The great renown which Newton had acquired, caused all his productions to be received with avidity. Hence it was that Whiston published in 1707, without the knowledge or consent of Newton, the "Arithmetica universalis," which appears to have been merely the text of the lectures on Algebra, that he delivered at Cambridge, written rapidly for his own use, and not intended for publication. Science, however, must congratulate itself on the transgression of confidence that has fortunately made this work known; for it were impossisible to see a more perfect model of the art by which geometrical or numerical questions may be submitted to algebraical calculation; whether we regard the happy choice of the unknown quantities, or the ingenious combination of analytical formulæ, employed in finding the simplest method of solution. A second and more complete edition was published in London in 1712, according to Gravesande, with the participation of Newton himself-a proof that this production of his youth appeared to him neither unworthy of his name nor of his

attention.

It was also, by the care of some other editor, but with his consent, that in 1711 a small treatise, entitled "Methodus differentialis," was published, in which he shows how to draw a parabolic curve through any given number of points-a determination which, when reduced into formulæ, is very useful in the interpolation of series, and in approximating to the quadratures of

curves.

In the same year, by other hands, was published the long-suppressed treatise, "Analysis per equationes numero terminorum infinitas," which he had composed in 1665, and in which, as we have already said, he had explained his first discoveries in fluxions, and in expansions, by means of infinite series. A copy of this dissertation had formerly been taken by Collins, from the original sent to him by Barrow; and having been found among his papers after his death, leave was obtained from Newton to publish it-a permission which he probably gave the more willingly, as the work being of old date, incontestably established his claims to the invention of the new method.

Newton formerly had prepared, on the same subject, a more extensive treatise, entitled "method of Fluxions," which he proposed to join as an introduction

to a treatise on algebra, by Kinckhuysen, of which he had undertaken to publish an edition in 1672: this, without doubt, would have been more valuable than the book itself, but his fear of scientific quarrels induced him then to keep his manuscript secret. Towards the close of his life, he again thought of publishing it, but it was not printed till after his death. The same apprehension had, as we have already said, prevented him from publishing his " Optical Lectures" delivered at Cambridge. Happily, however, he had entrusted copies to many persons, and among others, to Gregory, professor of astronomy at Oxford, one of which being printed three years after his death, has preserved to us this work. It presents a very detailed experimental exposition of the phenomena of the composition and decomposition of light, with their most usual applications: it is, in fact, the Optics without the most difficult part, viz. the theory of colours produced by thin plates; but, in the other parts, fully developed both by calculations and by numerous experiments. In this form, it was extremely proper for the use to which Newton intended it, and at this day it offers a most valuable model for an elementary exposition of phenomena by experiment.

Here would terminate our account of the works on which the fame of Newton reposes, had not a new literary dispute (about 1712), which, in fact, he did not provoke, and the existence of which, perhaps, he more than once regretted, completely revealed all the fertility of his wonderful genius, and assembled a multitude of analytical discoveries, which we find in the correspondence that ensued. We have seen that Newton, for a long time, obstinately guarded the secret of his discoveries, and particularly that of the method of fluxions, of which he justly foresaw the future utility in calculating the phenomena of nature. However, in 1676, Leibnitz having heard of the new results that Newton was said to have obtained by means of infinite series, testified to Oldenburg the desire he felt to become acquainted with them. The latter induced Newton not to refuse a communication which could not but be honourable to him. In consequence (23rd of June, 1676), Newton sent to Oldenburg a letter to be transmitted to Leibnitz, in which he gave expressions for the expansion in series of binomial powers, of the sine in terms of the arc, of the are

in terms of its sine, and of elliptical, circular, and hyperbolic functions, without, however, any demonstration or indication of the means he had used for obtaining these results; merely stating that he possessed a method by which, when these series were given, he could obtain the quadratures of the curves from which they were derived, as well as the surfaces and centres of gravity of the solids formed by their rotation. This may in fact be done by considering each term of these series as the ordinate of a particular curve, and by then applying the method previously given by Mercator, for squaring curves, of which the ordinates are expressed rationally in terms of the abscissa. This is precisely what Leibnitz remarked in his answer to Newton on the 27th of the following August, adding that he should be glad to know the demonstration of the theorems on which Newton founded his method of reducing into series; but that, for himself, though he recognized the utility of this method, he employed another, which consisted in decomposing the given curve into its superficial elements, and in transforming these infinitely small elements into others, equivalent to them, but belonging to a curve whose ordinate was expressed rationally in terms of the abscissa, so that the method of Mercator might be applied in squaring it. After giving different explanations of this method, he declares in express terms that he does not believe that "all problems, except those of Diophantes, can be resolved by it alone, or by series," as Newton had affirmed in his letter; and among the problems which elude these processes, he mentions the case of finding curves from their tangents; adding that he had already treated many questions of this sort by means of a direct analysis, and that the most difficult had been thus solved. This was more than enough to show Newton that Leibnitz was at least upon the track of the infinitesimal calculus, if he did not possess it already; and, therefore, in his answer (dated Oct. 24th, though apparently delivered to Leibnitz much later), after giving the explanations requested by Leibnitz on the formation of binomial series, and after stating to him the succession of ideas, by means of which he had discovered them, Newton hastens to declare that he possesses for drawing tangents to curves a method equally applicable to equations, whether disen

gaged or not of radical quantities; "but," he adds, " as I cannot push further the explication of this method, I have concealed the principle in this anagram." ***

He announced that he had established on this foundation many theorems for simplifying the quadrature of curves, and gave expressions for the areas in terms of the ordinates in several simple cases; but he enveloped both the method and the principle on which it rested in another anagram more complicated than the first.

The evident object of Newton, in this letter, was to place his claims to priority of invention in the hands of Leibnitz himself. The noble frankness of Leibnitz appears on this occasion to the greatest advantage: for in his answer to Newton (21st of June, 1677) he employs neither anagram nor evasion, but details simply and openly the method of the infinitesimal calculus, with the differential notation, the rules of differentiation, the formation of differential equations, and the applications of these processes to various questions in analysis and geometry; and, what mathematicians will consider as far from being unimportant, the figures employed in the exposition of these methods offer precisely the same letters, and the same method of notation, that Leibnitz had used in his first letter of the 14th of April the preceding year. Newton made no reply to this memorable letter, either because he no longer felt the wish, or because, from Oldenburg's death, (which happened in the autumn of the same year,) he had no longer an opportunity of doing so.

Leibnitz published his differential method in the Leipzig Acts for 1684, in a form exactly similar to that which he had sent to Newton. No claim was set up at that time to contest his right of discovery, and Newton himself, three years afterwards, eternalized that right by recognizing it in the Principia, in the following terms. "In a correspondence which took place about ten years ago, between that very celebrated mathematician G. Leibnitz and myself, I mentioned to him that I possessed a method (which I concealed in an anagram) for determining maxima and

minima, for drawing tangents, and for similar operations, which was equally applicable both to rational and irrational quantities: that illustrious man replied that he also had fallen on a method of the same kind (se quoque in ejusmodi methodum incidisse), and communicated to me his method, which scarcely differed from mine, except in the notation and the idea of the generation of quantities."

There is a curious ambiguity in the words, "he replied that he had fallen on a method of the same kind," which, to those who had not seen the letters that were interchanged, might convey the idea, that Leibnitz had discovered the key to Newton's anagram; but this meaning is not to be found in Leibnitz's letter; he only announces a supposition, honourable to his character, viz. that the concealed method of Newton has, perhaps, some connexion with that which he communicates to him. With this explanation, the above passage in the Principia is in truth a formal recognition of Leibnitz's claims. It was so considered by every one when it appeared, and during twenty years Leibnitz was allowed, without any dispute, to develope all the parts of the differential calculus, and to deduce from it an immense number of brilliant applications, which seemed to extend the power of mathematical analysis far beyond any preconceived limits. In this interval, Wallis, by publishing the above-mentioned letters between Leibnitz and Newton, only rendered, if possible, the claims of the former more complete and more incontestable in the eyes of every impartial person. It was not till 1699 that Nicholas Fatio de Duillier,* in a Memoir, in which he employed the infinitesimal calculus, claimed, in favour of Newton, the first invention of it; "and," added he, "with regard to what Mr. Leibnitz, the second inventor of this caculus may have borrowed from Newton, I refer to the judgment of those persons who have seen the letters and manuscripts relating to this business." Did Fatio really believe what he was writing, or did he wish to flatter the national pride of the country in which he lived? or was he not in some manner irritated at Leibnitz having rendered so little justice to the Principia, and at his appearing to arrogate to himself a sort of empire over all discoveries made by

• A Genevese settled in England.

the aid of the new calculus? These questions we do not pretend to decide; but the two latter suppositions are the most probable. Leibnitz replied, by stating the facts, and quoting his letters, and the testimony rendered to him by Newton himself. Fatio was silent; and thus the matter stood till 1704, when Newton published the Optics. In giving an account of the treatise on the quadrature of curves, which was joined to this work, the editor of the Leipzig Acts naturally mentioned the evident analogy that existed between Newton's method of fluxions and the differential calculus which had been published twenty years previously by Leibnitz, in the same Acts, and which had since become the means of making an infinity of analytical discoveries. In comparing the two methods, the editor (whom Newton supposes to have been Leibnitz himself) did not precisely say, that the method of fluxions was a mere transformation of the differential calculus; but he used terms which might bear such an interpretation. This was the signal for attack, on the part of the English writers: one of the most violent of them, Keil, professor of astronomy at Oxford, said, in a paper printed in the Philosophical Transac tions, not only that Newton was the first inventor of the method of fluxions, but also that Leibnitz had stolen it from him, by merely changing the name and the notation used by Newton. This produced an indignant reply from Leibnitz, who had the imprudence to submit the question to the judgment of the Royal Society, that is to say, of a tribunal which was presided over by his rival. The society, with scrupulous fidelity, collected all the original letters that could be found bearing on the matter in question, and thus, with regard to the facts, its conduct was unimpeachable; but the most important and delicate part of the business, viz. the discussion of those papers, and the consequences to be deduced by them, it referred to arbitrators chosen by itself, who were not known, and about whose appointment Leibnitz was not consulted. These arbitrators decided that Newton had indubitably been the first discoverer of the method of fluxions, a truth which is certainly incontestable in the sense that discovery and invention are synonimous terms; but they also added two assertions, which can only be considered as the expression of their personal opinion-first, that the differential and

fluxional methods are one and the same thing; and, secondly, that Leibnitz must have seen a letter of Newton's, (dated 10th December, 1672,) in which the method of fluxions is described in a manner sufficiently clear for any intelligent person to understand. Now of these two assertions, the second is not proved in any one of its parts, and the letter of Newton alluded to, appears, according to his custom, to have been more intended for establishing his right, than proper for indicating the manner of attaining his method. With regard to the first assertion, that the methods are absolutely identical, it may easily be refuted by the simple consideration, that if the method of fluxions alone existed at the present moment, the invention of the differential calculus with its notation, and its principle of decomposition into infinitely small elements, would still be an admirable discovery, and one which would immediately bring to light a number of applications, which we now possess, but which probably would not have been obtainable without its assistance. Admitting then, as certain, the priority of Newton's ideas on this subject, we think that the reserve he maintained regarding it left the field open to all other inventors; and that from the general tendency of the mathematical researches of that period, both Leibnitz and Newton might have separately arrived by different means at the knowledge of a method, the want of which was then so sensibly felt in all analytical researches. The quarrel between Newton and Leibnitz has not been without advantage to mathematical science; since it produced the precious collection of letters on infinitesimal analysis, collected by the Royal Society, and published in 1712, under the name of the Commercium Epistolicum. But as regards these two great men themselves, the bitterness with which it inspired the one against the other, became the torment and the misfortune of the remain der of their lives. Newton went so far as to affirm, that Leibnitz had deprived him of the differential calculus, and then that this calculus was identical with Barrow's method of tangents: an assertion of which he could not but have perceived the injustice, since, if he pretended, on the one hand, that the differential calculus and the method of fluxions were the same, he must have also admitted the method of fluxions to be identical with Barrow's method of tan

This

gents, an assertion which he was far from admitting. Newton suffered himself to be carried away so far as to pretend that the paragraph inserted in the Principia, by which he had so openly acknowledged the independent rights of Leibnitz, was by no means intended to render him that testimony, but, on the contrary, to establish the priority of the method of fluxions over that of the differential calculus. Newton's animosity was not even calmed by the death of Leibnitz, in 1716: for he immediately afterwards printed two manuscript letters of Leibnitz, written in the preceding year, accompanied with a bitter refutation. Six years later, (in 1722) he caused a new edition of the Commercium Epistolicum to be printed, at the head of which he placed a very partial extract from this Collection. was apparently made by himself, and had already appeared two years before the death of Leibnitz, in the Philosophi cal Transactions for 1715. Finally, Newton had the weakness to leave out, or allow to be left out, in the third edition of the Principia, published under his own inspection, 1725, the famous Scholium, in which he had admitted the rights of his rival. To render such conduct, not to say excusable, but even comprehensible, on the part of a man who must so well have known that the only tribunal that can decide on such causes is impartial posterity, it is necessary to say that Leibnitz, on his side, had neither been less passionate nor less unjust. Hurt by the unexpected publication of the Commercium Epistolicum, and irritated by a decision, given without his knowledge, by judges whom he had not appointed, and who had not waited for his defence, he summoned contrary testimonies in his support. Leibnitz had the misfortune to produce proofs equally exaggerated with those brought forward by Newton. He printed, and spread throughout Europe, an anonymous letter (since discovered to have been written by J. Bernoulli), extremely injurious to Newton, whom it represented as having fabricated his method of fluxions from the differential calculus. Leibnitz committed a still greater fault. He was in the habit of corresponding with the Princess of Wales, daughterin-law to George the First. This princess, endowed with a highly cultivated mind, had received Newton with extreme kindness, and was fond of conversing with him. She declared that she esteemed

herself happy in living at a time that enabled her to become acquainted with so great a genius. Leibnitz made use of his correspondence with the princess, to lower Newton in her eyes, and to represent his philosophy to her not only as physically false, but also as dangerous in a religious point of view; and, what is still more inconceivable, he founded these accusations on passages in the Principia, and in the Optics, which Newton had evidently composed and inserted with intentions sincerely religious, and as genuine professions of his firm belief in a divine Providence. For instance, in explaining the true method to be pursued in natural philosophy, Newton says, in his Twenty-eighth Query," the main business of this science is to argue from phenomena, without feigning hypotheses, and to deduce causes from effects, till we come to the very First Cause; which certainly is not mechanical: and not only to unfold the mechanism of the world, but chiefly to resolve these and such like questions. What is there in places almost empty of matter, and whence is it, that the sun and planets gravitate towards one another, without dense matter between them? Whence is it that nature doth nothing in vain, and whence arises all that order and beauty, which we see in the world? To what end are comets, and whence is it that planets move all one and the same way, in orbs concentric, while comets move all manner of ways in orbs very eccentric; and what hinders the fixed stars from falling upon one another? How came the bodies of animals to be contrived with so much art? and for what ends were their several parts? was the eye contrived without skill in optics, and the ear without knowledge of sounds? How do the motions of the body follow from the will, and whence is the instinct in animals? Is not the sensoryof animals that place to which the sensitive substance is present; and into which the sensible species of things are carried through the nerves and brain, that there they may be perceived, by their immediate presence to that substance? And these things being rightly dispatched, does it not appear from phenomena, that there is a Being incorporeal, living, intelligent, omnipresent, who in infinite space, as it were, in his sensory, sees the things themselves intimately, and thoroughly perceives them, and comprehends them wholly by their immediate presence to himself; and

which things, the images only, carried through the organs of sense into our little sensoriums, are there seen and beheld, by that which in us perceives and thinks; and though every true step made in this philosophy bring us not immediately to the knowledge of the First Cause, yet it brings us nearer to it, and on that account is to be highly valued?"

It is thus that Newton speaks of a Supreme Being; and even those who might dispute the arguments which he gives for such an existence, must still recognize, in this passage, the sentiments of a mind deeply imbued with religious feelings, and convinced of their true foundation. It was upon this ground, however, that Leibnitz attacked him in his correspondence with the princess: "it appears," says he, in one of his letters, "that natural religion is diminishing extremely in England;" and he cites as a proof the works of Locke, and the above passage from Newton; elsewhere he says, "that these principles are precisely those of the materialists." When we see a mind of the order of that of Leibnitz expressing itself with such blind contempt for the grand and incontrovertible discovery of universal gravitation, and employing such arguments in objecting to it, we are disposed to compassionate the occasional weakness of the finest intellects, and to deplore the petty passions which tarnish the splendour of genius. The rank of the person to whom this accusation was addressed increased its importance in those days. The king was informed of the matter, and expressed his expectation that Newton would reply. It would appear that it was this authority that determined Newton personally to enter the lists; but he only undertook the defence of the mathematical part of the question; the philosophical part he left to Dr. Clarke, who, though inferior as a mathematician, was a better metaphysician than himself. From this resulted a great number of letters, written by Clarke and Leibnitz to each other, which were all inspected by the princess. In the course of this correspondence, as often happens, the original question was lost amidst collateral disquisitions.* On reading these letters, it must excite surprise that a woman of rank could amuse herself with discussions of this sort,

⚫ These letters were published in France by Des maizeaux.