What light could Leibnitz derive from such an anagram? All we can conclude from such a letter is, that, at the time when it was written, Newton was in possession of the method of fluxions; by which, however, is to be understood simply the method of tangents and of quadratures; for the method of resolving differential equations was then out of the question, this not being invented till long after, as has been said above.

Leibnitz, in his letter to Oldenburg, begins with saying, that he, as well as Newton, had found Sluze's method for tangents to be imperfect. Then he explains openly and without mystery that of the differential calculus, affirming, that he had long employed it for drawing the tangents of curve lines. Here then we have the clear and positive solution of the problem, the possession of which Newton so carefully endeavoured to reserve to himself.

In

The scholia to the Principia say as follows: a correspondence in which I was engaged with the very learned geometrician Mr. Leibnitz ten years ago*, having informed him, that I was acquainted with a method of determining the maxima and minima, drawing tangents, and doing other similar things, which succeeded equally in rational equations and radical quantities, and having, concealed this method by transposing the letters of the words, which signified: an equation containing any number of flowing quantities being given, to find the fluxions, and inversely that celebrated gentleman answered, that

* Through the medium of Oldenburg.

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he

he had found a similar method; and this, which he communicated to me, differed from mine only in the enunciation and notation.' To this the edition of 1714 adds: and in the idea of the generation of quantities.' Is it possible, to say more expressly, that ·Leibnitz separately invented the method of fluxious, and that he had communicated it frankly, without involving himself in mystery like Newton?

From these three pieces therefore it is clear, that, if Newton first invented the method of fluxions, as is pretended to be proved by his letter of the 10th of december 1672, Leibnitz equally invented it on his part, without borrowing any thing from his rival. These two great men by the strength of their genius arrived at the same discovery through different paths: one, by considering fluxions as the simple relations of quantities, which rise or vanish at the same instant; the other, by reflecting, that, in a series of quantities which increase or decrease, the difference between two consecutive terms may become infinitely small, that is to say, less than any determinable finite magnitude.

This opinion, at present universally received except in England, was that of Newton himself, when he first published his Principia, as we see from the extract above given. At that time the truth was near it's source, and not yet altered by the passions. In vain did Newton afterward change his language, led away by the flattery of his countrymen and disciples; in vain did he pretend, that the glory of a discovery belongs entirely to the first inventor, and that second inventors ought not to be admitted to share it. In the first place, without discussing his pretended priority, it was replied, that two men, who separately

make

make the same important discovery, have an equal claim to admiration; and that he, who first makes it public, has the first claim to the public gratitude. It was then proved to him, that even his own principle did not justly apply here.

The design of stripping Leibnitz, and making him pass for a plagiary, was carried so far in England, that during the height of the dispute it was said (and Newton himself was not ashamed to support the objection), that the differential calculus of Leibnitz was nothing more than the method of Barrow. What are you thinking of, answered Leibnitz, to bring such a charge against me? Will you have the differential calculus to be nothing but the method of Barrow, when I claim it? and at the same time say it was invented by Mr. Newton, when you wish to rob me of it? Can you be so blinded by passion, as not to perceive this manifest contradiction? If the diffe rential calculus were really the method of Barrow (which you well know it is not), who would most deserve to be called a plagiary? Mr. Newton, who was the pupil and friend of Barrow, and had opportunities of gathering from his conversation ideas, which are not in his works? or I, who could be instructed only by his works, and never had any acquaintance with the author?

John Bernoulli, who in concert with his brother learned the analysis of infinites from the writings of Leibnitz, opposed to the Commercium epistolicum a letter, where he advances, not only that the method of fluxions did not precede the differential calculus, but that it might have originated from it; and that Newton

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Newton had not reduced it to general analytical operations in form of an algorithm, till the differential calculus was already disseminated through all the journals of Holland and Germany. His reasons are in substance, 1st, the Commercium epistolicum exhibits no vestige of Newton's having employed dotted letters, to denote fluxions, in the writings alleged: 2dly, in the Principia, where the author had so frequently occasion for employing this calculus, and giving it's algorithm, he has not done it; he proceeds every where by means of lines and figures, without any determinate analysis, and simply in the manner of Huygens, Roberval, Cavalleri, &c.: 3dly, the dotted letters first began to appear in the third volume of Wallis's Works, several years after the differential calculus was every where known: 4thly, the true method of differencing differences, or of taking the fluxions of fluxions, was unknown to Newton, since even in his treatise on quadratures, not published till 1704, the rule he gives at the end for determining the fluxions of all orders, by considering these fluxions as the terms of the power of a binomial formed of a variable quantity, and it's first fluxion, and treating this first fluxion as constant, is false, except simply for the term which answers to the first fluxion: 5thly, at the same period of 1704, Newton was not versed in the integral calculus of differential equations, which Leibnitz and the two Bernoullis had already carried so far; otherwise he would not have failed to treat this part of the analysis of infinites, the most difficult, and at least as worthy of

being

being promulgated and carried to perfection as the quadratures, on which he enlarged so much.

To this letter the english answered, that the notation did not constitute the method: that the principles of the calculus of fluxions were contained in Newton's great work, and in his letters: that the rule in the treatise on quadratures for finding the fluxions of all orders was true, suppressing the denominators of the terms of the series, and gave by consequence quantities proportional to the true fluxions. I do not find, that they gave any answer to the last objection.

The partizans of Leibnitz replied, that the advantages of an analytic method depend in great measure on the simplicity of the algorithm: that the Characteristic of Leibnitz had already occasioned the new analysis to make immense progress, at a time when scarcely any one had heard of Newton's book: that it was in vain to endeavour to deny or palliate the erroneousness of Newton's rule for finding the fluxions of all orders: and that it could not be said, that the terms of a series of fractions were proportional to the terms of another series of fractions, when the corresponding terms had different denominators, as was the case here,

Such were nearly the reasons alleged and contested between the two parties for more than four years. The death of Leibnitz, which happened in 1716, it may be supposed, should have put an end to the dispute: but the english, pursuing even the manes of that great man, published in 1726 an edition of the BB 4

Principia,