  Like his colleagues de Prony, Petit, and Poisson at the Ecole Polytechnique,
Cauchy used infinitesimals in the Leibniz-Euler tradition both in his research
and teaching. Cauchy applied infinitesimals in an 1826 work in differential
geometry where infinitesimals are used neither as variable quantities nor as
sequences but rather as numbers. He also applied infinitesimals in an 1832
article on integral geometry, similarly as numbers. We explore these and other
applications of Cauchy's infinitesimals as used in his textbooks and research
articles.
  An attentive reading of Cauchy's work challenges received views on Cauchy's
role in the history of analysis and geometry. We demonstrate the viability of
Cauchy's infinitesimal techniques in fields as diverse as geometric
probability, differential geometry, elasticity, Dirac delta functions,
continuity and convergence.
  Keywords: Cauchy--Crofton formula; center of curvature; continuity;
infinitesimals; integral geometry; limite; standard part; de Prony; Poisson
