NEWTON S PRINCIPIA. THE MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY, BY SIR ISAAC NEWTON *prt 2

one which hath its beginning and its end in the knowledge of Him whose glory the heavens declare, and whose handiwork the firmament showeth forth. The introduction of the pure and lofty doctrines of the PRINCIPIA was perseveringly resisted. Descartes, with his system of vortices, had sown plausibly to the imagination, and error had struck down deeply, and shot up luxuriantly, not only in the popular, but in the scientific mind. Besides the idea in itself so simple and so grand that the great masses of the planets were 38 LIFE OF SIR ISAAC NEWTON. suspended in empty space, and retained in their orbits by an in visible influence residing in the sun was to the ignorant a thing inconceivable, and to the learned a revival of the occult qualities of the ancient physics. This remark applies particularly to the continent. Leibnitz misapprehended ; Huygens in part rejected ; John Bernouilli opposed ; and Fontenelle never received the doc trines of the PRINCIPIA. So that, the saying of Voltaire is prob ably true, that though Newton survived the publication of his great work more than forty years, yet, at the time of his death, lie had not above twenty followers out of England. But in England, the reception of our author s philosophy was rapid and triumphant. His own labours, while Lucasian Pro fessor ; those of his successors in that Chair Whiston and Saunderson ; those of Dr. Samuel Clarke, Dr. Laughton, Roger Cotes, and Dr. Bentley ; the experimental lectures of Dr. Keill and Desaguliers ; the early and powerful exertions of David Gregory at Edinburgh, and of his brother James Gregory at St. Andrew s, tended to diffuse widely in England and Scotland a knowledge of, and taste for the truths of the PRINCIPIA. Indeed, its mathematical doctrines constituted, from the first, a regular part of academical instruction ; while its physical truths, given to the public in popular lectures, illustrated by experiments, had, before the lapse of twenty ) ( ar.s, become familiar to, and adopted by the general mind. Pemberton s popular " View of Sir Isaac Newton s Philosophy" was published, in 1728 ; and the year after ward, an English translation of the PRINCIPIA, and System of the World, by Andrew Motte. And since that period, the labours of Le Seur and Jacquier, of Thorpe, of Jebb, of Wright and others have greatly contributed to display the most hidden treasures of the PRINCIPIA. About the time of the publication of the Principia, James II., bent on re-establishing the Romish Faith, had, among other ille gal acts, ordered by mandamus, the University of Cambridge to confer the degree of Master of Arts upon an ignorant monk. Obedience to this mandate was resolutely refused. Newton was one of the nine delegates chosen to defend the independence of the University. They appeared before the High Court ; and LIFE OF SIR ISAAC NEWTON. 39 successfully : the king abandoned his design. The prominent part which our author took in these proceedings, and his eminence in the scientific world, induced his proposal as one of the parlia mentary representatives of the University. He was elected, in 1688, and sat in the Convention Parliament till its dissolution. After the first year, however, he seems to have given little or no attention to his parliamentary duties, being seldom absent from the University till his appointment in the Mint, in 1695. Newton began his theological researches sometime previous to 1691 ; in the prime of his years, and in the matured vigour of his intellectual powers. From his youth, as we have seen, he had devoted himself with an activity the most unceasing, and an energy almost superhuman to the discovery of physical truth ; giving to Philosophy a new foundation, and to Science a new temple. To pass on, then, from the consideration of the material, more directly to that of the spiritual, was a natural, nay, with so large and devout a soul, a necessary advance. The Bible was to him of inestimable worth. In the elastic freedom, which a pure and unswerving faith in Him of Nazareth gives, his mighty facul ties enjoyed the only completest scope for development. His original endowment, however great, combined with a studious application, however profound, would never, without this libera tion from the dominion of passion and sense, have enabled him to attain to that wondrous concentration and grasp of intellect, for which Fame has as yet assigned him no equal. Gratefully he owned, therefore, the same Author in the Book of Nature and the Book of Revelation. These were to him as drops of the same unfathomable ocean ; as outrayings of the same inner splendour ; as tones of the same ineffable voice ; as segments of the same infinite curve. "With great joy he had found himself enabled to proclaim, as an interpreter, from the hieroglyphs of Creation, the existence of a God : and now, with greater joy, and in the fulness of his knowledge, and in the fulness of his strength, he laboured to make clear, from the utterances of the inspired Word, the far mightier confirmations of a Supreme Good, in all its glorious amplitude of Being and of Attribute ; and to bring the infallible workings thereof plainly home to the understandings and the 40 LIFE OF SIR ISAAC NEWTON. affections of his fellow-men ; and finally to add the weight of his own testimony in favour of that Religion, whose truth is now. in deed, " girded with the iron and the rock of a ponderous and co lossal demonstration." His work, entitled, OBSERVATIONS UPON THE PROPHECIES OF HOLY WRIT, PARTICULARLY THE PROPHECIES OF DANIEL AND THE APOCALYPSE OF ST. JOHN, first published in London, in 1733 4to. consists of two parts : the one devoted to the Prophecies oi Daniel, and the other to the Apocalypse of St. John. In the first part, he treats concerning the compilers of the books of the Old Testament ; of the prophetic language ; of the vision of the four beasts ; of the kingdoms represented by the feet of the image composed of iron and clay ; of the ten kingdoms repre sented by the ten horns of the beast ; of the eleventh horn of Daniel s fourth beast ; of the power which should change times and laws ; of the kingdoms represented in Daniel by the ram and he-goat ; of the prophecy of the seventy weeks ; of the times of the birth and passion of Christ ; of the prophecy of the Scripture of Truth ; of the king who doeth according to his will, and magnified himself above every god, and honoured Mahuzzims, and regarded not the desire of women ; of the Mahuzzim, hon oured by the king who doeth according to his will. In the sec ond part, he treats of the time when the Apocalypse was written , of the scene of the vision, and the relation which the Apocalypse has to the book of the law of Moses, and to the worship of God in the temple ; of the relation which the Apocalypse has to the prophecies of Daniel, and of the subject of the prophecy itself Newton regards the prophecies as given, not for the gratification of man s curiosity, by enabling him to foreknow ; but for his con viction that the world is governed by Providence, by witnessing their fulfilment. Enough of prophecy, he thinks, has already been fulfilled to afford the diligent seeker abundant evidence of God s providence. The whole work is marked by profound erudition, sagacity and argument. And not less learning, penetration and masterly reasoning are conspicuous in his HISTORICAL ACCOUNT OF Two NOTABLE CORRUPTIONS OF SCRIPTURES IN A LETTER TO A FRIEND. This LIFE OF SIR ISAAC NEWTON. 41 Treatise, first accurately published in Dr. Horsley s edition of his works, relates to two texts : the one, 1 Epistle of St. John v. 7 ; the other, 1 Epistle of St. Paul to Timothy iii. 16. As this work had the effect to deprive the advocates of the doctrine of the Trinity of two leading texts, Newton has been looked upon as an Arian ; but there is absolutely nothing in his writings to warrant such a conclusion. His regaining theological works consist of the LEXICON PROPHETICUM, which was left incomplete ; a Latin Dissertation on the sacred cubit of the Jews, which was translated into English, and published, in 1737. among the Miscellaneous Works of John Greaves ; and FOUR LETTERS addressed to Dr. Bentlty, contain ing some arguments in proof of a Deity. These Letters were dated respectively : 10th December, 1692 ; 17th January, 1693 ; 25th February, 1693; and llth February, 1693 the fourth bearing an earlier date than the third. The best faculties and the profoundest acquirements of our author are convincingly manifest in these lucid and powerful compositions. They were published in 1756, and reviewed by Dr. Samuel Johnson. Newton s religious writings are distinguished by their absolute freedom from prejudice. Everywhere, throughout them, there glows the genuine nobleness of soul. To his whole life, indeed, we may here fitly extend the same observation. He was most richly imbued with the very spirit of the Scriptures which he so delighted to study and to meditate upon. His was a piety, so fervent, so sincere and practical, that it rose up like a holy incense from every thought and act. His a benevolence that not only willed, but endeavoured the best for all. His a philanthropy that held in the embracings of its love every brother-man. His a toleration of the largest and the truest ; condemning per secution in every, even its mildest form ; and kindly encouraging each striving after excellence : .1 toleration that came not of indifference for the immoral and the impious met with their quick rebuke but a toleration that came of the wise humbleness and the Christian charity, which see, in the nothingness of self and the almightiness of TRUTH, no praise for the ablest, and no blame for th^ feeblest in their strugglings upward to light and life. 42 LIFE OF SIR ISAAC NEWTON, Tn the winter of 1691-2, on returning from chapel, one morn ing, Newton foima tnat a favourite little dog, called Diamond, had overturned a lighted taper on his desk, and that several pa pers containing the results of certain optical experiments, were nearly consumed. His only exclamation, on perceiving his loss, was, " Oh Diamond, Diamond, little knowest thou the mischiel thou hast done," Dr. Brewster, in his life of our author, gives the following extract from the manuscript Diary of Mr. Abraham De La Pryme. a student in the University at the time of this oc currence. " 1692. February, 3. What I heard to-day I must relate. There is one Mr. Newton (whom I have very oft seen), Fellow of Trinity College, that is mighty famous for his learning, being a most excellent mathematician, philosopher, divine, &c. He has been Fellow of the Royal Society these many years ; and among other very learned books and tracts, he : s written one upon the mathe matical principles of philosophy, which has given him a mighty name, he having received, especially from Scotland, abundance of congratulatory letters for the same ; but of all the books he ever wrote, there was one of colours and light, established upon thou sands of experiments which he had been twenty years of making, and which had cost him many hundreds of pounds. This book which he vaiued so much, and which was so much talked of, had the ill luck to perish, and be utterly lost just when the learned author was almost at pitting a conclusion at the same, after this manner : In a winter s morning, leaving it among his other papers on his study table while he went to chapel, the candle, which he had unfortunately left burning there, too, catched hold by some means of other papers, and they fired the aforesaid book, and ut terly consumed it and several other valuable writings ; arid which is most wonderful did no further mischief. But when Mr. New ton came from chapel, and had seen what was done, every one thought he would have run mad, he was so troubled thereat that he was not himself for a month after. A long account of this his system of colours you may find in the Transactions of the Royal Society, which he had sent up to them long before this sad mis chance happened unto him." LIFE OF SIR ISAAC NEWTON. 43 It will be borne in mind that all of Newton s theological wri tings, with the exception of the Letters to Dr. Bentley, were composed before this event which, we must conclude, from Pryme s words, produced a serious impression upon our author for about a month. But M. Biot, in his Life of Newton, relying on a memorandum contained in a small manuscript Journal of Huygens, declares this occurrence to have caused a deran-gement of New ton s intellect. M. Blot s opinions and deductions, however, as well as those of La Place, upon this subject, were based upon erroneous data, and have been overthrown by the clearest proof. There is not, in fact, the least evidence that Newton s reason was, for a single moment, dethroned ; on the contrary, the testimony is conclusive that he was, at all times, perfectly capable of carry ing on his mathematical, metaphysical and astronomical inquiries. Loss of sleep, loss of appetite, and irritated nerves will disturb somewhat the equanimity of the most serene ; and an act done, or language employed, under such temporary discomposure, is not a just criterion of the general tone and strength of a man s mind. As to the accident itself, we may suppose, whatever might have been its precise nature, that it greatly distressed him, and, still further, that its shock may have originated the train of nervous derangements, which afflicted him, more or less, for two years afterward. Yet, during this very period of ill health, we find him putting forth his highest powers. In 1692, he prepared for, and transmitted to Dr. Wallis the first proposition of the Treatise on Quadratures, with examples of it in first, second and third flux ions. He investigated, in the same year, the subject of haloes ; making and recording numerous and important observations rela tive thereto. Those profound and beautiful Letters to Dr. Bentley were written at the close of this and the beginning of the next year. In October, 1693, Locke, who was then about publishing a second edition of his work on the Human Understanding, request ed Newton to reconsider his opinions on innate ideas. And in 1694, he was zealously occupied in perfecting his lunar theory ; visiting Flamstead, at the Royal Observatory of Greenwich, in September, and obtaining a series of lunar observations ; and 14 LIFE OF SIR ISAAC NEWTON. commencing, in October, a correspondence with that distinguished practical Astronomer, which continued till 1698. We now arrive at the period when Newton permanently with drew from the seclusion of a collegiate, and entered upon a more active and public life. He was appointed Warden of the Mint, in 1695, through the influence of Charles Montague, Chancellor of the Exchequer, and afterward Earl of Halifax. The current roin of the nation had been adulterated and debased, and Mon tague undertook a re-coinage. Our author s mathematical and chemical knowledge proved eminently useful in accomplishing this difficult and most salutary reform. In 1699, he was pro moted to the Mastership of the Mint an office worth twelve or fifteen hundred pounds per annum, and which he held during the remainder of his life. He wrote, in this capacity, an official Re port on the Coinage, which has been published ; he also prepared a Table of Assays of Foreign Coins, which was printed at the end of Dr. Arbuthnot s Tables of Ancient Coins, Weights, and Measures, in 1727. Newton retained his Professorship at Cambridge till 1703. But he had, on receiving the appointment of Master of the Mint, in 1699, made Mr. Whiston his deputy, with all the emoluments of the office ; and, on finally resigning, procured his nomination to the vacant Chair. In January 1697, John Bernouilli proposed to the most distin guished mathematicians of Europe two problems for solution. Leibnitz, admiring the beauty of one of them, requested the time for solving it to be extended to twelve months twice the period originally named. The delay was readily granted. Newton, how ever, sent in, the day after he received the problems, a solution of them to the President of the Royal Society. Bernouilli obtained solutions from Newton, Leibinitz and the Marquis De L Hopital ; but Newton s though anonymous, he immediately recognised " tanquam ungue leonem" as the lion is known by his claw. We may mention here the famous problem of the trajectories proposed by Leibnitz, in 1716, for the purpose of "feeling the pulse of the English Analysts." Newton received the problem about five o clock in the afternoon, as he was returning from the LIFE OF SIR ISAAC NEWTON. 45 Mint ; and though it was extremely difficult and he himself much fatigued, yet he completed its solution, the same evening before he went to bed. The history of these problems affords, by direct comparison, a striking illustration of Newton s vast superiority of mind. That amazing concentration and grasp of intellect, of which we have spoken, enabled him to master speedily, and, as it were, by a single effort, those things, for the achievement of which, the many would essay utterly in vain, and the very, very few attain only after long and renewed striving. And yet, with a modesty as unparalleled as his power, he attributed his successes, not to any extraordinary sagacity, but solely to industry and patient thought. Mr- kept the subject of consideration constantly before him, and waited till the first dawning opened gradually into a full and clear light ; never quitting, if possible, the mental process till the object of it were wholly gained. He never allowed this habit of meditation to appear in his intercourse with society ; but in the privacy of his own chamber, or in the midst of his own family, he gave himself up to the deepest abstraction. Occupied with some interesting investigation, he would often sit down on his bedside, after he rose, and remain there, for hours, partially dressed. Meal-time would frequently come and pass unheeded ; so that, unless urgently reminded, he would neglect to take the re quisite quantity of nourishment. But notwithstanding his anx iety to be left undisturbed, he would, when occasion required, turn aside his thoughts, though bent upon the most intricate re search, and then, when leisure served, again direct them to the very point where they ceased to act : and this he seemed to ac complish not so much by the force of his memory, as by the force of his inventive faculty, before the vigorous intensity of which, no subject, however abstruse, remained long unexplored. Me was elected a member of the Royal Academy of Sciences at Paris, in 1699, when that distinguished Body were empowered, by a new charter, to admit a small number of foreign associates. In 1700, he communicated to Dr. Halley a description of his re flecting instrument for observing the moon s distance from the fixed stars. This description was published in the Philosophical 46 LIFE OF SIR ISAAC NEWTON, Transactions, in 1742. The instrument was the same as that produced by Mr. Hadley, in 1731, and which, under the name of Hadley s Quadrant, has been of so great use in navigation. On the assembling of the new Parliament, in 1701, Newton was re- elected one of the members for the University of Cambridge. In 1703, he was chosen President of the Royal Society of London, to which office he was annually re-elected till the period of his decease about twenty-five years afterward. Our author unquestionably devoted more labour to, and, in many respects, took a greater pride in his Optical, than his other discoveries. This science he had placed on a new and indestruc tible basis ; and he wished not only to build, but to perfect the costly and glowing structure. He had communicated, before the publication of the PRINCIPIA, his most important researches on light to the Royal Society, in detached papers which were inserted in successive numbers of the Transactions ; but he did not pub lish a connected view of these labours till 1704, when they appeared under the title of OPTICS : OR, A TREATISE ON THE REFLEXIONS, REFRACTIONS, INFLEXIONS AND COLOURS OF LIGHT. To this, but to no subsequent edition, were added two Mathematical Trea tises, entitled, TRACTATUS DUO DE SPECIEBUS ET MAGNITUDINE FIGURARUM cuRViLiNEARUM ; the one bearing the title TRACTATUS DE QUADRATURA CuRVARUM ; and the other, that of ENUMERATIO LINEARUM TERTII ORDiNis. The publication of these Mathemati cal Treatises was made necessary in consequence of plagiarisms from the manuscripts of them loaned by the author to his friends. Dr. Samuel Clarke published a Latin translation of the Optics, in in 1706 ; whereupon he was presented by Newton, as a mark of his grateful approbation, with five hundred pounds, or one hun dred pounds for each of his children. The work was afterward translated into French. It had a remarkably wide circulation, and appeared, in several successive editions, both in England and on the Continent. There is displayed, particularly on this Opti cal Treatise, the author s talent for simplifying and communica ting the profoundest speculations. It is a faculty rarely united to that of the highest invention. Newton possessed both ; and thus that mental perfectness which enabled him to create, to combine, LIFE OF SIR ISAAC NEWTON. 47 and to teach, and so render himself, not the "ornament" cnly; but inconceivably more, the pre-eminent benefactor of his species. The honour of knighthood v/as conferred on our author in 1705. Soon afterward, he was a candidate again for the Repre sentation of the University, but was defeated by a large majority. It is thought that a more pliant man was preferred by both min isters and electors. Newton was always remarkable for simplicity of dress, and his only known departure from it was on this oc casion, when he is said to have appeared in a suit of laced clothes. The Algebraical Lectures which he had, Juring nine years, delivered at Cambridge, were published by Whiston, in 1707, under the title of ARITHMETICS UNIVERSALIS, SINE DE COMPOSI TIONS ET RESOLUTIONS ARITHMETICA LIBER. This publication is said to have been a breach of confidence on Whiston s part. Mr. Ralphson, not long afterward, translated the work into English ; and a second edition of it, with improvements by the author, was issued at London, 1712, by Dr. Machin. Subsequent editions, both in English and Latin, with commentaries, have been published. In June, 1709, Newton intrusted the superintendence of a second edition of the PRINCIPIA to Roger Cotes, Plumian Pro fessor of Astronomy at Cambridge. The first edition had been sold off for some time. Copies of the work had become very rare, and could only be obtained at several times their original cost. A great number of letters passed oetween the author and Mr. Cotes during the preparation of the edition, which finally appeared in May, 1713. It had many alterations and improve ments, and was accompanied by an admirable Preface from the pen of Cotes. Our author s early Treatise, entitled, ANALYSIS PER EQUATIONES NUMERO TERMINORUM INFINITAS, as well as a small Tract, Gearing the title of METHODUS DIFFERENTIALS, was published, witn nis consent, in 1711. The former of these, and the Treatise De Quadratura Curvarum, translated into Englisn, witn a .arge com mentary, appeared in 1745. His work, entitled. ARTIS ANA LYTICS SPECIMINA, VEL GEOMETRIA ANALYTICA, was iirs; given to the world in the edition of Dr. Horsley, 1779. 48 LIFE OF SIR ISAAC NEWTON. It is a notable fact, in Newton s history, that he never volun* tarily published any one of his purely mathematical writings The cause of this unwillingness in some, and, in other instances, of his indifference, or, at least, want of solicitude to put forth his works may be confidently sought for in his repugnance to every thing like contest or dispute. But, going deeper than this aver sion, we find, underlying his whole character and running parallel with all his discoveries, that extraordinary humility which always preserved him in a position so relatively just to the behests of time and eternity, that the infinite value of truth, and the utter worthlessness of fame, were alike constantly present to him. Judging of his course, however, in its more temporary aspect, as bearing upon his immediate quiet, it seemed the most unfortunate. For an early publication, especially in the case of his Method of Fluxions, would have anticipated all rivalry, and secured him from the contentious claims of Leibnitz. Still each one will solve the problem of his existence in his own way, and, with a manlike Newton, his own, as we conceive, could be no other than the best way. The conduct of Leibnitz in this affair is quite irreconcilable with the stature and strength of the man ; giant-like, and doing nobly, in many ways, a giant s work, yet cringing himself into the dimensions and performances of a common calumniator. Opening in 1699, the discussion in question continued till the close of Leibnitz s life, in 1716. We give the summary of the case as contained in the Report of the Committee of the Royal Society, the deliberately weighed opinion of which has been adopted as an authoritative decision in all countries. " We have consulted the letters and letter books in the custody of the Royal Society, and those found among the papers of Mr. John Collins, dated between the years 1669 and 1677, inclusive ; and showed them to such as knew and avouched the hands of Mr. Barrow, Mr. Collins, Mr. Oldenburg, and Mr. Leibnitz ; and compared those of Mr. Gregory with one another, and with copies of some of them taken in the hand of Mr. Collins ; and have extracted from them what relates to the matter referred to us : all which extracts, herewith delivered to you, we believe to be genuine and authentic. And by these letters and papers wf find: LIFE OF SIR ISAAC NEWTON. 49 " I. Mr. Leibnitz was in London in the beginning of the year 1673 ; and went thence in or about March, to Paris, where he kept a correspondence with Mr. Collins, by means of Mr. Olden burg, till about September, 1676, and then returned, by London and Amsterdam, to Hanover: and that Mr. Collins was very free in communicating to able mathematicians what he had received from Mr, Newton and Mr. Gregory. " II. That when Mr. Leibnitz was the first time in London, he contended for the invention of another differential method, properly so called ; and, notwithstanding he was shown by Dr. Pell that it was Newton? s method, persisted in maintaining it to be his own invention, by reason that he had found it by himself without knowing what Newton had done before, and had much improved it. And we find no mention of his having any other differential method than Newton s before his letter of the 21st of June, 1677, which was a year after a copy of Mr. Newton s letter of the 10th of December, 1672, had been sent to Paris to be communicated to him ; and above four years after Mr. Collins began to communicate that letter to his correspondents ; in which letter the method of fluxions was sufficiently described to any intelligent person. "III. That by Mr. Newton s letter, of the 13th of June, 1676 it appears that he had the method of fluxions above five years before the writing of that letter. And by his Analysis per ^Equationes numero Terminorum Infmitas, communicated by Dr. Barrow to Mr. Collins, in July, 1669, we find that he had invented the method before that time. "IV. That the differential method is one and the same with the method of fluxions, excepting the name and mode of notation ; Mr. Leibnitz calling those quantities differences which Mr. Newton calls moments, or fluxions ; and marking them with a letter d a mark not used by Mr. Newton. " And, therefore, we take the proper question to be, not who invented this or that method, but, who was the first inventor of the method ? And we believe that those who have reputed Mr. Leibnitz the first inventor knew little or nothing of his correspond ence with Mr. Collins and Mr. Oldenburg long before, nor of Mr. 50 LIFE OP SIR ISAAC NEWTON. Newton s hiving that method above fifteen years before Mr Leibnitz began to publish it in the Acta Eruditorum of Leipsic. " For which reason we reckon Mr. Newton the first inventor ; and are of opinion that Mr. Keill, in asserting the same, has been no ways injurious to Mr. Leibnitz. And we submit to the judg ment of the Society, whether the extract and papers, now pre sented to you, together with what is extant, to the same pur pose, in Dr. Wallis s third volume, may not deserve to be made public." This Report, with the collection of letters and manuscripts, under the title of COMMERCIUM EPISTOLICUM D. JOHANNIS COLLINS ET ALIORUM DE ANALYSI PROMOTA JuSSU SoCIETATIS REGIES EDITUM, appeared accordingly in the early part of 1713. Its publication seemed to infuse additional bitterness into the feelings of Leibnitz, who descended to unfounded charges and empty threats. He had been privy counsellor to the Elector of Han over, before that prince was elevated to the British throne ; and in his correspondence, in 1715 and 1716, with the Abbe Conti, then at the court of George L, and with Caroline, Princess of Wales, he attacked the doctrines of the PRINCIPIA, and indirectly its author, in a manner very discreditable to himself, both as a learned and as an honourable man. His assaults, however, were triumphantly met; and, to the complete overthrow of his rival pretensions, Newton was induced to give the finishing blow. The verdict is universal and irreversible that the English preceded the German philosopher, by at least ten years, in the invention of fluxions. Newton could not have borrowed from Leibnitz ; but Leibnitz might have borrowed from Newton. A new edition of the Commercium Epistolicum was published in 1722-5 (?) ; but neither in this, nor in the former edition, did our author take any part. The disciples, enthusiastic, capable and ready, effectually shielded, with the buckler of Truth, the character of the Master, whose own conduct throughout was replete with delicacy, dignity and justice. He kept aloof from the controversy in which Dr. Keill stood forth as the chief representative of the Newtonian side till the very last, when, for the satisfaction of the King, George L. rather than for his own, he consented to put forth his LIFE OF SI| L^.-vJ NEWTON. 5i hand and firmly secure his rights upon a certain and impregnable, basis. A petition to have inventions for promoting the discovery of the longitude at sea, suitably rewarded, was presented to the House of Commons, in 1714. A committee, having been appointed to investigate the subject, called upon Newton and others for their opinions. That of our author was given in writing, A report, favourable to the desired measure, was then taken up, and a bill for its adoption subsequently passed. On the ascension of George I., in 1714, Newton became an object of profound interest at court. His position under govern ment, his surpassing fame, his spotless character, and. above all, his deep and consistent piety, attracted the reverent regard of the Princess of Wales, afterward queen -consort to George II. She was a woman of a highly cultivated mind, and derived the greatest pleasure from conversing with Newton and corresponding with Leibnitz. One day, in conversation with her, our author men tioned and explained a new system of chronology, which he had composed at Cambridge, where he had been in the habit " of refreshing himself with history and chronology, when he wac weary with other studies." Subsequently, in the year 1718, she requested a copy of this interesting and ingenious work Newton, accordingly, drew up an abstract of the system from the separate papers in which it existed, and gave it to her on condition that it should riot be communicated to any other person. Sometime afterward she requested that the Abbe Conti might be allowed to have a copy of it The author consented: and the abbe received a copy of the manuscript, under the like injunction and promise of secrecy. This manuscript bore the title of " A short Chronicle, from the First Memory of Tilings in Europe, to the Conquest of Persia, by Alexander the Great." After Newton took up his residence in London, he lived in a style suited to his elevated position and rank. He kept his car riage, with an establishment of three male and three female serv ants. But to everything like vain show and luxury he was utterly averse. His household affairs, for the last twenty years of his life, were under the charge of his niece, Mrs. Catherine Barton, 52 LIFE OF SIR ISAAC NEWTON. wife and widow of Colonel Barton a woman of great beauty and accomplishment and subsequently married to John Conduit, Esq. At home Newton was distinguished by that dignified and gentle hospitality which springs alone from true nobleness. On all pro per occasions, he gave splendid entertainments, though without ostentation. In society, whether of the palace or the cottage, his manner was self-possessed and urbane ; his look benign and affable ; his speech candid and modest ; his whole air undisturb edly serene. He had none of what are usually called the singu larities of genius ; suiting himself easily to every company except that of the vicious and wicked ; and speaking of himself and others, naturally, so as never even to be suspected of vanity. There was in him, if we may be allowed the expression, a WHOLE NESS of nature, which did not admit of such imperfections and weakness the circle was too perfect, the law too constant, and the disturbing forces too slight to suffer scarcely any of those eccentricities which so interrupt and mar the movements of many bright spirits, rendering their course through the world more like that of the blazing meteor than that of the light and life-impart ing sun. In brief, the words GREATNESS and GOODNESS could not, humanly speaking, be more fitly employed than when applied as the pre-eminent characteristics of this pure, meek and vene rable sage. In the eightieth year of his age, Newton was seized with symptoms of stone in the bladder. His disease was pronounced incurable. He succeeded, however, by means of a strict regimen, and other precautions, in alleviating his complaint, and procuring long intervals of ease. His diet, always frugai, was now extremely temperate, consisting chiefly of broth, vegetables, and fruit, with, now and then, a little butcher meat. He gave up the use of his carriage, and employed, in its stead, when he went out, a chair. All invitations to dinner were declined ; and only small parties were received, occasionally, at his own house. In 1724 he wrote to the Lord Provost of Edinburgh, offering to contribute twenty pounds yearly toward the salary of Mr. Maclaurin, provided he accepted the assistant Professorship of Mathematics in the University of that place. Not only in the LIFE OP SIR ISAAC NEWTON. 53 cause of ingenuity and learning, but in that of religion in relieving the poor and .assisting his relations, Newton annually expended large sums. He was generous and charitable almost to a fault. Those, he would often remark, who gave away nothing till they died, never gave at all. His wealth had become considerable by a prudent economy ; but he regarded money in no other light than as one of the means wherewith he had been intrusted to do good, and he faithfully employed it accordingly. He experienced, in spite of all his precautionary measures, a return of his complaint in the month of August, of the same year, 1 724, when he passed a stone the size of pea ; it came from him in two pieces, the one at the distance of two day.s from the other. Tolerable good health then followed for some months. In Janu ary, 1725, however, he was taken with a violent cough and inflam mation of the lungs. In consequence of this attack, he was pre vailed upon to remove to Kensington, where his health greatly improved. In February following, he was attacked in both feet with the gout, of the approach of which he had received, a few years before, a slight warning, and the presence of which now produced a very beneficial change in his general health. Mr. Conduit, his nephew, has recorded a curious conversation which took place, at or near this time, between himself and Sir Isaac. "I was, on Sunday night, the 7th March, 1724-5, at Kensing ton, with Sir Isaac Newton, in his lodgings, just after he was out of a fit of the gout, which he had had in both of his feet, for the first time, in the eighty-third year of his age. He was better after it, and his head clearer and memory stronger than I had known them for some time. He then repeated to me, by way of dis course, very distinctly, though rather in answer to my queries, than in one continued narration, what he had often hinted to me before, viz. : that it was his conjecture (he would affirm nothing) that there was a sort of revolution in the heavenly bodies ; that the vapours and light, emitted by the sun, which had their sedi ment, as water and other matter, had gathered themselves, by degrees, into a body, and attracted more matter from the planets, and at last made a secondary planet (viz. : one of those that go round another planet), and then, by gathering to them, and 54 LIFE OF SIR ISAAC NEWTON. attracting more matter, became a primary planet ; and then, bf increasing still, became a comet, which, after certain revolutions, by coming nearer and nearer to the sun, had all its volatile parts condensed, and became a matter tit to recruit and replenish the sun (which must waste by the constant heat and light it emitted), as a faggot would this fire if put into it (we were sitting by a wood fire), and that that would probably be the effect of the comet of 1680, sooner or later ; for, by the observations made upon it, it appeared, before it came near the sun, with a tail only two or three degrees long ; but, by the heat it contracted, in going so near the sun, it seemed to have a tail of thirty or forty degrees when it went frpm it ; that he could not say when this comet would drop into the sun ; it might perhaps have five or six revo lutions more first, but whenever it did it would so much increase the heat of the sun that this earth would be burned, and no ani mals in it could live. That he took the three phenomena, seen by Hipparchus, Tycho Brahe, and Kepler s disciples, to have been of this kind, for he could not otherwise account for an extraor dinary light, as those were, appearing, all at once, among the the fixed stars (all which he took to be suns, enlightening other planets, as our sun does ours), as big as Mercury or Venus seems to us, and gradually diminishing, for sixteen months, and then sinking into nothing. He seemed to doubt whether there were not intelligent beings, superior to us, who superintended these revolutions of the heavenly bodies, by the direction of the Supreme Being. He appeared also to be very clearly of opinion that the inhabitants of this world were of short date, and alledged, as one reason for that opinion, that all arts, as letters, ships, printing, needle, &c., were discovered within the memory of history, which could not have happened if the world had been eternal ; and that there were visible marks of ruin upon it which could not be effected by flood only. When I asked him how this earth could have been repeopled if ever it had undergone the same fate it was threatened with hereafter, by the comet of 1680, he answered, that required the power of a Creator. He said he took all the planets to be composed of the same matter with this earth, viz. : earth, water, stones, &c.3 but variously concocted. J LIFE OP SIR ISAAC NEWTON. 55 asked him why he would not publish his conjectures, as conjec tures, and instanced that Kepler had communicated his ; and though he had not gone near so far as Kepler, yet Kepler s guesses were so just and happy that they had been proved and demonstrated by him. His answer was, " I do not deal in con jectures." But, on my talking to him about the four observations that had been made of the comet of 1680, at 574 years distance, and asking him the particular times, he opened his Principia, which laid on the table, and showed me the particular periods, viz.: 1st. The Julium Sidus, in the time of Justinian, in 1106, in 1680. " And I, observing that he said there of that comet, incidet in corpus solis, and in the next paragraph adds, stellae fixae refici possunt, told him I thought he owned there what we had been talking about, viz. : that the comet would drop into the sun, and that fixed stars were recruited and replenished by comets when they dropped into them ; and, consequently, that the sun would be recruited too ; and asked him why he would not own as fully what he thought of the sun as well as what he thought of the fixed stars. He said, that concerned us more; and, laugh ing, added, that he had said enough for people to know his meaning." In the summer of 1725, a French translation of the chronolo gical MS., of which the Abbe Conti had been permitted, some time previous, to have a copy, was published at Paris, in violation of all good faith. The Punic Abbe had continued true to his promise of secrecy while he remained in England ; but no sooner did he reach Paris than he placed the manuscript into the hands of M. Freret, a learned antiquarian, who translated the work, and accompanied it with an attempted refutation of the leading points of the system. In November, of the same year, Newton received a presentation copy of this publication, which bore the title of ABREGE DE CHRONOLOGIE DE M. LE CHEVALIER NEWTON, FAIT PAR LUI-MEME, ET TRADUIT SUR LE MANUSCRIPT ANGLAIS. Soon afterward a paper entitled, REMARKS ON TFE OBERVATIONS MADE ON A CHRONOLOGICAL INDEX OF SIR ISAAC NE.WTON, TRANSLATED INTO FRENCH BY THE OBSERVATOR, ANL PUBLISHED AT PARIS, 56 LIFE OF SIR ISAAC NEWTON, was drawn up by our author, and printed in the Philosophical Transactions for 1725. It contained a history of the whole matter, and a triumphant reply to the objections of M. Freret. This answer called into the field a fresh antagonist, Father Soueiet, whose five dissertations on this subject were chiefly remarkable for the want of knowledge and want of decorum, which they displayed. In consequence of these discussions, Newton was in duced to prepare his larger work for the press, and had nearly completed it at the time of his death. It was published in 1728, under the title of THE CHRONOLOGY OF THE ANCIENT KINGDOMS AMENDED, TO WHICH is PREFIXED A SHORT CHRONICLE FROM THE FIRST MEMORY OF THINGS IN EUROPE TO THE CONQUEST OF PERSIA BY ALEXANDER THE GREAT. It consists of six chap ters: 1. On the Chronology of the Greeks; according to Whiston, our author wrote out eighteen copies of this chapter with his own hand, differing little from one another. 2. Of the Empire of Egypt; 3. Of the Assyrian Empire; 4. Of the two contempo rary Empires of the Babylonians and Medes ; 5. A Description of the Temple of Solomon ; 6. Of the Empire of the Persians ; this chapter was not found copied with the other five, but as it was discovered among his papers, arid appeared to be a continu ation of the same work, the Editor thought proper to add it thereto. Newton s LETTER TO A PERSON OF DISTINCTION WHO HAD DESIRED HIS OPINION OF THE LEARNED BlSHO^ LLOYD S HYPOTHESIS CONCERNING THE FORM OF THE MOST ANCIENT ^EAR, closes this enumeration of his Chronological Writings. A ihird edition of the PRINCIPIA appeared in 1726, with many changes and additions. About four years were consumed in its preparation and publication, which were under the superintend- ance of Dr. Henry Pemberton, an accomplished mathematician, and the author of "A VIEW OF SIR ISAAC NEWTON S PHILO SOPHY." 1728. This gentleman enjoyed numerous opportunities of conversing with the aged and illustrious author. " I found," says Pemberton, " he had read fewer of the modern mathemati cians than one could have expected; but his own prodigious invention readily supplied him with what he might have an occa sion for in the pursuit of any subject he undertook. I have often LIFE OF SIR ISAAC NEWTON. 57 heard him censure the handling geometrical subjects ly algebraic calculations ; and his book of Algebra he called by the name of Universal Arithmetic, in opposition to the injudicious title of Geometry, which Descartes had given to the treatise, wherein he shows how the geometer may assist his invention by such kind of computations. He thought Huygens the most elegant of any mathematical writer of modern times, and the most just imitator of the ancients. Of their taste and form of demonstration, Sir Isaac always professed himself a great admirer. I have heard him even censure himself for not following them yet more closely than he did ; and speak with regret of his mistake at the begin ning of his mathematical studies, in applying himself to the works of Descartes and other algebraic writers, before he had considered the elements of Euclid with that attention which so excellent a writer deserves." " Though his memory was much decayed," continues Dr. Pemberton, "he perfectly understood his own writings." And even this failure of memory, we would suggest, might have been more apparent than real, or, in medical terms, more the result of func tional weakness than organic decay. Newton seems never to have confided largely to his memory : and as this faculty mani fests the most susceptibility to cultivation ; so, in the neglect of due exercise, it more readily and plainly shows a diminution of its powers. Equanimity and temperance had, indeed, preserved Newton singularly free from all mental and bodily ailment. His hair was, to the last, quite thick, though as white as silver. He never made use of spectacles, and lost but one tooth to the day of his death. He was of middle stature, well-knit, and, in the latter part of his life, somewhat inclined to be corpulent. Mr. Conduit says, " he had a very lively and piercing eye, a comely and gra cious aspect, with a fine head of hair, white as silver, without any baldness, and when his peruke was off was a venerable sight." According to Bishop Atterbury, "in the whole air of his face and make there was nothing of that penetrating sagacity which appears in his compositions. He had something rather languid in his look and manner which did not raise any great expectation 58 LIFE OF SIR ISAAC NEWTON. in those who did not know him." Hearne remarks, " Sir Isaac was a man of no very promising aspect. He was a short, wellset man. He was full of thought, and spoke very little in com pany, so that his conversation was not agreeable. When he rode in his coach, one arm would be out of his coach on one side and the other on the other." These different accounts we deem easily reconcilable. In the rooms of the Royal Society, in the street, or in mixed assemblages, Newton s demeanour always courteous, unassuming and kindly still had in it the overawings of a profound repose and reticency, out of which the communica tive spirit, and the "lively and piercing eye" would only gleam in the quiet and unrestrained freedom of his own fire-side. " But this I immediately discovered in him," adds Pemberton, still further, "which at once both surprised and charmed me. Neither his extreme great age, nor his universal reputation had rendered him stiff in opinion, or in any degree elated. Of this I had occasion to have almost daily experience. The remarks I continually sent him by letters on his Principia, were received with the utmost goodness. These were so far from being any ways displeasing to him, that, on the contrary, it occasioned him to speak many kind things of me to my friends, and to honour me with a public testimony of his good opinion." A modesty, open ness, and generosity, peculiar to the noble and comprehensive spirit of Newton. " Full of wisdom and perfect in beauty," yet not lifted up by pride nor corrupted by ambition. None, how ever, knew so well as himself the stupendousness of his discoveries in comparison with all that had been previously achieved ; and none realized so thoroughly as himself the littleness thereof in comparison with the vast region still unexplored. A short time before his death he uttered this memorable sentiment: " I do not know what I may appear to the world ; but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, while the great ocean of truth lay all undis covered before me." How few ever reach the shore even, much less find "a smoother pebble or a prettier shell!" Newton had now resided about two years at Kensington ; and LIFE OF SIR ISAAC NEWTON. 59 the air which he enjoyed there, and the state of absolute rest, proved of great benefit to him. Nevertheless he would occasion ally go to town. And on Tuesday, the 28th of February, 1727, he proceeded to London, for the purpose of presiding at a meeting of the Royal Society. At this time his health was considered, by Mr. Conduit, better than it had been for many years. But the unusual fatigue he was obliged to suffer, in attending the meeting, and in paying and receiving visits, speedily produced a violent return of the affection in the bladder. He returned to Kensington on Saturday, the 4th of March. Dr. Mead and Dr. Cheselden attended him ; they pronounced his disease to be the stone, and held out no hopes of recovery. On Wednesday, the 15th of March, he seemed a little better; and slight, though groundless, encouragement was felt that he might survive the attack. From the very first of it, his sufferings had been intense. Paroxysm followed paroxysm, in quick succession : large drops )f sweat rolled down his face ; but not a groan, not a complaint, not the least mark of peevishness or impatience escaped him : and during the short intervals of relief, he even smiled and con versed with his usual composure and cheerfulness. The flesh quivered, but the heart quaked not ; the impenetrable gloom was settling down : the Destroyer near ; the portals of the tomb opening, still, arnid this utter wreck and dissolution of the mortal, the immortal remained serene, unconquerable : the radiant light broke through the gathering darkness ; and Death yielded up its sting, and the grave its victory. On Saturday morning, 18th, he read the newspapers, and carried on a pretty long conversation with Dr. Mead. His senses and faculties were then strong and vigorous ; but at six o clock, the same evening, he became insen sible ; and in this state he continued during the whole of Sunday, and till Monday, the 20th, when he expired, between one and two o clock in the morning, in the eighty-fifth year of his age. And these were the last days of Isaac Newton. Thus closed the career of one of earth s greatest and best men. His mission was fulfilled. Unto the Giver, in many-fold addition, the talents were returned. While it was yet day he had worked ; and for the night that quickly cometh he was not unprepared. Full of 60 LIFE OF SIR ISAAC NEWTON. years, ind full of honours, the heaven-sent was recalled ; and, in the confidence of a " certain hope," peacefully he passed awa} into the silent depths of Eternity. His body was placed in Westminster Abbey, with the state and ceremonial that usually attended the interment of the most distinguished. In 1731, his relatives, the inheritors of his personal estate, erected a monument to his memory in the most conspicu ous part of the Abbey, which had often been refused by the dean and chapter to the greatest of England s nobility. During the same year a medal was struck at the Tower in his honour ; arid, in 1755, a full-length statue of him, in white marble, admirably executed, by Roubiliac, at the expense of Dr. Robert Smith, was erected in the ante-chamber of Trinity College, Cambridge. There is a painting executed in the glass of one of the windows of the same college, made pursuant to the will of Dr. Smith, who left five hundred pounds for that purpose. Newton left a personal estate of about thirty-two thousand pounds. It was divided among his four nephews and four nieces of the half blood, the grand-children of his mother, by the Reve rend Mr. Smith. The family estates of Woolsthorpe arid Sustern fell to John Newton, the heir-at-law, whose great grand-father was Sir Isaac s uncle. Before his death he made an equitable distribution of his two other estates : the one in Berkshire to the sons and daughter of a brother of Mrs. Conduit ; and the other, at Kensington, to Catharine, the only daughter of Mr. Conduit, and who afterward became Viscountess Lymington. Mr. Con duit succeeded to the offices of the Mint, the duties of which he had discharged during the last two years of Sir Isaac s life. Our author s works are found in the collection of Castilion, Berlin, 1744, 4to. 8 torn.; in Bishop Horsley s Edition, London, 1779, 4to. 5 vol.; in the Biographia Brittannica, &c. Newton also published Bern. Varcnii Geographia, &c., 1681, 8vo. There are, however, numerous manuscripts, letters, and other papers, which have never been given to the world: these are preserved, in various collections, namely, in the library of Trinity College, Cambridge ; in the library of Corpus Christi College, Oxford ; in the library of Lord Macclesfield : and, lastly arid LIFE OF SIR ISAAC NEWTON. 61 chiefly, in the possession of the family of the Earl of Portsmouth, through the Viscountess Lymington. Everything appertaining to Newton has been kept and che rished with peculiar veneration. Different memorials of him are preserved in Trinity College, Cambridge ; in the rooms of the Royal Society, of London : and in the Museum of the Royal Society of Edinburgh. The manor-house, at Woolsthorpe, was visited by Dr. Stuke ley, in October, 1721, who, in a letter to Dr. Mead, written in 1727, gave the following description of it: " Tis built of stone, as is the way of the country hereabouts, and a reasonably good one. They led me up stairs and showed me Sir Isaac s stud}-, where I supposed he studied, when in the country, in his younger days, or perhaps when he visited his mother from the University. I observed the shelves were of his own making, being pieces of deal boxes, which probably he sent his books and clothes down in on those occasions. There were, some years ago, two or threr hundred books in it of his father-in-law, Mr. Smith, which Sir Isaac gave to Dr. Newton, of our town." The celebrated appletree, the fall of one of the apples of which is said to have turned the attention of Newton to the subject of gravity, was destroyed by the wind about twenty years ago ; but it has been preserved in the form of a chair. The house itself has been protected with religious care. It was repaired in 1798, and a tablet of white marble put up in the room where our author was born, with the follow, ng inscription : " Sir Isaac Newton, son of John Newton, Lord of the Manor of Woolsthorpe, was born in this room, on the 25th of December, 1642." Nature and Nature s Laws wei-e hid in night, God said, " Let NEWTON be," and all was light. THE PEINCIPIA. THE AUTHOR S PREFACE SINCE the ancients (as we are told by Pappus), made great account oi the science of mechanics in the investigation of natural things : and the moderns, laying aside substantial forms and occult qualities, have endeav oured to subject the phenomena of nature to the laws of mathematics, I have in this treatise cultivated mathematics so far as it regards philosophy. The ancients considered mechanics in a twofold respect ; as rational, which proceeds accurately by demonstration ; and practical. To practical me chanics all the manual arts belong, from which mechanics took its name. Rut as artificers do not work with perfect accuracy, it comes to pass that mechanics is so distinguished from geometry, that what is perfectly accu rate is called geometrical , what is less so, is called mechanical. But the errors are not in the art, but in the artificers. He that works with less accuracy is an imperfect mechanic ; and if any could work with perfect accuracy, he would be the most perfect mechanic of all ; for the description if right lines and circles, upon which geometry is founded, belongs to me chanics. Geometry does not teach us to draw these lines, but requires them to be drawn ; for it requires that the learner should f.rst be taught to describe these accurately, before he enters upon geometry ; then it shows how by these operations problems may be solved. To describe right lines and circles are problems, but not geometrical problems. The solution of these problems is required from mechanics ; and by geometry the use of them, when so solved, is shown ; and it is the glory of geometry that from those few principles, brought from without, it is able to produce so many things. Therefore geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring. But since the manual arts are chiefly conversant in the moving of bodies, it comes to pass that geometry is commonly referred to their magnitudes, and mechanics to their motion. In this sense rational mechanics will be the science of motions resulting from any forces whatsoever, and of the forces required to produce any mo tions, accurately proposed and demonstrated. This part of mechanics was i:;vm THE AUTHOR & PREFACE. cultivated by the ancients in the five powers which relate to manual arts, who considered gravity (it not being a manual power), ho Otherwise than as it moved weights by those powers. Our design not respecting arts, but philosophy, and our subject not manual but natural powers, we consider chiefly those things which relate to gravity, levity, elastic force, the resist ance of fluids, and the like forces, whether attractive or impulsive ; and therefore we offer this work as the mathematical principles :f philosophy ; for all the difficulty of philosophy seems to consist in this from the phenom ena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena ; and to this end the general propositions in the first and second book are directed. In the third book we give an example of this in the explication of the System of the World : for by the propositions mathematically demonstrated in the former books, we in the third derive from the celestial phenomena the forces of gravity with which bodies tend to the sun and the several planets. Then from these forces, by other propositions which are also mathematical, we deduce the mo tions of the planets, the comets, the moon, and the sea. I wish we could do- rive the rest of the phenomena of nature by the same kind of reasoning from mechanical principles; for I am induced by many reasons to suspect that they may all depend upon certain forces by which the particles of bodies. by some causes hitherto unknown, are either mutually impelled towards each other, and cohere in regular figures, or are repelled and recede from each other; which forces being unknown, philosophers have hitherto at tempted the search of nature in vain ; but I hope the principles here laid down will afford some light either to this or some truer method of philosophy. In the publication of this work the most acute and universally learned Mr. Edmund Halley not only assisted me with his pains in correcting the press and taking care of the schemes, but it was to his solicitations that its becoming public is owing ; for when he had obtained of me my demonstra tions of the figure of the celestial orbits, he continually pressed me to com municate the same to the Royal Societ//, who afterwards, by their kind en couragement and entreaties, engaged me to think of publishing them. But after I had begun to consider the inequalities of the lunar motions, and had entered upon some other things relating to the laws and measures oi gravity, and other forces : and the figures that would be described by bodies attracted according to given laws ; and the motion of several bodies moving among themselves; the motion of bodies in resisting mediums; the forces, densities, and motions, of rn( Hums ; the orbits of the comets, and such like ; Ixix deferred that publication till I had made a searcli into those matters, and could put forth the whole together. What relates to the lunar motions (be ing imperfect), I have put all together in the corollaries of Prop. 66, to avoid being obliged to propose and distinctly demonstrate the several things there contained in a method more prolix than the subject deserved, and in terrupt the series of the several propositions. Some things, found out after the rest, I chose to insert in places less suitable, rather than change the number of the propositions and the citations. I heartily beg that what 1 have here done may be read with candour; and that the defects in a subject so difficult be not so much reprehended as kindly supplied, and in vestigated by new endeavours of mv readers. ISAAC NEWTON. Cambridge, Trinity Coupge May 8, liHB. In the second edition the second section of the first book was enlarged. In the seventh section of the second book the theory of the resistances of fluids was more accurately investigated, and confirmed by new experiments. In the third book the moon s theory and the profession of the equinoxes were more fully deduced from their principles ; and the theory of the comets was confirmed by more examples of the calculati >n of their orbits, done also with greater accuracy. In this third edition the resistance of mediums is somewhat more largely handled than before; and new experiments of the resistance of heavy bodies falling in air are added. In the third book, the argument to prove that the moon is retained in its orbit by the force of gravity is enlarged on ; and there are added new observations of Mr. Pound s of the proportion of the diameters of Ju.piter to each other : there are, besides, added Mr. Kirk s observations of the comet in 16SO ; the orbit of that comet com puted in an ellipsis by Dr. Halley ; and the ortit of the comet in computed by Mr. Bradley, OOK I. THE MATHEMATICAL PRINCIPLES OF NATURAL PHILOSOPHY DEFINITIONS. DEFINITION I. 77w? quantity of matter is the measure of the same, arising from its density and hulk conjutictly. THUS air of a double density, in a double space, is quadruple in quanti ty ; in a triple space, sextuple in quantity. The same thing is to be un derstood of snow, and fine dust or powders, that are condensed by compres sion or liquefaction and of all bodies that are by any causes whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts oi bodies. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body ; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter. DEFINITION II. The quantity of motion is the measure nf tlie same, arising from the velocity and quantity of matter corjunctly. The motion of the whole i<! the sum of the motions of all the parts ; and therefore in a body double in quantity, with equal velocity, the motion is iouble ; with twice the velocity, it is quadruple, DEFINITION III. The vis insita, or innate force of matter, is a power of resisting, hy which every body, as much as in it lies, endeavours to persevere in its present stale, whether it be of rest, or of moving uniformly forward in a right line. This force is ever proportional to the body whose force it is ; and differs nothing from the inactivity of the mass, but in our manner of conceiving T4 THE MATHEMATICAL PRINCIPLES it. A body, from the inactivity of matter, is not without difficulty put out of its state of rest or motion. Upon which account, this vis insita, may, by a most significant name, be called vis inertia, or force of inactivity. Hut a body exerts this force only, when another force, impressed upon it, endeavours to change its condition ; and the exercise of this force may bo considered both as resistance and impulse ; it is resistance, in so far as the body, for maintaining its present state, withstands the force impressed; it is impulse, in so far as the body, by not easily giving way to the impressed force of another, endeavours to change the state of that other. Resistance is usually ascribed to bodies at rest, and impulse to those in motion; but motion and rest, as commonly conceived, are only relatively distin guished ; nor are those bodies always truly at rest, which commonly are taken to be so. DKFLMTIOX IV. Ait impressed force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line. This force consists in the action only; and remains no longer in the body, when the action is over. For a body maintains every new state it acquires, by its vis inertice only. Impressed forces are of differe.it origins as from percussion, from pressure, from centripetal force. DEFINITION V. A centripetal force is that by which bodies are drawn or impelled, or any way tend, towards a point as to a centre. Of this sort is gravity, by which bodies tend to the centre of the earth magnetism, by which iron tends to the loadstone ; and that force, what ever it is, by which the planets are perpetually drawn aside from the rec tilinear motions, which otherwise they would pursue, and made to revolve in curvilinear orbits. A stone, whirled about in a sling, endeavours to re cede from the hand that turns it ; and by that endeavour, distends the sling, and that with so much the greater force, as it is revolved with the greater velocity, and as soon as ever it is let go, flies away. That force which opposes itself to this endeavour, and by which the sling perpetually draws back the stone towards the hand, and retains it in its orbit, because it is directed to the hand as the centre of the orbit, I call the centripetal force. And the same thing is to be understood of all bodies, revolved in any orbits. They all endeavour to recede from the centres of their orbits ; and wore it not for the opposition of a contrary force which restrains them to, and detains them in their orbits, which I therefore call centripetal, would tiy off in right lines, with an uniform motion. A projectile, if it was not for the force of gravity, would not deviate towards the earth, tut would OF NATUJIAL PHILOSOPHY. 7fl go off from it in a right line, and that with an uniform motion,, if the re sistance of the air was taken away. It is by its gravity that it is drawn aside perpetually from its rectilinear course, and made to deviate towards the earth, more or less, according to the force of its gravity, and the velo city of its motion. The less its gravity is, for the quantity of its matter, or the greater the velocity with which it is projected, the less will it devi ate from a rectilinear course, and the farther it will go. If a leaden balJ, projected from the top of a mountain by the force of gunpowder with a given velocity, and in a direction parallel to the horizon, is carried in a curve line to the distance of two miles before it falls to the ground ; the same, if the resistance of the air were taken away, with a double or decuple velocity, would fly twice or ten times as far. And by increasing the velo city, we may at pleasure increase the distance to which it might be pro jected, and diminish the curvature of the line, which it might describe, till at last it should fall at the distance of 10, 30, or 90 degrees, or even might go quite round the whole earth before it falls ; or lastly, so that it might never fall to the earth, but go forward into the celestial spaces, and pro ceed in its motion in iiifiuitum. And after the same manner that a pro jectile, by the force of gravity, may be made to revolve in an orbit, and go round the whole earth, the moon also, either by the force of gravity, if it is endued with gravity, or by any other force, that impels it towards the earth, may be perpetually drawn aside towards the earth, out of the r&tilinear way, which by its innate force it would pursue; and would be made to revolve in the orbit which it now describes ; nor could the moon with out some such force, be retained in its orbit. If this force was too small, it would not sufficiently turn the moon out of a rectilinear course : if it was too great, it would turn it too much, arid draw down the moon from its orbit towards the earth. It is necessary, that the force be of a just quantity, and it belongs to the mathematicians to find the force, that may serve exactly to retain a body in a given orbit, with a given velocity ; and vice versa, to determine the curvilinear way, into which a body projected from a given place, with a given velocity, may be made to deviate from its natural rectilinear way, by means of a given force. The quantity of any centripetal force may be considered as of three kinds; aboolu e, accelerative, and motive. DEFINITION VI. The absolute quantity of a centripetal force is the measure f >f the same proportional to the efficacy of the cause that propagates it from the cen tre, through the spaces round about. Thus the magnetic force is greater in one load-stone and less in another according to their sizes and strength of intensity. 76 THE MATHEMATICAL PRINCIPLES DEFINITION VII. The accelerative quantity of a centripetal force is the measure, of tht same, proportional to the velocity which it generates in a given time. Thus the force of the same load-stone is greater at a less distance, and less at a greater : also the force of gravity is greater in valleys, less on tops of exceeding high mountains ; and yet less (as shall hereafter be shown), at greater distances from the body of the earth ; but at equal distan ces, it is the same everywhere ; because (taking away, or allowing for, the resistance of the air), it equally accelerates all falling bodies, whether heavy or light, great or small. DEFINITION VIII. TJie motive quantity of a centripetal force, is the measure of the samt\ proportional to the motion which it generates in a given twip. Thus the weight is greater in a greater body, less in a less body ; and. in the same body, it is greater near to the earth, and less at remoter dis tances. This sort of quantity is the centripetency, or propension of the whole body towards the centre, or, as I may say, its weight ; and it is al ways known by the quantity of an equal and contrary force just sufficient to Ifinder the descent of the body. These quantities of forces, we may, for brevity s sake, call by the names of motive, accelerative, and absolute forces ; and, for distinction s sake, con sider them, with respect to the bodies that tend to the centre ; to the places of those bodies ; and to the centre of force towards which they tend ; that is to say, I refer the motive force to the body as an endeavour and propen sity of the whole towards a centre, arising from the propensities of the several parts taken together ; the accelerative force to the place of the body, as a certain power or energy diffused from the centre to all places around to move the bodies that are in them : and the absolute force to the centre, as endued with some cause, without which those motive forces would not be propagated through the spaces round about ; whether that cause be some central body (siuh as is the load-stone, in the centre of the magnetic force, or the earth in the centre of the gravitating force), or anything else that does not yet appear. For I here design only to give a mathematical notion of those forces, without considering their physical causes and seats. Wherefore the accelerative force will stand in the same relation to the motive, as celerity does to motion. For the quantity of motion arises from the celerity drawn into the quantity of matter : and the motive force arises from the accelerative force drawn into the same quantity of matter. For the sum of the actions of the accelerative force, upon the several ; articles of the body, is the motive force of the whole. Hence it is, that near the OF NATURAL PHILOSOPHY. 77 surface of the earth, where the accelerative gravity, or force productive of gravity, in all bodies is the same, the motive gravity or the weight is as the body : but if we should ascend to higher regions, where the accelerative gravity is less, the weight would be equally diminished, and would always be as the product of the body, by the accelerative gravity. So in those re gions, where the accelerative gravity is diminished into one half, the weight of a body two or three times less, will be four or six times less. I likewise call attractions and impulses, in the same sense, accelerative, and motive ; and use the words attraction, impulse or propensity of any sort towards a centre, promiscuously, and indifferently, one for another ; considering those forces not physically, but mathematically : wherefore, the reader is not to imagine, that by those words, I anywhere take upon me to define the kind, or the manner of any action, the causes or the physical reason thereof, or that I attribute forces, in a true and physical sense, to certain centres (which are only mathematical points) ; when at any time I happen to speak of centres as attracting, or as endued with attractive powers. SCHOLIUM. Hitherto I have laid down the definitions of such words as are less known, and explained the sense in which I would have them to be under stood in the following discourse. I do not define time, space, place and motion, as being well known to all. Only I must observe, that the vulgar conceive those quantities under no other notions but from the relation they bear to sensible objects. And thence arise certain prejudices, for the re moving of which, it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common. I. Absolute, true, and mathematical time, of itself, and from its own na ture flows equably without regard to anything external, and by another name is called duration : relative, apparent, and common time, is some sen sible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time ; such as an hour, a day, a month, a year. II. Absolute space, in its own nature, without regard to anything exter nal, remains always similar and immovable. Relative space is some mo vable dimension or measure of the absolute spaces ; which our senses de termine by its position to bodies ; and which is vulgarly taken for immo vable space ; such is the dimension of a subterraneous, an aereal, or celestial space, determined by its position in respect of the earth. Absolute and relative space, are the same in figure and magnitude ; but they do not re main always numerically the same. For if the earth, for instance, moves, a space of our air, which relatively and in respect of the earth remains al ways the same, will at one time be one part of the absolute space into which TS THE MATHEMATICAL PRINCIPLES the air passes ; at another time it will be another part of the same, and so. absolutely understood, it will be perpetually mutable. III. Place is a part of space which a body takes up, and is according to the space, either absolute or relative. I say, a part of space ; not the situation, nor the external surface of the body. For the places of equal solids are always equal ; but their superfices, by reason of their dissimilar figures, are often unequal. Positions properly have no quantity, nor are they so much the places themselves, as the properties of places. The motion of the whole is the same thing with the sum of the motions of the parts ; that is, the translation of the whole, out of its place, is the same thing with the sum of the translations of the parts out of their places ; and therefore the place of the whole is the same thing with the sum of the places of the parts, and for that reason, it is internal, and in the whole body. IV. Absolute motion is the translation of a body from one absolute place into another ; and relative motion, the translation from one relative place into another. Thus in a ship under sail, the relative place of a body is that part of the ship which the body possesses ; or that part of its cavity which the body fills, and which therefore moves together with the ship : and relative rest is the continuance of the body in the same part of the ship, or of its cavity. But real, absolute rest, is the continuance of the body in the same part of that immovable space, in which the ship itself, its cavity, and all that it contains, is moved. Wherefore, if the earth is really at rest, the body, which relatively rests in the ship, will really and absolutely move with the same velocity which the ship has on the earth. But if the earth also moves, the true and absolute motion of the body will arise, partly from the true motion of the earth, in immovable space ; partly from the relative motion of the ship on the earth ; and if the body moves also relatively in the ship ; its true motion will arise, partly from the true motion of the earth, in immovable space, and partly from the relative mo tions as well of the ship on the earth, as of the body in the ship ; and from these relative motions will arise the relative motion of the body on the earth. As if that part of the earth, where the ship is, was truly moved toward the east, with a velocity of 10010 parts; while the ship itself, with a fresh gale, and full sails, is carried towards the west, with a velocity ex pressed by 10 of those parts ; but a sailor walks in the ship towards the east, with 1 part of the said velocity ; then the sailor will be moved truly in immovable space towards the east, with a velocity of 10001 parts, and relatively on the earth towards the west, with a velocity of 9 of those parts. Absolute time, in astronomy, is distinguished from relative, by the equa tion or correction of the vulgar time. For the natural days are tr^y un equal, though they are commonly considered as equal, and used for a meas ure of time ; astronomers correct this inequality for their more accurate deducing of the celestial motions. It may be, that there is no such thing as an equable motion, whereby time may H accurately measured. All mo OF NATURAL PHILOSOPHY. 79 tions may be accelerated and retarded; but the true, or equable, progress of absolute time is liable to no change. The duration or perseverance of the existence of things remains the same, whether the motions are swift or slow, or none at all : and therefore it ought to be distinguished from what are only sensible measures thereof ; and out of which we collect it, by means of the astronomical equation. The necessity of which equation, for deter mining the times of a phamomenon, is evinced as well from the experiments of the pendulum clock, as by eclipses of the satellites of Jupiter. As the order of the parts of time is immutable, so also is the order of the parts of space. Suppose those parts to be moved out of their places, and they will be moved (if the expression may be allowed) out of themselves. For times and spaces are, as it were, the places as well of themselves as of all other things. All things are placed in time as to order of succession ; and in space as _to order of situation. It is from their essence or nature that they are places ; and that the primary places of things should be moveable, is absurd. These are therefore the absolute places ; and trans lations out of those places, are the only absolute motions. But because the parts of space cannot be seen, or distinguished from one another by our senses, therefore in their stead we use sensible measures of them. For from the positions and distances of things from any body con sidered as immovable, we define all places ; and then with respect to such places, we estimate all motions, considering bodies as transferred from some of those places into others. And so, instead of absolute places and motions, we use relative ones; and that without any inconvenience in common af fairs ; but in philosophical disquisitions, we ought to abstract from our senses, and consider things themselves, distinct from what are only sensible measures of them. For it may be that there is no body really at rest, to which the places and motions of others may be referred. But we may distinguish rest and motion, absolute and relative, one from the other by their properties, causes and effects. It is a property of rest, that bodies really at rest do rest in respect to one another. And therefore as it is possible, that in the remote regions of the fixed stars, or perhaps far beyond them, there may be some body absolutely at rest ; but impossi ble to know, from the position of bodies to one another in our regions whether any of these do keep the same position to that remote body; it follows that absolute rest cannot be determined from the position of bodies in our regions. It is a property of motion, that the parts, which retain given positions to their wholes, do partake of the motions of those wholes. For all the parts of revolving bodies endeavour to recede from the axis of motion ; and the impetus of bodies moving forward, arises from the joint impetus of all the parts. Therefore, if surrounding bodies are moved, those that are relatively at rest within them, will partake of their motion. Upon which account, the true and absolute motion of a body cannot be Jeter- 8C THE MATHEMATICAL PRINCIPLES mined by the translation of it from those which only seem to rest ; for the external bodies ought not only to appear at rest, but to be really at rest. For otherwise, all included bodies, beside their translation from near the surrounding ones, partake likewise of their true motions ; and though that translation were not made they would not be really at rest, but only seem to be so. For the surrounding bodies stand in the like relation to the surrounded as the exterior part of a whole does to the interior, or as the shell does to the kernel ; but, if the shell moves, the kernel will also move, as being part of the whole, without any removal from near the shell. A property, near akin to the preceding, is this, that if a place is moved, whatever is placed therein moves along with it ; and therefore a body, which is moved from a place in motion, partakes also of the motion of its place. Upon which account, all motions, from places in motion, are no other than parts of entire and absolute motions ; and every entire motion is composed of the motion of the body out of its first place, and the motion of this place out of its place ; and so on, until we come to some immovable place, as in the before-mentioned example of the sailor. Where fore, entire and absolute motions can be no otherwise determined than by immovable places : and for that reason I did before refer those absolute motions to immovable places, but relative ones to movable places. Now no other places are immovable but those that, from infinity to infinity, do all retain the same given position one to another ; and upon this account must ever remain unmoved ; and do thereby constitute immovable space. The causes by which true and relative motions are distinguished, one from the other, are the forces impressed upon bodies to generate motion. True motion is neither generated nor altered, but by some force impressed upon the body moved : but relative motion may be generated or altered without any force impressed upon the body. For it is sufficient only to impress some force on other bodies with which the former is compared, that by their giving way, that relation may be changed, in which the re lative rest or motion of this other body did consist. Again, true motion suffers always some change from any force impressed upon the moving body ; but relative motion docs not necessarily undergo any change by such forces. For if the same forces are likewise impressed on those other bodies, with which the comparison is made, that the relative position may be pre served, then that condition will be preserved in which the relative motion consists. And therefore any relative motion may be changed when the true motion remains unaltered, and the relative may be preserved when the true suffers some change. Upon which accounts; true motion does by no means consist in such relations. The effects whicli distinguish absolute from relative motion arc, the forces of receding from the axis of circular motion. For there are no such forces in a circular motion purely relative, but in a true and absolute cir cular motion., they are greater or less, according t the quantity of the OF NATURAL PHILOSOPHY. 1 motion. If a vessel, hung: by & }ong cord, is so often turned ubout that the cord is strongly twisted, then filled with water, and held at rest together with the water ; after, by the sudden action of another force, it is whirled about the contrary way, and while the cord is untwisting itself, the vessel continues for some time in this motion ; the surface of the water will at first be plain, as before the vessel began to move : but the vessel; by grad ually communicating its motion to the water, will make it begin sensibly ^to revolve, and recede by little and little from the middle, and ascend to the sides of the vessel, forming itself into a concave figure (as I have experi enced), and the swifter the motion becomes, the higher will the water rise, till at last, performing its revolutions in the same times with the vessel, it becomes relatively at rest in it. This ascent of the water shows its en deavour to recede from the axis of its motion ; and the true and absolute circular motion of the water, which is here directly contrary to the relativej discovers itself, and may be measured by this endeavour. At first, when the relative motion of the water in the vessel was greatest, it pro duced no endeavour to recede from the axis ; the water showed no tendency to the circumference, nor any ascent towards the sides of the vessel, but remained of a plain surface, and therefore its true circular motion had not yet begun. But afterwards, when the relative motion of the water had decreased, the ascent thereof towards the sides of the vessel proved its en deavour to recede from the axis ; and this endeavour showed the real cir cular motion of the water perpetually increasing, till it had acquired its greatest quantity, when the water rested relatively in the vessel. And therefore this endeavour does not depend upon any translation of the water in respect of the ambient bodies, nor can true circular motion be defined by such translation. There is only one real circular motion of any one revolving body, corresponding to only one power of endeavouring to recede from its axis of motion, as its proper and adequate effect ; but relative motions, in one and the same body, are innumerable, according to the various relations it bears to external bodies, and like other relations, arc altogether destitute of any real effect, any otherwise than they may perhaps par take of that one only true motion. And therefore in their system who suppose that our heavens, revolving below the sphere of the fixed stars, carry the planets along with them ; the several parts of those heavens, and the planets, which are indeed relatively at rest in their heavens, do yet really move. For they change their position one to another (which never happens to bodies truly at rest), and being carried together with their heavens, partake of their motions, and as parts of revolving wholes, endeavour to recede from the axis of their motions. Wherefore relative quantities are not the quantities themselves, whose names they bear, but those sensible measures of them (either accurate cr inaccurate), which arc commonly used instead of the measured quantities themselves. And if the meaning of words is to he determined bv their 82 THE MATHEMATICAL PRINCIPLES use, then by the names time, space, place and motion, their measures arv properly to be understood ; and the expression will be unusual, and purely mathematical, if the measured quantities themselves are meant. Upon which account, they do strain the sacred writings, who there interpret those words for the measured quantities. Nor do those less defile the purity of mathematical and philosophical truths, who confound real quan tities themselves with their relations and vulgar measures. It is indeed a matter of great difficulty to discover, and effectually to distinguish, the true motions of particular bodies from the apparent ; be cause the parts of that immovable space, in which those motions are per formed, do by no means come under the observation of our senses. Yet the thing is not altogether desperate : for we have some arguments to guide us, partly from the apparent motions, which are the differences of the true motions ; partly from the forces, which are the causes and effects of the true motions. For instance, if tAvo globes, kept at a given distance one from the other by means of a cord that connects them, were revolved about their common centre of gravity, we might, from the tension of the cord, discover the endeavour of the globes to recede from the axis of their motion, and from thence we might compute the quantity of their circular motions. And then if any equal forces should be impressed at once on the alternate faces of the globes to augment or diminish their circular motions, from the increase or decr ase of the tensicn of 1 le cord, we might infer the increment or decrement of their motions : and thence would be found on what faces those forces ought to be impressed, that the motions of the globes might be most augmented ; that is, we might discover their hindermost faces, or those which, in the circular motion, do follow. But the faces which follow being known, and consequently the opposite ones that precede, we should likewise know the determination of their motions. And thus we might find both the quantity and the determination of this circu lar motion, even in an immense vacuum, where there was nothing external or sensible with which the globes could be compared. But now, if in that space some remote bodies were placed that kept always a given position one to another, as the fixed stars do in our regions, we could not indeed determine from the relative translation of the globes among those bodies, whether the motion did belong to the globes or to the bodies. But if we observed the cord, and found that its tension was that very tension which the motions of the globes required, we might conclude the motion to be in the globes, and the bodies to be at rest ; and then, lastly, from the trans lation of the globes among the bodies, we should find the determination oi their motions. But how we are to collect the true motions from their causes, effects, and apparent differences ; and, vice versa, how from the mo tions, either true or apparent, we may come to the knowledge of theii causes and effects, shall be explained more at large in the following tra<;t For to this end it was that I composed it. OF NATURAL PHILOSOPHY. AXIOMS, OR LAWS OF MOTION. LAW I. Hvery body perseveres in its state of rest, or of uniform motion in a ri^ht line, unless it is compelled to change that state by forces impressed thereon. PROJECTILES persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is re tarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve then jDotions both pro gressive and circular for a much longer time. LAW II. The alteration of motion is ever proportional to the motive force impreus ed ; and is made in the direction of the right line in. which that force is impressed. If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subducted from the former motion, according as they directly conspire with or are directly contrary to each other ; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both. LAW III. To every action there is always opposed an equal reaction : or the mu tual actions of two bodies upon each other are always equal, and di rected to contrary parts. Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. 84 THE MATHEMATICAL PRINCIPLES If a body impinge upon another, and by its force change the motion of (It* other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, towards the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of bodies ; that is to say, if the bodies are not hindered by any other impediments. For, because the motions are equally changed, the changes of the velocities made towards contrary parts are reciprocally pro portional to the bodies. This law takes place also in attractions, as will be proved in the next scholium. COROLLARY I. A body by two forces conjoined will describe the diagonal of a parallelo gram, in the same time that it wovld describe the sides, by those forces apart. If a body in a given time, by the force M impressed apart in the place A, should with an uniform motion / be carried from A to B ; and by the force N impressed apart in the same place, should be carried from A to c ~\) C ; complete the parallelogram ABCD, and, by both forces acting together, it will in the same time be carried in the diagonal from A to D. For since the force N acts in the direction of the line AC, parallel to BD, this force (by the second law) will not at all alter the velocity generated by the other force M, by which the body is carried towards the line BD. The body therefore will arrive at the line BD in the same time, whether the rorce N be impressed or not ; and therefore at the end of that time it will he found somewhere in the line BD. By the same argument, at the end of the same time it AY ill be found somewhere in the line CD. Therefore it will be found in the point D, where both lines meet. But it will move in ;i right line from A to D, by Law I. COROLLARY II. And hence is explained the composition of any one direct force AD, out of any two oblique forces AC and CD ; and, on the contrary, the re solution of any one direct force AD into two oblique forces AC and CD : which composition and resolution are abundantly confirmed from, mechanics. As if the unequal radii OM and ON drawn from the centre O of any wheel, should sustain the weights A and P by the cords MA and NP ; and the forces of those weights to move the wheel were required. Through the rentre O draw the right line KOL, meeting the cords perpendicularly in A and L; and from the centre O, with OL the greater of the distances OF NATURAL PHILOSOPHY. OK arid OL, describe a circle, meeting the cord MA in D : and drawing OD, make AC paral- "^ lei and DC perpendicular thereto. Now, it being indifferent whether the points K, L, D, of the cords be lixed to the plane of the wheel or not, the weights will have the same effect whether they are suspended from the points K and L, or from D and L. Let the whole force of the weight A be represented by the line AD, and let it be resolved into the forces AC and CD ; of which the force AC, drawing the radius OD directly from the centre, will have no effect to move the wheel : but the other force DC, drawing the radius DO perpendicularly, will have the same effect as if it drew perpendicularly the radius OL equal to OD ; that is, it w ill have the same effect as the weight P, if that weight is to the weight A as the force DC is to the force DA ; that is (because of the sim ilar triangles ADC, DOK), as OK to OD or OL. Therefore the weights A and P, which are reciprocally as the radii OK and OL that lie in the same right line, will be equipollent, and so remain in equilibrio ; which is the well known property of the balance, the lever, and the wheel. If either weight is greater than in this ratio, its force to move the wheel will be so much greater. If the weight p, equal to the weight P, is partly suspended by the cord NJO, partly sustained by the oblique plane pG ; draw p}i, NH, the former perpendicular to the horizon, the latter to the plane pG ; and if the force of the weight p tending downwards is represented by the line /?H, it may be resolved into the forces joN, HN. If there was any plane /?Q, perpendicular to the cord y?N, cutting the other plane pG in a line parallel to the horizon, and the weight p was supported only by those planes pQ, pG, it would press those planes perpendicularly with the forces pN, HN; to wit, the plane joQ, with the force joN, and the plane pG with the force HN. And therefore if the plane pQ was taken away, so thnt the weight might stretch the cord, because the cord, now sustaining the weight, supplies the place of the plane that was removed, it will be strained by the same force joN which pressed upon the plane before. Therefore, the tension of this oblique cord joN will be to that of the other perpendic ular cord PN as jt?N to joH. And therefore if the weight p is to the weight A in a ratio compounded of the reciprocal ratio of the least distances of the cords PN, AM, from the centre of the wheel, and of the direct ratio of pH tojoN, the weights will have the same effect towards moving the wheel, and will therefore sustain each other : as any one may find by experiment. But the weight p pressing upon those two oblique planes, may be con sidered as a wedge between the two internal surfaces of a body split by it; and hence tlif ft IV.P* of th^ v, ^dge and the mallet may be determined; foi 8G THE MATHEMATICAL PRINCIPLES because the force with which the weight p presses the plane pQi is to the force with which the same, whether by its own gravity, or by the blow of a mallet, is impelled in the direction of the line joH towards both the planes, as joN to pH ; and to the force with which it presses the other plane pG, as joN to NH. And thus the force of the screw may be deduced from a like resolution of forces ; it being no other than a wedge impelled with the force of a lever. Therefore the use of this Corollary spreads far and wide, and by that diffusive extent the truth thereof is farther con firmed. For on what has been said depends the whole doctrine of mechan ics variously demonstrated by different authors. For from hence are easily deduced the forces of machines, which are compounded of wheels, pullics, levers, cords, and weights, ascending directly or obliquely, and other mechan ical powers ; as also the force of the tendons to move the bones of animals. COROLLARY III. The (/uaittity of motion, which is collected by taking the sum of the mo tions directed towards the same parts, and the difference of those that are directed to contrary parts, suffers no change from the action oj bodies among themselves. For action and its opposite re-action are equal, by Law III, and there fore, by Law II, they produce in the motions equal changes towards oppo site parts. Therefore if the motions are directed towards the same parts. whatever is added to the motion of the preceding body will be subducted from the motion of that which follows ; so that the sum will be the same as before. If the bodies meet, with contrary motions, there will be an equal deduction from the motions of both ; and therefore the difference of the motions directed towards opposite parts will remain the same. Thus if a spherical body A with two parts of velocity is triple of a spherical body B which follows in the same right line with ten parts of velocity, the motion of A will be to that of B as 6 to 10. Suppose, then, their motions to be of 6 parts and of 10 parts, and the sum will be 16 parts. Therefore, upon the meeting of the bodies, if A acquire 3, 4, or 5 parts of motion, B will lose as many ; and therefore after reflexion A will proceed With 9, 10, or 11 parts, and B with 7, 6, or 5 parts; the sum remaining always of 16 parts as before. If the body A acquire 9, 10, 11, or 12 parts of motion, and therefore after meeting proceed with 15, 16, 17, or 18 parts, the body B, losing so many parts as A has got, will either proceed with 1 part, having lost 9, or stop and remain at rest, as having lost its whole progressive motion of 10 parts ; or it will go back with 1 part, having not only lost its whole motion, but (if 1 may so say) one part more; or it will go back with 2 parts, because a progressive mo tion of 12 parts is taken off. And so the sums of the Conspiring motions 15 ,1, or 16-1-0, and the differences of the contrary i otions 17 1 and OF NATURAL PHILOSOPHY. [S 2, will always be equal to 16 parts, as they were before tie meeting and reflexion of the bodies. But, the motions being known with whicli the bodies proceed after reflexion, the velocity of either will be also known, by taking the velocity after to the velocity before reflexion, as the motion after is to the motion before. As in the last case, where the motion of tho body A was of parts before reflexion and of IS parts after, and the velocity was of 2 parts before reflexion, the velocity thereof after reflexion will be found to be of 6 parts ; by saying, as the parts of motion before to 18 parts after, so are 2 parts of velocity before reflexion to (5 parts after. But if the bodies are cither not spherical, or, moving in different right lines, impinge obliquely one upon the other, and their mot ons after re flexion are required, in those cases we are first to determine the position of the plane that touches the concurring bodies in the point of concourse , then the motion of each body (by Corol. II) is to be resolved into two, one perpendicular to that plane, and the other parallel to it. This done, be cause the bodies act upon each other in the direction of a line perpendicu lar to this plane, the parallel motions are to be retained the same after reflexion as before ; and to the perpendicular motions we are to assign equal changes towards the contrary parts ; in such manner that the sum of the conspiring and the difference of the contrary motions may remain the same as before. From such kind of reflexions also sometimes arise the circular motions of bodies about their own centres. But these are cases which I do not consider in what follows ; and it would be too tedious to demonstrate every particular that relates to this subject. COROLLARY IV. The common centre of gravity of two or more bodies does not alter its state of motion or rest by the actions of the bodies among themselves ; and therefore the common centre of gravity of all bodies acting upon each other (excluding outward actions and impediments) is either at rest, or moves uniformly in a right line. For if two points proceed with an uniform motion in right lines, and their distance be divided in a given ratio, the dividing point will be either at rest, or proceed uniformly in a right line. This is demonstrated here after in Lem. XXIII and its Corol., when the points are moved in the same plane ; and by a like way of arguing, it may be demonstrated when the points are not moved in the same plane. Therefore if any number of Kdies move uniformly in right lines, the common centre of gravity of any two of them is either at rest, or proceeds uniformly in a right line ; because the line which connects the centres of those two bodies so moving is divided at that common centre in a given ratio. In like manner the common centre of those two and that of a third body will be either at rest or moving uni formly in aright line because at that centre the distance 1 etween th? 88 THE MATHEMATICAL PRINCIPLES common centre of the two bodies, and the centre of this last, is divided in a given ratio. In like manner the common centre of these three, and of a fourth body, is either at rest, or moves uniformly in a right line ; because the distance between the common centre of the three bodies, and the centre of the fourth is there also divided in a given ratio, and so on m itifinitum. Therefore, in a system of bodies where there is neither any mutual action among themselves, nor any foreign force impressed upon them from without, and which consequently move uniformly in right lines, the common centre of gravity of them all is either at rest or moves uniformly forward in a right line. Moreover, in a system of two bodies mutually acting upon each other, since the distances between their centres and the common centre of gravity of both are reciprocally as the bodies, the relative motions of those bodies, whether of approaching to or of receding from that centre, will be equal among themselves. Therefore since the changes which happen to motions are equal and directed to contrary parts, the common centre of those bodies, by their mutual action between themselves, is neither promoted nor re tarded, nor suffers any change as to its state of motion or rest. But in a system of several bodies, because the common centre of gravity of any two acting mutually upon each other suffers no change in its state by that ac tion : and much less the common centre of gravity of the others with which that action does not intervene ; but the distance between those two centres is divided by the common centre of gravity of all the bodies into parts re ciprocally proportional to the total sums of those bodies whose centres they are : and therefore while those two centres retain their state of motion or rest, xhe common centre of all does also retain its state : it is manifest that the common centre of all never suffers any change in the state of its mo tion or rest from the actions of any two bodies between themselves. But in such & system all the actions of the bodies among themselves either hap pen between two bodies, or are composed of actions interchanged between some two bodies ; and therefore they do never produce any alteration in the comrrv n centre of alias to its state of motion or rest. Wherefore tiince that centre, when the bodies do not act mutually one upon another, Oilier is nt rest or moves uniformly forward in some right line, it will, :v\>U7ithst?nding the mutual actions of the bodies among themselves, always jAY-jevere in its state, either of rest, or of proceeding uniformly in a right liiv,, unless it is forced out of this state by the action of some power imprev^-d from without upon the whole system. And therefore the same law take*1 place in a system consisting of many bodies as in one single body, with wsgard to their persevering in their state of motion or of rest. For the pi \\jressive motion, whether of one single body, or of a whole system of bodies us always to be estimated from the motion of the centre of gravity. COROLLARY V. The motions cf bcdies included in a given space a ~e Ike same among OF NATURAL PHILOSOPHY. 89 themselves, whether that space is at rest, or moves uniformly forwards in a right line without any circular motion. For the differences of the motions tending towards the same parts, and the sums of those that tend towards contrary parts, are, at first (by sup position), in both cases the same ; and it is from those sums and differences that the collisions and impulses do arise with which the bodies mutually impinge one upon another. Wherefore (by Law II), the effects of those collisions will be equal in both cases ; and therefore the mutual motions of the bodies among themselves in the one case will remain equal to the mutual motions of the bodies among themselves in the other. A clear proof of which we have from the experiment of a ship ; where all motions happen after the same manner, whether the ship is at rest, or is carried uniformly forwards in a right line. COROLLARY VI. If bodies, any how moved among themselves, are urged in the direct-ton of parallel lines by equal accelerative forces, they will all continue to move among themselves, after the same manner as if they had been urged by no such forces. For these forces acting equally (with respect to the quantities of the DO dies to be moved), and in the direction of parallel lines, will (by Law II) move all the bodies equally (as to velocity), and therefore will never pro duce any change in the positions or motions of the bodies among themselves. SCHOLIUM. Hitherto I have laid down such principles as have been received by math ematicians, and are confirmed by abundance of experiments. By the first two Laws and the first two Corollaries, Galileo discovered that the de scent of bodies observed the duplicate ratio of the time, and that the mo tion of projectiles was in the curve of a parabola; experience agreeing with both, unless so far as these motions are a little retarded by the re sistance of the air. When a body is falling, the uniform force of its gravity acting equally, impresses, in equal particles of time, equal forces upon that body, and therefore generates equal velocities; and in the whole time impresses a whole force, and generates a whole velocity proportional to the time. And the spaces described in proportional times are as the velocities and the times conjunctly ; that is, in a duplicate ratio of the times. And when a body is thrown upwards, its uniform gravity im presses forces and takes off velocities proportional to the times ; and the times of ascending to the greatest heights are as the velocities to be taken off, and those heights are as the velocities and the times conjunetly, or ir, the duplicate ratio of the velocities. And if a body be projected in any direction, the motion arising from its projection jS compounded with the 90 THE MATHEMATICAL PRINCIPLES motion arising from its gravity. As if the body A by its motion of piojection alone could describe in a given time the right line AB, and with its motion of falling alone could describe in the same time the altitude AC ; complete the paralellogram ABDC, and the body by that compounded motion will at the end of the time be found in the place D ; and the curve line AED, which that body describes, will be a parabola, to which the right line AB will be a tangent in A ; and whose ordinate BD will be as the square of the line AB. On the same Laws and Corollaries depend those things which have been demon strated concerning the times of the vibration of pendulums, and are con firmed by the daily experiments of pendulum clocks. By the same, to gether with the third Law, Sir Christ. Wren, Dr. Wallis, and Mr. Huvgens, the greatest geometers of our times, did severally determine the rules of the congress and reflexion of hard bodies, and much about the same time communicated their discoveries to the Royal Society, exactly agreeing among themselves as to those rules. Dr. Wallis, indeed, was something more early in the publication ; then followed Sir Christopher Wren, and, lastly, Mr. Huygens. But Sir Christopher Wren confirmed the truth of the thing before the Royal Society by the experiment of pendulums, which Mr. Mariottc soon after thought fit to explain in a treatise entirely upon that subject. But to bring this experiment to an accurate agreement with the theory, we are to have a due regard as well to the resistance of the air bodies. Let the spherical bodies CD F II as to the clastic force of the concurrin A, B be suspended by the parallel and equal strings AC, Bl), from the centres C, D. About these centres, with those intervals, describe the semicircles EAF, GBH, bisected by the radii CA, DB. Bring the body A to any point R of the arc EAF, and (withdrawing the body B) let it go from thence, and after one oscillation suppose it to return to the point V : then RV will be the