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My understanding (from a talk by Rob Bradley) is that Cauchy is responsible for the now-standard $\varepsilon$ - $\delta$ formulation of calculus, introduced in his 1821 Cours d’analyse. Although perhaps instead it was introduced by Bolzano a few years earlier. My question is not about who was first with this notation, but rather:

Why were the symbols $\varepsilon$ and $\delta$ used?

Why not, say, $\alpha$ and $\beta$? (Imagine how different our mathematical discourse would be...) Are there appropriate (French) words beginning with 'e' and/or 'd' that determined the choice? Or perhaps Cauchy used up $\alpha,\beta,\gamma$ for other purposes prior to introducing $\delta,\varepsilon$? Does anyone know?

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    $\begingroup$ If I had to take a wild guess, I'd say $\delta$ for distance, but I don't have any supporting evidence. $\endgroup$ Commented Nov 30, 2011 at 19:07
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    $\begingroup$ I heard $\epsilon$ for error, but also have no evidence. $\endgroup$ Commented Nov 30, 2011 at 19:12
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    $\begingroup$ @Gerald: quote from J.V. Grabiner's Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus (maa.org/pubs/Calc_articles/ma002.pdf )"The epsilon corresponds to the initial letter in the word “erreur’’ (or “error’’), and Cauchy in fact used for “error’’ in some of his work on probability [31]." $\endgroup$ Commented Nov 30, 2011 at 19:38
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    $\begingroup$ @Jérôme: apparently, yes: gallica.bnf.fr/ark:/12148/bpt6k90196z/f47.image (Démonstration). $\endgroup$ Commented Dec 1, 2011 at 12:26
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    $\begingroup$ This doesn't exactly answer your question -- but I like to tell my calculus students that an "enemy" has challenged their claim of a limit, and they need to be able to "defend" their claim against any challenge. $\endgroup$ Commented Dec 1, 2011 at 14:47

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Thanks to H. M. Šiljak for finding the 1983 Amer. Math. Monthly paper by Judith Grabiner, which I feel settles the question (at least for $\epsilon$). Here is a longer quote encompassing that which H.M. excerpted:

Mathematicians are used to taking the rigorous foundations of the calculus as a completed whole. What I have tried to do as a historian is to reveal what went into making up that great achievement. This needs to be done, because completed wholes by their nature do not reveal the separate strands that go into weaving them—especially when the strands have been considerably transformed. In Cauchy's work, though, one trace indeed was left of the origin of rigorous calculus in approximations—the letter epsilon. The $\epsilon$ corresponds to the initial letter in the word "erreur" (or "error"), and Cauchy in fact used $\epsilon$ for "error" in some of his work on probability [31]. It is both amusing and historically appropriate that the "$\epsilon$," once used to designate the "error" in approximations, has become transformed into the characteristic symbol of precision and rigor in the calculus. As Cauchy transformed the algebra of inequalities from a tool of approximation to a tool of rigor, so he transformed the calculus from a powerful method of generating results to the rigorous subject we know today.

[31] Cauchy, Sur la plus grande erreur à craindre dans un résultat moyen, et sur le système de facteurs qui rend cette plus grande erreur un minimum, Comptes rendus 37, 1853; in Oeuvres, series 1, vol. 12, pp. 114–124.

A further finding by H. M. Šiljak (linked in a comment above), verifying that Cauchy did indeed use both $\epsilon$ and $\delta$:
         Cauchy title
      alt text

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    $\begingroup$ So it's $\epsilon$ for error in the answer, and presumably $\delta$ is in reference to difference in the input variables. $\endgroup$ Commented Nov 30, 2011 at 21:37
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    $\begingroup$ Or possibly that $\delta$ is just the next letter over... $\endgroup$ Commented Nov 30, 2011 at 22:00
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    $\begingroup$ @Simon: And given the choice between $\delta$ and $\zeta$, the two adjacent letters, it somehow seems natural to choose $\delta$. $\endgroup$ Commented Dec 1, 2011 at 2:06
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    $\begingroup$ If we have $f(x) = y$, then $&\epsilon$ is a difference in $y$-values and $\delta$ is a difference in $x$-values. $\delta$ comes before $\epsilon$ alphabetically, as $x$ does before $y$. $\endgroup$ Commented Dec 1, 2011 at 14:22
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    $\begingroup$ Her evidence is provided in reference "[31]" above. $\endgroup$ Commented May 14 at 14:10
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The claim that

"Cauchy is responsible for the now-standard ε - δ formulation of calculus, introduced in his 1821 Cours d’analyse"

is ahistorical. Cauchy never gave an epsilon-delta definition of either limit or continuity. For example, he defined continuity in terms of

"infinitesimal $\alpha$ always produces an infinitesimal change $f(x+\alpha)-f(x)$ in the function".

Cauchy did develop some arguments that carry the seeds of epsilontic arguments (both in his 1821 text and in his 1823 text) that would be developed by the followers of Weierstrass half a century later. An analysis of both such arguments and his infinitesimal mathematics can be found in this recent publication:

Katz, M. "Episodes from the history of infinitesimals." British Journal for the History of Mathematics (2025). https://doi.org/10.1080/26375451.2025.2474811, https://arxiv.org/abs/2503.04313

Notably, he sometimes used $\epsilon$ in the "epsilontic" sense, and sometimes as an infinitesimal (though the definition of continuity is stated in terms of $\alpha$ as above, not $\epsilon$).

Today, his "epsilontic" arguments would be found unconvincing and probably would not get an "A" on modern calculus texts, because of a lack of emphasis on alternating quantifiers and the order of quantifiers.

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