45 



rem not previously demonstrated. Cauchy obtained it in LSI 2. 

 Legendre reported on it to the Academy with an enthusiasm very 

 foreign to his character. " We only intended," he said, " to give 

 an idea of this demonstration, and have extracted almost the whole 

 of it. We have thus furnished a new proof of the sagacity with 

 which this young geometer has succeeded in conquering a difficulty 

 which had arrested the progress of the masters of the art, and 

 which it was of importance to solve in order to complete the theory 

 of the solid bodies." These first two memoirs of Cauchy seemed 

 to foretell a peculiar and exclusive aptitude for pure geometry ; but 

 it was soon discovered that his genius had a much wider range. 

 In the years 1813 and 1814 he produced two remarkable analy- 

 tical memoirs, and in 1815 he presented a memoir on the theory of 

 numbers, in which he proved and extended a theorem enunciated by 

 Fermat, a theorem some particular cases of which only had been 

 established by the most able writers in that department of mathe- 

 matical science, Legendre and Gauss. He published an elegant 

 theorem on the number of values which a function can assume, when 

 the letters which it contains are interchanged. Twenty years later, 

 this theorem enabled the celebrated Abel to prove the impossibility 

 of solving algebraic equations of the fifth or higher degrees. In 

 the same year, the Academy proposed, as the subject of the great 

 mathematical prize, the investigation of the theory of the propaga- 

 tion of waves on the surface of a heavy fluid of indefinite depth. 

 Cauchy gave a complete solution of the problem. His memoir, 

 which obtained the prize in 1816, has for its motto the line of 

 Virgil- 



" Nosse quot lonii veniant ad littora fluctus." Georg. ii. 



A peculiarly happy quotation, as the line may be said to contain a 

 striking enunciation of the problem proposed. 



This fertility in a young man of seven-and-twenty would have 

 secured for him the first place which became vacant in the Mathe- 

 matical Section of the Institute. He was admitted into it under 

 circumstances much to be regretted. After the short crisis of a 

 hundred days, a royal ordinance, dated March 21, 1816, re-esta- 

 blished the old Academies under their original names, the Academy 

 of France, of Sciences, of Inscriptions and Belles Lettres, of the 



