CHAP. VII. Continuation of the progress of geometry. Solutions of various problems. Two very curious problems, proposed by Hermann in the Leipsic Transactions for 1719, employed the geometricians for some time, with much utility. The first consisted in finding a curve, the area of which should be equal to a certain proposed function of the coordinates. The second, far more difficult, was to determine an algebraic curve, such that the indefinite expression of it's length should include the quadrature of a given algebraic curve, plus or minus a given number of algebraic quantities. Nicholas Bernoulli, the son, resolved the first: Act. Lips. 1720. As to the second he confessed, though he wrote under his father's eye, that he could not resolve it without certain suppositions, which restricted it's generality. Hermann gave the general solution, by a very ingenious method, founded on the theory of evolutes, ib. 1723; and on this occasion he had the advantage over the Bernoullis. The next year John Bernoulli returned to the same question, and treated it in a more direct and analytical manner, giving it at the same time still greater extent. One general observation may be made on all the problems thus depending on the analysis of infinites. cc 4 Commonly Commonly they can be formed into equations with tolerable ease: but the principal difficulty is to resolve these equations, which is frequently such, as to elude all the powers of analysis. Accordingly the greatest geometricians have turned their attention to the improvement of the integral calculus, or the resolution of differential equations of all orders. With this view count James Riccati, having fallen on a differential equation of the first order with two variable quantities, apparently very simple, but which notwithstanding he was unable to resolve generally, proposed the question to all geometricians, in the Leipsic Transactions for 1725. No one could completely accomplish the object: but a great number of cases were pointed out, in which the indeterminates are separable, and in which the equation may consequently be resolved by the quadratures of curves. The authors of these elegant discoveries were Riccati himself, Nicholas Bernoulli the nephew, Nicholas Bernoulli the son, his brother Daniel, and Goldebach. They all obtained the same results by different methods. The equation in question is commonly called Riccati's, though it had been already considered by James Bernoulli, who had resolved certain particular cases of it. It is much the same in the analysis of infinites as the quadrature of the circle is in elementary geometry. When an equation is reduced to this, the problem is considered as resolved. If the equation do not fall under separable cases, we have no other resource, but to resolve it by the methods of approx imation. The The celebrated Euler, who was born in 1707, and died in 1783, a man destined to produce a revolution in the analytical science, made himself known at this time by various researches; and among others by a very elegant solution of the problem of reciprocal trajectories, which appeared in the Leipsic Transactions for 1727, and which he afterward extended and improved. He had acquired his first knowledge of mathematics under John Bernoulli, who, at the end of his own solution of the problem just mentioned, predicted what such a pupil would one day become. At the foundation of the Academy of Petersburg, the example that Ptolemy Philadelphus had given at the Museum at Alexandria was revived: a colony of geometricians, astronomers, natural philosophers, &c., was invited from all parts of Europe. Among them were Nicholas Bernoulli and his brother Daniel, Euler, Leutmann, Bulfinger, &c, and beside these resident members, the academy had several illustrious foreign associates, as John Bernoulli, Wolf, Poleni, Michelotti, &c. All these men, ardent, laborious, and eminent for their genius, were eager to enrich the collections of the academy. In the first volume of it's Memoirs, published in 1726, we find two or three excellent papers by Nicholas Bernoulli, who was unhappily cut off almost at his entrance on his career. The two persons who contributed most to the glory of geometry in this establishment, both at it's commencement, and during it's progress, were Daniel Bernoulli and Euler. Most of the problems, that had been attempted at the first effervescence of the new geometry, had for their object particular theories, to which all the extent they were susceptible of had not been given. Daniel Bernoulli and Euler generalized many of these old problems, such as that of the catenary curves, and isoperimetrical figures: they likewise treated others, that were absolutely new and very difficult, as, for example, the determination of the oscillatory motions of a heavy chain suspended vertically, the investigation of the tones given by an elastic slip of metal when struck, the motions resulting from the eccentric percussions of bodies, &c. All these questions required great natural sagacity, and a profound knowledge of analysis. Our two geometricians resolved them each separately; and we ought not to forget the rare example of moderation and honour which they then exhibited, and from which they never once deviated. They mutually proposed problems to each other, and worked on the same subjects, without their rivalry of talents, or difference of opinion on certain points connected with physics, diminishing that strict friendship, which they had contracted in their youth. Each frankly and without hesitation did justice to the other: in the science of analysis Daniel Bernoulli struck to Euler, whom he called his admiral; but in questions that required more acuteness of intellect than profound geometry, Daniel Bernoulli took the lead: in fact he had a peculiar talent for applying geometry to physics, and subjecting phenomena, which were known only in a general and vague manner, to precise calculation. Το To Pascal has been attributed the project of making men submit to the yoke of religion, by the force of eloquence and reasoning. It seems as if Euler had been in like manner desirous of rendering analysis paramount over all parts of mathematics. We find him continually busied in improving this grand instrument, and showing the art of handling it with dexterity. Scarcely had he attained the age of twentyone, when he gave a new and general method of resolving whole classes of differential equations of the second order, subject to certain conditions. Mem. of the Ac. of Petersburg for 1728. This had been accomplished before only in certain particular cases; and then rather by the sagacity of the analyst, than by uniform and determinate methods. In Italy, Gabriel Manfredi published, from time to time, ingenious papers on geometry and analysis, in the Commentaries of the Institute of Bologna, and in the journals. Another geometrician of the same nation, the count de Fagnani, opened a field of new problems of a very attractive kind. He taught how to determine arcs of the ellipsis, or hyperbola, the difference of which is an algebraic quantity. Leibnitz and John Bernoulli, who had attempted this research, judged, that it could not be subjected to the new calculi. They had merely resolved the question for the parabola, and by employing the common algebraic calculus. It is likewise resolved by the same means in the marquis de l'Hopital's treatise on Conic Sections. Fagnani very dexterously applied the integral calculus to the arcs of the ellipsis and hyperbola, thus including the |