Biographj of M. Le Comte Lagrange. 247 



geometer, of that order, he often employed very fine 

 theorems which he owed to his opponent ; D'Alembert, on 

 his side, added to the researches of Lagrange. " Your 

 problem appeared to me so fine," wrote he to him, '* that I 

 have sought another solution for it ; I have found a more 

 simple method of arriving at your elegant formula." These 

 examples, which it would be easy to multiply, prove with 

 what courtesy these celebrated rivals corresponded. Vying 

 with each other incessantly, conquered as well as conquerors, 

 they found at every moment, in their discussions them- 

 selves, reasons to esteem one another the more, and each 

 supplied for his antagonist opportunities that were to lead 

 him to new triumphs. 



The Academy of sciences of Paris, had proposed for the 

 subject of one of its prizes, the theory of the libration 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 tlie exception of some variations observed by astrono- 

 mers, and of which Cassini I. had well explained the 

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

 the phenomena, and of deducing them analytically from 

 the principle of universal gravitation. Such a chance was 

 an appeal to the genius of Lagrange ; one, which was held 

 out to him of applying bis 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 ; jai lu avec autant de jjlaisir que de fruit voire belle 

 piece sur la libration, si digne du prix quelle a remportce. 



About this time he turned his attention to the theorems 

 of Fermat, on the properties of numbers. Many geometers, 

 undoubtedly, practised upon the theorems of Fermat, but 

 not one ever succeeded. Euler alone had made some pro- 

 gress in this difficult path, wherein have since distinguished 

 themselves M. Legendre and M. Gauss. Lagrange, upon 

 demonstrating or correcting some attentive glimpses of 

 Euler, resolved a problem which appeared to be the knot 

 of all the rest, and from which he derived a useful result, 

 that is to say, the complete resolution of equations of 

 the second degree, with two unknown quantities which 



