The new calculus was not at first cultivated with that attention which its importance deserved; and, therefore, in order to rouse the attention of mathematicians, Leib nitz, in 1687, proposed the following problem : " 'ro de termine the curve a heavy body ought to describe, in or der to descend equally in equal times." Huygens was the first that sheaved what was the nature of the curve, but he did not indicate his method of solution. James Bernoulli also resolved the problem by the differential calculus, and published his analysis in the Leipsic Acts of 1690. About the same time, John Bernoulli, a younger brother of James, began his career as a mathematician : he studied the science, aided by his brother's instructions, and he con tracted a friendship for Leibnitz, which continued until the death of the latter, in 1716. He made the calculus known in France, and gave lessons on the subject to the Marquis de Leibnitz and the Bernoullis re solved many new and difficult problems, which they pro posed as challenges to the geometers of that period. They also determined the nature of the catenaria, (or curve formed by a chain or cord which hangs freely, but is fas tened at its extremities,) and the curve of swiftest descent, which had proved too difficult for Galileo, and the mathe matical theories known in his time. A spirit of rivalship was excited between the two Bernoullis, and they waged a war of problems, each endeavouring to puzzle the other; this, although carried on with a degree of animosity on the part of John not at all becoming, was yet of advan tage to the science, as it produced the celebrated isoperi metrical problems, a class more difficult than any that had previously engaged the attention of mathematicians ; al though, indeed, Newton had resolved a problem of this kind in his Principia, when treating of the solid of least resistance. The calculus went on improving continual ly ; it was applied to the theory of evolutes, one of the most beautiful discoveries made by Huygens; but, with the ex ception of some pieces in the Leipsic Acts, there was as yet no work professedly on the subject; at length, the Mar quis de published his ylnalyse des infiniment pe tits, in 1699. John Bernoulli claimed the invention of the principal methods in this work, confidentially to Leibnitz in l'llopital's lifetime, and publicly after his death. In deed, L'Ilopital acknowledges in the preface his obliga tions to the two Bernoullis and Leibnitz, allowing them to claim as much of it as they pleased, and professing that lie would be content with the remainder. L'Hopital's book treats only of one part of the theory, viz. the diffe rential calculus, which answers to the direct method of fluxions. He says lie had intended to give a work on the integral calculus ; that is, the inverse method of fluxions ; but Leibnitz had informed him, that he was then prepay.
ing a treatise De Scientia injinhti, which he did not wish to anticipate. This work, however, never appeared. The first general theory in this part of the subject related to the integration of rational fractions, which John Bernoulli gave in 1702 : but, indeed, he had indicated the method of integrating differential equations, by separating the varia ble quantities as far back as 1694. In 1707, Gabriel Man fredi, an Italian, gave an entire work, entitled, De Con structione requationum differentialium primi gradus, which contained all that had been done down to that time relating to the integral calculus.
John Bernoulli composed a series of lectures on the in tegral calculus, for the use of.his scholar and patron, L'Ho pital; this was when he came to Paris in the year 1692.
These are curious, as the earliest essays in this branch of the calculus, and valuable by their intrinsic merit. They would have formed an excellent sequel to L'Hopital's work, but they were not published until 1742, when they appeared in the third volume of Bernoulli's works.
It is to be regretted that Newton did not accomplish a design he had formed in 1671, of publishing his method of fluxions, and its application; for, with the exception of what he himself had done, hardly any thing appeared in England on the subject before the end of the century. Da vid Gregory explained some of its principles and applica tions, in a treatise, De dimensions figurarurn, printed in 1684. John Craig published a treatise, De curvarunz qua draturis, in 1693, which he afterwards enlarged and pub lished again in 1718, with the title De calculo fluentium. De Moivre and Fatio gave solutions in the Philosophical Transactions of the problem concerning the solid of least resistance ; the latter in 1695, and the former in 1699.
In the year 1703, George Cheyne, a Scottish mathema tician and physician, published his Methodus Fluxionum inversa, Edin. 1703. The author committed some mis takes which were pointed out by De Moivre : He had also been wanting in justice to the mathematicians on the con tinent, and this exposed him to the animadversions of John Bernoulli. In the year 1704, a treatise of fluxions was published by Charles Hayes, Gent. This, we believe, was the earliest work on the subject that was written in the English language.
It is remarkable that Newton himself should have been so slow in publishing any thing relating to his calculus. The year 1699 must be considered as the epoch at which his numerous analytical inventions were first made gene rally known ; but this was in the second volume of the works of Wallis. At length, however, in the year 1704, when he printed his Optics, he added to it, Tractatus de Quadratura Curvarum, in which he explains the principles of his method, applying it to quadratures. Besides, he composed the work he had originally intended, On the me !hod of Fluxions and Infinite Series, with its application to the Geometry of Curve Lines. It was written in Latin, and Dr Pemberton once intended to have published it, with the author's consent, in his lifetime : This, however, was not clone; and it was not printed until 1736, many years after Newton's death, when Colson translated it into Eng lish, and added to it a comment.
In enumerating the early improvers of the fluxional cal culus, Cotes deserves particularly to be mentioned : He discovered a very elegant property of the circle, by which the fluents of a certain class of rational fractions were de termined by means of the trigonometrical tables and loga rithms. Unfortunately for science this excellent mathe matician died early in life. Newton had formed great ex pectations from him. His theorem forms the basis of his posthumous work, Harmonia Mensurarum, published in 1722, by his friend Dr Smith. The inventions of Cotes were extended and completed by De Moivre, in his Ms cellanea Analytica, published in 1730. Dr Brook Taylor also holds a distinguished place in the higher class of those who extended the calculus. Methodus Incrementorum, printed in 1715, contains in the second part many appli cations of fluxions to physico-mathematical problems. His theorem for the developement of any function of a binomial, leads to many beautiful applications of Iluxions ; and one of the greatest mathematicians in modern times, the late Lagrange, has made it the foundation of his theory of the calculus.