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Charles Bossut on Leibniz and Newton
Charles Bossut wrote A General History of Mathematics from the Earliest Times to the Middle of the Eighteenth Century which was published in 1802. It was translated into English by J Bonnycastle, Royal Military Academy in Woolwich, and the English translation was published in London in 1803.
Over the years historians have greatly increased our understanding of past events. We would not expect accounts written closer to the period they describe to show the insights that come later. However this does not make such accounts less interesting for they are reflecting views from a period when often memories of passions that were aroused are still present. We present below a version of Bossut's account of the Leibniz-Newton controversy over their priority in inventing the calculus.
This was written around 100 years after the events it describes but is interesting in that Bossut does more than give facts; he analyses them and attempts to draw logical conclusions.
We give below a version of Chapters V and VI of Bossut's A General History of Mathematics from the Earliest Times to the Middle of the Eighteenth Century.
Click on a link below to go to that section
Over the years historians have greatly increased our understanding of past events. We would not expect accounts written closer to the period they describe to show the insights that come later. However this does not make such accounts less interesting for they are reflecting views from a period when often memories of passions that were aroused are still present. We present below a version of Bossut's account of the Leibniz-Newton controversy over their priority in inventing the calculus.
This was written around 100 years after the events it describes but is interesting in that Bossut does more than give facts; he analyses them and attempts to draw logical conclusions.
We give below a version of Chapters V and VI of Bossut's A General History of Mathematics from the Earliest Times to the Middle of the Eighteenth Century.
Click on a link below to go to that section
An Examination of the Claims of Leibniz and Newton to the Invention of the Analysis of Infinites.
The productions of genius being of an order infinitely superior to all other objects of human ambition, we need not be surprised at the warmth with which Leibniz and Newton disputed the discovery of the new geometry. These two illustrious rivals, or rather Germany and England, contended in some respects for the empire of science.
The first spark of the war was excited by Nicholas Facio de Duillier, a Genevese retired to England; the same who afterwards exhibited a strange instance of madness, by attempting publicly to resuscitate a dead body in St Paul's church, but who was at that time in his sound senses, and enjoyed some reputation among geometricians. Urged on the one hand by the English, and on the other by personal resentment against Leibniz, from whom he professed not to have received the marks of esteem he conceived to be his due, he 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 said this for the sake of truth and his own conscience; and that he left to others the task of determining what Leibniz, the second inventor, had borrowed from the English geometrician.
Leibniz, 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 when they had both coincided in some geometrical inventions, Newton himself had declared in his Principia that neither had borrowed anything 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 work of an English writer, in which the 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, etc.
The assertion of Facio, being altogether destitute of proof, was forgotten for several years. In 1708, Keil, perhaps excited by Newton, or at last secure of not being disavowed by him, renewed the same accusation. Leibniz observed that Keil, whom he notwithstanding termed a learned man, was too young to pass a decided judgement on things that had occurred several years before; and he repeated what he had before said, that he rested on the candour and justice of Newton himself. Keil returned the charge; and in 1711, in a letter to Sir Hans Sloane, secretary to the Royal Society, he was not contented with saying that Newton was the first inventor; but plainly intimated that Leibniz, 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.
Leibniz, 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 Leibniz. 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; perhaps therefore delicacy should have induced him to lay the cause before another tribunal. It is true that Fontenelle has said, in his eulogy of Leibniz, that 'Newton did not appear in the business, but left the care of his glory to his countrymen, who were sufficiently zealous.' Leibniz, however, was dead and Newton living when he said this: and no doubt he had been deceived by false documents, for in the course of the dispute Newton wrote two very sharp letters against Leibniz in which we may be a little surprised to perceive too much art and ingenuity employed for the purpose of revoking or weakening the testimonies of high esteem which he had formerly expressed for him on different occasions, particularly in the celebrated scholia to the 7th proposition of the 2nd book of the Principia.
It appears that the Royal Society, while it hastened the publication of the documents that made against Leibniz, without waiting for those which he promised in his defence, was sensible that it would be accused of partiality or precipitation: for it took care soon after to declare that it had no intention of passing judgment on the cause, but left all the world at liberty to discuss it and give its opinion. I beg leave, therefore, to go into this examination to which I will pay all attention in my power. To me Leibniz and Newton are both indifferent: I have received from neither of them, if I may use an expression of Tacitus, either benefit or injury. The sublime genius of both demands profound homage; but it is incumbent on us still more to respect the truth.
Newton, gifted by nature with superior intellect, was 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 his genius was toward studies of this kind, and the success he obtained was prodigious. Fontenelle has applied to him what Lucan said of the Nile: that mankind had not been permitted to see his feeble beginnings. It is affirmed that he laid the foundations of the grand theories, by which he afterwards obtained so much fame, at the age of twenty-five.
Leibniz, 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, etc. 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, Leibniz would have completely gained his cause: but this is not sufficient on the present occasion to enable us to pronounce with confidence. The inventor may have long kept the secret to himself; he may have allowed some hints to escape him on which another may have seized. If possible, therefore, let us trace it to it's source and endeavour to discover the beneficent being who, to adopt the fine simile of Fontenelle, like Prometheus in the fable stole fire from the gods to impart it to mankind.
The Commercium epistolicum contains in the first place, to date from the year 1669, several analytical discoveries of Newton. In the piece entitled De Analysi per Aequationes Numero Terminorum infinitas besides the method for resolving equations by approximation, which has nothing to do with us here, Newton teaches how to square curves, the ordinates of which are expressed by monomials or sums of monomials; and when the ordinates contain complex radicals, he reduces the question to the former case by evolving the ordinate into an infinite series of simple terms by means of the binomial theorem, which no one had done before. Sluze and Gregory had each separately found a method for tangents; and Newton, in a letter to Collins dated December the 10th, 1672, proves that he had likewise found one; he applies it to an example without adding the demonstration; and he afterwards says that it is only a corollary of another general method which he has for drawing tangents, squaring curves, finding their lengths and centres of gravity, etc., without being stopped by the radical quantities, as Hudde was in his method for maxima and minima.
In these two writings of Newton the English have clearly perceived the method of fluxions; after it had been made known throughout Europe, however, by the writings of Leibniz and the two Bernoullis: but the geometricians of other countries have not seen with exactly the same eyes. While they agree that the evolution of radicals into series is a considerable step made by Newton, they immediately perceive, without the assistance of any subsequent and conjectural light, that the methods of Fermat, Wallis, and Barrow, might have been employed to find the results concerning quadratures which Newton contents himself with enunciating; since, after the evolution of radicals, if there be any, nothing more is necessary but to sum up the monomial quantities. They confess that the two pieces in question contain a vague indication of the method of fluxions, if you will: an indication perhaps sufficient to show that Newton was then in possession of the first principles of the method; but too obscure to make the reader at all acquainted with it.
What renders this conjecture very probable is that Mr Oldenburg, secretary to the Royal Society, sending a copy of Sluze's Method of Tangents, which had been printed at London, to the author on the 10th of July 1673, encloses him an extract from a letter of Newton's; in which, after having observed that this method justly belongs to Sluze, Newton goes on thus: 'as to the methods,' (he is speaking of that of Sluze and his own) 'they are the same, though I believe they are derived from different principles. I know not, however, whether the principles of Mr Sluze be as fertile as mine, which extends to the affected equations of irrational terms, without it's being necessary to change their form.' If he had then possessed the method of fluxions in such an advanced state as has since been pretended, would he not have spoken with so much reserve, instead of saying plainly that the method of Sluze and that of fluxions were different? Will it be supposed that he expressed himself thus out of modesty? Surely the truth may be spoken without any infringement of the laws of modesty, even when it is to our own advantage.
All these considerations appear to me to evince that, if the piece De Analysi per Aequationes and the letter of 1672 contain the method of fluxions, it was at least enveloped in great darkness. But whether it were or not, I shall proceed to demonstrate that Leibniz either had no knowledge of these two pieces before he discovered his differential calculus, or derived no information from them. This is an important point which his defenders have not sufficiently established and on which I hope to leave no doubt remaining.
In 1672 Leibniz quitted the universities of Germany and came to France, where he was chiefly occupied in the study of the law of nations and history. He was already initiated into mathematics, however, as in 1666 he had published a little tract on some properties of numbers. In the beginning of 1673 he went to London where he saw Oldenburg, with whom he commenced an epistolary correspondence. In one of his letters to Oldenburg, written even while he was in London, Leibniz says that having discovered a method of summing up certain series by means of their differences, this method was shown to him already published in a book by Mouton, canon of St Paul's at Lyon, On the Diameters of the Sun and Moon: that he then invented another method, which he explains, of forming the differences and thence deducing the sums of the series: that he is capable of summing up a series of fractions of which the numerators are unity and the denominators either the terms of the series of natural numbers, those of the series of triangular numbers, or those of the series of pyramidal numbers, etc. All these researches are ingenious and seem to have at least a remote relation to the calculus of differences. The English have never asserted, and at any rate there exists not the least proof, that Leibniz had seen the two pieces by Newton above mentioned during this first visit to England.
After staying some months in London, Leibniz returned to Paris where he formed an acquaintance with Huygens, who laid open to him the sanctuary of the profoundest geometry. He soon found the approximate quadrature of the circle by a series analogous to that which Mercator had given for the approximate quadrature of the hyperbola. This series he communicated to Huygens by whom it was highly applauded; and to Oldenburg, who answered him that Newton had already invented similar things not only for the circle but for other curves of which he sent him sketches. In fact the theory of series was already far advanced in England at that time; and though Leibniz had likewise penetrated deeply into it, he always acknowledged that the English, and Newton in particular, had preceded and surpassed him in that branch of analysis: but this is not the differential calculus, and the English have shown too evident a partiality in their endeavours to connect these two objects together,
Let us hear and examine the history which Leibniz gives of his discovery of the differential calculus. He relates that, on combining his old remarks on the differences of numbers with his recent meditations on geometry, he hit upon this calculus about the year 1676; that he made astonishing applications of it to geometry; that being obliged to return to Hanover about the same time he could not entirely follow the thread of his meditations; that endeavouring nevertheless to bring forward his new discovery, he went by the way of England and Holland; that he stayed some days in London where he became acquainted with Collins who showed him several letters from Gregory, Newton, and other mathematicians, which turned chiefly on series.
According to this account, it would appear that Leibniz, wishing to spread abroad his new discovery, must then have made known in England the differential calculus. Let us add that in a letter from Collins to Newton, dated the 5th of March 1677, it is said that Leibniz, having spent a week in London in October 1676, had put into Collin's hands some papers [This passage and several other large fragments of the same letter were suppressed in the Commercium epistolicum], of which extracts or copies should be sent to Newton immediately. Collins says nothing of the nature of these papers, and we find no trace of them in the Commercium epistolicum. But if the account given by Leibniz be just, or if his memory did not deceive him, when he said he was in possession of the differential calculus before his second visit to England, no doubt some private reason then occurred to induce him to keep his discovery secret, contrary to the design he had first formed of bringing it forward: for in this very letter Collins mentions another from Leibniz to Oldenburg written from Amsterdam the 28th of November 1676 in which Leibniz proposes the construction of tables of formulas tending to improve the method of Sluze, instead of explaining the differential calculus or at least pointing it out as much more expeditious and more convenient.
The English therefore are justified in saying that Leibniz, when he passed through London in 1676, did not teach them the differential calculus: but they ought to acknowledge that the same letter conclusively proves that he likewise learnt nothing from them on the subject. In fact if, as has been asserted since, a knowledge of the method of fluxions had then been imparted to him, must he not have been out of his senses to propose a month after to the secretary of the Royal Society, a man of great skill in these matters, means of improving the method of Sluze itself, without saying a single word of another method much more simple which had just been taught him in England?
Thus I believe I may decidedly conclude either that Leibniz did not see the work De Analysi and Newton's letter when he was in England in October 1676; or, if he did see them, that he derived no assistance from them any more than the learned geometricians of England who had had all that time to mediate on them, and besides were at hand to apply to the author for every necessary illustration. The English have never formally declared that he had seen the book De Analysi; they content themselves with positively asserting that he had seen the letter of 1672. But supposing this to be true, we can draw no inference against Leibniz from it; for beside that the letter contains only results, without any demonstration, it is not very clear that it indicates a method essentially different from that of Sluze as the reader may have remarked from Newton's words already quoted.
In the whole of this business there are but three pieces truly decisive; first, a letter from Newton to Oldenburg dated the 24th of October 1676 which was communicated to Leibniz the year following: second, the answer which Leibniz returned to Oldenburg respecting this letter, June the 21st, 1677: third, the scholia to Newton's Principia already quoted, published towards the end of 1686. Let us briefly analyse these three pieces.
Newton's letter, exclusive of different researches concerning series which are here to be left out of the question, contains several theorems that have the method of fluxions for their basis, but the author keeps his demonstrations secret. He contents himself with saying that he has deduced them from the solution of a general problem which he expresses enigmatically by transposing the letters and the sense of which, as explained after the business was known, is 'an equation containing flowing quantities being given, to find fluxions and inversely.' What light could Leibniz derive from such an anagram? All we can conclude from such a letter is that at the time it was written Newton was in possession of the method of fluxions; by which, however, is to be understood simply the method of tangents and quadratures; for the method of resolving differential equations was then out of the question, this not being invented till long after as has been said above.
Leibniz, in his letter to Oldenburg, begins with saying that he, as well as Newton, had found Sluze's method for tangents to be imperfect. Then he explains openly and without mystery that of the differential calculus, affirming that he had long employed it for drawing the tangents of curve lines. Here then we have the clear and positive solution of the problem, the possession of which Newton so carefully endeavoured to reserve to himself.
The scholia to the Principia says as follows: 'In a correspondence in which I was engaged with the very learned geometrician Mr Leibniz ten years ago [through the medium of Oldenburg], having informed him that I was acquainted with a method of determining the maxima and minima, drawing tangents, and doing other similar things, which succeeded equally in rational equations and radical quantities, and having concealed this method by transposing the letters of the words which signify: 'an equation containing any number of flowing quantities being given, to find fluxions and inversely': that celebrated gentleman answered that he had found a similar method; and this, which he communicated to me differed from mine only in the enunciation and notation.' To this the edition of 1714 adds: 'and in the idea of the generation of quantities.' Is it possible to say more expressly that Leibniz separately invented the method of fluxions and that he had communicated it frankly without involving himself in mystery like Newton?
From these three pieces therefore it is clear that if Newton first invented the method of fluxions, as is pretended to be proved by his letter of the 10th of December 1672, Leibniz equally invented it on his part, without borrowing anything 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, as we see from the extract given above. At that time the truth was near it's source and not yet altered by the passions. In vain did Newton afterwards 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. In the first place, without discussing his pretended priority, it was replied that two men, who separately make the same important discovery, have an equal claim to admiration; and that he who first makes it public has the first claim to the public gratitude. It was then proved to him that even his own principle did not justly apply here.
The design of stripping Leibniz and making him pass for a plagiary was carried so far in England that during the height of the dispute it was said (and Newton himself was not ashamed to support the objection) that the differential calculus of Leibniz was nothing more than the method of Barrow. What are you thinking of, answered Leibniz, to bring such a charge against me? Will you have the differential calculus to be nothing but the method of Barrow when I claim it? and at the same time say it was invented by Mr Newton when you wish to rob me of it? Can you be so blinded by passion as not to perceive this manifest contradiction? 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 a pupil and friend of Barrow and had the 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?
Johann Bernoulli, who in concert with his brother learned the analysis of infinities from the writings of Leibniz, opposed to the Commercium epistolicum a letter where he 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 though all the journals of Holland and Germany. His reasons are in substance, first, the Commercium epistolicum exhibits no vestige of Newton's having employed dotted letters to denote fluxions in the writings alleged; secondly, in the Principia, where the author had so frequently occasion for employing this calculus and giving it's algorithm, he has not done it; he proceeds everywhere by means of lines and figures without any determinate analysis, and simply in the manner of Huygens, Roberval, Cavalleri, etc,: thirdy, the dotted letters first began to appear in the third volume of Wallis's Works, several years after the differential calculus was everywhere known; fourthly, the true method of differencing differences, or of taking the fluxions of fluxions, was unknown to Newton, since even in his treatise on quadratures, not published till 1704, the rule he gives at the end for determining the fluxions of all orders by considering these fluxions as the terms of the power of a binomial formed of a variable quantity, and it's first fluxion, and treating the first fluxion as constant, is false except simply for the term which answers to the first fluxion: fifthly, at the same period of 1704 Newton was not versed in the integral calculus of differential equations which Leibniz and the two Bernoullis had already carried so far; otherwise he would not have failed to treat this part of the analysis of infinities, the most difficult, and at least as worthy of being promulgated and carried to perfection as the quadratures on which he enlarged so much.
To this letter the English answered that the notation did not constitute the method: that the principles of the calculus of fluxions were contained in Newton's great work and in his letters: that the rule in the treatise on quadratures for finding the fluxions of all orders was true, suppressing the denominators of the terms of the series and gave by consequence quantities proportional to the true fluxions. I do not find that they gave any answer to the last objection.
The partisans of Leibniz replied that the advantages of an analytic method depend in great measure on the simplicity of the algorithm: that the Characteristic of Leibniz had already occasioned the new analysis to make immense progress at a time when scarcely anyone had heard of Newton's book: that it was in vain to endeavour to deny or palliate the erroneousness of Newton's rule for finding the fluxions of all orders: and that it could not be said that the terms of a seris of fractions were proportional to the terms of another series of fractions when the corresponding terms had different denominators, as was the case here.
Such were the reasons alleged and contested between the two parties for more than four years. The death of Leibniz, 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 scolium relating to Leibniz 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 more unjust?
In later times there have been geometricians who, without taking a decided part between Newton and Leibniz, have objected to the latter, 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 Leibniz might answer: 'I have proposed the existence of infinitely small quantities only as subsidiary, or as a simple hypothesis, serving to abridge the calculus and reasonings on which it is founded. I have no need for the existence of infinitely small quantities: it is enough for my purpose, as I have said in several of my works, that the differences are less than any finite quantity you please to assign; and consequently the error which may result from my supposition is less than any determinable error, 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. Mr de Fontenelle, who however meant me well, was wrong when he contented himself with saying at the beginning of his Géométrie de l'Infini, that, after having at first admitted infinitely small quantities, I had at length receded so far as to reduce the infinities of different orders to mere incomparables in the sense in which a grain of sand would be incomparable to the globe of the Earth. He should have added that I use this similitude only for the purpose of presenting a general and sensible idea of my differences to the imagination of certain readers; and that in the paper to which he alludes, I conclude with remarking expressly that instead of an infinitely large or small quantity, quantities should be assumed as large or as small as is necessary to render the error less than a given error. 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 anyone; and whatever may have been urged in this respect my rival has no real advantage over me.
Lastly it has been said that, notwithstanding Newton's affection of employing synthesis alone in his Principia, at present it cannot be doubted that he discovered a great number of propositions by the analytic method of fluxions: that this application of it to multitude of objects so important implies a long series of meditation: and that at least, according to all appearance, he must have been in possession of the method of fluxions before Leibniz, for he must have spent many years in writing his book. Let us examine the consequences to be drawn from this induction.
Perhaps there never existed a man more highly endowed than Newton with that strength of intellect and vigour of mind which are capable of conceiving, pursuing, and executing a vast design. Leibniz has published no single work that can be compared with the Principia for importance and connection of subject. Too much carried away by the vivacity of his genius, and the multitude and variety of his occupations, journeys, and literary correspondence with most of the learned in all parts of the World, he could not confine himself to delving long at one subject, or pursuing in detail all the consequences of a grand principle: but the Collection of his Works, and his Epistolary Correspondence with Johann Bernoulli, bear in every part the strongest marks of invention. Everywhere he disseminates new ideas, and germs of theories, the development of which would sometimes produce whole treatises. He has the advantage over Newton of having invented and carried to a great length the integral calculus of differential equations. If he be not equally profound with the English geometrician, he appears to surpass him in that rapid penetration and sharpness of intellect which seize on the most subtle and striking questions in any subject. The one has left us a greater mass of geometrical truths: the other more accelerated the progress of science in his time by the simple and commodious notation of his calculus, the applications he made of it himself, or enabled others of the learned to make, the encouragements he gave them, and the new paths he was continually opening to their meditations. To conclude, whatever length of time the completion of the Principia may have required, we ought not to forget that this work did not appear till two or three years after Leibniz had published his differential calculus, and some sketches of the integral.
Continuation of the dispute. War of problems between Johann Bernoulli and the English.
In this long dispute the mutual respect which the laws of decorum exact from all men was too frequently forgotten: yet it had at least the advantage of exciting a very active emulation among the greatest geometricians of the time. It produced challenges of very difficult problems, the solution of which gave rise to new theories, and considerably extended the domain of geometry.
Some time before his death, Leibniz, wishing to feel the pulse of the English, as he expressed himself, caused the celebrated problem of orthogonal trajectories to be proposed to them, which consists in finding the curve that cuts a series of given curves at a constant angle, or at an angle varying according to a given law. It is said that Newton, returning home at 4 o'clock greatly fatigued, received the problem and did not go to bed till he had solved it. His method may be reduced to the following few words. 'The nature of the curves to be intersected gives their tangents at the points of intersection: the angles of intersection give the perpendiculars of the intersected curves: two adjacent perpendiculars give by their points of concurrence the centre of curvature of the intersecting curve. Let the axis of the abscisses be conveniently placed, and assume the first fluxion of the absciss as unity: the position of the perpendicular will give the first fluxion of the ordinate of the required curve, and the curvature of this same curve will give the second fluxion of the ordinate: thus the problem will always be reduced to an equation. As to the resolution of the equation,' he adds, 'this belongs to another method.'
The English already triumphed: but Johann Bernoulli, taking up the cause of Leibniz who had just died, laughed at this scheme of a solution. He maintained that nothing was more easy than to come to the equation of the trajectory: that several particular questions of the kind had been handled with success even long before: that the essential part of the business was to resolve the differential equation of the trajectory when it could be done, either precisely or by the quadratures of curves: and that this resolution, far from being foreign to the problem, was the necessary completion of it: whence he concluded that Newton, having no method for it, had only eluded the difficulties of the question and by no means surmounted them.
Nicholas [Nicolaus (II) Bernoulli in the notation of our archive], the son of Johann Bernoulli, resolved in a very elegant manner the particular case in which the intersected curves are hyperbolas with the same centre and the same vertex. His cousin Nicholas Bernoulli [Nicolaus (I) Bernoulli in the notation of our archive] and Jacob Hermann treated the question more generally by methods which came to the same thing, although they had no communication with each other on the subject. These methods easily applied to all cases where the intersected curves are geometrical and even to some transcendental curves. Hermann, wishing to extend the formulas farther than they would bear, fell into some mistakes which were pointed out by the Bernoullis. However, they all agreed in considering Newton's solution as insufficient and of no use.
It appears that Newton completely left the field from this time: but some of his friends or disciples continued the war with ardour. Dr Taylor distinguished himself in it the most. Without stopping to develop Newton's solution, he gave one of his own in the Philosophical Transactions for 1717 which answered the question as proposed by Leibniz in it's full extent. Had he contented himself with this he would have merited only praise: but, urged on by his resentment against Johann Bernoulli who had spoken a little slightingly of him on another occasion, he prefixed to his solution some insulting reflections on the partisans of Leibniz, having principally in view Johann Bernoulli their leader. Among other things, he said that if they did not perceive how Newton's solution led to the equations of the problem, it must be attributed to their ignorance: illorum imperitiae tribuendum. The man to whom this strange insult was addressed was by no means inclined to forbearance; and he avenged himself in a manner the most humiliating to Taylor's vanity.
In a dissertation on orthogonal trajectories in the Leipzig Transactions for 1718, composed jointly by Johann Bernoulli and his son Nicholas, it was agreed that Dr Taylor's solution was accurate, and even evinced some sagacity; but then it was shown that it was far from being sufficiently general and that there existed a great number of resolvable cases to which it could not be applied. At the same time Johann Bernoulli gave another method which, to the advantage of being incomparablely more simple, added that of embracing all the geometrical curves, all the mechanical curves completely similar, and lastly a great number of mechanical curves incompletely similar. These discoveries were the product of a profound, new, and delicate analysis. The author had in his hands an instrument which he managed with dexterity, the method of differencing de curva in curvam. His victory was unequivocal: and Dr Taylor, notwithstanding the tone of self-sufficiency he at first assumed, was forced to acknowledge here a superior.
I shall observe, by the by, that the authors of this dissertation mention on the same subject a little piece of Nicholas Bernoulli the nephew's, in which we find for the first time the celebrated theorem of condition on which depends the reality of differential equations of the first order with three variable quantities: a theorem which some modern geometricians have endeavoured to arrogate to themselves.
While the question of trajectories was agitating, Dr Taylor proposed various problems on the integration of rational functions, at that time new and very difficult. Johann Bernoulli, who had already made some attempts in this direction in the Memoirs of the Academy of Sciences for 1702, easily solved all these problems in the Leipzig Transactions for 1719; and from the results he obtained he formed a series of curious theorems, the development and demonstration of which were useful exercises for his son and nephew.
We ought not to omit here for the honour of England, that Roger Cotes, professor of mathematics at Cambridge, had treated the same subject and reduced the integration of rational fractions to general and very commodious formulas in his celebrated work entitled Harmonia Mensurarum: but this was not published until six years after his death, which happened in 1716; no doubt therefore Taylor and the Bernoullis were unacquainted with its contents. In the same work of Cotes there are several other very useful discoveries such as his method of estimating errors in applied mathematics, his remarks on the differential method of Newton, his celebrated theorem for the resolution of certain equations, etc. Cotes was but thirty-four years of age when he died. Newton esteemed him highly and often said. 'if Mr Cotes had lived, he would have taught us all something.'
The animosity between Dr Taylor and Johann Bernoulli increased daily. In 1715 the Dr published his Methodus Incrementorum directa et inversa; a profound work, but a little obscure, in which he treated several problems that had been already resolved without quoting any person. In 1716 a letter appeared in the Leipzig Transactions commending Johann Bernoulli and openly treating Taylor as a plagiary. Of this he complained with acrimony; and at the same time retorted the accusation by showing that Johann Bernoulli, in his last solution of the isoperimetrical problem, had travestied the solution of his brother, and that all the simplifications he had made in it had not changed it's nature. From that time Johann Bernoulli kept no measures with him; and he published under the name of one Burcard, a schoolmaster in Basle, an answer to Taylor which was full of insult and ridicule, among which however we meet with some useful truths.
The problem of orthogonal trajectories gave birth to that of reciprocal trajectories, which was proposed at the end of the dissertation of the Bernoullis. They required the curves, which being constructed in two contrary directions on one axis of a given position, and then moving parallel to themselves with unequal velocities, should constantly intersect each other at a given angle. This was a fresh subject of analytical difficulties to be surmounted and for extending the science. It was a long time agitated between Johann Bernoulli and an anonymous Englishman, who was afterwards known to be Dr Pemberton, the particular friend of Newton. We are again obliged to say that Johann Bernoulli retained his superiority here by the simplicity and elegance of his solutions.
The English geometricians had formed a league against Johann Bernoulli, and attacked him on subjects of every kind. Alone, says Fontenelle, like another Horatius Cocles, he sustained on the bridge the efforts of their whole army. Keill, a soldier more bold than puissant, imagined he had found an opportunity of perplexing him. The theory of the resistance of mediums to the motion of bodies passing through them formed a considerable part of the Principia. Newton had determined the curve described by a projectile in a medium resisting in the ratio of the simple velocity: but had not touched on the case, at that time more difficult, where the resistance of the medium is as the square of the velocity. This case Keill proposed to Johann Bernoulli, who not only resolved it in a very short time, but extended the solution to the general hypothesis in which the resistance of the medium should be as any power of the velocity of the projectile. When he had discovered this theory, he offered repeatedly to send it to a confidential person in London on condition that Keill would give up his solution likewise; but Keill, though strongly urged, maintained a profound silence. The reason for this is not difficult to conjecture: he had not resolved his own problem. When he proposed it, he expected that no one would discover what had escaped the sagacity of Newton. In this conjecture he was cruelly mistaken: and his challenge, which was something more than indiscreet, drew on him a reprimand from the Swiss geometrician that was so much the more poignant as the only mode of answering it satisfactorily was by a solution of the problem which he could neither effect by his own skill nor by the assistance of his friends. Johann Bernoulli's triumph was complete. In the first intoxication of victory he indulged himself in sarcasms and jests against his rivals, not commended for their elegance but certainly pardonable on a man of a frank and honest disposition insidiously attacked, and who had to avenge affronts offered not only to himself but also to an illustrious friend whose loss he still lamented.
These learned contests drew the attention of all geometricians; and notwithstanding the acrimony infused into them by the passions, they stimulated men's minds and produced new proselytes to mathematics on all sides.
Miscellaneous articles.
I shall now step back a little and resume some other subjects which I have been obliged to leave in arrear.
In 1711 appeared the Analysis of Games of Chance by Remond de Montmort: a work abounding with acute and profound ideas the object of which is to subject probabilities to calculation; to estimate chances; to regulate wagers, etc. It does not properly belong to the new geometry, yet it contributed to it's progress by stimulating the spirit of combination in general, and by the extent which the author gave to the theory of series, a happy supplement to the imperfection of the rigorous methods in all branches of mathematics.
Three years afterwards de Moivre published a little treatise on the same subject entitled Mensura Sortis, chiefly remarkable for containing the elements of the theory of recurrent series and some very ingenious applications of it. This Essay, gradually increased by the reflections of the author, has grown up into a considerable work admired by all geometricians. The best edition is that of 1738 in English under the title of the Doctrine of Chances. De Moivre was a French geometrician whom the revocation of the edict of Nantes had obliged to quit his country, and who had fled to London. Born with superior talents for geometry the narrowness of his fortune obliged him to teach mathematics for a livelihood. Newton had the highest esteem for him. It is reported that during the last ten or twelve years of Newton's life, when a person came to ask him for an explanation of any part of his works he used to say: 'Go to Mr de Moivre; he knows all these things better than I do.'
Nicholas Bernoulli, the nephew, came to Paris in 1711. His great reputation and mild and easy manners gained him many illustrious friends. Among the number of these was Montmort, with whom he formed a strict intimacy in consequence of the similarity of their dispositions and taste for the analysis of probabilities. They spent three whole months together in the country solely employed in resolving the most difficult problems on this subject. All these new researches, and the elucidations arising from them, produced a second edition of Montmort's book in 1714, much superior to the first.
I have already mentioned Dr Taylor's Methodus Incrementorum, but this work, celebrated even in the present day, deserves more particular notice.
The author gives the name increments or decrements of variable quantities to the differences, whether finite or infinitely small, their calculus, either direct or inverse, belongs to the Leibnizian analysis or the method of fluxions; and Dr Taylor resolves a great number of problems of this kind. But when the differences are finite the method of finding the relations they bear to the quantities that produce them forms a new kind of calculus, the first principles of which were given by Dr Taylor; and in this respect the book has the merit of originality. In this manner he has summed up some very curious series.
The extreme conciseness, or rather obscurity, with which this work is written, long retarded the success which was due to it. Nicole, however, a very distinguished French geometrician, was able to understand it: he very clearly unfolded the method for resolving finite differences and added several new series of his own invention. The two excellent papers which he published on this subject in the Memoirs of the Academy of Sciences for 1717 and 1728 may be considered as the first methodical and luminous elementary treatise on the integral calculus with finite differences that ever appeared.
Several other works of the time might be mentioned but I must be brief. I would request the reader, therefore, to consult the periodical publications of Germany, France, England, Italy, etc. with those by the different academies, where he will find a number of valuable papers on every branch of mathematics.
It has been observed that the Royal Society of London and the Academy of Sciences at Paris arose nearly at the same time, or about the year 1660. The Academy of Berlin, the establishment of which was projected in 1700, took a regular and legal form in 1710, under the auspices of Frederick III, elector of Brandenburg, and the first king of Prussia, and Leibniz was appointed perpetual president. The Institute of Bologna was founded in 1713 through the means of the celebrated count de Marsigli to whom natural history is so much indebted. In 1726 Catharine I, the widow of Peter the Great, founded the Academy of St Petersburg. Several other learned societies have since been formed which it would take up too much room to particularise. All these establishments have been of extreme utility to the progress of the sciences.
Last Updated August 2007