Who gave you the epsilon?

Who gave you the epsilon? is the title of an article by J. Grabiner on Cauchy from the 1980s, and the implied answer is “Cauchy”. On the other hand, historian I. Grattan-Guinness points out in his book that “Weierstrass undoubtedly saw himself as Cauchy’s heir in analysis and so helped to create the belief that Cauchy’s achievements included ideas that were actually his own” (The development of the foundations of mathematical analysis from Euler to Riemann, page 120). In fact, Cauchy apparently never gave an epsilon-delta definition of either limit or continuity. His infinitesimal definition of continuity (“infinitesimal $x$-increment always produces infinitesimal $y$-increment”) is reproduced in a number of secondary studies, for example in J. Gray (Plato’s Ghost, page 64). The question is then, who in fact gave you epsilon-delta: Cauchy or Weierstrass? Editors are requested to refrain from answers based purely on opinion, but rather to base themselves on published sources supporting either of the two views. In particular, a primary source in Cauchy giving an epsilon-delta definition of continuity would be welcome.

Note 1. Grabiner’s example cited by @Brian illustrates the issue well. Cauchy infrequently uses preliminary forms of epsilon, delta arguments (notice the opening “let delta, epsilon be very small numbers” which would certainly not make it into calculus texts today) in some of his proofs, but he never gave an epsilon, delta definition. This example suggests that Cauchy never gave such a definition, for if he did, Grabiner would have likely cited it. Cauchy’s definition of continuity is “infinitesimal $x$-increment always produces an infinitesimal change in $y$”. This is closer to the modern infinitesimal definition of continuity than to the modern epsilon, delta definition; in fact it looks identical to the modern hyperreal definition at the syntactic level.

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It was Bernard Bolzano. According to Wikipedia, “In calculus, the (ε, δ)-definition of limit … is a formalization of the notion of limit. It was first given by Bernard Bolzano in 1817, followed by a less precise form by Augustin-Louis Cauchy.”

In fact Grabiner cites an example of Cauchy’s use of the $\epsilon,\delta$ formulation of continuity in a proof. I copy the relevant passage from the original, which may be found here under SEPTIÈME LEÇON:

THÉORÈME. — Si, la fonction $f(x)$ étant continue entre les limites $x=x_0$, $x=X$, on désigne par $A$ la plus petite, et par $B$ la plus grande des valeurs que la fonction dérivée $f\,'(x)$ reçoit dans cet intervalle, le rapport aux différences finies $$\frac{f(X)-f(x_0)}{X-x_0}$$ sera nécessairement compris entre $A$ et $B$.

Démonstration. — Désignons par $\delta,\epsilon$ deux nombres très petits, le premier étant choisi de telle sorte que, pour des valeurs numériques de $i$ inférieures à $\delta$, et pour une valeur quelconque de $x$ comprise entre les limites $x_0,X$, le rappart $$\frac{f(x+i)-f(x)}i$$ reste toujours supérieur à $f\,'(x)-\epsilon$ et inférieur à $f\,'(x)+\epsilon$.

The proof continues for another three-quarters of a page or so, but that is clearly the familiar formalization.

She notes that he did not incorporate this formalism into his purely verbal definition of limit, but it’s clear from this example that he had it in mind.

To complement Brian M. Scott’s thoughtful answer, I would like to present a view based closely on primary sources in Cauchy, and specifically on his Cours d’Analyse (1821). An analysis of this book suggests a predominance of an infinitesimal approach in Cauchy’s foundations for analysis. Already in the introduction, Cauchy states that he found it impossible to present analysis without infinitesimals. He proceeds to present two definitions of continuity, both based on infinitesimals (and none based on epsilon, delta). Following such definitions, Cauchy presents a detailed discussion including as many as eight separate propositions, analyzing the properties of infinitesimals, various degrees of infinitesimals, some structure theorems for infinitesimals of polynomial form with respect to a base infinitesimal, etc.

By comparison, note that the key piece of evidence presented by Brian (following Grabiner) for a beginning of an epsilon, delta approach are buried in “lesson 7” of a later book; the technique appears in a discussion of the derivative rather than being presented as a definition; and moreover such discussions are comparatively infrequent in Cauchy as compared to the predominance of an approach using infinitesimals.

In fact, as late as 1853, in an article on a notion related to uniform convergence, Cauchy again presents his infinitesimal definition of continuity nearly identical to the one given in 1821 (many intermediate works present the same definition, as well).

Note also Cauchy’s definition of limit: ‘When the successively attributed values of the same variable indefinitely approach a fixed value, so that finally they differ from it by as little as desired, the last is called the limit of all the others’. As Grabiner points out, this is very close to definitions appearing from Leibniz onward. To the best of my knowledge, there is no epsilon, delta definition of limit anywhere in Cauchy. These issues were studied in a recent text by Borovik and myself here.

The evidence suggests that, while both (A) the beginnings of epsilon, delta arguments and (B) infinitesimal foundations for analysis are present in Cauchy, the latter predominate. This is in contrast with what appears to be a prevailing view in the textbooks as well as historical works by Boyer, Grabiner, and others. Reading the latter, one would never have guessed that Cauchy spoke of infinitesimals at all. The evidence suggests further that, to follow Grattan-Guinness’s comment, the true beginnings of the infuence of the epsilon, delta approach are in Weierstrass (Bolzano having not been influential in any broad sense, according to most historians).