DEVELOPMENT OF MATHEMATICAL THOUGHT. 649 



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



Revision 



slowly if in many quarters, but it did not become gener- offunda- 

 ally recognised till late in the century, when thinkers of 



general point of view. " Abel," as 

 Monsieur L. Sylow says ('Memorial 

 des Etudes d'Abel,' p. 14), "e"tait 

 avant tout algebriste. II a dit lui- 

 meme que la the'orie des Equations 

 e"taitson sujet favori, ce quid'ailleurs 

 apparait clairement dans ses osuvres. 

 Dans ses travaux sur les fonctions 

 elliptiques, le traitement des di- 

 verses Equations alge"briques dont 

 cette the'orie abonde est mis forte- 

 ment en Evidence, et dans le premier 

 de ces travaux, la resolution de ces 

 Equations est meme indiquee comme 

 e"tant le sujet principal. Qui plus 

 est, la the'orie des equations etait 

 entre ses mains 1'instrument le plus 

 e'fficace. Ce fut ainsi sans aucun 

 doute la resolution de 1'equation de 

 division des fonctions elliptiques qui 

 tout d'abord le conduisit a la the'orie 

 de la transformation. Elle joue 

 encore un role capitale dans sa de"- 

 monstration du the'oreme dit the'o- 

 reme d'Abel, et dans les recherches 

 generates sur les inte'grales des diffe'r- 

 entielles alge"briques qui se trouvent 

 dans son dernier memoire le ' Precis 

 d'une The'orie 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 years before Abel 

 and Jacobi entered the field from a 

 still more general point of view. 

 Already, in 1798, when he was only 

 twenty-one, he must have recognised 

 the necessity of enlarging and defin- 

 ing the fundamental conceptions of 

 algebra and of functionality or math- 

 ematical dependence ; and it is very 

 likely that the magnitude of the 



undertaking, for which his astron- 

 omical labours left him no time, 

 debarred him from publishing the 

 important results which 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 research (see 

 Konigsberger, 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 Cauchy, 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 explained 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 had been presented to the Acad- 

 emy. (See the above-mentioned 

 correspondence between Legendre 

 and Jacobi, 1829 ; also Sylow, p. 

 31). 



