68 G. FLEURI. 



by their methods as the terrible array of radicals in Warren's 

 treatise sufficiently proves." 



Argand did succeed, and so far nothing has been changed in 

 his representation* by a line of a product of two lines, each 

 represented by a symbol a + hi. But where Argand failed was in 

 his attempt to represent by similar methods the points of a space 

 of three dimensions. The matter was much more difficult, and it 

 was only thirty-seven years afterwards that Hamilton succeeded 

 in that representation. 



Argand's work was not sold but only distributed to a few friends, 

 so that his ideas were but little spread when seven years afterwards 

 a young artillery officer named Frangais, sent to the " Annales" 

 of Gergonne the sketch of a theory whose first ieda — as he 

 mentioned — was found by him in a letter from Legendre to his 

 brother. In that letter, Legendre stated that the idea had not 

 come from him, but from another mathematician, whose name 

 he did not give. The paper of Frangais fell into Argand's hands, 

 who sent immediately a note to Gergonne making himself known 

 as the author of the work spoken of by Legendre and giving a 

 pretty complete account of his pamphlet of 1806. 



Frangais and Argand's papers gave rise in Gergonne's Annales 

 to a discussion between Frangais, Gergonne and Servois, which 

 was finally settled by a remarkable paper of Argand, where he 

 explained in a more satisfactory manner several parts of his theory 

 amongst which we notice the demonstration of the fundamental 

 proposition of algebraical equations : 



"Any algebraical equation has, at least, a root of form a + bi,''' 

 a demonstration since reproduced by Cauchy (twenty-two years 

 afterwards) under an analytical but certainly less striking form 

 than Argand's. Fourteen years afterwards, Warren in England 

 and Mourey in France were still working at the same question of 

 calculus of complex quantities without seeming to have any 

 knowledge of Argand's work. But they were soon forgotten — 



* The algebraical expression was already known. 



