DEVELOPxMENT OF MATHEMATICAL THOUGHT. 049 



That such a revision had become necessary was seen, 17. 



1 1 • i> • 1 • 1 • 1 U'-vision 



slowly if lu many quarters, l)ut it tlid not become gener- 'jffu"<ia- 

 ally recognised till lute in the century, when thinkers of 



iiientalfi. 



general point of view. " Abel," as 

 Monsieur L. Sylow says ( ' Memorial 

 des dtudes d'Abel,' j). 14), "otait 

 avant tout algebriste. II a dit lui- 

 mcuie (jue la theorie des equations 

 (jtaitson sujet favori,cequid'ailleui-s 

 apparait claireinent dans ses omvre.s. 

 Dans ses travaux sur les fonctious 

 elliptiques, le traitenient des di- 

 verses Equations algdbriques dont 

 cette theorie abonde est mis forte- 

 meiit en ovidouce, et dans le premier 

 de ces travaux, la resolution de ces 

 equations est meme indiquee comme 

 etant le sujet principal. Qui plus 

 est, la theorie des wiuations etait 

 entre ses mains I'instrument le plus 

 dfficace. Ce fut ainsi sans aucun 

 doute la resolution de IV'quation de 

 division des fonctions elliptiques qui 

 tout d'abord le couduisit a la theorie 

 de la transformation. Elle joue 

 encore un role capitale dans sa de- 

 monstration du thcoreme dit theo- 

 rcme d'Abel, et dans les recherehes 

 gendrales surles integrates des diffcr- 

 entielles alg(5briques qui se trouvent 

 dans son dernier memoire le ' Prdcis 

 d'une Theorie des fonctions ellip- 

 tiques.' " But whilst Abel certainly 

 took a much more general view 

 than either Legendre or Jacobi, both 

 of whom came to a kind of dead- 

 lock on the roads they had chosen 

 (Jacobi, when he attempted to ex- 

 tend the theory of the periodicity 

 of functions), it is now quite clear 

 that Gauss viewed the whole sub- 

 ject almost thirty ye.ars before Abel 

 and Jacobi entered the field from a 

 still more general jMjint of view. 

 Already, in 1798, when he was only 

 twenty-one, he must have recognised 

 the necessity of eidarging and defin- 

 ing the fundamental concej)tions of 

 algebra and of functionality or math- 

 ematical dependence ; and it is very 

 likely that the magnitude of the 



uudeitaking, fur which his astron- 

 omical labours left him no time, 

 debarred him from publishing the 

 important results wliich he had 

 already attained, and which covered 

 to a great extent the field cultivated 

 in the meantime by Abel and 



' Jacobi, leaving only the celebrated 

 theorem of the former (referring to 

 the algebraical comparison of the 

 higher non - algebraical functions) 

 and the discovery of a new 

 function on the part of Jacobi 

 (his Theta function) as the two 

 great additions which we owe to 

 them in this line of re.search (see 

 Kiiuigsberger, loc. cit., p. 104). 

 In this recognition of the funda- 



! mental change which mathematical 

 science demanded, and its bearing 

 upon these special problems here 

 referred to. Gauss must have for a 

 long time stood alone ; for his great 

 rival (.'aucliy, to whom we are 

 mainly indebted for taking the first 

 steps in this direction, did not for 

 many years apply his fundamental 

 and novel ideas to the theory of 

 elliptic functions, which up to the 

 year 1844, when Hermite entered 

 the field, were almost exclusively 

 cultivated by German and Scandi- 

 navian writers (see R. L. Ellis, 

 " Report on the recent Progress of 

 Analysis," Brit. Assoc, 1846 ; re- 

 printed in ' Mathematical and other 

 Writings,' p. 311). Nor could it 

 otherwise be exidained how Cauchy 

 could keep the manuscript of Abel's 

 great memoir without ever occupy- 

 ing himself with it. and thus delay 

 its publication for fifteen years after 

 it held been presented to the .\cad- 

 emy. (See the above - mentioned 

 corrcs])ondence Ijctween Legendre 

 and Jacobi, 1829 ; also Sylow, p. 

 31). 



