Lagrange' s Memoirs. 71 



The Academy of sciences of Paris, had proposed for the subject 

 of one of its prizes, the theory of the hbration of the moon : that is 

 to say, it asked the cause why the moon, in turning around the Earth 

 always shows the same face, with the exception of some variations 

 observed by astronomers, and of which Cassini I. had well explain- 

 ed the mechanism. The point was, to find the means of calculating 

 the phenomena, and of deducing them analytically from the princi- 

 ple of universal gravitation. Such a chance was an appeal to the 

 genius of Lagrange ; one, which was held out to him of applying his 

 principles and his analytical discoveries. The hope of D'Alembert 

 was not blasted. The piece of Lagrange is one of his highest titles 

 of glory. Therein are seen the first developments of his ideas and 

 the germ of the Mecanique Analytique. D'Alembert wrote to him ; 

 fai lu avec autant de plaisir que de fruit voire belle piece sur la 

 Hbration, si digne du prix qu'elle a remportee. 



His success* inspired in the Academy the confidence of proposing 

 the theory of the satellites of Jupiter. Euler, Clairaut, and D'Alem- 

 bert had engaged about the problems of the three bodies on account 

 of the motions of the moon. Bailly applied then the theory of Clair- 

 aut to the problems of satellites. It led him to results strikingly in- 

 teresting ; yet this theory was insufficient. The earth has but one 

 moon, Jupiter four, which must be reciprocally troubled and deranged 

 in their orbit. The problem seemed that of six bodies, the Sun, Ju- 

 piter, and the four moons. M. Lagrange attacked the difficulty in 

 front, triumphed over it happily, demonstrated the cause of inequali- 

 ties observed by astronomers, and pointed out some others too feeble to 

 be noticed by observations. The shortness of time fixed for the ses- 

 sion, and the immense calculations, both analytical and numerical did 

 not allow the subject to be wholly exhausted in a first memoir. The 

 author himself announced this," promising farther researches to which 

 other labors and perhaps his own taste, always prevented him from 

 devoting himself. Twenty four years after. M. le Compte Laplace 

 assumed this difficult theory, made in it some interesting discoveries 

 which completed it and put astronomers in the situation of banish- 

 ing all empiricism from their tables. 



About the same time a problem of a totally different kind attracted 

 the attention of M. Lagrange. Fermat, one of the greatest geom- 

 eters of France and of his time had left, on the properties of num- 

 bers, some very remarkable theorems to which he had arrived per- 

 haps by means of induction, but of which he had promised demon- 



