Continuity between Cauchy and Bolzano: Issues of antecedents and priority

TL;DR

This paper reevaluates the historical development of the concept of continuity, arguing that Cauchy's initial ideas predate Bolzano's and clarifying their respective influences and misunderstandings.

Contribution

It challenges previous assumptions by showing Cauchy's early work on continuity predates Bolzano's and clarifies the historical relationship between their ideas.

Findings

Cauchy's notion of continuity dates back to at least 1817.

Bolzano's discussion of continuity was influenced by earlier authors, not solely by Cauchy.

The plagiarism hypothesis linking Cauchy and Bolzano is refuted.

Abstract

In a paper published in 1970, Grattan-Guinness argued that Cauchy, in his 1821 book Cours d'Analyse, may have plagiarized Bolzano's book Rein analytischer Beweis (RB), first published in 1817. That paper was subsequently discredited in several works, but some of its assumptions still prevail today. In particular, it is usually considered that Cauchy did not develop his notion of the continuity of a function before Bolzano developed his in RB, and that both notions are essentially the same. We argue that both assumptions are incorrect, and that it is implausible that Cauchy's initial insight into that notion, which eventually evolved to an approach using infinitesimals, could have been borrowed from Bolzano's work. Furthermore, we account for Bolzano's interest in that notion and focus on his discussion of a definition by K\"astner (in Section 183 of his 1766 book), which the former seems to have misrepresented at least partially. Cauchy's treatment of continuity goes back at least to his 1817 course summaries, refuting a key component of Grattan-Guinness' plagiarism hypothesis (that Cauchy may have lifted continuity from RB after reading it in a Paris library in 1818). We explore antecedents of Cauchy and Bolzano continuity in the writings of K\"astner and earlier authors.