DEVELOPMENT OF MATHEMATICAL THOUGHT. G47 



mathenuitics in j^eneral. This was done about fifteen or 

 twenty years after Gauss had begun to puljlish his 

 isolated memoirs, in a comprehensive treatment of the 

 subject by Cauchy, who, before 1820, delivered lectures 

 on Analysis at the J^cole Polytechnique and in (jther lo. 



Caucliy's 



colleges, and commenced their puldication in 1821. In Anaiy>.is. 

 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.^ 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 



' The earliest labours of Cauchy cotnme des inductions projires h. 



were geonietrical, and he evidently faire pre.-*.sentir quelque fois la 



ac([uired through them an insight verite, mais t[ui s'accordent peu 



into the contrast between the avec I'exautitude si vantee des 



rigour of the older geonietrical sciences niath(5matiques. On doit 



and the looseness of the modern meme observer qu'elles tendent a 



algebraical methods. In tliis re- faire attribuer aux fcjrmules al- 



gard he says : "J'ai cherchc a leur ' gcljriques une i^tendue indctinie, 



donner toute la rigueur (|u'on tandis ([ue, dans la rcalitt-, la plu- 



exige en goonietrie, de manifcre a part de ces fornmles subsistent 



ue jamais recourir aux raisons uniquenient sous certaines condi- 



tirees de la goneralite de I'algcbre. titms, et pour certaines valeurs des 



Les raisons de cette espece, quoitjue ((uantiti's (ju'elles renferment. Kti 



assez communement admises, sur- doteriainant ces conditions et ces 



tout dans le passage des series con- valeurs, et en iixant dune maniore 



vergenles aux series divergentes, precise le sens des notations dont 



et des (luautitos reelles aux ex- ' je me sers, je fais disparaitre toute 



presbions iniaginaires ne peuvent 1 incertitude" ('Coura d'Analyse,' 



etre consider($s, ce nie senible, que I 1821, Introd., p. ii). 



