646 



SCIENTIFIC THOUGHT. 



of this kind was given by Brook Taylor, and somewhat 

 modified by Maclaurin. It embraced all then known and 

 many new series, and was employed without hesitation 

 by Euler and other great analysts. In the beginning of 

 the centur}", Poisson, Gauss, and Abel drew attention to 

 the necessity of investigating systematically what is 

 termed the convergency ^ of a series. As a specimen 

 of this kind of research, Gauss published, in 1812, an 

 investigation of a series of very great generality and 

 importance." We can say that through these two isolated 

 memoirs of Gauss, the first of the three on equations, 

 published in 1799, and the memoir on the series of 

 1812, a new and more rigorous treatment of the in- 

 finite and the continuous as mathematical conceptions 

 was introduced into analysis, and that in both he showed 

 the necessity of extending the system of numbering and 

 measviriug by the conception of the complex quantity. 

 But it cannot be maintained that Gauss succeeded in 

 impressing the new line of thought upon the science of 



^ A very good account ot the 

 gradual evolution of the idea of 

 the convergency of a series will be 

 found in Dr R. Reiff's ' Geschichte 

 der unendlichen Reihen" (Tiibin- 

 gen, 1899, p. 118, &c.) Also in 

 the preface to Joseph Bertrand's 

 'Traite de Calcul Differeutiel' 

 (Paris, 1864, p. xxix, &c.) Accord- 

 ing to the latter Leibniz seems to 

 have been the first to demand 

 definite rules for the convergency 

 of Infinite Series, for he wrote to 

 Hermann in 1705 as follows: 

 " Je ne demande pas que Ton 

 trouve la valeur d"uue serie quel- 

 conque sous forme finie ; un tel 

 probleme surpasserait les forces 

 des geometres. Je voudrais seule- 

 ment que Ton trouvat moyen de 



decider si la valeur exprimee par 

 une serie est possible, c'est-a-dire 

 convergente, et cela sans connaitre 

 I'origiue de la serie. II est neces- 

 saire, pour qu'une serie indefinie 

 represente une quantite finie, que 

 Ton puisse demontrer sa converg- 

 ence, et que Ton s'assure qu'en la 

 prolongeant suflfisamment Terreur 

 devient aussi petite que I'on veut.'' 

 In spite of this, Leibniz, through 

 his treatment of the series of 

 Grandi, l-l + l-l,&c., the sum 

 of which he declared to be |, seems 

 to have exerted a baneful influence 

 on his successors, including Euler 

 (See Reiff, loc. cit., pp. 118, 158). 



- The memoir on the Hypergeo- 

 metrical series. 



