A two-track tour of Cauchy's Cours

Cauchy published his Cours d’Analyse 200 years ago. We analyze Cauchy’s take on the concepts of rigor, continuity, and limit, and explore a pair of approaches in the literature to the meaning of his infinitesimal analysis and his sum theorem on the convergence of series of continuous functions. 1. Rigor then and now Building upon pioneering work by Kepler, Fermat, Cavalieri, Gregory, Wallis, Barrow, and others, Isaac Newton and Gottfried Wilhelm von Leibniz invented the calculus in the 17th century. While immediately acquiring an enthusiastic following, the new methods proved to be controversial in the eyes of some of their contemporaries, who employed the more traditional methods of their predecessors. One of the controversial aspects of the new technique was Leibniz’s distinction between assignable and inassignable quantities (including infinitesimals and infinite quantities; see [23], [1], [2]). At the French Academy, the opposition to the new calculus was led by Michel Rolle, and across the Channel, by George Berkeley. The scientific success of the new methods ultimately silenced the opposition, but lingering doubts persisted (fed in part by doctrinal theological issues; see [6]). A new era was ushered in by Augustin-Louis Cauchy’s textbook Cours d’Analyse, addressed to the students of the Ecole Polytechnique in Paris. Cauchy published his Cours d’Analyse (CDA) 200 years ago. The book was of fundamental importance for the development of both real and complex analysis. Hans Freudenthal mentioned in his essay on Cauchy (for the Dictionary of Scientific Biography) [15] that Niels Henrik Abel described the CDA as “an excellent work which should be read by every analyst who loves mathematical rigor.” But what did rigor mean to Abel and Cauchy? In the early 19th century context, the term rigor referred to the standard of mathematical precision set by the geometry of Euclid. This 2020 Mathematics Subject Classification. Primary 01A55, Secondary 26E35, 01A85 .