ISH.] M. Lagrange. 325 



esteem were very flattering to a young man of 24 years of age, they 

 do no less honour to the great man, who at that time swayed the 

 sceptre of mathematics, and who thus accurately estimated the 

 merit of a work that announced to him a successor. 



But these praises are to be found in a letter. It may be sup- 

 posed that the great and good Euler has indulged in some of those 

 exaggerations wliich the epistolary style permits. Let us see then 

 how he has expressed himself in the dissertation which his letter 

 announced. It begins as follows : 



" After having fatigued myself for a long time and to no purpose 

 in endeavouring to find this integral, what was my astonishment 

 when I learnt that in the Turin Memoirs the problem was resolved 

 with as much facility as felicity. This fine discovery produced in 

 me so much the more admiration, as it is very diiferent from the 

 methods which I had given, and far surpasses them all in sim- 

 plicity." 



It is thus that Euler begins the memoir in which he explains 

 with his usual clearness the foundation of the method of his young 

 rival, and the theory of the new calculus, which he called the cal- 

 culus of variations. 



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 Lagrange, such as he stated 

 them himself two days before his death. 



The first attempts to determine the maximnm and minimum in 

 all indefinite integral formulas, were made upon the occasion of 

 the curve of swiftest descent, and the isoperimetres of Bernoulli. 

 Euler had brought them to a general method, in an original work, 

 in which the profoundest knowledge of the calculus is conspicuous. 

 But however ingenious his method was, it had not all the simplicity 

 which one would wish to see in a work of pure analysis. The 

 author admitted this himself. He allowed the necessity of a de- 

 monstration independent of geometry. He appeared to doubt the 

 resources of analysis, and terminated his work by saying, " If my 

 principle he not sufficiently demonstrated, yet as it is conformable to 

 truth, I have no doubt that by means of a rigid metaphysical ex- 

 planation it may be put in the clearest light, and I leave that task to 

 the metaphysicians." 



This appeal, to which the metaphysicians paid no attention, was 

 listened to by Lagrange, and excited his emulation, in a short 

 time the young man found the solution of which Euler had despaired. 

 He Jbiind it by analysis. And in giving an account of the way in 

 which he had been led to that discovery, he said expressly, and as 

 it were in answer to Euler's doubt, that he regarded it not as a 

 metaphysical principle, but as a necessary result of the laws of 

 mechanics, as a sim])le corollary from a more general la>v, which 

 he afterwards made the foundation of his Mechunitiue Analy ique. 

 (See that work, page IS'J of the first edition). 

 This noble emulation wliich excited him to triumph over diffi- 



