  19th century real analysis received a major impetus from Cauchy's work.
Cauchy mentions variable quantities, limits, and infinitesimals, but the
meaning he attached to these terms is not identical to their modern meaning.
  Some Cauchy historians work in a conceptual scheme dominated by an assumption
of a teleological nature of the evolution of real analysis toward a preordained
outcome. Thus, Gilain and Siegmund-Schultze assume that references to limite in
Cauchy's work necessarily imply that Cauchy was working with an Archi-medean
continuum, whereas infinitesimals were merely a convenient figure of speech,
for which Cauchy had in mind a complete justification in terms of Archimedean
limits. However, there is another formalisation of Cauchy's procedures
exploiting his limite, more consistent with Cauchy's ubiquitous use of
infinitesimals, in terms of the standard part principle of modern infinitesimal
analysis.
  We challenge a misconception according to which Cauchy was allegedly forced
to teach infinitesimals at the Ecole Polytechnique. We show that the debate
there concerned mainly the issue of rigor, a separate one from infinitesimals.
A critique of Cauchy's approach by his contemporary de Prony sheds light on the
meaning of rigor to Cauchy and his contemporaries. An attentive reading of
Cauchy's work challenges received views on Cauchy's role in the history of
analysis, and indicates that he was a pioneer of infinitesimal techniques as
much as a harbinger of the Epsilontik.
