To make the motives of this admiration which Euler bestowed with so much frankness better understood, it will not be useless to go back to the origin of the researches of La Grange, such as he stated them himself two days be fore his death.
The first attempts to determine the maximum and mini mum in all indefinite integral formula, were made upon the occasion of' the curve of swiftest descent, and the iso perimeters of B:!rnoulli. Euler had brought them to a general method, in an original work, in which the pro foundest knowledge of the calculus is conspicuous. But however ingenious his method was, it had not all the sim plicity which one would wish to see in a work of pure analysis. The author admitted this himself. Ile allowed the necessity of a demonstration independent of geometry. He appeared to doubt the resources of analysis, and ter minated his work by saying, " If my principle be not suf ficiently demonstrated, yet as it is conformable to truth, I have no doubt that, by means of a rigid metaphysical expla nation, it may be put in the clearest light, and I leave that task to the metaphysicians." This appeal, to which the metaphysicians paid no atten tion, was listened to by La Grange, and excited his emula tion. In a short time the young man found the solution of which Euler had despaired. He found it by analysis. And in giving an account of the way in which he had been Ied to that discovery, he said expressly, and as it were in answer to Euler's doubt, that he regarded it not as a me taphysical principle, but as a necessary result of the laws of mechanics, as a simple corollary from a more general law, which he afterwards made the foundation of his Me chanique 4nalytique. (See that work, page 189 of the first edition.) This noble emulation, which excited him to triumph over difficulties considered as insurmountable, and to rec tify or complete theories remaining imperfect, appears to have always directed M. la Grange in the choice of his subjects.
D'Alembert had considered it as impossible to subject to calculation the motions of a fluid inclosed in a vessel, unless this vessel had a certain figure. La Grange demon strates the contrary ; except in the case when the fluid di vides itself into different masses. But even then we may determine the places where the fluid divides itself into dif ferent portions, and ascertain the motion of each as if it were alone.
D'Alembert had thought that in a fluid mass, such ar; the earth may have been at its origin, it was not necessary for the different beds to be on a level. La Grange shows that the equations of D'Alembert are themselves equa tions of beds on a level.
In combating D'Alembert with all the delicacy due to a mathematician of his rank, he often employs very beauti ful theorems, for which he was indebted to his adversary. D'Alembert on his side added to the researches of La Grange. " Your problem appeared to me so beautiful,"
says he in a lettter to La Grange, " that I have sought for another solution of it. I have found a simpler me thod of arriving at your elegant formula." These exam ples, which it would be easy to multiply, prove with what politeness these celebrated rivals corresponded, who, op posing each other without intermission, whether conquer ors or conquered, constantly found in their discussion rea sons for esteeming each other more, and furnished to their antagonist occasions which might Lead them to new triumphs.
The academy of sciences of Paris had proposed, as the subject of a prize, the theory of the libration of the moon. That is to say, they demanded the cause why the moon, in revolving round the earth, always turns the same face to it, some variations excepted, observed by astronomers, and of which Cassini had first explained the phenomena. The point was to calculate all the phenomena, and to deduce them from the principle of universal gravitation. Such a subject was an appeal to the genius of Grange, an op portunity furnished to apply his analytical principles and discoveries. The attempt of D'Alembull was not disap pointed. The memoir of La Grange is one of his finest pieces. We see in it th,_ first development of his ideas, and the germ of his illecaniqzte 4nalytique. D'Alembert wrote to him : " I have read with as :nucn pleasure as ad vantage your excellent paper on the Libration, so worthy of the prize which it obtained." This success encouraged the academy to propose, as a prize, the theory of the satellites of Jupiter. Euler, Clairaut, and D'Alembert had employed themselves about the problem of three bodies, as connected with the lunar motions. Bailly then applied the theory of Clairaut to the problem of the satellites, and it had led him to very in teresting results. But this theory was insufficient. The earth has only one moon, while Jupiter has four, which ought continually to act upon each other, and alter their positions in their revolutions. The problem was that of six bodies. La Grange attacked the difficulty and over came it, demonstrated the cause of the inequalities observ ed by astronomers, and pointed out some others too feeble to be ascertained by observations. The shortness of the time allowed, and the immensity of the calculations, both analytical and numerical, did not permit him to exhaust the subject entirely in a first memoir. He was sensible of tl;is himself, and promised further results, which his other labours always prevented him from giving. Twenty four years after, M. La Place took up that difficult theory, and made important discoveries in it, which completed it, and put it in the power of astronomers to banish empiri cism from their tables.