spoken with so much reserve, instead of saying plainly, that the method of Sluze and that of fluxions were different? Will it be supposed, that he expressed himself thus out of modesty? Surely the truth may be spoken without any infringement of the laws of modesty, even when it is to our own advantage.

All these considerations appear to me to evince, that, if the piece De Analysi per Equationes, and the letter of 1672, contain the method of fluxions, it was at least enveloped in great darkness. But whether it were or not, I shall proceed to demonstrate, that Leibnitz either had no knowledge of these two pieces before he discovered his differential calculus, or derived no information from them. This is a grand point, which his defenders have not sufficiently established, and on which I hope to leave no doubt remaining.

In 1672 Leibnitz quitted the universities of Germany, and came to France, where he was chiefly occupied in the study of the law of nations and history. He was already initiated into mathematics, however, as in 1666 he had published a little tract on some properties of numbers. In the beginning of 1673 he went to London, where he saw Oldenburg, with whom he commenced an epistolary correspondence. In one of his letters to Oldenburg, written even while he was at London, Leibnitz says, that, having discovered a method of summing up certain series by means of their differences, this method was shown to him already published in a book by Mouton, canon of St. Paul's at Lyons, On the Diameters of the Sun

and

and Moon' that he then invented another method, which he explains, of forming the differences, and thence deducing the sums of the series: that he is capable of summing up a series of fractions, of which the numerators are unity, and the denominators either the terms of the series of natural numbers, those of the series of triangular numbers, or those of the series of pyramidal numbers, &c. All these researches are ingenious, and seem to have at least a remote relation to the calculation of differences. The english have never asserted, and at any rate there exists not the least proof, that Leibnitz had seen the two pieces by Newton abovementioned during this first visit to England.

After staying some months in London, Leibnitz returned to Paris, where he formed an acquaintance with Huygens, who laid open to him the sanctuary of the profoundest geometry. He soon found the approximate quadrature of the circle by a series analogous to that, which Mercator had given for the approximate quadrature of the hyperbola. This series he communicated to Huygens, by whom it was highly applauded; and to Oldenburg, who answered him, that Newton had already invented similar things, not only for the circle, but for other curves, of which he sent him sketches. In fact the theory of series was already far advanced in England at that time; and though Leibnitz had likewise penetrated deeply into it, he always acknowledged, that the english, and Newton particularly, had preceded and surpassed him in that branch of analysis: but this is not the differential calculus, and the english have shown too evident

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evident a partiality in their endeavours to connect these two objects together.

Let us hear and examine the history, which Leibnitz gives of his discovery of the differential calculus. He relates, that, on combining his old remarks on the differences of numbers with his recent meditations on geometry, he hit upon this calculus about the year 1676; that he made astonishing applications of it to geometry; that being obliged to return to Hanover about the same time, he could not entirely follow the thread of his meditations; that endeavouring ne vertheless to bring forward [à faire valoir] his new discovery, he went by the way of England and Holland; that he staid some days at London, where he became acquainted with Collins, who showed him se veral letters from Gregory, Newton, and other mathe maticians, which turned chiefly on series.

According to this account, it would appear, that Leibnitz, wishing to spread abroad his new discovery, must then have made known in England the differential calculus. Let us add, that in a letter from Collins to Newton, dated the 5th of march 1677, it is said, that Leibnitz, having spent a week in London in october 1676, had put into Collins's hands some papers*, of which extracts or copies should be sent to Newton immediately. Collins says nothing of the nature of these papers, and we find no trace of them in the Commercium epistolicum. But if the account given. by Leibnitz be just, or if his memory did not

* This passage, and several other large fragments of the same letter, were suppressed in the Commercium epistolicum. See it complete in Wallis's Works, Vol. III. p. 646.

deceive

deceive him, when he said he was in possession of the differential calculus before his second visit to England, no doubt some private reason then occurred, to induce him to keep his discovery secret, contrary to the design he had first formed of bringing it forward: for in this very letter Collins mentions another from Leibnitz to Oldenburg, written from Amsterdam the 28th of November 1676, in which Leibnitz proposes the construction of tables of formulæ tending to improve the method of Sluze, instead of explaining the differential calculus, or at least pointing it out as much more expeditious and more convenient.

The english therefore are justified in saying, that Leibnitz, when he passed through London in 1676, did not teach them the differential calculus: but they ought to acknowledge, that the same letter con-clusively proves, that he likewise learned nothing from them on the subject. In fact, if, as has been asserted since, a knowledge of the method of fluxions had then been imparted to him, must he not have been out of his senses, to propose a month after to the secretary of the Royal Society, a man of great skill in those matters, means of improving the method of Sluze itself, without saying a single word of another method much more simple, which had just been taught him in England?

Thus I believe I may decidedly conclude, either that Leibnitz did not see the work De Analysi and Newton's letter, when he was in England in october 1676; or, if he did see them, that he derived no assistance from them, any more than the learned geometricians of England, who had had all that time to meditate

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ditate on them, and besides were at hand to apply to the author for every necessary illustration. The english have never formally declared, that he had scen the book De Analysi; they contented themselves with positively asserting, that he had seen the letter of 1672. But supposing this to be true, we can draw no inference against Leibnitz from it: for beside that the letter contains only results, without any demonstration, it is not very clear, that it indicates a method essentially different from that of Sluze, as the reader may have remarked from Newton's words already quoted.

In the whole of this business there are but three pieces truly decisive: 1st, a letter from Newton to Oldenburg, dated the 24th of october, 1676, which was communicated to Leibnitz the year following: 2d, the answer which Leibnitz returned to Oldenburg respecting this letter, june the 21st, 1677: 3d, the scholia to Newton's Principia already quoted, published toward the end of 1686. Let us briefly analyze these three pieces.

Newton's letter, exclusive of different researches concerning series, which are here to be left out of the question, contains several theorems, that have the method of fluxions for their basis, but the author keeps his demonstrations secret. He contents himself with saying, that he has deduced them from the solution of a general problem, which he expresses enigmatically by transposing the letters, and the sense of which, as explained after the business was known, is an equation containing flowing quantities being given, to find the fluxions, and inversely.'

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