History of calculus

"Algebra made an enormous difference to geometry. Whereas Archimedes had to make an ingenious new approach to each new figure... calculus dealt with a great variety of figures in the same way, via their equations. That was the whole point. Calculus was a method of calculating results, rather than proving them. If pressed, mathematicians could justify their calculations by the method of exhaustion, but it seemed impractical if not unnecessary... Huygens was probably the only major mathematician who stuck to the 'methods of the ancients.' The methods of calculus were so much more powerful and efficient that rigour became secondary. ...By the middle of the eighteenth century, calculus had solved almost all the problems of classical geometry, and new ones the ancients had not dreamed of. It had also revealed the secrets of the heavens, explaining the motions of the moons and planets with uncanny precision."

"The "exhaustion method" (the term "exhaust" appears first in , 1647) was the Platonic school's answer to Zeno. It avoided the pitfalls of the infinitesimals by simply discarding them... by reducing problems... to... formal logic only. ...This indirect method... the standard Greek and Renaissance mode of strict proof in area and volume computation was quite rigorous, ...It had the disadvantage that the result... must be known in advance, so that the mathematician finds it first by another less rigorous and more tentative method. ...a letter from Archimedes to Eratosthenes... described a nonrigorous but fertile way of finding results ...known as the "Method." It has been suggested... that it represented a school of mathematical reasoning competing with Eudoxus... In Democritus' school, according to the theory of Luria, the notion of a "geometrical atom" was introduced. ...several mathematicians before Newton, notably Kepler, used essentially the same conceptions... our modern limit conceptions have made it possible to build this... into a theory as rigorous as... "exhaustion"... The advantage of the "atom method" over the "exhaustion method" was that it facilitated the finding of new results. Antiquity had thus the choice between a rigorous but relatively sterile, and a loosely-founded but far more fertile method. ...in practically all classical texts the first [the exhaustion] method was used. This... may be connected with the fact that mathematics had become a hobby of the leisure class which was based on slavery, indifferent to invention, and interested in contemplation. It may also be a reflection of the victory of Platonic idealism over Democritian materialism in the realm of mathematical philosophy."

"Fermat is... honored with the invention of the differential calculus on account of his method of maxima and minima and of tangents, which, of the prior processes, is in reality the nearest to the algorithm of Leibniz; one could with equal justice, attribute to him the invention of the integral calculus; his treatise De æquationum localium transmutatione, etc., gives indeed the method of integration by parts as well as rules of integration, except the general powers of variables, their sines and powers thereof. However, it must be remarked that one does not find in his writings a single word on the main point, the relation between the two branches of the infinitesimal calculus."

"The calculus was the first achievement of modern mathematics, and it is difficult to overestimate its importance. ...it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking."

"The origins of calculus are clearly empirical. Kepler's first attempts at integration were formulated as "dolichometry"—measurement of kegs—that is, volumetry for bodies with curved surfaces. This is... post-Euclidean geometry, and... nonaxiomatic, empirical geometry. Of this, Kepler was fully aware. The main effort and... discoveries, those of Newton and Leibniz, were of an explicitly physical origin. Newton invented the calculus "of fluxions" essentially for the purpose of mechanics—in fact... calculus and mechanics were developed by him more or less together. The first formulations of the calculus were not even mathematically rigorous. An inexact, semiphysical formulation was the only one available for over a hundred and fifty years after Newton! And yet, some of the most important advances of analysis took place during this period... ! Some of the leading mathematical spirits... were clearly not rigorous, like Euler; but others, in the main, were, like Gauss or Jacobi. The development was as confused and ambiguous as can be, and its relation to empiricism was certainly not according to our present (or Euclid's) ideas of abstraction and rigor. Yet... that period produced mathematics as first class as ever existed! And even after the reign of rigor was... re-established with Cauchy, a... relapse into semiphysical methods took place with Riemann."

"Riemann gave a rigorous definition of the integral by enclosing it between... the "lower sum"... the sum of the areas of the rectangles below the curve, and the "upper sum"... the sum of rectangles of somewhat greater height, which cover the area. The treatise on conoids and spheroids shows that Archimedes was familiar with this method of inclusion and... used it for the determination of volumes. But... one cannot say that he was familiar with the concept of the integral. His integrals always remained tied to a definite geometric interpretation, as volumes or as areas of plane figures. We have no evidence that he understood that one single concept is the foundation of all these geometric interpretations... he bases his rigorous proofs on totally different methods... Nevertheless, his rigorous determination of areas and volumes make Archimedes the precursor of the modern integral calculus."

"On a novel plan, I have combined the historical progress with the scientific developement of the subject; and endeavoured to lay down and inculcate the principles of the Calculus, whilst I traced its gradual and successive improvements. ...there is little doubt, the student's curiosity and attention will be more excited and sustained, when he finds history blended with science, and the demonstration of formulae accompanied with the object and the causes of their invention, than by a mere analytical exposition of the principles of the subject. He will have an opportunity of observing how a calculus, from simple beginnings, by easy steps, and seemingly the slightest improvements, is advanced to perfection; his curiosity too, may be stimulated to an examination of the works of the contemporaries of Newton; works once read and celebrated: yet the writings of the Bernoullis are not antiquated from loss of beauty, nor deserve neglect..."

"Are there indivisible lines? And, generally, is there a simple unit in every class of quanta? §1. Some people maintain this thesis on the following grounds:— (i) If we recognize the validity of the predicates 'big' and 'great', we must equally recognize the validity of their opposites 'little' and 'small'. Now that which admits practically an infinite number of divisions, is 'big' not 'little' . Hence, the 'little' quantum and the 'small' quantum will clearly admit only a finite number of divisions. But if the divisions are finite in number, there must be a simple magnitude. Hence in all classes of quanta there will be found a simple unit, since in all of them the predicates 'little' and 'small' apply. (ii) Again, if there is an Idea of line, and if the Idea is first of the things called by its name:—then, since the parts are by nature prior to their whole, the Ideal Line must be indivisible. And on the same principle, the Ideal Square, the Ideal Triangle, and all the other Ideal Figures—and, generalizing, the Ideal Plane and the Ideal Solid—must be without parts: for otherwise it will result that there are elements prior to each of them. (iii) Again, if Body consists of elements, and if there is nothing prior to the elements, Fire and, generally, each of the elements which are the constituents of Body must be indivisible: for the parts are prior to their whole. Hence there must be a simple unit in the objects of sense as well as in the objects of thought. (iv) Again, Zeno's argument proves that there must be simple magnitudes. For the body, which is moving along a line, must reach the half-way point before it reaches the end. And since there always is a half-way point in any 'stretch' which is not simple, motion—unless there be simple magnitudes—involves that the moving body touches successively one-by-one an infinite number of points in a finite time: which is impossible. But even if the body which is moving along the line, does touch the infinity of points in a finite time, an absurdity results. For since the quicker the movement of the moving body, the greater the 'stretch' which it traverses in an equal time: and since the movement of thought is quickest of all movements:—it follows that thought too will come successively into contact with an infinity of objects in a finite time. And since 'thought's coming into contact with objects one-by-one' is counting, we must admit that it is possible to count the units of an infinite sum in a finite time. But since this is impossible there must be such a thing as an indivisible line. ..."

"Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out."

"You may find this work (if I judge rightly) quite new. For I see no reason why I should not proclaim it; nor do I believe that others will take it wrongly. ...it teaches all by a new method, introduced by me for the first time into geometry, and with such clarity that in these more abstruse problems no-one (as far as I know) has used..."

"This method of mine takes its beginnings where Cavalieri ends his Method of indivisibles. ...for as his was the Geometry of indivisibles, so I have chosen to call my method the Arithmetic of infinitesimals."

"Around 1650 I came across the mathematical writings of Torricelli, where among other things, he expounds the geometry of indivisibles of Cavalieri. ...His method, as taught by Torricelli... was indeed all the more welcome to me because I do not know that anything of that kind was observed in the thinking of almost any mathematician I had previously met."

"It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. ...the art of making discoveries should be extended by considering noteworthy examples of it."

"Among the most renowned discoveries of the times must be considered that of a new kind of mathematical analysis, known by the name of the differential calculus; and of this... the origin and the method of the discovery are not yet known to the world at large."

"Its author invented it nearly forty years ago, and nine years later (nearly thirty years ago) published it in a concise form; and from that time it has... been a method of general employment; while many splendid discoveries have been made by its assistance... so that it would seem that a new aspect has been given to mathematical knowledge arising out of its discovery."

"Now there never existed any uncertainty as to the name of the true inventor, until recently, in 1712, certain upstarts... acted with considerable shrewdness, in that they put off starting the dispute until those who knew the circumstances, Huygens, Wallis, Tschirnhaus, and others, on whose testimony they could have been refuted, were all dead."

"They have changed the whole point of the issue, for... they have set forth their opinion... as to give a dubious credit to Leibniz, they have said very little about the calculus; instead every other page is made up of what they call infinite series. Such things were first given as discoveries by of Holstein who obtained them by the process of division, and Newton gave the more general form by extraction of roots binomial expansion by the interpolation method of Wallis]. This is certainly a useful discovery, for by it arithmetical approximations are reduced to an analytical reckoning; but it has nothing at all to do with the differential calculus. Moreover, even in this they make use of fallacious reasoning; for whenever this rival works out a quadrature by the addition of the parts by which a figure is gradually increased, at once they hail it as the use of the differential calculus... By the selfsame argument, Kepler (in his Stereometria Doliorum), Cavalieri, Fermat, Huygens, and Wallis used the differential calculus; and indeed, of those who dealt with "indivisibles" or the "infinitely small," who did not use it? But Huygens, who as a matter of fact had some knowledge of the method of fluxions as far as they are known and used, had the fairness to acknowledge that a new light was shed upon geometry by this calculus, and that knowledge of things beyond the province of that science was wonderfully advanced by its use."

"On his return from England to France in the year 1673... at the instigation of Huygens he began to work at Cartesian analysis (which afore-time had been beyond him), and in order to obtain an insight into the geometry of quadratures he consulted the Synopsis Geometriae of Honoratus Fabri, Gregory St. Vincent, and a little book by Dettonville (i.e., Pascal [letters to M. de Carcavi]). Later on from one example given by Dettonville, a light suddenly burst upon him, which strange to say Pascal himself had not perceived in it. For when he proves the theorem of Archimedes for measuring the surface of a sphere or parts of it, he used a method in which the whole surface of the solid formed by a rotation round any axis can be reduced to an equivalent plane figure. From it our young friend made out for himself the following general theorem. Portions of a straight line normal to a curve, intercepted between the curve and an axis, when taken in order and applied at right angles to the axis give rise to a figure equivalent to the moment of the curve about the axis. When he showed this to Huygens the latter praised him highly and confessed to him that by the help of this very theorem he had found the surface of parabolic s and others of the same sort, stated without proof many years before in his work on the pendulum clock. Our young friend, stimulated by this and pondering on the fertility of this point of view, since previously he had considered infinitely small things such as the intervals between the ordinates in the method of Cavalieri and such only, studied the triangle... which he called the Characteristic Triangle..."

"To find the area of a given figure, another figure is sought such that its subnormals are respectively equal to the ordinates of the given figure, and then this second figure is the of the given one; and thus from this extremely elegant consideration we obtain the reduction of the areas of surfaces described by rotation to plane quadratures, as well as the rectification of curves; at the same time we can reduce these quadratures of figures to an inverse problem of tangents. From these results, our young friend [Leibniz] wrote down a large collection of theorems (among which in truth there were many that were lacking in elegance) of two kinds. For in some of them only definite magnitudes were dealt with, after the manner not only of Cavalieri, Fermat, Honoratus Fabri, but also of Gregory St. Vincent, Guldinus, and Dettonville; others truly depended on infinitely small magnitudes, and advanced to a much greater extent. But later our young friend did not not trouble to go on with these matters, when he noticed that the same method had been brought into use and perfected by not only Huygens, Wallis, van Huraet, and Neil, but also by James Gregory and Barrow."

"The Method of Fluxions is the general Key, by help whereof the modern Mathematicians unlock the secrets of Geometry, and consequently of Nature. And as it is that which hath enabled them so remarkably to outgo the Ancients in discovering Theorems and solving Problems, the exercise and application thereof is become the main, if not sole, employment of all those who in this Age pass for profound Geometers. But whether this Method be clear or obscure, consistent or repugnant, demonstrative or precarious, as I shall inquire with the utmost impartiality, so I submit my inquiry to your own Judgment, and that of every candid Reader."

"It is said, that the minutest Errors are not to be neglected in Mathematics: that the Fluxions are Celerities, not proportional to the finite Increments though ever so small; but only to the Moments or nascent Increments, whereof the Proportion alone, and not the Magnitude, is considered. And of the aforesaid Fluxions there be other Fluxions, which Fluxions of Fluxions are called second Fluxions. And the Fluxions of these second Fluxions are called third Fluxions; and soon, fourth, fifth, sixth, &c. ad infinitum. Now as our Sense is strained and puzzled with the perception of Objects extremely minute, even so the Imagination, which Faculty derives from Sense, is very much strained and puzzled to frame clear Ideas of the least Particles of time, or the least Increments generated therein: and much more so to comprehend the Moments, or those Increments of the flowing Quantities in statu nascenti, in their very first origin or beginning to exist, before they become finite Particles."

"And it seems still more difficult, to conceive the abstracted Velocities of such nascent imperfect Entities. But the Velocities of the Velocities, the second, third, fourth and fifth Velocities, &c. exceed, if I mistake not, all Humane Understanding. The further the Mind analyseth and pursueth these fugitive Ideas, the more it is lost and bewildered; the Objects, at first fleeting and minute, soon vanishing out of sight."

"[T]o conceive a Part of such infinitely small Quantity, that shall be still infinitely less than it, and consequently though multiply'd infinitely shall never equal the minutest finite Quantity, is, I suspect, an infinite Difficulty to any Man whatsoever; and will be allowed such by those who candidly say what they think; provided they really think and reflect, and do not take things upon trust."

"[O]ur modem Analysts are not content to consider only the Differences of finite Quantities: they also consider the Differences of those Differences, and the Differences of the Differences of the first Differences. And so on ad infinitum. That is, they consider Quantities infinitely less than the least discernible Quantity; and others infinitely less than those infinitely small ones; and still others infinitely less than the preceding Infinitesimals, and so on without end or limit."

"Insomuch that we are to admit an infinite succession of Infinitesimals... in an infinite Progression towards nothing, which you still approach and never arrive at."

"All these Points, I fay, are supposed and believed by... Men who pretend to believe no further than they can see. ...But he who can digest a second or third Fluxion, a second or third Difference, need not, methinks, be squeamish about any Point in Divinity. ...{W]ith what appearance of Reason shall any Man presume to say, that Mysteries may not be Objects of Faith, at the fame time that he himself admits such obscure Mysteries to be the Object of Science?"

"[T]he modern Mathematicians... scruple not to say, that by the help of these new Analytics they can penetrate into Infinity itself: That they can even extend their Views beyond Infinity: that their Art comprehends not only Infinite, but Infinite of Infinite (as they express it) or an Infinity of Infinites."

"But, notwithstanding all these Assertions and Pretensions, it may be justly questioned whether, as other Men in other Inquiries are often deceived by Words or Terms, so they likewise are not wonderfully deceived and deluded by their own peculiar Signs, Symbols, or Species."

"But if we remove the Veil and look underneath, if laying aside the Expressions we set ourselves attentively to consider the things themselves... we shall discover much Emptiness, Darkness, and Confusion; nay, if I mistake not, direct Impossibilities and Contradictions."

"The ancients drew tangents to the conic sections, and to the other geometrical curves of their invention, by particular methods, derived in each case from the individual properties of the curve in question. Archimedes determined in a similar manner the tangents of the spiral, a mechanical curve. Among the moderns, des Cartes, Fermat, Roberval, Barrow, Sluze, and others, had invented uniform methods, of more or less simplicity, for drawing tangents to geometrical curves, which was a great step: but it was previously necessary, that the equations of the curves should be freed from radical quantities, if they contained any; and this operation sometimes required immense, if not absolutely impracticable calculations. The tangent of the , a modern mechanical curve, had been determined only by some artifices founded on it's nature, and from which we could derive no light in other cases. A general method, applicable indifferently to curves of all kinds, geometrical or mechanical, without the necessity of making their radical quantities disappear in any case, remained to be discovered. This sublime discovery, the first step in the method of fluxions, was published by Leibnitz in the Leipsic Transactions for the month of October, 1684. The ever memorable paper that contained it is entitled: 'A New Method for Maxima and Minima, and likewise for Tangents, which is affected neither by Fractions nor irrational Quantities; and a peculiar Kind of Calculus for them.' In this we find the method of differencing all kinds of quantities, rational, fractional, or radical, and the application of these calculi to a very complicated case, which points out the mode for all cases. The author afterward resolves a problem de maximis et minimis, the object of which is to find the path, in which an atom of light must traverse two different mediums, in order to pass from one point to another with most facility. The result of the solution is, that the sines of the angles of incidence and refraction must be to each other in the inverse ratio of the resistances of the two mediums. Lastly he applies his new calculus to a problem, which Beaune had formerly proposed to des Cartes, from whom he obtained only an imperfect solution of it. ...Leibnitz showed in a couple of lines the required curve to be ...the common logarithmic curve."

"In two small tracts on the quadratures of curves, which appeared in 1685, Leibniz] published the first ideas of the calculus summatorius, or inverse method of fluxions. These are farther developed in another tract, entitled, 'Of recondite Geometry, and the Analysis of Indivisibles, and Infinites,' published the following year. In this Leibnitz gives the fundamental rule of the integral calculus; and explains in what the problems of the inverse method of tangents consist, which have since been varied in so many ways. ...and he observes generally, that all the problems of quadratures, before given by geometricians, might be resolved without any difficulty by his method."

"While Leibnitz was in possession of all these treasures, Newton had yet published nothing, from which the world could learn, that he on his part had arrived at similar results. But toward the end of the year 1686, his Philosophiæ naturalis Principia mathematica issued from the press: a vast and profound work ...the key of the most difficult problems resolved in it is the method of fluxions, or analysis of infinites, but exhibited in a form which disguised it, and rendered the author difficult to follow. Accordingly at first it had not all the success it deserved: it was charged with obscurity, with demonstrations derived from sources too remote, and an affected use of the synthetic method of the ancients, while analysis would much better have made known the spirit and progress of the invention. ...mathematicians did Newton the justice to acknowledge, that, at the period when his book was published, he was master of the method of fluxions to a high degree, at least with respect to that part which concerns the quadratures of curves."

"Two illustrious men, who adopted his method with such ardour, rendered it so completely their own, and made so many elegant applications of it that Leibnitz several times published in the journals, with a disinterestedness worthy of so great a man, that it was as much indebted to them as to himself. ...I am speaking of the two brothers James and John Bernoulli."

"Every branch of the new geometry proceeded with rapidity. Problems issued from all quarters; and the periodical publications became a kind of learned amphitheatre, in which the greatest geometricians of the time, Huygens, Leibnitz, the Bernoullis, and the marquis de l'Hopital combated with bloodless weapons; the honour of France being ably supported by the marquis for several years."

"The following problem, proposed by John Bernoulli, in 1693, contributed greatly to the progress of the methods for summing up differences. To find a curve such that the tangents terminating at the axis shall be in a given ratio with the parts of the axis comprised between the curve and these tangents. This was resolved by Huygens, Leibniz, James Bernoulli, and the marquis de l'Hopital. On this occasion Huygens passed on the new methods an encomium so much the more honourable, as this great man, having made several sublime discoveries without them, might have been dispensed from proclaiming their advantages. He confessed, that he beheld 'with surprise and admiration the extent and fertility of this art; that, wherever he turned his eyes, it presented new uses to his view; and that it's progress would be as unbounded as it's speculations.' How unfortunate, that science was bereft of him at an age, when with this new instrument he might still have rendered it so many important services!"

"We find an excellent tract by James Bernoulli concerning the elastic curve, isochronous curves, the path of mean direction in the course of a vessel, the inverse method of tangents, &c. On most of these subjects he had treated already; but here he has given them with additions, corrections, and improvements. His scientific discussions are interspersed with some historical circumstances, which will be read with pleasure. Here for the first time he repels the unjust and repeated attacks of his brother; and exhorts him to moderate his pretensions; to attach less importance to discoveries, which the instrument, with which they were both furnished, rendered easy; and to acknowledge, that, 'as quantities in geometry increase by degrees, so every man, furnished with the same instrument, would find by degrees the same results.' Very modest and remarkable expressions from the pen of one of the greatest geometricians, that ever lived."

"In 1696 a great number of works appeared which gave a new turn to the analysis of infinites. ...and above all the celebrated work of the marquis de l'Hopital, entitled: 'The Analysis of Infinites, for the understanding of curve Lines,'... Such a work had long been a desideratum. 'Hitherto,' says Fontenelle, in his eulogy on the marquis, 'the new geometry had been only a kind of mystery, a cabbalistic science, confined to five or six persons. Frequently solutions were given in the public journals, while the method, by which they had been obtained, was concealed: and even when it was exhibited, it was but a faint gleam of the science breaking out from those clouds, which quickly closed upon it again. The public, or, to speak more properly, the small number of those who aspired to the higher geometry, were struck with useless admiration, by which they were not enlightened; and means were found to obtain their applause, while the information, with which it should have been repaid, was withheld.' The work of the marquis de l'Hopital, completely unveiling the science of the differential calculus, was received with universal encomiums, and still retains it's place among the classical works on the subject. But the time was not yet arrived for treating in the same manner the inverse method of fluxions, which is immense in it's detail, and which, notwithstanding the great progress it has made, is still far from being entirely completed. Leibnitz promised a work, which, under the title of Scientia Infiniti, was to comprise both the direct and inverse methods of fluxions: but this, which would have been of great utility at that time, never appeared."

"The marquis de l'Hopital had given in his work on the analysis of Infinites a very ingenious rule... No person thought proper to dispute his title to this while he lived; but about a month after his death, John Bernoulli, remarking that this rule was incomplete, made a necessary addition to it, and thence took occasion to declare himself it's author. Several of the marquis de l'Hopital's friends complained loudly... Instead of retracting his assertion, John Bernoulli went much farther; and by degrees he claimed as his own every thing of most importance in the Analysis of Infinites. The reader will indulge me in a brief examination of his pretensions. In 1692 John Bernoulli came to Paris. He was received with great distinction by the marquis de l'Hopital, who soon after carried him to his country seat at Ourques in Touraine, where they spent four months in studying together the new geometry. Every attention, and every substantial mark of acknowledgment, were lavished on the learned foreigner. Soon after, the marquis de l'Hopital found himself enabled, by persevering and excessive labour which totally ruined his health, to solve the grand problems, that were proposed to each other by the geometricians of the time. From the year 1693 he made one in the lists of mathematical science, in which he distinguished himself till his death. At this period he was ranked among the first geometricians of Europe; and it is particularly to be observed, that John Bernoulli was one of his most zealous panegyrists. Perhaps he was exalted too high during his lifetime: but the accusation brought against him by John Bernoulli after his death forms too weighty a counterpoise, and justice ought to restore the true balance. ... The extracts of letters, which John Bernoulli has brought forward, are far from proving what he has asserted. ...It is true we find from them, that John Bernoulli had composed lessons in geometry for the marquis de l'Hopital, but by no means that these lessons were the Analysis of Infinites... We see too in these extracts, that the marquis, while at work on his book, solicited from John Bernoulli, with the confidence of friendship, explanations relative to certain questions, which are treated in it... Amid all these uncertainties, it is most equitable and prudent, to adhere to the general declaration made by the marquis in his preface, that he was greatly indebted to John Bernoulli [aux lumiéres de J. B.]; and to presume, that if he had any obligations to him of a particular nature, he would not have ventured to mask them in the expressions of vague and general acknowledgment. If... any one should think proper to credit John Bernoulli on his bare word, when he gives himself out for the author of the Analysis of infinites, the code of morality... will never absolve him, for having disturbed the ashes of a generous benefactor, in order to gratify a paltry love of self, so much the less excusable, as he possessed sufficient scientific wealth besides."

"Toward the end of the year 1704, Newton gave to the World in one volume his Optics in english, an enumeration of lines of the third order, and a treatise on the quadrature of curves, both in latin. ...the treatise on quadratures, belongs to the new geometry. The particular object of this treatise is the resolution of differential formulæ of the first order, or of a single variable quantity; on which depends the precise, or at least the approximate, quadrature of curves. With great address Newton forms series, by means of which he refers the resolution of certain complicated formulæ to those of more simple ones; and these series, suffering an interruption in certain cases, then give the fluents in finite terms. The development of this theory affords a long chain of very elegant propositions, where among other curious problems we remark the method of resolving rational fractions, which was at that time difficult, particularly when the roots are equal. Such an important and happy beginning makes us regret, that the author has given only the first principles of the analysis of differential equations. It is true he teaches us to take the fluxions, of any given order, of an equation with any given number of variable quantities, which belongs to the differential calculus: but he does not inform us, how to solve the inverse problem; that is to say, he has pointed out no means of resolving differential equations, either immediately, or by the separation of the indeterminate quantities, or by the reduction into series, &c. This theory however had already made very considerable progress in Germany, Holland, and France, as may be concluded from the problems of the catenarian, isochronous, and elastic curves, and particularly by the solution which James Bernoulli had given of the isoperimetrical problem. Newton's opponents have argued from his treatise on quadratures, that, when this work appeared, the author was perfectly acquainted only with that branch of the inverse method of fluxions which relates to quadratures, and not with the resolution of differential equations. Newton almost entirely melted down the treatise of Quadratures into another entitled, the Method of Fluxions, and of Infinite Series. This contains only the simple elements of the geometry of infinite, that is to say, the methods of determining the tangents of curve lines, the common maxima and minima, the lengths of curves, the areas they include, some easy problems on the resolution of differential equations, &c. The author had it in contemplation several times to print this work, but he was always diverted from it by some reason or other, the chief of which was no doubt, that it could neither add to his fame, nor even contribute to the advancement of the higher geometry. In 1736, nine years after Newton's death, Dr. Pemberton gave it to the world in english. In 1740 it was translated into french, and a preface was prefixed to it, in which the merits of Leibnitz are depreciated so excessively, and in such a decided tone as might impose on some readers, if the writer of this preface Buffon] had not sufficiently blunted his own criticisms, by betraying how little knowledge of the subject he possessed."

"Nicholas Facio de Duillier... thought proper to say, in a little tract 'on the curve of swiftest descent, and the solid of least resistance,' which appeared in 1699, that Newton was the first inventor of the new calculus... and that he left to others the task of determining what Leibnitz, the second inventor, had borrowed from the english geometrician. Leibnitz, justly feeling himself hurt by this priority of invention ascribed to Newton, and the consequence maliciously insinuated, answered with great moderation, that Facio no doubt spoke solely on his own authority; that he could not believe it was with Newton's approbation; that he would not enter into any dispute with that celebrated man, for whom he had the profoundest veneration, as he had shown on all occasions; that when they had both coincided in some geometrical inventions, Newton himself had declared in his Principia, that neither had borrowed any thing from the other; that when he published his differential calculus in 1684, he had been master of it about eight years; that about the same time, it was true, Newton had informed him, but without any explanation, of his knowing how to draw tangents by a general method, which was not impeded by irrational quantities; but that he could not judge whether this method were the differential calculus, since Huygens, who at that time was unacquainted with this calculus, equally affirmed himself to be in possession of a method, which had the same advantages; that the first work of an english writer, in which the differential calculus was explained in a positive manner, was the preface to Wallis's Algebra, not published till 1693; that, relying on all these circumstances, he appealed entirely to the testimony and candour of Newton, &c."

"In 1708, Keil... renewed the same accusation. ...Keil returned to the charge; and in 1711, in a letter to sir , secretary to the Royal Society, he was not contented with saying, that Newton was the first inventor; but plainly intimated, that Leibnitz, after having taken his method from Newton's writings, had appropriated it to himself, merely employing a different notation; which was charging him in other words with plagiarism. Leibnitz, indignant at such an accusation, complained loudly to the Royal Society; and openly required it to suppress the clamours of an inconsiderate man, who attacked his fame and his honour. The Royal Society appointed a committee, to examine all the writings that related to this question, and in 1712 it published these writings, with the report of the committee, under the following title: Commercium epistolicum de Analysi promota. Without being absolutely affirmative, the conclusion of the report is, that Keil had not calumniated Leibnitz. The work was dispersed over all Europe with profusion. Newton was at that time president of the Royal Society, where he enjoyed the highest respect and most ample power..."

"Newton, gifted by nature with superiour intellect, and born at a time when Harriot, Wren, Wallis, Barrow, and others, had already rendered the mathematical sciences flourishing in England, enjoyed likewise the advantage of receiving lessons from Barrow in his early youth at Cambridge. The whole bent of early youth was toward studies of this kind, and the success he obtained was prodigious. ... Leibnitz, who was four years younger, found but moderate assistance in his studies in Germany. He formed himself alone. His vast and devouring genius, aided by an extraordinary memory, took in every branch of human knowledge; literature, history, poetry, the law of nations, the mathematical sciences, natural philosophy, &c. This multiplicity of pursuits necessarily checked the rapidity of his progress in each; and accordingly he did not appear as a great mathematician till seven or eight years after Newton. Both these great men were in possession of the new analysis long before they made it known to the world. If priority of publication determined priority of discovery, Leibnitz would have completely gained his cause: but this is not sufficient..."

"If Newton first invented the method of fluxions, as is pretended to be proved by his letter of the 10th of december 1672, Leibnitz equally invented it on his part, without borrowing any thing from his rival. These two great men by the strength of their genius arrived at the same discovery through different paths: one, by considering fluxions as the simple relations of quantities, which rise or vanish at the same instant; the other, by reflecting, that, in a series of quantities which increase or decrease, the difference between two consecutive terms may become infinitely small, that is to say, less than any determinable finite magnitude. This opinion, at present universally received except in England, was that of Newton himself, when he first published his Principia... At that time the truth was near it's source, and not yet altered by the passions. In vain did Newton afterward change his language, led away by the flattery of his countrymen and disciples; in vain did he pretend, that the glory of a discovery belongs entirely to the first inventor, and that second inventors ought not to be admitted to share it. ...two men, who separately make the same important discovery, have an equal claim to admiration; and... he who first makes it public, has the first claim to the public gratitude."

"The design of stripping Leibnitz, and making him pass for a plagiary, was carried so far in England, that during the height of the dispute it was said... that the differential calculus of Leibnitz was nothing more than the method of Barrow. What are you thinking of, answered Leibnitz, to bring such a charge against me? ...If the differential calculus were really the method of Barrow (which you well know it is not) who would most deserve to be called a plagiary? Mr. Newton, who was the pupil and friend of Barrow, and had opportunities of gathering from his conversation ideas, which are not in his works? or I, who could be instructed only by his works, and never had any acquaintance with the author?"

"John Bernoulli who... learned the analysis of infinites from the writings of Leibnitz, ... advances not only that the method of fluxions did not precede the differential calculus, but that it might have originated from it; and that Newton had not reduced it to general analytical operations in form of an algorithm, till the differential calculus was already disseminated through all the journals of Holland and Germany."

"The death of Leibnitz, which happened in 1716, it may be supposed, should have put an end to the dispute: but the english, pursuing even the manes of that great man, published in 1726 an edition of the Principia in which the scholium relating to Leibnitz was omitted. This was confessing his discovery in a very authentic and awkward manner. Must they not be aware, that the chimerical design of annihilating the testimony, which an honourable emulation had formerly rendered to truth, would be ascribed to national prejudice, or to a sentiment even still more unjust?"

"In later times there have been geometricians, who... have objected... that the metaphysics of his method were obscure, or even defective; that there are no quantities infinitely small; and that there remain doubts concerning the accuracy of a method, into which such quantities are introduced. But Leibnitz might answer: ...I have no need of the existence of infinitely small quantities: it is enough for my purpose, as I have said in several of my works, that my differences are less than any finite quantity you please to assign; and that consequently the errour, which may result from my supposition, is less than any determinable errour, which is the same as absolutely nothing. The manner in which Archimedes demonstrates the proportion of the sphere to the cylinder, has a similar principle for it's basis. ...The metaphysics of my calculation, therefore, are perfectly conformable to those of the method of exhaustion of the ancients, the certainty of which has never been questioned by any one."

"It would seem from Fermat's correspondence with Descartes as if he had thought out the principles of analytical geometry for himself before reading Descartes' Discours, and had realized that from the equation of a curve (or as he calls it the "specific property") all its properties could be deduced. His extant papers on this subject deal however only with the application of infinitesimals to geometry; it seems probable that these papers are a revision of his original manuscripts (which he destroyed) and were written about 1663, but he was certainly in possession of the general idea of his method for finding maxima and minima as early as 1628 or 1629. Kepler had already remarked that the values of a function immediately adjacent to and on either side of a maximum (or minimum) value must be equal. Fermat applied this to a few examples. Thus to find the maximum value of x(a - x) he took a consecutive value of x, namely x - e where e is very small, and put x(a - x) = (x - e) (a - x + e). Simplifying and ultimately putting e = 0 he got x = \frac{1}{2}a. This value of x makes the given expression a maximum. [This] is the principle of Fermat's method, but his analysis is more involved."

"[Fermat] obtained the to the ellipse, cycloid, cissoid, conchoid, and quadratrix by making the ordinates of the curve and a straight line the same for two points whose abscissae were x and x - e; but there is nothing to indicate that he was aware that the process was general, and though in the course of his work he used the principle, it is probable that he never separated it, so to speak, from the symbols of the particular problem he was considering. The first definite statement of the method was due to Barrow and was published in 1669."

"In 1669 [Isaac Barrow] issued his Lectiones opticæ et geometricæ: this, which is his only important work, was republished with a few minor alterations in 1674. A complete edition of all Barrow's lectures was edited for Trinity College by W. Whewell, Cambridge, 1860. It is said in the preface to the Lectiones opticæ et geometricæ that Newton revised and corrected these lectures adding matter of his own, but it seems probable from Newton's remarks in the fluxional controversy that the additions were confined to the parts which dealt with optics. ... The geometrical lectures contain some new ways of determining the areas and tangents of curves. The most celebrated of these is the method given for the determination of tangents to curves. Fermat had observed that the tangent at a point P on a curve was determined if one other point besides P on it [the tangent line] was known; hence if the length of the MT could be found (thus determining the point T) then the line TP would be the required tangent. Now Barrow remarked that if the abscissa and ordinate at a point Q adjacent to P were drawn he got a small triangle PQR (which he called the differential triangle because, its sides PR and PQ were the differences of the abscissas and ordinates of P and Q) so thatTM : MP = QR : RP.To find QR : RP he supposed that x,y were the coordinates of P and x - e, y - a those of Q. ...Using the equation of the curve and neglecting the squares and higher powers of e and a as compared with their first powers he obtained e : a The ratio a/e was subsequently (in accordance with a suggestion made by de Sluze) termed the angular coefficient of the tangent at the point. Barrow applied this method to the following curves (i) x^2 (x^2 + y^2) = r^2y^2; (ii) x^3 + y^3 = r^3; (iii) x^3 + y^3 = rxy, called la galande; (iv) y = (r - x) tan\frac{\pi x}{2r}, the quadratrix; and (v) y = r \tan \frac{\pi\,x}{2r}. ...take as an illustration the simpler case of the parabola y^2 = px. Using the notation given above we have for the point P, y^2 = px; and for the point Q, (y - a)^2 = p(x - e). Subtracting we get 2ay - a^2 = pe. But if a is an infinitesimal quantity, a^2 must be infinitely smaller and may therefore be neglected: hence e : a = 2y : p. Therefore TM : y = e : a = 2y : p. That is TM = \frac{2y^2}{p} = 2x. This is exactly the procedure of the differential calculus, except that we there have a rule by which we can get the ratio \frac{a}{e} or dy \over dx directly without the labour of going through a calculation similar to the above for every separate case."