## 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.

Original language | English |
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Pages (from-to) | 127-149 |

Number of pages | 23 |

Journal | Real Analysis Exchange |

Volume | 45 |

Issue number | 1 |

DOIs | |

State | Published - 2020 |

### Bibliographical note

Funding Information:We thank Peter Fletcher, Elías Fuentes Guillén, Karel Hrbacek, Taras Kudryk, David Pierce, and David Sherry for helpful suggestions. V. Kanovei was partially supported by RFBR Grant 17-01-00705.

Funding Information:

We thank Peter Fletcher, El?as Fuentes Guill?n, Karel Hrbacek, Taras Kudryk, David Pierce, and David Sherry for helpful suggestions. V. Kanovei was partially supported by RFBR Grant 17-01-00705.

Publisher Copyright:

© 2020 Michigan State University Press. All rights reserved.

## Keywords

- Cauchy–Crofton formula
- Center of curvature
- Continuity
- De Prony
- Infinitesimals
- Integral geometry
- Limite
- Poisson
- Standard part