CHAPTER XII.

  Doctrine of Infinite Quantities—Labours of Pappus—Kepler—Cavaleri—Roberval—Fermat—Wallis—Newton discovers the Binomial Theorem—and the Doctrine of Fluxions in 1666—His Manuscript Work containing this Doctrine communicated to his Friends—His Treatise on Fluxions—His Mathematical Tracts—His Universal Arithmetic—His Methodus Differentialis—His Geometria Analytica—His Solution of the Problems proposed by Bernouilli and Leibnitz—Account of the celebrated Dispute respecting the Invention of Fluxions—Commercium Epistolicum—Report of the Royal Society—General View of the Controversy.

Previous to the time of Newton, the doctrine of infinite quantities had been the subject of profound study. The ancients made the first step in this curious inquiry by a rude though ingenious attempt to determine the area of curves. The method of exhaustions which was used for this purpose consisted in finding a given rectilineal area to which the inscribed and circumscribed polygonal figures continually approached by increasing the number of their sides. This area was obviously the area of the curve, and in the case of the parabola it was found by Archimedes to be two-thirds of the area169 formed by multiplying the ordinate by the abscissa. Although the synthetical demonstration of the results was perfectly conclusive, yet the method itself was limited and imperfect.

The celebrated Pappus of Alexandria followed Archimedes in the same inquiries; and in his demonstration of the property of the centre of gravity of a plane figure, by which we may determine the solid formed by its revolution, he has shadowed forth the discoveries of later times.

In his curious tract on Stereometry, published in 1615, Kepler made some advances in the doctrine of infinitesimals. Prompted to the task by a dispute with the seller of some casks of wine, he studied the measurement of solids formed by the revolution of a curve round any line whatever. In solving some of the simplest of these problems, he conceived a circle to be formed of an infinite number of triangles having all their vertices in the centre, and their infinitely small bases in the circumference of the circle, and by thus rendering familiar the idea of quantities infinitely great and infinitely small, he gave an impulse to this branch of mathematics. The failure of Kepler, too, in solving some of the more difficult of the problems which he himself proposed roused the attention of geometers, and seems particularly to have attracted the notice of Cavaleri.

This ingenious mathematician was born at Milan in 1598, and was Professor of Geometry at Bologna. In his method of Indivisibles, which was published in 1635, he considered a line as composed of an infinite number of points, a surface of an infinite number of lines, and a solid of an infinite number of surfaces; and he lays it down as an axiom that the infinite sums of such lines and surfaces have the same ratio when compared with the linear or superficial unit, as the surfaces and solids which are to be determined. As it is not true that an infinite170 number of infinitely small points can make a line, or an infinite number of infinitely small lines a surface, Pascal removed this verbal difficulty by considering a line as composed of an infinite number of infinitely short lines, a surface as composed of an infinite number of infinitely narrow parallelograms, and a solid of an infinite number of infinitely thin solids. But, independent of this correction, the conclusions deduced by Cavaleri are rigorously true, and his method of ascertaining the ratios of areas and solids to one another, and the theorems which he deduced from it may be considered as forming an era in mathematics.

By the application of this method, Roberval and Toricelli showed that the area of the cycloid is three times that of its generating circle, and the former extended the method of Cavaleri to the case where the powers of the terms of the arithmetical progression to be summed were fractional.

In applying the doctrine of infinitely small quantities to determine the tangents of curves, and the maxima and minima of their ordinates, both Roberval and Fermat made a near approach to the invention of fluxions—so near indeed that both Lagrange and Laplace56 have pronounced the latter to be the true inventer of the differential calculus. Roberval supposed the point which describes a curve to be actuated by two motions, by the composition of which it moves in the direction of a tangent; and had he possessed the method of fluxions, he could, in every case, have determined the relative velocities of these motions, which depend on the nature of the curve, and consequently the direction of the tangent which he assumed to be in the diagonal of a parallelogram whose sides had the171 same ratio as the velocities. But as he was able to determine these velocities only in the conic sections, &c. his ingenious method had but few applications.

The labours of Peter Fermat, a counsellor of the parliament of Toulouse, approached still nearer to the fluxionary calculus. In his method of determining the maxima and minima of the ordinates of curves, he substitutes x + e for the independent variable x in the function which is to become a maximum, and as these two expressions should be equal when e becomes infinitely small or 0, he frees this equation from surds and radicals, and after dividing the whole by e, e is made = 0, and the equation for the maximum is thus obtained. Upon a similar principle he founded his method of drawing tangents to curves. But though the methods thus used by Fermat are in principle the same with those which connect the theory of tangents and of maxima and minima with the analytical method of exhibiting the differential calculus, yet it is a singular example of national partiality to consider the inventer of these methods as the inventer of the method of fluxions.

“One might be led,” says Mr. Herschel, “to suppose by Laplace’s expression that the calculus of finite differences had then already assumed a systematic form, and that Fermat had actually observed the relation between the two calculi, and derived the one from the other. The latter conclusion would scarcely be less correct than the former. No method can justly be regarded as bearing any analogy to the differential calculus which does not lay down a system of rules (no matter on what considerations founded, by what names called, or by what extraneous matter enveloped) by means of which the second term of the development of any function of x + e in powers of e, can be correctly calculated, ‘qu? extendet se,’ to use Newton’s expression,172 ‘citra ullum molestum calculum in terminis surdis ?que ac in integris procedens.’ It would be strange to suppose Fermat or any other in possession of such a method before any single surd quantity had ever been developed in a series. But, in point of fact, his writings present no trace of the kind; and this, though fatal to his claim, is allowed by both the geometers cited. Hear Lagrange’s candid avowal. ‘Il fait disparaitre dans cette equation,’ that of the maximum between x and e, ‘les radicaux et les fractions s’il y en à.’ Laplace, too, declares that ‘il savoit etendre son calcul aux fonctions irrationelles en se debarrassant des irrationalités par l’elevation des radicaux aux puissances.’ This is at once giving up the point in question. It is allowing unequivocally that Fermat in these processes only took a circuitous route to avoid a difficulty which it is one of the most express objects of the differential calculus to face and surmount. The whole claim of the French geometer arises from a confusion (too often made) of the calculus and its applications, the means and the end, under the sweeping head of ‘nouveaux calculs’ on the one hand, and an assertion somewhat too unqualified, advanced in the warmth and generality of a preface, on the other.”57

The discoveries of Fermat were improved and simplified by Hudde, Huygens, and Barrow; and by the publication of the Arithmetic of Infinites by Dr. Wallis, Savilian professor of geometry at Oxford, mathematicians were conducted to the very entrance of a new and untrodden field of discovery. This distinguished author had effected the quadrature of all curves whose ordinates can be expressed by any direct integral powers; and though he had extended his conclusions to the cases where the ordinates are expressed by the inverse or fractional powers, yet173 he failed in its application. Nicolas Mercator (Kauffman) surmounted the difficulty by which Wallis had been baffled, by the continued division of the numerator by the denominator to infinity, and then applying Wallis’s method to the resulting positive powers. In this way he obtained, in 1667, the first general quadrature of the hyperbola, and, at the same time, gave the regular development of a function in series.

In order to obtain the quadrature of the circle, Dr. Wallis considered that if the equations of the curves of which he had given the quadrature were arranged in a series, beginning with the most simple, these areas would form another series. He saw also that the equation of the circle was intermediate between the first and second terms of the first series, or between the equation of a straight line and that of a parabola, and hence he concluded, that by interpolating a term between the first and second term of the second series, he would obtain the area of the circle. In pursuing this singularly beautiful thought, Dr. Wallis did not succeed in obtaining the indefinite quadrature of the circle, because he did not employ general exponents; but he was led to express the entire area of the circle by a fraction, the numerator and denominator of which are each obtained by the continued multiplication of a certain series of numbers.

Such was the state of this branch of mathematical science, when Newton, at an early age, directed to it the vigour of his mind. At the very beginning of his mathematical studies, when the works of Dr. Wallis fell into his hands, he was led to consider how he could interpolate the general values of the areas in the second series of that mathematician. With this view he investigated the arithmetical law of the coefficients of the series, and obtained a general method of interpolating, not only the series above referred to, but also other series. These174 were the first steps taken by Newton, and, as he himself informs us, they would have entirely escaped from his memory if he had not, a few weeks before,58 found the notes which he made upon the subject. When he had obtained this method, it occurred to him that the very same process was applicable to the ordinates, and, by following out this idea, he discovered the general method of reducing radical quantities composed of several terms into infinite series, and was thus led to the discovery of the celebrated Binomial Theorem. He now neglected entirely his methods of interpolation, and employed that theorem alone as the easiest and most direct method for the quadratures of curves, and in the solution of many questions which had not even been attempted by the most skilful mathematicians.

After having applied the Binomial theorem to the rectification of curves, and to the determination of the surfaces and contents of solids, and the position of their centres of gravity, he discovered the general principle of deducing the areas of curves from the ordinate, by considering the area as a nascent quantity, increasing by continual fluxion in the proportion of the length of the ordinate, and supposing the abscissa to increase uniformly in proportion to the time. In imitation of Cavalerius, he called the momentary increment of a line a point, though it is not a geometrical point, but an infinitely short line; and the momentary increment of an area or surface he called a line, though it is not a geometrical line, but an infinitely narrow surface. By thus regarding lines as generated by the motion of points, surfaces by the motions of lines, and solids by the motion of surfaces, and by considering that the ordinates, absciss?, &c. of curves thus formed, vary according to a regular law depending on the equation of the175 curve, he deduces from this equation the velocities with which these quantities are generated; and by the rules of infinite series he obtains the ultimate value of the quantity required. To the velocities with which every line or quantity is generated, Newton gave the name of Fluxions, and to the lines or quantities themselves that of Fluents. This method constitutes the doctrine of fluxions which Newton had invented previous to 1666, when the breaking out of the plague at Cambridge drove him from that city, and turned his attention to other subjects.

But though Newton had not communicated this great invention to any of his friends, he composed his treatise, entitled Analysis per equationes numero terminorum infinitas, in which the principle of fluxions and its numerous applications are clearly pointed out. In the month of June, 1669, he communicated this work to Dr. Barrow, who mentions it in a letter to Mr. Collins, dated the 20th June, 1669, as the production of a friend of his residing at Cambridge, who possesses a fine genius for such inquiries. On the 31st July, he transmitted the work to Collins; and having received his approbation of it, he informs him that the name of the author of it was Newton, a fellow of his own college, and a young man who had only two years before taken his degree of M.A. Collins took a copy of this treatise, and returned the original to Dr. Barrow; and this copy having been found among Collins’s papers by his friend Mr. William Jones, and compared with the original manuscript borrowed from Newton, it was published with the consent of Newton in 1711, nearly fifty years after it was written.

Though the discoveries contained in this treatise were not at first given to the world, yet they were made generally known to mathematicians by the correspondence of Collins, who communicated them to James Gregory; to MM. Bertet and Vernon in176 France; to Slusius in Holland; to Borelli in Italy; and to Strode, Townsend, and Oldenburg, in letters dated between 1669 and 1672.

Hitherto the method of fluxions was known only to the friends of Newton and their correspondents; but, in the first edition of the Principia, which appeared in 1687, he published, for the first time, the fundamental principle of the fluxionary calculus, in the second lemma of the second book. No information, however, is here given respecting the algorithm or notation of the calculus; and it was not till 1693–5[?] that it was communicated to the mathematical world in the second volume of Dr. Wallis’s works, which were published in that year. This information was extracted from two letters of Newton written in 1692.

About the year 1672, Newton had undertaken to publish an edition of Kinckhuysen’s Algebra, with notes and additions. He therefore drew up a treatise, entitled, A Method of Fluxions, which he proposed as an introduction to that work; but the fear of being involved in disputes about this new discovery, or perhaps the wish to render it more complete, or to have the sole advantage of employing it in his physical researches, induced him to abandon this design. At a later period of his life he again resolved to give it to the world; but it did not appear till after his death, when it was translated into English, and published in 1736, with a commentary by Mr. John Colson, Professor of Mathematics in Cambridge.59

To the first edition of Newton’s Optics, which appeared in 1704, there were added two mathematical177 treatises, entitled, Tractatus duo de speciebus et magnitudine figurarum curvilinearum, the one bearing the title of Tractatus de Quadratura Curvarum, and the other Enumeratio linearum tertii ordinis. The first contains an explanation of the doctrine of fluxions, and of its application to the quadrature of curves; and the second a classification of seventy-two curves of the third order, with an account of their properties. The reason for publishing these two tracts in his Optics (in the subsequent editions of which they are omitted) is thus stated in the advertisement:—“In a letter written to M. Leibnitz in the year 1679, and published by Dr. Wallis, I mentioned a method by which I had found some general theorems about squaring curvilinear figures on comparing them with the conic sections, or other the simplest figures with which they might be compared. And some years ago I lent out a manuscript containing such theorems; and having since met with some things copied out of it, I have on this occasion made it public, prefixing to it an introduction, and joining a scholium concerning that method. And I have joined with it another small tract concerning the curvilineal figures of the second kind, which was also written many years ago, and made known to some friends, who have solicited the making it public.”

In the year 1707, Mr. Whiston published the algebraical lectures which Newton had, during nine years, delivered at Cambridge, under the title of Arithmetica Universalis, sive de Compositione et Resolutione Arithmetica Liber. We are not accurately informed how Mr. Whiston obtained possession of this work; but it is stated by one of the editors of the English edition, that “Mr. Whiston thinking it a pity that so noble and useful a work should be doomed to a college confinement, obtained leave to make it public.” It was soon afterward translated into English by Mr. Ralphson; and a second edition of it, with improvements by the author, was published at London178 in 1712, by Dr. Machin, secretary to the Royal Society. With the view of stimulating mathematicians to write annotations on this admirable work, the celebrated S’Gravesande published a tract, entitled, Specimen Commentarii in Arithmeticam Universalem; and Maclaurin’s Algebra seems to have been drawn up in consequence of this appeal.

Among the mathematical works of Newton we must not omit to enumerate a small tract entitled, Methodus Differentialis, which was published with his consent in 1711. It consists of six propositions, which contain a method of drawing a parabolic curve through any given number of points, and which are useful for constructing tables by the interpolation of series, and for solving problems depending on the quadrature of curves.

Another mathematical treatise of Newton’s was published for the first time in 1779, in Dr. Horsley’s edition of his works.60 It is entitled, Artis Analytic? Specimina, vel Geometria Analytica. In editing this work, which occupies about 130 quarto pages, Dr. Horsley used three manuscripts, one of which was in the handwriting of the author; another, written in an unknown hand, was given by Mr. William Jones to the Honourable Charles Cavendish; and a third, copied from this by Mr. James Wilson, the editor of Robins’s works, was given to Dr. Horsley by Mr. John Nourse, bookseller to the king. Dr. Horsley has divided it into twelve chapters, which treat of infinite series; of the reduction of affected equations; of the specious resolution of equations; of the doctrine of fluxions; of maxima and minima; of drawing tangents to curves; of the radius of curvature; of the quadrature of curves; of the area of curves which are comparable with the conic sections; of the construction of mechanical problems, and on finding the lengths of curves.

179 In enumerating the mathematical works of our author, we must not overlook his solutions of the celebrated problems proposed by Bernouilli and Leibnitz. On the Kalends of January, 1697, John Bernouilli addressed a letter to the most distinguished mathematicians in Europe,61 challenging them to solve the two following problems:

1. To determine the curve line connecting two given points which are at different distances from the horizon, and not in the same vertical line, along which a body passing by its own gravity, and beginning to move at the upper point, shall descend to the lower point in the shortest time possible.

2. To find a curve line of this property that the two segments of a right line drawn from a given point through the curve, being raised to any given power, and taken together, may make every where the same sum.

On the day after he received these problems, Newton addressed to Mr. Charles Montague, the President of the Royal Society, a solution of them both. He announced that the curve required in the first problem must be a cycloid, and he gave a method of determining it. He solved also the second problem, and he showed that by the same method other curves might be found which shall cut off three or more segments having the like properties. Leibnitz, who was struck with the beauty of the problem, requested Bernouilli, who had allowed six months for its solution, to extend the period to twelve months. This delay was readily granted, solutions were obtained from Newton, Leibnitz, and the Marquis de L’Hopital; and although that of Newton was anonymous, yet Bernouilli recognised in it his powerful mind, “tanquam,” says he, “ex ungue leonem,” as the lion is known by his claw.

The last mathematical effort of our author was180 made with his usual success, in solving a problem which Leibnitz proposed in 1716, in a letter to the Abbé Conti, “for the purpose, as he expressed it, of feeling the pulse of the English analysts.” The object of this problem was to determine the curve which should cut at right angles an infinity of curves of a given nature, but expressible by the same equation. Newton received this problem about five o’clock in the afternoon, as he was returning from the Mint; and though the problem was extremely difficult, and he himself much fatigued with business, yet he finished the solution of it before he went to bed.

Such is a brief account of the mathematical writings of Sir Isaac Newton, not one of which were voluntarily communicated to the world by himself. The publication of his Universal Arithmetic is said to have been a breach of confidence on the part of Whiston; and, however this may be, it was an unfinished work, never designed for the public. The publication of his Quadrature of Curves, and of his Enumeration of Curve Lines, was rendered necessary, in consequence of plagiarisms from the manuscripts of them which he had lent to his friends, and the rest of his analytical writings did not appear till after his death. It is not easy to penetrate into the motives by which this great man was on these occasions actuated. If his object was to keep possession of his discoveries till he had brought them to a higher degree of perfection, we may approve of the propriety, though we cannot admire the prudence of such a step. If he wished to retain to himself his own methods, in order that he alone might have the advantage of them in prosecuting his physical inquiries, we cannot reconcile so selfish a measure with that openness and generosity of character which marked the whole of his life. If he withheld his labours from the world in order to avoid the disputes and contentions to which they might give rise,181 he adopted the very worst method of securing his tranquillity. That this was the leading motive under which he acted, there is little reason to doubt. The early delay in the publication of his method of fluxions, after the breaking out of the plague at Cambridge, was probably owing to his not having completed the algorithm of that calculus; but no apology can be made for the imprudence of withholding it any longer from the public. Had he published this noble discovery even previous to 1673, when his great rival had not even entered upon those studies which led him to the same method, he would have secured to himself the undivided honour of the invention, and Leibnitz could have aspired to no other fame but that of an improver of the doctrine of fluxions. But he unfortunately acted otherwise. He announced to his friends that he possessed a method of great generality and power; he communicated to them a general account of its principles and applications; and the information which was thus conveyed directed the attention of mathematicians to subjects to which they might not have otherwise applied their powers. In this way the discoveries which he had previously made were made subsequently by others; and Leibnitz, in place of appearing in the theatre of science as the disciple and the follower of Newton, stood forth with all the dignity of a rival; and, by the early publication of his discoveries had nearly placed himself on the throne which Newton was destined to ascend.

It would be inconsistent with the popular nature of a work like this, to enter into a detailed history of the dispute between Newton and Leibnitz respecting the invention of fluxions. A brief and general account of it, however, is indispensable.

In the beginning of 1673, Leibnitz came to London in the suite of the Duke of Hanover, and he became acquainted with the great men who then adorned the capital of England. Among these was Oldenburg,182 a countryman of his own, who was then secretary to the Royal Society. About the beginning of March, in the same year, Leibnitz went to Paris, where, with the assistance of Huygens, he devoted himself to the study of the higher geometry. In the month of July he renewed his correspondence with Oldenburg, and he communicated to him some of the discoveries which he had made relative to series, particularly the series for a circular arc in terms of the tangent. Oldenburg informed him in return of the discoveries on series which had been made by Newton and Gregory; and in 1676 Newton communicated to him, through Oldenburg, a letter of fifteen closely printed quarto pages, containing many of his analytical discoveries, and stating that he possessed a general method of drawing tangents, which he thought it necessary to conceal in two sentences of transposed characters. In this letter neither the method of fluxions nor any of its principles are communicated; but the superiority of the method over all others is so fully described, that Leibnitz could scarcely fail to discover that Newton possessed that secret of which geometers had so long been in quest.

Had Leibnitz at the time of receiving this letter been entirely ignorant of his own differential method, the information thus conveyed to him by Newton could not fail to stimulate his curiosity, and excite his mightiest efforts to obtain possession of so great a secret. That this new method was intimately connected with the subject of series was clearly indicated by Newton; and as Leibnitz was deeply versed in this branch of analysis, it is far from improbable that a mind of such strength and acuteness might attain his object by direct investigation. That this was the case may be inferred from his letter to Oldenburg (to be communicated to Newton) of the 21st June, 1677, where he mentions that he had for some time been in possession of a method of drawing183 tangents more general than that of Slusius, namely, by the differences of ordinates. He then proceeds with the utmost frankness to explain this method, which was no other than the differential calculus. He describes the algorithm which he had adopted, the formation of differential equations, and the application of the calculus to various geometrical and analytical questions. No answer seems to have been returned to this letter either by Newton or Oldenburg, and, with the exception of a short letter from Leibnitz to Oldenburg, dated 12th July, 1677, no further correspondence seems to have taken place. This, no doubt, arose from the death of Oldenburg in the month of August, 1677,62 when the two rival geometers pursued their researches with all the ardour which the greatness of the subject was so well calculated to inspire.

In the hands of Leibnitz the differential calculus made rapid progress. In the Acta Eruditorum, which was published at Leipsic in November, 1684, he gave the first account of it, describing its algorithm in the same manner as he had done in his letter to Oldenburg, and pointing out its application to the drawing of tangents, and the determination of maxima and minima. He makes a remote reference to the similar calculus of Newton, but lays no claim to the sole invention of the differential method. In the same work for June, 1686, he resumes the subject; and when Newton had not published a single word upon184 fluxions, and had not even made known his notation, the differential calculus was making rapid advances on the Continent, and in the hands of James and John Bernouilli had proved the means of solving some of the most important and difficult problems.

The silence of Newton was at last broken, and in the second lemma of the second book of the Principia, he explained the fundamental principle of the fluxionary calculus. His explanation, which occupied only three pages, was terminated with the following scholium:—“In a correspondence which took place about ten years ago between that very skilful geometer, G. G. Leibnitz, and myself, I announced to him that I possessed a method of determining maxima and minima, of drawing tangents, and of performing similar operations which was equally applicable to rational and irrational quantities, and concealed the same in transposed letters involving this sentence, (data equatione quotcunque fluentes quantitates involvente, fluxiones invenire et vice versa). This illustrious man replied that he also had fallen on a method of the same kind, and he communicated to me his method which scarcely differed from mine except in the notation [and in the idea of the generation of quantities.”]63 This celebrated scholium, which is so often referred to in the present controversy, has, in our opinion, been much misapprehended. While M. Biot considers it as “eternalizing the rights of Leibnitz by recognising them in the Principia,” Professor Playfair regards it as containing “a highly favourable opinion on the subject of the discoveries of Leibnitz.” To us it appears to be nothing more than the simple statement of the fact, that the method communicated by Leibnitz was nearly the same as his own; and this much he might have said, whether he believed that Leibnitz had seen the fluxionary calculus among the185 papers of Collins, or was the independent inventor of his own. It is more than probable, indeed, that when Newton wrote this scholium he regarded Leibnitz as a second inventor; but when he found that Leibnitz and his friends had showed a willingness to believe, and had even ventured to throw out the suspicion, that he himself had borrowed the doctrine of fluxions from the differential calculus, he seems to have altered the opinion which he had formed of his rival, and to have been willing in his turn to retort the charge.

This change of opinion was brought about by a series of circumstances over which he had no control. M. Nicolas Fatio de Duillier, a Swiss mathematician, resident in London, communicated to the Royal Society, in 1699, a paper on the line of quickest descent, which contains the following observations:—“Compelled by the evidence of facts, I hold Newton to have been the first inventor of this calculus, and the earliest by several years; and whether Leibnitz, the second inventor, has borrowed any thing from the other, I would prefer to my own judgment that of those who have seen the letters and other copies of the same manuscripts of Newton.” This imprudent remark, which by no means amounts to a charge of plagiarism, for Leibnitz is actually designated the second inventor, may be considered as showing that the English mathematicians had been cherishing suspicions unfavourable to Leibnitz, and there can be no doubt that a feeling had long prevailed that this mathematician either had, or might have seen, among the papers of Collins, the “Analysis per Equationes, &c.,” which contained the principles of the fluxionary method. Leibnitz replied to the remark of Duillier with much good feeling. He appealed to the facts as exhibited in his correspondence with Oldenburg; he referred to Newton’s scholium as a testimony in his favour; and, without disputing or acknowledging the priority of Newton’s186 claim, he asserted his own right to the invention of the differential calculus. Fatio transmitted a reply to the Leipsic Acts; but the editor refused to insert it. The dispute, therefore, terminated, and the feelings of the contending parties continued for some time in a state of repose, though ready to break out on the slightest provocation.

When Newton’s Optics appeared in 1704, accompanied by his Treatise on the Quadrature of Curves, and his enumeration of lines of the third order, the editor of the Leipsic Acts (whom Newton supposed to be Leibnitz himself) took occasion to review the first of these tracts. After giving an imperfect analysis of its contents, he compared the method of fluxions with the differential calculus, and, in a sentence of some ambiguity, he states that Newton employed fluxions in place of the differences of Leibnitz, and made use of them in his Principia in the same manner as Honoratus Fabri, in his Synopsis of Geometry, had substituted progressive motion in place of the indivisibles of Cavaleri.64 As Fabri, therefore, was not the inventor of the method which is here referred to, but borrowed it from Cavaleri, and only changed the mode of its expression, there can be no doubt that the artful insinuation contained in the above passage was intended to convey the impression that Newton had stolen his method of fluxions from Leibnitz. The indirect character of this attack, in place of mitigating its severity, renders it doubly odious; and we are persuaded that no candid reader can peruse the passage without a strong conviction that it justifies, in the fullest manner,187 the indignant feelings which it excited among the English philosophers. If Leibnitz was the author of the review, or if he was in any way a party to it, he merited the full measure of rebuke which was dealt out to him by the friends of Newton, and deserved those severe reprisals which doubtless imbittered the rest of his days. He who dared to accuse a man like Newton, or indeed any man holding a fair character in society, with the odious crime of plagiarism, placed himself without the pale of the ordinary courtesies of life, and deserved to have the same charge thrown back upon himself. The man who conceives his fellow to be capable of such intellectual felony, avows the possibility of himself committing it, and almost substantiates the weakest evidence of the worst accusers.

Dr. Keill, as the representative of Newton’s friends, could not brook this base attack upon his countryman. In a letter printed in the Philosophical Transactions for 1708, he maintained that Newton was “beyond all doubt” the first inventor of fluxions. He referred for a direct proof of this to his letters published by Wallis; and he asserted “that the same calculus was afterward published by Leibnitz, the name and the mode of notation being changed.” If the reader is disposed to consider this passage as retorting the charge of plagiarism upon Leibnitz, he will readily admit that the mode of its expression is neither so coarse nor so insidious as that which is used by the writer in the Leipsic Acts. In a letter to Hans Sloane, dated March, 1711, Leibnitz complained to the Royal Society of the treatment he had received. He expressed his conviction that Keill had erred more from rashness of judgment than from any improper motive, and that he did not regard the accusation as a calumny; and he requested that the society would oblige Mr. Keill to disown publicly the injurious sense which his words might bear. When this letter was read to the188 society, Keill justified himself to Sir Isaac Newton and the other members by showing them the obnoxious review of the Quadrature of Curves in the Leipsic Acts. They all agreed in attaching the same injurious meaning to the passage which we formerly quoted, and authorized Keill to explain and defend his statement. He accordingly addressed a letter to Sir Hans Sloane, which was read at the society on the 24th May, 1711, and a copy of which was ordered to be sent to Leibnitz. In this letter, which is one of considerable length, he declares that he never meant to state that Leibnitz knew either the name of Newton’s method or the form of notation, and that the real meaning of the passage was, “that Newton was the first inventor of fluxions or of the differential calculus, and that he had given, in two letters to Oldenburg, and which he had transmitted to Leibnitz, indications of it sufficiently intelligible to an acute mind, from which Leibnitz derived, or at least might derive, the principles of his calculus.”

The charge of plagiarism which Leibnitz thought was implied in the former letter of his antagonist is here greatly modified, if not altogether denied. Keill expresses only an opinion that the letter seen by Leibnitz contained intelligible indications of the fluxionary calculus. Even if this opinion were correct, it is no proof that Leibnitz either saw these indications or availed himself of them, or if he did perceive them, it might have been in consequence of his having previously been in possession of the differential calculus, or having enjoyed some distant view of it. Leibnitz should, therefore, have allowed the dispute to terminate here; for no ingenuity on his part, and no additional facts, could affect an opinion which any other person as well as Keill was entitled to maintain.

Leibnitz, however, took a different view of the subject, and wrote a letter to Sir Hans Sloane, dated December 19, 1711, which excited new feelings,189 and involved him in new embarrassments. Insensible to the mitigation which had been kindly impressed upon the supposed charge against his honour, he alleges that Keill had attacked his candour and sincerity more openly than before;—that he acted without any authority from Sir Isaac Newton, who was the party interested;—and that it was in vain to justify his proceedings by referring to the provocation in the Leipsic Acts, because in that journal no injustice had been done to any party, but every one had received what was his due. He branded Keill with the odious appellation of an upstart, and one little acquainted with the circumstances of the case;65 he called upon the society to silence his vain and unjust clamours,66 which, he believed, were disapproved by Newton himself, who was well acquainted with the facts, and who, he was persuaded, would willingly give his opinion on the matter.

This unfortunate letter was doubtless the cause of all the rancour and controversy which so speedily followed, and it placed his antagonist in a new and a more favourable position. It may be correct, though few will admit it, that Keill’s second letter was more injurious than the first; but it was not true that Keill acted without the authority of Newton, because Keill’s letter was approved of and transmitted by the Royal Society, of which Newton was the president, and therefore became the act of that body. The obnoxious part, however, of Leibnitz’s letter consisted in his appropriating to himself the opinions of the reviewer in the Leipsic Acts, by declaring that, in a review which charged Newton with plagiarism, every person had got what was his due. The whole character of the controversy was now changed: Leibnitz places himself in the190 position of the party who had first disturbed the tranquillity of science by maligning its most distinguished ornament; and the Royal Society was imperiously called upon to throw all the light they could upon a transaction which had exposed their venerable president to so false a charge. The society, too, had become a party to the question, by their approbation and transmission of Keill’s second letter, and were on that account alone bound to vindicate the step which they had taken.

When the letter of Leibnitz, therefore, was read, Keill appealed to the registers of the society for the proofs of what he had advanced; Sir Isaac also expressed his displeasure at the obnoxious passage in the Leipsic Review, and at the defence of it by Leibnitz, and he left it to the society to act as they thought proper. A committee was therefore appointed on the 11th March, consisting of Dr. Arbuthnot, Mr. Hill, Dr. Halley, Mr. Jones, Mr. Machin, and Mr. Burnet, who were instructed to examine the ancient registers of the society, to inquire into the dispute, and to produce such documents as they should find, together with their own opinions on the subject. On the 24th April the committee produced the following report:—

“We have consulted the letters and letter-books in the custody of the Royal Society, and those found among the papers of Mr. John Collins, dated between the years 1669 and 1677, inclusive; and showed them to such as knew and avouched the hands of Mr. Barrow, Mr. Collins, Mr. Oldenburg, and Mr. Leibnitz; and compared those of Mr. Gregory with one another, and with copies of some of them taken in the hand of Mr. Collins; and have extracted from them what relates to the matter referred to us; all which extracts herewith delivered to you we believe to be genuine and authentic. And by these letters and papers we find,—

“I. Mr. Leibnitz was in London in the beginning191 of the year 1673; and went thence, in or about March, to Paris, where he kept a correspondence with Mr. Collins by means of Mr. Oldenburg, till about September, 1676, and then returned by London and Amsterdam to Hanover: and that Mr. Collins was very free in communicating to able mathematicians what he had received from Mr. Newton and Mr. Gregory.

“II. That when Mr. Leibnitz was the first time in London, he contended for the invention of another differential method properly so called; and, notwithstanding that he was shown by Dr. Pell that it was Newton’s method, persisted in maintaining it to be his own invention, by reason that he had found it by himself without knowing what Newton had done before, and had much improved it. And we find no mention of his having any other differential method than Newton’s before his letter of the 21st of June, 1677, which was a year after a copy of Mr. Newton’s letter of the 10th of December, 1672, had been sent to Paris to be communicated to him; and above four years after, Mr. Collins began to communicate that letter to his correspondent; in which letter the method of fluxions was sufficiently described to any intelligent person.

“III. That by Mr. Newton’s letter of the 13th of June, 1676, it appears that he had the method of fluxions above five years before the writing of that letter. And by his Analysis per ?quationes numero Terminorum Infinitas, communicated by Dr. Barrow to Mr. Collins in July, 1669, we find that he had invented the method before that time.

“IV. That the differential method is one and the same with the method of fluxions, excepting the name and mode of notation; Mr. Leibnitz calling those quantities differences which Mr. Newton calls moments or fluxions; and marking them with the letter d—a mark not used by Mr. Newton.

“And therefore we take the proper question to be192 not who invented this or that method, but who was the first inventor of the method. And we believe that those who have reputed Mr. Leibnitz the first inventor knew little or nothing of his correspondence with Mr. Collins and Mr. Oldenburg long before, nor of Mr. Newton’s having that method above fifteen years before Mr. Leibnitz began to publish it in the Acta Eruditorum of Leipsic.

“For which reason we reckon Mr. Newton the first inventor; and are of opinion that Mr. Keill, in asserting the same, has been no ways injurious to Mr. Leibnitz. And we submit to the judgment of the society whether the extract and papers now presented to you, together with what is extant to the same purpose in Dr. Wallis’s third volume, may not deserve to be made public.”

This report being read, the society unanimously ordered the collection of letters and manuscripts to be printed, and appointed Dr. Halley, Mr. Jones, and Mr. Machin to superintend the press. Complete copies of it, under the title of Commercium Epistolicum D. Johannis Collins et aliorum de analysi promota, were laid before the society on the 8th January, 1713, and Sir Isaac Newton, as president, ordered a copy to be delivered to each person of the committee appointed for that purpose, to examine it before its publication.

Leibnitz received information of the appearance of the Commercium Epistolicum when he was at Vienna; and “being satisfied,” as he expresses it, “that it must contain malicious falsehoods, I did not think proper to send for it by post, but wrote to M. Bernouilli to give me his sentiments. M. Bernouilli wrote me a letter dated at Basle, June 7th, 1713, in which he said that it appeared probable that Sir Isaac Newton had formed his calculus after having seen mine.”67 This letter was published by a friend of193 Leibnitz, with reflections, in a loose sheet entitled Charta Volans, and dated July 29, 1713. It was widely circulated without either the name of the author, printer, or place of publication, and was communicated to the Journal Literaire by another friend of Leibnitz, who added remarks of his own, and stated, that when Newton published the Principia in 1687, he did not understand the true differential method; and that he took his fluxions from Leibnitz.

In this state of the controversy, Mr. Chamberlayne conceived the design of reconciling the two distinguished philosophers; and in a letter dated April 28, 1714,68 he addressed himself to Leibnitz, who was still at Vienna. In replying to this letter, Leibnitz declared that he had given no occasion for the dispute; “that Newton procured a book to be published, which was written purposely to discredit him, and sent it to Germany, &c. as in the name of the society;” and he stated that there was room to doubt whether Newton knew his invention before he had it of him. Mr. Chamberlayne communicated this letter to Sir Isaac Newton, who replied that Leibnitz had attacked his reputation in 1705, by intimating that he had borrowed from him the method of fluxions; that if Mr. C. could point out to him any thing in which he had injured Mr. Leibnitz, he would give him satisfaction; that he would not retract things which he knew to be true; and that he believed that the Royal Society had done no injustice by the publication of the Commercium Epistolicum.

The Royal Society, having learned that Leibnitz complained of their having condemned him unheard, inserted a declaration in their journals on the 20th May, 1714, that they did not pretend that the report of their committee should pass for a decision of the society. Mr. Chamberlayne sent a copy of this to Leibnitz, along with Sir Isaac’s letter, and Dr. Keill’s194 answer to the papers inserted in the Journal Literaire. After perusing these documents, M. Leibnitz replied, “that Sir Isaac’s letter was written with very little civility; that he was not in a humour to put himself in a passion against such people; that there were other letters among those of Oldenburg and Collins which should have been published; and that on his return to Hanover, he would be able to publish a Commercium Epistolicum which would be of service to the history of learning.” When this letter was read to the Royal Society, Sir Isaac remarked, that the last part of it injuriously accused the society of having made a partial selection of papers for the Commercium Epistolicum; that he did not interfere in any way in the publication of that work, and had even withheld from the committee two letters, one from Leibnitz in 1693, and another from Wallis in 1695, which were highly favourable to his cause. He stated that he did not think it right for M. Leibnitz himself, but that, if he had letters to produce in his favour, that they might be published in the Philosophical Transactions, or in Germany.

About this time the Abbé Conti, a noble Venetian, came to England. He was a correspondent of Leibnitz, and in a letter which he had received soon after his arrival,69 he enters upon his dispute with Newton. He charges the English “with wishing to pass for almost the only inventors.” He declares “that Bernouilli had judged rightly in saying that Newton did not possess before him the infinitesimal characteristic and algorithm.” He remarks that Newton preceded him only in series; and he confesses that during his second visit to England, “Collins showed him part of his correspondence,” or, as he afterward expresses it, he saw “some of the letters of Newton at Mr. Collins’s.” He then attacks195 Sir Isaac’s philosophy, particularly his opinions about gravity and vacuum, the intervention of God for the preservation of his creatures; and he accuses him of reviving the occult qualities of the schools. But the most remarkable passage in this letter is the following: “I am a great friend of experimental philosophy, but Newton deviates much from it when he pretends that all matter is heavy, or that each particle of matter attracts every other particle.”

The above letter to the Abbé Conti was generally shown in London, and came to be much talked of at court, in consequence of Leibnitz having been privy counsellor to the Elector of Hanover when that prince ascended the throne of England. Many persons of distinction, and particularly the Abbé Conti, urged Newton to reply to Leibnitz’s letter, but he resisted all their solicitations. One day, however, King George I. inquired when Sir Isaac Newton’s answer to Leibnitz would appear; and when Sir Isaac heard this, he addressed a long reply to the Abbé Conti, dated February 26th, O. S. 1715–16. This letter, written with dignified severity, is a triumphant refutation of the allegations of his adversary; and the following passage deserves to be quoted, as connected with that branch of the dispute which relates to Leibnitz’s having seen part of Newton’s letters to Mr. Collins. “He complains of the committee of the Royal Society, as if they had acted partially in omitting what made against me; but he fails in proving the accusation. For he instances in a paragraph concerning my ignorance, pretending that they omitted it, and yet you will find it in the Commercium Epistolicum, p. 547, lines 2, 3, and I am not ashamed of it. He saith that he saw this paragraph in the hands of Mr. Collins when he was in London the second time, that is in October, 1676. It is in my letter of the 24th of October, 1676, and therefore he then saw that letter. And196 in that and some other letters writ before that time, I described my method of fluxions; and in the same letter I described also two general methods of series, one of which is now claimed from me by Mr. Leibnitz.” The letter concludes with the following paragraph: “But as he has lately attacked me with an accusation which amounts to plagiary; if he goes on to accuse me, it lies upon him by the laws of all nations to prove his accusations, on pain of being accounted guilty of calumny. He hath hitherto written letters to his correspondents full of affirmations, complaints, and reflections, without proving any thing. But he is the aggressor, and it lies upon him to prove the charge.”

In transmitting this letter to Leibnitz, the Abbé Conti informed him that he himself had read with great attention, and without the least prejudice, the Commercium Epistolicum, and the little piece70 that contains the extract; that he had also seen at the Royal Society the original papers of the Commercium Epistolicum, and some other original pieces relating to it. “From all this,” says he, “I infer, that, if all the digressions are cut off, the only point is, whether Sir Isaac Newton had the method of fluxions or infinitesimals before you, or whether you had it before him. You published it first, it is true, but you have owned also that Sir Isaac Newton had given many hints of it in his letters to Mr. Oldenburg and others. This is proved very largely in the Commercium, and in the extract of it. What answer do you give? This is still wanting to the public, in order to form an exact judgment of the affair.” The Abbé adds, that Mr. Leibnitz’s own friends waited for his answer with great impatience, and that they thought he could not dispense with answering, if not Dr. Keill, at least Sir Isaac Newton197 himself, who had given him a defiance in express terms.

Leibnitz was not long in complying with this request. He addressed a letter to the Abbé Conti, dated April 9th, 1716, but he sent it through M. Ramond at Paris, to communicate it to others. When it was received by the Abbé Conti, Newton wrote observations upon it, which were communicated only to some of his friends, and which, while they placed his defence on the most impregnable basis, at the same time threw much light on the early history of his mathematical discoveries.

The death of Leibnitz on the 14th November, 1716, put an end to this controversy, and Newton some time afterward published the correspondence with the Abbé Conti, which had hitherto been only privately circulated among the friends of the disputants.71

In 1722, a new edition of the Commercium Epistolicum was published, and there was prefixed to it a general review of its contents, which has been falsely ascribed to Newton.72 When the third edition198 of the Principia was published in 1725, the celebrated scholium which we have already quoted, and in which Leibnitz’s differential calculus was mentioned, was struck out either by Newton or by the editor. This step was perhaps rash and ill-advised; but as the scholium had been adduced by Leibnitz and others as a proof that Newton acknowledged him to be an independent inventor of the calculus,—an interpretation which it does not bear, and which Newton expressly states he never intended it to bear,—he was justified in withdrawing a passage which had been so erroneously interpreted, and so greatly misapplied.

In viewing this controversy, at the distance of more than a century, when the passions of the individual combatants have been allayed, and national jealousies extinguished, it is not difficult to form a correct estimate of the conduct and claims of the two rival analysts. By the unanimous verdict of all nations, it has been decided that Newton invented fluxions at least ten years before Leibnitz. Some of the letters of Newton which bore reference to this great discovery were perused by the German mathematician; but there is no evidence whatever that he borrowed his differential calculus from these letters. Newton was therefore the first inventor, and Leibnitz the second. It was impossible that the former could have been a plagiarist; but it was possible for the latter. Had the letters of Newton contained even stronger indications than they do of the new calculus, no evidence short of proof could have justified any allegation against Leibnitz’s honour. The talents which he displayed in the improvement of the calculus showed that he was capable of inventing it; and his character stood sufficiently high to repel every suspicion of his integrity. But if it would have been criminal to charge Leibnitz with plagiarism, what must we think of those who dared to accuse Newton of borrowing199 his fluxions from Leibnitz? This odious accusation was made by Leibnitz himself, and by Bernouilli; and we have seen that the former repeated it again and again, as if his own good name rested on the destruction of that of his rival. It was this charge against Newton that gave rise to the attack of Keill, and the publication of the Commercium Epistolicum; and, notwithstanding this high provocation, the committee of the Royal Society contented themselves with asserting Newton’s priority, without retorting the charge of plagiarism upon his rival.

Although an attempt has been recently made to place the conduct of Leibnitz on the same level with that of Newton, yet the circumstances of the case will by no means justify such a comparison. The conduct of Newton was at all times dignified and just. He knew his rights, and he boldly claimed them. Conscious of his integrity, he spurned with indignation the charge of plagiarism with which an ungenerous rival had so insidiously loaded him; and if there was one step in his frank and unhesitating procedure which posterity can blame it is his omission, in the third edition of the Principia, of the references to the differential calculus of Leibnitz. This omission, however, was perfectly just. The scholium which he had left out was a mere historical statement of the fact, that the German mathematician had sent him a method which was the same as his own; and when he found that this simple assertion had been held by Leibnitz and others as a recognition of his independent claim to the invention, he was bound either to omit it altogether, or to enter into explanations which might have involved him in a new controversy.

The conduct of Leibnitz was not marked with the same noble lineaments. That he was the aggressor is universally allowed. That he first dared to breathe the charge of plagiarism against Newton, and that he often referred to it, has been sufficiently200 apparent; and when arguments failed him he had recourse to threats—declaring that he would publish another Commercium Epistolicum, though he had no appropriate letters to produce. All this is now matter of history; and we may find some apology for it in his excited feelings, and in the insinuations which were occasionally thrown out against the originality of his discovery; but for other parts of his conduct we seek in vain for an excuse. When he assailed the philosophy of Newton in his letters to the Abbé Conti, he exhibited perhaps only the petty feelings of a rival; but when he dared to calumniate that great man in his correspondence with the Princess of Wales, by whom he was respected and beloved; when he ventured to represent the Newtonian philosophy as physically false, and as dangerous to religion; and when he founded these accusations on passages in the Principia and the Optics glowing with all the fervour of genuine piety, he cast a blot upon his name, which all his talents as a philosopher, and all his virtues as a man, will never be able to efface.