From these facts, and they are all that bear directly on the question concerning the invention of the infinitesimal analysis, if they be fairly and dispassionately examined, I think that no doubt can remain, that Newton was the first inventor of that analysis, which he called by the name of Fluxions; but that, in the communications made by him, or his friends, to Leibnitz, there was nothing that could convey any idea of the principle on which that analysis was founded, or of the algorithm which it involved. The things stated were merely results; and though some of those relating to the tan gents of curves might show the author to be in possession of a method of investigation different from infinite series, yet they afforded no indication of the nature of that me thod, or the principles on which it proceeded.
In what manner Newton's communications, in the two letters already referred to, may have acted in stimulating the curiosity and extending or even directing the views of such a man as Leibnitz, I shall not presume to decide (nor even, if such effect be ad mitted, will it take from the originality of his discoveries); but that in the authenticated communications which took place between these philosophers, there was nothing which could make known the nature of the fluxionary calculus, I consider as a fact most fully established.
Of the new or infinitesimal analysis, we are, therefore, to consider Newton as the first inventor, Leibnitz as the second ; his discovery, though posterior in time, having been made independently of the other, and having no less claim to origin silty. It had the advantage also of being first made known to the world ; an ac count of it, and of its peculiar algorithm, having been inserted in the first volume of the Acta Eruditorwm, in 1684. Thus, while Newton's discovery remained a secret, communicated only to a few friends, the geometry of Leibnitz was spreading with great rapidity over the Continent. Two most able coadjutors, the brothers James and John Bernoulli, joined their talents to those of the original inventor, and illustrated the new methods by the solution of a great variety of difficult and interesting problems. The re serve of Newton still kept his countrymen ignorant of his geometrical discoveries, and the first book that appeared in England on the new geometry was that of Craig, who professedly derived his knowledge from the writings of Leibnitz and his friends. No thing, however,. like rivalship or hostility between these inventors had yet appeared ;' each seemed willing to admit the originality of the other's discoveries ; and Newton, in the passage of the Principia just referred to, gave a highly favourable opinion on the subject of the discoveries of Leibnitz.
The quiet, however, that now prevailed between the English and German philoso phers, was clearly of a nature to be easily disturbed. With the English was convic tion, and, as we have seen, a well grounded conviction, that the first discovery of the Infinitesimal Analysis was the property of Newton ; but the analysis thus discovered was yet unknown to the public, and was in the hands of the inventor and his friends. With the Germans, there• was the conviction, also well founded, that the invention of their countrymin was perfectly original; and they had the satisfaction to see his cal culus everywhere adopted, and himself considered all over the Continent as the sole in ventor. The friends of Newton could not but resist this latter claim, and the friends of Leibnitz, seeing that their master had become the great teacher of the new calcu lus, could not easily bring themselves to acknowledge that he was not the first disco verer. The tranquillity that existed under such circumstances, if once disturbed, was
not likely to be speedily restored.
Accordingly, a remark of Fatio de Duillier, a mathematician, not otherwise very remarkable, was sufficient to light up a flame which a whole century has been hardly sufficient to extinguish. In a paper on the line of swiftest descent, which he pre sented to the Royal Society in 1699, was this sentence : " I hold Newton to have been the first inventor of this calculus, and the earliest, by several years, induced by the evidence of facts; and whether Leibnitz, the second inventor, has any thing from the other, I leave to the judgment of those who have seen the letters and manuscripts of Newton." Leibnitz replied to this charge in the Leipsic Journal, without any.asperity, simply stating himself to have been, as well as Newton, the in ventor; neither contesting nor acknowledging Newton's claim to priority, but asserting his own to the first publication of the calculus.
Not long after this, the publication of Newton's Quadrature of Curves, and his Enumeration of the lines of the third order (1705), afforded the same jonrnalists an op portunity of showing their determination to retort the insinuations of Dnillier, and to carry the war into the country of the enemy. After giving a very imperfect synopsis of the first of these books, they add : " Pro diferentiis igitur Leibtaitianis D. New tonus adhibet, semperque adhibuit fiuxiones ; quce suet proxime ut fluentium aug menta equalibus temporis particulis quam minimis genita ; iisque tuna in suis Prin cipiis Naturce Mathematicis, tuna in alas post editis, eleganter est uses; quatnadmo dutn Honoratus Fabrius in sua Synopsi Geometricc2 1110i2SUSIE progressus Cava lieriance methodo substituit.' In spite of the politeness and ambiguity' of this passage, the most obvious meaning appeared to be, that Newton had been led to the notion of fluxion by the differentials of Leibnitz, just as Honoratus Fabri had been led to substitute the idea of progressive motion for the indivisibles of Cavalieri. A charge so entirely unfounded, so incon tistent with acknowledged facts, and so little consonant to declarations that had for merly come from the same quarter, could not but call forth the indignation of Newton and his friends, especially as it was known, that these journalists spoke the language of Leibnitz and Bernoulli. In that indignation they were perfectly justified; but when the minds of contending parties have become irritated in a certain degree, it often happens that the injustice of one side is retaliated by an equal injustice from the opposite. Accordingly, Keill, who, with more zeal than judgment, undertook the defence of Newton's claims, instead of endeavouring to establish the priority of his dis coveries, by an appeal to facts and to dates that could be accurately ascertained (in which be would have been completely successful), undertook to prove, that the commu nications of Newton to Leibnitz, were sufficient to put the latter in possession of the principles of the new analysis, after which be bad only to substitute the notion of dif ferentials for that of fluxion. In support of a charge which it would have required the clearest and most irresistible evidence to justify, he had, however, nothing to offer but equivocal. facts and overstrained arguments, such as could only convince those who were already disposed to believe. They were, accordingly, received as sound reasoning in England, rejected as absurd in Germany, and read with no effect by the mathematicians of France and Italy.