Episodes from the history of infinitesimals

TL;DR

This historical survey traces the fate of infinitesimals from d'Alembert's chimeric critique through 19th‑century debates (Moigno, Noël, Poisson, Riemann) to Cantor’s set‑theoretic upheavals and the Peano‑driven revival. It analyzes how infinitesimals moved from controversial, sometimes genuine quantities (as in $dx$, $d\tau$) to objects scrutinized under the lens of formalism, yet ultimately laid groundwork for modern infinitesimal frameworks such as nonstandard analysis. The work highlights key methodological shifts—epsilon‑style reasoning, infinite proximity, and geometric encodings of curvature—and illuminates how historical critiques and reconceptualizations have shaped contemporary foundations. The historical perspective underscores the lasting significance of infinitesimal reasoning in differential geometry and its ongoing development in rigorous infinitesimal analysis.

Abstract

Infinitesimals have seen ups and downs in their tumultuous history. In the 18th century, d'Alembert set the tone by describing infinitesimals as chimeras. Some adversaries of infinitesimals, including Moigno and Connes, picked up on the term. We highlight the work of Cauchy, Noël, Poisson and Riemann. We also chronicle reactions by Moigno, Lamarle and Cantor, and signal the start of a revival with Peano.