investigations and claim the invention. Lagrange did quite as much as Euler towards the creation of the Calculus of Variations. As it came from Euler it lacked an analytic foundation, and this Lagrange supplied. He separated the principles of this calculus from geometric considerations by which his predecessor had derived them. Euler had assumed as fixed the limits of the integral, i.e. the extremities of the curve to be determined, but Lagrange removed this restriction and allowed all co-ordinates of the curve to vary at the same time. Euler introduced in 1766 the
name "calculus of variations," and did much to improve this science along the lines marked out by Lagrange.
Another subject engaging the attention of Lagrange at Turin was the propagation of sound. In his papers on this subject in the Miscellanea Taurinensia, the young mathematician appears as the critic of Newton, and the arbiter between
Euler and D'Alembert. By considering only the particles
which are in a straight line, he reduced the problem to the same partial differential equation that represents the motions of vibrating strings. The general integral of this was found by
D'Alembert to contain two arbitrary functions, and the question
now came to be discussed whether an arbitrary function may be discontinuous. D'Alembert maintained the negative against Euler, Daniel Bernoulli, and finally Lagrange,–-arguing
that in order to determine the position of a point of the chord at a time , the initial position of the chord must be continuous. Lagrange settled the question in the affirmative.
By constant application during nine years, Lagrange, at the age of twenty-six, stood at the summit of European fame. But his intense studies had seriously weakened a constitution never robust, and though his physicians induced him to take rest and exercise, his nervous system never fully recovered its tone, and he was thenceforth subject to fits of melancholy.
In 1764 the French Academy proposed as the subject of a prize the theory of the libration of the moon. It demanded an explanation, on the principle of universal gravitation, why the moon always turns, with but slight variations, the same face to the earth. Lagrange secured the prize. This success encouraged the Academy to propose as a prize the theory of the four satellites of Jupiter,–-a problem of six bodies, more difficult than the one of three bodies previously solved by Clairaut, D'Alembert, and Euler. Lagrange overcame the
difficulties, but the shortness of time did not permit him to
exhaust the subject. Twenty-four years afterwards it was completed by Laplace. Later astronomical investigations of
Lagrange are on cometary perturbations (1778 and 1783), on Kepler's problem, and on a new method of solving the problem
of three bodies.
Being anxious to make the personal acquaintance of leading mathematicians, Lagrange visited Paris, where he enjoyed the stimulating delight of conversing with Clairaut, D'Alembert, Condorcet, the Abbé Marie,