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 century, Poisson, Gauss, and Abel drew attention to 

 the necessity of investigating systematically what is 

 termed the convergency l 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. 2 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 

 measuring 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 



1 A very good account of the decider si la valeur exprimee par 



gradual evolution of the idea of une serie est possible, c'est-a-dire 



the convergency of a series will be convergente, et cela sans connaitre 



found in Dr R. Reiff's ' Geschichte j 1'origine de la serie. II est ne'ces- 



der unendlichen Reihen ' (Tiibin- saire, pour qu'une seVie inde'finie 



gen, 1899, p. 118, &c. ) Also in represents une quantite" nine, que 



the preface to Joseph Bertrand's 1'on puisse demontrer sa converg- 



' Traitd de Calcul Differentiel ' ence, et que Ton s'assure qu'en la 



(Paris, 1864, p. xxix, &c.) Accord- I prolongeant suffisamment 1'erreur 



ing to the latter Leibniz seems to devient aussi petite que 1'ou veut." 



have been the first to demand j In spite of this, Leibniz, through 



definite rules for the convergency his treatment of the series of 



of Infinite Series, for he wrote to Grandi, 1-1 + 1-1, &c., the sum 



Hermann in 1705 as follows : j of which he declared to be J, seems 



" Je ne demande pas que Ton i to have exerted a baneful influence 



trouve la valeur d'une serie quel- 

 conque sous forme finie ; un tel 

 problems surpasserait les forces 

 des ge"ometres. Je voudrais seule- 

 ment que 1'on trouvat moyen de 



on his successors, including Euler 

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



2 The memoir on the Hypergeo- 

 metrical series. 



