Chapter XIV

<A>

MEMOIRS
OF THE
LIFE AND WRITINGS OR SIR ISAAC NEWTON.

CHAPTER XIV.

HISTORY OF THE INFINITESIMAL CALCULUS — ARCHIMEDES — PAPPUS — NAPIER — EDWARD WRIGHT — KEPLER'S TREATISE ON STEREOMETRY — CAVALIERI'S GEOMETRIA INDIVISIBILIUM — ROBERVAL — TORICELLI — FERMAT — WALLI'S ARITHMETICA INFINITORUM — HUDDE — GREGORY — SLUSIUS — NEWTON'S DISCOVERY OF FLUXIONS IN 1655 — GENERAL ACCOUNT OF THE METHOD, AND OF ITS APPLICATIONS — HIS ANALYSIS PER EQUATIONES, ETC. — HIS DISCOVERIES COMMUNICATED TO ENGLISH AND FOREIGN MATHEMATICIANS — THE METHOD OF FLUXIONS AND QUADRATURES — ACCOUNT OF HIS OTHER MATHEMATICAL WRITINGS — HE SOLVES THE PROBLEMS PROPOSED BY BERNOULLI AND LEIBNITZ — LEIBNITZ VISITS LONDON, AND CORRESPONDS WITH THE ENGLISH MATHEMATICIANS, AND WITH NEWTON THROUGH OLDENBURG — HE DISCOVERS THE DIFFERENTIAL CALCULUS, AND COMMUNICATES IT TO NEWTON — NOTICE OF OLDENBURG — CELEBRATED SCHOLIUM RESPECTING FLUXIONS IN THE PRINCIPIA — ACCOUNT OF THE CHANGES UPON IT — LEIBNITZ'S MANUSCRIPTS IN HANOVER.

IN the history of Newton's optical and astronomical discoveries, which we have given in the preceding chapters, we have seen him involved in disputes with his own countrymen as well as with foreigners, in reference to the value and the priority of his labours. Such extreme sensitiveness as that with which he felt the criticisms and discussed the claims of his opponents, has been seldom <2> exhibited in the annals of science; and so great was his dread of controversy, and so feeble his love of wealth and of fame, that, but for the importunities of his friends, his most important researches would have been withheld from the world. If he had been warned of the dangers of a scientific career by the troubled lives of Galileo and of Kepler, he must have learned from their history that great truths have never been received with implicit submission, and that in every age and every state of society the newest and the highest must undergo more than one ordeal — the ordeal of the ignorant, whose capacity they transcend — the ordeal of philosophy, by which they are to be tested and confirmed, and the ordeal of personal jealousy and rival schools, by which they are to be misrepresented and condemned. The discoveries of Newton were tried by all these tests: they emerged purer and greater from the opposition of the Dutch savans: they were placed on a firmer basis by the skilful analysis of Hooke and of Huygens; and they were more warmly received and more widely extended after they had triumphed over the rival speculations of the followers of Aristotle and Descartes.

In the history of Newton's mathematical discoveries, which the same dread of controversy had induced him to withhold from the world, we shall find him involved in more exciting discussions, — in what may even be called quarrels, in which both the temper and the character of the disputants were severely tried. In the controversy respecting the discovery of fluxions, or of the differential calculus, Newton took up arms in his own cause, and though he never placed himself in the front rank of danger, he yet combated with all the ardour of his comrades. Hitherto it had been his lot to contend with <3> individuals unknown to science, or with the philosophers of his own country who were occupied with the same studies; but interests of a larger kind, and feelings of a higher class, sprung up around him. National sympathies mingled themselves with the abstractions of number and of quantity. The greatest mathematicians of the age took the field, and statesmen and princes contributed an auxiliary force to the settlement of questions upon which, after the lapse of nearly 200 years, a verdict has not yet been pronounced.

Painful as the sight must always be when superior minds are brought into collision, society gains from the contest more than the parties lose. We are too apt to regard great men, of the order of Newton and Leibnitz, as exempt from the common infirmities of our nature, and to worship them as demigods more than to admire them as sages. In the history upon which we are about to enter we shall see distinguished philosophers upon the stage, superior, doubtless, to their fellows, but partaking in all the frailties of temper, and exposed to all the suspicions of injustice, which embitter the controversies of ordinary life.

Although the honour of having invented the calculus of fluxions, or the differential calculus, has been conferred upon Newton and Leibnitz, yet, as in every other great invention, they were but the individuals who combined the scattered lights of their predecessors, and gave a method, a notation, and a name, to the doctrine of quantities infinitely small.

By an ingenious attempt to determine the area of curves the ancients made the first step in this interesting inquiry. Their principles were sound, but their want of an organized method of operation prevented them from <4> even forming a calculus. The method of exhaustions which they employed for this purpose consisted in making the curve a limiting area, to which the inscribed and circumscribed polygonal figures continually approached by increasing the number of their sides. The area thus obtained was obviously the area of the curve. In the case of the parabola Archimedes shewed that its area is two-thirds of its circumscribing rectangle, or of the product of the ordinate and the abscissa; and he proved that the superficies of the sphere was equal to the convex superficies of the circumscribing cylinder, or to four times one of its great circles, and that the solidity of the sphere was two-thirds of that of the cylinder. His writings abound in trains of thouhght, which are strictly conducted on the principles of the modern calculus, but in place of this calculus we have only an imperfect arithmetic.

Pappus of Alexandria, who flourished about the close of the fourth century, followed Archimedes in the same inquiries, and his celebrated theorems on the centre of gravity^[1] is the only fruit which sprung from the seed sown by the Greek geometer till we reach the commencement of the seventeenth century. We search in vain the writings of Cardan, Tartaglia, Vieta, and Stevinus, for any proof of their power to employ the infinitesimal principle.

Our countryman, John Napier of Merchiston, and his contemporary, Edward Wright, were not only the first to revive the use of the infinitesimal principle, but the first who applied it in an arithmetical form. They <5> distinctly apprehended the idea of a sufficient approach to the calculation of gradual change by the substitution of small and discontinuous changes. In this way Napier arrived at the representation of the results of arithmetical and geometrical progression taking place continuously in two different magnitudes, and associated the logarithm of any quantity with its primitive. In this manner, too, Wright exhibited what we now call an integration by quadrature, in his celebrated construction of the meridional parts. Both of these geometers fully conceived the idea, as it was embodied in their several problems; and though we cannot ascribe to either a distinct conception of it, we cannot withhold from them the honour of being the first of modern writers who assisted their successors in its conception.

In his treatise on Stereometry, published in 1615, Kepler made some advances in the doctrine of infinitesimals. In consequence of a dispute with a wine-merchant he studied the mensuration of round solids, or those which are formed by the revolution of the conic sections round any line whatever within or without the section. He considered the circle as consisting of an infinite number of triangles, having their vertices in the centre, and their infinitely small bases in the circumference. In like manner, he considered the cone as composed of an infinite number of pyramids, and the cylinder of an infinite number of prisms, 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 in solving some of the more difficult problems which he himself proposed, drew the attention of geometers to the subject of infinitely small <6> quantities, and seems particularly to have attracted the attention of Cavalieri. This celebrated mathematician, who was the friend as well as the disciple of Galileo, was born at Milan in 1598, and was professor of geometry at Bologna. Although he had invented his method of indivisibles so early as 1629, his work entitled Geometria Indivisibilium did not appear till 1635, nor his Exercitationes, containing his most remarkable results, till 1647. He considers 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 assumes as an axiom, that the infinite sums of such lines and surfaces have the same ratio, existing in equal numbers in different surfaces or solids, as the surfaces or solids to be determined. As it is not true that an infinite number of infinitely small points can make a line, nor an infinite number of infinitely small lines a surface, Pascal proposed to return to the idea of Kepler 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. If Cavalieri had been more advanced in algebra he might, perhaps, have gone farther; but he was undoubtedly the first who applied the algebraical process to the quadrature of parabolas of an integer order; and his chief instrument, as it was afterwards that of Wallis, was the theorem, that $1^{\mathit{n}}+2^{\mathit{n}}\mathrm{\dots }{\mathit{x}}^{\mathit{n}}$, divided by ${\mathit{x}}^{\mathit{n}}+{\mathit{x}}^{\mathit{n}}\dots {\mathit{x}}^{\mathit{n}}$, is $1:(\mathrm{n}+1)$ when x is infinite.

Previous to the publication of Cavalieri's work, Roberval had adopted the same principle, and proved that the area of the cycloid was equal to three times that of its generating circle. He determined also the centre of gravity <7> of its area, and the solids formed by its revolution about its axis or its base. We owe to the same mathematician a general method of drawing tangents to certain curves, mechanical and geometrical, which was in some respects similar to that of fluxions. Regarding every curve as described by a point, Roberval^[2] considered the point as influenced 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 have determined in every case 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 the diagonal of a parallelogram whose sides were as the velocities.

Without knowing what had been done by Roberval, Toricelli, a pupil of Galileo, published, in 1644, a solution of the cycloidal problems; but though the demonstrations were so different as to prove that he had not seen those of Roberval, and though his character and talents might have protected him from so ungenerous a charge, the French mathematician did not scruple to accuse him of plagiarism. Toricelli made much use of the infinitesimal methods, and was one of those who most clearly foresaw the approach of a new calculus.

The methods of Peter Fermat, counsellor of the Parliament of Toulouse, for obtaining maxima and minima, and for drawing tangents to curves, had such a striking resemblance to those of the differential calculus, that Laplace, and, in a more qualified manner, Lagrange, have pronounced Fermat^[3] to be the inventor. We need not <8> say that this is an exaggeration: Fermat and others came so close to the calculus as actually to invent cases of it; but none before Newton and Leibnitz ever imagined, far less organized, a general method which should combine the scattered cases of their predecessors into a uniform and extensible system.

As the time for the real invention approached, the anticipatory cases were multiplied. The Arithmetica Infinitorum of Wallis, (1655), not to speak of any other of his writings, applied and extended the ideas of Cavalieri, and produced an ample field of results. It appears, in modern language, like a treatise on $\mathit{\int }{\mathit{x}}^{\mathrm{n}}\mathit{dx}$ for all values of n except $-1$, and on $\mathit{\int }{({\mathit{a}}^2-{\mathit{x}}^2)}^{\mathit{n}}$ for all integer values of n. It gives the first description of the method of rectifying a curve. In the work before cited, Schooten publishes a letter from Henry Van Heuraet, written in 1659, giving the algebraic rectification of every parabola of the form ${\mathit{y}}^{\mathit{n}}=\mathrm{a}{\mathit{x}}^{\mathit{n}+1}$, except in the case of $\mathit{n}=1$, which case is shown to depend on the quadrature of the hyperbola. This had been completed a year or two before, about the same time at which William Neile communicated to Wallis his rectification of the semicubical parabola. Fermat also did the same as Neile, under the forms of the old geometry. Descartes, in 1648, showed that he had made progress in a method of finding areas, centres of gravity, and tangents; and he afterwards determined the character of a curve by what we should now call a transformation of a differential equation.

In his Commentary on Descartes, Schooten published two letters of John Hudde, the second of which is dated <9> January 27, 1658. It shows how to make a rational function integral or fractional, a maximum or minimum, and even treats the case in which the function and its variable are connected by an unsolved rational equation. The rules are, for the first time, extricated from algebraical process, and presented in calcular form. These very remarkable results were well known to both Newton and Leibnitz, and are freely cited by both.

James Gregory, in 1668, gave two of what we should now call integrations of trigonometrical functions. He demonstrated the connexion which had been observed between Wright's meridional parts and the logarithms of cotangents.

The methods of drawing tangents, invented by Barrow and by Slusius, were published in 1670 and in 1673. Such methods were then common; and Barrow, in announcing his, says he scarcely perceives the use of publishing it, because several similar methods were well known. But both these methods obtained an undue importance in the great controversy, and this probably arose from their being both published in England.

Such are the methods which Newton and Leibnitz received from their predecessors, and, were we obliged to describe them in modern terms, we should call them isolated instances of differentiation and integration, of calcular rules of differentiation, of quadrature, rectification, and determination of centres of gravity, of determination of maxima and minima, both of explicit and implicit functions, &c. But we can hardly permit ourselves to invite the reader to look back under general terms, because he can hardly use the general terms without having the idea of a general system. Some will almost be inclined to ask what was left for Newton and Leibnitz to <10> do? The best answer is, that it was left for them to put the querist in a position to ask the question. Had it not been for Newton and Leibnitz, that is, supposing their place had never been supplied, the close approach of the investigators to each other, and to a common method, would never have been visible.

We have already seen^[4] that the attention of Newton had been directed to these subjects so early as 1663 and 1664. Upon reading Dr. Wallis' work in the winter of 1664-5, he obtained an expression in series for the area of circular sectors; and from the consideration that the arch has the same proportion to its sector that an arch of 90° has to the whole quadrant, he found an expression for the length of the arch. At the same time he determined the area of the rectangular hyperbola intercepted between the curve, its asymptote, and two ordinates parallel to the other asymptote; and it was by this series that he computed the area of the hyperbola to fifty-two figures, when the plague had, in the summer of 1669, driven him from Cambridge to Boothby. At the same time he was led, by the happy thought of substituting indefinite indices of powers for definite ones, to give a more general form to the 59th proposition of Dr. Wallis's Arithmetic of Infinites. In the beginning of 1665, he likewise discovered a method of tangents similar to those of Hudde, Gregory, and Slusius, and a method of finding the curvature of curve lines at any given point; and, continuing to pursue the method of interpolation, he found the quadrature of all curves whose ordinates are the powers of binomials affected with indices whole, fractional, or surd, affirmative or negative; together with a rule for reducing any power of a binomial into an <11> approximating or converging series. In the spring of the same year he discovered a method of doing the same thing by the continual division and extraction of roots; and he soon after extended the method to the extraction of the roots of adfected equations in species.

Having met with an example of the method of Fermat, in Schooten's Commentary on the Second Book of Descartes, Newton succeeded in applying it to adfected equations, and determining the proportion of the increments of indeterminate quantities. These increments he called moments, and to the velocities with which the quantities increase he gave the names of motions, velocities of increase, and fluxions. He considered quantities not as composed of indivisibles, but as generated by motion; and as the ancients considered rectangles as generated by drawing one side into the other, that is, by moving one side upon the other to describe the area of the rectangle, so Newton regarded the areas of curves as generated by drawing the ordinate into the abscissa, and all indeterminate quantities as generated by continual increase. Hence, from the flowing of time and the moments thereof, he gave the name of flowing quantities to all quantities which increase in time, that of fluxions to the velocities of their increase, and that of moments to their parts generated in moments of time. He considered time as flowing uniformly, and represented it by any other quantity, which is regarded as flowing uniformly, and its fluxion by a unit. These moments of time, or of its exponent, he considers as equal to one another, and represents by the letter ο, or by any other mark multiplied by unity. The other flowing quantities are represented by any letters or marks, but most commonly by the letters at the end of the alphabet; while <12> their fluxions are represented by any other letters or marks, or by the same letters in a different form or size, and their moments by their fluxions multiplied by a moment of time.

In a manuscript, dated 13th November 1665, the direct method of fluxions is described with examples, and the following problem is resolved: — "An equation being given expressing the relation of two or more lines, x, y, z, &c., described in the same time by two or more moving bodies, A, B, C, &c., to find the relation of their velocities, p, q, r, &c., with which these lines are described". In the same manuscript we find an application of this method to the drawing of tangents, by determining the motion of any point which describes the curve, and also to the determination of the radius of curvature of any curve line, by making the perpendicular to the curve move upon it at right angles, and finding that point of the perpendicular which is in least motion, for that point will be the centre of curvature of the curve at that point upon which the perpendicular stands. On another leaf of the same book, dated May 20, 1665, the same method is given, but in different words, and fluxions are represented with dots superfixed. In another leaf, dated May 16, 1666, there is given a general method, consisting of seven propositions, of solving problems by motion, the seventh proposition being the same, though differently expressed, from that in the paper of November 13, 1665.

In a small tract, written in October 1666, we find the same method in the same number of propositions; but the seventh is improved by shewing how to proceed in equations involving fractions and surds, and such quantities as were afterwards called transcendent. To this tract <13> there is added an eighth proposition, containing the inverse method of fluxions, in so far as he had then attained it, namely, by the method of quadratures, and by most of the theorems in the Scholium to the tenth proposition of his Book of Quadratures, which with many more are contained in this tract. Newton then proceeds to shew the application of the propositions to the solution of the twelve following problems, many of which were at that time entirely new:

" 1. To draw tangents to curve lines.

" 2. To find the quantity of the crookedness of lines.

" 3. To find the points distinguishing between the concave and convex portions of curved lines.

" 4. To find the point at which lines are most or least curved.

" 5. To find the nature of the curve line whose area is expressed by any given equation.

" 6. The nature of any curve line being given, to find other lines whose areas may be compared to the area of that given line.

" 7. The nature of any curve line being given, to find its area when it may be done; or two curved lines being given, to find the relation of their areas when it may be.

" 8. To find such curved lines whose lengths may be found, and also to find their lengths.

" 9. Any curve line being given, to find other lines whose lengths may be compared to its length, or to its area, and to compare them.

" 10. To find curve lines whose areas shall be equal, or have any given relations to the length of any given curve line drawn into a given right line.

" 11. To find the length of any curve line when it may be.

<14>

" 12. To find the nature of a curve line whose length is expressed by any given equation when it may be done."

Such were the improvements in the higher geometry which Newton had made before the end of 1666. His analysis, consisting of the method of series and fluxions combined, was so universal as to apply to almost all kinds of problems. He had not only invented the method of fluxions in 1665, in which the motions or velocities of flowing quantities increase or decrease, but he had considered the increase or decrease of these motions or velocities themselves, to which he afterwards gave the name of second fluxions, — using sometimes letters with one or two dots, to represent first and second fluxions.

It does not appear that Newton imparted any of these methods to his mathematical friends; but in order to communicate some of his results, he composed his treatise entitled Analysis per Equationes Numero Terminorum Infinitas, in which the method of fluxions and its applications are supposed by some to be explained; while others are of opinion, that it treats only of moments or infinitely small increments, and exhibits the algebraical processes involved in their use. In June 1669, he communicated this work to Dr. Barrow, who, in letters to Collins of the 20th June, the 31st July, and the 20th August, mentions it, as we have already seen,^[5] as the production of Newton, a young man of great genius. Having taken a copy of this treatise, Collins returned the original to Dr. Barrow, from whom it again passed into the hands of Newton. At the death of Collins, Mr. William Jones found the copy among his papers; and having compared it with the original given him by Newton, it was published, along with some other analytical <15> tracts of the same author, in 1711, nearly fifty years after it was composed.

Although the discoveries contained in this treatise were not at first given to the world, yet they were generally known to mathematicians by the correspondence of Collins, who communicated them to James Gregory in Scotland; to M. Bertet, and an English gentleman, Francis Vernon, secretary to the English ambassador in Paris; to Slusius in Holland; to Borelli in Italy; and to Thomas Strode, Oldenburg, Dary, and others in England, in letters dated between 1669 and 1672.

In the years 1669 and 1670, Newton had prepared for the press a new and enlarged edition of Kinckhuysen's Introduction to Algebra.^[6] He at first proposed to add to it, as an introduction, a treatise entitled, a Method of Fluxions and Quadratures; but the fear of being involved in disputes as annoying as those into which his optical discoveries had led him, and which were not yet concluded, prevented him from giving this treatise to the world. At a later period of our author's life, Dr. Pemberton had prevailed upon him to publish it, and for this purpose had examined all the calculations and prepared the diagrams. The latter part of the treatise, however, in which he intended to shew the manner of resolving problems which cannot be reduced to quadratures, was <16> never finished; and when Newton was about to supply this defect, his death put a stop to the plan.^[7] It was therefore not till the year 1 ~3 6 that a translation of the work appeared, with a commentary by Mr. John Colson, Professor of Mat~.ematies in Cambridge.^[8]

Between the years 167l and 1676, Newton did not pursue his mathematical studies. His optical researches, and the disputes in which they involved him, occupied all his time; and there is reason to believe, that as soon as these disputes were over, he directed the whole energy of his mind to those researches which constitute the Principia.

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 one of the most important rules of the fluxionary calculus, which forms the Second Lemma of the Second Book, and points out the method of finding the moment of the products of any power whatsoever.

In writing the Principia, Newton made great use of both the direct and the inverse method of fluxions; but though all the difficult propositions in that work were invented by the aid of the calculus, yet the calculations were not put down, and the propositions were demonstrated by the method of the ancients, shortened by the substitution of the doctrine of limits for that of exhaustions. No information, however, is given in the Principia respecting the algorithm or notation of the calculus; and it was not till 1693 that it was communicated to the mathematical world, in the Second <17> Volume of Dr. Wallis's Works, which was published in that year. The friends of Newton in Holland had informed Dr. Wallis that Newton's "Method of Fluxions" had passed there with great applause by the name of Leibnitz's Calculus Differentialis. The Doctor, who was at that time printing the Preface to his First Volume, inserted in it a brief notice of Newton's claim to the discovery of fluxions, and published in his second volume some extracts from the Quadratura Curvarum, with which Newton had furnished him.^[9]

To the first edition of Newton's Optics, which appeared in 1704, there were added two mathematical treatises, entitled, Tractatus duo de speciebus et magnitudine figurarum curvilinearum, the one bearing the title of Tractatus de Quadratura Curvarum,^[10] and the other Enumeratio linearum tertii ordinis.^[11] 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, all 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 <18> 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 delivered at Cambridge, under the title of Arithmetica Universalis, sive de Compositione et Resolutione Arithmetica Liber.^[12] We are not accurately informed how Mr. Whiston obtained possession of this work;^[13] 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 afterwards translated into English by Mr. Raphson; and a second edition of it, with improvements by the author, was published at London 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 <19> 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 was published for the first time in 1799, in Dr. Horsley's edition of his works. It is entitled, Artis Analyticœ Specimina, {illeg}l Geometria Analytica.^[14] In editig 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 which are comparable with the conic sections; of the construction of mechanical problems, and on finding the lengths of curves.

In enumerating the mathematical works of our author, we must no overlook his solutions of the celebrated problems proposed by John Bernoulli and Leibnitz. In June 1696, John Bernoulli addressed a letter to the most distinguished mathematicians in Europe,^[15] 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 <20> 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 everywhere the same sum.^[16]

This challenge was first made in the Leipsic Acts, for June 1696.^[17] Six months were allowed by Bernoulli for the solution of the problem, and in the event of none being sent to him he promised to publish his own. The six months, however, elapsed without any solution being produced; but he received a letter from Leibnitz, stating that he had "cut the knot of the most beautiful of these problems," and requesting that the period for their solution should be extended to Christmas next, that the French and Italian mathematicians might have no reason to complain of the shortness of the period. Bernoulli adopted the suggestion, and publicly announced the prorogation for the information of those who might not see the Leipsic Acts.

On the 29th January 1696-7, Newton received from France two copies of the printed paper containing the problems, and on the following day he transmitted a solution of them to Charles Montague, Chancellor of the Exchequer, and then President of the Royal Society.^[18]

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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. Solutions were also obtained from Leibnitz and the Marquis de l'Hôpital; and although that of Newton was anonymous, yet Bernoulli recognised in it his powerful mind; " tanquam," says he, "ex ungue leonem" as the lion is known by his claw.

One of the last mathematical efforts of our author was 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 difficult, and he himself fatigued with business, he reduced it to a fluxional equation before he went to bed.

In his reply to Leibnitz,^[19] Conti does not even mention the solution of Newton; but as if such a problem had been beneath the notice of the English geometers, he says: — "Your problem was very easily resolved, and in a short time. Several geometers, both in London and Oxford, have given the solution. It is general, and extends to all sorts of curves, whether geometrical or mechanical. The problem is proposed somewhat equivocally; but I believe that M. De Moivre is not wrong when he says that we must fix the idea of a series of <22> curves, and suppose, for example, that they have the same subtangent for the same abscissa, which would correspond not only with the conic sections, but with an infinity of other curves, both geometrical and mechanical."

Such is a brief account of the mathematical writings of Sir Isaac Newton, not one of which was voluntarily communicated to the world by himself. The publication of his Universal Arithmetic is said to have been made by Whiston against his will; 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 in Newton's opinion 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 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, nor with the communications which he so freely made to Barrow, Collins, and others. If he withheld his labours from the world in order to avoid the disputes and contentions to which they might give rise, 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 <23> plague at Cambridge, was probably owing to his not having completed the whole of his design; but no apology can be made for the imprudence of withholding it any longer from the public, — an imprudence which is the more inexplicable, as he was repeatedly urged by Wallis, Halley, and his other friends, to present it to the world.^[20] Had he published this noble discovery previous to 1673, when his great rival had made but little progress in 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, might have directed the attention of mathematicians to subjects to which they would not have otherwise applied their powers. The discoveries which he had previously made were made subsequently by others; and Leibnitz, instead of appearing on the theatre of science as the disciple and the follower of Newton, stood forth with all the dignity of a second inventor; 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 nature of this work 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, when Leibnitz came to <24> London in the suite of the Duke of Hanover, he became acquainted with the great men who then adorned the capital of England. Among these was Henry Oldenburg, a countryman of his own, who was at that time Secretary to the Royal Society. Leibnitz had not then, as he himself assures us,^[21] entered upon the study of the higher geometry, but he eagerly embraced the opportunity which was now offered to him of learning the discoveries of the English mathematician. With this view he kept up a correspondence with Oldenburg, communicating to him freely certain arithmetical and analytical methods of his own, and receiving in return an account of the discoveries in series made by James Gregory and Newton. In the two letters^[22] written in London to Oldenburg, and in the first four which he addressed to him from Paris,^[23] he refers only to certain properties of numbers which he had discovered; but in those of a subsequent date, he mentions a theorem of his own for expressing the area of a circle, or of any given sector of it, by an infinite series of rational numbers;^[24] and of deducing, by the same method, the arc of a circle from its sine.^[25] In reply to these letters, Oldenburg acquainted him with the previous discoveries of Newton, and transmitted to him a communication from Collins, describing several series which had been sent to him by Gregory on the 15th February 1671. Leibnitz stated in reply,^[26] that he was so much distracted with business, that he had not time to compare these series with his own; and he promises to communicate <25> his opinion to Oldenburg as soon as he has made the comparison. In continuing his correspondence with Oldenburg, Leibnitz requested farther information respecting the analytical discoveries recently made in England; and it was in compliance with this request that Newton, at the pressing solicitation of Oldenburg and Collins, wrote a long letter, dated 13th June 1676, to be communicated to Leibnitz.

This letter, which was sent to Leibnitz in Paris, along with extracts from Gregory's letters, on the 26th June, contained Newton's method of series, and, after describing it, he added, "that analysis, by the assistance of infinite equations of this kind, extends to almost all problems except some numerical ones like those of Diophantus, but does not become altogether universal without some farther methods of reducing problems to infinite equations, and infinite equations to finite ones, when it might be done."

Leibnitz answered this letter on the 27th August, and, in return for Newton's method of series, he sent to Olddenburg a theorem for transmuting figures into one another; and thus demonstrated the series of Gregory for finding the arch from its tangent. In consequence of Leibnitz having requested still farther information, Newton addressed to Oldenburg his celebrated letter of the 24th October 1676. In this letter he gave an account of his discovery of the method of series before the plague in the summer of 1665. He stated, that on the publication of Mercator's Logarithmotechnia, he had communicated a compendium of this method through Dr. Barrow to Mr. Collins, and, that five years after, he had, at the suggestion of the latter, written a large tract on the same subject, joining with it a method from which the de <26> termination of maxima and minima, and the method of tangents of Slusius and some others flowed. "This method," he continued, "was not limited to surds, but was founded upon the following proposition, which he communicated enigmatically in a series of transposed letters, Data equatione quotcunque fluentes quantitates involvente, fluxiones invenire, et vice versa. This proposition," he added, "facilitated the quadrature of curves, and afforded him infinite series, which broke off and became finite when the curve was capable of being squared by a finite equation." In the conclusion of this letter, Newton stated that his method extended to inverse problems of tangents, and others more difficult, and that in solving these he used two methods, one more general than the other, which he expressed enigmatically in transposed letters, which formed the following sentence: — "Una methodus consistit in extractione fluentis quantitatis ex equatione simul involvente fluxionem ejus: altera tantum in assumptione seriei pro quantitate quâlibet incognita, ex quâ cetera commodè derivari possunt, et in collatione terminorum homologorum æquationis resultantis, ad eruendos terminos assumptæ seriei."

This letter, though dated 24th October, had not been forwarded to Leibnitz on the 5th March 1677. At the time Newton was writing it, Leibnitz spent a week in London, on his return from Paris to Germany; but it must have reached him in the spring of that year, as he sent an answer to it dated June 21, 1677.

In this remarkable letter he frankly describes his differential calculus and its algorithm. He says that he agrees with Newton in the opinion that Slusius's method of tangents is not absolute, and that he himself had long ago (a multo tempore) treated the subject of tangents <27> much more generally by the differences of ordinates. He gives an example of drawing tangents, and shews how to proceed, as Newton expresses it, "without sticking at {illeg}rds." He then expresses the opinion, that the method of drawing tangents, which Newton wished to conceal, does not differ from his; and he regards this opinion as confirmed by the statement of Newton, that his method facilitated the quadrature of curves.

No answer seems to have been returned to this communication either by Oldenburg or Newton, and, with the exception of a short letter from Leibnitz to the former, dated 12th July 1677, no farther correspondence between them seems to have taken place. This no doubt arose from the death of Oldenburg in the month of August 1678;^[27] and the two rival geometers, having through him become acquainted with each other's labours, were left to pursue them with all the ardour which the importance of the subject could not fail to inspire.

<28>

In the hands of Leibnitz, the differential calculus made rapid progress. In the Acta Eruditorum, which appeared at Leipsic in October 1684, he describes its algorithm in the same manner as he had done in his letter to Oldenburg. He points out its application to the drawing of tangents, and the determination of maxima and minima;^[28]and he adds, that these are only the beginnings of a much more sublime geometry, applicable to the most difficult and beautiful problems even of mixed mathematics, which, without his differential calculus, or one SIMILAR to it, could not be treated with equal facility. The suppression of Newton's name in this reference to a similar calculus, which was obviously that of Newton, indicated in the letters of 1676, was the first false step in the fluxionary controversy, and may be regarded as its com mencement.

While Leibnitz was thus making known the principles of his Calculus, Newton was occupied in preparing his Principia for the press. In the autumn of 1684, he had sent the principal propositions of his work to the Royal Society; but it would appear from his letter to Halley of the 20th June, 1686, that the second book of the Principia had not then been sent to him. He must therefore have been acquainted with the paper of Leibnitz in the Acta Eruditorum, before he sent the manuscript of the second book to press; and it was doubtless from this cause that he was led to compose the second lemma of that book, in which he, for the first time, explains the fundamental principle of the fluxionary calculus. This lemma, which occupies only three pages, was terminated <29> with the following scholium, which has been the subject of such angry discussion.

"The correspondence which took place about ten years ago, between that very skilful geometer G. G. Leibnitz and myself, when I had 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 surds and to rational quantities, and concealed the same in transposed letters, involving this sentence, (Data Æquatione 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 his method, which scarcely differed from my own,^[29] except in the forms of words and notation, (and in the idea of the generation of quantities.^[30]) The fundamental principle of both is contained in this lemma."

This celebrated scholium has been viewed in different lights by Leibnitz and his followers. Leibnitz asserts,^[31] that Newton "has accorded to him in this scholium the invention of the differential calculus independently of his own;" and M. Biot considers the scholium as "eternalizing the rights of Leibnitz by recognising them in the Principia." But the scholium has no such meaning, and it was not the intention of the author that it should be thus understood. It is a statement of the simple fact, that Leibnitz communicated to him a method which was nearly <30> the same as his own, — a sentiment which he might have expressed whether he believed that Leibnitz was an independent inventor of his calculus, or had derived it from his communication and correspondence with his friend.^[32]

The manuscripts of Newton furnish us with some curious information on this subject, and place it beyond a doubt that he regarded the silence of Leibnitz, in his communication of 1684, as an aggressive movement, which he was bound to repel. "After seven years," says Newton,^[33] "viz., in October 1684, he published the elements of this method, (the method mentioned to Leibnitz in his letter of October 24, 1676,) as his own, without referring to the correspondence which he formerly had with the English about these matters. He mentioned, indeed, a methodus similis, but whose that method was, and what he knew of it, he did not say, as he should have done. And thus his silence put me upon a necessity of writing the scholium upon the second lemma of the second Book of Principles, lest it should be thought that I borrowed that lemma from Mr. Leibnitz. In my letter of 24th October 1676, when I had been speaking of the Method of Fluxions, I added, Fundamentum harum operationum, satis obvium quidem, quoniam non possum explicationem ejus prosequi, sic potius celavi 6æccdæ 13eff 7i 3l 9n 4o <31> 4qrr 4s 9t 12vx. And in the said scholium I opened this enigma, saying, that it contained the sentence, Data æquatione quotcunque, fluentes quantitates involvente, fluxiones invenire, et vice versa; and was written in the year 1676, for I looked upon this as a sufficient security, without entering into a wrangle; but Mr. Leibnitz was of another opinion."

In 1724, when the third edition of the Principia was preparing for the press, Newton had resolved to substantiate his claims to the first, if not the sole invention, of the new calculus, and we have found several rough draughts of the changes which he intended to have made upon the scholium. In one of these^[34] he gives an account of the fundamental principle of the fluxionary calculus, and distinctly states that it "might have been easily collected even from the letter which he wrote to Collins on the 1Oth December 1672,^[35] a copy of which was sent to Leibnitz in 1676."^[36]

<32>

In another folio sheet, we have the scholium in three different forms, including the substance of the one previously published.^[37] In all of them it is distinctly stated that Newton's letter to Collins, of the 1Oth December 1672, containing the method of drawing tangents, with an example, had been sent to Leibnitz in June 1676, and that on his return from France through England to Germany, he had consulted Newton's letters in the hands of Collins, and had not long after this fallen upon a similar method. We have not succeeded in finding a copy of the scholium, as <33> it was published in the first edition of the Principia,^[38] or any traces of the grounds upon which he omitted the historical details in the original draughts of it.

It would be interesting to know why these contemplated additions to the scholium were not adopted, and a single paragraph from the letter of December 10, 1672, substituted for the original scholium. In the letters of Pemberton to Newton, in 1724 and 1725, I have found no reference to this change upon the scholium.

It appears, therefore, that Newton had resolved to overlook the agressive movement of Leibnitz in 1684; and on another occasion, when he believed his rights to be invaded, he exercised the same forbearance.^[39] Circumstances, however, now occurred which induced his friends to come forward in his cause. Having learned, as we have seen, that Newton's "notions of Fluxions passed there by the name of Leibnitz's Differential Calculus," Dr. Wallis <34> stopped the printing of the Preface to the first volume of his Works, in order to claim for Newton the invention of Fluxions, as contained in the letters of June and October 1676, which had been sent to Leibnitz. In intimating to Newton what he had done, he said, "You are not so kind to your reputation (and that of the nation) as you might be, when you let things of worth lie by you so long, till others carry away the reputation which is due to you."^[40]

Early in the year 1691, the celebrated James Bernoulli "spoke contemptuously" of the Differential Calculus, maintaining that it differed from that of Barrow only in notation, and in an abridgment of the operation;^[41] but it nevertheless "grew into reputation," and made great progress after the Marquis de l'Hospital had published, in 1696, his excellent work on the Analysis of Infinitesimals. The claims of the two rival geometers increased in value with the stake for which they contended, and an event soon occurred which placed them in open combat. Hitherto neither Newton nor Lelbnitz <35> had claimed to himself the merit of being the sole inventor of the new calculus. Newton was acknowledged even by his rival as the first inventor, and in his scholium he was supposed to have allowed Leibnitz in return the merit of a second inventor. Newton, however, had always believed, without publicly avowing it, that Leibnitz had derived his calculus from the communications made to him by Oldenburg; and Leibnitz, though he had repeatedly declared that he and Newton had borrowed nothing from each other, was yet inclined to consider his rival as a plagiarist.

This celebrated controversy, rendered interesting by the transcendent talents of its promoters, and instructive by the moral frailties with which it was stained, will form the subject of the following chapter.