Why do we use $\varepsilon$ and $\delta$?

Question

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?

If I had to take a wild guess, I'd say $\delta$ for distance, but I don't have any supporting evidence. — Thierry Zell, Nov 30 '11 at 19:07
@Gerald: quote from J.V. Grabiner's Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus (http://www.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]." — Harun Šiljak, Nov 30 '11 at 19:38
@Mariano: You are saying that this has a provable answer? That's absurd. My own guess is that $\alpha, \beta, \gamma$ were used since the greeks for various angles of the triangle, so $\epsilon, \delta were the first letters with no reserved use, but I do not expect to be able to prove this. — Igor Rivin, Nov 30 '11 at 20:07
In France, most of the time (in my undergraduate studies at least) we use $\varepsilon$ and $\eta$ instead of $\varepsilon$ and $\delta$. So I would be surprised if you found a reason for $\delta$. Is this $\delta$ really in Cauchy's work? — Jérôme Poineau, Dec 01 '11 at 08:14
@Jérôme: apparently, yes: http://gallica.bnf.fr/ark:/12148/bpt6k90196z/f47.image (Démonstration). — Harun Šiljak, Dec 01 '11 at 12:26
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. — Frank Thorne, Dec 01 '11 at 14:47
@Igor: The accepted answer below comes about as close to being a provable answer as one is going to get for historical questions. I don't think community wiki automatically applies to a question just because it's a historical or terminological question rather than a mathematical one. It's true that some questions, such as your hypothetical one, might be almost surely unanswerable, but then my reaction would be to close the question rather than to make it community wiki. — Timothy Chow, Dec 01 '11 at 15:15
I remember vaguely a story about an MIT TA who decided to switch the standard role of epsilon and delta and also of 'n' and 'N' — Gil Kalai, Dec 08 '11 at 15:16
@FrankThorne That's really good, as it fits into a narrative of "games", as discussed for example at https://mathoverflow.net/a/88553 and at https://mathoverflow.net/a/38882 — Todd Trimble, Aug 20 '17 at 23:59
Traditionally, $\ \alpha\ \beta\ \gamma\ \phi\ \psi\ $ stand for angles. Now, not much is left, first of all $\ \delta\ $ and $\ \epsilon\ $ are left. — Wlod AA, Oct 25 '20 at 08:47
To confuse Israeli students: $\delta$ looks like a script Lamed, which would correspond to $\lambda$, which looks like a Gimel (unlike $\gimel$ which looks not very much like one), which in turn corresponds to $\gamma$, which looks like a script A'yin. So all in all, very confusing. — Asaf Karagila, Oct 25 '20 at 16:30

score 30 · Accepted Answer · edited Mar 10 '17 at 09:42

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

So it's $\epsilon$ for error in the answer, and presumably $\delta$ is in reference to difference in the input variables. — Ryan Budney, Nov 30 '11 at 21:37
@Simon: And given the choice between $\delta$ and $\zeta$, the two adjacent letters, it somehow seems natural to choose $\delta$. — Joseph O'Rourke, Dec 01 '11 at 02:06
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 $y$. — Michael Lugo, Dec 01 '11 at 14:22

Why do we use $\varepsilon$ and $\delta$?

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