70 Lagtange's Memoirs. 



moyen des princlpes d'une saine metapbyssique on ne puisse lui 

 donner la plus grande evidence, et j'en laisse le soin a ceux qui font 

 leur etat de la metaphysique. 



This appeal to which metaphysicians did not answer, was under- 

 stood by Lagrange who excited their jealousy. 



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 which had led him to this discovery, he said positively, 

 to answer the doubts of Euler, that he viewed it, not as a metaphys- 

 ical principle, but as a necessary result of the laws of mechanics, as 

 a simple corollary of a more general law, which he afterwards made 

 the base of his Mecanique AnaJytique. (See this work, page 246 of 

 the second edition, or 189 of the first.) 



This noble spirit that excited him to triumph over difficulties re-, 

 garded as insurmountable, and to rectify or complete theories still im- 

 perfect, appeared to have constantly directed Lagrange in the choice 

 of his subject. 



D'Alembert had thought it impossible to submit to the calculus 

 the motions of a fluid contained in a vessel, if this vessel had not a 

 certain figure. Lagrange demonstrated on the contrary that there 

 would be no difficult}'^ except in the case when the fluid is divided 

 into many portions. Yet then we can determine the places where 

 the fluid ought to be divided into many portions of which we can de- 

 termine the motions as if they were isolated. 



D'Alembert had thought that in a fluid mass such as the earth 

 might have been originally, it was not necessary that the different 

 layers should be on a level : Lagrange shews that the equations of 

 D'Alembert were themselves only those of strata on a level. 



In opposing D'Alembert with all the respect due to a 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 for it another solution ; I have found a 

 more simple method to arrive at your elegant formula." These ex- 

 amples, which it would be easy to multiply, prove with what cour- 

 tesy these celebrated rivals corresponded. Vying with each other 

 incessantly, conquered as well as conquerors, they found at every 

 moment in their discussions themselves, reasons to esteem one an- 

 other the more, and each supplied for his antagonist opportunities 

 that were to lead him to new triumphs. 



