punctorum classes ; quorum priora sint in priore piano AB, posteriora in posteriore CD, quae quidem nihil rem turbant : nam prioris classis FIG. 36. A THEORY OF NATURAL PHILOSOPHY 191 a far wider application than to the single cases of mass endowed with equal & parallel forces such as we have assumed to exist in our heavy bodies ; & indeed such do not exist accurately even in the latter. Hence, taking the definition given above, which is independent of gravity & the nature of equilibrium of weights, I will proceed to deduce from it certain corollaries, which will enable us to demonstrate the properties of the centre of gravity. 242. First of all, then, if there should be any plane such that the two sums of the General corollary perpendicular distances of all the points on either side of it taken together are equal to O f ^h^d^ances'of one another, then the sums of the distances taken together in any other given direction, ail the points of a that is the same for all of them, will also be equal to one another. For, any perpendicular ^slng hrVu'g'h distance will evidently be in the same ratio to the corresponding distance inclined at a the centre of grav- given angle. Hence the sums of the former distances will bear the same ratio to the sums e^he/sicfeoTit! of the latter distances ; & therefore the equality of the sums in either of the two cases will involve the equality of the sums for the other also. Consequently, in what follows, whenever I speak of distances, I intend in general distances in any given direction, unless I expressly say that they are perpendicular distances. 243. If now we take any other plane parallel to the plane for which the sums of the Two . theorems distances on either side are equal, then the sum of the distances of all the points lying p e ara iief o P the on the one side of it will exceed the sum for those lying on the other side by an amount P| ane o f equal equal to the distance between the two planes measured in the like direction multiplied demonstrations, by the number of all the points. Conversely, if there are two parallel planes, & if the excess of the sum of the distances from one of them over the sum of the distances from the other is equal to the distance between the planes multiplied by the number of the points, then the second plane will have the sums of the opposite distances equal to one another. This is easily seen to be true ; for, if the plane of equal distances is assumed to be moved towards the other plane by a parallel motion in the direction in which the distances the measured, then as the plane is moved each of the distances on the one side increase, & those on the other side decrease by just the amount through which the plane is moved ; & should any distance vanish in the meantime, there will be an increase on the other side of just the same amount. Thus, it is evident that the excess of all the distances on the near side above the sum of all the distances on the far side will be equal to the distance through which the plane has been moved, taken as many times as there are points. On the other hand, when the plane is moved back again, this excess is destroyed, namely exactly the amount that was produced as the plane moved forward, & consequently equality will be restored. But to give a more rigorous demonstration, let the straight line AB, in Fig. 36, represent the plane of equal distances, & let CD represent a plane parallel to it. Then all the points can be grouped into three classes ; let the first of these be that in which we have every point that lies on the near side of both the planes, as E ; let the second be that in which every point lies between the two planes, as F ; & the third, every point lying on the far side of both planes, as G. Let straight lines, drawn in any given direction whatever, through the points meet AB in M, H, K, & the straight line CD in N, I, L ; also let any straight line, drawn in the same direction, meet AB, CD in O & P. Then it is clear that OP will be equal to MN, HI, or KL. Now, let us denote the sum of all the points of the first class, like E, by the letter E, & the sum of all the distances like EM by the letter e ; & those of the second class by the letters F & / ; those of the third class by G & g ; & the distance OP by O. Then it is evident that the sum of all the MN's will be E X O ; the sum of all the Hi's will be F X O ; the sum of all the KL's will be G X O ; also in every case, EN = EM + MN, FI = HI FH, & GL = KG KL. Hence the sum of the EN's will be e + E X O, the sum of the FI's will be F X O /, & the sum of the GL's will be g G X O. Hence, the sum of all the distances of the points lying on the near side of the plane CD, that is to say, those belonging to the first & second classes, will be equal to<?+ExO + Fx O / ; & the sum of all those lying on the far side, that is, of the third class, will be equal to g G X O. Hence, the excess of the former over the latter will be equal to ^+ExO+FxO / g+GxO. Therefore, if at first we had e = / + g, then, on omitting e f g, we have the total excess equal to ExO+FxO + GxO,or(E+F-fG)xO, i.e., the sum of all the points multiplied by the distance between the planes. Conversely, if the excess with respect to the second plane CD were equal to this sum multiplied by the distance O, it must be that e f g is equal to nothing, & thus e = f -\- g; in other words the sum of the distances with respect to the first plane AB must be equal on one side & the other. 244. If any of the points should be in one or other of the two planes, these may also C ^ le s tio " s t f ^ e be included in the foregoing formulae, if we suppose that the distance for each of them ^^ e ' a u a posib"e is zero distance from the plane in which they lie. Then these cases may also be included cases. by considering that there are two fresh classes of points ; namely, first those lying in the first plane AB, & secondly those lying in the second plane CD ; & these classes will in 192 PHILOSOPHIC NATURALIS THEORIA Theoremata pro piano posito ultra omnia pun eta: eorum extensio ad qua; vis plana. Cuivis piano in- veniri posse paral- lelum planum dis- tantiarum aequa- lium. Thoorema prseci- puum si tria plana distantiarum aequa- lium habeant uni- cum punctum commune ; rcliqua ornnia por id tran- seuntia erunt ejus- modi. Demonstratio ejus- dem. distantiae a priore piano erunt omnes simul zero, & a posteriore sequabuntur distantiae O ductae in eorum numerum, quae summa accedit priori summae punctorum jacentium citra ; posterioris autem classis distantiaa a priore erant prius simul aequales summaa ipsorum ductae itidem in O, & deinde fiunt nihil ; adeoque [115] summse distantiarum punctorum jacentium ultra, demitur horum posteriorum punctorum summa itidem ducta in O, & proinde excessui summse citeriorum supra summam ulteriorum accedit summa omnium punctorum harum duarum classium ducta in eandem O. 245. Quod si planum parallelum piano distantiarum aequalium jaceat ultra omnia puncta ; jam habebitur hoc theorema : Summa omnium distantiarum punctorum omnium ab eo piano cequabitur distantly planorum ducta in omnium punctorum summam, & si fuerint duo plana parallela ejusmodi, ut alterum jaceat ultra omnia puncta, fcsf summa omnium distanti- arum ab ipso cequetur distantice planorum ductce in omnium punctorum numerum ; alterum illud planum erit planum distantiarum cequalium. Id sane patet ex eo, quod jam secunda sum- ma pertinens ad puncta ulteriora, quae nulla sunt, evanescat, & excessus totus sit sola prior summa. Quin immo idem theorema habebit locum pro quovis piano habente etiam ulteriora puncta, si citeriorum distantiae habeantur pro positivis, & ulteriorum pro negativis ; cum nimirum summa constans positivis, & negativis sit ipse excessus positivorum supra negativa ; quo quidem pacto licebit considerare planum distantiarum aequalium, ut planum, in quo summa omnium distantiarum sit nulla, negativis nimirum distantiis elidentibus positivas. 246. Hinc autem facile jam patet, data cuivis piano haberi aliquod planum parallelum, quod sit planum distantiarum cequalium ; quin immo data positione punctorum, & piano illo ipso, facile id alterum definitur. Satis est ducere a singulis punctis datis rectas in data directione ad planum datum, quae dabuntur ; turn a summa omnium, quae jacent ex parte altera, demere summam omnium, si quae sunt, jacentium ex opposita, ac residuum dividere per numerum punctorum. Ad earn distantiam ducto piano priori parallelo, id erit planum quaesitum distantiarum aequalium. Patet autem admodum facile & illud ex eadem demonstratione, & ex solutione superioris problematis, dato cuivis piano non nisi unicum esse posse planum distantiarum aequalium, quod quidem per se satis patet. 247. Hisce accuratissime demonstratis, atque explicatis, progrediar ad demonstrandum haberi aliquod gravitatis centrum in quavis punctorum congerie, utcunque dispersorum, & in quotcunque massas ubicunque sitas coalescentium. Id net ope sequentis theorematis ; si per quoddam punctum tran- seant tria plana distantiarum cequalium se non in eadem communi aliqua recta secan- tia ; omnia alia plana transeuntia per illud idem punctum erunt itidem distantiarum esqualium plana. Sit enimin fig. 37, ejus- modi punctum C, per quod transeant tria plana GABH, XABY, ECDF, qua; om- nia sint plana distantiarum aequalium, ac sit quodvis aliud planum KICL tran- [i i6]-siens itidem per C, ac secans pri- mum ex iis recta CI quacunque ; opor- tet ostendere, hoc quoque fore planum distantiarum aequalium, si ilia priora ejusmodi sint. Concipiaturquodcunque punctum P ; & per ipsum P concipiatur tria plana parallela planis DCEF, ABYX, GABH, quorum sibi priora duo mutuo occurrant in recta PM, postrema duo in recta PV, primum cum tertio in recta PO : ac primum occurrat piano GABH in MN, secundum vero eidem in MS, piano DCEF in QR, ac piano CIKL in SV, ducaturque ST parallela rectis QR, MP, quas, utpote parallelorum planorum intersectiones, patet fore itidem parallelas inter se, uti & MN, PO, DC inter se, ac MS, PTV, BA inter se. 248. Jam vero summa omnium dis antiarum a piano KICL secundum datam direc- tionem BA erit summa omnium PV, quae resolvitur in tres summas, omnium PR, omnium RT, omnium TV, sive eae, ut figura exhibet in unam colligendss sint, sive, quod in aliis plani novi inclinationibus posset accidere, una ex iis demenda a reliquis binis, ut habeatur omnium PV summa. Porro quaevis PR est distantia a piano DCEF secundum eandem earn directionem ; quaevis RT est aequalis QS sibi respondenti, quae ob datas directiones laterum trianguli SCQ est ad CQ, aequalem MN, sive PO, distantiae a piano XABY secundum A THEORY OF NATURAL PHILOSOPHY 193 no way cause any difficulty. For the distances of the points of the first class from the first plane, all together, will be zero, & their distances from the second plane will, all together, be equal to the distance O multiplied by the number of them ; & this sum is to be added to the former sum for the points lying on the near side. Again, the distances of the points of the second class from the first plane were, all together, at first equal to the distance O multiplied by their number, & then are nothing for the second plane. Hence from the sum of the distances of the points lying on the far side, we have to take away the sum of these last points also multiplied by the distance O ; & thus, to the excess of the sum of the points on the near side over the sum of the points on the far side we have to add the sum of all the points in these two classes multiplied by the same distance O. 245. Now, if the plane parallel to the plane of equal distances should lie on the far Theorems for a side of all the points then the following theorem is obtained. The sum of all the distances P 1 , 3 ;" 6 l v in s beyond / M i i 7 i -77 7 77- 7 7 i , . , . , , a 1 1 t h e points ; of all the -points from this plane will be equal to the distance between the planes multiplied by extension of these the sum of all the points ; y if there were two parallel planes, such that one of them lies beyond theorems to ^any all the points, ff if the sum of all the distances from this plane is equal to the distance between the planes multiplied by the number of points, then the other plane will be the plane of equal distances. This is perfectly clear from the fact that in this case the second sum relating to the points that lie beyond the planes vanishes, for there are no such points, & the whole excess corresponds to the first sum alone. Further, the same theorem holds good for any plane even if there are points beyond it, if the distances of points on the near side of it are reckoned as positive & those on the far side as negative ; for the sum formed from the positives &am