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Newton*, (dated Cambridge, 26th June, 1686,) especially as it will enable us to trace more clearly the progress and developement of his ideas throughout this important research.

"In order to let you know the case between Mr. Hooke and me, I give you an account of what passed between us in our letters, so far as I could remember; for 'tis long since they were writ, and I do not know that I have seen them since. I am almost confident by circumstances, that Sir Christopher Wren knew the duplicate proportion when I gave him a visit; and then Mr. Hooke, by his book Cometa, written afterwards (1678), will prove the last of us three that knew it. I intended in this letter to let you understand the case fully, but it being a frivolous business, I shall content myself to give you the heads of it in short, viz. that I never extended the duplicate proportion lower than to the superficies of the earth, and before a certain demonstration I found the last year, have suspected it not to reach accurately enough down so low; and therefore in the doctrine of projectiles never used it, nor considered the motion of the heavens, and consequently Mr. Hooke could not, from my letters, which were about projectiles, and the regions descending hence to the centre, conclude me ignorant of the theory of the heavens. That what he told me of the duplicate proportion was erroneous, namely, that it reaches down from hence to the centre of the earth-that it is not candid to require me now to confess myself in print then ignorant of the duplicate proportion in the heavens, for no other reason but because he had told it me in the case of projectiles, and so upon mistaken grounds accused me of that ignorance; that, in my answer to his first letter, I refused his correspondence; told him I had laid philosophy aside, sent him only the experiment of projectiles (rather shortly hinted, than carefully described) in compliment, to sweeten my answer, expected to hear no further from him, could scarce persuade myself to answer his second letter, did not answer his third, was upon other things, thought no further of philosophical matters than his letters put me upon it, and therefore may be allowed not to have had my thoughts about me so well at that time. That, by the same

This latter is printed in the Biographia Brit.

tannica, Art. Hooke,

reason, he concluded me ignorant of the rest of that theory I had read before in his books. That, in one of my papers, writ (I cannot say what year, but I am sure some time before I had any correspondence with Mr. Oldenburg, and that's above fifteen years ago) the proportion of the forces of the planets to the sun reciprocally duplicate to their distances from him, and the proportion of our gravity to the moon's conatus recedendi a centro terræ is calculated, though not accurately enough.—That, when Huygenius put out his treatise de Horologio Oscillatorio, a copy being presented to me, in my letter of thanks to him I gave those rules in the end thereof a particular commendation for their usefulness in computing the forces of the moon from the earth, and the earth from the sun, in determining a problem about the moon's phase, and putting a limit to the parallax, which shews that I had then my eye upon the forces of the planets arising from their circular motion, and understood it; so that a while after, when Mr. Hooke propounded the problem solemnly in the end of his Attempt to prove the motion of the earth, if I had not known the duplicate proportion before, I could not but have found it now. Between ten and eleven years ago, there is an hypothesis of mine registered in your books, wherein I hinted a cause of gravity towards the earth, sun, and planets, with the dependence of the celestial motions thereon; in which the proportion of the decrease of gravity from the superficies of the planet (though for brevity sake not there expressed) can be no other than reciprocally duplicate of the distance from the centre; and I hope I shall not be urged to declare in print that I understood not the obvious mathematical conditions of my own hypothesis; but grant I received it afterwards from Mr. Hooke, yet have I as great a right to it as to the ellipsis. For as Kepler knew the orb to be not circular but oval, I guessed it to be elliptical; so Mr. Hooke, without knowing what I have found out since his letters to me, can know no more but that the proportion was duplicate quam proxime at great distances from the centre, and only guessed it to be so accurately, and guessed amiss in extending that proportion down to the very centre; whereas Kepler guessed right at the ellipsis, and so Hooke found less of the proportion than Kepler did of the

ellipse, there is so strong an objection against the accurateness of this proportion, that without my demonstrations, to which Hooke is yet a stranger, it cannot be believed by a judicious philosopher to be anywhere accurate. And so, in stating this business, I do pretend to have done for the proportion as for the ellipse, and to have as much right to the one from Hooke and all men, as to the other from Kepler, and, therefore, on this account also, he must, at least, moderate his pretences. The proof you sent me I like very well: I designed the whole to consist of three books; the second was finished last summer, being short, and only wants transcribing, and drawing the cuts fairly. Some new proportions I have since thought of, which I can as well let alone. The third wants the theory of comets. In autumn last, I spent two months in calculations to no purpose, for want of a good method, which made me afterwards return to the first book, and enlarge it with divers propositions, some relating to comets, others to other things found out last winter. The third I now design to suppress. Philosophy is - such an impertinently litigious lady, that a man had as good be engaged in lawsuits, as have to do with her. I found it so formerly, and now I am no sooner come near her again, but she gives me warning. The two first books, without the third, will not bear so well the title of Philosophic Naturalis Principia Mathematica; and, therefore, I had altered it to this, De Motú corporum libri duo; but, upon second thoughts, I retain the former title, 'twill help the sale of the book, which I ought not to diminish now 'tis yours."

Newton then adds, in a postscript, "Sinee my writing this letter, I am told by one who had it from another lately present at one of your meetings, how that Mr. Hooke should make a great stir, pretending I had all from him, and desiring they would see that he had justice done him. This carriage to wards me is very strange and undeserved; so that I cannot forbear in stating the point of justice, to tell you further that he has published Borelli's hypothesis in his own name; and the asserting of this to himself, and completing it as his own, seems to me the ground of all the stir he makes. Borelli did something and wrote modestly. He has done nothing, and yet written in such a way, as if he knew, and had suf

ficiently hinted all but what remained to be determined by the drudgery of calculations and observations, excusing himself from that labour, by reason of his other business; whereas he should rather have excused himself by reason of his inability-for it is very plain, by his words, he knew not how to go about it. Now is not this very fine? Mathematicians that find out, settle, and do all the business, must content themselves with being nothing but dry calculators and drudges; and another that does nothing but pretend and grasp at all things, must carry away all the invention, as well of those that were to follow him, as those that went before. Much after the same manner were his letters writ to me, telling me that gravity in descent from hence to the centre of the earth was reciprocally in a duplicate ratio of the altitude that the figure described by projectiles in that region would be an ellipsis, and that all the motions of the heavens were thus to be accounted for; and this he did in such a way, as if he had found out all, and knew it most certainly. And upon this information, I must now acknowledge, in print, I had all from him, and so did nothing myself but drudge in calculating, demonstrating, and writing upon the inventions of this great man; and yet, after all, the first of these three things he told me is false, and very unphilosophical; the second is as false; and the third was more than he knew, or could affirm me ignorant of, by anything that passed between us in our letters. Nor do I understand by what right he claims it as his own; for as Borelli wrote long before him, that, by a tendency of the planets towards the sun, like that of gravity or magnetism, the planets would move in ellipses: so Bullialdus wrote, that all force respecting the sun as its centre, and depending upon matter, must be in a reciprocally duplicate ratio of the distance from the centre, and used that very argument for it, by which you, Sir, in the last Transactions, have proved this ratio in gravity."

The remainder of this letter offering no other historical details, we will not continue the quotation; but the extremely curious reply of Halley to Newton is well worthy of attention. It is dated 20th June, 1686. Halley begins by encouraging Newton not to heed the effects of Hooke's expostulations with the Royal Society, and then continues,

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According to your desire, I waited upon Sir C. Wren, to inquire of him, if he had the first notion of the reciprocal duplicate proportion from Mr. Hooke? his answer was, that he himself, very many years since, had had his thoughts upon making out the planet's motions by a composition of a descent towards the sun and an impressed motion; but that at length he gave over, not finding the means of doing it. Since which time Mr. Hooke had frequently told him that he had done it, and attempted to make it out to him, but that he never was satisfied that his demonstrations were cogent. And this I know to be true, that in January, 168, I having, from the sesquialterate proportion of Kepler, concluded that the centripetal force decreased in the proportion of the squares of the distance reciprocally, came on Wednesday to town, from Islington, where I met with Sir C. Wren and Mr. Hooke, and falling in discourse about it, Mr. Hooke affirmed, that upon that principle all the laws of the celestial motions were to be demonstrated, and that he himself had done it. I declared the ill success of my attempts; and Sir Christopher, to encourage the inquiry, said, that he would give Mr. Hooke, or me, two months time to bring him a convincing demonstration thereof; and besides the honour, he of us that did' it should have from him a present of a book of forty shillings. Mr. Hooke then said he had it, but that he would conceal it for some time, that others, trying and failing, might know how to value it, when he should make it public. However, I remember that Sir Christopher Wren was little satisfied that he could do it; and though Mr. Hooke then promised to show it to him, I do not find that, in that particular, he has been so good as his word. The August following, when I did myself the honour to visit you, I then learned the good news, that you had brought this demonstration to perfection, and you were pleased to promise me a copy thereof, which I received with great satisfaction; and thereupon took another journey to Cambridge, on purpose to confer with you about it, since which time it has been entered upon the register-books of the society. Mr. Hooke, according to the philosophically ambitious temper he is of, would, had he been master of a like demonstration, no longer have concealed it, the reason he told Sir Christopher and me now ceasing.

But now he says that it is but one small part of an excellent system of nature, which he has conceived but has not yet completely made out; so that he thinks not fit to publish one part without the other. But I have plainly told him, unless he produce another differing demonstration, and let the world judge of it, neither I nor any one else can believe it.. After the meeting of the Royal Society, at which your book was presented, being adjourned to the Coffee-house, Mr. Hooke did there endeavour to gain belief, that he had some such things by him, and that he gave you the first hint of this invention; but I found they were all of opinion that nothing thereof appearing in print, nor on the books of the Society, you ought to be considered as the inventor. And if in truth he knew it before you, he ought not to blame any one but himself, for having taken no more care to secure a discovery which he puts so much value on." Halley concludes, by conjuring Newton, in the name of science, not to suppress the third volume through disgust at the conduct of an envious rival. Happily he succeeded, and Newton has, in a scholium,* generously mentioned Wren, Hooke, and Halley, as having all three recognized in the celestial motions the existence of an attraction reciprocally proportional to the square of the distance.

Newton's Principia appeared complete in 1687. We may form some idea of the novelty and profundity of the discoveries which it contained, on learning that, when it was first published, not more than two or three among Newton's contemporaries were capable of understanding it; that Huygens himself, a man whose mind was particularly suited to appreciate its merit, only in part adopted the idea of gravitation, and that merely as regarded the heavenly bodies, while he rejected its influence between the separate particles of matter-being preoccupied by the hypothetical ideas he had formed respecting the cause of gravity; that Leibnitz, perhaps through rivalry, or perhaps by a prepossession in favour of his own metaphysical system, completely mistook the beauty and the certainty of the method employed by Newton in this work, and even went so far as to publish a dissertation, in which he endeavoured to demonstrate the same truths on different principles;

Book 1, Prop. 4..

that even many years after the publication of the Principia, several most profound mathematicians (John Bernoulli, for instance) opposed it, and that Fonenelle, though in advance of his age on most subjects of philosophy, expressed somewhat more than doubts concerning the law of attraction, and persisted, during his whole life, in upholding the vortices of Descartes; and in fine, that more than fifty years elapsed before the great physical truth contained and demonstrated in the Principia was, we do not say followed up and developed, but even understood by the generality of learned men. Whatever difficulty, however, the just appreciation of such a work may present, we can here give a brief account of it with entire confidence, by translating the words of that illustrious man, whose genius has so much contributed to Newton's glory, in having by his own discoveries subjected all the movements of the celestial bodies to the law of universal gravitation. After having exhibited him as setting out from the laws of Kepler, in order to discover the nature and the law of the force that governs the motions of the planets and the satellites in their orbits, and afterwards generalizing this idea according to the phenomena that presented themselves until he had ascended to the certain and mathematical knowledge of universal gravitation, "Newton," says LAPLACE, "having arrived at this point, saw all the great phenomena of the universe flow from the principle he had discovered. By considering gravity at the surface of the heavenly bodies as the result of the attractions of all their particles, he discovered this remarkable and characteristical property of a law of attraction reciprocal to the square of the distance, namely, that two spheres formed of concentric layers, and with densities varying according to any law whatever, attract each other mutually, as if their masses were united at their centres. Thus the bodies of the solar system act upon each other, and upon the bodies placed at their surfaces, very nearly as if they were so many centres of attraction-a result which contributes to the regularity of their movements, and which made this illustrious mathematician recognize the gravity of the earth in the force that retains the moon in her orbit. He proved that the

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Exposition du Système du Monde, par Mons. le Compte LAPLACE. Paris, 1813. 4to. pp. 413, 426.

earth's movement in rotation must have flattened it at the poles; and he determined the laws of gravitation in the degrees of the meridian, and in the force of gravity at the earth's surface. He saw that the attractions of the sun and moon excite and maintain in the ocean those oscillations which are there observed under the name of tides. He recognized several inequalities in the moon's motion and the retrograde motion of her nodes to be owing to the action of the sun. Afterwards, considering the excess of matter in the terrestrial spheroid at the equator, as a system of satellites adhering to its surface, he

found that the combined actions of the sun and of the moon tend to cause a retrogradation, in the nodes of the circles they describe round the axis of the earth; and that the sum of these tendencies being communicated to the whole mass of the planet, ought to produce in the intersection of its equator with the eclip tic that slow retrogradation known by the name of the precession of the equinoxes. The true cause of this great phenomenon could not have even been suspected before the time of Newton, since he was the first who made known the two leading facts on which it depends. Kepler himself, urged by an active imagination to explain every thing by hypothesis, was constrained to avow in this instance the failure of his efforts. But, with the exception of the theory of the elliptical motions of the planets and comets, the attraction of spheres, the ratio of the masses of the planets accompanied by satellites to that of the sun, all the other discoveries respecting the motions and figures of the heavenly bodies were left by him in an incomplete state. His theory of the figures of the planets is limited, by supposing them to be homogeneous. His solution of the problem of the precession of the equinoxes, though very ingenious, and notwithstanding the apparent agreement of its result with observations, is defective in many particulars. Among the numerous perturbations in the motions of the heavenly bodies, he has only considered those of the moon, the greatest of which, viz. evection, has wholly escaped his researches. Newton has well established the existence of the principle he had the merit of discovering; but the development of its consequences and advantages has been the work of the successors of this great mathematician. The imperfection of the infinitesimal calculus when

first discovered, did not allow him completely to resolve the difficult problems which the theory of the universe offers; and he was oftentimes forced to give mere hints, which were always uncertain till confirmed by rigorous analysis. Notwithstanding these unavoidable defects, the importance and the generality of his discoveries respecting the system of the universe, and the most interesting points of natural philosophy, the great number of profound and original views which have been the origin of the most brilliant discoveries of the mathematicians of the last century, which were all presented with much elegance, will insure to the Principia a lasting preeminence over all other productions of the human mind."

The great results that Newton has amassed in the Principia are almost all presented in a synthetical form, like that used in the writings of the ancients. Nevertheless we may assert, that he did not discover them by means of synthesis, which is neither sufficiently easy of application, nor sufficiently fertile in results to be employed in discovering such complicated truths, or for foreseeing consequences so remote from their first principle. It is hence evident, from this very impossibility, that Newton attained these great results by the help of analytical methods, of which he had himself so much increased the power; and this conclusion acquires certainty from the correspondence between Newton and Cotes, relating to the second edition* of the Principia, for in it we find Cotes, the pupil of Newton, employing the analytical form either in submitting to Newton the difficulties he met with, or in solving them himself. It remains to be explained why Newton preferred setting forth his discoveries by a different method, thus depriving himself of the increase of glory he would infallibly have obtained, by giving to the world the several analytical inventions with which he must have been acquainted in solving the questions he has treated. Among these we may mention the principle of the calculus of variations, which must have been necessary to him in determining the solid of the least resistance. It were difficult to say with certainty what decided him to make such a sacrifice, but if we may hazard a conjecture, it may not be impossible that,

M. Biot examined this correspondence at Cambridge.

from the excessive apprehension which he laboured under of having his results attacked, he preferred the synthetical form, as being a severer method of demonstration, and as being likely to inspire more confidence in those who should read his work at a time when the methods of the infinitesimal analysis were still but little known; and when, from their novelty, they might appear less convincing to many of his readers. Whilst the Principia were preparing for the press, chance produced an incident that drew Newton from his studious retreat, and brought him on the theatre of public affairs. King James II. desiring to re-establish catholicism in England, and thinking fit to attack the usages and rights of the Protestants, had, among other measures, commanded the University of Cambridge to confer the degree of M. A. on Francis, a Benedictine Monk, without requiring of him the oath prescribed by the statutes against the catholic religion. The University asserted its privileges; and Newton (who had shown himself one of the most ardent in encouraging resistance) was one of the delegates sent to maintain their rights before the High Commission Court. These delegates made so firm and unexpected a defence, that the king thought proper to drop the affair. It was this circumstance, perhaps, as much as the personal merit of Newton, that induced the University to elect him, the following year, as their representative to serve in the Convention Parliament, which declared the throne vacant, and called William to the crown. He sat in this parliament until its dissolution, but without acting a remarkable part. C. Montague, afterwards Earl of Halifax, was a member at the same time, and having been educated at Cambridge, was able to appreciate the merit of the genius who formed the glory of the University. Hence, when Halifax, having become Chancellor of the Exchequer, in 1696, conceived the design of a general recoinage, he demanded and obtained for Newton the honourable and lucrative employment of Warden of the Mint, which was at once an act of kindness, and a choice influenced by discernment. In fact, Newton rendered very signal service in executing the important measure which the statesman had determined on ; being

• Vide Burnet, History of hisOwn Time, vol. i. p. 698,

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