Lagrange's Memoirs. 69 



<b' ""£> 



If these delicate procedures, and the testimonies of the highest es- 

 teem should flatter a young man who was not twenty four years old, 

 they do no less honor to a great man, who, holding then the sceptre 

 of mathematics, knew how to receive in this manner the work which 

 pointed to him his successor. 



But these eulogies are contained in a letter : hence we might think 

 that the great and good Euler may have suffered himself to go on in 

 some of the exaggeration permitted in the epistolary style ; let us 

 see then how he afterwards expressed himself in the dissertation 

 which his letter announced. Here is the beginning, 



" After I had long and vainly fatigued myself in seeking for this 



integral, (postquam diu et multum desudassem nequicquam in- 



quisivissem) what was my astonishment (penitus obstupui) when I 

 learned that in the Memoirs of Turin, this problem is found resolved 

 with as much ease as excellence. This fine discovery caused me 



the more admiration as it is the more different from the meth- 



■ 



ods which I have given, and as it surpasses them considerably in 

 simplicity." It is thus that Euler begins the memoirs in which he 

 explains with his usual clearness, the reasons of the method of his 

 young rival, and the theory of this new calculus, which he has called 

 the calculus of variations. 



To render more sensible all the different motives which gave birth 

 to the admiration that Euler showed with so noble a candor, it will 

 not be useless to recur to the origin of the different researches of 

 Lagrange, such as he gave it himself two days before his death. 



The first attempt to determine the maximum and minimum in all 

 indefinite integral formulas, had been made on account of the curve 

 of the swiftest descent, and the isoperimeters of Bernouilli. Euler 

 had reduced them to a general method, in an original work, where- 

 in shines throughout a deep knowledge of the calculus ; but, how- 

 ever ingenious was his method, it had not all the simplicity which 

 we can desire in a work of pure analysis. The author concluded so 

 himself; he perceived the necessity of a demonstration independent 

 of geometry and of analysis. 



In an appendix to the volume having for its title du Mount- 

 ment des projectiles dans un milieu non resistant, he seemed 

 wholly to distrust the resources of analysis, and finishes by saying 

 Si mon principe (it is that which Lagrange has since named the 

 principle of the last action) n'est pas suffisamment demon tre, com- 

 me cependant it est conforme a la verite, je ne doute pas qu'an 



