Cauchy's Work on Integral Geometry, Centers of Curvature, and Other Applications of Infinitesimals

• Published in: Real analysis exchange Vol. 45; no. 1; pp. 127 - 150
• Main Authors: Bair, Jacques, Błaszczyk, Piotr, Heinig, Peter, Kanovei, Vladimir, Katz, Mikhail G., McGaffey, Thomas
• Format: Journal Article
• Language: English
• Published: East Lansing Michigan State University Press 01.01.2020
• Subjects: Cauchy problems; Continuity (mathematics); Delta function; Differential geometry; Integrals; Mathematical analysis; Mathematical functions; Textbooks; Theorems
• ISSN: 0147-1937; 1930-1219
• DOI: 10.14321/realanalexch.45.1.0127

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. Mathematical Reviews subject classification: Primary: 01A55; Secondary: 26E35, 01A85, 03A05, 53C65 Key words: Cauchy–Crofton formula, center of curvature, continuity, infinitesimals, integral geometry, limite, standard part, de Prony, Poisson