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 wa............