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 ficole Poly technique 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 i comme des inductions propres a 



were geometrical, and he evidently faire pressentir quelque fois la 



acquired through them an insight verite, mais qui s'accordent peu 



into the contrast between the avec 1'exactitude si vante'e des 



rigour of the older geometrical sciences mathe'matiques. On doit 



and the looseness of the modern ; meme observer qu'elles tendent ;'i 



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



gard he says : "J'ai cherche" a 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 jatnais recourir aux raisons uniquement sous certaines condi- 



tirees de la generalite de 1'algebre. tions, et pour certaines valeurs des 



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



assez commuue'inent admises, sur- determinant ces conditions et ces 



tout dans le passage des series con- valeurs, et en fixant d'une maniere 



vergentes aux series divergentes, precise le sens des notations dont 



et des quantites re'elles aux ex- je me sers, je fais disparaitre toute 



pressions imaginaires ne peuvent incertitude" ('Cours d'Analyse,' 



etre considers, ce me semble, que 1821, Introd., p. ii). 



