Editing Augustin-Louis Cauchy (section)

===''Cours d'analyse''===
{{Main|Cours d'analyse}}
[[Image:Cauchy.jpg|left|thumb|The title page of a textbook by Cauchy.]] In his book ''Cours d'analyse'' Cauchy stressed the importance of rigor in analysis. ''Rigor'' in this case meant the rejection of the principle of ''[[Generality of algebra]]'' (of earlier authors such as Euler and Lagrange) and its replacement by geometry and [[infinitesimal]]s.{{sfn|Borovik|Katz|2012|pp=245–276}} Judith Grabiner wrote Cauchy was "the man who taught rigorous analysis to all of Europe".{{sfn|Grabiner|1981}}  The book is frequently noted as being the first place that inequalities, and <math>\delta-\varepsilon</math> arguments were introduced into calculus.  Here Cauchy defined continuity as follows: ''The function f(x) is continuous with respect to x between the given limits if, between these limits, an infinitely small increment in the variable always produces an infinitely small increment in the function itself.''

M. Barany claims that the École mandated the inclusion of infinitesimal methods against Cauchy's better judgement.{{sfn|Barany|2011}}  Gilain notes that when the portion of the curriculum devoted to ''Analyse Algébrique'' was reduced in 1825, Cauchy insisted on placing the topic of continuous functions (and therefore also infinitesimals) at the beginning of the Differential Calculus.{{sfn|Gilain|1989}} Laugwitz (1989) and Benis-Sinaceur (1973) point out that Cauchy continued to use infinitesimals in his own research as late as 1853.

Cauchy gave an explicit definition of an infinitesimal in terms of a sequence tending to zero.  There has been a vast body of literature written about Cauchy's notion of "infinitesimally small quantities", arguing that they lead from everything from the usual "epsilontic" definitions or to the notions of [[non-standard analysis]]. The consensus is that Cauchy omitted or left implicit the important ideas to make clear the precise meaning of the infinitely small quantities he used.{{sfn|Barany|2013}}