DEVELOPMENT OF MATHEMATICAL THOUGHT. 647 



mathematics in general. This was done about fifteen or 

 twenty years after Gauss had begun to publish his 

 isolated memoirs, in a comprehensive treatment of the 

 subject by Cauchy, who, before 1820, delivered lectures 

 on Analysis at the cole Polytechnique and in other ie. 



Cauchy's 



colleges, and commenced their publication in 1821. In Analysis. 

 this course of lectures the discussion of the notions of the 

 infinite, of the continuous, of the convergence of series, 

 and of the extension of our conception of quantity 

 beyond the ordinary or real quantities of algebra, is 

 put in the foreground, and the illicit habit of using the 

 generalisations of algebra without defining the conditions 

 of their validity severely criticised. 1 It is also evident, 

 from the extensive notes which Cauchy added to the 

 " cours " of 1821, that he felt the necessity of a revision 

 of the fundamental notions of algebra. The publication 

 of 1821 was followed by others on the Calculus, and it 

 is through these treatises mainly that a new spirit was 

 infused into general mathematical literature, first in 



1 The earliest labours of Cauchy j comme des inductions propres 



were geometrical, and he evidently 

 acquired through them an insight 

 into the contrast between the 

 rigour of the older geometrical 

 and the looseness of the modern 



faire pressentir quelque fois la 

 ve'rite', mais qui s'accordent peu 

 avec 1'exactitude si vant^e des 

 sciences mathematiques. On doit 

 meme observer qu'elles tendent 



algebraical methods. In this re- j faire attribuer aux formules al- 



gard he says : " J : ai cherche' h leur ' gebriques une etendue indefinie, 



donner toute la rigueur qu'on | tandis que, dans la realite, la plu- 



exige en geometric, de maniere a part de ces formules subsistent 



ne jamais recourir aux raisons I uniquement sous certaines condi- 



tirdes de la generality de 1'algebre. j tions, et pour certaines valeurs des 



Les raisons de cette espece, quoique quantites qu'elles renferment. En 



assez communement admiaes, sur- | determinant ces conditions et ces 



tout dans le passage des series con- 

 vergentes aux series divergentes, 

 et des quantite*s reelles aux ex- 

 pressions imaginaires ne peuvent 

 etre considered, ce me semble, que 



valeurs, et en fixant d'une maniere 

 precise le sens des notations dont 

 je me sers, je fais disparaitre toute 

 incertitude" ('Cours d'Analyse,' 

 1821, Introd., p. ii). 



