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 xmyn = I, and obtains the integral - - xy + A, where A is the n—m constant of integration. He states that, if n is greater than m, A = o, being the initial area at the origin. He is then faced with the necessity of making the area infinite when n = m, and still more infinite when n<jn. He says : " The area is infinite, when n = m, because this makes the divisor zero ; and thus the area becomes still more infinite if n<^m." Put into symbols, the argument is : Since «-OT<O, >- > oo . The historically interesting n— m o point about this is that it represents the persistance of an error originally made by Wallis in his Ariihmetica Infinitorum (it was Wallis who invented the sign oc to stand for " simple infinity," the value of i/o, and hence of «/o). Wallis had justification for his error, if indeed it was an error in his case ; for his exponents were characteristics of certain infinite series, and he could make his own laws about these so that they suited the geometrical problems to which they were applied ; it was not necessary that they should obey the laws of inequality that were true for ordinary numbers. Boscovich's mistake is, of course, that of assuming that the constant is zero in every case ; and in this he is probably deceived by using the formula xy -f- A, instead of ^B/("-*l) -}- A, for the area. From the latter n—m n — m it is easily seen that since the initial area is zero, we must have A = ow/("~m). If n is m— n equal to or greater than m, the constant A is indeed zero ; but if n is less than m, the constant is infinite. The persistence of this error for so long a time, from 1655 to 175%> during which we have the writings of Newton, Leibniz, the Bernoullis and others on the calculus, seems to lend corroboration to a doubt as to whether the integral sign was properly understood as a summation between limits, and that this sum could be expressed as the difference of two values of the same function of those limits. It appears to me that this point is one of very great importance in the history of the development of mathematical thought. Some idea of how prolific Boscovich was as an author may be gathered from the catalogue of his writings appended at the end of this volume. This catalogue has been taken from the end of the original first Venetian edition, and brings the list up to the date of its publication, 1763. It was felt to be an impossible task to make this list complete up to the time of the death of Boscovich ; and an incomplete continuation did not seem desirable. Mention must however be made of one other work of Boscovich at least ; namely, a work in five quarto volumes, published in 1785, under the title of Opera pertinentia ad Opticam et Astronomiam. Finally, in order to bring out the versatility of the genius of Boscovich, we may mention just a few of his discoveries in science, which seem to call for special attention. In astro- nomical science, he speaks of the use of a telescope filled with liquid for the purpose of measuring the aberration of light ; he invented a prismatic micrometer contemporaneously with Rochon and Maskelyne. He gave methods for determining the orbit of a comet from three observations, and for the equator of the sun from three observations of a " spot " ; he carried out some investigations on the orbit of Uranus, and considered the rings of Saturn. In what was then the subsidiary science of optics, he invented a prism with a variable angle for measuring the refraction and dispersion of different kinds of glass ; and put forward a theory of achromatism for the objectives and oculars of the telescope. In mechanics and geodesy, he was apparently the first to solve the problem of the " body of greatest attraction " ; he successfully attacked the question of the earth's density ; and perfected the apparatus and advanced the theory of the measurement of the meridian. In mathematical theory, he seems to have recognized, before Lobachevski and Bolyai, the impossibility of a proof of Euclid's " parallel postulate " ; and considered the theory of the logarithms of negative numbers. J. M. C. N.B. — The page numbers on the left-hand pages of the index are the pages of the original Latin Edition of 1763 ; they correspond with the clarendon numbers inserted throughout the Latin text of this edition. CORRIGENDA Attention is called to the following important corrections, omissions, and alternative renderings ; misprints involving a single letter or syllable only are given at the end of the volume. p. 27, 1. 8, for in one plane read in the same direction p. 47, 1. 62, literally on which ... is exerted p. 49, 1. 33, for just as ... is read so that . . . may be P- S3> 1- 9> after a line add but not parts of the line itself p. 61, Art. 47, Alternative rendering: These instances make good the same point as water making its way through the pores of a sponge did for impenetrability ; p. 67, 1. 5, for it is allowable for me read I am disposed ; unless in the original libet is taken to be a misprint for licet p. 73, 1. 26, after nothing add in the strict meaning of the term p. 85, 1. 27, after conjunction add of the same point of space p. 91, 1. 25, Alternative rendering : and these properties might distinguish the points even in the view of the followers of Leibniz 1. 5 from bottom, Alternative rendering : Not to speak of the actual form of the leaves present in the seed p. 115, 1. 25, after the left add but that the two outer elements do not touch each other 1. 28, for two little spheres read one little sphere p. 117, 1. 41, for precisely read abstractly p. 125, 1. 29, for ignored read urged in reply p. 126, 1. 6 from bottom, it is -possible that acquirere is intended for acquiescere, with a corresponding change in the translation p. 129, Art. 162, marg. note, for on what they may be founded read in what it consists, p. 167, Art. 214, 1. 2 of marg. note, transpose by and on footnote, 1. I, for be at read bisect it at p. 199, 1. 24, for so that read just as p. 233, 1. 4 from bottom, for base to the angle read base to the sine of the angle last line, after vary insert inversely p. 307, 1. 5 from end, for motion, as (with fluids) takes place read motion from taking place p. 323, 1. 39, for the agitation will read the fluidity will P- 345» 1- 32> for described read destroyed p. 357, 1. 44, for others read some, others of others 1. 5 from end, for fire read a fiery and insert a comma before substance XIX THEORIA PHILOSOPHIC NATURALIS TYPOGRAPHUS VENETUS LECTORI PUS, quod tibi offero, jam ab annis quinque Viennse editum, quo plausu exceptum sit per Europam, noveris sane, si Diaria publica perlegeris, inter quse si, ut omittam caetera, consulas ea, quae in Bernensi pertinent ad initium anni 1761 ; videbis sane quo id loco haberi debeat. Systema continet Naturalis Philosophise omnino novum, quod jam ab ipso Auctore suo vulgo Boscovichianum appellant. Id quidem in pluribus Academiis jam passim publice traditur, nee tantum in annuis thesibus, vel disserta- tionibus impressis, ac propugnatis exponitur, sed & in pluribus elementaribus libris pro juventute instituenda editis adhibetur, exponitur, & a pluribus habetur pro archetype. Verum qui omnem systematis compagem, arctissimum partium nexum mutuum, fcecun- ditatem summam, ac usum amplissimum ac omnem, quam late patet, Naturam ex unica simplici lege virium derivandam intimius velit conspicere, ac contemplari, hoc Opus consulat, necesse est. Haec omnia me permoverant jam ab initio, ut novam Operis editionem curarem : accedebat illud, quod Viennensia exemplaria non ita facile extra Germaniam itura videbam, & quidem nunc etiam in reliquis omnibus Europse partibus, utut expetita, aut nuspiam venalia prostant, aut vix uspiam : systema vero in Italia natum, ac ab Auctore suo pluribus hie apud nos jam dissertationibus adumbratum, & casu quodam Viennae, quo se ad breve tempus contulerat, digestum, ac editum, Italicis potissimum typis, censebam, per univer- sam Europam disseminandum. Et quidem editionem ipsam e Viennensi exemplari jam turn inchoaveram ; cum illud mihi constitit, Viennensem editionem ipsi Auctori, post cujus discessum suscepta ibi fuerat, summopere displicere : innumera obrepsisse typorum menda : esse autem multa, inprimis ea, quas Algebraicas formulas continent, admodum inordinata, & corrupta : ipsum eorum omnium correctionem meditari, cum nonnullis mutationibus, quibus Opus perpolitum redderetur magis, & vero etiam additamentis. Illud ergo summopere desideravi, ut exemplar acquirerem ab ipso correctum, & auctum ac ipsum edition! praesentem haberem, & curantem omnia per sese. At id quidem per hosce annos obtinere non licuit, eo universam fere Europam peragrante ; donee demum ex tarn longa peregrinatione redux hue nuper se contulit, & toto adstitit editionis tempore, ac praeter correctores nostros omnem ipse etiam in corrigendo diligentiam adhibuit ; quanquam is ipse haud quidem sibi ita fidit, ut nihil omnino effugisse censeat, cum ea sit humanas mentis conditio, ut in eadem re diu satis intente defigi non possit. Haec idcirco ut prima quaedam, atque originaria editio haberi debet, quam qui cum Viennensi contulerit, videbit sane discrimen. E minoribus mutatiunculis multae pertinent ad expolienda, & declaranda plura loca ; sunt tamen etiam nonnulla potissimum in pagin- arum fine exigua additamenta, vel mutatiunculas exiguae factae post typographicam constructionem idcirco tantummodo, ut lacunulae implerentur quae aliquando idcirco supererant, quod plures ph'ylirae a diversis compositoribus simul adornabantur, & quatuor simul praela sudabant; quod quidem ipso praesente fieri facile potuit, sine ulla pertur- batione sententiarum, & ordinis. THE PRINTER AT VENICE TO THE READER ! OU will be well aware, if you have read the public journals, with what applause the work which I now offer to you has been received throughout Europe since its publication at Vienna five years ago. Not to mention others, if you refer to the numbers of the Berne Journal for the early part of the year 1761, you will not fail to see how highly it has been esteemed. It contains an entirely new system of Natural Philosophy, which is already commonly known as the Boscovichian theory, from the name of its author, As a matter of fact, it is even now a subject of public instruction in several Universities in different parts ; it is expounded not only in yearly theses or dissertations, both printed & debated ; but also in several elementary books issued for the instruction of the young it is introduced, explained, & by many considered as their original. Any one, however, who wishes to obtain more detailed insight into the whole structure of the theory, the close relation that its several parts bear to one another, or its great fertility & wide scope for the purpose of deriving the whole of Nature, in her widest range, from a single simple law of forces ; any one who wishes to make a deeper study of it must perforce study the work here offered. All these considerations had from the first moved me to undertake a new edition of the work ; in addition, there was the fact that I perceived that it would be a matter of some difficulty for copies of the Vienna edition to pass beyond the confines of Germany — indeed, at the present time, no matter how diligently they are inquired for, they are to be found on sale nowhere, or scarcely anywhere, in the rest of Europe. The system had its birth in Italy, & its outlines had already been sketched by the author in several dissertations pub- lished here in our own land ; though, as luck would have it, the system itself was finally put into shape and published at Vienna, whither he had gone for a short time. I therefore thought it right that it should be disseminated throughout the whole of Europe, & that preferably as the product of an Italian press. I had in fact already commenced an edition founded on a copy of the Vienna edition, when it came to my knowledge that the author was greatly dissatisfied with the Vienna edition, taken in hand there after his departure ; that innumerable printer's errors had crept in ; that many passages, especially those that contain Algebraical formulae, were ill-arranged and erroneous ; lastly, that the author himself had in mind a complete revision, including certain alterations, to give a better finish to the work, together with certain additional matter. That being the case, I was greatly desirous of obtaining a copy, revised & enlarged by himself ; I also wanted to have him at hand whilst the edition was in progress, & that he should superintend the whole thing for himself. This, however, I was unable to procure during the last few years, in which he has been travelling through nearly the whole of Europe ; until at last he came here, a little while ago, as he returned home from his lengthy wanderings, & stayed here to assist me during the whole time that the edition was in hand. He, in addition to our regular proof-readers, himself also used every care in cor- recting the proof ; even then, however, he has not sufficient confidence in himself as to imagine that not the slightest thing has escaped him. For it is a characteristic of the human mind that it cannot concentrate long on the same subject with sufficient attention. It follows that this ought to be considered in some measure as a first & original edition ; any one who compares it with that issued at Vienna will soon see the difference between them. Many of the minor alterations are made for the purpose of rendering certain passages more elegant & clear ; there are, however, especially at the foot of a page, slight additions also, or slight changes made after the type was set up, merely for the purpose of filling up gaps that were left here & there — these gaps being due to the fact that several sheets were being set at the same time by different compositors, and four presses were kept hard at work together. As he was at hand, this could easily be done without causing any disturbance of the sentences or the pagination. 4 TYPOGRAPHUS VENETUS LECTORI Inter mutationes occurret ordo numerorum mutatus in paragraphis : nam numerus 82 de novo accessit totus : deinde is, qui fuerat 261 discerptus est in 5 ; demum in Appendice post num. 534 factse sunt & mutatiunculse nonnullae, & additamenta plura in iis, quse pertinent ad sedem animse. Supplementorum ordo mutatus est itidem ; quse enim fuerant 3, & 4, jam sunt I, & 2 : nam eorum usus in ipso Opere ante alia occurrit. Illi autem, quod prius fuerat primum, nunc autem est tertium, accessit in fine Scholium tertium, quod pluribus numeris complec- titur dissertatiunculam integram de argumento, quod ante aliquot annos in Parisiensi Academia controversise occasionem exhibuit in Encyclopedico etiam dictionario attactum, in qua dissertatiuncula demonstrat Auctor non esse, cur ad vim exprimendam potentia quaepiam distantiae adhibeatur potius, quam functio. Accesserunt per totum Opus notulae marginales, in quibus eorum, quae pertractantur argumenta exponuntur brevissima, quorum ope unico obtutu videri possint omnia, & in memoriam facile revocari. Postremo loco ad calcem Operis additus est fusior catalogus eorum omnium, quse hue usque ab ipso Auctore sunt edita, quorum collectionem omnem expolitam, & correctam, ac eorum, quse nondum absoluta sunt, continuationem meditatur, aggressurus illico post suum regressum in Urbem Romam, quo properat. Hie catalogus impressus fuit Venetisis ante hosce duos annos in reimpressione ejus poematis de Solis ac Lunae defectibus. Porro earn, omnium suorum Operum Collectionem, ubi ipse adornaverit, typis ego meis excudendam suscipiam, quam magnificentissime potero. Haec erant, quae te monendum censui ; tu laboribus nostris fruere, & vive felix. THE PRINTER AT VENICE TO THE READER 5 Among the more Important alterations will be found a change in the order of numbering the paragraphs. Thus, Art. 82 is additional matter that is entirely new ; that which was formerly Art. 261 is now broken up into five parts ; &, in the Appendix, following Art. 534, both some slight changes and also several additions have been made in the passages that relate to the Seat of the Soul. The order of the Supplements has been altered also : those that were formerly num- bered III and IV are now I and II respectively. This was done because they are required for use in this work before the others. To that which was formerly numbered I, but is now III, there has been added a third scholium, consisting of several articles that between them give a short but complete dissertation on that point which, several years ago caused a controversy in the University of Paris, the same point being also discussed in the Dictionnaire Encydopedique. In this dissertation the author shows that there is no reason why any one power of the distance should be employed to express the force, in preference to a function. Short marginal summaries have been inserted throughout the work, in which the arguments dealt with are given in brief ; by the help of these, the whole matter may be taken in at a glance and recalled to mind with ease. Lastly, at the end of the work, a somewhat full catalogue of the whole of the author's publications up to the present time has been added. Of these publications the author intends to make a full collection, revised and corrected, together with a continuation of those that are not yet finished ; this he proposes to do after his return to Rome, for which city he is preparing to set out. This catalogue was printed in Venice a couple of years ago in connection with a reprint of his essay in verse on the eclipses of the Sun and Moon. Later, when his revision of them is complete, I propose to undertake the printing of this complete collection of his works from my own type, with all the sumptuousness at my command. Such were the matters that I thought ought to be brought to your notice. May you enjoy the fruit of our labours, & live in happiness. EPISTOLA AUCTORIS DEDICATORIA EDITIONIS VIENNENSIS AD CELSISSIMUM TUNC PRINCIPEM ARCHIEPISCOPUM VIENNENSEM, NUNC PR^TEREA ET CARDINALEM EMINENTISSIMUM, ET EPISCOPUM VACCIENSEM CHRISTOPHORUM E COMITATIBUS DE MIGAZZI IA.BIS veniam, Princeps Celsissime, si forte inter assiduas sacri regirninis curas importunus interpellator advenio, & libellum Tibi offero mole tenuem, nee arcana Religionis mysteria, quam in isto tanto constitutus fastigio adminis- tras, sed Naturalis Philosophise principia continentem. Novi ego quidem, quam totus in eo sis, ut, quam geris, personam sustineas, ac vigilantissimi sacrorum Antistitis partes agas. Videt utique Imperialis haec Aula, videt universa Regalis Urbs, & ingenti admiratione defixa obstupescit, qua dili- gentia, quo labore tanti Sacerdotii munus obire pergas. Vetus nimirum illud celeberrimum age, quod agis, quod ab ipsa Tibi juventute, cum primum, ut Te Romas dantem operam studiis cognoscerem, mihi fors obtigit, altissime jam insederat animo, id in omni reliquo amplissimorum munerum Tibi commissorum cursu haesit firmissime, atque idipsum inprimis adjectum tarn multis & dotibus, quas a Natura uberrime congestas habes, & virtutibus, quas tute diuturna Tibi exercitatione, atque assiduo labore comparasti, sanc- tissime observatum inter tarn varias forenses, Aulicas, Sacerdotales occupationes, istos Tibi tarn celeres dignitatum gradus quodammodo veluti coacervavit, & omnium una tarn populorum, quam Principum admirationem excitavit ubique, conciliavit amorem ; unde illud est factum, ut ab aliis alia Te, sublimiora semper, atque honorificentiora munera quodammodo velut avulsum, atque abstractum rapuerint. Dum Romse in celeberrimo illo, quod Auditorum Rotae appellant, collegio toti Christiano orbi jus diceres, accesserat Hetrusca Imperialis Legatio apud Romanum Pontificem exercenda ; cum repente Mech- liniensi Archiepiscopo in amplissima ilia administranda Ecclesia Adjutor datus, & destinatus Successor, possessione prsestantissimi muneris vixdum capta, ad Hispanicum Regem ab Augustissima Romanorum Imperatrice ad gravissima tractanda negotia Legatus es missus, in quibus cum summa utriusque Aulae approbatione versatum per annos quinque ditissima Vacciensis Ecclesia adepta est ; atque ibi dum post tantos Aularum strepitus ea, qua Christianum Antistitem decet, & animi moderatione, & demissione quadam, atque in omne hominum genus charitate, & singular! cura, ac diligentia Religionem administras, & sacrorum exceres curam ; non ea tantum urbs, atque ditio, sed universum Hungariae Regnum, quanquam exterum hominem, non ut civem suum tantummodo, sed ut Parentem aman- tissimum habuit, quern adhuc ereptum sibi dolet, & angitur ; dum scilicet minore, quam unius anni intervallo ab Ipsa Augustissima Imperatrice ad Regalem hanc Urbem, tot Imperatorum sedem, ac Austriacae Dominationis caput, dignum tantis dotibus explicandis theatrum, eocatum videt, atque in hac Celsissima Archiepiscopali Sede, accedente Romani Pontificis Auctoritate collocatum ; in qua Tu quidem personam itidem, quam agis, diligen- tissime sustinens, totus es in gravissimis Sacerdotii Tui expediendis negotiis, in iis omnibus, quae ad sacra pertinent, curandis vel per Te ipsum usque adeo, ut saepe, raro admodum per AUTHOR'S EPISTLE DEDICATING THE FIRST VIENNA EDITION TO CHRISTOPHER, COUNT DE MIGAZZI, THEN HIS HIGHNESS THE PRINCE ARCHBISHOP OF VIENNA, AND NOW ALSO IN ADDITION HIS EMINENCE THE CARDINAL, BISHOP OF VACZ OU will pardon me, Most Noble Prince, if perchance I come to disturb at an inopportune moment the unremitting cares of your Holy Office, & offer you a volume so inconsiderable in size ; one too that contains none of the inner mysteries of Religion, such as you administer from the highly exalted position to which you are ordained ; one that merely deals with the prin- ciples of Natural Philosophy. I know full well how entirely your time is taken up with sustaining the reputation that you bear, & in performing the duties of a highly conscientious Prelate. This Imperial Court sees, nay, the whole of this Royal City sees, with what care, what toil, you exert yourself to carry out the duties of so great a sacred office, & stands wrapt with an overwhelming admiration. Of a truth, that well-known old saying, " What you do, DO," which from your earliest youth, when chance first allowed me to make your acquaintance while you were studying in Rome, had already fixed itself deeply in your mind, has remained firmly implanted there during the whole of the remainder of a career in which duties of the highest importance have been committed to your care. Your strict observance of this maxim in particular, joined with those numerous talents so lavishly showered upon you by Nature, & those virtues which you have acquired for yourself by daily practice & unremitting toil, throughout your whole career, forensic, courtly, & sacerdotal, has so to speak heaped upon your shoulders those unusually rapid advances in dignity that have been your lot. It has aroused the admiration of all, both peoples & princes alike, in every land ; & at the same time it has earned for you their deep affection. The consequence was that one office after another, each ever more exalted & honourable than the preceding, has in a sense seized upon you & borne you away a captive. Whilst you were in Rome, giving judicial decisions to the whole Christian world in that famous College, the Rota of Auditors, there was added the duty of acting on the Tuscan Imperial Legation at the Court of the Roman Pontiff. Sud- denly you were appointed coadjutor to the Archbishop of Malines in the administration of that great church, & his future successor. Hardly had you entered upon the duties of that most distinguished appointment, than you were despatched by the August Empress of the Romans as Legate on a mission of the greatest importance. You occupied yourself on this mission for the space of five years, to the entire approbation of both Courts, & then the wealthy church of Vacz obtained your services. Whilst there, the great distractions of a life at Court being left behind, you administer the offices of religion & discharge the sacred rights with that moderation of spirit & humility that befits a Christian prelate, in charity towards the whole race of mankind, with a singularly attentive care. So that not only that city & the district in its see, but the whole realm of Hungary as well, has looked upon you, though of foreign race, as one of her own citizens ; nay, rather as a well beloved father, whom she still mourns & sorrows for, now that you have been taken from her. For, after less than a year had passed, she sees you recalled by the August Empress herself to this Imperial City, the seat of a long line of Emperors, & the capital of the Dominions of Austria, a worthy stage for the display of your great talents ; she sees you appointed, under the auspices of the authority of the Roman Pontiff, to this exalted Archiepiscopal see. Here too, sustaining with the utmost diligence the part you play so well, you throw your- self heart and soul into the business of discharging the weighty duties of your priesthood, or in attending to all those things that deal with the sacred rites with your own hands : so much so that we often see you officiating, & even administering the Sacraments, in our 8 EPISTOLA AUCTORIS DEDICATORIA PRI1VLE EDITIONIS VIENNENSIS haec nostra tempora exemplo, & publico operatum, ac ipsa etiam Sacramenta administrantem videamus in templis, & Tua ipsius voce populos, e superiore loco docentum audiamus, atque ad omne virtutum genus inflammantem. Novi ego quidem haec omnia ; novi hanc indolem, hanc animi constitutionem ; nee sum tamen inde absterritus, ne, inter gravissimas istas Tuas Sacerdotales curas, Philosophicas hasce meditationes meas, Tibi sisterem, ac tantulae libellum molis homini ad tantum culmen evecto porrigerem, ac Tuo vellem Nomine insignitum. Quod enim ad primum pertinet caput, non Theologicas tantum, sed Philosophicas etiam perquisitiones Christiano Antistite ego quidem dignissimas esse censeo, & universam Naturae contemplationem omnino arbitror cum Sacerdotii sanctitate penitus consentire. Mirum enim, quam belle ab ipsa consideratione Naturae ad caslestium rerum contemplationem disponitur animus, & ad ipsum Divinum tantae molis Conditorem assurgit, infinitam ejus Potentiam Sapientiam, Providentiam admiratus, quae erumpunt undique, & utique se produnt. Est autem & illud, quod ad supremi sacrorum Moderatoris curam pertinet providere, ne in prima ingenuae juventutis institutione, quae semper a naturalibus studiis exordium ducit, prava teneris mentibus irrepant, ac perniciosa principia, quae sensim Religionem corrumpant, & vero etiam evertant penitus, ac eruant a fundamentis ; quod quidem jam dudum tristi quodam Europae fato passim evenire cernimus, gliscente in dies malo, ut fucatis quibusdam, profecto perniciosissimis, imbuti principiis juvenes, turn demum sibi sapere videantur, cum & omnem animo religionem, & Deum ipsum sapientissimum Mundi Fabricatorem, atque Moderatorem sibi mente excusserint. Quamobrem qui veluti ad tribunal tanti Sacerdotum Principis Universae Physicae Theoriam, & novam potissimum Theoriam sistat, rem is quidem praestet sequissimam, nee alienum quidpiam ab ejus munere Sacerdotali offerat, sed cum eodem apprime consentiens. Nee vero exigua libelli moles deterrere me debuit, ne cum eo ad tantum Principem accederem. Est ille quidem satis tenuis libellus, at non & tenuem quoque rem continet. Argumentum pertractat sublime admodum, & nobile, in quo illustrando omnem ego quidem industriam coUocavi, ubi si quid praestitero, si minus infiliclter me gessero, nemo sane me impudentiae arguat, quasi vilem aliquam, & tanto indignam fastigio rem offeram. Habetur in eo novum quoddam Universae Naturalis Philosophiae genus a receptis hue usque, usi- tatisque plurimam discrepans, quanquam etiam ex iis, quae maxime omnium per haec tempora celebrantur, casu quodam praecipua quasque mirum sane in modum compacta, atque inter se veluti coagmentata conjunguntur ibidem, uti sunt simplicia atque inextensa Leibnitian- orum elementa, cum Newtoni viribus inducentibus in aliis distantiis accessum mutuum, in aliis mutuum recessum, quas vulgo attractiones, & repulsiones appellant : casu, inquam : neque enim ego conciliandi studio hinc, & inde decerpsi quaedam ad arbitrium selecta, quae utcumque inter se componerem, atque compaginarem : sed omni praejudicio seposito, a principiis exorsus inconcussis, & vero etiam receptis communiter, legitima ratiocinatione usus, & continue conclusionum nexu deveni ad legem virium in Natura existentium unicam, simplicem , continuam, quae mihi & constitutionem elementorum materiae, & Mechanicae leges, & generales materiae ipsius proprietates, & praecipua corporum discrimina, sua applicatione ita exhibuit, ut eadem in iis omnibus ubique se prodat uniformis agendi ratio, non ex arbitrariis hypothesibus, & fictitiis commentationibus, sed ex sola continua ratio- cinatione deducta. Ejusmodi autem est omnis, ut eas ubique vel definiat, vel adumbret combinationes elementorum, quae ad diversa prasstanda phaenomena sunt adhibendas, ad quas combinationes Conditoris Supremi consilium, & immensa Mentis Divinae vis ubique requiritur, quae infinites casus perspiciat, & ad rem aptissimos seligat, ac in Naturam inducat. Id mihi quidem argumentum est operis, in quo Theoriam meam expono, comprobo, vindico : turn ad Mechanicam primum, deinde ad Physicam applico, & uberrimos usus expono, ubi brevi quidem libello, sed admodum diuturnas annorum jam tredecim medita- tiones complector meas, eo plerumque tantummodo rem deducens, ubi demum cum AUTHOR'S EPISTLE DEDICATING THE FIRST VIENNA EDITION 9 churches (a somewhat unusual thing at the present time), and also hear you with your own voice exhorting the people from your episcopal throne, & inciting them to virtue of every kind. I am well aware of all this ; I know full well the extent of your genius, & your con- stitution of mind ; & yet I am not afraid on that account of putting into your hands, amongst all those weighty duties of your priestly office, these philosophical meditations of mine ; nor of offering a volume so inconsiderable in bulk to one who has attained to such heights of eminence ; nor of desiring that it should bear the hall-mark of your name. With regard to the first of these heads, I think that not only theological but also philosophical investigations are quite suitable matters for consideration by a Christian prelate ; & in my opinion, a contemplation of all the works of Nature is in complete accord with the sanctity of the priesthood. For it is marvellous how exceedingly prone the mind becomes to pass from a contemplation of Nature herself to the contemplation of celestial, things, & to give honour to the Divine Founder of such a mighty structure, lost in astonishment at His infinite Power & Wisdom & Providence, which break forth & disclose themselves in all directions & in all things. There is also this further point, that it is part of the duty of a religious superior to take care that, in the earliest training of ingenuous youth, which always takes its start from the study of the wonders of Nature, improper ideas do not insinuate themselves into tender minds ; or such pernicious principles as may gradually corrupt the belief in things Divine, nay, even destroy it altogether, & uproot it from its very foundations. This is what we have seen for a long time taking place, by some unhappy decree of adverse fate, all over Europe ; and, as the canker spreads at an ever increasing rate, young men, who have been made to imbibe principles that counterfeit the truth but are actually most pernicious doc- trines, do not think that they have attained to wisdom until they have banished from their minds all thoughts of religion and of God, the All- wise Founder and Supreme Head of the Universe. Hence, one who so to speak sets before the judgment-seat of such a prince of the priesthood as yourself a theory of general Physical Science, & more especially one that is new, is doing nothing but what is absolutely correct. Nor would he be offering him anything inconsistent with his priestly office, but on the contrary one that is in complete harmony with it. Nor, secondly, should the inconsiderable size of my little book deter me from approach- ing with it so great a prince. It is true that the volume of the book is not very great, but the matter that it contains is not unimportant as well. The theory it develops is a strik- ingly sublime and noble idea ; & I have done my very best to explain it properly. If in this I have somewhat succeeded, if I have not failed altogether, let no one accuse me of presumption, as if I were offering some worthless thing, something unworthy of such dis- tinguished honour. In it is contained a new kind of Universal Natural Philosophy, one that differs widely from any that are generally accepted & practised at the present time ; although it so happens that the principal points of all the most distinguished theories of the present day, interlocking and as it were cemented together in a truly marvellous way, are combined in it ; so too are the simple unextended elements of the followers of Leibniz, as well as the Newtonian forces producing mutual approach at 'some distances & mutual separation at others, usually called attractions and repulsions. I use the words " it so happens " because I have not, in eagerness to make the whole consistent, selected one thing here and another there, just as it suited me for the purpose of making them agree & form a connected whole. On the contrary, I put on one side all prejudice, & started from fundamental principles that are incontestable, & indeed are those commonly accepted ; I used perfectly sound arguments, & by a continuous chain of deduction I arrived at a single, simple, continuous law for the forces that exist in Nature. The application of this law explained to me the constitution of the elements of matter, the laws of Mechanics, the general properties of matter itself, & the chief characteristics of bodies, in such a manner that the same uniform method of action in all things disclosed itself at all points ; being deduced, not from arbitrary hypotheses, and fictitibus explanations, but from a single con- tinuous chain of reasoning. Moreover it is in all its parts of such a kind as defines, or suggests, in every case, the combinations of the elements that must be employed to produce different phenomena. For these combinations the wisdom of the Supreme Founder of the Universe, & the mighty power of a Divine Mind are absolutely necessary ; naught but one that could survey the countless cases, select those most suitable for the purpose, and introduce them into the scheme of Nature. This then is the argument of my work, in which I explain, prove & defend my theory ; then I apply it, in the first instance to Mechanics, & afterwards to Physics, & set forth the many advantages to be derived from it. Here, although the book is but small, I yet include the well-nigh daily meditations of the last thirteen years, carrying on my conclu- io EPISTOLA AUCTORIS DEDICATORIA PRIM.£ EDITIONIS VIENNENSIS communibus Philosophorum consentio placitis, & ubi ea, quae habemus jam pro compertis, ex meis etiam deductionibus sponte fluunt, quod usque adeo voluminis molem contraxit. Dederam ego quidem dispersa dissertatiunculis variis Theorise meae qusedam velut specimina, quae inde & in Italia Professores publicos nonnullos adstipulatores est nacta, & jam ad exteras quoque gentes pervasit ; sed ea nunc primum tota in unum compacta, & vero etiam plusquam duplo aucta, prodit in publicum, quern laborem postremo hoc mense, molestiori- bus negotiis, quae me Viennam adduxerant, & curis omnibus exsolutus suscepi, dum in Italiam rediturus opportunam itineri tempus inter assiduas nives opperior, sed omnem in eodem adornando, & ad communem mediocrum etiam Philosophorum captum accommo- dando diligentiam adhibui. Inde vero jam facile intelliges, cur ipsum laborem meum ad Te deferre, & Tuo nuncupare Nomini non dubitaverim. Ratio ex iis, quae proposui, est duplex : primo quidem ipsum argumenti genus, quod Christianum Antistitem non modo non dedecet, sed etiam apprime decet : turn ipsius argumenti vis, atque dignitas, quae nimirum confirmat, & erigit nimium fortasse impares, sed quantum fieri per me potuit, intentos conatus meos ; nam quidquid eo in genere meditando assequi possum, totum ibidem adhibui, ut idcirco nihil arbitrer a mea tenuitate proferri posse te minus indignum, cui ut aliquem offerrem laborum meorum fructum quantumcunque, exposcebat sane, ac ingenti clamore quodam efnagitabat tanta erga me humanitas Tua, qua jam olim immerentem complexus Romae, hie etiam fovere pergis, nee in tanto dedignatus fastigio, omni benevolentiae significatione prosequeris. Accedit autem & illud, quod in hisce terris vix adhuc nota, vel etiam ignota penitus Theoria mea Patrocinio indiget, quod, si Tuo Nomine insignata prodeat in publicum, obtinebit sane validissimum, & secura vagabitur : Tu enim illam, parente velut hie orbatam suo, in dies nimirum discessuro, & quodammodo veluti posthumam post ipsum ejus discessum typis impressam, & in publicum prodeuntem tueberis, fovebisque. Haec sunt, quae meum Tibi consilium probent, Princeps Celsissime : Tu, qua soles humanitate auctorem excipere, opus excipe, & si forte adhuc consilium ipsum Tibi visum fuerit improbandum ; animum saltern aequus respice obsequentissimum Tibi, ac devinct- issimum. Vale. Dabam Viennce in Collegia Academico Soc. JESU Idibus Febr. MDCCLFIIL AUTHOR'S EPISTLE DEDICATING THE FIRST VIENNA EDITION 11 sions for the most part only up to the point where I finally agreed with the opinions com- monly held amongst philosophers, or where theories, now accepted as established, are the natural results of my deductions also ; & this has in some measure helped to diminish the size of the volume. I had already published some instances, so to speak, of my general theory in several short dissertations issued at odd times ; & on that account the theory has found some supporters amongst the university professors in Italy, & has already made its way into foreign countries. But now for the first time is it published as a whole in a single volume, the matter being indeed more than doubled in amount. This work I have carried out during the last month, being quit of the troublesome business that brought me to Vienna, and of all other cares ; whilst I wait for seasonable time for my return journey through the everlasting snow to Italy. I have however used my utmost endeavours in preparing it, and adapting it to the ordinary intelligence of philosophers of only moderate attainments. From this you will readily understand why I have not hesitated to bestow this book of mine upon you, & to dedicate it to you. My reason, as can be seen from what I have said, was twofold ; in the first place, the nature of my theme is one that is not only not unsuitable, but is suitable in a high degree, for the consideration of a Christian priest ; secondly, the power & dignity of the theme itself, which doubtless gives strength & vigour to my efforts — perchance rather feeble, but, as far as in me lay, earnest. What- ever in that respect I could gain by the exercise of thought, I have applied the whole of it to this matter ; & consequently I think that nothing less unworthy of you can be pro- duced by my poor ability ; & that I should offer to you some such fruit of my labours was surely required of me, & as it were clamorously demanded by your great kindness to me ; long ago in Rome you had enfolded my unworthy self in it, & here now you continue to be my patron, & do not disdain, from your exalted position, to honour me with every mark of your goodwill. There is still a further consideration, namely, that my Theory is as yet almost, if not quite, unknown in these parts, & therefore needs a patron's support ; & this it will obtain most effectually, & will go on its way in security if it comes before the public franked with your name. For you will protect & cherish it, on its publication here, bereaved as it were of that parent whose departure in truth draws nearer every day ; nay rather posthumous, since it will be seen in print only after he has gone. Such are my grounds for hoping that you will approve my idea, most High Prince. I beg you to receive the work with the same kindness as you used to show to its author ; &, if perchance the idea itself should fail to meet with your approval, at least regard favourably the intentions of your most humble & devoted servant. Farewell. University College of the Society of Jesus, VIENNA, February i$th, 1758. AD LECTOREM EX EDITIONS VIENNENSI amice Lector, Philosophic Naturalis Theoriam ex unica lege virium deductam, quam & ubi jam olim adumbraverim, vel etiam ex parte explica- verim, y qua occasione nunc uberius pertractandum, atque augendam etiam, susceperim, invenies in ipso -primes •partis exordia. Libuit autem hoc opus dividere in partes tres, quarum prima continet explicationem Theories ipsius, ac ejus analyticam deductionem, & vindicationem : secunda applicationem- satis uberem ad Mechanicam ; tertia applicationem ad Physicam. Porro illud inprimis curandum duxi, ut omnia, quam liceret, dilucide exponerentur, nee sublimiore Geometria, aut Calculo indigerent. Et quidem in prima, ac tertia parte non tantum nullcs analyticee, sed nee geometries demonstrations occurrunt, paucissimis qiiibusdam, quibus indigeo, rejectis in adnotatiunculas, quas in fine paginarum quarundam invenies. Queedam autem admodum pauca, quce majorem Algebra, & Geometries cognitionem requirebant, vel erant complicatiora aliquando, & alibi a me jam edita, in fine operis apposui, quce Supplementorum appellavi nomine, ubi W ea addidi, quce sentio de spatio, ac tempore, Theories mece consentanea, ac edita itidem jam alibi. In secunda parte, ubi ad Mechanicam applicatur Theoria,a geome- tricis, W aliquando etiam ab algebraicis demonstrationibus abstinere omnino non potui ; sed ece ejusmodi sunt, ut vix unquam requirant aliud, quam Euclideam Geometriam, & primas Trigonometries notiones maxime simplices, ac simplicem algorithmum. In prima quidem parte occurrunt Figures geometricce complures, quce prima fronte vide- buntur etiam complicate? rem ipsam intimius non perspectanti ; verum ece nihil aliud exhibent, nisi imaginem quandam rerum, quce ipsis oculis per ejusmodi figuras sistuntur contemplandce. Ejusmodi est ipsa ilia curva, quce legem virium exhibet. Invenio ego quidem inter omnia materice puncta vim quandam mutuam, quce a distantiis pendet, £5" mutatis distantiis mutatur ita, ut in aliis attractiva sit, in aliis repulsiva, sed certa quadam, y continua lege. Leges ejusmodi variationis binarum quantitatum a se invicem pendentium, uti Jiic sunt distantia, y vis, exprimi possunt vel per analyticam formulam, vel per geometricam curvam ; sed ilia prior expressio & multo plures cognitiones requirit ad Algebram pertinentes, & imaginationem non ita adjuvat, ut heec posterior, qua idcirco sum usus in ipsa prima operis parte, rejecta in Supplementa formula analytica, quce y curvam, & legem virium ab ilia expressam exhibeat. Porro hue res omnis reducitur. Habetur in recta indefinita, quce axis dicitur, punctum quoddam, a quo abscissa ipsius rectce segmenta referunt distantias. Curva linea protenditur secundum rectam ipsam, circa quam etiam serpit, y eandem in pluribus secat punctis : rectce a fine segmentorum erectce perpendiculariter usque ad curvam, exprimunt vires, quce majores sunt, vel minores, prout ejusmodi rectce sunt itidem majores, vel minores ; ac eesdem ex attrac- tivis migrant in repulsivis, vel vice versa, ubi illce ipsce perpendiculares rectce directionem mutant, curva ab alter a axis indefiniti plaga migrante ad alter am. Id quidem nullas requirit geometricas demonstrations, sed meram cognitionem vocum quarundam, quce vel ad prima per- tinent Geometries elementa, y notissimce sunt, vel ibi explicantur, ubi adhibentur. Notissima autem etiam est significatio vocis Asymptotus, unde & crus asymptoticum curvce appellatur ; dicitur nimirum recta asymptotus cruris cujuspiam curvce, cum ipsa recta in infinitum producta, ita ad curvilineum arcum productum itidem in infinitum semper accedit magis, ut distantia minuatur in infinitum, sed nusquam penitus evanescat, illis idcirco nunquam invicem con- venientibus. Consider atio porro attenta curvce propositce in Fig. I, &rationis, qua per illam exprimitur 12 THE PREFACE TO THE READER THAT APPEARED IN THE VIENNA EDITION EAR Reader, you have before you a Theory of Natural Philosophy deduced from a single law of Forces. You will find in the opening paragraphs of the first section a statement as to where the Theory has been already published in outline, & to a certain extent explained ; & also the occasion that led me to undertake a more detailed treatment & enlargement of it. For I have thought fit to divide the work into three parts ; the first of these contains the exposition of the Theory itself, its analytical deduction & its demonstration ; the second a fairly full application to Mechanics ; & the third an application to Physics. The most important point, I decided, was for me to take the greatest care that every- thing, as far as was possible, should be clearly explained, & that there should be no need for higher geometry or for the calculus. Thus, in the first part, as well as in the third, there are no proofs by analysis ; nor are there any by geometry, with the exception of a very few that are absolutely necessary, & even these you will find relegated to brief notes set at the foot of a page. I have also added some very few proofs, that required a knowledge of higher algebra & geometry, or were of a rather more complicated nature, all of which have been already published elsewhere, at the end of the work ; I have collected these under the heading Supplements ; & in them I have included my views on Space & Time, which are in accord with my main Theory, & also have been already published elsewhere. In the second part, where the Theory is applied to Mechanics, I have not been able to do without geometrical proofs altogether ; & even in some cases I have had to give algebraical proofs. But these are of such a simple kind that they scarcely ever require anything more than Euclidean geometry, the first and most elementary ideas of trigonometry, and easy analytical calculations. It is true that in the first part there are to be found a good many geometrical diagrams, which at first sight, before the text is considered more closely, will appear to be rather complicated. But these present nothing else but a kind of image of the subjects treated, which by means of these diagrams are set before the eyes for contemplation. The very curve that represents the law of forces is an instance of this. I find that between all points of matter there is a mutual force depending on the distance between them, & changing as this distance changes ; so that it is sometimes attractive, & sometimes repulsive, but always follows a definite continuous law. Laws of variation of this kind between two quantities depending upon one another, as distance & force do in this instance, may be represented either by an analytical formula or by a geometrical curve ; but the former method of representation requires far more knowledge of algebraical processes, & does not assist the imagination in the way that the latter does. Hence I have employed the latter method in the first part of the work, & relegated to the Supplements the analytical formula which represents the curve, & the law of forces which the curve exhibits. The whole matter reduces to this. In a straight line of indefinite length, which is called the axis, a fixed point is taken ; & segments of the straight line cut off from this point represent the distances. A curve is drawn following the general direction of this straight line, & winding about it, so as to cut it in several places. Then perpendiculars that are drawn from the ends of the segments to meet the curve represent the forces ; these forces are greater or less, according as such perpendiculars are greater or less ; & they pass from attractive forces to repulsive, and vice versa, whenever these perpendiculars change their direction, as the curve passes from one side of the axis of indefinite length to the other side of it. Now this requires no geometrical proof, but only a knowledge of certain terms, which either belong to the first elementary principles of "geometry, & are thoroughly well known, or are such as can be defined when they are used. The term Asymptote is well known, and from the same idea we speak of the branch of a curve as being asymptotic ; thus a straight line is said to be the asymptote to any branch of a curve when, if the straight line is indefinitely produced, it approaches nearer and nearer to the curvilinear arc which is also prolonged indefinitely in such manner that the distance between them becomes indefinitely diminished, but never altogether vanishes, so that the straight line & the curve never really meet. A careful consideration of the curve given in Fig. I, & of the way in which the relation 14 AD LECTOREM EX EDITIONE VIENNENSI nexus inter vires, y distantias, est utique admodum necessaria ad intelligendam Theoriam ipsam, cujus ea est prcecipua qucedam veluti clavis, sine qua omnino incassum tentarentur cetera ; sed y ejusmodi est, ut tironum, & sane etiam mediocrium, immo etiam longe infra mediocritatem collocatorum, captum non excedat, potissimum si viva accedat Professoris vox mediocriter etiam versati in Mechanica, cujus ope, pro certo habeo, rem ita patentem omnibus reddi posse, ut ii etiam, qui Geometric penitus ignari sunt, paucorum admodum explicatione vocabulorum accidente, earn ipsis oculis intueantur omnino perspicuam. In tertia parte supponuntur utique nonnulla, quce demonstrantur in secunda ; sed ea ipsa sunt admodum pauca, & Us, qui geometricas demonstrationes fastidiunt, facile admodum exponi possunt res ipsce ita, ut penitus etiam sine ullo Geometries adjumento percipiantur, quanquam sine Us ipsa demonstratio baberi non poterit ; ut idcirco in eo differre debeat is, qui secundam partem attente legerit, & Geometriam calleat, ab eo, qui earn omittat, quod ille primus veritates in tertia parte adhibitis, ac ex secunda erutas, ad, explicationem Physicce, intuebitur per evi- dentiam ex ipsis demonstrationibus haustam, hie secundus easdem quodammodo per fidem Geo- metris adhibitam credet. Hujusmodi inprimis est illud, particulam compositam ex punctis etiam homogeneis, prceditis lege virium proposita, posse per solam diversam ipsorum punctorum dispositionem aliam particulam per certum intervallum vel perpetuo attrahere, vel perpetuo repellere, vel nihil in earn agere, atque id ipsum viribus admodum diversis, y quce respectu diver- sarum particularum diver see sint, & diver see respectu partium diver sarum ejusdem particulce, ac aliam particulam alicubi etiam urgeant in latus, unde plurium phcenomenorum explicatio in Physica sponte fluit. Verum qui omnem Theories, y deductionum compagem aliquanto altius inspexerit, ac diligentius perpenderit, videbit, ut spero, me in hoc perquisitionis genere multo ulterius progressum esse, quam olim Newtonus ipse desideravit. Is enim in postremo Opticce questione prolatis Us, quce per vim attractivam, & vim repulsivam, mutata distantia ipsi attractive suc- cedentem, explicari poterant, hcec addidit : " Atque hcec quidem omnia si ita sint, jam Natura universa valde erit simplex, y consimilis sui, perficiens nimirum magnos omnes corporum ccelestium motus attractione gravitatis, quce est mutua inter corpora ilia omnia, & minores fere omnes particularum suarum motus alia aliqua vi attrahente, & repellente, qua est inter particulas illas mutua" Aliquanto autem inferius de primigeniis particulis agens sic habet : " Porro videntur mihi hce particulce primigenice non modo in se vim inertice habere, motusque leges passivas illas, quce ex vi ista necessario oriuntur ; verum etiam motum perpetuo accipere a certis principiis actuosis, qualia nimirum sunt gravitas, £ff causa fermentationis, & cohcerentia corporum. Atque hcec quidem principia considero non ut occultas qualitates, quce ex specificis rerum formis oriri fingantur, sed ut universales Naturce leges, quibus res ipsce sunt formatce. Nam principia quidem talia revera existere ostendunt phenomena Naturce, licet ipsorum causce quce sint, nondum fuerit explicatum. Affirmare, singulas rerum species specificis prceditas esse qualita- tibus occultis, per quas eae vim certam in agenda habent, hoc utique est nihil dicere : at ex phcenomenis Naturce duo, vel tria derivare generalia motus principia, & deinde explicare, quemadmodum proprietates, & actiones rerum corporearum omnium ex istis principiis conse- quantur, id vero magnus esset factus in Philosophia progressus, etiamsi principiorum istorum causce nondum essent cognitce. Quare motus principia supradicta proponere non dubito, cum per Naturam universam latissime pateant" Hcec ibi Newtonus, ubi is quidem magnos in Philosophia progressus facturum arbitratus est eum, qui ad duo, vel tria generalia motus principia ex Naturce phcenomenis derivata pheeno- menorum explicationem reduxerit, & sua principia protulit, ex quibus inter se diversis eorum aliqua tantummodo explicari posse censuit. Quid igitur, ubi tf? ea ipsa tria, & alia prcecipua quceque, ut ipsa etiam impenetrabilitas, y impulsio reducantur ad principium unicum legitima ratiocinatione deductum ? At id -per meam unicam, & simplicem virium legemprcestari, patebit sane consideranti operis totius Synopsim quandam, quam hie subjicio ; sed multo magis opus ipsum diligentius pervolventi. THE PRINTER AT VENICE TO THE READER \ OU will be well aware, if you have read the public journals, with what applause the work which I now offer to you has been received throughout Europe since its publication at Vienna five years ago. Not to mention others, if you refer to the numbers of the Berne Journal for the early part of the year 1761, you will not fail to see how highly it has been esteemed. It contains an entirely new system of Natural Philosophy, which is already commonly known as the Boscovichian theory, from the name of its author, As a matter of fact, it is even now a subject of public instruction in several Universities in different parts ; it is expounded not only in yearly theses or dissertations, both printed & debated ; but also in several elementary books issued for the instruction of the young it is introduced, explained, & by many considered as their original. Any one, however, who wishes to obtain more detailed insight into the whole structure of the theory, the close relation that its several parts bear to one another, or its great fertility & wide scope for the purpose of deriving the whole of Nature, in her widest range, from a single simple law of forces ; any one who wishes to make a deeper study of it must perforce study the work here offered. All these considerations had from the first moved me to undertake a new edition of the work ; in addition, there was the fact that I perceived that it would be a matter of some difficulty for copies of the Vienna edition to pass beyond the confines of Germany — indeed, at the present time, no matter how diligently they are inquired for, they are to be found on sale nowhere, or scarcely anywhere, in the rest of Europe. The system had its birth in Italy, & its outlines had already been sketched by the author in several dissertations pub- lished here in our own land ; though, as luck would have it, the system itself was finally put into shape and published at Vienna, whither he had gone for a short time. I therefore thought it right that it should be disseminated throughout the whole of Europe, & that preferably as the product of an Italian press. I had in fact already commenced an edition founded on a copy of the Vienna edition, when it came to my knowledge that the author was greatly dissatisfied with the Vienna edition, taken in hand there after his departure ; that innumerable printer's errors had crept in ; that many passages, especially those that contain Algebraical formulae, were ill-arranged and erroneous ; lastly, that the author himself had in mind a complete revision, including certain alterations, to give a better finish to the work, together with certain additional matter. That being the case, I was greatly desirous of obtaining a copy, revised & enlarged by himself ; I also wanted to have him at hand whilst the edition was in progress, & that he should superintend the whole thing for himself. This, however, I was unable to procure during the last few years, in which he has been travelling through nearly the whole of Europe ; until at last he came here, a little while ago, as he returned home from his lengthy wanderings, & stayed here to assist me during the whole time that the edition was in hand. He, in addition to our regular proof-readers, himself also used every care in cor- recting the proof ; even then, however, he has not sufficient confidence in himself as to imagine that not the slightest thing has escaped him. For it is a characteristic of the human mind that it cannot concentrate long on the same subject with sufficient attention. It follows that this ought to be considered in some measure as a first & original edition ; any one who compares it with that issued at Vienna will soon see the difference between them. Many of the minor alterations are made for the purpose of rendering certain passages more elegant & clear ; there are, however, especially at the foot of a page, slight additions also, or slight changes made after the type was set up, merely for the purpose of filling up gaps that were left here & there — these gaps being due to the fact that several sheets were being set at the same time by different compositors, and four presses were kept hard at work together. As he was at hand, this could easily be done without causing any disturbance of the sentences or the pagination. 14 AD LECTOREM EX EDITIONE VIENNENSI nexus inter vires, & distantias, est utique admodum necessaria ad intelligendam Theoriam ipsam, cujus ea est prcecipua queedam veluti clavis, sine qua omnino incassum tentarentur cetera ; sed y ejusmodi est, ut tironum, & sane etiam mediocrium, immo etiam longe infra mediocritatem collocatorum, captum non excedat, potissimum si viva accedat Professoris vox mediocriter etiam versati in Mechanics, cujus ope, pro certo habeo, rem ita patentem omnibus reddi posse, ut ii etiam, qui Geometric? penitus ignari sunt, paucorum admodum explicatione vocabulorum accidente, earn ipsis oculis intueantur omnino perspicuam, In tertia parte supponuntur utique nonnulla, que? demonstrantur in secunda ; sed ea ipsa sunt admodum pauca, & Us, qui geometricas demonstrationes fastidiunt, facile admodum exponi possunt res ipsee ita, ut penitus etiam sine ullo Geometric adjumento percipiantur, quanquam sine Us ipsa demonstratio haberi non poterit ; ut idcirco in eo differre debeat is, qui secundam partem attente legerit, y Geometriam calleat, ab eo, qui earn omittat, quod ille primus veritates in tertia parte adhibitis, ac ex secunda erutas, ad explicationem Physics, intuebitur per evi- dentiam ex ipsis demonstrationibus baustam, hie secundus easdem quodammodo per fidem Geo- metris adhibitam credet. Hujusmodi inprimis est illud, particulam compositam ex punctis etiam bomogeneis, preeditis lege virium proposita, posse per solam diversam ipsorum punctorum dispositionem aliam particulam per cerium intervallum vel perpetuo attrahere, vel perpetuo repellere, vel nihil in earn agere, atque id ipsum viribus admodum diversis, y que? respectu diver- sarum particularum diver see sint, y diverse? respectu partium diver sarum ejusdem particulce, ac aliam particulam alicubi etiam urgeant in latus, unde plurium pheenomenorum explicatio in Physica sponte ftuit. Ferum qui omnem Theorie?, y deductionum compagem aliquanto altius inspexerit, ac diligentius perpenderit, videbit, ut spero, me in hoc perquisitionis genere multo ulterius progressum esse, quam olim Newtonus ipse desideravit. Is enim in postremo Opticce questione prolatis Us, qua per vim attractivam, y vim repulsivam, mutata distantia ipsi attractive? suc- cedentem, explicari poterant, he?c addidit : " Atque he?c quidem omnia si ita sint, jam Natura universa valde erit simplex, y consimilis sui, perficiens nimirum magnos omnes corporum ccelestium motus attractione gravitatis, quee est mutua inter corpora ilia omnia, y minores fere omnes particularum suarum motus alia aliqua vi attrabente, y repellente, quiz est inter particulas illas mutua." Aliquanto autem inferius de primigeniis particulis agens sic habet : " Porro videntur mihi he? particule? primigeniee non modo in se vim inertice habere, motusque leges passivas illas, que? ex vi ista necessario oriuntur ; verum etiam motum perpetuo accipere a certis principiis actuosis, qualia nimirum sunt gravitas, y causa fermentationis, y cohcerentia corporum. Atque heec quidem principia considero non ut occultas qualitates, que? ex specificis rerum formis oriri fingantur, sed ut universales Nature? leges, quibus res ipse? sunt formates. Nam principia quidem talia revera existere ostendunt phenomena Nature?, licet ipsorum cause? que? sint, nondum fuerit explicatum. Affirmare, singulas rerum species specificis preeditas esse qualita- tibus occultis, per quas eae vim certam in agenda habent, hoc utique est nihil dicere : at ex phcenomenis Nature? duo, vel tria derivare generalia motus principia, y deinde explicare, quemadmodum proprietates, y actiones rerum corporearum omnium ex istis principiis conse- quantur, id vero magnus esset factus in Philosophia progressus, etiamsi principiorum istorum cause? nondum essent cognite?. Quare motus principia supradicta proponere non dubito, cum per Naturam universam latissime pateant" Hc?c ibi Newtonus, ubi is quidem magnos in Philosophia progressus facturum arbitratus est eum, qui ad duo, vel tria generalia motus principia ex Nature? pheenomenis derivata phe?no- menorum explicationem reduxerit, y sua principia protulit, ex quibus inter se diversis eorum aliqua tantummodo explicari posse censuit. Quid igitur, ubi y ea ipsa tria, y alia preecipua quczque, ut ipsa etiam impenetrabilitas, y impulsio reducantur ad principium unicum legitima ratiocinatione deductum ? At id per meam unicam, y simplicem virium legempr<zstari,patebit sane consideranti operis totius Synopsim quandam, quam hie subjicio ; sed multo magis opus ipsum diligentius pervolventi. THE PRINTER AT VENICE TO THE READER |JOU will be well aware, if you have read the public journals, with what applause the work which I now offer to you has been received throughout Europe since its publication at Vienna five years ago. Not to mention others, if you refer to the numbers of the Berne Journal for the early part of the year 1761, you will not fail to see how highly it has been esteemed. It contains an entirely new system of Natural Philosophy, which is already commonly known as the Boscovicbian theory, from the name of its author, As a matter of fact, it is even now a subject of public instruction in several Universities in different parts ; it is expounded not only in yearly theses or dissertations, both printed & debated ; but also in several elementary books issued for the instruction of the young it is introduced, explained, & by many considered as their original. Any one, however, who wishes to obtain more detailed insight into the whole structure of the theory, the close relation that its several parts bear to one another, or its great fertility & wide scope for the purpose of deriving the whole of Nature, in her widest range, from a single simple law of forces ; any one who wishes to make a deeper study of it must perforce study the work here offered. All these considerations had from the first moved me to undertake a new edition of the work ; in addition, there was the fact that I perceived that it would be a matter of some difficulty for copies of the Vienna edition to pass beyond the confines of Germany — indeed, at the present time, no matter how diligently they are inquired for, they are to be found on sale nowhere, or scarcely anywhere, in the rest of Europe. The system had its birth in Italy, & its outlines had already been sketched by the author in several dissertations pub- lished here in our own land ; though, as luck would have it, the system itself was finally put into shape and published at Vienna, whither he had gone for a short time. I therefore thought it right that it should be disseminated throughout the whole of Europe, & that preferably as the product of an Italian press. I had in fact already commenced an edition founded on a copy of the Vienna edition, when it came to my knowledge that the author was greatly dissatisfied with the Vienna edition, taken in hand there after his departure ; that innumerable printer's errors had crept in ; that many passages, especially those that contain Algebraical formulae, were ill-arranged and erroneous ; lastly, that the author himself had in mind a complete revision, including certain alterations, to give a better finish to the work, together with certain additional matter. That being the case, I was greatly desirous of obtaining a copy, revised & enlarged by himself ; I also wanted to have him at hand whilst the edition was in progress, & that he should superintend the whole thing for himself. This, however, I was unable to procure during the last few years, in which he has been travelling through nearly the whole of Europe ; until at last he came here, a little while ago, as he returned home from his lengthy wanderings, & stayed here to assist me during the whole time that the edition was in hand. He, in addition to our regular proof-readers, himself also used every care in cor- recting the proof ; even then, however, he has not sufficient confidence in himself as to imagine that not the slightest thing has escaped him. For it is a characteristic of the human mind that it cannot concentrate long on the same subject with sufficient attention. It follows that this ought to be considered in some measure as a first & original edition ; any one who compares it with that issued at Vienna will soon see the difference between them. Many of the minor alterations are made for the purpose of rendering certain passages more elegant & clear ; there are, however, especially at the foot of a page, slight additions also, or slight changes made after the type was set up, merely for the purpose of filling up gaps that were left here & there — these gaps being due to the fact that several sheets were being set at the same time by different compositors, and four presses were kept hard at work together. As he was at hand, this could easily be done without causing any disturbance of the sentences or the pagination. 4 TYPOGRAPHUS VENETUS LECTORI Inter mutationes occurret ordo numerorum mutatus in paragraphis : nam numerus 82 de novo accessit totus : deinde is, qui fuerat 261 discerptus est in 5 ; demum in Appendice post num. 534 factae sunt & mutatiunculae nonnullae, & additamenta plura in iis, quae pertinent ad sedem animse. Supplementorum ordo mutatus est itidem ; quae enim fuerant 3, & 4, jam sunt i, & 2 : nam eorum usus in ipso Opere ante alia occurrit. UK autem, quod prius fuerat primum, nunc autem est tertium, accessit in fine Scholium tertium, quod pluribus numeris complec- titur dissertatiunculam integrant de argumento, quod ante aliquot annos in Parisiensi Academia controversiae occasionem exhibuit in Encyclopedico etiam dictionario attactum, in qua dissertatiuncula demonstrat Auctor non esse, cur ad vim exprimendam potentia quaepiam distantice adhibeatur potius, quam functio. Accesserunt per totum Opus notulae marginales, in quibus eorum, quae pertractantur argumenta exponuntur brevissima, quorum ope unico obtutu videri possint omnia, & in memoriam facile revocari. Postremo loco ad calcem Operis additus est fusior catalogus eorum omnium, quae hue usque ab ipso Auctore sunt edita, quorum collectionem omnem expolitam, & correctam, ac eorum, quse nondum absoluta sunt, continuationem meditatur, aggressurus illico post suum regressum in Urbem Romam, quo properat. Hie catalogus impressus fuit Venetisis ante hosce duos annos in reimpressione ejus poematis de Solis ac Lunae defectibus. Porro earn omnium suorum Operum Collectionem, ubi ipse adornaverit, typis ego meis excudendam suscipiam, quam magnificentissime potero. Haec erant, quae te monendum censui ; tu laboribus nostris fruere, & vive felix. THE PREFACE TO THE READER THAT APPEARED IN THE VIENNA EDITION Reader, you have before you a Theory of Natural Philosophy deduced from a single law of Forces. You will find in the opening paragraphs of the first section a statement as to where the Theory has been already published in outline, & to a certain extent explained ; & also the occasion that led me to undertake a more detailed treatment & enlargement of it. For I have thought fit to divide the work into three parts ; the first of these contains the exposition of the Theory itself, its analytical deduction & its demonstration ; the second a fairly full application to Mechanics ; & the third an application to Physics. The most important point, I decided, was for me to take the greatest care that every- thing, as far as was possible, should be clearly explained, & that there should be no need for higher geometry or for the calculus. Thus, in the first part, as well as in the third, there are no proofs by analysis ; nor are there any by geometry, with the exception of a very few that are absolutely necessary, & even these you will find relegated to brief notes set at the foot of a page. I have also added some very few proofs, that required a knowledge of higher algebra & geometry, or were of a rather more complicated nature, all of which have been already published elsewhere, at the end of the work ; I have collected these under the heading Supplements ; & in them I have included my views on Space & Time, which are in accord with my main Theory, & also have been already published elsewhere. In the second part, where the Theory is applied to Mechanics, I have not been able to do without geometrical proofs altogether ; & even in some cases I have had to give algebraical proofs. But these are of such a simple kind that they scarcely ever require anything more than Euclidean geometry, the first and most elementary ideas of trigonometry, and easy analytical calculations. It is true that in the first part there are to be found a good many geometrical diagrams, which at first sight, before the text is considered more closely, will appear to be rather complicated. But these present nothing else but a kind of image of the subjects treated, which by means of these diagrams are set before the eyes for contemplation. The very curve that represents the law of forces is an instance of this. I find that between all points of matter there is a mutual force depending on the distance between them, & changing as this distance changes ; so that it is sometimes attractive, & sometimes repulsive, but always follows a definite continuous law. Laws of variation of this kind between two quantities depending upon one another, as distance & force do in this instance, may be represented either by an analytical formula or by a geometrical curve ; but the former method of representation requires far more knowledge of algebraical processes, & does not assist the imagination in the way that the latter does. Hence I have employed the latter method in the first part of the work, & relegated to the Supplements the analytical formula which represents the curve, & the law of forces which the curve exhibits. The whole matter reduces to this. In a straight line of indefinite length, which is called the axis, a fixed point is taken ; & segments of the straight line cut off from this point represent the distances. A curve is drawn following the general direction of this straight line, & winding about it, so as to cut it in several places. Then perpendiculars that are drawn from the ends of the segments to meet the curve represent the forces ; these forces are greater or less, according as such perpendiculars are greater or less ; & they pass from attractive forces to repulsive, and vice versa, whenever these perpendiculars change their direction, as the curve passes from one side of the axis of indefinite length to the other side of it. Now this requires no geometrical proof, but only a knowledge of certain terms, which either belong to the first elementary principles of geometry, & are thoroughly well known, or are such as can be defined when they are used. The term Asymptote is well known, and from the same idea we speak of the branch of a curve as being asymptotic ; thus a straight line is said to be the asymptote to any branch of a curve when, if the straight line is indefinitely produced, it approaches nearer and nearer to the curvilinear arc which is also prolonged indefinitely in such manner that the distance between them becomes indefinitely diminished, but never altogether vanishes, so that the straight line & the curve never really meet. A careful consideration of the curve given in Fig. I, & of the way in which the relation 13 i4 AD LECTOREM EX EDITIONE VIENNENSI nexus inter vires, & distantias, est utique ad.rn.odum necessaria ad intelligendam Theoriam ipsam, cujus ea est prescipua qucsdam veluti clavis, sine qua omnino incassum tentarentur cetera ; sea y ejusmodi est, ut tironum, & sane etiam mediocrium, immo etiam longe infra mediocritatem collocatorum, captum non excedat, potissimum si viva accedat Professoris vox mediocriter etiam versati in Mechanica, cujus ope, pro certo habeo, rem ita patentem omnibus reddi posse, ut ii etiam, qui Geometries penitus ignari sunt, paucorum admodum explicatione vocabulorum accidente, earn ipsis oculis intueantur omnino perspicuam. In tertia parte supponuntur utique nonnulla, ques demonstrantur in secunda ; sed ea ipsa sunt admodum pauca, & Us, qui geometricas demonstrationes fastidiunt, facile admodum exponi possunt res ipsez ita, ut penitus etiam sine ullo Geometries adjumento percipiantur, quanquam sine Us ipsa demonstratio haberi non poterit ; ut idcirco in eo differre debeat is, qui secundam partem attente legerit, y Geometriam calleat, ab eo, qui earn omittat, quod ille primus veritates in tertia parte adbibitis, ac ex secunda erutas, ad explicationem Physices, intuebitur per evi- dentiam ex ipsis demonstrationibus haustam, hie secundus easdem quodammodo per fidem Geo- metris adhibitam credet. Hujusmodi inprimis est illud, particulam compositam ex punctis etiam homogeneis, presditis lege virium proposita, posse per solam diversam ipsorum punctorum dispositionem aliam particulam per certum intervallum vel perpetuo attrahere, vel perpetuo repellere, vel nihil in earn agere, atque id ipsum viribus admodum diversis, y qua respectu diver- sarum particularum diver see sint, & diver see respectu partium diver sarum ejusdem particules, ac aliam particulam alicubi etiam urgeant in latus, unde plurium phesnomenorum explicatio in Physica sponte ftuit. Verum qui omnem Theories, y deductionum compagem aliquanto altius inspexerit, ac diligentius perpenderit, videbit, ut spero, me in hoc perquisitionis genere multo ulterius progressum esse, quam olim Newtonus ipse desideravit. Is enim in postremo Optices questione prolatis Us, ques per vim attractivam, & vim repulsivam, mutata distantia ipsi attractives suc- cedentem, explicari poterant, hesc addidit : " Atque h<sc quidem omnia si ita sint, jam Natura universa valde erit simplex, y consimilis sui, perficiens nimirum magnos omnes corporum ceslestium motus attractione gravitatis, qucs est mutua inter corpora ilia omnia, & minores fere omnes particularum suarum motus alia aliqua vi attrahente, y repellente, ques est inter particulas illas mutua." Aliquanto autem inferius de primigeniis particulis agens sic habet : " Porro videntur mihi hce particules primigenics non modo in se vim inerties habere, motusque leges passivas illas, ques ex vi ista necessario oriuntur ; verum etiam motum perpetuo accipere a certis principiis actuosis, qualia nimirum sunt gravitas, y causa fermentationis, y cohesrentia corporum. Atque hesc quidem principia considero non ut occultas qualitates, ques ex specificis rerum formis oriri fingantur, sed ut universales Natures leges, quibus res ipscs sunt formates. Nam principia quidem talia revera existere ostendunt phesnomena Natures, licet ipsorum causes ques sint, nondum fuerit explicatum. Affirmare, singulas rerum species specificis presditas esse qualita- tibus occultis, per quas eae vim certam in agenda habent, hoc utique est nihil dicere : at ex phesnomenis Natures duo, vel tria derivare generalia motus principia, y deinde explicare, quemadmodum proprietates, y actiones rerum corporearum omnium ex istis principiis conse- quantur, id vero magnus esset factus in Philosophia progressus, etiamsi principiorum istorum causes nondum essent cognites. Quare motus principia supradicta proponere non dubito, cum per Naturam universam latissime pateant." Hcsc ibi Newtonus, ubi is quidem magnos in Philosophia progressus facturum arbitratus est eum, qui ad duo, vel tria generalia motus principia ex Natures phesnomenis derivata phesno- menorum explicationem reduxerit, y sua principia protulit, ex quibus inter se diversis eorum aliqua tantummodo explicari posse censuit. Quid igitur, ubi y ea ipsa tria, y alia prcscipua quesque, ut ipsa etiam impenetrabilitas, y impulsio reducantur ad principium unicum legitima ratiocinatione deductum ? At id per meam unicam, y simplicem virium legem presstari, patebit sane consideranti operis totius Synopsim quandam, quam hie subjicio ; sed multo magis opus ipsum diligentius pervolventi. PREFACE TO READER THAT APPEARED IN THE VIENNA EDITION 15 between the forces & the distances is represented by it, is absolutely necessary for the under- standing of the Theory itself, to which it is as it were the chief key, without which it would be quite useless to try to pass on to the rest. But it is of such a nature that it does not go beyond the capacity of beginners, not even of those of very moderate ability, or of classes even far below the level of mediocrity ; especially if they have the additional assistance of a teacher's voice, even though he is only moderately familiar with Mechanics. By his help, I am sure, the subject can be made clear to every one, so that those of them that are quite ignorant of geometry, given the explanation of but a few terms, may get a perfectly good idea of the subject by ocular demonstration. In the third part, some of the theorems that have been proved in the second part are certainly assumed, but there are very few such ; &, for those who do not care for geo- metrical proofs, the facts in question can be quite easily stated in such a manner that they can be completely understood without any assistance from geometry, although no real demonstration is possible without them. There is thus bound to be a difference between the reader who has gone carefully through the second part, & who is well versed in geo- metry, & him who omits the second part ; in that the former will regard the facts, that have been proved in the second part, & are now employed in the third part for the ex- planation of Physics, through the evidence derived from the demonstrations of these facts, whilst the second will credit these same facts through the mere faith that he has in geome- tricians. A specially good instance of this is the fact, that a particle composed of points quite homogeneous, subject to a law of forces as stated, may, merely by altering the arrange- ment of those points, either continually attract, or continually repel, or have no effect at all upon, another particle situated at a known distance from it ; & this too, with forces that differ widely, both in respect of different particles & in respect of different parts of the same particle ; & may even urge another particle in a direction at right angles to the line join- ing the two, a fact that readily gives a perfectly natural explanation of many physical phenomena. Anyone who shall have studied somewhat closely the whole system of my Theory, & what I deduce from it, will see, I hope, that I have advanced in this kind of investigation much further than Newton himself even thought open to his desires. For he, in the last of his " Questions " in his Opticks, after stating the facts that could be explained by means of an attractive force, & a repulsive force that takes the place of the attractive force when the distance is altered, has added these words : — " Now if all these things are as stated, then the whole of Nature must be exceedingly simple in design, & similar in all its parts, accom- plishing all the mighty motions of the heavenly bodies, as it does, by the attraction of gravity, which is a mutual force between any two bodies of the whole system ; and Nature accomplishes nearly all the smaller motions of their particles by some other force of attrac- tion or repulsion, which is mutual between any two of those particles." Farther on, when he is speaking about elementary particles, he says : — " Moreover, it appears to me that these elementary particles not only possess an essential property of inertia, & laws of motion, though only passive, which are the necessary consequences of this property ; but they also constantly acquire motion from the influence of certain active principles such as, for instance, gravity, the cause of fermentation, & the cohesion of solids. I do not consider these principles to be certain mysterious qualities feigned as arising from characteristic forms of things, but as universal laws of Nature, by the influence of which these very things have been created. For the phenomena of Nature show that these principles do indeed exist, although their nature has not yet been elucidated. To assert that each & every species is endowed with a mysterious property characteristic to it, due to which it has a definite mode in action, is really equivalent to saying nothing at all. On the other hand, to derive from the phenomena of Nature two or three general principles, & then to explain how the pro- perties & actions of all corporate things follow from those principles, this would indeed be a mighty advance in philosophy, even if the causes of those principles had not at the time been discovered. For these reasons I do not hesitate in bringing forward the principles of motion given above, since they are clearly to be perceived throughout the whole range of Nature." These are the words of Newton, & therein he states his opinion that he indeed will have made great strides in philosophy who shall have reduced the explanation of phenomena to two or three general principles derived from the phenomena of Nature ; & he brought forward his own principles, themselves differing from one another, by which he thought that some only of the phenomena could be explained. What then if not only the three he mentions, but also other important principles, such as impenetrability & impul- sive force, be reduced to a single principle, deduced by a process of rigorous argument ! It will be quite clear that this is exactly what is done by my single simple law of forces, to anyone who studies a kind of synopsis of the whole work, which I add below ; but it will be iar more clear to him who studies the whole work with some earnestness, SYNOPSIS TOTIUS OPERIS EX EDITIONE VIENNENSI PARS I sex numeris exhibeo, quando, & qua occasione Theoriam meam invenerim, ac ubi hucusque de ea egerim in dissertationibus jam editis, quid ea commune habeat cum Leibnitiana, quid cum Newtoniana Theoria, in quo ab utraque discrepet, & vero etiam utrique praestet : addo, quid alibi promiserim pertinens ad aequilibrium, & oscillationis centrum, & quemadmodum iis nunc inventis, ac ex unico simplicissimo, ac elegant- issimo theoremate profluentibus omnino sponte, cum dissertatiunculam brevem meditarer, jam eo consilio rem aggressus ; repente mihi in opus integrum justse molis evaserit tractatio. 7 Turn usque ad num. II expono Theoriam ipsam : materiam constantem punctis prorsus simplicibus, indivisibilibus, & inextensis, ac a se invicem distantibus, quae puncta habeant singula vim inertiae, & praeterea vim activam mutuam pendentem a distantiis, ut nimirum, data distantia, detur & magnitude, & directio vis ipsius, mutata autem distantia, mutetur vis ipsa, quae, imminuta distantia in infinitum, sit repulsiva, & quidem excrescens in infinitum : aucta autem distantia, minuatur, evanescat, mutetur in attrac- tivam crescentem primo, turn decrescentem, evanescentem, abeuntem iterum in repul- sivam, idque per multas vices, donee demum in majoribus distantiis abeat in attractivam decrescentem ad sensum in ratione reciproca duplicata distantiarum ; quern nexum virium cum distantiis, & vero etiam earum transitum a positivis ad negativas, sive a repulsivis ad attractivas, vel vice versa, oculis ipsis propono in vi, qua binae elastri cuspides conantur ad es invicem accedere, vel a se invicem recedere, prout sunt plus justo distractae, vel con- tractae. II Inde ad num. 16 ostendo, quo pacto id non sit aggregatum quoddam virium temere coalescentium, sed per unicam curvam continuam exponatur ope abscissarum exprimentium distantias, & ordinatarum exprimentium vires, cujus curvae ductum, & naturam expono, ac ostendo, in quo differat ab hyperbola ilia gradus tertii, quae Newtonianum gravitatem exprimit : ac demum ibidem & argumentum, & divisionem propono operis totius. 1 6 Hisce expositis gradum facio ad exponendam totam illam analysim, qua ego ad ejusmodi Theoriam deveni, & ex qua ipsam arbitror directa, & solidissima ratiocinatione deduci totam. Contendo nimirum usque ad numerum 19 illud, in collisione corporum debere vel haberi compenetrationem, vel violari legem continuitatis, velocitate mutata per saltum, si cum inaequalibus velocitatibus deveniant ad immediatum contactum, quae continuitatis lex cum (ut evinco) debeat omnino observari, illud infero, antequam ad contactum deveniant corpora, debere mutari eorum velocitates per vim quandam, quae sit par extinguendse velocitati, vel velocitatum differentiae, cuivis utcunque magnae. 19 A num. 19 ad 28 expendo effugium, quo ad eludendam argumenti mei vim utuntur ii, qui negant corpora dura, qua quidem responsione uti non possunt Newtoniani, & Corpus- culares generaliter, qui elementares corporum particulas assumunt prorsus duras : qui autem omnes utcunque parvas corporum particulas molles admittunt, vel elasticas, difficultatem non effugiunt, sed transferunt ad primas superficies, vel puncta, in quibus committeretur omnino saltus, & lex continuitatis violaretur : ibidem quendam verborum lusum evolvo, frustra adhibitum ad eludendam argumenti mei vim. * Series numerorum, quibus tractari incipiunt, quae sunt in textu, 16 SYNOPSIS OF THE WHOLE WORK (FROM THE VIENNA EDITION) PART I N the first six articles, I state the time at which I evolved my Theory, what i * led me to it, & where I have discussed it hitherto in essays already pub- lished : also what it has in common with the theories of Leibniz and Newton ; in what it differs from either of these, & in what it is really superior to them both. In addition I state what I have published else- where about equilibrium & the centre of oscillation ; & how, having found out that these matters followed quite easily from a single theorem of the most simple & elegant kind, I proposed to write a short essay thereon ; but when I set to work to deduce the matter from this principle, the discussion, quite unexpectedly to me, developed into a whole work of considerable magnitude. From this .until Art. II, I explain the Theory itself : that matter is unchangeable, 7 and consists of points that are perfectly simple, indivisible, of no extent, & separated from one another ; that each of these points has a property of inertia, & in addition a mutual active force depending on the distance in such a way that, if the distance is given, both the magnitude & the direction of this force are given ; but if the distance is altered, so also is the force altered ; & if the distance is diminished indefinitely, the force is repulsive, & in fact also increases indefinitely ; whilst if the distance is increased, the force will be dimin- ished, vanish, be changed to an attractive force that first of all increases, then decreases, vanishes, is again turned into a repulsive force, & so on many times over ; until at greater distances it finally becomes an attractive force that decreases approximately in the inverse ratio of the squares of the distances. This connection between the forces & the distances, & their passing from positive to negative, or from repulsive to attractive, & conversely, I illustrate by the force with which the two ends of a spring strive to approach towards, or recede from, one another, according as they are pulled apart, or drawn together, by more than the natural amount. From here on to Art. 1 6 I show that it is not merely an aggregate of forces combined n haphazard, but that it is represented by a single continuous curve, by means of abscissse representing the distances & ordinates representing the forces. I expound the construction & nature of this curve ; & I show how it differs from the hyperbola of the third degree which represents Newtonian gravitation. Finally, here too I set forth the scope of the whole work & the nature of the parts into which it is divided. These statements having been made, I start to expound the whole of the analysis, by 16 which I came upon a Theory of this kind, & from which I believe I have deduced the whole of it by a straightforward & perfectly rigorous chain of reasoning. I contend indeed, from here on until Art. 19, that, in the collision of solid bodies, either there must be compene- tration, or the Law of Continuity must be violated by a sudden change of velocity, if the bodies come into immediate contact with unequal velocities. Now since the Law of Continuity must (as I prove that it must) be observed in every case, I infer that, before the bodies reach the point of actual contact, their velocities must be altered by some force which is capable of destroying the velocity, or the difference of the velocities, no matter how great that may be. From Art. 19 to Art. 28 I consider the artifice, adopted for the purpose of evading the 19 strength of my argument by those who deny the existence of hard bodies ; as a matter of fact this cannot be used as an argument against me by the Newtonians, or the Corpuscular- ians in general, for they assume that the elementary particles of solids are perfectly hard. Moreover, those who admit that all the particles of solids, however small they may be, are soft or elastic, yet do not escape the difficulty, but transfer it to prime surfaces, or points ; & here a sudden change would be made & the Law of Continuity violated. In the same connection I consider a certain verbal quibble, used in a vain attempt to foil the force of my reasoning. * These numbers are the numbers of the articles, in which the matters given in the text are first discussed. 17 C 1 8 SYNOPSIS TOTIUS OPERIS 28 Sequentibus num. 28 & 29 binas alias responsiones rejicio aliorum, quarum altera, ut mei argument! vis elidatur, affirmat quispiam, prima materiae elementa compenetrari, alter dicuntur materiae puncta adhuc moveri ad se invicem, ubi localiter omnino quiescunt, & contra primum effugium evinco impenetrabilitatem ex inductione ; contra secundum expono aequivocationem quandam in significatione vocis motus, cui aequivocationi totum innititur. 30 Hinc num. 30, & 31 ostendo, in quo a Mac-Laurino dissentiam, qui considerata eadem, quam ego contemplatus sum, collisione corporum, conclusit, continuitatis legem violari, cum ego eandem illaesam esse debere ratus ad totam devenerim Theoriam meam. 32 Hie igitur, ut meae deductionis vim exponam, in ipsam continuitatis legem inquire, ac a num. 32 ad 38 expono, quid ipsa sit, quid mutatio continua per gradus omnes intermedios, quae nimirum excludat omnem saltum ab una magnitudine ad aliam sine transitu per 39 intermedias, ac Geometriam etiam ad explicationem rei in subsidium advoco : turn earn probo primum ex inductione, ac in ipsum inductionis principium inquirens usque ad num. 44, exhibeo, unde habeatur ejusdem principii vis, ac ubi id adhiberi possit, rem ipsam illustrans exemplo impenetrabilitatis erutae passim per inductionem, donee demum ejus vim 45 applicem ad legem continuitatis demonstrandam : ac sequentibus numeris casus evolvo quosdam binarum classium, in quibus continuitatis lex videtur laedi nee tamen laeditur. 48 Post probationem principii continuitatis petitam ab inductione, aliam num. 48 ejus probationem aggredior metaphysicam quandam, ex necessitate utriusque limitis in quanti- tatibus realibus, vel seriebus quantitatum realium finitis, quae nimirum nee suo principio, nee suo fine carere possunt. Ejus rationis vim ostendo in motu locali, & in Geometria 52 sequentibus duobus numeris : turn num. 52 expono difficultatem quandam, quas petitur ex eo, quod in momento temporis, in quo transitur a non esse ad esse, videatur juxta ejusmodi Theoriam debere simul haberi ipsum esse, & non esse, quorum alterum ad finem praecedentis seriei statuum pertinet, alterum ad sequentis initium, ac solutionem ipsius fuse evolvo, Geometria etiam ad rem oculo ipsi sistendam vocata in auxilium. 63 Num. 63, post epilogum eorum omnium, quae de lege continuitatis sunt dicta, id principium applico ad excludendum saltum immediatum ab una velocitate ad aliam, sine transitu per intermedias, quod & inductionem laederet pro continuitate amplissimam, & induceret pro ipso momento temporis, in quo fieret saltus, binas velocitates, ultimam nimirum seriei praecedentis, & primam novas, cum tamen duas simul velocitates idem mobile habere omnino non possit. Id autem ut illustrem, & evincam, usque ad num. 72 considero velocitatem ipsam, ubi potentialem quandam, ut appello, velocitatem ab actuali secerno, & multa, quae ad ipsarum naturam, ac mutationes pertinent, diligenter evolvo, nonnullis etiam, quae inde contra meae Theoriae probationem objici possunt, dissolutis. His expositis conclude jam illud ex ipsa continuitate, ubi corpus quodpiam velocius movetur post aliud lentius, ad contactum immediatum cum ilia velocitatum inaequalitate deveniri non posse, in quo scilicet contactu primo mutaretur vel utriusque velocitas, vel alterius, per saltum, sed debere mutationem velocitatis incipere ante contactum ipsum. 73 Hinc num. 73 infero, debere haberi mutationis causam, quae appelletur vis : turn num. 74 74 hanc vim debere esse mutuam, & agere in partes contrarias, quod per inductionem evinco, 75 & inde infero num. 75, appellari posse repulsivam ejusmodi vim mutuam, ac ejus legem exquirendam propono. In ejusmodi autem perquisitione usque ad num. 80 invenio illud, debere vim ipsam imminutis distantiis crescere in infinitum ita ut par sit extinguendae velocitati utcunque magnse ; turn & illud, imminutis in infinitum etiam distantiis, debere in infinitum augeri, in maximis autem debere esse e contrario attractivam, uti est gravitas : inde vero colligo limitem inter attractionem, & repulsionem : turn sensim plures, ac etiam plurimos ejusmodi limites invenio, sive transitus ab attractione ad repulsionem, & vice versa, ac formam totius curvae per ordinatas suas exprimentis virium legem determino. SYNOPSIS OF THE WHOLE WORK 19 In the next articles, 28 & 29, I refute a further pair of arguments advanced by others ; 28 in the first of these, in order to evade my reasoning, someone states that there is compene- tration of the primary elements of matter ; in the second, the points of matter are said to be moved with regard to one another, even when they are absolutely at rest as regards position. In reply to the first artifice, I prove the principle of impenetrability by induc- tion ; & in reply to the second, I expose an equivocation in the meaning of the term motion, an equivocation upon which the whole thing depends. Then, in Art. 30, 31, I show in what respect I differ from Maclaurin, who, having 30 considered the same point as myself, came to the conclusion that in the collision of bodies the Law of Continuity was violated ; whereas I obtained the whole of my Theory from the assumption that this law must be unassailable. At this point therefore, in order that the strength of my deductive reasoning might 32 be shown, I investigate the Law of Continuity ; and from Art. 32 to Art. 38, I set forth its nature, & what is meant by a continuous change through all intermediate stages, such as to exclude any sudden change from any one magnitude to another except by a passage through intermediate stages ; & I call in geometry as well to help my explanation of the matter. Then I investigate its truth first of all by induction ; &, investigating the prin- 39 ciple of induction itself, as far as Art. 44, 1 show whence the force of this principle is derived, & where it can be used. I give by way of illustration an example in which impenetrability is derived entirely by induction ; & lastly I apply the force of the principle to demonstrate the Law of Continuity. In the articles that follow I consider certain cases of two kinds, 45 in which the Law of Continuity appears to be violated, but is not however really violated. After this proof of the principle of continuity procured through induction, in Art. 48, 48 I undertake another proof of a metaphysical kind, depending upon the necessity of a limit on either side for either real quantities or for a finite series of real quantities ; & indeed it is impossible that these limits should be lacking, either at the beginning or the end. I demonstrate the force of this reasoning in the case of local motion, & also in geometry, in the next two articles. Then in Art. 52 I explain a certain difficulty, which is derived from the S2 fact that, at the instant at which there is a passage from non-existence to existence, it appears according to a theory of this kind that we must have at the same time both existence and non-existence. For one of these belongs to the end of the antecedent series of states, & the other to the beginning of the consequent series. I consider fairly fully' the solution of this problem ; and I call in geometry as well to assist in giving a visual representation of the matter. In Art. 63, after summing up all that has been said about the Law of Continuity, I 63 apply the principle to exclude the possibility of any sudden change from one velocity to another, except by passing through intermediate velocities ; this would be contrary to the very full proof that I give for continuity, as it would lead to our having two velocities at the instant at which the change occurred. That is to say, there would be the final velocity of the antecedent series, & the initial velocity of the consequent series ; in spite of the fact that it is quite impossible for a moving body to have two different velocities at the same time. Moreover, in order to illustrate & prove the point, from here on to Art. 72, I consider velocity itself ; and I distinguish between a potential velocity, as I call it, & an actual velocity ; I also investigate carefully many matters that relate to the nature of these velocities & to their changes. Further, I settle several difficulties that can be brought up in opposition to the proof of my Theory, in consequence. This done, I then conclude from the principle of continuity that, when one body with a greater velocity follows after another body having a less velocity, it is impossible that there should ever be absolute contact with such an inequality of velocities ; that is to say, a case of the velocity of each, or of one or the other, of them being changed suddenly at the instant of contact. I assert on the other hand that the change in the velocities must begin before contact. Hence, in Art. 73, I infer that there must be a cause for this change : 73 which is to be called " force." Then, in Art. 74, I prove that this force is a mutual one, & 74 that it acts in opposite directions ; the proof is by induction. From this, in Art. 75, I 75 infer that such a mutual force may be said to be repulsive ; & I undertake the investigation of the law that governs it. Carrying on this investigation as far as Art. 80, I find that this force must increase indefinitely as the distance is diminished, in order that it may be capable of destroying any velocity, however great that velocity may be. Moreover, I find that, whilst the force must be indefinitely increased as the distance is indefinitely decreased, it must be on the contrary attractive at very great distances, as is the case for gravitation. Hence I infer that there must be a limit-point forming a boundary between attraction & repulsion ; & then by degrees I find more, indeed very many more, of such limit-points, or points of transition from attraction to repulsion, & from repulsion to attraction ; & I determine the form of the entire curve, that expresses by its ordinates the law of these forces. 20 SYNOPSIS TOTIUS OPERIS 8 1 Eo usque virium legem deduce, ac definio ; turn num. 81 eruo ex ipsa lege consti- tutionem elementorum materiae, quae debent esse simplicia, ob repulsionem in minimis distantiis in immensum auctam ; nam ea, si forte ipsa elementa partibus constarent, nexum omnem dissolveret. Usque ad num. 88 inquire in illud, an hasc elementa, ut simplicia esse debent, ita etiam inextensa esse debeant, ac exposita ilia, quam virtualem extensionem appellant, eandem exclude inductionis principio, & difficultatem evolvo turn earn, quae peti possit ab exemplo ejus generis extensionis, quam in anima indivisibili, & simplice per aliquam corporis partem divisibilem, & extensam passim admittunt : vel omnipraesentiae Dei : turn earn, quae peti possit ab analogia cum quiete, in qua nimirum conjungi debeat unicum spatii punctum cum serie continua momentorum temporis, uti in extensione virtuali unicum momentum temporis cum serie continua punctorum spatii conjungeretur, ubi ostendo, nee quietem omnimodam in Natura haberi usquam, nee adesse semper omnimodam inter 88 tempus, & spatium analogiam. Hie autem ingentem colligo ejusmodi determinationis fructum, ostendens usque ad num. 91, quantum prosit simplicitas, indivisibilitas, inextensio elementorum materiae, ob summotum transitum a vacuo continue per saltum ad materiam continuam, ac ob sublatum limitem densitatis, quae in ejusmodi Theoria ut minui in infinitum potest, ita potest in infinitum etiam augeri, dum in communi, ubi ad contactum deventum est, augeri ultra densitas nequaquam potest, potissimum vero ob sublatum omne continuum coexistens, quo sublato & gravissimae difficultates plurimse evanescunt, & infinitum actu existens habetur nullum, sed in possibilibus tantummodo remanet series finitorum in infinitum producta. 91 His definitis, inquire usque ad num. 99 in illud, an ejusmodi elementa sint censenda homogenea, an heterogenea : ac primo quidem argumentum pro homogeneitate saltern in eo, quod pertinet ad totam virium legem, invenio in homogenietate tanta primi cruris repulsivi in minimis distantiis, ex quo pendet impenetrabilitas, & postremi attractivi, quo gravitas exhibetur, in quibus omnis materia est penitus homogenea. Ostendo autem, nihil contra ejusmodi homogenietatem evinci ex principio Leibnitiano indiscernibilium, nihil ex inductione, & ostendo, unde tantum proveniat discrimen in compositis massulis, ut in frondibus, & foliis ; ac per inductionem, & analogiam demonstro, naturam nos ad homo- geneitatem elementorum, non ad heterogeneitatem deducere. 100 Ea ad probationem Theoriae pertinent ; qua absoluta, antequam inde fructus colli- gantur multiplices, gradum hie facio ad evolvendas difficultates, quae vel objectae jam sunt, vel objici posse videntur mihi, primo quidem contra vires in genere, turn contra meam hanc expositam, comprobatamque virium legem, ac demum contra puncta ilia indivisibilia, & inextensa, quae ex ipsa ejusmodi virium lege deducuntur. 101 Primo quidem, ut iis etiam faciam satis, qui inani vocabulorum quorundam sono perturbantur, a num. 101 ad 104 ostendo, vires hasce non esse quoddam occultarum qualitatum genus, sed patentem sane Mechanismum, cum & idea earum sit admodum distincta, & existentia, ac lex positive comprobata ; ad Mechanicam vero pertineat omnis 104 tractatio de Motibus, qui a datis viribus etiam sine immediate impulsu oriuntur. A num. 104 ad 106 ostendo, nullum committi saltum in transitu a repulsionibus ad attractiones, 1 06 & vice versa, cum nimirum per omnes inter medias quantitates is transitus fiat. Inde vero ad objectiones gradum facio, quae totam curvas formam impetunt. Ostendo nimirum usque ad num. 116, non posse omnes repulsiones a minore attractione desumi ; repulsiones ejusdem esse seriei cum attractionibus, a quibus differant tantummodo ut minus a majore, sive ut negativum a positivo ; ex ipsa curvarum natura, quae, quo altioris sunt gradus, eo in pluribus punctis rectam secare possunt, & eo in immensum plures sunt numero ; haberi potius, ubi curva quaeritur, quae vires exprimat, indicium pro curva ejus naturae, ut rectam in plurimis punctis secet, adeoque plurimos secum afferat virium transitus a repulsivis ad attractivas, quam pro curva, quae nusquam axem secans attractiones solas, vel solas pro distantiis omnibus repulsiones exhibeat : sed vires repulsivas, & multiplicitatem transituum esse positive probatam, & deductam totam curvas formam, quam itidem ostendo, non esse ex arcubus natura diversis temere coalescentem, sed omnino simplicem, atque earn ipsam SYNOPSIS OF THE WHOLE WORK 21 So far I have been occupied in deducing and settling the law of these forces. Next, in Art. 8r, I derive from this law the constitution of the elements of matter. These must be 81 quite simple, on account of the repulsion at very small distances being immensely great ; for if by chance those elements were made up of parts, the repulsion would destroy all connections between them. Then, as far as Art. 88, I consider the point, as to whether these elements, as they must be simple, must therefore be also of no extent ; &, having ex- plained what is called " virtual extension," I reject it by the principle of induction. I then consider the difficulty which may be brought forward from an example of this kind of extension ; such as is generally admitted in the case of the indivisible and one-fold soul pervading a divisible & extended portion of the body, or in the case of the omnipresence of GOD. Next I consider the difficulty that may be brought forward from an analogy with rest ; for here in truth one point of space must be connected with a continuous series of instants of time, just as in virtual extension a single instant of time would be connected with a continuous series of points of space. I show that there can neither be perfect rest any- gg where in Nature, nor can there be at all times a perfect analogy between time and space. In this connection, I also gather a large harvest from such a conclusion as this ; showing, as far as Art. 91, the great advantage of simplicity, indivisibility, & non-extension in the elements of matter. For they do away with the idea of a passage from a continuous vacuum to continuous matter through a sudden change. Also they render unnecessary any limit to density : this, in a Theory like mine, can be just as well increased to an indefinite extent, as it can be indefinitely decreased : whilst in the ordinary theory, as soon as contact takes place, the density cannot in any way be further increased. But, most especially, they do away with the idea of everything continuous coexisting ; & when this is done away with, the majority of the greatest difficulties vanish. Further, nothing infinite is found actually existing ; the only thing possible that remains is a series of finite things produced inde- finitely. These things being settled, I investigate, as far as Art. 99, the point as to whether QJ elements of this kind are to be considered as being homogeneous or heterogeneous. I find my first evidence in favour of homogeneity — at least as far as the complete law of forces is concerned — in the equally great homogeneity of the first repulsive branch of my curve of forces for very small distances, upon which depends impenetrability, & of the last attrac- tive branch, by which gravity is represented. Moreover I show that there is nothing that can be proved in opposition to homogeneity such as this, that can be derived from either the Leibnizian principle of " indiscernibles," or by induction. I also show whence arise those differences, that are so great amongst small composite bodies, such as we see in boughs & leaves ; & I prove, by induction & analogy, that the very nature of things leads us to homogeneity, & not to heterogeneity, for the elements of matter. These matters are all connected with the proof of my Theory. Having accomplished IOo this, before I start to gather the manifold fruits to be derived from it, I proceed to consider the objections to my theory, such as either have been already raised or seem to me capable of being raised ; first against forces in general, secondly against the law of forces that I have enunciated & proved, & finally against those indivisible, non-extended points that are deduced from a law of forces of this kind. First of all then, in order that I may satisfy even those who are confused over the 101 empty sound of certain terms, I show, in Art. 101 to 104, that these forces are not some sort of mysterious qualities ; but that they form a readily intelligible mechanism, since both the idea of them is perfectly distinct, as well as their existence, & in addition the law that governs them is demonstrated in a direct manner. To Mechanics belongs every dis- cussion concerning motions that arise from given forces without any direct impulse. In Art. 104 to 106, I show that no sudden change takes place in passing from repulsions to 104 attractions or vice versa ; for this transition is made through every intermediate quantity. Then I pass on to consider the objections that are made against the whole form of my 106 curve. I show indeed, from here on to Art. 116, that all repulsions cannot be taken to come from a decreased attraction ; that repulsions belong to the self-same series as attrac- tions, differing from them only as less does from more, or negative from positive. From the very nature of the curves (for which, the higher the degree, the more points there are in which they can intersect a right line, & vastly more such curves there are), I deduce that there is more reason for assuming a curve of the nature of mine (so that it may cut a right line in a large number of points, & thus give a large number of transitions of the forces from repulsions to attractions), than for assuming a curve that, since it does not cut the axis anywhere, will represent attractions alone, or repulsions alone, at all distances. Further, I point out that repulsive forces, and a multiplicity of transitions are directly demonstrated, & the whole form of the curve is a matter of deduction ; & I also show that it is not formed of a number of arcs differing in nature connected together haphazard ; 22 SYNOPSIS TOTIUS OPERIS simplicitatem in Supplementis cvidentissime demonstro, exhibens methodum, qua deveniri possit ad aequationem ejusmodi curvse simplicem, & uniformem ; licet, ut hie ostendo, ipsa ilia lex virium possit mente resolvi in plures, quae per plures curvas exponantur, a quibus tamen omnibus ilia reapse unica lex, per unicam illam continuant, & in se simplicem curvam componatur. 121 A num. 121 refello, quae objici possunt a lege gravitatis decrescentis in ratione reciproca duplicata distantiarum, quae nimirum in minimis distantiis attractionem requirit crescentem in infinitum. Ostendo autem, ipsam non esse uspiam accurate in ejusmodi ratione, nisi imaginarias resolutiones exhibeamus ; nee vero ex Astronomia deduci ejusmodi legem prorsus accurate servatam in ipsis Planetarum, & Cometarum distantiis, sed ad summum ita 124 proxime, ut differentia ab ea lege sit perquam exigua : ac a num. 124 expendo argumentum, quod pro ejusmodi lege desumi possit ex eo, quod cuipiam visa sit omnium optima, & idcirco electa ab Auctore Naturae, ubi ipsum Optimismi principium ad trutinam revoco, ac exclude, & vero illud etiam evinco, non esse, cur omnium optima ejusmodi lex censeatur : in Supplementis vero ostendo, ad qua; potius absurda deducet ejusmodi lex, & vero etiam aliae plures attractionis, quae imminutis in infinitum distantiis excrescat in infinitum. 131 Num. 131 a viribus transeo ad elementa, & primum ostendo, cur punctorum inexten- sorum ideam non habeamus, quod nimirum earn haurire non possumus per sensus, quos solae massae, & quidem grandiores, afficiunt, atque idcirco eandem nos ipsi debemus per reflexionem efformare, quod quidem facile possumus. Ceterum illud ostendo, me non inducere primum in Physicam puncta indivisibilia, & inextensa, cum eo etiam Leibnitianae monades recidant, sed sublata extensione continua difficultatem auferre illam omnem, quae jam olim contra Zenonicos objecta, nunquam est satis soluta, qua fit, ut extensio continua ab inextensis effici omnino non possit. 140 Num. 140 ostendo, inductionis principium contra ipsa nullam habere vim, ipsorum autem existentiam vel inde probari, quod continuitas se se ipsam destruat, & ex ea assumpta probetur argumentis a me institutis hoc ipsum, prima elementa esse indivisibilia, & inextensa, 143 nee ullum haberi extensum continuum. A num. 143 ostendo, ubi continuitatem admittam, nimirum in solis motibus ; ac illud explico, quid mihi sit spatium, quid tempus, quorum naturam in Supplementis multo uberius expono. Porro continuitatem ipsam ostendo a natura in solis motibus obtineri accurate, in reliquis affectari quodammodo ; ubi & exempla quaedam evolvo continuitatis primo aspectu violatae, in quibusdam proprietatibus luminis, ac in aliis quibusdam casibus, in quibus quaedam crescunt per additionem partium, non (ut ajunt) per intussumptionem. \ 153 A num. 153 ostendo, quantum haec mea puncta a spiritibus differant ; ac illud etiam evolvo, unde fiat, ut in ipsa idea corporis videatur includi extensio continua, ubi in ipsam idearum nostrarum originem inquire, & quae inde praejudicia profluant, expono. Postremo 165 autem loco num. 165 innuo, qui fieri possit, ut puncta inextensa, & a se invicem distantia, in massam coalescant, quantum libet, cohaerentem, & iis proprietatibus praeditam, quas in corporibus experimur, quod tamen ad tertiam partem pertinet, ibi multo uberius pertrac- tandum ; ac ibi quidem primam hanc partem absolve. PARS II 166 Num. 166 hujus partis argumentum propono ; sequenti vero 167, quae potissimum in curva virium consideranda sint, enuncio. Eorum considerationem aggressus, primo quidem 1 68 usque ad num. 172 in ipsos arcus inquire, quorum alii attractivi, alii repulsivi, alii asym- ptotici, ubi casuum occurrit mira multitudo, & in quibusdam consectaria notatu digna, ut & illud, cum ejus formae curva plurium asymptotorum esse possit, Mundorum prorsus similium seriem posse oriri, quorum alter respectu alterius vices agat unius, & indissolubilis SYNOPSIS OF THE WHOLE WORK 23 but that it is absolutely one-fold. This one-fold character I demonstrate in the Supple- ments in a very evident manner, giving a method by which a simple and uniform equation may be obtained for a curve of this kind. Although, as I there point out, this law of forces may be mentally resolved into several, and these may be represented by several correspond- ing curves, yet that law, actually unique, may be compounded from all of these together by means of the unique, continuous & one-fold curve that I give. In Art. 121, I start to give a refutation of those objections that may be raised from I2i a consideration of the fact that the law of gravitation, decreasing in the inverse duplicate ratio of the distances, demands that there should be an attraction at very small distances, & that it should increase indefinitely. However, I show that the law is nowhere exactly in conformity with a ratio of this sort, unless we add explanations that are merely imaginative ; nor, I assert, can a law of this kind be deduced from astronomy, that is followed with per- fect accuracy even at the distances of the planets & the comets, but one merely that is at most so very nearly correct, that the difference from the law of inverse squares is very slight. From Art. 124 onwards,! examine the value of the argument that can be drawn 124 in favour of a law of this sort from the view that, as some have thought, it is the best of all, & that on that account it was selected by the Founder of Nature. In connection with this I examine the principle of Optimism, & I reject it ; moreover I prove conclusively that there is no reason why this sort of law should be supposed to be the best of all. Fur- ther in the Supplements, I show to what absurdities a law of this sort is more likely to lead ; & the same thing for other laws of an attraction that increases indefinitely as the distance is diminished indefinitely. In Art. 131 I pass from forces to elements. I first of all show the reason why we may 1*1 not appreciate the idea of non-extended points ; it is because we are unable to perceive them by means of the senses, which are only affected by masses, & these too must be of considerable size. Consequently we have to build up the idea by a process of reasoning ; & this we can do without any difficulty. In addition, I point out that I am not the first to introduce indivisible & non-extended points into physical science ; for the " monads " of Leibniz practically come to the same thing. But I show that, by rejecting the idea of continuous extension, I remove the whole of the difficulty, which was raised against the disciples of Zeno in years gone by, & has never been answered satisfactorily ; namely, the difficulty arising from the fact that by no possible means can continuous extension be made up from things of no extent. In Art. 140 I show that the principle of induction yields no argument against these 140 indivisibles ; rather their existence is demonstrated by that principle, for continuity is self-contradictory. On this assumption it may be proved, by arguments originated by myself, that the primary elements are indivisible & non-extended, & that there does not exist anything possessing the property of continuous extension. From Art. 143 onwards, j ., I point out the only connection in which I shall admit continuity, & that is in motion. I state the idea that I have with regard to space, & also time : the nature of these I explain much more fully in the Supplements. Further, I show that continuity itself is really a property of motions only, & that in all other things it is more or less a false assumption. Here I also consider some examples in which continuity at first sight appears to be violated, such as in some of the properties of light, & in certain other cases where things increase by addition of parts, and not by intussumption, as it is termed. From Art. 153 onwards, I show how greatly these points of mine differ from object- 153 souls. I consider how it comes about that continuous extension seems to be included in the very idea of a body ; & in this connection, I investigate the origin of our ideas & I explain the prejudgments that arise therefrom. Finally, in Art. 165, I lightly 165 sketch what might happen to enable points that are of no extent, & at a distance from one another, to coalesce into a coherent mass of any size, endowed with those properties that we experience in bodies. This, however, belongs to the third part ; & there it will be much more fully developed. This finishes the first part. PART II In Art. 1 66 I state the theme of this second part ; and in Art. 167 I declare what 166 matters are to be considered more especially in connection with the curve of forces. Com- ing to the consideration of these matters, I first of all, as far as Art. 172, investigate the 168 arcs of the curve, some of which are attractive, some repulsive and some asymptotic. Here a marvellous number of different cases present themselves, & to some of them there are noteworthy corollaries ; such as that, since a curve of this kind is capable of possessing a considerable number of asymptotes, there can arise a series of perfectly similar cosmi, each of which will act upon all the others as a single inviolate elementary system. From Art. 172 24 SYNOPSIS TOTIUS OPERIS 172 element!. Ad. num. 179 areas contemplor arcubus clausas, quae respondentes segmento axis cuicunque, esse possunt magnitudine utcunque magnae, vel parvae, sunt autem mensura 179 incrementi, vel decrement! quadrat! velocitatum. Ad num. 189 inquire in appulsus curvse ad axem, sive is ibi secetur ab eadem (quo casu habentur transitus vel a repulsione ad attractionem, vel ab attractione ad repulsionem, quos dico limites, & quorum maximus est in tota mea Theoria usus), sive tangatur, & curva retro redeat, ubi etiam pro appulsibus considero recessus in infinitum per arcus asymptoticos, & qui transitus, sive limites, oriantur inde, vel in Natura admitti possint, evolvo. 189 Num. 189 a consideratione curvae ad punctorum combinationem gradum facio, ac primo quidem usque ad num. 204 ago de systemate duorum punctorum, ea pertractans, quas pertinent ad eorum vires mutuas, & motus, sive sibi relinquantur, sive projiciantur utcunque, ubi & conjunctione ipsorum exposita in distantiis limitum, & oscillationibus variis, sive nullam externam punctorum aliorum actionem sentiant, sive perturbentur ab eadem, illud innuo in antecessum, quanto id usui futurum sit in parte tertia ad exponenda cohaesionis varia genera, fermentationes, conflagrationes, emissiones vaporum, proprietates luminis, elasticitatem, mollitiem. 204 Succedit a Num. 204 ad 239 multo uberior consideratio trium punctorum, quorum vires generaliter facile definiuntur data ipsorum positione quacunque : verum utcunque data positione, & celeritate nondum a Geometris inventi sunt motus ita, ut generaliter pro casibus omnibus absolvi calculus possit. Vires igitur, & variationem ingentem, quam diversae pariunt combinationes punctorum, utut tantummodo numero trium, persequor 209 usque ad num. 209. Hinc usque ad num. 214 quaedam evolvo, quae pertinent ad vires ortas in singulis ex actione composita reliquorum duorum, & quae tertium punctum non ad accessum urgeant, vel recessum tantummodo respectu eorundem, sed & in latus, ubi & soliditatis imago prodit, & ingens sane discrimen in distantiis particularum perquam exiguis ac summa in maximis, in quibus gravitas agit, conformitas, quod quanto itidem ad Naturae 214 explicationem futurum sit usui, significo. Usque ad num. 221 ipsis etiam oculis contem- plandum propono ingens discrimen in legibus virium, quibus bina puncta agunt in tertium, sive id jaceat in recta, qua junguntur, sive in recta ipsi perpendiculari, & eorum intervallum secante bifariam, constructis ex data primigenia curva curvis vires compositas exhibentibus : 221 turn sequentibus binis numeris casum evolvo notatu dignissimum, in quo mutata sola positione binorum punctorum, punctum tertium per idem quoddam intervallum, situm in eadem distantia a medio eorum intervallo, vel perpetuo attrahitur, vel perpetuo repellitur, vel nee attrahitur, nee repellitur ; cujusmodi discrimen cum in massis haberi debeat multo 222 majus, illud indico, num. 222, quantus inde itidem in Physicam usus proveniat. 223 Hie jam num. 223 a viribus binorum punctorum transeo ad considerandum totum ipsorum systema, & usque ad num. 228 contemplor tria puncta in directum sita, ex quorum mutuis viribus relationes quaedam exurgunt, quas multo generaliores redduntur inferius, ubi in tribus etiam punctis tantummodo adumbrantur, quae pertinent ad virgas rigidas, flexiles, elasticas, ac ad vectem, & ad alia plura, quae itidem inferius, ubi de massis, multo generaliora 228 fiunt. Demum usque ad num. 238 contemplor tria puncta posita non in directum, sive in aequilibrio sint, sive in perimetro ellipsium quarundam, vel curvarum aliarum ; in quibus mira occurrit analogia limitum quorundam cum limitibus, quos habent bina puncta in axe curvae primigeniae ad se invicem, atque ibidem multo major varietas casuum indicatur pro massis, & specimen applicationis exhibetur ad soliditatem, & liquationem per celerem 238 intestinum motum punctis impressum. Sequentibus autem binis numeris generalia quaedam expono de systemate punctorum quatuor cum applicatione ad virgas solidas, rigidas, flexiles, ac ordines particularum varies exhibeo per pyramides, quarum infimae ex punctis quatuor, superiores ex quatuor pyramidibus singulae coalescant. 24° A num. 240 ad massas gradu facto usque a num. 264 considero, quae ad centrum gravi- tatis pertinent, ac demonstro generaliter, in quavis massa esse aliquod, & esse unicum : ostendo, quo pacto determinari generaliter possit, & quid in methodo, quae communiter adhibetur, desit ad habendam demonstrationis vim, luculenter expono, & suppleo, ac SYNOPSIS OF THE WHOLE WORK 25 to Art. 179, I consider the areas included by the arcs; these, corresponding to different 172 segments of the axis, may be of any magnitude whatever, either great or small ; moreover they measure the increment or decrement in the squares of the velocities. Then, on as 179 far as Art. 189, 1 investigate the approach of the curve to the axis ; both when the former is cut by the latter, in which case there are transitions from repulsion to attraction and from attraction to repulsion, which I call ' limits,' & use very largely in every part of my Theory ; & also when the former is touched by the latter, & the curve once again recedes from the axis. I consider, too, as a case of approach, recession to infinity along an asymp- totic arc ; and I investigate what transitions, or limits, may arise from such a case, & whether such are admissible in Nature. In Art. 189, I pass on from the consideration of the curve to combinations of points. l%9 First, as far as Art. 204, I deal with a system of two points. I work out those things that concern their mutual forces, and motions, whether they are left to themselves or pro- jected in any manner whatever. Here also, having explained the connection between these motions & the distances of the limits, & different cases of oscillations, whether they are affected by external action of other points, or are not so disturbed, I make an antici- patory note of the great use to which this will be put in the third part, for the purpose of explaining various kinds of cohesion, fermentations, conflagrations, emissions of vapours, the properties of light, elasticity and flexibility. There follows, from Art. 204 to Art. 239, the much more fruitful consideration of a 204 system of three points. The forces connected with them can in general be easily deter- mined for any given positions of the points ; but, when any position & velocity are given, the motions have not yet been obtained by geometricians in such a form that the general calculation can be performed for every possible case. So I proceed to consider the forces, & the huge variation that different combinations of the points beget, although they are only three in number, as far as Art. 209. From that, on to Art. 214, I consider certain 209 things that have to do with the forces that arise from the action, on each of the points, of the other two together, & how these urge the third point not only to approach, or recede from, themselves, but also in a direction at right angles ; in this connection there comes forth an analogy with solidity, & a truly immense difference between the several cases when the distances are very small, & the greatest conformity possible at very great distances such as those at which gravity acts ; & I point out what great use will be made of this also in explaining the constitution of Nature. Then up to Art. 221, I give ocular demonstra- 214 tions of the huge differences that there are in the laws of forces with which two points act upon a third, whether it lies in the right line joining them, or in the right line that is the perpendicular which bisects the interval between them ; this I do by constructing, from the primary curve, curves representing the composite forces. Then in the two articles 221 that follow, I consider the case, a really important one, in which, by merely changing the position of the two points, the third point, at any and the same definite interval situated at the same distance from the middle point of the interval between the two points, will be either continually attracted, or continually repelled, or neither attracted nor repelled ; & since a difference of this kind should hold to a much greater degree in masses, I point out, in Art. 222, the great use that will be made of this also in Physics. 222 At this point then, in Art. 223, I pass from the forces derived from two points to the 223 consideration of a whole system of them ; and, as far as Art. 228, I study three points situated in a right line, from the mutual forces of which there arise certain relations, which I return to later in much greater generality ; in this connection also are outlined, for three points only, matters that have to do with rods, either rigid, flexible or elastic, and with the lever, as well as many other things ; these, too, are treated much more generally later on, when I consider masses. Then right on to Art. 238, I consider three points that do not lie in a right line, whether they are in equilibrium, or moving in the perimeters of certain ellipses or other curves. Here we come across a marvellous analogy between certain limits and the limits which two points lying on the axis of the primary curve have with respect to each other ; & here also a much greater variety of cases for masses is shown, & an example is given of the application to solidity, & liquefaction, on account of a quick internal motion being impressed on the points of the body. Moreover, in the two articles that then follow, I state some general propositions with regard to a system of four points, together with their application to solid rods, both rigid and flexible ; I also give an illus- tration of various classes of particles by means of pyramids, each of which is formed of four points in the most simple case, & of four of such pyramids in the more complicated cases. From Art. 240 as far as Art. 264, I pass on to masses & consider matters pertaining to 24° the centre of gravity ; & I prove that in general there is one, & only one, in any given mass. I show how it can in general be determined, & I set forth in clear terms the point that is lacking in the usual method, when it comes to a question of rigorous proof ; this deficiency 26 SYNOPSIS TOTIUS OPERIS exemplum profero quoddam ejusdem generis, quod ad numerorum pertinet multiplica- tionem, & ad virium compositionem per parallelogramma, quam alia methodo generaliore exhibeo analoga illi ipsi, qua generaliter in centrum gravitatis inquire : turn vero ejusdem ope demonstro admodum expedite, & accuratissime celebre illud Newtoni theorema de statu centri gravitatis per mutuas internas vires numquam turbato. 264 Ejus tractionis fructus colligo plures : conservationem ejusdem quantitatis motuum in 265 Mundo in eandem plagam num. 264, sequalitatem actionis, & reactionis in massis num. 265, 266 collisionem corporum, & communicationem motus in congressibus directis cum eorum 276 legibus, inde num. 276 congressus obliques, quorum Theoriam a resolutione motuum reduce 277, 278 ad compositionem num. 277, quod sequent! numero 278 transfero ad incursum etiam in 270 planum immobile ; ac a num. 279 ad 289 ostendo nullam haberi in Natura veram virium, aut motuum resolutionem, sed imaginariam tantummodo, ubi omnia evolvo, & explico casuum genera, quae prima fronte virium resolutionem requirere videntur. 289 A num. 289 ad 297 leges expono compositionis virium, & resolutionis, ubi & illud notissimum, quo pacto in compositione decrescat vis, in resolutione crescat, sed in ilia priore conspirantium summa semper maneat, contrariis elisis ; in hac posteriore concipiantur tantummodo binae vires contrarise adjectas, quse consideratio nihil turbet phenomena ; unde fiat, ut nihil inde pro virium vivarum Theoria deduci possit, cum sine iis explicentur omnia, ubi plura itidem explico ex iis phsenomenis, quse pro ipsis viribus vivis afferri solent. 2Q7 A num. 297 occasione inde arrepta aggredior qusedam, quae ad legem continuitatis pertinent, ubique in motibus sancte servatam, ac ostendo illud, idcirco in collisionibus corporum, ac in motu reflexo, leges vulgo definitas, non nisi proxime tantummodo observari, & usque ad num. 307 relationes varias persequor angulorum incidentisa, & reflexionis, sive vires constanter in accessu attrahant, vel repellant constanter, sive jam attrahant, jam repellant : ubi & illud considero, quid accidat, si scabrities superficiei agentis exigua sit, quid, si ingens, ac elementa profero, quae ad luminis reflexionem, & refractionem explican- dam, definiendamque ex Mechanica requiritur, relationem itidem vis absolutae ad relativam in obliquo gravium descensu, & nonnulla, quae ad oscillationum accuratiorem Theoriam necessaria sunt, prorsus elementaria, diligenter expono. 307 A num. 307 inquire in trium massarum systema, ubi usque ad num. 313 theoremata evolvo plura, quae pertinent ad directionem virium in singulis compositarum e binis reliquarum actionibus, ut illud, eas directiones vel esse inter se parallelas, vel, si utrinque 313 indefinite producantur, per quoddam commune punctum transire omnes : turn usque ad 321 theoremata alia plura, quae pertinent ad earumdem compositarum virium rationem ad se invicem, ut illud & simplex, & elegans, binarum massarum vires acceleratrices esse semper in ratione composita ex tribus reciprocis rationibus, distantise ipsarum a massa tertia, sinus anguli, quern singularum directio continet cum sua ejusmodi distantia, & massae ipsius earn habentis compositam vim, ad distantiam, sinum, massam alteram ; vires autem motrices habere tantummodo priores rationes duas elisa tertia. 321 Eorum theorematum fructum colligo deducens inde usque ad num. 328, quae ad aequilibrium pertinent divergentium utcumque virium, & ipsius aequilibrii centrum, ac nisum centri in fulcrum, & quae ad prseponderantiam, Theoriam extendens ad casum etiam, quo massae non in se invicem agant mutuo immediate, sed per intermedias alias, quse nexum concilient, & virgarum nectentium suppleant vices, ac ad massas etiam quotcunque, quarum singulas cum centro conversionis, & alia quavis assumpta massa connexas concipio, unde principium momenti deduce pro machinis omnibus : turn omnium vectium genera evolvo, ut & illud, facta suspensione per centrum gravitatis haberi aequilibrium, sed in ipso centro debere sentiri vim a fulcro, vel sustinente puncto, sequalem summae ponderum totius systematis, unde demum pateat ejus ratio, quod passim sine demonstratione assumitur, nimirum systemate quiescente, & impedito omni partium motu per aequilibrium, totam massam concipi posse ut in centro gravitatis collectam. SYNOPSIS OF THE WHOLE WORK 27 I supply, & I bring forward a certain example of the same sort, that deals with the multi- plication of numbers, & to the composition of forces by the parallelogram law ; the latter I prove by another more general method, analogous to that which I use in the general investigation for the centre of gravity. Then by its help I prove very expeditiously & with extreme rigour that well-known theorem of Newton, in which he affirmed that the state of the centre of gravity is in no way altered by the internal mutual forces. I gather several good results from this method of treatment. In Art. 264, the con- 264 servation of the same quantity of motion in the Universe in one plane ; in Art. 265 the 265 equality of action and reaction amongst masses ; then the collision of solid bodies, and the 266 communication of motions in direct impacts & the laws that govern them, & from that, 276 in Art. 276, oblique impacts ; in Art. 277 I reduce the theory of these from resolution of 277 motions to compositions, & in the article that follows, Art. 278, I pass to impact on to a 278 fixed plane; from Art. 279 to Art. 289 I show that there can be no real resolution of forces 279 or of motions in Nature, but only a hypothetical one ; & in this connection I consider & explain all sorts of cases, in which at first sight it would seem that there must be resolution. From Art. 289 to Art. 297, 1 state the laws for the composition & resolution of forces ; 289 here also I give the explanation of that well-known fact, that force decreases in composition, increases in resolution, but always remains equal to the sum of the parts acting in the same direction as itself in the first, the rest being equal & opposite cancel one another ; whilst in the second, all that is done is to suppose that two equal & opposite forces are added on, which supposition has no effect on the phenomena. Thus it comes about that nothing can be deduced from this in favour of the Theory of living forces, since everything can be explained without them ; in the same connection, I explain also many of the phenomena, which are usually brought forward as evidence in favour of these ' living forces.' In Art. 297, I seize the opportunity offered by the results just mentioned to attack 207 certain matters that relate to the law of continuity, which in all cases of motion is strictly observed ; & I show that, in the collision of solid bodies, & in reflected motion, the laws, as usually stated, are therefore only approximately followed. From this, as far as Art. 307, I make out the various relations between the angles of incidence & reflection, whether the forces, as the bodies approach one another, continually attract, or continually repel, or attract at one time & repel at another. I also consider what will happen if the roughness of the acting surface is very slight, & what if it is very great. I also state the first principles, derived from mechanics, that are required for the explanation & determination of the reflection & refraction of light ; also the relation of the absolute to the relative force in the oblique descent of heavy bodies ; & some theorems that are requisite for the more accurate theory of oscillations ; these, though quite elementary, I explain with great care. From Art. 307 onwards, I investigate the system of three bodies ; in this connection, as far as Art. 313, I evolve several theorems dealing with the direction of the forces on each one of the three compounded from the combined actions of the other two ; such as the theorem, that these directions are either all parallel to one another, or all pass through some one common point, when they are produced indefinitely on both sides. Then, as ^j, far as Art. 321, I make out several other theorems dealing with the ratios of these same resultant forces to one another ; such as the following very simple & elegant theorem, that the accelerating forces of two of the masses will always be in a ratio compounded of three reciprocal ratios ; namely, that of the distance of either one of them from the third mass, that of the sine of the angle which the direction of each force makes with the corresponding distance of this kind, & that of the mass itself on which the force is acting, to the corre- sponding distance, sine and mass for the other : also that the motive forces only have the first two ratios, that of the masses being omitted. I then collect the results to be derived from these theorems, deriving from them, as far ,2I as Art. 328, theorems relating to the equilibrium of forces diverging in any manner, & the centre of equilibrium, & the pressure of the centre on a fulcrum. I extend the theorem relating to preponderance to the case also, in which the masses do not mutually act upon one another in a direct manner, but through others intermediate between them, which connect them together, & supply the place of rods joining them ; and also to any number of masses, each of which I suppose to be connected with the centre of rotation & some other assumed mass, & from this I derive the principles of moments for all machines. Then I consider all the different kinds of levers ; one of the theorems that I obtain is, that, if a lever is suspended from the centre of gravity, then there is equilibrium ; but a force should be felt in this centre from the fulcrum or sustaining point, equal to the sum of the weights of the whole system ; from which there follows most clearly the reason, which is every- where assumed without proof, why the whole mass can be supposed to be collected at its centre of gravity, so long as the system is in a state of rest & all motions of its parts are pro- hibited by equilibrium. 28 SYNOPSIS TOTIUS OPERIS 328 A num. 328 ad 347 deduce ex iisdem theorematis, quae pertinent ad centrum oscilla- tionis quotcunque massarum, sive sint in eadem recta, sive in piano perpendiculari ad axem rotationis ubicunque, quse Theoria per systema quatuor massarum, excolendum aliquanto diligentius, uberius promoveri deberet & extendi ad generalem habendum solidorum nexum, 344 qua re indicata, centrum itidem percussionis inde evolve, & ejus analogiam cum centre oscillationis exhibeo. 347 Collecto ejusmodi fructu ex theorematis pertinentibus ad massas tres, innuo num. 347, quae mihi communia sint cum ceteris omnibus, & cum Newtonianis potissimum, pertinentia ad summas virium, quas habet punctum, vel massa attracta, vel repulsa a punctis singulis 348 alterius massae ; turn a num. 348 ad finem hujus partis, sive ad num. 358, expono quasdam, quae pertinent ad fluidorum Theoriam, & primo quidem ad pressionem, ubi illud innuo demonstratum a Newtono, si compressio fluidi sit proportionalis vi comprimenti, vires repulsivas punctorum esse in ratione reciproca distantiarum, ac vice versa : ostendo autem illud, si eadem vis sit insensibilis, rem, praeter alias curvas, exponi posse per Logisticam, & in fluidis gravitate nostra terrestri prseditis pressiones haberi debere ut altitudines ; deinde vero attingo ilia etiam, quae pertinent ad velocitatem fluidi erumpentis e vase, & expono, quid requiratur, ut ea sit sequalis velocitati, quae acquiretur cadendo per altitudinem ipsam, quemadmodum videtur res obtingere in aquae efHuxu : quibus partim expositis, partim indicatis, hanc secundam partem conclude. PARS III 358 Num. 358 propono argumentum hujus tertise partis, in qua omnes e Theoria mea 360 generales materis proprietates deduce, & particulares plerasque : turn usque ad num. 371 ago aliquanto fusius de impenetrabilitate, quam duplicis generis agnosco in meis punctorum inextensorum massis, ubi etiam de ea apparenti quadam compenetratione ago, ac de luminis trarlsitu per substantias intimas sine vera compenetratione, & mira quaedam phenomena 371 hue pertinentia explico admodum expedite. Inde ad num. 375 de extensione ago, quae mihi quidem in materia, & corporibus non est continua, sed adhuc eadem praebet phaeno- menae sensibus, ac in communi sententia ; ubi etiam de Geometria ago, quae vim suam in 375 mea Theoria retinet omnem : turn ad num. 383 figurabilitatem perseqUor, ac molem, massam, densitatem singillatim, in quibus omnibus sunt quaedam Theoriae meae propria 383 scitu non indigna. De Mobilitate, & Motuum Continuitate, usque ad num. 388 notatu 388 digna continentur : turn usque ad num. 391 ago de aequalitate actionis, & reactionis, cujus consectaria vires ipsas, quibus Theoria mea innititur, mirum in modum conformant. Succedit usque ad num. 398 divisibilitas, quam ego ita admitto, ut quaevis massa existens numerum punctorum realium habeat finitum tantummodo, sed qui in data quavis mole possit esse utcunque magnus ; quamobrem divisibilitati in infinitum vulgo admissae sub- stituo componibilitatem in infinitum, ipsi, quod ad Naturae phenomena explicanda 398 pertinet, prorsus aequivalentem. His evolutis addo num. 398 immutabilitatem primorum materiae elementorum, quse cum mihi sint simplicia prorsus, & inextensa, sunt utique immutabilia, & ad exhibendam perennem phasnomenorum seriem aptissima. 399 A num. 399 ad 406 gravitatem deduco ex mea virium Theoria, tanquam ramum quendam e communi trunco, ubi & illud expono, qui fieri possit, ut fixae in unicam massam 406 non coalescant, quod gravitas generalis requirere videretur. Inde ad num. 419 ago de cohaesione, qui est itidem veluti alter quidam ramus, quam ostendo, nee in quiete con- sistere, nee in motu conspirante, nee in pressione fluidi cujuspiam, nee in attractione maxima in contactu, sed in limitibus inter repulsionem, & attractionem ; ubi & problema generale propono quoddam hue pertinens, & illud explico, cur massa fracta non iterum coalescat, cur fibrae ante fractionem distendantur, vel contrahantur, & innuo, quae ad cohaesionem pertinentia mihi cum reliquis Philosophis communia sint. 419 A cohacsione gradum facio num. 419 ad particulas, quae ex punctis cohaerentibus efformantur, de quibus .ago usque ad num. 426. & varia persequor earum discrimina : SYNOPSIS OF THE WHOLE WORK 29 From Art. 328 to Art. 347, I deduce from these same theorems, others that relate to 328 the centre of oscillation of any number of masses, whether they are in the same right line, or anywhere in a plane perpendicular to the axis of rotation ; this theory wants to be worked somewhat more carefully with a system of four bodies, to be gone into more fully, & to be extended so as to include the general case of a system of solid bodies ; having stated this, I evolve from it the centre of percussion, & I show the analogy between it & the centre 344 of oscillation. I obtain all such results from theorems relating to three masses. After that, in Art. 347 347, I intimate the matters in which I agree with all others, & especially with the followers of Newton, concerning sums of forces, acting on a point, or an attracted or repelled mass, due to the separate points of another mass. Then, from Art. 348 to the end of this part, 348 i.e., as far as Art. 359, I expound certain theorems that belong to the theory of fluids ; & first of all, theorems with regard to pressure, in connection with which I mention that one which was proved by Newton, namely, that, if the compression of a fluid is proportional to the compressing force, then the repulsive forces between the points are in the reciprocal ratio of the distances, & conversely. Moreover, I show that, if the same force is insen- sible, then the matter can be represented by the logistic & other curves ; also that in fluids subject to our terrestrial gravity pressures should be found proportional to the depths. After that, I touch upon those things that relate to the velocity of a fluid issuing from a vessel ; & I show what is necessary in order that this should be equal to the velocity which would be acquired by falling through the depth itself, just as it is seen to happen in the case of an efflux of water. These things in some part being explained, & in some part merely indicated, I bring this second part to an end. PART III In Art. 358, I state the theme of this third part ; in it I derive all the general & most 358 of the special, properties of matter from my Theory. Then, as far as Art. 371, I deal some- 360 what more at length with the subject of impenetrability, which I remark is of a twofold kind in my masses of non-extended points ; in this connection also, I deal with a certain apparent case of compenetrability, & the passage of light through the innermost parts of bodies without real compenetration ; I also explain in a very summary manner several striking phenomena relating to the above. From here on to Art. 375, I deal with exten- 371 sion ; this in my opinion is not continuous either in matter or in solid bodies, & yet it yields the same phenomena to the senses as does the usually accepted idea of it ; here I also deal with geometry, which conserves all its power under my Theory. Then, as far 375 as Art. 383, I discuss figurability, volume, mass & density, each in turn ; in all of these subjects there are certain special points of my Theory that are not unworthy of investi- gation. Important theorems on mobility & continuity of motions are to be found from here on to Art. 388 ; then, as far as Art. 391, I deal with the equality of action & reaction, & my conclusions with regard to the subject corroborate in a wonderful way the hypothesis of those forces, upon which my Theory depends. Then follows divisibility, as far as Art. 39 1 398 ; this principle I admit only to the extent that any existing mass may be made up of a number of real points that are finite only, although in any given mass this finite number may be as great as you please. Hence for infinite divisibility, as commonly accepted, I substitute infinite multiplicity ; which comes to exactly the same thing, as far as it is concerned with the explanation of the phenomena of Nature. Having considered these subjects I add, in Art. 398, that of the immutability of the primary elements of matter ; 398 according to my idea, these are quite simple in composition, of no extent, they are every- where unchangeable, & hence are splendidly adapted for explaining a continually recurring set of phenomena. From Art. 399 to Art. 406, 1 derive gravity from my Theory of forces, as if it were a 399 particular branch on a common trunk ; in this connection also I explain how it can happen that the fixed stars do not all coalesce into one mass, as would seem to be required under 406 universal gravitation. Then, as far as Art. 419, I deal with cohesion, which is also as it were another branch ; I show that this is not dependent upon quiescence, nor on motion that is the same for all parts, nor on the pressure of some fluid, nor on the idea that the attraction is greatest at actual contact, but on the limits between repulsion and attraction. I propose, & solve, a general problem relating to this, namely, why masses, once broken, do not again stick together, why the fibres are stretched or contracted before fracture takes place ; & I intimate which of my ideas relative to cohesion are the same as those held by other philosophers. In Art. 419, 1 pass on from cohesion to particles which are formed from a number of 4J9 cohering points ; & I consider these as far as Art. 426, & investigate the various distinctions 30 SYNOPSIS TOTIUS OPERIS ostendo nimirum, quo pacto varias induere possint figuras quascunque, quarum tenacissime sint ; possint autem data quavis figura discrepare plurimum in numero, & distributione punctorum, unde & oriantur admodum inter se diversae vires unius particulae in aliam, ac itidem diversae in diversis partibus ejusdem particulae respectu diversarum partium, vel etiam respectu ejusdem partis particulse alterius, cum a solo numero, & distributione punctorum pendeat illud, ut data particula datam aliam in datis earum distantiis, & superficierum locis, vel attrahat, vel repellat, vel respectu ipsius sit prorsus iners : turn illud addo, particulas eo dimcilius dissolubiles esse, quo minores sint ; debere autem in gravitate esse penitus uniformes, quaecunque punctorum dispositio habeatur, & in aliis proprietatibus plerisque debere esse admodum (uti observamus) diversas, quae diversitas multo major in majoribus massis esse debeat. 426 A num. 426 ad 446 de solidis, & fluidis, quod discrimen itidem pertinet ad varia cohaesionum genera ; & discrimen inter solida, & fluida diligenter expono, horum naturam potissimum repetens ex motu faciliori particularum in gyrum circa alias, atque id ipsum ex viribus circumquaque aequalibus ; illorum vero ex inaequalitate virium, & viribus quibusdam in latus, quibus certam positionem ad se invicem servare debeant. Varia autem distinguo fluidorum genera, & discrimen profero inter virgas rigidas, flexiles, elasticas, fragiles, ut & de viscositate, & humiditate ago, ac de organicis, & ad certas figuras determinatis corporibus, quorum efformatio nullam habet difficultatem, ubi una particula unam aliam possit in certis tantummodo superficiei partibus attrahere, & proinde cogere ad certam quandam positionem acquirendam respectu ipsius, & retinendam. Demonstro autem & illud, posse admodum facile ex certis particularum figuris, quarum ipsae tenacissimae sint, totum etiam Atomistarum, & Corpuscularium systema a mea Theoria repeti ita, ut id nihil sit aliud, nisi unicus itidem hujus veluti trunci foecundissimi ramus e diversa cohaesionis ratione prorumpens. Demum ostendo, cur non quaevis massa, utut constans ex homogeneis punctis, & circa se maxime in gyrum mobilibus, fluida sit ; & fluidorum resistentiam quoque attingo, in ejus leges inquirens. 446 A num. 446 ad 450 ago de iis, quae itidem ad diversa pertinent soliditatis genera, nimirum de elasticis, & mollibus, ilia repetens a magna inter limites proximos distantia, qua fiat, ut puncta longe dimota a locis suis, idem ubique genus virium sentiant, & proinde se ad priorem restituant locum ; hasc a limitum frequentia, atque ingenti vicinia, qua fiat, ut ex uno ad alium delata limitem puncta, ibi quiescant itidem respective, ut prius. Turn vero de ductilibus, & malleabilibus ago, ostendens, in quo a fragilibus discrepent : ostendo autem, haec omnia discrimina a densitate nullo modo pendere, ut nimirum corpus, quod multo sit altero densius, possit tarn multo majorem, quam multo minorem soliditatem, & cohaesionem habere, & quaevis ex proprietatibus expositis aeque possit cum quavis vel majore, vel minore densitate componi. 450 Num. 450 inquire in vulgaria quatuor elementa ; turn a num. 451 ad num. 467 persequor 452 chemicas operationes ; num. 452 explicans dissolutionem, 453 praecipitationem, 454, & 455 commixtionem plurium substantiarum in unam : turn num. 456, & 457 liquationem binis methodis, 458 volatilizationem, & effervescentiam, 461 emissionem efHuviorum, quae e massa constanti debeat esse ad sensum constans, 462 ebullitionem cum variis evaporationum generibus ; 463 deflagrationem, & generationem aeris ; 464 crystallizationem cum certis figuris ; ac demum ostendo illud num. 465, quo pacto possit fermentatio desinere ; & num. 466, quo pacto non omnia fermentescant cum omnibus. 467 A fermentatione num. 467 gradum facio ad ignem, qui mihi est fermentatio quaedam substantiae lucis cum sulphurea quadam substantia, ac plura inde consectaria deduce usque 471 ad num. 471 ; turn ab igne ad lumen ibidem transeo, cujus proprietates praecipuas, ex 472 quibus omnia lucis phaenomena oriuntur, propono num. 472, ac singulas a Theoria mea deduce, & fuse explico usque ad num. 503, nimirum emissionem num. 473, celeritatem 474, propagationem rectilineam per media homogenea, & apparentem tantummodo compene- trationem a num. 475 ad 483, pellucidatem, & opacitatem num. 483, reflexionem ad angulos aequales inde ad 484, refractionem ad 487, tenuitatem num. 487, calorem, & ingentes intestines motus allapsu tenuissimae lucis genitos, num. 488, actionem majorem corporum eleosorum, & sulphurosorum in lumen num. 489 : turn num. 490 ostendo, nullam resist- SYNOPSIS OF THE WHOLE WORK 31 between them. I show how it is possible for various shapes of all sorts to be assumed, which offer great resistance to rupture ; & how in a given shape they may differ very greatly in the number & disposition of the points forming them. Also that from this fact there arise very different forces for the action of one particle upon another, & also for the action of different parts of this particle upon other different parts of it, or on the same part of another particle. For that depends solely on the number & distribution of the points, so that one given particle either attracts, or repels, or is perfectly inert with regard to another given particle, the distances between them and the positions of their surfaces being also given. Then I state in addition that the smaller the particles, the greater is the diffi- culty in dissociating them ; moreover, that they ought to be quite uniform as regards gravitation, no matter what the disposition of the points may be ; but in most other properties they should be quite different from one another (which we observe to be the case) ; & that this difference ought to be much greater in larger masses. From Art. 426 to Art. 446, 1 consider solids & fluids, the difference between which is 426 also a matter of different kinds of cohesion. I explain with great care the difference between solids & fluids ; deriving the nature of the latter from the greater freedom of motion of the particles in the matter of rotation about one another, this being due to the forces being nearly equal ; & that of the former from the inequality of the forces, and from certain lateral forces which help them to keep a definite position with regard to one another. I distinguish between various kinds of fluids also, & I cite the distinction between rigid, flexible, elastic & fragile rods, when I deal with viscosity & humidity ; & also in dealing with organic bodies & those solids bounded by certain fixed figures, of which the formation presents no difficulty ; in these one particle can only attract another particle in certain parts of the surface, & thus urge it to take up some definite position with regard to itself, & keep it there. I also show that the whole system of the Atomists, & also of the Corpus- cularians, can be quite easily derived by my Theory, from the idea of particles of definite shape, offering a high resistance to deformation ; so that it comes to nothing else than another single branch of this so to speak most fertile trunk, breaking forth from it on account of a different manner of cohesion. Lastly, I show the reason why it is that not every mass, in spite of its being constantly made up of homogeneous points, & even these in a high degree capable of rotary motion about one another, is a fluid. I also touch upon the resistance of fluids, & investigate the laws that govern it. From Art. 446 to Art. 450, I deal with those things that relate to the different kinds 446 of solidity, that is to say, with elastic bodies, & those that are soft. I attribute the nature of the former to the existence of a large interval between the consecutive limits, on account of which it comes about that points that are far removed from their natural positions still feel the effects of the same kind of forces, & therefore return to their natural positions ; & that of the latter to the frequency & great closeness of the limits, on account of which it comes about that points that have been moved from one limit to another, remain there in relative rest as they were to start with. Then I deal with ductile and malleable solids, pointing out how they differ from fragile solids. Moreover I show that all these differ- ences are in no way dependent on density ; so that, for instance, a body that is much more dense than another body may have either a much greater or a much less solidity and cohesion than another ; in fact, any of the properties set forth may just as well be combined with any density either greater or less. In Art. 450 I consider what are commonly called the " four elements " ; then from 450 Art. 451 to Art. 467, I treat of chemical operations ; I explain solution in Art. 452, preci- 452 pitation in Art. 453, the mixture of several substances to form a single mass in Art. 454, 455, liquefaction by two methods in Art. 456, 457, volatilization & effervescence in Art. 458, emission of effluvia (which from a constant mass ought to be approximately constant) in. Art. 461, ebullition & various kinds of evaporation in Art. 462, deflagration & generation of gas in Art. 463, crystallization with definite forms of crystals in Art. 464 ; & lastly, I show, in Art. 465, how it is possible for fermentation to cease, & in Art. 466, how it is that any one thing does not ferment when mixed with any other thing. From fermentation I pass on, in Art. 467, to fire, which I look upon as a fermentation 467 of some substance in light with some sulphureal substance ; & from this I deduce several propositions, up to Art. 471. There I pass on from fire to light, the chief properties of 471 which, from which all the phenomena of light arise, I set forth in Art. 472 ; & I deduce 472 & fully explain each of them in turn as far as Art. 503. Thus, emission in Art. 473, velo- city in Art. 474, rectilinear propagation in homogeneous media, & a compenetration that is merely apparent, from Art. 475 on to Art. 483, pellucidity & opacity in Art. 483, reflec- tion at equal angles to Art. 484, & refraction to Art. 487, tenuity in Art. 487, heat & the great internal motions arising from the smooth passage of the extremely tenuous light in Art. 488, the greater action of oleose & sulphurous bodies on light in Art. 489. Then I 32 SYNOPSIS TOTIUS OPERIS entiam veram pati, ac num. 491 explico, unde sint phosphora, num. 492 cur lumen cum majo e obliquitate incidens reflectatur magis, num. 493 & 494 unde diversa refrangibilitas ortum ducat, ac num. 495, & 496 deduce duas diversas dispositiones ad asqualia redeuntes intervalla, unde num. 497 vices illas a Newtono detectas facilioris reflexionis, & facilioris transmissus eruo, & num. 498 illud, radios alios debere reflecti, alios transmitti in appulsu ad novum medium, & eo plures reflecti, quo obliquitas incidentise sit major, ac num. 499 & 500 expono, unde discrimen in intervallis vicium, ex quo uno omnis naturalium colorum pendet Newtoniana Theoria. Demum num. 501 miram attingo crystalli Islandicse proprietatem, & ejusdem causam, ac num. 502 diffractionem expono, quse est quaedam inchoata refractio, sive reflexio. 503 Post lucem ex igne derivatam, quse ad oculos pertinet, ago brevissime num. 503 de 504 sapore, & odore, ac sequentibus tribus numeris de sono : turn aliis quator de tactu, ubi 507 etiam de frigore, & calore : deinde vero usque ad num. 514 de electricitate, ubi totam 511 Franklinianam Theoriam ex meis principiis explico, eandem ad bina tantummodo reducens principia, quse ex mea generali virium Theoria eodem fere pacto deducuntur, quo prsecipi- 514 tationes, atque dissolutiones. Demum num. 514, ac 515 magnetismum persequor, tam directionem explicans, quam attractionem magneticam. 516 Hisce expositis, quas ad particulares .etiam proprietates pertinent, iterum a num. 516 ad finem usque generalem corporum complector naturam, & quid materia sit, quid forma, quse censeri debeant essentialia, quse accidentialia attributa, adeoque quid transformatio sit, quid alteratio, singillatim persequor, & partem hanc tertiam Theorise mesa absolve. De Appendice ad Metaphysicam pertinente innuam hie illud tantummodo, me ibi exponere de anima illud inprimis, quantum spiritus a materia differat, quern nexum anima habeat cum corpore, & quomodo in ipsum agat : turn de DEO, ipsius & existentiam me pluribus evincere, quae nexum habeant cum ipsa Theoria mea, & Sapientiam inprimis, ac Providentiam, ex qua gradum ad revelationem faciendum innuo tantummodo. Sed hsec in antecessum veluti delibasse sit satis. SYNOPSIS OF THE WHOLE WORK 33 show, in Art. 490, that it suffers no real resistance, & in Art. 491 I explain the origin of bodies emitting light, in Art. 492 the reason why light that falls with greater obliquity is reflected more strongly, in Art. 493, 494 the origin of different degrees of refrangibility, & in Art. 495, 496 I deduce that there are two different dispositions recurring at equal intervals ; hence, in Art. 497, I bring out those alternations, discovered by Newton, of easier reflection & easier transmission, & in Art. 498 I deduce that some rays should be reflected & others transmitted in the passage to a fresh medium, & that the greater the obli- quity of incidence, the greater the number of reflected rays. In Art. 499, 500 I state the origin of the difference between the lengths of the intervals of the alternations ; upon this alone depends the whole of the Newtonian theory of natural colours. Finally, in Art. 501, I touch upon the wonderful property of Iceland spar & its cause, & in Art. 502 I explain diffraction, which is a kind of imperfect refraction or reflection. After light derived from fire, which has to do with vision, I very briefly deal with taste & smell in Art. 503, £ of sound in the three articles that follow next. Then, in the S°3 next four articles, I consider touch, & in connection with it, cold & heat also. After that, 5°4 as far as Art. 514, I deal with electricity ; here I explain the whole of the Franklin theory 5°7 by means of my principles ; I reduce this theory to two principles only, & these are 5 1 1 derived from my general Theory of forces in almost the same manner as I have already derived precipitations & solutions. Finally, in Art. 514, 515, I investigate magnetism, explaining 5H both magnetic direction £ attraction. These things being expounded, all of which relate to special properties, I once more consider, in the articles from 516 to the end, the general nature of bodies, what matter is, 516 its form, what things ought to be considered as essential, & what as accidental, attributes ; and also the nature of transformation and alteration are investigated, each in turn ; & thus I bring to a close the third part of my Theory. I will mention here but this one thing with regard to the appendix on Metaphysics ; namely, that I there expound more especially how greatly different is the soul from matter, the connection between the soul & the body, & the manner of its action upon it. Then with regard to GOD, I prove that He must exist by many arguments that have a close con- nection with this Theory of mine ; I especially mention, though but slightly, His Wisdom and Providence, from which there is but a step to be made towards revelation. But I think that I have, so to speak, given my preliminary foretaste quite sufficiently. [I] PHILOSOPHIC NATURALIS THEORIA In quo conveniat cum systemate Newtoniano, & Leibnitiano. Cujusmodi systema> Theoria exhibeat. PARS I Theorice expositio, analytica deductio^ & vindicatio. lRIUM mutuarum Theoria, in quam incidi jam ab Anno 1745, dum e notissimis principiis alia ex aliis consectaria eruerem, & ex qua ipsam simplicium materise elementorum constitutionem deduxi, systema exhibet medium inter Leibnitianum, & Newtonianum, quod nimirum & ex utroque habet plurimum, & ab utroque plurimum dissidet ; at utroque in immensum simplicius, proprietatibus corporum generalibus sane omnibus, & [2] peculiaribus quibusque praecipuis per accuratissimas demonstrationes deducendis est profecto mirum in modum idoneum. 2. Habet id quidem ex Leibnitii Theoria elementa prima simplicia, ac prorsus inex- tensa : habet ex Newtoniano systemate vires mutuas, quae pro aliis punctorum distantiis a se invicem aliae sint ; & quidem ex ipso itidem Newtono non ejusmodi vires tantummodo, quse ipsa puncta determinent ad accessum, quas vulgo attractiones nominant ; sed etiam ejusmodi, quae determinent ad recessum, & appellantur repulsiones : atque id ipsum ita, ut, ubi attractio desinat, ibi, mutata distantia, incipiat repulsio, & vice versa, quod nimirum Newtonus idem in postrema Opticse Quaestione proposuit, ac exemplo transitus a positivis ad negativa, qui habetur in algebraicis formulis, illustravit. Illud autem utrique systemati commune est cum hoc meo, quod quaevis particula materiae cum aliis quibusvis, utcunque remotis, ita connectitur, ut ad mutationem utcunque exiguam in positione unius cujusvis, determinationes ad motum in omnibus reliquis immutentur, & nisi forte elidantur omnes oppositas, qui casus est infinities improbabilis, motus in iis omnibus aliquis inde ortus habeatur. In quo differat a Leibnitiano & ipsi praestet. 3. Distat autem a Leibnitiana Theoria longissime, turn quia nullam extensionem continuam admittit, quae ex contiguis, & se contingentibus inextensis oriatur : in quo quidem dirficultas jam olim contra Zenonem proposita, & nunquam sane aut soluta satis, aut solvenda, de compenetratione omnimoda inextensorum contiguorum, eandem vim adhuc habet contra Leibnitianum systema : turn quia homogeneitatem admittit in elementis, omni massarum discrimine a sola dispositione, & diversa combinatione derivato, ad quam homogeneitatem in elementis, & discriminis rationem in massis, ipsa nos Naturae analogia ducit, ac chemicae resolutiones inprimis, in quibus cum ad adeo pauciora numero, & adeo minus inter se diversa principiorum genera, in compositorum corporum analysi deveniatur, id ipsum indicio est, quo ulterius promoveri possit analysis, eo ad majorem simplicitatem, & homogeneitatem devenire debere, adeoque in ultima demum resolutione ad homogenei- tatem, & simplicitatem summam, contra quam quidem indiscernibilium principium, & principium rationis sufficients usque adeo a Leibnitianis depraedicata, meo quidem judicio, nihil omnino possunt. in quo differat a A Distat itidem a Newtoniano systemate quamplunmum, turn in eo, quod ea, quae Newtoniano & ipsi XT . . r\ • r\ • r praestet. Newtonus in ipsa postremo (Juaestione (Jpticae conatus est expncare per tna pnncipia, gravitatis, cohsesionis, fermentationis, immo & reliqua quamplurima, quae ab iis tribus principiis omnino non pendent, per unicam explicat legem virium, expressam unica, & ex pluribus inter se commixtis non composita algebraica formula, vel unica continua geometrica curva : turn in eo, quod in mi-[3]-nimis distantiis vires admittat non positivas, sive attractivas, uti Newtonus, sed negativas, sive repulsivas, quamvis itidem eo majores in 34 A THEORY OF NATURAL PHILOSOPHY PART I Exposition ^ ^Analytical Derivation & Proof of the Theory I. ' ^i ^^ HE following Theory of mutual forces, which I lit upon as far back as the year The kind of sys- 1745, whilst I was studying various propositions arising from other very p^ents.6 well-known principles, & from which I have derived the very constitu- tion of the simple elements of matter, presents a system that is midway between that of Leibniz & that of Newton ; it has very much in common with both, & differs very much from either ; &, as it is immensely more simple than either, it is undoubtedly suitable in a marvellous degree for deriving all the general properties of bodies, & certain of the special properties also, by means of the most rigorous demonstrations. 2. It indeed holds to those simple & perfectly non-extended primary elements upon what there is in which is founded the theory of Leibniz ; & also to the mutual forces, which vary as the * s£^"0"{ to$^ distances of the points from one another vary, the characteristic of the theory of Newton ; ton *& Leibniz. in addition, it deals not only with the kind of forces, employed by Newton, which oblige the points to approach one another, & are commonly called attractions ; but also it considers forces of a kind that engender recession, & are called repulsions. Further, the idea is introduced in such a manner that, where attraction ends, there, with a change of distance, repulsion begins ; this idea, as a matter of fact, was suggested by Newton in the last of his ' Questions on Optics ', & he illustrated it by the example of the passage from positive to negative, as used in algebraical formulas. Moreover there is this common point between either of the theories of Newton & Leibniz & my own ; namely, that any particle of matter is connected with every other particle, no matter how great is the distance between them, in such a way that, in accordance with a change in the position, no matter how slight, of any one of them, the factors that determine the motions of all the rest are altered ; &, unless it happens that they all cancel one another (& this is infinitely impro- bable), some motion, due to the change of position in question, will take place in every one of them. 3. But my Theory differs in a marked degree from that of Leibniz. For one thing, How it differs from, because it does not admit the continuous extension that arises from the idea of consecutive, non-extended points touching one another ; here, the difficulty raised in times gone by in opposition to Zeno, & never really or satisfactorily answered (nor can it be answered), with regard to compenetration of all kinds with non-extended consecutive points, still holds the same force against the system of Leibniz. For another thing, it admits homogeneity amongst the elements, all distinction between masses depending on relative position only, & different combinations of the elements ; for this homogeneity amongst the elements, & the reason for the difference amongst masses, Nature herself provides us with the analogy. Chemical operations especially do so ; for, since the result of the analysis of compound substances leads to classes of elementary substances that are so comparatively few in num- ber, & still less different from one another in nature ; it strongly suggests that, the further analysis can be pushed, the greater the simplicity, & homogeneity, that ought to be attained ; thus, at length, we should have, as the result of a final decomposition, homogeneity & simplicity of the highest degree. Against this homogeneity & simplicity, the principle of indiscernibles, & the doctrine of sufficient reason, so long & strongly advocated by the followers of Leibniz, can, in my opinion at least, avail in not the slightest degree. 4. My Theory also differs as widely as possible from that of Newton. For one thing, HOW it differs from, because it explains by means of a single law of forces all those things that Newton himself, * surpasses, the i i i i. . X • f-\ • , i • i theory of Newton. in the last of his Questions on Uptics , endeavoured to explain by the three principles of gravity, cohesion & fermentation ; nay, & very many other things as well, which do not altogether follow from those three principles. Further, this law is expressed by a single algebraical formula, & not by one composed of several formulae compounded together ; or by a single continuous geometrical curve. For another thing, it admits forces that at very small distances are not positive or attractive, as Newton supposed, but negative or repul- 35 missum. 36 PHILOSOPHIC NATURALIS THEORIA infinitum, quo distantise in infinitum decrescant. Unde illud necessario consequitur, ut nee cohaesio a contactu immediate oriatur, quam ego quidem longe aliunde desumo ; nee ullus immediatus, &, ut ilium appellare soleo, mathematicus materiae contactus habeatur, quod simplicitatem, & inextensionem inducit elementorum, quae ipse variarum figurarum voluit, & partibus a se invicem distinctis composita, quamvis ita cohasrentia, ut nulla Naturae vi dissolvi possit compages, & adhaesio labefactari, quas adhaesio ipsi, respectu virium nobis cognitarum, est absolute infinita. Ubi de ipsa ctum 5. Quae ad ejusmodi Theoriam pertinentia hucusque sunt edita, continentur disserta- ante ; & quid pro- tionibus meis, De viribus vivis, edita Anno 1741;, De Lumine A. 1748, De Leee Continuitatis ml«<mm " . T r^ ... . • . rj . ... A. 1754, De Lege virium in natura existentium A. 1755, De divisibihtate materite, C5 principiis corporum A. 1757, ac in meis Supplementis Stayanae Philosophiae versibus traditae, cujus primus Tomus prodiit A. 1755 : eadem autem satis dilucide proposuit, & amplissimum ipsius per omnem Physicam demonstravit usum vir e nostra Societate doctissimus Carolus Benvenutus in sua Physics Generalis Synopsi edita Anno 1754. In ea Synopsi proposuit idem & meam deductionem aequilibrii binarum massarum, viribus parallelis animatarum, quas ex ipsa mea Theoria per notissimam legem compositionis virium, & aequalitatis inter actionem, & reac- tionem, fere sponte consequitur, cujus quidem in supplementis illis § 4. ad lib. 3. mentionem feci, ubi & quae in dissertatione De centra Gravitatis edideram, paucis proposui ; & de centre oscillationis agens, protuli aliorum methodos praecipuas quasque, quae ipsius determinationem a subsidiariis tantummodo principiis quibusdam repetunt. Ibidem autem de sequilibrii centre agens illud affirmavi : In Natura nullce sunt rigidce virgce, infiexiles, & omni gravitate, ac inertia carentes, adeoque nee revera ullce leges pro Us conditcz ; & si ad genuina, & simpli- cissima natures principia, res exigatur, invenietur, omnia pendere a compositione virium, quibus in se invicem agunt particula materice ; a quibus nimirum viribus omnia Natures pb&nomena proficiscuntur. Ibidem autem exhibitis aliorum methodis ad centrum oscillationis perti- nentibus, promisi, me in quarto ejusdem Philosophiae tomo ex genuinis principiis investiga- turum, ut aequilibrii, sic itidem oscillationis centrum. Qua occasione hoc 6. Porro cum nuper occasio se mihi praebuisset inquirendi in ipsum oscillationis centrum turn 'opus." Cnp ex meis principiis, urgente Scherffero nostro viro doctissimo, qui in eodem hoc Academico Societatis Collegio nostros Mathesim docet ; casu incidi in theorema simplicisimum sane, & admodum elegans, quo trium massarum in se mutuo agentium comparantur vires, [4] quod quidem ipsa fortasse tanta sua simplicitate effugit hucusque Mechanicorum oculos ; nisi forte ne effugerit quidem, sed alicubi jam ab alio quopiam inventum, & editum, me, quod admodum facile fieri potest, adhuc latuerit, ex quo theoremate & asquilibrium, ac omne vectium genus, & momentorum mensura pro machinis, & oscillationis centrum etiam pro casu, quo oscillatio fit in latus in piano ad axem oscillationis perpendiculari, & centrum percussionis sponte fluunt, & quod ad sublimiores alias perquisitiones viam aperit admodum patentem. Cogitaveram ego quidem initio brevi dissertatiuncula hoc theorema tantummodo edere cum consectariis, ac breve Theoriae meae specimen quoddam exponere ; sed paullatim excrevit opusculum, ut demum & Theoriam omnem exposuerim ordine suo, & vindicarim, & ad Mechanicam prius, turn ad Physicam fere universam applicaverim, ubi & quae maxima notatu digna erant, in memoratis dissertationibus ordine suo digessi omnia, & alia adjeci quamplurima, quae vel olim animo conceperam, vel modo sese obtulerunt scribenti, & omnem hanc rerum farraginem animo pervolventi. eiementa in- 7. Prima elementa materiae mihi sunt puncta prorsus indivisibilia, & inextensa, quae in imrftenso vacuo ita dispersa sunt, ut bina quaevis a se invicem distent per aliquod intervallum, quod quidem indefinite augeri potest, & minui, sed penitus evanescere non potest, sine conpenetratione ipsorum punctorum : eorum enim contiguitatem nullam admitto possi- bilem ; sed illud arbitror omnino certum, si distantia duorum materiae punctorum sit nulla, idem prorsus spatii vulgo concept! punctum indivisibile occupari ab utroque debere, & A THEORY OF NATURAL PHILOSOPHY 37 sive ; although these also become greater & greater indefinitely, as the distances decrease indefinitely. From this it follows of necessity that cohesion is not a consequence of imme- diate contact, as I indeed deduce from totally different considerations ; nor is it possible to get any immediate or, as I usually term it, mathematical contact between the parts of matter. This idea naturally leads to simplicity & non-extension of the elements, such as Newton himself postulated for various figures ; & to bodies composed of parts perfectly distinct from one another, although bound together so closely that the ties could not be broken or the adherence weakened by any force in Nature ; this adherence, as far as the forces known to us are concerned, is in his opinion unlimited. 5. What has already been published relating to this kind of Theory is contained in my when & where I dissertations, De Viribus vivis, issued in 1745, De Lumine, 1748, De Lege Continuitatis, *£££ th^theory'* 1754, De Lege virium in natura existentium, 1755, De divisibilitate materia, y principiis & a promise that i corporum, 1757, & in my Supplements to the philosophy of Benedictus Stay, issued in verse, made> of which the first volume was published in 1755. The same theory was set forth with considerable lucidity, & its extremely wide utility in the matter of the whole of Physics was demonstrated, by a learned member of our Society, Carolus Benvenutus, in his Physics Generalis Synopsis published in 1754. In this synopsis he also at the same time gave my deduction of the equilibrium of a pair of masses actuated by parallel forces, which follows quite naturally from my Theory by the well-known law for the composition of forces, & the equality between action & reaction ; this I mentioned in those Supplements, section 4 of book 3, & there also I set forth briefly what I had published in my dissertation De centra Gravitatis. Further, dealing with the centre of oscillation, I stated the most note- worthy methods of others who sought to derive the determination of this centre from merely subsidiary principles. Here also, dealing with the centre of equilibrium, I asserted : — " In Nature there are no rods that are rigid, inflexible, totally devoid of weight & inertia ; y so, neither are there really any laws founded on them. If the matter is worked back to the genuine W simplest natural principles, it will be found that everything depends on the com- position of the forces with which the particles of matter act upon one another ; y from these very forces, as a matter of fact, all phenomena of Nature take their origin." Moreover, here too, having stated the methods of others for the determination of the centre of oscillation, I promised that, in the fourth volume of the Philosophy, I would investigate by means of genuine principles, such as I had used for the centre of equilibrium, the centre of oscillation as well. 6. Now, lately I had occasion to investigate this centre of oscillation, deriving it from The occasion that my own principles, at the request of Father Scherffer, a man of much learning, who teaches |^ mathematics in this College of the Society. Whilst doing this, I happened to hit upon a matter. really most simple & truly elegant theorem, from which the forces with which three masses mutually act upon one another are easily to be found ; this theorem, perchance owing to its extreme simplicity, has escaped the notice of mechanicians up till now (unless indeed perhaps it has not escaped notice, but has at some time previously been discovered & published by some other person, though, as may very easily have happened, it may not have come to my notice). From this theorem there come, as the natural consequences, the equilibrium & all the different kinds of levers, the measurement of moments for machines, the centre of oscillation for the case in which the oscillation takes place sideways in a plane perpendicular to the axis of oscillation, & also the centre of percussion ; it opens up also a beautifully clear road to other and more sublime investigations. Initially, my idea was to publish in a short esssay merely this theorem & some deductions from it, & thus to give some sort of brief specimen of my Theory. But little by little the essay grew in length, until it ended in my setting forth in an orderly manner the whole of the theory, giving a demonstration of its truth, & showing its application to Mechanics in the first place, and then to almost the whole of Physics. To it I also added not only those matters that seemed to me to be more especially worth mention, which had all been already set forth in an orderly manner in the dissertations mentioned above, but also a large number of other things, some of which had entered my mind previously, whilst others in some sort pb truded themselves on my notice as I was writing & turning over in my mind all this conglomer- ation of material. 7. The primary elements of matter are in my opinion perfectly indivisible & non- The primary eie- extended points ; they are so scattered in an immense vacuum that every two of them are ^biVnon^xtended separated from one another by a definite interval ; this interval can be indefinitely & they are not increased or diminished, but can never vanish altogether without compenetration of the c points themselves ; for I do not admit as possible any immediate contact between them. On the contrary I consider that it is a certainty that, if the distance between two points of matter should become absolutely nothing, then the very same indivisible point of space, according to the usual idea of it, must be occupied by both together, & we have true 38 PHILOSOPHIC NATURALIS THEORIA haberi veram, ac omnimodam conpenetrationem. Quamobrem non vacuum ego quidem admitto disseminatum in materia, sed materiam in vacuo disseminatam, atque innatantem. Eorum inertias vis g jn n;sce punctis admitto determinationem perseverandi in eodem statu quietis, vel cujusmodi. . r . r. ,. , . . , . J . . • i * XT ' motus umiormis in directum l«) m quo semel sint posita, si seorsum smgula in JNatura existant ; vel si alia alibi extant puncta, componendi per notam, & communem metho- dum compositionis virium, & motuum, parallelogrammorum ope, praecedentem motum cum mo-[5]-tu quern determinant vires mutuae, quas inter bina quaevis puncta agnosco a distantiis pendentes, & iis mutatis mutatas, juxta generalem quandam omnibus com- munem legem. In ea determinatione stat ilia, quam dicimus, inertiae vis, quae, an a libera pendeat Supremi Conditoris lege, an ab ipsa punctorum natura, an ab aliquo iis adjecto, quodcunque, istud sit, ego quidem non quaere ; nee vero, si velim quasrere, in- veniendi spem habeo ; quod idem sane censeo de ea virium lege, ad quam gradum jam facio. Eorundem vires g Censeo igitur bina quaecunque materiae puncta determinari asque in aliis distantiis mutuae in alus , y •,.. , -1 . . . distantiis attrac- ad mutuum accessum, in alns ad recessum mutuum, quam ipsam determinationem appello tivae, in aliis re- vim, in priore casu attractivam, in posteriore repulsivam, eo nomine non agendi modum, sed pulsivae : v i n u m . ,r . . . , '. . . ejusmodi exempia. ipsam determinationem expnmens, undecunque provemat, cujus vero magnitude mutatis distantiis mutetur & ipsa secundum certam legem quandam, quae per geometricam lineam curvam, vel algebraicam formulam exponi possit, & oculis ipsis, uti moris est apud Mechanicos repraesentari. Vis mutuae a distantia pendentis, & ea variata itidem variatae, atque ad omnes in immensum & magnas, & parvas distantias pertinentis, habemus exemplum in ipsa Newtoniana generali gravitate mutata in ratione reciproca duplicata distantiarum, qua; idcirco numquam e positiva in negativam migrare potest, adeoque ab attractiva ad repul- sivam, sive a determinatione ad accessum ad determinationem ad recessum nusquam migrat. Verum in elastris inflexis habemus etiam imaginem ejusmodi vis mutuae variatae secundum distantias, & a determinatione ad recessum migrantis in determinationem ad accessum, & vice versa. Ibi enim si duae cuspides, compresso elastro, ad se invicem accedant, acquirunt determinationem ad recessum, eo majorem, quo magis, compresso elastro, distantia decrescit ; aucta distantia cuspidum, vis ad recessum minuitur, donee in quadam distantia evanescat, & fiat prorsus nulla ; turn distantia adhuc aucta, incipit determinatio ad accessum, quae perpetuo eo magis crescit, quo magis cuspides a se invicem recedunt : ac si e contrario cuspidum distantia minuatur perpetuo ; determinatio ad accessum itidem minuetur, evanescet, & in determinationem ad recessum mutabitur. Ea determinatio oritur utique non ab immediata cuspidum actione in se invicem, sed a natura, & forma totius intermediae laminae plicatae ; sed hie physicam rei causam non merer, & solum persequor exemplum determinationis ad accessum, & recessum, quae determinatio in aliis distantiis alium habeat nisum, & migret etiam ab altera in alteram. virium earundero 10. Lex autem virium est ejusmodi, ut in minimis distantiis sint repulsivae, atque eo majores in infmitum, quo distantiae ipsae minuuntur in infinitum, ita, ut pares sint extinguen- [6]-dae cuivis velocitati utcunque magnae, cum qua punctum alterum ad alterum possit accedere, antequam eorum distantia evanescat ; distantiis vero auctis minuuntur ita, ut in quadam distantia perquam exigua evadat vis nulla : turn adhuc, aucta distantia, mutentur in attractivas, prime quidem crescentes, turn decrescentes, evanescentes, abeuntes in repulsivas, eodem pacto crescentes, deinde decrescentes, evanescentes, migrantes iterum in attractivas, atque id per vices in distantiis plurimis, sed adhuc perquam exiguis, donee, ubi ad aliquanto majores distantias ventum sit, incipiant esse perpetuo attractivae, & ad sensum reciproce (a) Id quidem respectu ejus spatii, in quo continemur nos, W omnia quis nostris observari sensibus possunt, corpora ; quod quiddam spatium si quiescat, nihil ego in ea re a reliquis differo ; si forte moveatur motu quopiam, quern motum ex hujusmodi determinatione sequi debeant ipsa materia puncta ; turn bcec mea erit quiedam non absoluta, sed respectiva inertia: vis, quam ego quidem exposui W in dissertatione De Maris aestu fcf in Supplementis Stayanis Lib. I. § 13 ; ubi etiam illud occurrit, quam oh causam ejusmodi respectivam inertiam excogitarim, & quibus rationihus evinci putem, absolutam omnino demonstrari non posse ; sed ea hue non pertinent. A THEORY OF NATURAL PHILOSOPHY 39 compenetration in every way. Therefore indeed I do not admit the idea of vacuum interspersed amongst matter, but I consider that matter is interspersed in a vacuum & floats in it. 8. As an attribute of these points I admit an inherent propensity to remain in the The nat.ure ?f the same state of rest, or of uniform motion in a straight line, («) in which they are initially the" possess.1* set, if each exists by itself in Nature. But if there are also other points anywhere, there is an inherent propensity to compound (according to the usual well-known composition of forces & motions by the parallelogram law), the preceding motion with the motion which is determined by the mutual forces that I admit to act between any two of them, depending on the distances & changing, as the distances change, according to a certain law common to them all. This propensity is the origin of what we call the ' force of inertia ' ; whether this is dependent upon an arbitrary law of the Supreme Architect, or on the nature of points itself, or on some attribute of them, whatever it may be, I do not seek to know ; even if I did wish to do so, I see no hope of finding the answer ; and I truly think that this also applies to the law of forces, to which I now pass on. 9. I therefore consider that any two points of matter are subject to a determination The mutual forces to approach one another at some distances, & in an equal degree recede from one another at Stw^*^™!* other distances. This determination I call ' force ' ; in the first case ' attractive ', in the distances & repui- second case ' repulsive ' ; this term does not denote the mode of action, but the propen- ^mpies sity itself, whatever its origin, of which the magnitude changes as the distances change ; this kind, this is in accordance with a certain definite law, which can be represented by a geometrical curve or by an algebraical formula, & visualized in the manner customary with Mechanicians. We have an example of a force dependent on distance, & varying with varying distance, & pertaining to all distances either great or small, throughout the vastness of space, in the Newtonian idea of general gravitation that changes according to the inverse squares of the distances : this, on account of the law governing it, can never pass from positive to nega- tive ; & thus on no occasion does it pass from being attractive to being repulsive, i.e., from a propensity to approach to a propensity to recession. Further, in bent springs we have an illustration of that kind of mutual force that varies according as the distance varies, & passes from a propensity to recession to a propensity to approach, and vice versa. For here, if the two ends of the spring approach one another on compressing the spring, they acquire a propensity for recession that is the greater, the more the distance diminishes between them as the spring is compressed. But, if the distance between the ends is increased, the force of recession is diminished, until at a certain distance it vanishes and becomes absolutely nothing. Then, if the distance is still further increased, there begins a propensity to approach, which increases more & more as the ends recede further & further away from one another. If now, on the contrary, the distance between the ends is con- tinually diminished, the propensity to approach also diminishes, vanishes, & becomes changed into a propensity to recession. This propensity certainly does not arise from the imme- diate action of the ends upon one another, but from the nature & form of the whole of the folded plate of metal intervening. But I do not delay over the physical cause of the thing at this juncture ; I only describe it as an example of a propensity to approach & recession, this propensity being characterized by one endeavour at some distances & another at other distances, & changing from one propensity to another. 10. Now the law of forces is of this kind ; the forces are repulsive at very small dis- The Iaw .of forces tances, & become indefinitely greater & greater, as the distances are diminished indefinitely, for the pomts- in such a manner that they are capable of destroying any velocity, no matter how large it may be, with which one point may approach another, before ever the distance between them vanishes. When the distance between them is increased, they are diminished in such a way that at a certain distance, which is extremely small, the force becomes nothing. Then as the distance is still further increased, the forces are change-d to attractive forces ; these at first increase, then diminish, vanish, & become repulsive forces, which in the same way first increase, then diminish, vanish, & become once more attractive ; & so on, in turn, for a very great number of distances, which1 are all still very^ minute : until, finally, when we get to comparatively great distances, they begin to be continually attractive & approxi- (a) This indeed holds true for that space in which we, and all bodies that can be observed by our senses, are contained. Now, if this space is at rest, I do not differ from other philosophers with regard to the matter in question ; but if perchance space itself moves in some way or other, what motion ought these points of matter to comply with owing to this kind of propensity ? In that case Ms force of inertia that I postulate is not absolute, but relative ; as indeed I explained both in the dissertation De Maris Aestu, and also in the Supplements to Stay's Philosophy, book I, section 13. Here also will be found the conclusions at which I arrived with regard to relative inertia of this sort, and the arguments by which I think it is proved that it is impossible to show that it is generally abxlute. But these things do not concern us at present. 4° PHILOSOPHI/E NATURALIS THEORIA proportionales quadratis distantiarum, atque id vel utcunque augeantur distantiae etiam in infinitum, vel saltern donee ad distantias deveniatur omnibus Planetarum, & Cometarum distantiis longe majores. Leg is simpiicitas ii. Hujusmodi lex primo aspectu videtur admodum complicata, & ex diversis legibus exprimibihs per temere jnter se coagmentatis coalescens ; at simplicissima, & prorsus incomposita esse potest, COIlLlIlUtllTl CUf VclIUi • t i • • • 1*1* A 1 1 " J" 1 expressa videlicet per unicam contmuam curvam, vel simphcem Algebraicam iormulam, uti innui superius. Hujusmodi curva linea est admodum apta ad sistendam oculis ipsis ejusmodi legem, nee requirit Geometram, ut id praestare possit : satis est, ut quis earn intueatur tantummodo, & in ipsa ut in imagine quadam solemus intueri depictas res qualescunque, virium illarum indolem contempletur. In ejusmodi curva eae, quas Geometrae abscissas dicunt, & sunt segmenta axis, ad quern ipsa refertur curva, exprimunt distantias binorum punctorum a se invicem : illae vero, quae dicuntur ordinatae, ac sunt perpendiculares lineee ab axe ad curvam ductae, referunt vires : quae quidem, ubi ad alteram jacent axis partem, exhibent vires attractivas ; ubi jacent ad alteram, rcpulsivas, & prout curva accedit ad axem, vel recedit, minuuntur ipsae etiam, vel augentur : ubi curva axem secat, & ab altera ejus parte transit ad alteram, mutantibus directionem ordinatis, abeunt ex positivis in negativas, vel vice versa : ubi autem arcus curvae aliquis ad rectam quampiam axi perpendicularem in infinitum productam semper magis accedit ita ultra quoscumque limites, ut nunquam in earn recidat, quern arcum asymptoticum appellant Geometrae, ibi vires ipsae in infinitum excrescunt. Forma curvae ips- ius. 12. Ejusmodi curvam exhibui, & exposui in dissertationibus De viribus vivis a Num. 51, De Lumine Num. 5, De Lege virium in Naturam existentium a Num. 68, & in sua Synopsi Physics Generalis P. Benvenutus eandem protulit a Num. 108. En brevem quandemejus ideam. In Fig. i, Axis C'AC habet in puncto A asymptotum curvae rectilineam AB indefinitam, circa quam habentur bini curvae rami hinc, & inde aequales, prorsus inter se, & similes, quorum alter DEFGHIKLMNOPQRSTV habet inprimis arcum ED [7] asympto- ticum, qui nimirum ad partes BD, si indefinite producatur ultra quoscunque limites, semper magis accedit ad rectam AB productam ultra quoscunque limites, quin unquam ad eandem deveniat ; hinc vero versus DE perpetuo recidit ab eadam recta, immo etiam perpetuo versus V ab eadem recedunt arcus reliqui omnes, quin uspiam recessus mutetur in accessum. Ad axem C'C perpetuo primum accedit, donee ad ipsum deveniat alicubi in E ; turn eodem ibi secto progreditur, & ab ipso perpetuo recedit usque ad quandam distantiam F, postquam recessum in accessum mutat, & iterum ipsum axem secat in G, ac flexibus continuis contor- quetur circa ipsum, quern pariter secat in punctis quamplurimis, sed paucas admodum ejusmodi sectiones figura exhibet, uti I, L, N, P, R. Demum is arcus desinit in alterum crus TpsV, jacens ex parte opposita axis respectu primi cruris, quod alterum crus ipsum habet axem pro asymptoto, & ad ipsum accedit ad sensum ita, ut distantiae ab ipso sint in ratione reciproca duplicata distantiarum a recta BA. Abscissae exprimen- d!nateStaexprimen- tes vires. 13. Si ex quovis axis puncto a, b, d, erigatur usque ad curvam recta ipsi perpendicularis aS> ^r' ^h , segmentum axis Aa, Ab, Ad, dicitur abscissa, & refert distantiam duorum materiae punctorum quorumcunque a se invicem ; perpendicularis ag, br, db , dicitur ordinata, & exhibet vim repulsivam, vel attractivam, prout jacet respectu axis ad partes D, vel oppositas. Mutationes ordina- tarum, & virium iis expressarum. 14. Patet autem, in ea curvae forma ordinatam ag augeri ultra quoscunque limites, si abscissa Aa, minuatur pariter ultra quoscunque limites ; quae si augeatur, ut abeat in Ab, ordinata minuetur, & abibit in br, perpetuo imminutam in accessu b ad E, ubi evanescet : turn aucta abscissa in Ad, mutabit ordinata directionem in dh , ac ex parte opposita augebitur prius usque ad F, turn decrescet per il usque ad G, ubi evanescet, & iterum mutabit directionem regressa in mn ad illam priorem, donee post evanescentiam, & directionis mutationem factam in omnibus sectionibus I, L, N, P, R, fiant ordinatas op, vs, directionis constantis, & decrescentes ad sensum in ratione reciproca duplicata abscissarum Ao, Av. Quamobrem illud est manifestum, per ejusmodi curvam exprimi eas ipsas vires, initio A THEORY OF NATURAL PHILOSOPHY 0 PHILOSOPHIC NATURALIS THEORIA o A THEORY OF NATURAL PHILOSOPHY 43 mately inversely proportional to the squares of the distances. This holds good as the distances are increased indefinitely to any extent, or at any rate until we get to distances that are far greater than all the distances of the planets & comets. 11. A law of this kind will seem at first sight to be very complicated, & to be the result The simplicity of of combining together several different laws in a haphazard sort of way ; but it can be of the law can ^ re~ ^.t. • i 1 i • j o v j • i v i • 1 i r presented by means the simplest kind & not complicated in the slightest degree ; it can be represented for of a continuous instance by a single continuous curve, or by an algebraical formula, as I intimated above. curve- A curve of this sort is perfectly adapted to the .graphical representation of this sort of law, & it does not require a knowledge of geometry to set it forth. It is sufficient for anyone merely to glance at it, & in it, just as in a picture we are accustomed to view all manner of things depicted, so will he perceive the nature of these forces. In a curve of this kind, those lines, that geometricians call abscissae, namely, segments of the axis to which the curve is referred, represent the distances of two points from one another ; & those, which we called ordinates, namely, lines drawn perpendicular to the axis to meet the curve, repre- sent forces. These, when they lie on one side of the axis represent attractive forces, and, when they lie on the other side, repulsive forces ; & according as the curve approaches the axis or recedes from it, they too are diminished or increased. When the curve cuts the axis & passes from one side of it to the other, the direction of the ordinates being changed in consequence, the forces pass from positive to negative or vice versa. When any arc of the curve approaches ever more closely to some straight line perpendicular to the axis and indefinitely produced, in such a manner that, even if this goes on beyond all limits, yet the curve never quite reaches the line (such an arc is called asymptotic by geometricians), then the forces themselves will increase indefinitely. 12. I set forth and explained a curve of this sort in my dissertations De Firibus vivis The form of the (Art. 51), De Lumine (Art. 5), De lege virium in Natura existentium (Art. 68) ; and Father curve- Benvenutus published the same thing in his Synopsis Physicce Generalis (Art. 108). This will give you some idea of its nature in a few words. In Fig. i the axis C'AC has at the point A a straight line AB perpendicular to itself, which is an asymptote to the curve ; there are two branches of the curve, one on each side of AB, which are equal & similar to one another in every way. Of these, one, namely DEFGHIKLMNOPQRSTV, has first of all an asymptotic arc ED ; this indeed, if it is produced ever so far in the direction ED, will approach nearer & nearer to the straight line AB when it also is produced indefinitely, but will never reach it ; then, in the direction DE, it will continually recede from this straight line, & so indeed will all the rest of the arcs continually recede from this straight line towards V. The first arc continually approaches the axis C'C, until it meets it in some point E ; then it cuts it at this point & passes on, continually receding from the axis until it arrives at a certain distance given by the point F ; after that the recession changes to an approach, & it cuts the axis once more in G ; & so on, with successive changes of curvature, the curve winds about the axis, & at the same time cuts it in a number of points that is really large, although only a very few of the intersections of this kind, as I, L, N, P, R, are shown in the diagram. Finally the arc of the curve ends up with the other branch TpsV, lying on the opposite side of the axis with respect to the first branch ; and this second branch has the axis itself as its asymptote, & approaches it approximately in such a manner that the distances from the axis are in the inverse ratio of the squares of the distances from the straight line AB. 13. If from any point of the axis, such as a, b, or d, there is erected a straight line per- The abscissae re- pendicular to it to meet the curve, such as ag, br, or db then the segment of the axis, Aa, £res^Jg Ab, or Ad, is called the abscissa, & represents the distance of any two points of matter from forces, one another ; the perpendicular, ag, br, or dh, is called the ordinate, & this represents the force, which is repulsive or attractive, according as the ordinate lies with regard to the axis on the side towards D, or on the opposite side. 14. Now it is clear that, in a curve of this form, the ordinate ag will be increased Change in the or- beyond all bounds, if the abscissa Aa is in the same way diminished beyond all bounds ; & fbat tlfey reprSent! if the latter is increased and becomes Ab, the ordinate will be diminished, & it will become br, which will continually diminish as b approaches to E, at which point it will vanish. Then the abscissa being increased until it becomes Ad, the ordinate will change its direction as it becomes db, & will be increased in the opposite direction at first, until the point F is reached, when it will be decreased through the value il until the point G is attained, at which point it vanishes ; at the point G, the ordinate will once more change its direction as it returns to the position mn on the same side of the axis as at the start. Finally, after vanishing & changing direction at all points of intersection with the axis, such as I, L, N, P, R, the ordinates take the several positions indicated by op, vs : here the direction remains unchanged, & the ordinates decrease approximately in the inverse ratio of the squares of the abscissae Ao, Av. Hence it is perfectly evident that, by a curve of this kind, we can 44 PHILOSOPHIC NATURALIS THEORIA Discrimen hu us legis virium a gravitate N e w- toniana : ejus usus in Physica : ordo pertractandorum. Occasio inveniendae Theories ex consid- eraticine impulsus. V repulsivas, & imminutis in infinitum distantiis auctas in infinitum, auctis imminutas, turn evanescentes, abeuntes, mutata directione, in attractivas, ac iterum evenescentes, mutatasque per vices : donee demum in satis magna distantia evadant attractive ad sensum in ratione reciproca duplicata distantiarum. 15. Haec virium lex a Newtoniana gravitate differt in ductu, & progressu curvae earn exprimentis quse nimirum, ut in fig. 2, apud Newtonum est hyperbola DV gradus tertii, jacens tota citra axem, quern nuspiam secat, jacentibus omni-[8]-bus ordinatis vs, op, bt, ag ex parte attractiva, ut idcirco nulla habeatur mutatio e positivo ad negativum, ex attractione in repulsi- onem, vel vice versa ; caeterum utraque per ductum exponitur curvae continue habentis duo crura infinita asymptotica in ramis singulis utrinque in infinitum productis. Ex hujusmodi autem virium lege, & ex solis principiis Mechanicis notissimis, nimirum quod ex pluribus viribus, vel motibus componatur vis, vel motus quidam ope parallelogrammorum, quorum latera exprimant vires, vel mo- tus componentes, & quod vires ejusmodi in punctis singulis, tempusculis singulis aequalibus, inducant velocitates, vel motus proportion- ales sibi, omnes mihi profluunt generales, & praecipuae quacque particulars proprietates cor- porum,uti etiam superius innui, nee ad singulares proprietates derivandas in genere afHrmo, eas haberi per diversam combinationem, sed combinationes ipsas evolvo, & geometrice demon- stro, quae e quibus combinationibus phasnomena, & corporum species oriri debeant. Verum antequam ea evolvo in parte secunda, & tertia, ostendam in hac prima, qua via, & quibus positivis rationibus ad earn virium legem devenerim, & qua ratione illam elementorum materiae simplicitatem eruerim, turn quas difHcultatem aliquam videantur habere posse, dissolvam. 1 6. Cum anno 1745 De Viribus vivis dissertationem conscriberem, & omnia, quse a viribus vivis repetunt, qui Leibnitianam tuentur sententiam, & vero etiam plerique ex iis, qui per solam velocitatem vires vivas metiuntur, repeterem immediate a sola velocitate genita per potentiarum vires, quae juxta communem omnium Mechanicorum sententiam velocitates vel generant, vel utcunque inducunt proportionales sibi, & tempusculis, quibus agunt, uti est gravitas, elasticitas, atque aliae vires ejusmodi ; ccepi aliquant: o diligentius inquirere in earn productionem velocitatis, quae per impulsum censetur fieri, ubi tota velocitas momento temporis produci creditur ab iis, qui idcirco percussionis vim infinities majorem esse censent viribus omnibus, quae pressionem solam momentis singulis exercent. Statim illud mihi sese obtulit, alias pro percussionibus ejusmodi, quee nimirum momento temporis finitam velocitatem inducant, actionum leges haberi debere. FIG origo ejusdem ex 17. Verum re altius considerata, mihi illud incidit, si recta utamur ratiocinandi methodo, susTmrnedUatTalin eum agendi modum submovendum esse a Natura, quae nimirum eandem ubique virium lege Continuitatis. legem, ac eandem agendi rationem adhibeat : impulsum nimirum immediatum alterius corporis in alterum, & immediatam percussionem haberi non posse sine ilia productione finitse velocitatis facta momento temporis indivisibili, & hanc sine saltu quodam, & Isesione illius, quam legem Continuitatis appellant, quam quidem legem in Natura existere, & quidem satis [9] valida ratione evinci posse existimabam. En autem ratiocinationem ipsam, qua turn quidem primo sum usus, ac deinde novis aliis, atque aliis meditationibus illustravi, ac confirmavi. minus velox. Laesio legis Continu- 18. Concipiantur duo corpora aequalia, quae moveantur in directum versus eandem cOTpus^efocruTim- plagam> & id, quod praecedit, habeat gradus velocitatis 6, id vero, quod ipsum persequitur mediate incurrat in gradus 12. Si hoc posterius cum sua ilia velocitate illaesa deveniat ad immediatum contactum cum illo priore ; oportebit utique, ut ipso momento temporis, quo ad contactum devenerint, illud posterius minuat velocitatem suam, & illud primus suam augeat, utrumque per saltum, abeunte hoc a 12 ad 9, illo a 6 ad 9, sine ullo transitu per intermedios gradus n, & 7 ; 10, & 8 ; 9^, & 8i, &c. Neque enim fieri potest, ut per aliquam utcunque exiguam continui A THEORY OF NATURAL PHILOSOPHY 45 represent the forces in question, which are initially repulsive & increase indefinitely as the distances are diminished indefinitely, but which, as the distances increase, are first of all diminished, then vanish, then become changed in direction & so attractive, again vanish, & change their direction, & so on alternately ; until at length, at a distance comparatively great they finally become attractive & are sensibly proportional to the inverse squares of the distance. ic. This law of forces differs from the law of gravitation enunciated by Newton in Difference between . J -nii r i i • i i • • this 'aw °f forces the construction & development or the curve that represents it ; thus, the curve given in & Newton's law of Fie. 2, which is that according to Newton, is DV, a hyperbola of the third degree, lying gravitation ; i t s ii • i r i • i • i • i • nil'6 use ln Physics ; altogether on one side of the axis, which it does not cut at any point ; all the ordmates, the order in which such as vs, op, bt, ag lie on the side of the axis representing attractive forces, & there- ^ets^ects are to fore there is no change from positive to negative, i.e., from attraction to repulsion, or vice versa. On the other hand, each of the laws is represented by the construction of a continuous curve possessing two infinite asymptotic branches in each of its members, if produced to infinity on both sides. Now, from a law of forces of this kind, & with the help of well-known mechanical principles only, such as that a force or motion can be com- pounded from several forces or motions by the help of parallelograms whose sides represent the component forces or motions, or that the forces of this kind, acting on single points for single small equal intervals of time, produce in them velocities that are proportional to themselves ; from these alone, I say, there have burst forth on me in a regular flood all the general & some of the most important particular properties of bodies, as I intimated above. Nor, indeed, for the purpose of deriving special properties, do I assert that they ought to be obtained owing to some special combination of points ; on the contrary I consider the combinations themselves, & prove geometrically what phenomena, or what species of bodies, ought to arise from this or that combination. Of course, before I come to consider, both in the second part and in the third, all the matters mentioned above, I will show in this first part in what way, & by what direct reasoning, I have arrived at this law of forces, & by what argument I have made out the simplicity of the elements of matter ; then I will give an explanation of every point that may seem to present any possible difficulty. 16. In the year 1745, I was putting together my dissertation De Firibus vivis, & had The occasion that derived everything that they who adhere to the idea of Leibniz, & the greater number of o^my^L^Trom those who measure ' living forces ' by means of velocity only, derive from these ' living the consideration forces ' ; as, I say I had derived everything directly & solely from the velocity generated by of imPulsive action, the forces of those influences, which, according to the generally accepted view taken by all Mechanicians, either generate, or in some way induce, velocities that are proportional to themselves & the intervals of time during which they act ; take, for instance, gravity, elasticity, & other forces of the same kind. I then began to investigate somewhat more carefully that production of velocity which is thought to arise through impulsive action, in which the whole of the velocity is credited with being produced in an instant of time by those, who think, because of that, that the force of percussion is infinitely greater than all forces which merely exercise pressure for single instants. It immediately forced itself upon me that, for percussions of this kind, which really induce a finite velocity in an instant of time, laws for their actions must be obtained different from the rest. 17. However, when I considered the matter more thoroughly, it struck me that, if The cause of we employ a straightforward method of argument, such a mode of action must be with- w^s the^pposftion drawn from Nature, which in every case adheres to one & the same law of forces, & the raised to the Law same mode of action. I came to the conclusion that really immediate impulsive action of °he idea' one body on another, & immediate percussion, could not be obtained, without the pro- impulse, duction of a finite velocity taking place in an indivisible instant of time, & this would have to be accomplished without any sudden change or violation of what is called the Law of Continuity ; this law indeed I considered as existing in Nature, & that this could be shown to be so by a sufficiently valid argument. The following is the line of argument that I employed initially ; afterwards I made it clearer & confirmed it by further arguments & fresh reflection. 1 8. Suppose there are two equal bodies, moving in the same straight line & in the violation of the same direction ; & let the one that is in front have a degree of velocity represented by ^ tod^movrng1 6, & the one behind a degree represented by 12. If the latter, i.e., the body that was be- more swiftly comes hind, should ever reach with its velocity undiminished, & come into absolute contact with, J"*° with^another the former body which was in front, then in every case it would be necessary that, at the body moving more very instant of time at which this contact happened, the hindermost body should diminish slowlv- its velocity, & the foremost body increase its velocity, in each case by a sudden change : one of them would pass from 12 to 9, the other from 6 to 9, without any passage through the intermediate degrees, n & 7, 10 & 8, 9$ & 8f, & so on. For it cannot possibly happen 46 PHILOSOPHIC NATURALIS THEORIA temporis particulam ejusmodi mutatio fiat per intermedios gradus, durante contactu. Si enim aliquando alterum corpus jam habuit 7 gradus velocitatis, & alterum adhuc retinet 1 1 ; toto illo tempusculo, quod effluxit ab initio contactus, quando velocitates erant 12, & 6, ad id tempus, quo sunt n, & 7, corpus secundum debuit moveri cum velocitate majore, quam primum, adeoque plus percurrere spatii, quam illud, £ proinde anterior ejus superficies debuit transcurrere ultra illius posteriorem superficiem, & idcirco pars aliqua corporis sequentis cum aliqua antecedentis corporis parte compenetrari debuit, quod cum ob impenetrabilitatem, quam in materia agnoscunt passim omnes Physici, & quam ipsi tri- buendam omnino esse, facile evincitur, fieri omnino non possit ; oportuit sane, in ipso primo initio contactus, in ipso indivisibili momento temporis, quod inter tempus continuum praecedens contactum, & subsequens, est indivisibilis limes, ut punctum apud Geometras est limes indivisibilis inter duo continue lineae segmenta, mutatio velocitatum facta fuerit per saltum sine transitu per intermedias, laesa penitus ilia continuitatis lege, quae itum ab una magnitudine ad aliam sine transitu per intermedias omnino vetat. Quod autem in corporibus aequalibus diximus de transitu immediato utriusque ad 9 gradus velocitatis, recurrit utique in iisdem, vel in utcunque inaequalibus de quovis alio transitu ad numeros quosvis. Nimirum ille posterioris corporis excessus graduum 6 momento temporis auferri debet, sive imminuta velocitate in ipso, sive aucta in priore, vel in altero imminuta utcunque, & aucta in altero, quod utique sine saltu, qui omissis infinitis intermediis velocitatibus habeatur, obtineri omnino non poterit. Objectio petita a ig. Sunt, qui difficultatem omnem submoveri posse censeant, dicendo, id quidem ita se cofporum.dl ' habere debere, si corpora dura habeantur, quae nimirum nullam compressionem sentiant, nullam mutationem figurae ; & quoniam hsec a multis excluduntur penitus a Natura ; dum se duo globi contingunt, introcessione, [10] & compressione partium fieri posse, ut in ipsis corporibus velocitas immutetur per omnes intermedios gradus transitu facto, & omnis argumenti vis eludatur. Ea uti non posse, 2O fa mprjmis ea responsione uti non possunt, quicunque cum Newtono, & vero etiam qui admittunt ele- _, \ . . r . j • o menta soiida, & cum plerisquc veterum Pnilosopnorum pnma elementa matenae omnino dura admittunt, & dura- soiida, cum adhaesione infinita, & impossibilitate absoluta mutationis figurae. Nam in primis elementis illis solidis, & duris, quae in anteriore adsunt sequentis corporis parte, & in praece- dentis posteriore, quae nimirum se mutuo immediate contingunt, redit omnis argumenti vis prorsus illaesa. Extensionem con- 2i. Deinde vero illud omnino intelligi sane non potest, quo pacto corpora omnia partes primoT pores,1™*! aliquas postremas circa superficiem non habeant penitus solidas, quae idcirco comprimi parietes soiidos, ac ornnino non possint. In materia quidem, si continua sit, divisibilitas in infinitum haberi potest, & vero etiam debet ; at actualis divisio in infinitum difficultates secum trahit sane inextricablies ; qua tamen divisione in infinitum ii indigent, qui nullam in corporibus admittunt particulam utcunque exiguam compressionis omnis expertem penitus, atque incapacem. Ii enim debent admittere, particulam quamcunque actu interpositis poris distinctam, divisamque in plures pororum ipsorum velut parietes, poris tamen ipsis iterum distinctos. Illud sane intelligi non potest, qui fiat, ut, ubi e vacuo spatio transitur ad corpus, non aliquis continuus haberi debeat alicujus in se determinatae crassitudinis paries usque ad primum porum, poris utique carens ; vel quomodo, quod eodem recidit, nullus sit extimus, & superficiei externae omnium proximus porus, qui nimirum si sit aliquis, parietem habeat utique poris expertem, & compressionis incapacem, in quo omnis argumenti superioris vis redit prorsus illaesa. legis Con- 22. At ea etiam, utcunque penitus inintelligibili, sententia admissa, redit omnis eadem iprimis su^r™ argument! vis in ipsa prima, & ultima corporum se immediate contingentium superficie, vel debus, vel punctis. s{ nullae continuae superficies congruant, in lineis, vel punctis. Quidquid enim sit id, in quo contactus fiat, debet utique esse aliquid, quod nimirum impenetrabilitati occasionem praestet, & cogat motum in sequente corpore minui, in prascedente augeri ; id, quidquid est, in quo exeritur impenetratibilitatis vis, quo fit immediatus contactus, id sane velocitatem mutare debet per saltum, sine transitu per intermedia, & in eo continuitatis lex abrumpi A THEORY OF NATURAL PHILOSOPHY 47 that this kind of change is made by intermediate stages in some finite part, however small, of continuous time, whilst the bodies remain in contact. For if at any time the one body then had 7 degrees of velocity, the other would still retain 1 1 degrees ; thus, during the whole time that has passed since the beginning of contact, when the velocities were respectively 12 Si 6, until the time at which they are 1 1 & 7, the second body must be moved with a greater velocity than the first ; hence it must traverse a greater distance in space than the other. It follows that the front surface of the second body must have passed beyond the back surface of the first body ; & therefore some part of the body that follows behind must be penetrated by some part of the body that goes in front. Now, on account of impenetrability, which all Physicists in all quarters recognize in matter, & which can be easily proved to be rightly attributed to it, this cannot possibly happen. There really must be, in the commencement of contact, in that indivisible instant of time which is an indivisible limit between the continuous time that preceded the contact & that subsequent to it (just in the same way as a point in geometry is an indivisible limit between two seg- ments of a continuous line), a change of velocity taking place suddenly, without any passage through intermediate stages ; & this violates the Law of Continuity, which absolutely denies the possibility of a passage from one magnitude to another without passing through intermediate stages. Now what has been said in the case of equal bodies concerning the direct passing of both to 9 degrees of velocity, in every case holds good for such equal bodies, or for bodies that are unequal in any way, concerning any other passage to any numbers. In fact, the excess of velocity in the hindmost body, amounting to 6 degrees, has to be got rid of in an instant of time, whether by diminishing the velocity of this body, or by increasing the velocity of the other, or by diminishing somehow the velocity of the one & increasing that of the other ; & this cannot possibly be done in any case, without the sudden change that is obtained by omitting the infinite number of intermediate velocities. 19. There are some people, who think that the whole difficulty can be removed by An objection de- saying that this is just as it should be, if hard bodies, such as indeed experience no com- ^edexr^ncenyilo1 pression or alteration of shape, are dealt with ; whereas by many philosophers hard bodies hard bodies. are altogether excluded from Nature ; & therefore, so long as two spheres touch one another, it is possible, by introcession & compression of their parts, for it to happen that in these bodies the velocity is changed, the passage being made through all intermediate stages ; & thus the whole force of the argument will be evaded. 20. Now in the first place, this reply can not be used by anyone who, following New- This reP'y cannot ton, & indeed many of the ancient philosophers as well, admit the primary elements of ^"admit^oiid0* matter to be absolutely hard & solid, possessing infinite adhesion & a definite shape that it hard elements. is perfectly impossible to alter. For the whole force of my argument then applies quite unimpaired to those solid and hard primary elements that are in the anterior part of the body that is behind, & in the hindmost part of the body that is in front ; & certainly these parts touch one another immediately. 21. Next it is truly impossible to understand in the slightest degree how all bodies do Continuous exten- not have some of their last parts just near to the surface perfectly solid, & on that account mary ^resT* walls altogether incapable of being compressed. If matter is continuous, it may & must be sub- bounding them, ject to infinite divisibility ; but actual division carried on indefinitely brings in its train difficulties that are truly inextricable ; however, this infinite division is required by those who do not admit that there are any particles, no matter how small, in bodies that are perfectly free from, & incapable of, compression. For they must admit the idea that every particle is marked off & divided up, by the action of interspersed pores, into many boundary walls, so to speak, for these pores ; & these walls again are distinct from the pores them- selves. It is quite impossible to understand why it comes about that, in passing from empty vacuum to solid matter, we are not then bound to encounter some continuous wall of some definite inherent thickness from the surface to the first pore, this wall being everywhere devoid of pores ; nor why, which comes to the same thing in the end, there does not exist a pore that is the last & nearest to the external surface ; this pore at least, if there were one, certainly has a wall that is free from pores & incapable of compression ; & here then the whole force of the argument used above applies perfectly unimpaired. 22. Moreover, even if this idea is admitted, although it may be quite unintelligible, Violation of the then the whole force of the same argument applies to the first or last surface of the bodies ta^s'piace^any that are in immediate contact with one another ; or, if there are no continuous surfaces rate, in prime sur- congruent, then to the lines or points. For, whatever the manner may be in which contact takes place, there must be something in every case that certainly affords occasion for impenetrability, & causes the motion of the body that follows to be diminished, & that of the one in front to be increased. This, whatever it may be, from which the force of impene- trability is derived, at the instant at which immediate contact is obtained, must certainly change the velocity suddenly, & without any passage through intermediate stages ; & by 48 PHILOSOPHIC NATURALIS THEORIA debet, atque labefactari, si ad ipsum immediatum contactum illo velocitatum discrimine deveniatur. Id vero est sane aliquid in quacunque e sententiis omnibus continuam extensionem tribuentibus materise. Est nimirum realis affectio qusedam corporis, videlicet ejus limes ultimus realis, superficies, realis superficiei limes linea, realis lineae limes punctum, qua affectiones utcunque in iis sententiis sint prorsus inseparabiles [n] ab ipso corpore, sunt tamen non utique intellectu confictae, sed reales, quas nimirum reales dimensiones aliquas habent, ut superficies binas, linea unam, ac realem motum, & translationem cum ipso corpore, cujus idcirco in iis sententiis debent, esse affectiones quaedam, vel modi. Objectio petita a 27. Est, qui dicat, nullum in iis committi saltum idcirco, quod censendum sit, nullum vucemassa, &,.J r . .. , ,, i\/r x motns. quae super- habere motum, superficiem, Imeam, punctum, quae massam habeant nullam. Motus, mquit, ficiebus, & punctis a Mechanicis habet pro mensura massam in velocitatem ductam : massa autem est super- non convemant. _.. , . • • «• • i • j • • • • /-^ ficies baseos ducta in crassitudmem, sive altitudmem, ex. gr. m pnsmatis. Quo minor est ejusmodi crassitude, eo minor est massa, & motus, ac ipsa crassitudine evanescente, evanescat oportet & massa, & motus. Kesponsionis ini- 24. Verum qui sic ratiocinatur, inprimis ludit in ipsis vocibus. Massam vulgo appellant tacam.^punctmn! quantitatem materiae, & motum corporum metiuntur per massam ejusmodi, ac velocitatem. posita extensione At quemadmodum in ipsa geometrica quantitate tria genera sunt quantitatum, corpus, vel contmua, e - ^11^^ qUO(J trinam dimensionem habet, superficies quae binas, linae, quae unicam, quibus accedit linese limes punctum, omni dimensione, & extensione carens ; sic etiam in Physica habetur in communi corpus tribus extensionis speciebus praeditum ; superficies realis extimus corporis limes, praedita binis ; linea, limes realis superficiei, habens unicam; & ejusdem lineae indivisibilis limes punctum. Utrobique alterum alterius est limes, non pars, & quatuor diversa genera constituunt. Superficies est nihil corporeum, sed non & nihil superficial, quin immo partes habet, & augeri potest, & minui ; & eodem pacto linea in ratione quidem superficiei est nihil, sed aliquid in ratione linese ; ac ipsum demum punctum est aliquid in suo genere, licet in ratione lineae sit nihil. QUO pacto nomen 25. Hinc autem in iis ipsis massa quaedam considerari potest duarum dimensionum, vel motus 'debeat8 con- unius, vel etiam nullius continuae dimensionis, sed numeri punctorum tantummodo, uti venire superficie- quantitas ejus genere designetur ; quod si pro iis etiam usurpetur nomen massae generaliter, bus, imeis, punctis. motus quantitas definiri poterit per productum ex velocitate, & massa ; si vero massae nomen tribuendum sit soli corpori, turn motus quidem corporis mensura erit massa in velocitatem ducta ; superficiei, lineae, punctorum quotcunque motus pro mensura habebit quantitatem superficiei, vel lineae, vel numerum punctorum in velocitatem ducta ; sed motus utique iis omnibus speciebus tribuendus erit, eruntque quatuor motuum genera, ut quatuor sunt quantitatum, solidi, superficiei, lineae, punctorum ; ac ut altera harum erit nihil in alterius ratione, non in sua ; ita alterius motus erit nihil in ratione alterius sed erit sane aliquid in ratione sui, non purum nihil. Fore, ut ea laedatur saltern in velocitate punctorum. Motum passim rI2i 2Q- gt quidem jpSj Mechanici vulgo motum tribuunt & superficiebus & lineis, & tnbui punctis; ,'•..* , . '. -m • • j fore, lit in eo ixda- punctis, ac centri gravitatis motum ubique nommant rhysici, quod centrum utique punctum i^r Continuitatis est aliquod, non corpus trina praeditum dimensione, quam iste ad motus rationem, & appellationem requirit, ludendo, ut ajebam, in verbis. Porro in ejusmodi motibus exti- marum saltern superficierum, vel linearum, vel punctorum, saltus omnino committi debet, si ea ad contactum immediatum deveniant cum illo velocitatum discrimine, & continuitatis lex violari. 27. Verum hac omni disquisitione omissa de notione motus, & massae, si factum ex velocitate, & massa, evanescente una e tribus dimensionibus, evanescit ; remanet utique velocitas reliquarum dimensionum, quae remanet, si eae reapse remanent, uti quidem omnino remanent in superficie, & ejus velocitatis mutatio haberi deberet per saltum, ac in ea violari continuitatis lex jam toties memorata. -, ti°exin?P<!ne- 28. Haec quidem ita evidentia sunt, ut omnino dubitari non possit, quin continuitatis trabilitate admissa ,.,..,/ .-KT • » • • i « i • • j- • • j •• in minimis parti- lex infnngi debeat, & saltus m Naturam induci, ubi cum velocitatis discrimine ad se invicem cuiis. & ejus confu- accedant corpora, & ad immediatum contactum deveniant, si modo impenetrabilitas corporibus tribuenda sit, uti revera est. Earn quidem non in integris tantummodo corpori- bus, sed in minimis etiam quibusque corporum particulis, atque elementis agnoverunt Physici universi. Fuit sane, qui post meam editam Theoriam, ut ipsam vim mei argument} A THEORY OF NATURAL PHILOSOPHY 49 that the Law of Continuity must be broken & destroyed, if immediate contact is arrived at with such a difference of velocity. Moreover, there is in truth always something of this sort in every one of the ideas that attribute continuous extension to matter. There is some real condition of the body, namely, its last real boundary, or its surface, a real boundary of a surface, a line, & a real boundary of a line, a point ; & these conditions, however insepar- able they may be in these theories from the body itself, are nevertheless certainly not fictions of the brain, but real things, having indeed certain real dimensions (for instance, a surface has two dimensions, & a line one) ; they also have real motion & movement of trans- lation along with the body itself ; hence in these theories they must be certain conditions or modes of it. 23. Someone may say that there is no sudden change made, because it must be con- Objection derived sidered that a surface, a line or a point, having no mass, cannot have any motion. He may 1™™ mo/io^w^idi say that motion has, according to Mechanicians, as its measure, the mass multiplied by the do not accord with velocity ; also mass is the surface of the base multiplied by the thickness or the altitude, surfaces & P°mis- as for instance in prisms. Hence the less the thickness, the less the mass & the motion ; thus, if the thickness vanishes, then both the mass & therefore the motion must vanish as well. 24. Now the man who reasons in this manner is first of all merely playing with words. Commencement of Mass is commonly called quantity of matter, & the motion of bodies is measured by mass the answer to tl?ls : . i • * • i « . , •*• —- . * , . c • -_* cl SUrlclCC, OF ii 11116, of this kind & the velocity. But, just as in a geometrical quantity there are three kinds of or a point, is some- quantities, namely, a body or a solid having three dimensions, a surface with two, & a line \ with one : to which is added the boundary of a line, a point, lacking dimensions altogether, is supposed to ex- & of no extension. So also in Physics, a body is considered to be endowed with three lst' species of extension ; a surface, the last real boundary of a body, to be endowed with two ; a line, the real boundary of a surface, with one ; & the indivisible boundary of the line, to be a point. In both subjects, the one is a boundary of the other, & not a part of it ; & they form four different kinds. There is nothing solid about a surface ; but that does not mean that there is also nothing superficial about it ; nay, it certainly has parts & can be increased or diminished. In the same way a line is nothing indeed when compared with a surface, but a definite something when compared with a line ; & lastly a point is a definite something in its own class, although nothing in comparison with a line. 25. Hence also in these matters, a mass can be considered to be of two dimensions, or The manner in of one, or even of no continuous dimension, but only numbers of points, just as quantity of wn'^ma^and^the this kind is indicated. Now, if for these also, the term mass is employed in a generalized term motus is bound sense, we shall be able to define the quantity of motion by the product of the velocity & !:°;^p^'to?ujfaces' 1 Ii • ' • 1 1 1 • • • 1 1 * 1 1 1 i HOPS, <X pOintS. the mass. But if the term mass is only to be used in connection with a solid body, then indeed the motion of a solid body will be measured by the mass multiplied by the velocity ; but the motion of a surface, or a line, or any number of points will have as their measure the quantity of the surface, or line, or the number- of the points, multiplied by the velocity. Motion at any rate will be ascribed in all these cases, & there will be four kinds of motion, as there are four kinds of quantity, namely, for a solid, a surface, a line, or for points ; and, as each class of the latter will be as nothing compared with the class before it, but something in its own class, so the motion of the one will be as nothing compared with the motion of the other, but yet really something, & not entirely nothing, compared with those of its own class. 26. Indeed, Mechanicians themselves commonly ascribe motion to surfaces, lines & Motion is ascribed points, & Physicists universally speak of the motion of the centre of gravity ; this centre is minateiy3 the'i^w undoubtedly some point, & not a body endowed with three dimensions, which the objector of Continuity is vio- demands for the idea & name of motion, by playing with words, as I said above. On the ' other hand, in this kind of motions of ultimate surfaces, or lines, or points, a sudden change must certainly be made, if they arrive at immediate contact with a difference of velocity as above, & the Law of Continuity must be violated. 27. But, omitting all debate about the notions of motion & mass, if the product of it is at least a fact the velocity & the mass vanishes when one of the three dimensions vanish, there will still fated^tf^the^idea remain the velocity of the remaining dimensions ; & this will persist so long as the dimen- of the velocity of sions persist, as they do persist undoubtedly in the case of a surface. Hence the change P°mts- in its velocity must have been made suddenly, & thereby the Law of Continuity, which I have already mentioned so many times, is violated. 28. These things are so evident that it is absolutely impossible to doubt that the Law objection derived /./-!•••• r- i „ i j j i . . j j . »T iv from the admission of Continuity is infringed, & that a sudden change is introduced into Nature, when bodies Of impenetrability approach one another with a difference of velocity & come into immediate contact, if only in verv small Par- we are to ascribe impenetrability to bodies, as we really should. And this property too, tion. ' not in whole bodies only, but in any of the smallest particles of bodies, & in the elements as well, is recognized by Physicists universally. There was one, I must confess, who, after I 50 PHILOSOPHIC NATURALIS THEORIA infringeret, affirmavit, minimas corporum particulas post contactum superficierum com- penetrari non nihil, & post ipsam compenetrationem mutari velocitates per gradus. At id ipsum facile demonstrari potest contrarium illi inductioni, & analogiae, quam unam habemus in Physica investigandis generalibus naturae legibus idoneam, cujus inductionis vis quae sit, & quibus in locis usum habeat, quorum locorum unus est hie ipse impenetrabilitatis ad minimas quasque particulas extendendae, inferius exponam. Objectio a voce motus assumpta pro mutatione; confutatio ex reahtate motus 2Q. Fuit itidem e Leibnitianorum familia, qui post evulgatam Theoriam meam cen- . ' ,./>- •, • j- • j« j j -i • suerit, dimcultatem ejusmodi amoveri posse dicendo, duas monades sibi etiam mvicem occurrentes cum velocitatibus quibuscunque oppositis aequalibus, post ipsum contactum ..... . i, . . .' r ..... pergere moven sine locali progressione. Ham progressionem, ajebat, revera omnmo nihil esse, si a spatio percurso sestimetur, cum spatium sit nihil ; motum utique perseverare, & extingui per gradus, quia per gradus extinguatur energia ilia, qua in se mutuo agunt, sese premendo invicem. Is itidem ludit in voce motus, quam adhibet pro mutatione quacunque, & actione, vel actionis modo. Motus locaiis, & velocitas motus ipsius, sunt ea, quse ego quidem adhibeo, & quae ibi abrumpuntur per saltum. Ea, ut evidentissime constat, erant aliqua ante contactum, & post contactum mo-[i3]-mento temporis in eo casu abrumpuntur ; nee vero sunt nihil ; licet spatium pure imaginarium sit nihil. Sunt realis affectio rei mobilis fundata in ipsis modis localiter existendi, qui modi etiam relationes inducunt dis- tantiarum reales utique. Quod duo corpora magis a se ipsis invicem distent, vel minus ; quod localiter celerius moveantur, vel lentius ; est aliquid non imaginarie tantummodo, sed realiter diversum ; in eo vero per immediatum contactum saltus utique induceretur in eo casu, quo ego superius sum usus. Qui Continuitatu, 30. Et sane summus nostri aevi Geometra, & Philosophus Mac-Laurinus, cum etiam ipse jegem summover- conisjonem corporum contemplatus vidisset, nihil esse, quod continuitatis legem in collisione corporum facta per immediatum contactum conservare, ac tueri posset, ipsam continuitatis legem deferendam censuit, quam in eo casu omnino violari affirmavit in eo opere, quod de Newtoni Compertis inscripsit, lib. I, cap. 4. Et sane sunt alii nonnulli, qui ipsam con- tinuitatis legem nequaquam admiserint, quos inter Maupertuisius, vir celeberrimus, ac de Republica Litteraria optime meritus, absurdam etiam censuit, & quodammodo inexplica- bilem. Eodem nimirum in nostris de corporum collisione contemplationibus devenimus Mac-Laurinus, & ego, ut viderimus in ipsa immediatum contactum, atque impulsionem cum continuitatis lege conciliari non posse. At quoniam de impulsione, & immediate corporum contactu ille ne dubitari quidem posse arbitrabatur, (nee vero scio, an alius quisquam omnem omnium corporum immediatum contactum subducere sit ausus antea, utcunque aliqui aeris velum, corporis nimirum alterius, in collisione intermedium retinuerint) continuitatis legem deseruit, atque infregit. Theorise exortus, t^'t Uf fien 31. Ast ego cum ipsam continuitatis legem aliquanto diligentius considerarim, & , quibus ea innititur, perpenderim, arbitratus sum, ipsam omnino e Natura submoveri non posse, qua proinde retenta contactum ipsum immediatum submovendum censui in collisionibus corporum, ac ea consectaria persecutus, quae ex ipsa continuitate servata sponte profluebant, directa ratiocinatione delatus sum ad earn, quam superius exposui, virium mutuarum legem, quae consectaria suo quaeque ordine proferam, ubi ipsa, quae ad continuitatis legem retinendam argumenta me movent, attigero. Lex Continuitatis 32. Continuitatis lex, de qua hie agimus, in eo sita est, uti superius innui, ut quaevis quid sit : discn- • j i • i- T i- • • r men inter status, quantitas, dum ab una magmtudme ad aliam migrat, debeat transire per omnes intermedias & incrementa. ejusdem generis magnitudines. Solet etiam idem exprimi nominandi transitum per gradus intermedios, quos quidem gradus Maupertuisius ita accepit, quasi vero quaedam exiguae accessiones fierent momento temporis, in quo quidem is censuit violari jam necessario legem ipsam, quae utcunque exiguo saltu utique violatur nihilo minus, quam maximo ; cum nimi-[l4]-rum magnum, & parvum sint tantummodo respectiva ; & jure quidem id censuit ; si nomine graduum incrementa magnitudinis cujuscunque momentanea intelligerentur. A THEORY OF NATURAL PHILOSOPHY 51 had published my Theory, endeavoured to overcome the force of the argument I had used by asserting that the minute particles of the bodies after contact of the surfaces were subject to compenetration in some measure, & that after compenetration the velocities were changed gradually. But it can be easily proved that this is contrary to that induction & analogy, such as we have in Physics, one peculiarly adapted for the investigation of the general laws of Nature. What the power of this induction is, & where it can be used (one of the cases is this very matter of extending impenetrability to the minute particles of a body), I will set forth later. 29. There was also one of the followers of Leibniz who, after I had published my Objection to the Theory, expressed his opinion that this kind of difficulty could be removed by saying that used for°a"change^ two monads colliding with one another with any velocities that were equal & opposite refutation from the ,,,., .. . .-I , , • TT reality of local mo- would, alter they came into contact, go on moving without any local progression, rle tion. added that that progression would indeed be absolutely nothing, if it were estimated by the space passed over, since the space was nothing ; but the motion would go on & be destroyed by degrees, because the energy with which they act upon one another, by mutual pressure, would be gradually destroyed. He also is playing with the meaning of the term motus, which he uses both for any change, & for action & mode of action. Local motion, & the velocity of that motion are what I am dealing with, & these are here broken off suddenly. These, it is perfectly evident, were something definite before contact, & after contact in an instant of time in this case they are broken off. Not that they are nothing ; although purely imaginary space is nothing. They are real conditions of the movable thing depending on its modes of extension as regards position ; & these modes induce relations between the distances that are certainly real. To account for the fact that two bodies stand at a greater distance from one another, or at a less ; or for the fact that they are moved in position more quickly, or more slowly ; to account for this there must be some- thing that is not altogether imaginary, but real & diverse. In this something there would be induced, in the question under consideration, a sudden change through immediate contact. 30. Indeed the finest geometrician & philosopher of our times, Maclaurin, after he too There are some who had considered the collision of solid bodies & observed that there is nothing which could i^doi continuity5 maintain & preserve the Law of Continuity in the collision of bodies accomplished by immediate contact, thought that the Law of Continuity ought to be abandoned. He asserted that, in general in the case of collision, the law was violated, publishing his idea in the work that he wrote on the discoveries of Newton, bk. i, chap. 4. True, there are some others too, who would not admit the Law of Continuity at all ; & amongst these, Mauper- tuis, a man of great reputation & the highest merit in the world of letters, thought it was senseless, & in a measure inexplicable. Thus, Maclaurin came to the same conclusion as myself with regard to our investigations on the collision of bodies ; for we both saw that, in collision, immediate contact & impulsive action could not be reconciled with the Law of Continuity. But, whereas he came to the conclusion that there could be no doubt about the fact of impulsive action & immediate contact between the bodies, he impeached & abrogated the Law of Continuity. Nor indeed do I know of anyone else before me, who has had the courage to deny the existence of all immediate contact for any bodies whatever, although there are some who would retain a thin layer of air, (that is to say, of another body), in between the two in collision. 31. But I, after considering the Law of Continuity somewhat more carefully, & The origin of my pondering over the fundamental ideas on which it depends, came to the conclusion that this°Law, as'shouid it certainly could not be withdrawn altogether out of Nature. Hence, since it had to be be done, retained, I came to the conclusion that immediate contact in the collision of solid bodies must be got rid of ; &, investigating the deductions that naturally sprang from the conservation of continuity, I was led by straightforward reasoning to the law that I have set forth above, namely, the law of mutual forces. These deductions, each set out in order, I will bring forward when I come to touch upon those arguments that persuade me to retain the Law of Continuity. 32. The Law of Continuity, as we here deal with it, consists in the idea that, as I Jhe nature of the . j , ..''... . , , Law of Continuity ; intimated above, any quantity, in passing from one magnitude to another, must pass through distinction between all intermediate magnitudes of the same class. The same notion is also commonly expressed stat<~s & incre- , , ° .,,. ,. -11 ments. by saying that the passage is made by intermediate stages or steps ; these steps indeed Maupertuis accepted, but considered that they were very small additions made in an instant of time. In this he thought that the Law of Continuity was already of necessity violated, the law being indeed violated by any sudden change, no matter how small, in no less a degree than by a very great one. For, of a truth, large & small are only relative terms ; & he rightly thought as he did, if by the name of steps we are to understand momentaneous PHILOSOPHIC NATURALIS THEORIA Geometriae usus ad earn exponendam : momenta punctis, tempera continua lineis expressa. Fluxus ordinatae transeuntis per m agnit u d i nes omnes intermedias. Idem in quantitate variabili expressa : aequivocatio in voce gradus. FKMH K' M' D' FIG. 3. Verum id ita intelligendum est ; ut singulis momentis singuli status respondeant ; incre- menta, vel decrementa non nisi continuis tempusculis. 33. Id sane admodum facile concipitur ope Geometriae. Sit recta quaedam AB in fig. 3, ad quam referatur quaedam alia linea CDE. Exprimat prior ex iis tempus, uti solet utique in ipsis horologiis circularis peripheria ab indicis cuspide denotata tempus definire. Quemadmodum in Geometria in lineis puncta sunt indivisibiles limites continuarum lineas partium, non vero partes linese ipsius ; ita in tempore distinguenda; erunt partes continui temporis respondentes ipsis lines partibus, continue itidem & ipsas, a mo- mentis, quae sunt indivisibiles earum partium limites, & punctis respondent ; nee inpos- terum alio sensu agens de tempore momenti nomen adhibebo, quam eo indivisibilis limitis ; particulam vero temporis utcunque exiguam, & habitam etiam pro infinitesima, tempusculum appellabo. 34. Si jam a quovis puncto rectae AB, ut F, H, erigatur ordinata perpendicularis FG, HI, usque ad lineam CD ; ea poterit repraesentare quantitatem quampiam continuo variabilem. Cuicunque momento temporis F, H, respondebit sua ejus quantitatis magnitudo FG, HI ; momentis autem intermediis aliis K, M, aliae magnitudines, KL, MN, respondebunt ; ac si a puncto G ad I continua, & finita abeat pars linese CDE, facile patet & accurate de- monstrari potest, utcunque eadem contorqueatur, nullum fore punctum K intermedium, cui aliqua ordinata KL non respondeat ; & e converse nullam fore ordinatam magnitu- dinis intermediae inter FG, HI, quae alicui puncto inter F, H intermedio non respondeat. 35. Quantitas ilia variabilis per hanc variabilem ordinatam expressa mutatur juxta continuitatis legem, quia a magnitudine FG, quam habet momento temporis F, ad magni- tudinem HI, quae respondet momento temporis H, transit per omnes intermedias magnitu- dines KL, MN, respondentes intermediis momentis K, M, & momento cuivis respondet determinata magnitudo. Quod si assumatur tempusculum quoddam continuum KM utcunque exiguum ita, ut inter puncta L, N arcus ipse LN non mutet recessum a recta AB in accessum ; ducta LO ipsi parallela, habebitur quantitas NO, quas in schemate exhibito est incrementum magnitudinis ejus quantitatis continuo variatae. Quo minor est ibi temporis particula KM, eo minus est id incrementum NO, & ilia evanescente, ubi congruant momenta K, M, hoc etiam evanescit. Potest quaevis magnitudo KL, MN appellari status quidam variabilis illius quantitatis, & gradus nomine deberet potius in-[i5]-telligi illud incrementum NO, quanquam aliquando etiam ille status, ilia magnitudo KL nomine gradus intelligi solet, ubi illud dicitur, quod ab una magnitudine ad aliam per omnes intermedios gradus transeatur ; quod quidem aequivocationibus omnibus occasionem exhibuit. status singuios 36. Sed omissis aequivocationibus ipsis, illud, quod ad rem facit, est accessio incremen- menta^vero'utcun" torum facta non momento temporis, sed tempusculo continuo, quod est particula continui que parva tem- temporis. Utcunque exiguum sit incrementum ON, ipsi semper respondet tempusculum respondereC°ntinuis q.u°ddam KM continuum. Nullum est in linea punctum M ita proximum puncto K, ut sit primum post ipsum ; sed vel congruunt, vel intercipiunt lineolam continua bisectione per alia intermedia puncta perpetuo divisibilem in infinitum. Eodem pacto nullum est in tempore momentum ita proximum alteri praecedenti momento, ut sit primum post ipsum, sed vel idem momentum sunt, vel inter jacet inter ipsa tempusculum continuum per alia intermedia momenta divisibile in infinitum ; ac nullus itidem est quantitatis continuo variabilis status ita proximus praecedenti statui, ut sit primus post ipsum accessu aliquo momentaneo facto : sed differentia, quae inter ejusmodi status est, debetur intermedio continuo tempusculo ; ac data lege variationis, sive natura lineae ipsam exprimentis, & quacunque utcunque exigua accessione, inveniri potest tempusculum continuum, quo ea accessio advenerit. Transitus sine sal- 37- Atque sic quidem intelligitur, quo pacto fieri possit transitus per intermedias tu, etiamapositivis magnitudines omnes, per intermedios status, per gradus intermedios, quin ullus habeatur ad negativa perm- ,° . r -, . ... , ' " hiium, quod tamen saltus utcunque exiguus momento temporis factus. Notari mud potest tantummodo, m°" eSstedVereu'ida1m mutati°nem neri alicubi per incrementa, ut ubi KL abit, in MN per NO ; alicubi per reaiis status,1"0 ' decrementa, ut ubi K'L' abeat in N'M' per O'N' ; quin immo si linea CDE, quse legem A THEORY OF NATURAL PHILOSOPHY 53 increments of any magnitude whatever. But the idea should be interpreted as follows : single states correspond to single instants of time, but increments or decrements only to small intervals of continuous time. 33. The idea can be very easily assimilated by the help of geometry. Explanation by the Let AB be any straight line (Fig. 3), to which as axis let any other line CDE be referred. "nsseta°tfs ^eTes^ Let the first of them represent the time, in the same manner as it is customary to specify ted by points, con- the time in the case of circular clocks by marking off the periphery with the end of a pointer. 1™°^ "^s*** °f Now, just as in geometry, points are the indivisible boundaries of the continuous parts of a line, so, in time, distinction must be made between parts of continuous time, which cor- respond to these parts of a line, themselves also continuous, & instants of time, which are the indivisible boundaries of those parts of time, & correspond to points. In future I shall not use the term instant in any other sense, when dealing with time, than that of the indivisible boundary ; & a small part of time, no matter how small, even though it is considered to be infinitesimal, I shall term a tempuscule, or small interval of time. 34. If now from any points F,H on the straight line AB there are erected at right angles T.he flux °.f the or~ to it ordinates FG, HI, to meet the line CD ; any of these ordinates can be taken to repre- through^ ail *interS sent a quantity that is continuously varying. To any instant of time F, or H, there will mediate values, correspond its own magnitude of the quantity FG, or HI ; & to other intermediate instants K, M, other magnitudes KL, MN will correspond. Now, if from the point G, there pro- ceeds a continuous & finite part of the line CDE, it is very evident, & it can be rigorously proved, that, no matter how the curve twists & turns, there is no intermediate point K, to which some ordinate KL does not correspond ; &, conversely, there is no ordinate of magnitude intermediate between FG & HI, to which there does not correspond a point intermediate between F & H. 35. The variable quantity that is represented by this variable ordinate is altered in The same holds accordance with the Law of Continuity ; for, from the magnitude FG, which it has at able1 quantity w the instant of time F, to the magnitude HI, which corresponds to the instant H, it passes represented ; equi- through all intermediate magnitudes KL, MN, which correspond to the intermediate oUhe1(term Itep^ instants K, M ; & to every instant there corresponds a definite magnitude. But if we take a definite small interval of continuous time KM, no matter how small, so that between the points L & N the arc LN does not alter from recession from the line AB to approach, & draw LO parallel to AB, we shall obtain the quantity NO that in the figure as drawn is the increment of the magnitude of the continuously varying quantity. Now the smaller the interval of time KM, the smaller is this increment NO ; & as that vanishes when the instants of time K, M coincide, the increment NO also vanishes. Any magnitude KL, MN can be called a state of the variable quantity, & by the name step we ought rather to under- stand the increment NO ; although sometimes also the state, or the magnitude KL is accustomed to be called by the name step. For instance, when it is said that from one magnitude to another there is a passage through all intermediate stages or steps ; but this indeed affords opportunity for equivocations of all sorts. 36. But, omitting all equivocation of this kind, the point is this : that addition of single states cor- .' 1-11 • • <• • i • 11 . respond to instants, increments is accomplished, not m an instant 01 time, but in a small interval of con- but increments tinuous time, which is a part of continuous time. However small the increment ON may however sma11 to i i i Tru if mi •»•> intervals of con- DC, there always corresponds to it some continuous interval KM. 1 here is no point M tinuous time. in the straight line AB so very close to the point K, that it is the next after it ; but either the points coincide, or they intercept between them a short length of line that is divisible again & again indefinitely by repeated bisection at other points that are in between M & K. In the same way, there is no instant of time that is so near to another instant that has gone before it, that it is the next after it ; but either they are the same instant, or there lies between them a continuous interval that can be divided indefinitely at other inter- mediate instants. Similarly, there is no state of a continuously varying quantity so very near to a preceding state that it is the next state to it, some momentary addition having been made ; any difference that exists between two states of the same kind is due to a continuous interval of time that has passed in the meanwhile. Hence, being given the law of variation, or the nature of the line that represents it, & any increment, no matter how small, it is possible to find a small interval of continuous time in which the increment took place. 37. In this manner we can understand how it is possible for a passage to take place Passages without through all intermediate magnitudes, through intermediate states, or through intermediate from^positive1 8 to stages, without any sudden change being made, no matter how small, in an instant of time, negative through T' 11 1111 • i 111- /i zero : zero how- It can merely be remarked that change in some places takes place by increments (as when ever ;s not realiy KL becomes MN by the addition of NO), in other places by decrements (as when K'L' nothing, but acer- ' tain real state. 54 PHILOSOPHIC NATURALIS THEORIA variationis exhibit, alicubi secet rectam, temporis AB, potest ibidem evanescere magnitude, ut ordinata M'N', puncto M' allapso ad D evanesceret, & deinde mutari in negativam PQ, RS, habentem videlicet directionem contrariam, quae, quo magis ex oppositae parte crescit, eo minor censetur in ratione priore, quemadmodum in ratione possessionis, vel divitiarum, pergit perpetuo se habere pejus, qui iis omnibus, quae habebat, absumptis, aes alienum contrahit perpetuo majus. Et in Geometria quidem habetur a positivo ad negativa transitus, uti etiam in Algebraicis formulis, tarn transeundo per nihilum, quam per innnitum, quos ego transitus persecutus sum partim in dissertatione adjecta meis Sectionibus Conicis, partim in Algebra § 14, & utrumque simul in dissertatione De Lege Continuitatis ; sed in Physica, ubi nulla quantitas in innnitum excrescit, is casus locum non habet, & non, nisi transeundo per nihilum, transitus fit a positi-[i6]-vis ad negativa, ac vice versa ; quanquam, uti inferius innuam, id ipsum sit non nihilum revera in se ipso, sed realis quidem status, & habeatur pro nihilo in consideration quadam tantummodo, in qua negativa etiam, qui sunt veri status, in se positivi, ut ut ad priorem seriem pertinentes negative quodam modo, negativa appellentur. Proponitur pro- ,§_ Exposita hoc pacto, & vindicata continuitatis lege, earn in Natura existere plerique banda existentia _, ., J . . r . . .... ... P . . ,-, r. legis Continuitat.s. Philosophi arbitrantur, contradicentibus nonnullis, uti supra mnui. Ego, cum in earn primo inquirerem, censui, eandem omitti omnino non posse ; si earn, quam habemus unicam, Naturae analogiam, & inductionis vim consulamus, ope cujus inductionis earn demonstrare conatus sum in pluribus e memoratis dissertationibus, ac eandem probationem adhibet Benvenutus in sua Synopsi Num. 119; in quibus etiam locis, prout diversis occasionibus conscripta sunt, repetuntur non nulla. Ejus probatio ab ,g Longum hie esset singula inde excerpere in ordinem redacta : satis erit exscribere mductione satis ,. Jy . °_ , ~ . P . r ,-, -n • i • • j ampia. dissertatioms De lege Continuitatis numerum 138. Post mductionem petitam praecedente numero a Geometria, quae nullum uspiam habet saltum, atque a motu locali, in quo nunquam ab uno loco ad alium devenitur, nisi ductu continue aliquo, unde consequitur illud, dis- tantiam a dato loco nunquam mutari in aliam, neque densitatem, quae utique a distantiis pendet particularum in aliam, nisi transeundo per intermedias ; fit gradus in eo numero ad motuum velocitates, & ductus, quas magis hie ad rem faciunt, nimirum ubi de velocitate agimus non mutanda per saltum in corporum collisionibus. Sic autem habetur : " Quin immo in motibus ipsis continuitas servatur etiam in eo, quod motus omnes in lineis continuis fiunt nusquam abruptis. Plurimos ejusmodi motus videmus. Planetae, & cometse in lineis continuis cursum peragunt suum, & omnes retrogradationes fiunt paullatim, ac in stationibus semper exiguus quidem motus, sed tamen habetur semper, atque hinc etiam dies paullatim per auroram venit, per vespertinum crepusculum abit, Solis diameter non per saltum, sed continuo motu supra horizontem ascendit, vel descendit. Gravia itidem oblique projecta in lineis itidem pariter continuis motus exercent suos, nimirum in parabolis, seclusa ^aeris resistentia, vel, ea considerata, in orbibus ad hyperbolas potius accedentibus, & quidem semper cum aliqua exigua obliquitate projiciuntur, cum infinities infinitam improbabilitatem habeat motus accurate verticalis inter infinities infinitas inclinationes, licet exiguas, & sub sensum non cadentes, fortuito obvenienfe, qui quidem motus in hypothesi Telluris^motae a parabolicis plurimum distant, & curvam continuam exhibent etiam pro casu projectionis accurate verticalis, quo, quiescente penitus Tellure, & nulla ventorum vi deflectente motum, haberetur [17] ascensus rectilineus, vel descensus. Immo omnes alii motus a gravitate pendentes, omnes ab elasticitate, a vi magnetica, continuitatem itidem servant ; cum earn servent vires illse ipsae, quibus gignuntur. Nam gravitas, cum decrescat in ratione reciproca duplicata distantiarum, & distantise per saltum mutari non possint, mutatur per omnes intermedias magnitudines. Videmus pariter, vim magneticam a distantiis pendere lege continua ; vim elasticam ab inflexione, uti in laminis, vel a distantia, ut in particulis aeris compressi. In iis, & omnibus ejusmodi viribus, & motibus, quos gignunt, continuitas habetur semper, tarn in lineis quae describuntur, quam in velocitatibus, quae pariter per omnes intermedias magnitudines mutantur, ut videre est in pendulis, in ascensu corporum gravium, A THEORY OF NATURAL PHILOSOPHY 55 becomes N'M' by the subtraction of O'N') ; moreover, if the line CDE, which represents the law of variation, cuts the straight AB, which is the axis of time, in any point, then the magnitude can vanish at that point (just as the ordinate M'N' would vanish when the point M' coincided with D), & be changed into a negative magnitude PQ, or RS, that is to say one having an opposite direction ; & this, the more it increases in the opposite sense, the less it is to be considered in the former sense (just as in the idea of property or riches, a man goes on continuously getting worse off, when, after everything he had has been taken away from him, he continues to get deeper & deeper into debt). In Geometry too we have this passage from positive to negative, & also in algebraical formulae, the passage being made not only through nothing, but also through infinity ; such I have discussed, the one in a dissertation added to my Conic Sections, the other in my Algebra (§ 14), & both of them together in my essay De Lege Continuitatis ; but in Physics, where no quantity ever increases to an infinite extent, the second case has no place ; hence, unless the passage is made through the value nothing, there is no passage from positive to negative, or vice versa. Although, as I point out below, this nothing is not really nothing in itself, but a certain real state ; & it may be considered as nothing only in a certain sense. In the same sense, too, negatives, which are true states, are positive in themselves, although, as they belong to the first set in a certain negative way, they are called negative. 38. Thus explained & defended, the Law of Continuity is considered by most philoso- I propose to prove phers to exist in Nature, though there are some who deny it, as I mentioned above. I, LaVof^Continuity6 when first I investigated the matter, considered that it was absolutely impossible that it should be left out of account, if we have regard to the unparalleled analogy that there is with Nature & to the power of induction ; & by the help of this induction I endeavoured to prove the law in several of the dissertations that I have mentioned, & Benvenutus also used the same form of proof in his Synopsis (Art. 119). In these too, as they were written on several different occasions, there are some repetitions. 39. It would take too long to extract & arrange in order here each of the passages in Proof by induction these essays ; it will be sufficient if I give Art. 138 of the dissertation De Lege Continuitatis. s~^^ for the After induction derived in the preceding article from geometry, in which there is no sudden change anywhere, & from local motion, in which passage from one position to another never takes place unless by some continuous progress (the consequence of which is that a distance from any given position can never be changed into another distance, nor the density, which depends altogether on the distances between the particles, into another density, except by passing through intermediate stages), the step is made in that article to the velocities of motions, & deductions, which have more to do with the matter now in hand, namely, where we are dealing with the idea that the velocity is not changed suddenly in the collision of solid bodies. These are the words : " Moreover in motions themselves continuity is preserved also in the fact that all motions take place in continuous lines that are not broken anywhere. We see a great number of motions of this kind. The planets & the comets pursue their courses, each in its own continuous line, & all retrogradations are gradual ; & in stationary positions the motion is always slight indeed, but yet there is always some ; hence also daylight comes gradually through the dawn, & goes through the evening twilight, as the diameter of the sun ascends above the horizon, not suddenly, but by a continuous motion, & in the same manner descends. Again heavy bodies projected obliquely follow their courses in lines also that are just as continuous ; namely, in para- bolae, if we neglect the resistance of the air, but if that is taken into account, then in orbits that are more nearly hyperbolae. Now, they are always projected with some slight obli- quity, since there is an infinitely infinite probability against accurate vertical motion, from out of the infinitely infinite number of inclinations (although slight & not capable of being observed), happening fortuitously. These motions are indeed very far from being para- bolae, if the hypothesis that the Earth is in motion is adopted. They give a continuous curve also for the case of accurate vertical projection, in which, if the Earth were at rest, & no wind-force deflected the motion, rectilinear ascent & descent would be obtained. All other motions that depend on gravity, all that depend upon elasticity, or magnetic force, also preserve continuity ; for the forces themselves, from which the motions arise, preserve it. For gravity, since it diminishes in the inverse ratio of the squares of the dis- tances, & the distances cannot be changed suddenly, is itself changed through every inter- mediate stage. Similarly we see that magnetic force depends on the distances according to a continuous law ; that elastic force depends on the amount of bending as in plates, or according to distance as in particles of compressed air. In these, & all other forces of the sort, & in the motions that arise from them, we always get continuity, both as regards the lines which they describe & also in the velocities which are changed in similar manner through all intermediate magnitudes ; as is seen in pendulums, in the ascent of heavy 56 PHILOSOPHISE NATURALIS THEORIA & in aliis mille ejusmodi, in quibus mutationes velocitatis fiunt gradatim, nee retro cursus reflectitur, nisi imminuta velocitate per omnes gradus. Ea diligentissime continuitatem servat omnia. Hinc nee ulli in naturalibus motibus habentur anguli, sed semper mutatio directionis fit paullatim, nee vero anguli exacti habentur in corporibus ipsis, in quibus utcunque videatur tennis acies, vel cuspis, microscopii saltern ope videri solet curvatura, quam etiam habent alvei fluviorum semper, habent arborum folia, & frondes, ac rami, habent lapides quicunque, nisi forte alicubi cuspides continuae occurrant, vel primi generis, quas Natura videtur affectare in spinis, vel secundi generis, quas videtur affectare in avium unguibus, & rostro, in quibus tamen manente in ipsa cuspide unica tangente continuitatem servari videbimus infra. Infinitum esset singula persequi, in quibus continuitas in Natura observatur. Satius est generaliter provocare ad exhibendum casum in Natura, in quo eontinuitas non servetur, qui omnino exhiberi non poterit." Duplex inductionis 40. Inductio amplissima turn ex hisce motibus, ac velocitatibus, turn ex aliis pluribus vimhabeatittductio exemPn's> <lU3e habemus in Natura, in quibus ea ubique, quantum observando licet depre- incompieta. hendere, continuitatem vel observat accurate, vel affcctat, debet omnino id efficere, ut ab ea ne in ipsa quidem corporum collisione recedamus. Sed de inductionis natura, & vi, ac ejusdem usu in Physica, libet itidem hie inserere partem numeri 134, & totum 135, disserta- tionis De Lege Continuitatis. Sic autem habent ibidem : " Inprimis ubi generales Naturae leges investigantur, inductio vim habet maximam, & ad earum inventionem vix alia ulla superest via. Ejus ope extensionem, figurabilitem, mobilitatem, impenetrabilitatem corporibus omnibus tribuerunt semper Philosophi etiam veteres, quibus eodem argumento inertiam, & generalem gravitatem plerique e recentioribus addunt. Inductio, ut demon- strationis vim habeat, debet omnes singulares casus, quicunque haberi possunt percurrere. Ea in Natu-[i8]-rae legibus stabiliendis locum habere non potest. Habet locum laxior qusedam inductio, quae, ut adhiberi possit, debet esse ejusmodi, ut inprimis in omnibus iis casibus, qui ad trutinam ita revocari possunt, ut deprehendi debeat, an ea lex observetur, eadem in iis omnibus inveniatur, & ii non exiguo numero sint ; in reliquis vero, si quse prima fronte contraria videantur, re accuratius perspecta, cum ilia lege possint omnia conciliari ; licet, an eo potissimum pacto concilientur, immediate innotescere, nequaquam possit. Si eae conditiones habeantur ; inductio ad legem stabiliendam censeri debet idonea. Sic quia videmus corpora tarn multa, quae habemus prae manibus, aliis corporibus resistere, ne in eorum locum adveniant, & loco cedere, si resistendo sint imparia, potius, quam eodem perstare simul ; impenetrabilitatem corporum admittimus ; nee obest, quod qusedam corpora videamus intra alia, licet durissima, insinuari, ut oleum in marmora, lumen in crystalla, & gemmas. Videmus enim hoc phsenomenum facile conciliari cum ipsa impene- trabilitate, dicendo, per vacuos corporum poros ea corpora permeare. (Num. 135). Praeterea, qusecunque proprietates absolutae, nimirum quae relationem non habent ad nostros sensus, deteguntur generaliter in massis sensibilibus corporum, easdem ad quascunque utcunque exiguas particulas debemus transferre ; nisi positiva aliqua ratio obstet, & nisi sint ejusmodi, quae pendeant a ratione totius, seu multitudinis, contradistincta a ratione partis. Primum evincitur ex eo, quod magna, & parva sunt respectiva, ac insensibilia dicuntur ea, quse respectu nostrae molis, & nostrorum sensuum sunt exigua. Quare ubi agitur de proprietatibus absolutis non respectivis, quaecunque communia videmus in iis, quse intra limites continentur nobis sensibiles, ea debemus censere communia etiam infra eos limites : nam ii limites respectu rerum, ut sunt in se, sunt accidentales, adeoque siqua fuisset analogise Isesio, poterat ilia multo facilius cadere intra limites nobis sensibiles, qui tanto laxiores sunt, quam infra eos, adeo nimirum propinquos nihilo. Quod nulla ceciderit, indicio est, nullam esse. Id indicium non est evidens, sed ad investigationis principia pertinet, quae si juxta A THEORY OF NATURAL PHILOSOPHY 57 bodies, & in a thousand other things of the same kind, where the changes of velocity occur gradually, & the path is not retraced before the velocity has been diminished through all degrees. All these things most strictly preserve continuity. Hence it follows that no sharp angles are met with in natural motions, but in every case a change of direction occurs gradually ; neither do perfect angles occur in bodies themselves, for, however fine an edge or point in them may seem, one can usually detect curvature by the help of the microscope if nothing else. We have this gradual change of direction also in the beds of rivers, in the leaves, boughs & branches of trees, & stones of all kinds ; unless, in some cases perchance, there may be continuous pointed ends, either of the first kind, which Nature is seen to affect in thorns, or of the second kind, which she is seen to do in the claws & the beak of birds ; in these, however, we shall see below that continuity is still preserved, since we are left with a single tangent at the extreme end. It would take far too long to mention every single thing in which Nature preserves the Law of Continuity ; it is more than sufficient to make a general statement challenging the production of a single case in Nature, in which continuity is not preserved ; for it is absolutely impossible for any such case to be brought forward." 40. The effect of the very complete induction from such motions as these & velocities, induction of a two- as well as from a large number of other examples, such as we have in Nature, where Nature *old , kil\d '• when e c ,...-..& why incomplete in every case, as far as can be gathered from direct observation, maintains continuity or induction has vaii- tries to do so, should certainly be that of keeping us from neglecting it even in the case of collision of bodies. As regards the nature & validity of induction, & its use in Physics, I may here quote part of Art. 134 & the wjiole of Art. 135 from my dissertation De Lege Continuitatis, The passage runs thus : " Especially when we investigate the general laws of Nature, induction has very great power ; & there is scarcely any other method beside it for the discovery of these laws. By its assistance, even the ancient philosophers attributed to all bodies extension, figurability, mobility, & impenetrability ; & to these properties, by the use of the same method of reasoning, most of the later philosophers add inertia & universal gravitation. Now, induction should take account of every single case that can possibly happen, before it can have the force of demonstration ; such induction as this has no place in establishing the laws of Nature. But use is made of an induction of a less rigorous type ; in order that this kind of induction may be employed, it must be of such a nature that in all those cases particularly, which can be examined in a manner that is bound to lead to a definite conclusion as to whether or no the law in question is followed, in all of them the same result is arrived at ; & that these cases are not merely a few. Moreover, in the other cases, if those which at first sight appeared to be contradictory, on further & more accurate investigation, can all of them be made to agree with the law ; although, whether they can be made to agree in this way better than in any other whatever, it is impossible to know directly anyhow. If such conditions obtain, then it must be considered that the induction is adapted to establishing the law. Thus, as we see that so many of the bodies around us try to prevent other bodies from occupying the position which they themselves occupy, or give way to them if they are not capable of resisting them, rather than that both should occupy the same place at the same time, therefore we admit the impenetrability of bodies. Nor is there anything against the idea in the fact that we see certain bodies penetrating into the innermost parts of others, although the latter are very hard bodies ; such as oil into marble, & light into crystals & gems. For we see that this phenomenon can very easily be reconciled with the idea of impenetrability, by supposing that the former bodies enter and pass through empty pores in the latter bodies (Art. 135). In addition, whatever absolute properties, for instance those that bear no relation to our senses, are generally found to exist in sensible masses of bodies, we are bound to attribute these same properties also to all small parts whatsoever, no matter how small they may be. That is to say, unless some positive reason prevents this ; such as that they are of such a nature that they depend on argument having to do with a body as a whole, or with a group of particles, in contradistinction to an argument dealing with a part only. The proof comes in the first place from the fact that great & small are relative terms, & those things are called insensible which are very small with respect to our own size & with regard to our senses. Therefore, when we consider absolute, & not relative, properties, whatever we perceive to be common to those contained within the limits that are sensible to us, we should consider these things to be still common to those beyond those limits. For these limits, with regard to such matters as are self-contained, are accidental ; & thus, if there should be any violation of the analogy, this would be far more likely to happen between the limits sensible to us, which are more open, than beyond them, where indeed they are so nearly nothing. Because then none did happen thus, it is a sign that there is none. This sign is not evident, but belongs to the principles of investigation, which generally proves successful if it is carried out in accordance with certain definite wisely 5 8 PHILOSOPHIC NATURALIS THEORIA quasdam prudentes regulas fiat, successum habere solet. Cum id indicium fallere possit ; fieri potest, ut committatur error, sed contra ipsum errorem habebitur praesumptio, ut etiam in jure appellant, donee positiva ratione evincatur oppositum. Hinc addendum fuit, nisi ratio positiva obstet. Sic contra hasce regulas peccaret, qui diceret, corpora quidem magna compenetrari, ac replicari, & inertia carere non posse, compenetrari tamen posse, vel replicari, vel sine inertia esse exiguas eorum partes. At si proprietas sit respectiva, respectu nostrorum sensuum, ex [19] eo, quod habeatur in majoribus massis, non debemus inferre, earn haberi in particulis minoribus, ut est hoc ipsum, esse sensibile, ut est, esse coloratas, quod ipsis majoribus massis competit, minoribus non competit ; cum ejusmodi magnitudinis discrimen, accidentale respectu materiae, non sit accidentale respectu ejus denominationis sensibile, coloratum. Sic etiam siqua proprietas ita pendet a ratione aggregati, vel totius, ut ab ea separari non possit ; nee ea, ob rationem nimirum eandem, a toto, vel aggregate debet transferri ad partes. Est de ratione totius, ut partes habeat, nee totum sine partibus haberi potest. Est de ratione figurabilis, & extensi, ut habeat aliquid, quod ab alio distet, adeoque, ut habeat partes ; hinc eae proprietates, licet in quovis aggregate particularum materiae, sive in quavis sensibili massa, inveniantur, non debent inductionis vi transferri ad particulas quascunque." Et impenetrabili- 41. Ex his patet, & impenetrabilitatem, & continuitatis legem per ejusmodi inductionis ultatem tvtad""pCT genus abunde probari, atque evinci, & illam quidem ad quascunque utcunque exiguas inductionem : "* ad particulas corporum, hanc ad gradus utcunque exiguos momento temporis adjectos debere ipsam quid requu-a- exten(jj< Requiritur autem ad hujusmodi inductionem primo, ut ilia proprietas, ad quam probandam ea adhibetur, in plurimis casibus observetur, aliter enim probabilitas esset exigua ; & ut nullus sit casus observatus, in quo evinci possit, earn violari. Non est necessarium illud, ut in iis casibus, in quibus primo aspectu timeri possit defectus proprietatis ipsius, positive demonstretur, earn non deficere ; satis est, si pro iis casibus haberi possit ratio aliqua conciliandi observationem cum ipsa proprietate, & id multo magis, si in aliis casibus habeatur ejus conciliationis exemplum, & positive ostendi possit, eo ipso modo fieri aliquando conciliationem. Ejus appiicatio ad 42. Id ipsum fit, ubi per inductionem impenetrabilitas corporum accipitur pro generali impenetrab;htatem. jege ]sjaturaEi Nam impenetrabilitatem ipsam magnorum corporum observamus in exemplis sane innumeris tot corporum, quae pertractamus. Habentur quidem & casus, in quibus earn violari quis credent, ut ubi oleum per ligna, & marmora penetrat, atque insinuatur, & ubi lux per vitra, & gemmas traducitur. At praesto est conciliatio phasnomeni cum impenetra- bilitate, petita ab eo, quod ilia corpora, in quse se ejusmodi substantiae insinuant, poros habeant, quos 633 permeent. Et quidem haec conciliatio exemplum habet manifestissimum in spongia, quae per poros ingentes aqua immissa imbuitur. Poros marmorum illorum, & multo magis vitrorum, non videmus, ac multo minus videre possumus illud, non insinuari eas substantias nisi per poros. Hoc satis est reliquae inductionis vi, ut dicere debeamus, eo potissimum pacto se rem habere, & ne ibi quidem violari generalem utique impenetrabilitatis legem. simiiisad continu- [2O] 43- Eodem igitur pacto in lege ipsa continuitatis agendum est. Ilia tarn ampla itatem : duo cas- inductio, quam habemus, debet nos movere ad illam generaliter admittendam etiam pro iis quibus ean<videatur casibus, in quibus determinare immediate per observations non possumus, an eadem lacdi- habeatur, uti est collisio corporum ; ac si sunt casus nonnulli, in quibus eadem prima fronte violari videatur ; ineunda est ratio aliqua, qua ipsum phsenomenum cum ea lege conciliari possit, uti revera potest. Nonnullos ejusmodi casus protuli in memoratis dissertationibus, quorum alii ad geometricam continuitatem pertinent, alii ad physicam. In illis prioribus non immorabor ; neque enim geometrica continuitas necessaria est ad hanc physicam propugnandam, sed earn ut exemplum quoddam ad confirmationem quandam inductionis majoris adhibui. Posterior, ut saepe & ilia prior, ad duas classes reducitur ; altera est eorum casuum, in quibus saltus videtur committi idcirco, quia nos per saltum omittimus intermedias quantitates : rem exemplo geometrico illustro, cui physicum adjicio. A THEORY OF NATURAL PHILOSOPHY 59 chosen rules. Now, since the indication may possibly be fallacious, it may happen that an error may be made ; but there is presumption against such an error, as they call it in law, until direct evidence to the contrary can be brought forward. Hence we should add : unless some positive argument is against it. Thus, it would be offending against these rules to say that large bodies indeed could not suffer compenetration, or enfolding, or be deficient in inertia, but yet very small parts of them could suffer penetration, or enfolding, or be without inertia. On the other hand, if a property is relative with respect to our senses, then, from a result obtained for the larger masses we cannot infer that the same is to be obtained in its smaller particles ; for instance, that it is the same thing to be sensible, as it is to be coloured, which is true in the case of large masses, but not in the case of small particles ; since a distinction of this kind, accidental with respect to matter, is not accidental with respect to the term sensible or coloured. So also if any property depends on an argu- ment referring to an aggregate, or a whole, in such a way that it cannot be considered apart from the whole, or the aggregate ; then, neither must it (that is to say, by that same argument), be transferred from the whole, or the aggregate, to parts of it. It is on account of its being a whole that it has parts ; nor can there be a whole without parts. It is on account of its being figurable & extended that it has some thing that is apart from some other thing, & therefore that it has parts. Hence those properties, although they are found in any aggregate of particles of matter, or in any sensible mass, must not however be transferred by the power of induction to each & every particle." 41. From what has been said it is quite evident that both impenetrability & the Law Both impenetra- of Continuity can be proved by a kind of induction of this type ; & the former must be c^bf dTm""^ extended to all particles of bodies, no matter how small, & the latter to all additional steps, strated by indue- however small, made in an instant of time. Now, in the first place, to use this kind of quired for thisSpur- induction, it is required that the property, for the proof of which it is to be used, must be pose. observed in a very large number of cases ; for otherwise the probability would be very small. Also it is required that no case should be observed, in which it can be proved that it is violated. It is not necessary that, in those cases in which at first sight it is feared that there may be a failure of the property, that it should be directly proved that there is no failure. It is sufficient if in those cases some reason can be obtained which will make the observation agree with the property ; & all the more so, if in other cases an example of reconciliation can be obtained, & it can be positively proved that sometimes reconciliation can be obtained in that way. 42. This is just what does happen, when the impenetrability of solid bodies is accepted Application of in- as a law of Nature through inductive reasoning. For we observe this impenetrability of tr'abiuty.*0 lmpene" large bodies in innumerable examples of the many bodies that we consider. There are indeed also cases, in which one would think that it was violated, such as when oil penetrates wood and marble, & works its way through them, or when light passes through glasses & gems. But we have ready a means of making these phenomena agree with impenetrability, derived from the fact that those bodies, into which substances of this kind work their way, possess pores which they can permeate. There is a very evident example of this recon- ciliation in a sponge, which is saturated with water introduced into it by means of huge pores. We do not see the pores of the marble, still less those of glass ; & far less can we see that these substances do not penetrate except by pores. It satisfies the general force of induction if we can say that the matter can be explained in this way better than in any other, & that in this case there is absolutely no contradiction of the general law of impene- trability. 43. In the same way, then, we must deal with the Law of Continuity. The full Similar application " induction that we possess should lead us to admit in general this law even in those cases in ^sisses "oT cases which it is impossible for us to determine directly by observation whether the same law which there seems holds good, as for instance in the collision of bodies. Also, if there are some cases in which * the law at first sight seems to be violated, some method must be followed, through which each phenomenon can be reconciled with the law, as is in every case possible. I brought forward several cases of this kind in the dissertations I have mentioned, some of which pertained to geometrical continuity, & others to physical continuity. I will not delay over the first of these : for geometrical continuity is not necessary for the defence of the physical variety ; I used it as an example in confirmation of a wider induction. The latter, as well as very frequently the former, reduces to two classes ; & the first of these classes is that class in which a sudden change seems to have been made on account of our having omitted the intermediate quantities with a jump. I give a geometrical illustration, and then add one in physics. 6o PHILOSOPHISE NATURALIS THEORIA Exemplum geome- tricum primi gene- ris, ubi nos inter- mcdias magnitu- dines omittimus. Quando id accidat exempla physica dierum, & oscilla- tionum consequen- tium. 44. In axe curvae cujusdam in fig. 4. sumantur segmenta AC, CE, EG aequalia, & erigantur ordinatae AB, CD, EF, GH. Area; BACD, DCEF, FEGH videntur continue cujusdam seriei termini ita, ut ab ilia BACD acl DCEF, & inde ad FEGH immediate transeatur, & tamen secunda a prima, ut & tertia a secunda, differunt per quanti- tates finitas : si enim capiantur CI, EK sequales BA, DC, & arcus BD transferatur in IK ; area DIKF erit incrementum se- cundae supra primam, quod videtur imme- diate advenire totum absque eo, quod unquam habitum sit ejus dimidium, vel quaevis alia pars incrementi ipsius ; ut idcirco a prima ad secundam magnitudinem areae itum sit sine transitu per intermedias. At ibi omittuntur a nobis termini intermedii, qui continuitatem servant ; si enim ac aequalis FIG. 4. AC motu continue feratur ita, ut incipiendo ab AC desinat in CE ; magnitude areae BACD per omnes intermedias bacd abit in magnitu- dinem DCEF sine ullo saltu, & sine ulla violatione continuitatis. 45. Id sane ubique accidit, ubi initium secundae magnitudinis aliquo intervallo distal ab initio primas ; sive statim veniat post ejus finem, sive qua vis alia lege ab ea disjungatur. Sic in pliysicis, si diem concipiamus intervallum temporis ab occasu ad occasum, vel etiam ab ortu ad occasum, dies praecedens a sequent! quibusdam anni temporibus differt per plura secunda, ubi videtur fieri saltus sine ullo intermedio die, qui minus differat. At seriem quidem continuam ii dies nequaquam constituunt. Concipiatur parallelus integer Telluris, in quo sunt continuo ductu disposita loca omnia, quae eandem latitudinem geographicam habent ; ea singula loca suam habent durationem diei, & omnium ejusmodi dierum initia, ac fines continenter fluunt ; donee ad eundem redeatur locum, cujus pr£e-[2i]-cedens dies est in continua ilia serie primus, & sequens postremus. Illorum omnium dierum magni- tudines continenter fluunt sine ullo saltu : nos, intermediis omissis, saltum committimus non Natura. Atque huic similis responsio est ad omnes reliquos casus ejusmodi, in quibus initia, & fines continenter non fluunt, sed a nobis per saltum accipiuntur. Sic ubi pendulum oscillat in acre ; sequens oscillatio per finitam magnitudinem distat a praecedente ; sed & initium & finis ejus finite intervallo temporis distat a prascedentis initio, & fine, ac intermedii termini continua serie fluente a prima oscillatione ad secundam essent ii, qui haberentur, si primae, & secundae oscillationis arcu in aequalem partium numerum diviso, assumeretur via confecta, vel tempus in ea impensum, inter jacens inter fines partium omnium proportion- alium, ut inter trientem, vel quadrantem prioris arcus, & trientem,vel quadrantem posterioris, quod ad omnes ejus generis casus facile transferri potest, in quibus semper immediate etiam demonstrari potest illud, continuitatem nequaquam violari. Exempla secundi 46. Secunda classis casuum est ea, in qua videtur aliquid momento temporis peragi, atne«iOTimeUtSsed & tamen peragitur tempore successive, sed perbrevi. Sunt, qui objiciant pro violatione non momento' tem- continuitatis casum, quo quisquam manu lapidem tenens, ipsi statim det velocitatem quandam finitam : alius objicit aquae e vase effluentis, foramine constitute aliquanto infra superficiem ipsius aquae, velocitatem oriri momento temporis finitam. At in priore casu admodum evidens est, momento temporis velocitatem finitam nequaquam produci. Tempore opus est, utcunque brevissimo, ad excursum spirituum per nervos, & musculos, ad fibrarum tensionem, & alia ejusmodi : ac idcirco ut velocitatem aliquam sensibilem demus lapidi, manum retrahimus, & ipsum aliquandiu, perpetuo accelerantes, retinemus. Sic etiam, ubi tormentum bellicum exploditur, videtur momento temporis emitti globus, ac totam celeritatem acquirere ; at id successive fieri, patet vel inde, quod debeat inflammari tota massa pulveris pyrii, & dilatari aer, ut elasticitate sua globum acceleret, quod quidem fit omnino per omnes gradus. Successionem multo etiam melius videmus in globe, qui ab elastro sibi relicto propellatur : quo elasticitas est major, eo citius, sed nunquam momento temporis velocitas in globum inducitur. AppUcatio ipsorum 47. Hsec exempla illud praestant, quod aqua per pores spongiae ingressa respectu ad emuxum1IaquK impenetrabilitatis, ut ea responsione uti possimus in aliis casibus omnibus, in quibus accessio e vase. aliqua magnitudinis videtur fieri tota momento temporis ; ut nimirum dicamus fieri tempore A THEORY OF NATURAL PHILOSOPHY 61 44. In the axis of any curve (Fig. 4) let there be taken the segments AC, CE, EG equal Geometrical ex- to one another ; & let the ordinates AB, CD, EF, GH be erected. The areas BACD, DCEF, kind where6 Ivl FEGH seem to be terms of some continuous series such that we can pass directly from BACD omit . intermediate to DCEF and then on to FEGH, & yet the second differs from the first, & also the third from the second, by a finite quantity. For if CI, EK are taken equal to BA, DC, & the arc BD is transferred to the position IK ; then the area DIKF will be the increment of the second area beyond the first ; & this seems to be directly arrived at as a whole without that which at any one time is considered to be the half of it, or indeed any other part of the increment itself : so that, in consequence, we go from the first to the second magnitude of area without passing through intermediate magnitudes. But in this case we omit intermediate terms which maintain the continuity ; for if ac is equal to AC, & this is carried by a continuous motion in such a way that, starting from the position AC it ends up at the position CE, then the magnitude of the area BACD will pass through all intermediate values such as bacd until it reaches the magnitude of the area DCEF without any sudden change, & hence without any breach of continuity. 45. Indeed this always happens when the beginning of the second magnitude is distant when this will by a definite interval from the beginning of the first ; whether it comes immediately after happen = physical the end of the first or is disconnected from it by some other law. Thus in physics, if we casTof Consecutive look upon the day as the interval of time between sunset & sunset, or even between sunrise day^ OI. consecutive & sunset, the preceding day differs from that which follows it at certain times of the year by several seconds ; in which case we see that there is a sudden change made, without there being any intermediate day for which the change is less. But the fact is that these days do not constitute a continuous series. Let us consider a complete parallel of latitude on the Earth, along which in a continuous sequence are situated all those places that have the same geographical latitude. Each of these places has its own duration of the day, & the begin- nings & ends of days of this kind change uninterruptedly ; until we get back again to the same place, where the preceding day is the first of that continuous series, & the day that fol- lows is the last of the series. The magnitudes of all these days continuously alter without there being any sudden change : it was we who, by omitting the intermediates, made the sudden change, & not Nature. Similar to this is the answer to all the rest of the cases of the same kind, in which the beginnings & the ends do not change uninterruptedly, but are observed by us discontinuously. Similarly, when a pendulum oscillates in air, the oscillation that follows differs from the oscillation that has gone before by a finite magnitude. But both the begin- ning & the end of the second differs from the beginning & the end of the first by a finite inter- val of time ; & the intermediate terms in a continuously varying series from the first oscillation to the second would be those that would be obtained, if the arcs of the first & second oscilla- tions were each divided into the same number of equal parts, & the path traversed (or the time spent in traversing the path) is taken between the ends of all these proportional paths ; such as that between the third or fourth part of the first arc & the third or fourth part of the second arc. This argument can be easily transferred so as to apply to all cases of this kind ; & in such cases it can always be directly proved that there is no breach of continuity. 46. The second class of cases is that in which something seems to have been done in an Examples of the instant of time, but still it is really done in a continuous, but very short, interval of time. ^iS? the^chan'e There are some who bring forward, as an objection in favour of a breach of continuity, the is very rapid, but case in which a man, holding a stone in his hand, gives to it a definite velocity all at once ; f^an^nstant^of another raises an objection that favours a breach of continuity, in the case of water flowing time. from a vessel, where, if an opening is made below the level of the surface of the water, a finite velocity is produced in an instant of time. But in the first case it is perfectly clear that a finite velocity is in no wise produced in an instant of time. For there is need of time, although this is exceedingly short, for the passage of cerebral impulses through the nerves and muscles, for the tension of the fibres, and other things of that sort ; and therefore, in order to give a definite sensible velocity to the stone, we draw back the hand, and then retain the stone in it for some time as we continually increase its velocity forwards. So too when an engine of war is exploded, the ball seems to be driven forth and to acquire the whole of its speed in an instant of time. But that it is done continuously is clear, if only from the fact that the whole mass of the gunpowder has to be inflamed and the gas has to be expanded in order that it may accelerate the ball by its elasticity ; and this latter certainly takes place by degrees. The continuous nature of this is far better seen in the case of a ball propelled by releasing a spring ; here the stronger the elasticity, the greater the speed ; but in no case is the speed imparted to the ball in an instant of time. 47. These examples are superior to that of water entering through the pores of a sponge, Application of which we employed in the matter of impenetrability ; so that we can make use of this reply *£.gss.e particularly in all other cases in which some addition to a magnitude seems to have taken place entirely in to the flow of water an instant of time. Thus, without doubt we may say that it takes place in an exceedingly from a vesse1' 62 PHILOSOPHIC NATURALIS THEORIA brevissimo, utique per omnes intermedias magnitudines, ac illsesa penitus lege continuitatis. Hinc & in aquae effluentis exemplo res eodem redit, ut non unico momento, sed successive aliquo tempore, & per [22] omnes intermedias magnitudines progignatur velocitas, quod quidem ita se habere optimi quique Physici affirmant. Et ibi quidem, qui momento temporis omnem illam velocitatem progigni, contra me affirmet, principium utique, ut ajunt, petat, necesse est. Neque enim aqua, nisi foramen aperiatur, operculo dimoto, effluet ; remotio vero operculi, sive manu fiat, sive percussione aliqua, non potest fieri momento temporis, sed debet velocitatem suam acquirere per omnes gradus ; nisi illud ipsum, quod quaerimus, supponatur jam definitum, nimirum an in collisione corporum communicatio motus fiat momento temporis, an per omnes intermedios gradus, & magni- tudines. Verum eo omisso, si etiam concipiamus momento temporis impedimentum auferri, non idcirco momento itidem temporis omnis ilia velocitas produceretur ; ilia enim non a percussione aliqua, sed a pressione superincumbentis aquae orta, oriri utique non potest, nisi per accessiones continuas tempusculo admodum parvo, sed non omnino nullo : nam pressio tempore indiget, ut velocitatem progignat, in communi omnium sententia. Transitus ad meta- continuis ut in Geometria. 48. Illaesa igitur esse debet continuitatis lex, nee ad earn evertendam contra inductionem, tam uberem quidquam poterunt casus allati hucusque, vel iis similes. At ejusdem con- umcus, tinuitatis aliam metaphysicam rationem adinveni, & proposui in dissertatione De Lege Continuitatis, petitam ab ipsa continuitatis natura, in qua quod Aristoteles ipse olim notaverat, communis esse debet limes, qui praecedentia cum consequentibus conjungit, qui idcirco etiam indivisibilis est in ea ratione, in qua est limes. Sic superficies duo solida dirimens & crassitudine caret, & est unica, in qua immediatus ab una parte fit transitus ad aliam ; linea dirimens binas superficiei continuae partes latitudine caret ; punctum continuae lineae segmenta discriminans, dimensione omni : nee duo sunt puncta contigua, quorum alterum sit finis prioris segmenti, alterum initium sequentis, cum duo contigua indivisibilia, & inextensa haberi non possint sine compenetratione, & coalescentia quadam in unum. idem in tempore 49. Eodem autem pacto idem debet accidere etiam in tempore, ut nimirum inter tempus «> ti ua^'evide"6 contmuum praecedens, & continuo subsequens unicum habeatur momentum, quod sit tius in quibusdam. indivisibilis terminus utriusque ; nee duo momenta, uti supra innuimus, contigua esse possint, sed inter quodvis momentum, & aliud momentum debeat intercedere semper continuum aliquod tempus divisibile in infinitum. Et eodem pacto in quavis quantitate, quae continuo tempore duret, haberi debet series quasdam magnitudinum ejusmodi, ut momento temporis cuivis respondeat sua, quae praecedentem cum consequente conjungat, & ab ilia per aliquam determinatam magnitudinem differat. Quin immo in illo quantitatum genere, in quo [23] binae magnitudines simul haberi non possunt, id ipsum multo evidentius conficitur, nempe nullum haberi posse saltum immediatum ab una ad alteram. Nam illo momento temporis, quo deberet saltus fieri, & abrumpi series accessu aliquo momentaneo, deberent haberi duae magnitudines, postrema seriei praecedentis, & prima seriei sequentis. Id ipsum vero adhuc multo evidentius habetur in illis rerum statibus, in quibus ex una parte quovis momento haberi debet aliquis status ita, ut nunquam sine aliquo ejus generis statu res esse possit ; & ex alia duos simul ejusmodi status habere non potest. inde cur motus ip- ro> \& quidem satis patebit in ipso locali motu, in quo habetur phsenomenum omnibus calls non fiat, nisi •> . . * , . r r. ...... \ ,. . , . . per Hneam contin- sane notissimum, sed cujus ratio non ita facile ahunde redditur, inde autem patentissima est, Corpus a quovis loco ad alium quemvis devenire utique potest motu continuo per lineas quascunque utcunque contortas, & in immensum productas quaquaversum, quae numero infinities infinitae sunt : sed omnino debet per continuam aliquam abire, & nullibi inter- ruptam. En inde rationem ejus rei admodum manifestam. Si alicubi linea motus abrum- peretur ; vel momentum temporis, quo esset in primo puncto posterioris lineae, esset posterius eo momento, quo esset in puncto postremo anterioris, vel esset idem, vel anterius ? In primo, & tertio casu inter ea momenta intercederet tempus aliquod continuum divisibile in infinitum per alia momenta intermedia, cum bina momenta temporis, in eo sensu accepta, in quo ego hie ea accipio, contigua esse non possint, uti superiusexposui. Quamobrem in A THEORY OF NATURAL PHILOSOPHY 63 short interval of time, and certainly passes through every intermediate magnitude, and that the Law of Continuity is not violated. Hence also in the case of water flowing from a vessel it reduces to the same example : so that the velocity is generated, not in a single instant, but in some continuous interval of time, and passes through all intermediate magni- tudes ; and indeed all the most noted physicists assert that this is what really happens. Also in this matter, should anyone assert in opposition to me that the whole of the speed is produced in an instant of time, then he must use a •petitio principii, as they call it. For the water can-not flow out, unless the hole is opened, & the lid removed ; & the removal of the lid, whether done by hand or by a blow, cannot be effected in an instant of time, but must acquire its own velocity by degrees ; unless we suppose that the matter under investi- gation is already decided, that is to say, whether in collision of bodies communication of motion takes place in an instant of time or through all intermediate degrees and magnitudes. But even if that is left out of account, & if also we assume that the barrier is removed in an instant of time, none the more on that account would the whole of the velocity also be produced in an instant of time ; for it is impossible that such velocity can arise, not from some blow, but from a pressure arising from the superincumbent water, except by continuous additions in a very short interval of time, which is however not absolutely nothing ; for pressure requires time to produce velocity, according to the general opinion of everybody. 48. The Law of Continuity ought then to be subject to no breach, nor will the cases Passing to a meta- hitherto brought forward, nor others like them, have any power at all to controvert this haveT'smrie'iinUt law in opposition to induction so copious. Moreover I discovered another argument, a in the case of con- metaphysical one, in favour of this continuity, & published it in my dissertation De Lege g'^n^iy1"11^' &S "* Continuitatis, having derived it from the very nature of continuity ; as Aristotle himself long ago remarked, there must be a common boundary which joins the things that precede to those that follow ; & this must therefore be indivisible for the very reason that it is a boundary. In the same way, a surface of separation of two solids is also without thickness & is single, & in it there is immediate passage from one side to the other ; the line of separation of two parts of a continuous surface lacks any breadth ; a point determining segments of a continuous line has no dimension at all ; nor are there two contiguous points, one of which is the end of the first segment, & the other the beginning of the next ; for two contiguous indivisibles, of no extent, cannot possibly be considered to exist, unless there is compenetration & a coalescence into one. 49. In the same way, this should also happen with regard to time, namely, that between similarly for time a preceding continuous time & the next following there should be a single instant, which ^£y. mor^evi- is the indivisible boundary of either. There cannot be two instants, as we intimated above, dent in some than contiguous to one another ; but between one instant & another there must always intervene m others- some interval of continuous time divisible indefinitely. In the same way, in any quantity which lasts for a continuous interval of time, there must be obtained a series of magnitudes of such a kind that to each instant of time there is its corresponding magnitude ; & this magnitude connects the one that precedes with the one that follows it, & differs from the former by some definite magnitude. Nay even in that class of quantities, in which we cannot have two magnitudes at the same time, this very point can be deduced far more clearly, namely, that there cannot be any sudden change from one to another. For at that instant, when the sudden change should take place, & the series be broken by some momen- tary definite addition, two magnitudes would necessarily be obtained, namely, the last of the first series & the first of the next. Now this very point is still more clearly seen in those states of things, in which on the one hand there must be at any instant some state so that at no time can the thing be without some state of the kind, whilst on the other hand it can never have two states of the kind simultaneously. 50. The above will be sufficiently clear in the case of local motion, in regard to which Hence the reason the phenomenon is perfectly well known to all ; the reason for it, however, is not so easily ^Jj^ Recurs™;:!10" derived from any other source, whilst it follows most clearly from this idea. A body can continuous line, get from any one position to any other position in any case by a continuous motion along any line whatever, no matter how contorted, or produced ever so far in any direction ; these lines being infinitely infinite in number. But it is bound to travel by some continuous line, with no break in it at any point. Here then is the reason of this phenomenon quite clearly explained. If the motion in the line should be broken at any point, either the instant of time, at which it was at the first point of the second part of the line, would be after the instant, at which it was at the last point of the first part of the line, or it would be the same instant, or before it. In the first & third cases, there would intervene between the two instants some definite interval of continuous time divisible indefinitely at other intermediate instants ; for two instants of time, considered in the sense in which I have PHILOSOPHIC NATURALIS THEORIA primo casu in omnibus iis infinitis intermediis momentis nullibi esset id corpus, in secundo casu idem esset eodem illo memento in binis locis, adeoque replicaretur ; in terio haberetur replicatio non tantum respectu eorum binorum momentorum, sed omnium etiam inter- mediorum, in quibus nimirum omnibus id corpus esset in binis locis. Cum igitur corpus existens nee nullibi esse possit, nee simul in locis pluribus ; ilia vias mutatio, & ille saltus haberi omnino non possunt. 51. Idem ope Geometric magis adhuc oculis ipsis subjicitur. Exponantur per rectam AB tempora, ac per ordinatas ad lineas CD, EF, abruptas alicubi, diversi status rei cujuspiam. e metaphysica, Ductis ordinatis DG, EH, vel punctum H iaceret post G, ut in Fie. c : vel cum ipso ibus exemphs . • /• i • ij . r T . o J • r congrueret, ut in 6 ; vel ipsum prsccederet, ut in 7. In pnmo casu nulla responderet ordinata omnibus punctis rectae GH ; in secundo binae responderent GD, & HE eidem puncto G ; in tertio vero binae HI, & HE puncto H, binas GD, GK puncto G, & binae LM, LN Illustratio ejus i ex Geo- ratiocina- tione pluribus exempl D E. D G H FIG. 5. B A GH FIG. 6. H L G FIG. 7. puncto cuivis intermedio L ; nam ordinata est relatio quaedam distantly, quam habet punctum curvae cum puncto axis sibi respondente, adeoque ubi jacent in recta eadem perpendiculari axi bina curvarum puncta, habentur binae ordinatae respondentes eidem puncto axis. Quamobrem si nee o-[24]-mni statu carere res possit, nee haberi possint status simul bini ; necessario consequitur, saltum ilium committi non posse. Saltus ipse, si deberet accidere, uti vulgo fieri concipitur, accideret binis momentis G, & H, quae sibi in fig. 6 immediate succederent sine ullo immediato hiatu, quod utique fieri non potest ex ipsa limitis ratione, qui in continuis debet esse idem, & antecedentibus, & consequentibus communis, uti diximus. Atque idem in quavis reali serie accidit ; ut hie linea finita sine puncto primo, & postremo, quod sit ejus limes, & superficies sine linea esse non potest ; unde fit, ut in casu figurae 6 binae ordinatae necessario respondere debeant eidem puncto : ita in quavis finita reali serie statuum primus terminus, & postremus haberi necessario debent ; adeoque si saltus fit, uti supra de loco diximus ; debet eo momento, quo saltus confici dicitur, haberi simul status duplex ; qui cum haberi non possit : saltus itidem ille haberi omnino non potest. Sic, ut aliis utamur exemplis, distantia unius corporis ab alio mutari per saltum non potest, nee densitas, quia dux simul haberentur distantiae, vel duae densitates, quod utique sine replicatione haberi non potest ; caloris itidem, & frigoris mutatio in thermometris, ponderis atmosphaerae mutatio in barometris, non fit per saltum, quia binae simul altitudines mercurii in instrumento haberi deberent eodem momento temporis, quod fieri utique non potest ; cum quovis momento determinate unica altitude haberi debeat, ac unicus determinatus caloris gradus, vel frigoris ; quae quidem theoria innumeris casibus pariter aptari potest. 52. Contra hoc argumentum videtur primo aspectu adesse aliquid, quod ipsum pforsus non esse conjun- evertat, & tamen ipsi illustrando idoneum est maxime. Videtur nimirum inde erui, gend s in creatione • •« M • • • • o • • £»• • • J & annihiiatione, ac impossibilem esse & creationem rei cujuspiam, Scintentum. 01 enim conjungendus est ejus soiutio. postremus terminus praecedentis seriei cum primo sequentis ;" in ipso transitu a non esse ad esse, vel vice versa, debebit utrumque conjungi, ac idem simul erit, & non erit, quod est absurdum. Responsio in promptu est. Seriei finita; realis, & existentis, reales itidem, & existentes termini esse debent ; non vero nihili, quod nullas proprietates habet, quas exigat, Hinc si realium statuum seriei altera series realium itidem statuum succedat, quae non sit communi termino conjuncta ; bini eodem momento debebuntur status, qui nimirum sint bini limites earundem. At quoniam non esse est merum nihilum ; ejusmodi series limitem nullum extremum requirit, sed per ipsum esse immediate, & directe excluditur. Quamobrem primo, & postremo momento temporis ejus continui, quo res est, erit utique, nee cum hoc esse suum non esse conjunget simul ; at si densitas certa per horam duret, turn momento temporis in aliam mutetur duplam, duraturam itidem per alteram sequentem horam ; momento temporis, [25] quod horas dirimit, binae debebunt esse densitates simul, nimirum & simplex, & dupla, quae sunt reales binarum realium serierum termini. Objectio ab esse, & A THEORY OF NATURAL PHILOSOPHY 65 considered them, cannot be contiguous, as I explained above. Wherefore in the first case, at all those infinite intermediate instants the body would be nowhere at all ; in the second case, it would be at the same instant in two different places & so there would be replication. In the third case, there would not only occur replication in respect of these two instants but for all those intermediate to them as well, in all of which the body would forsooth be in two places at the same time. Since then a body that exists can never be nowhere, nor in several places at one & the same time, there can certainly be no alteration of path & no sudden change. 51. The same thing can be visualized better with the aid of Geometry. illustration of this Let times be represented by the straight line AB, & diverse states of any thing by SSyT^STS ordinates drawn to meet the lines CD, EF, which are discontinuous at some point. If the reasoning being ordmates DG, EH are drawn, either the point H will fall after the point G, as in Fig. 5 ; or it will coincide with it, as in Fig. 6 ; or it will fall before it, as in Fig. 7. In the first case, no ordinate will correspond to any one of the points of the straight line GH ; in the second case, GD and HE would correspond to the same point G ; in the third case, two ordinates, HI, HE, would correspond to the same point H, two, GD, GK, to the same point G, and two, LM, LN, to any intermediate point L. Now the ordinate is some relation as regards distance, which a point on the curve bears to the point on the axis that corresponds with it ; & thus, when two points of the curve lie in the same straight line perpendicular to the axis, we have two ordinates corresponding to the same point of the axis. Wherefore, if the thing in question can neither be without some state at each instant, nor is it possible that there should be two states at the same time, then it necessarily follows that the sudden change cannot be made. For this sudden change, if it is bound to happen, would take place at the two instants G & H, which immediately succeed the one the other without any direct gap between them ; this is quite impossible, from the very nature of a limit, which should be the same for,& common to, both the antecedents & the consequents in a continuous set, as has been said. The same thing happens in any series of real things ; as in this case there cannot be a finite line without a first & last point, each to be a boundary to it, neither can there be a surface without a line. Hence it comes about that in the case of Fig. 6 two ordinates must necessarily correspond to the same point. Thus, in any finite real series of states, there must of necessity be a first term & a last ; & so if a sudden change is made, as we said above with regard to position, there must be at the instant, at which the sudden change is said to be accomplished, a twofold state at one & the same time. Now since this can never happen, it follows that this sudden change is also quite impossible. Similarly, to make use of other illustrations, the distance of one body from another can never be altered suddenly, no more can its density ; for there would be at one & the same time two distances, or two densities, a thing which is quite impossible without replication. Again, the change of heat, or cold, in thermometers, the change in the weight of the air in barometers, does not happen suddenly ; for then there would necessarily be at one & the same time two different heights for the mercury in the instrument ; & this could not possibly be the case. For at any given instant there must be but one height, & but one definite degree of heat, & but one definite degree of cold ; & this argument can be applied just as well to innu- merable other cases. 52. Against this argument it would seem at first sight that there is something ready to hand which overthrows it altogether ; whilst as a matter of fact it is peculiarly fitted to together of existence exemplify it. It seems that from this argument it follows that both the creation of any * non-existence a.t •> • „ • i • • -11 rf >r T i <• -i i • the time of creation thing, & its destruction, are impossible, r or, it the last term of a series that precedes is to Or annihilation ; & be connected with the first term of the series that follows, then in the passage from a state its solution. of existence to one of non-existence, or vice versa, it will be necessary that the two are connected together ; & then at one & the same time the same thing will both exist & not exist, which is absurd. The answer to this is immediate. For the ends of a finite series that is real & existent must themselves be real & existent, not such as end up in absolute nothing, which has no properties. Hence, if to one series of real states there succeeds another series of real states also, which is not connected with it by a common term, then indeed there must be two states at the same instant, namely those which are their two limits. But since non-existence is mere nothing, a series of this kind requires no last limiting term, but is immediately & directly cut off by fact of existence. Wherefore, at the first & at the last instant of that continuous interval of time, during which the matter exists, it will certainly exist ; & its non-existence will not be connected with its existence simultaneously. On the other hand if a given density persists for an hour, & then is changed in an instant of time into another twice as great, which will last for another hour ; then in that instant of time which separates the two hours, there would have to be two densities at one & the same time, the simple & the double, & these are real terms of two real series. 66 PHILOSOPHIC NATURALIS THEORIA Unde hue transfer- enda solutio ipsa. Solutio petita ex geometrico exem- plo. Solutio physica atione. ex meta- consider- Illustratio ulterior geometrica. Applicatio ad crea- tionem, & annihi- lationem. D F i \ F D f m m* \ G G' P L 5 \ MJVI, ' A B C E H H'E'C7 FIG. 8. 53. Id ipsum in dissertatione De lege virium in Natura existentium satis, ni fallor, luculenter exposui, ac geometricis figuris illustravi, adjectis nonnullis, quae eodem recidunt, & quae in applicatione ad rem, de qua agimus, & in cujus gratiam haec omnia ad legem con- tinuitatis pertinentia allata sunt, proderunt infra ; libet autem novem ejus dissertationis numeros hue transferre integros, incipiendo ab octavo, sed numeros ipsos, ut & schematum numeros mutabo hie, ut cum superioribus consentiant. 54. " Sit in fig. 8 circulus GMM'wz, qui referatur ad datam rectam AB per ordinatas HM ipsi rectae perpendiculares ; uti itidem perpendiculares sint binae tangentes EGF, E'G'F'. Concipiantur igitur recta quaedam indefinita ipsi rectse AB perpendicularis, motu quodam continuo delata ab A ad B. Ubi ea habuerit, positionem quamcumque GD, quae praecedat tangentem EF, vel C'D', quae consequatur tangentem E'F' ; ordinata ad circulum nulla erit, sive erit impossibilis, & ut Geometrae loquuntur, imaginaria. Ubicunque autem ea sit inter binas tangentes EGF, E'G'F', in HI, HT, occurret circulo in binis punctis M, m, vel M', m', & habebitur valor ordinate HM, HOT, vel H'M', H'm'. Ordinata quidem ipsa respondet soli inter- vallo EE' : & si ipsa linea AB referat tempus ; momentum E est limes inter tempus praecedens continuum AE, quo ordinata non est, & tempus continuum EE' subsequens, quo ordinata est ; punc- tum E' est limes inter tempus praecedens EE', quo ordinata est, & subsequens E'B, quo non est. Vita igitur quaedam ordinatae est tempus EE' ; ortus habetur in E, interitus in E'. Quid autem in ipso ortu, & interitu ? Habetur-ne quoddam esse ordinatas, an non esse ? Habetur utique esse, nimi- rum EG, vel E'G', non autem non esse. Oritur tota finitae magnitudinis ordinata EG, interit tota finite magnitudinis E'G', nee tamen ibi conjungit esse, & non esse, nee ullum absurdum secum trahit. Habetur momento E primus terminus seriei sequentis sine ultimo seriei praecedentis, & habetur momento E' ultimus terminus seriei praecedentis sine primo termino seriei sequentis." 55. " Quare autem id ipsum accidat, si metaphysica consideratione rem perpendimus, statim patebit. Nimirum veri nihili nullae sunt verae proprietates : entis realis verae, & reales proprietates sunt. Quaevis realis series initium reale debet, & finem, sive primum, & ultimum terminum. Id, quod non est, nullam habet veram proprietatem, nee proinde sui generis ultimum terminum, aut primum exigit. Series praecedens ordinatae nullius, ultimum terminum non [26] habet, series consequens non habet primum : series realis contenta intervallo EE', & primum habere debet, & ultimum. Hujus reales termini terminum ilium nihili per se se excludunt, cum ipsum esse per se excludat non esse." 56. " Atque id quidem manifestum fit magis : si consideremus seriem aliquam praecedentem realem, quam exprimant ordinatae ad lineam continuam PLg, quae respondeat toti tempori AE ita, ut cuivis momento C ejus temporis respondeat ordinata CL. Turn vero si momento E debeat fieri saltus ab ordinata Eg ad ordinatam EG : necessario ipsi momento E debent respondere binae ordinatae EG, Eg. Nam in tota linea PLg non potest deesse solum ultimum punctum g ; cum ipso sublato debeat adhuc ilia linea terminum habere suum, qui terminus esset itidem punctum : id vero punctum idcirco fuisset ante contiguum puncto g, quod est absurdum, ut in eadem dissertatione De Lege Continuitatis demonstravimus. Nam inter quodvis punctum, & aliud punctum linea aliqua interjacere debet ; quae si non inter jaceat ; jam ilia puncta in unicum coalescunt. Quare non potest deesse nisi lineola aliqua gL ita, ut terminus seriei praecedentis sit in aliquo momento C praecedente momentum E, & disjuncto ab eo per tempus quoddam continuum, in cujus temporis momentis omnibus ordi'nata sit nulla." 57. " Patet igitur discrimen inter transitum a vero nihilo, nimirum a quantitate imaginaria, ad esse, & transitum ab una magnitudine ad aliam. In primo casu terminus nihili non habetur ; habetur terminus uterque seriei veram habentis existentiam, & potest quantitas, cujus ea est series, oriri, vel occidere quantitate finita, ac per se excludere non esse. In secundo casu necessario haberi debet utriusque seriei terminus, alterius nimirum postre- mus, alterius primus. Quamobrem etiam in creatione, & in annihilatione potest quantitas oriri, vel interire magnitudine finita, & primum, ac ultimum esse erit quoddam esse, quod secum non conjunget una non esse. Contra vero ubi magnitude realis ab una quantitate ad A THEORY OF NATURAL PHILOSOPHY 67 c*. I explained this very point clearly enough, if I mistake not, in my dissertation The s0"166 from n i • • • JIT- . • ' . • a T -11 j v i ... • i A 'IT ^ which the solution D,? lege vmum in Natura existentium, & 1 illustrated it by geometrical figures ; also I made u to be borrowed. some additions that reduced to the same thing. These will appear below, as an application to the matter in question ; for the sake of which all these things relating to the Law of Continuity have been adduced. It is allowable for me to quote in this connection the whole of nine articles from that dissertation, beginning with Art. 8 ; but I will here change the numbering of the articles, & of the diagrams as well, so that they may agree with those already given. 54. " In Fig. 8, let GMM'm be a circle, referred to a given straight line AB as axis, by Sotoion derived means of ordinates HM drawn perpendicular to that straight line ; also let the two tan- exampief" gents EGF, E'G'F' be perpendiculars to the axis. Now suppose that an unlimited straight line perpendicular to the axis AB is carried with a continuous motion from A to B. When it reaches some such position as CD preceding the tangent EF, or as C'D' subsequent to the tangent E'F', there will be no ordinate to the circle, or it will be impossible &, as the geometricians call it, imaginary. Also, wherever it falls between the two tangents EGF, E'G'F', as at HI or HT, it will meet the circle in two points, M, m or M', m' ; & for the value of the ordinate there will be obtained HM & Hm, or H'M' & H'm'. Such an ordinate will correspond to the interval EE' only ; & if the line AB represents time, the instant E is the boundary between the preceding continuous time AE, in which the ordinate does not exist, £ the subsequent continuous time EE', in which the ordinate does exist. The point E' is the boundary between the preceding time EE', in which the ordinate does exist, & the subsequent time E'B, in which it does not ; the lifetime, as it were, of the ordinate, is EE' ; its production is at E & its destruction at E'. But what happens at this production & destruction ? Is it an existence of the ordinate, or a non-existence I Of a truth there is an existence, represented by EG & E'G', & not a non-existence. The whole ordinate EG of finite magnitude is produced, & the whole ordinate E'G' of finite magnitude is destroyed; & yet there is no connecting together of the states of existence & non-existence, nor does it bring in anything absurd in its train. At the instant E we get the first term of the sub- sequent series without the last term of the preceding series ; & at the instant E' we have the last term of the preceding series without the first term of the subsequent series." 55. " The reason why this should happen is immediately evident, if we consider the Sol«tion from a matter metaphysically. Thus, to absolute nothing there belong no real properties ; but Sderatwn!* the properties of a real absolute entity are also real. Any real series must have a real beginning & end, or a first term & a last. That which does not exist can have no true property ; & on that account does not require a last term of its kind, or a first. The preceding series, in which there is no ordinate, does not have a last term ; & the subsequent series has likewise no first term ; whilst the real series contained within the interval EE' must have both a first term & a last term. The real terms of this series of themselves exclude the term of no value, since the fact of existence of itself excludes non-existence" 56. " This indeed will be still more evident, if we consider some preceding series of Further illustration i • • 11 i • i i i • T.T r „ i i • by geometry. real quantities, expressed by the ordinates to the curved line PLg ; & let this curve correspond to the whole time AE in such a way that to every instant C of the time there corresponds an ordinate CL. Then, if at the instant E there is bound to be a sudden change from the ordinate Eg to the ordinate EG, to that instant E there must of necessity correspond both the ordinates EG, Eg. For it is impossible that in the whole line PLg the last point alone should be missing ; because, if that point is taken away, yet the line is Bound to have an end to it, & that end must also be a point ; hence that point would be before & contiguous to the point g ; & this is absurd, as we have shown in the same dissertation De Lege Continuitatis. For between any one point & any other point there must lie some line ; & if such a line does not intervene, then those points must coalesce into one. Hence nothing can be absent, except it be a short length of line gL, so that the end of the series that precedes occurs at some instant, C, preceding the instant E, & separated from it by an interval of continuous time, at all instants of which there is no ordinate." 157. "Evidently, then, there is a distinction between passing from absolute nothing, Application to crea- •f' . '' . ... °. , . Y tion& annihilation. i.e., from an imaginary quantity, to a state of existence, & passing from one magnitude to another. In the first case the term which is naught is not reckoned in ; the term at either end of a series which has real existence is given, & the quantity, of which it is the series, can be produced or destroyed, finite in amount ; & of itself it will exclude non- existence. In the second case, there must of necessity be an end to either series, namely the last of the one series & the first of the other. Hence, in creation & annihilation, a quantity can be produced or destroyed, finite in magnitude ; & the first & last state of existence will be a state of existence of some kind ; & this will not associate with itself a state of non-existence. But, on the other hand, where a real magnitude is bound 68 PHILOSOPHIC NATURALIS THEORIA Aliquando videri nihtium id, quod est aliquid. Ordinatam nullam, ut & distantiam nullam existentium esse compenetra- tionem. Ad idem pertinere seriei realis genus earn distan t i a m nullam, & aliquam. Alia, quje videntur nihil, & sunt ali- quid : discrimen inter radicem ima- ginariam, & zero. aliam transire debet per saltum ; momento temporis, quo saltus committitur, uterque terminus haberi deberet. Manet igitur illaesum argumentum nostrum metaphysicum pro exclusione saltus a creatione & annihilatione, sive ortu, & interitu." 58. "At hie illud etiam notandum est ; quoniam ad ortum, & interitum considerandum geometricas contemplationes assumpsimus, videri quidem prima fronte, aliquando etiam realis seriei terminum postremum esse nihilum ; sed re altius considerata, non erit vere nihilum ; sed status quidam itidem realis, & ejusdem generis cum prsecedentibus, licet alio nomine insignitus." [27] 59. " Sit in Fig. 9. Linea AB, ut prius, ad quam linea qusedam PL deveniat in G (pertinet punctum G ad lineam PL, E ad AB continuatas, & sibi occurrentes ibidem), & sive pergat ultra ipsam in GM', sive retro resiliat per GM'. Recta CD habebit ordinatam CL, quas evanescet, ubi puncto C abeunte in E, ipsa CD abibit in EF, turn in positione ulteriori rectse perpendicularis HI, vel abibit in nega- tivam HM, vel retro positiva regredietur in HM'. Ubi linea altera cum altera coit, & punctum E alterius cum alterius puncto G congreditur, ordinata CL videtur abire in nihilum ita, ut nihilum, quemadmodum & supra innuimus, sit limes quidam inter seriem ordinatarum positivarum CL, & negativarum HM ; vel positivarum CL, & iterum posi- tivarum HM'. Sed, si res altius considere- tur ad metaphysicum conceptum reducta, in situ EF non habetur verum nihilum. In situ CD, HI habetur distantia quaedam punctorum C, L ; H, M : in situ EF habetur eorundem punctorum compene- tratio. Distantia est relatio quaedam FJG binorum modorum, quibus bina puncta existunt ; compenetratio itidem est relatio binorum modorum, quibus ea existunt, quae compenetratio est aliquid reale ejusdem prorsus generis, cujus est distantia, constituta nimirum per binos reales existendi modos." 60. " Totum discrimen est in vocabulis, quae nos imposuimus. Bini locales existendi modi infinitas numero relationes possunt constituere, alii alias. Hae omnes inter se & differunt, & tamen simul etiam plurimum conveniunt ; nam reales sunt, & in quodam genere congruunt, quod nimirum sint relationes ortae a binis localibus existendi modis. Diversa vero habent nomina ad arbitrarium instituta, cum alise ex ejusmodi relationibus, ut CL, dicantur distantiae positivae, relatio EG dicatur compenetratio, relationes HM dicantur distantiae negativse. Sed quoniam, ut a decem palmis distantiae demptis 5, relinquuntur 5, ita demptis aliis 5, habetur nihil (non quidem verum nihil, sed nihil in ratione distantiae a nobis ita appellatae, cum remaneat compenetratio) ; ablatis autem aliis quinque, remanent quinque palmi distantiae negativae ; ista omnia realia sunt, & ad idem genus pertinent ; cum eodem prorsus modo inter se differant distantia palmorum 10 a distantia palmorum 5, haec a distantia nulla, sed reali, quas compenetrationem importat, & haec a distantia negativa palmorum 5. Nam ex prima ilia quantitate eodem modo devenitur ad hasce posteriores per continuam ablationem palmorum 5. Eodem autem pacto infinitas ellipses, ab infinitis hyperbolis unica interjecta parabola discriminat, quae quidem unica nomen peculiare sortita est, cum illas numero infinitas, & a se invicem admodum discrepantes unico vocabulo com- plectamur ; licet altera magis oblonga ab altera minus oblonga plurimum itidem diversa sit." [28] 61. " Et quidem eodem pacto status quidam realis est quies, sive perseverantia in eodem modo locali existendi ; status quidam realis est velocitas nulla puncti existentis. nimirum determinatio perseverandi in eodem loco ; status quidam realis puncti existentis est vis nulla, nimirum determinatio retinendi praecedentem velocitatem, & ita porro ; plurimum haec discrepant a vero non esse. Casus ordinatae respondentis lineae EF in fig. 9, differt plurimum a casu ordinatae circuli respondentis lineae CD figurae 8 : in prima existunt puncta, sed compenetrata, in secunda alterum punctum impossible est. Ubi in solutione problematum devenitur ad quantitatem primi generis, problema determinationem peculiarem accipit ; ubi devenitur ad quantitatem secundi generis, problema evadit impossible ; usque adeo in hoc secundo casu habetur verum nihilum, omni reali proprietate carens ; in illo primo habetur aliquid realibus proprietatibus praeditum, quod ipsis etiam solutionibus problematum, & constructionibus veras sufficit, & reales determinationes ; cum realis, non imaginaria sit radix equationis cujuspiam, quae sit = o, sive nihilo aequalis." A THEORY OF NATURAL PHILOSOPHY 69 to pass suddenly from one quantity to another, then at the instant in which the sudden change is accomplished, both terms must be obtained. Hence, our argument on metaphysical grounds in favour of the exclusion of a sudden change from creation or annihilation, or production & destruction, remains quite unimpaired." 58. " In this connection the following point must be noted. As we have used geometrical Sometimes what is ideas for the consideration of production & destruction, it seems also that sometimes reallysomethingap- the last term of a real series is nothing. But if we go deeper into the matter, we find that it is not in reality nothing, but some state that is also real and of the same kind as those that precede it, though designated by another name." 59. " In Fig. 9, let AB be a line, as before, which some line PL reaches at G (where the When the ordinate point G belongs to the line PL, & E to the line AB, both being produced to meet one whe^thT'dlst^n'13 another at this point) ; & suppose that PL either goes on beyond the point as GM, or between two exis- recoils along GM'. Then the straight line CD will contain the ordinate CL, which will ^ tJ1™gs .u no" _ & „, . . . ' . thing, there is com- vanish when, as the point L, gets to H, L-D attains the position r,r ; & after that, in the penetration. further position of the perpendicular straight line HI, will either pass on to the negative ordinate HM or return, once more positive, to HM'. Now when the one line meets the other, & the point E of the one coincides with the point G of the other, the ordinate CL seems to run off into nothing in such a manner that nothing, as we remarked above, is a certain boundary between the series of positive ordinates CL & the negative ordinates HM, or between the positive ordinates CL & the ordinates HM' which are also positive. But if the matter is more deeply considered & reduced to a metaphysical concept, there is not an absolute nothing in the position EF. In the position CD, or HI, we have given a certain distance between the points C,L, or H,M ; in the position EF, there is compenetration of these points. Now distance is a relation between the modes of existence of two points ; also compenetration is a relation between two modes of existence ; & this compenetration is something real of the very same nature as distance, founded as it is on two real modes of existence." 60. " The whole difference lies in the words that we have given to the things in question. ™s ' no ' distance Two local modes of existence can constitute an infinite number of relations, some of one kmdT^f °series "of sort & some of another. All of these differ from one another, & yet agree with one real quantities as •i • i • i j r ia • • j • i • • j j ' some ' distance. another in a high degree ; ior they are real & to a certain extent identical, since indeed they are all relations arising from a pair of local modes of existence. But they have different names assigned to them arbitrarily, so that some of the relations of this kind, as CL, are called positive distances, the relation EG is called compenetration, & relations like HM are called negative distances. But, just as when five palms of distance are taken away from ten palms, there are left five palms, so when five more are taken away, there is nothing left (& yet not really nothing, but nothing in comparison with what we usually call distance ; for compenetration is left). Again, if we take away another five, there remain five palms of negative distance. All of these are real & belong to the same class ; for they differ amongst themselves in exactly the same way, namely, the distance of ten palms from the distance of five palms, the latter from ' no ' distance (which however is something real that denotes compenetration), & this again from a negative distance of five palms. For starting with the first quantity, the others that follow are obtained in the same manner, by a continual subtraction of five palms. In a similar manner a single intermediate parabola discriminates between an infinite number of ellipses & an infinite number of hyperbolas ; & this single curve receives a special name, whilst under the one term we include an infinite number of them that to a certain extent are all different from one another, although one that is considerably elongated may be very different from another that is less elongated." 61. "In the same way, rest, i.e., a perseverance in the same mode of local existence, other things that is some real state ; so is ' no ' velocity a real state of an existent point, namely, a propensity ^ndVet^re^eaJi^ to remain in the same place ; so also is ' no ' force a real state of an existent point, namely, something ; d i s- a propensity to retain the velocity that it has already; & so on. All these differ from a'~" a state of non-existence in the highest degree. The case of the ordinate corresponding & zero/ to the line EF in Fig. 9 differs altogether from the case of the ordinate of the circle corresponding to the line CD in Fig. 8. In the first there exist two points, but there is compenetration of these points ; in the other case, the second point cannot possibly exist. When, in the solution of problems, we arrive at a quantity of the first kind, the problem receives a special sort of solution ; but when the result is a quantity of the second kind, the problem turns out to be incapable of solution. So much indeed that, in this second case, there is obtained a true nothing that lacks every real property ; in the first case, we get something endowed with real properties, which also supplies true & real values to the solutions & constructions of the problems. For the root of any equation that = o, or is equal to nothing, is something that is real, & is not an imaginary thing." 70 PHILOSOPHIC NATURALIS THEORIA Conciusip prosolu- fa. " Firmum igitur manebit semper. & stabile, seriem realem quamcunque. quas tione ejus objec- . ~ °. , , , v » „ ... a i • r contmuo tempore finito duret, debere habere £ primum prmcipium, & ultimum nnem realem, sine ullo absurdo, & sine conjunctione sui esse cum non esse, si forte duret eo solo tempore : dum si prascedenti etiam exstitit tempore, habere debet & ultimum terminum seriei praecedentis, & primum sequentis, qui debent esse unicus indivisibilis communis limes, ut momentum est unicus indivisibilis limes inter tempus continuum praecedens, & subsequens. Sed haec de ortu, & interitu jam satis." Apphcatio leg is ft- ijt igitur contrahamus iam vela, continuitatis lex & inductione, & metaphysico contmuitatis ad J , ° , . . • i • • • • • . .. . . r ' . coiiisionem corpo- argumento abunde nititur, quas idcirco etiam in velocitatis commumcatione retmeri omnmo rum- debet, ut nimirum ab una velocitate ad aliam numquam transeatur, nisi per intermedias velocitates omnes sine saltu. Et quidem in ipsis motibus, & velocitatibus inductionem habuimus num. 39, ac difficultates solvimus num. 46, & 47 pertinentes ad velocitates, quae videri possent mutatse per saltum. Quod autem pertinet ad metaphysicum argumentum, si toto tempore ante contactum subsequentis corporis superficies antecedens habuit 12 gradus velocitatis, & sequenti 9, saltu facto momentaneo ipso initio contactus ; in ipso momento ea tempora dirimente debuisset habere & 12, & 9 simul, quod est absurdum. Duas enim velocitates simul habere corpus non potest, quod ipsum aliquanto diligentius demonstrabo. DUO velocitatum g, Velocitatis nomen, uti passim usurpatur a Mechanicis, asquivocum est; potest genera, potentials, T r r . T. . r & actuaiis. enim sigmncare velocitatem actuaiem, quas nimirum est relatio quaedam in motu asquabm spatii percursi divisi per tempus, quo percurritur ; & potest significare [29] quandam, quam apto Scholiasticorum vocabulo potentialem appello, quae nimirum est determinatio, ad actuaiem, sive determinatio, quam habet mobile, si nulla vis mutationem inducat, percur- rendi motu asquabili determinatum quoddam spatium quovis determinato tempore, quas quidem duo & in dissertatione De Viribus Fivis, & in Stayanis Supplements distinxi, distinctione utique .necessaria ad aequivocationes evitandas. Prima haberi non potest momento temporis, sed requirit tempus continuum, quo motus fiat, & quidem etiam motum aequabilem requirit ad accuratam sui mensuram ; secunda habetur etiam momento quovis determinata ; & hanc alteram intelligunt utique Mechanici, cum scalas geometricas effor- mant pro motibus quibuscunque difformibus, sive abscissa exprimente tempus, & ordinata velocitatem, utcunque etiam variata