208 LA GRANGE motions of vibrating strings, and demonstrated the insuffi ciency of the methods employed by both his great contem poraries in dealing with the latter subject. He further treated in a masterly manner of echoes and the mixture of sounds, and explained the phenomenon of grave harmonics as due to the occurrence of beats so rapid as to generate a musical note. This was followed, in the second volume of the Miscellanea Taurinensia (1762) by his " Essai d uno nouvelle methode pour determiner les maxima et les minima des formules intdgrales inde"finies," together with the application of this important development of analysis to the solution of several dynamical problems, as well as to the demonstration of the mechanical principle of "least action." The essential point in his advance on Euler s mode of investigating curves of maximum or minimum consisted in his purely analytical conception of the subject. He not only freed it from all trammels of geometrical construction, but by the introduction of the symbol 8 gave it the efficacy of a new calculus. He is thus justly re garded as the inventor of the " method of variations "- a name supplied by Euler in 17GG. By these performances Lagrange found himself, at the ag3 of twenty-six, on the summit of European fame. But fmcli a height had not been reached without cost. Intense application during early youth had checked his growth, and weakened a constitution never robust. Accesses of feverish exaltation culminated, in tho spring of 1761, in an attack of bilious hypochondria, which permanently lowered the tone of his nervous system, and rendered him liable, throughout his life, to recurrences of the same complaint at the same time of year. Rest and exercise, however, temporarily restored his health, and lie gave proof of the undiminished vigour of his powers by carrying off, in 1764, the prize offered by the Paris Academy of Sciences for the best essay on the libration of the moon. His treatise was remarkable, not only as offering a satis factory explanation of the coincidence between the lunar periods of rotation and revolution, but as containing the jirst employment of his radical formula of mechanics, obtained by combining with the principle of D Alembert that of virtual velocities. His success encouraged the Academy to propose, in 1766, as a theme for competition, the hitherto unattempted theory of the Jovian system. The prize was again awarded to Lagrange ; and he subsequently eirned the same distinction with essays on the problem of three bodies in 1772, on the secular equation of the moon in 1774, and in 1778 on the theory of cometary perturba tions. He hid in the meantime gratified a long felt desire by a visit to Paris, where he enjoyed the keen and stimulating delight of conversing with such mathematicians as Clairaut, D Alembert, Condorcst, and the Abbe Marie. An attack of illness frustrated his design of extending his journey to London, and ho returned, though not for long, to the comparative isolation of the Piedmontese capital. The post of director of the mathematical department of the Berlin Academy (of which he had been a member since 1759) becoming vacant by the removal of Euler to St Petersburg, both he and D Alembert united, by unpreme ditated concert, to recommend Lagrange as his successor. Euler s eulogium was enhanced by his desire to quit Berlin, D Alembert s by his dread of a royal command to repair thither ; and the result was that an invitation, conveying the wish of the " greatest king in Europe " to have the " greatest mathematician " at his court, was sent to Turin. On November 6, 1766, Lagrange was installed in his new position, with a salary of 6000 francs, ample leisure for scientific research, and an amount of royal favour sufficient to secure him respect without exciting envy. The national jealousy of foreigners, it is true, was at first a source of annoyance to him; but such prejudices were gradually disarmed by the mild inoffensiveness of his demeanour, and by his strict adherence to a policy of non-intervention outside his own immediate domain. We are told that the universal example of his colleagues, rather than any desire for female society, impelled him to matrimony ; an excess of home-sickness, however, probably directed his choice towards a lady of the Contl family (related to his own by a previous alliance), who, by his request, joined him at Berlin. The experiment was cut short by a lingering illness, during which he devoted all his time, and a con siderable store of medical knowledge, to the care of the dying woman. The long series of memoirs sonic of them complete treatises of great moment in the history of science communicated by Lagrange to the Berlin Academy between the years 1767 and 1787 were not the only fruits of his exile on the banks of the Spree. His JIccaniqite Analytiyue, the production in which his genius most fully and characteristically displayed itself, was clue to the same period. This great work was the perfect realization of a design present to the mind of its author almost from boy hood, and of which lie had given a clear though concise sketch in his first published essay. 1 Its scope may bo briefly described as the reduction of the theory of mechanics to certain general formulae, from the simple development of which should be derived the equations necessary for the solution of each separate problem. 2 From the fundamental principle of virtual velocities, which thus acquired a new significance, Lagrange deduced, with the aid of the calculus of variations, the whole system of mechanical truths, by processes so elegant, lucid, and harmonious as to constitute, in Sir William Hamilton s words, "a kind of scientific poem." This unification of method was one of matter also. By his mode of regarding a liquid as a material system characterized by the unshackled mobility of its minutest parts, the separation between the mechanics of matter in different forms of aggregation finally disappeared, and the fundamental equation of forces was for the first time extended to hydrostatics and hydrodynamics. 3 Thus a universal science of matter and motion was derived, by an unbroken sequence of deduction, from one radical principle ; and analytical mechanics assumed the clear and complete form of logical perfection which it now wears. A publisher having with some difficulty been found, the book appeared in Paris, under the supervision of Legendre, in 1788. But before that time Lagrange himself was on the spot. After the death of Frederick the Great. his presence was competed for by the courts of France, Spain, and Naples, and a residence in Berlin having ceased to possess any attraction for him, he removed to Paris in 1787. His reception was most flattering. Marie Antoinette warmly patronized him. He was lodged in the Louvre, received the grant of an income equal to that hitherto enjoyed by him, and, with the title of "veteran pensioner " in lieu of that of " foreign associate " (conferred in 1772), the right of voting at the deliberations of the Academy. In the midst of these distinctions, a profound melancholy seized upon him. His mathematical enthu siasm, hitherto the happiness of his life, was for the time completely quenched, and during two years the printed volume of his Mecaniqne, which he had seen only in manuscript, lay unopened beside him. He relieved his dejection with miscellaneous studies, especially with that of chemistry, which, in the new form given to it by Lavoisier, he found " aisee comme 1 algebre." The dis- 1 CEt/vrcs, i. p. 15. 2 Mec. An., Advertisement to 1st ed. 3 Diihring, Kritische Gesch. dcr Mechanik, pp. 220, 367; Lagrange, Mec. An., i. pp. 166-72, 3d ed.
