Cauchy's Cours d'analyse. An annotated translation
type: Item
instance of: scholarly article
title: Cauchy's Cours d'analyse. An annotated translation
zbMATH Open document ID: 1189.26001
DOI: 10.1007/978-1-4419-0549-9
author: Robert E. Bradley
author: C. Edward Sandifer
published in: Sources and Studies in the History of Mathematics and Physical Sciences
publication date: 2009-05-26T00:00:00Z
review text: Augustin-Louis Cauchy's (1789--1857) famous ``Cours d'Analyse de l'École Royale Polytechnique:\ 1.re Partie.\ Analyse Algébrique'' (Paris 1821) has now been translated into English for the first time (the English translation will henceforth be quoted as ET). The Cours d'Analyse is based on lectures which Cauchy held at the École Polytechnique in Paris from 1815. It is a well-known fact that Cauchy was not allowed to teach from this text-book when it appeared in 1821, because the authorities found it superfluously rigorous for the use of students. A second part of the Cours d'Analyse with a similar foundational approach never appeared. It is, however, clear that the Cours d'Analyse was enormously influential in the history of mathematics. This was due to Cauchy's -- in his prolific research -- and other mathematicians' references to and use of the fundamental definitions and notions introduced in Cauchy's book. It is sufficient here to mention Cauchy's rigorous introduction of limits, his use of the \(\varepsilon\)-\(\delta\)-formalism in the more complicated proofs, the intermediate value theorem for continuous functions, his insistence that divergent series have no sums, the Cauchy criterion for convergence of series, etc. As an example of Cauchy's style of presentation I quote here his definition of continuity and discontinuity from the translation ET which follows -- except for not reproducing italics for the word ``discontinuous'' which are in the original -- exactly Cauchy's wording and symbolism: ``Let \(f(x)\) be a function of the variable~\(x\), and suppose that for each value of \(x\) between two given limits the function always takes a unique finite value. If, beginning with a value of \(x\) contained between these limits, we add to the variable an infinitely small increment~\(\alpha\), the function itself is incremented by the difference \[ f(x+\alpha) - f(x), \] which depends both on the new variable \(\alpha\) and on the value of~\(x\). Given this, the …
reviewed by: Reinhard Siegmund-Schultze
Mathematics Subject Classification ID: 26-01
Mathematics Subject Classification ID: 01A75
Mathematics Subject Classification ID: 01A55
zbMATH DE Number: 5557088
Wikidata QID: Q55886864
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-1-4419-0549-9
OpenAlex ID: W1654683418
description: scientific article; zbMATH DE number 5557088