The consequences of the Newton-Leibniz controversy, are given in Chapter V of Bossut's book. A version is given here: Bossut Chapter V.
We give below Chapter VI of Bossut's A General History of Mathematics from the Earliest Times to the Middle of the Eighteenth Century on the later consequences of the Newton-Leibniz controversy.
Continuation of the dispute. War of problems between Johann Bernoulli and the English. Miscellaneous articles.
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 it's 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.
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.