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 Chapter V of Bossut's A General History of Mathematics from the Earliest Times to the Middle of the Eighteenth Century which begins a study of the Newton-Leibniz controversy.
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.