The science received considerable improvement from the mathematicians in Germany, particularly in that branch which relates to the fluents of fluxions, containing several variable quantities. The two Nicolas Bernonllis, one a son of James and the other a son of John Bernoulli, and Daniel Bernoulli, another son of John, gave, in the Leipsic Acts, and in the Petersburg Memoirs, a multitude of profound disquisitions relating to the calculus ; and to these may be added the labours of their countryman Herman.
The dispute between the schools of Newton and Leib nitz tended to the improvement of the calculus, by the problems which each party proposed as challenges to the other. Thus Leibnitz, in order to feel the pulse of the Eng lish, as he expressed it, a shot t time before his death, pro posed the problem of orthogonal trajectories, that is, curves which cut in a given angle a series of curves of the same gi ven kind. Newton resolved the problem on the day he re ceived it ; but John Bernoulli did not consider his solution as complete, because he had not integrated the fluxional equation, but taken for granted that the manner of doing it was known. On the other side, Neill challenged Bernoul li, by name, to find the path of a projectile, moving in a me dium, in which the resistance was as the square of the ve locity. Bernoulli quickly resolved the problem, not only in that particular case, but also when the resistance was as any power whatever of the velocity. He then required that Keill should produce his own solution ; but Keill had not resolved his problem himself, and in fact found it too hard. He therefore maintained a profound silence, and Bernoulli obtained a complete triumph over the English philosopher.
Taylor also proposed, as a challenge, a fluxion of a par ticular form to be integrated ; this was addressed to all ma thematicians not English. As John Bernoulli was under stood to be particularly aimed at, he offered to wager fifty guineas that he could resolve Taylor's problem, and fifty more that he would propose a problem which Taylor should not resolve, but winch he could resolve himself. Taylor did not accept this challenge : Bernoulli gave a so lution of Taylor's problem in the Leipsic Acts.
The new calculi excited a controversy of a different kind, respecting the accuracy of their principles. These were
attacked on the continent by Niewentiit, and Rolle ; and de fended by Leibnitz, Varignon, and Saurin. In England, Dr Berkeley, Bishop of Cloyne, called in question, not only the logical accuracy of tne reasoning employed to establish the theory of fluxions, but also the faith of mathematicians in general, in regard to matters of religion. He began the controversy in his work entitled the Minute Philosopher. But the principal attack was made in 1734, in The Analyst, or a discourse addressed to an Infidel Mathematician, (un derstood to be Dr Halley,) wherein it is examined whether the object. principles, and inferences of the modern analysis are more distinctly conceived than religious mysteries and _points of faith. One of the best answers to the Bishop came from the pen of Benjamin Robins, in A discourse con cerning the nature and certainty of Sir Isaac Newton's me thod of Fluxions,and of Prime and Ultimate Ratios. Berke.
ley, however, had some reason for his objections. The very concise manner in which the great inventor had pro mulgated his discovery, might leave room for a dispute about the accuracy of the terms. Instead of defending these, it wis better to adopt a mode of explanation more intelligible, and consonant to the common methods of ma thematical reasoning. This was done by Maclaurin, in his Treatise of Fluxions, (1742.) He has there placed the principles of the method upon the firm basis of geometri cal demonstration ; but his demonstrations are tedious, so that we fear few have patience enough to study the sub ject, as delivered in the first part of his work. The se cond, in which the subject is considered in the usual man ner, and algebraic characters arc employed, is very valua ble, and indeed the whole work abounds with original views of the theories connected with fluxions, and it proves the author to belong to the highest class of mathematicians.
Before the publication of Maclaurin's treatise, Mr Tho mas Simpson had given the first edition of his Xew Trea tise of Fluxions, (1737.) He new modelled the work, and published it again in 1750. This was a very valuable work at the time it appeared, and, as far as it goes, is at the pre sent time one of the best introductions to the method of fluxions in the English language.