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2020 Cauchy’s Work on Integral Geometry, Centers of Curvature, and Other Applications of Infinitesimals
Jacques Bair, Piotr Błaszczyk, Peter Heinig, Vladimir Kanovei, Mikhail G. Katz, Thomas McGaffey
Real Anal. Exchange 45(1): 127-150 (2020). DOI: 10.14321/realanalexch.45.1.0127

Abstract

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.

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Jacques Bair. Piotr Błaszczyk. Peter Heinig. Vladimir Kanovei. Mikhail G. Katz. Thomas McGaffey. "Cauchy’s Work on Integral Geometry, Centers of Curvature, and Other Applications of Infinitesimals." Real Anal. Exchange 45 (1) 127 - 150, 2020. https://doi.org/10.14321/realanalexch.45.1.0127

Information

Published: 2020
First available in Project Euclid: 9 May 2020

zbMATH: 07211607
Digital Object Identifier: 10.14321/realanalexch.45.1.0127

Subjects:
Primary: 01A55
Secondary: 01A85 , 03A05 , 26E35 , 53C65

Keywords: Cauchy--Crofton formula , center of curvature , continuity , de Prony , infinitesimals , integral geometry , Poisson

Rights: Copyright © 2020 Michigan State University Press

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Vol.45 • No. 1 • 2020
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