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Good epigraphs may attract more readers. Sometimes it is necessary. Usually epigraphs are interesting but not intriguing.

To pick up an epigraph is some kind of nearly mathematical problem: it should be unexpectedly relevant to the content.

What successful solutions are known for you? What epigraphs attracted your attention?

Please post only epigraphs because quotes were collected in Famous mathematical quotes.

There are certain common Privileges of a Writer, the Benefit whereof, I hope, there will be no Reason to doubt; Particularly, that where I am not understood, it shall be concluded, that something very useful and profound is coucht underneath. (JONATHAN SWIFT, Tale of a Tub, Preface 1704)

[Taken from Knuth, D. E. The art of computer programming. Volume 3: Sorting and searching.]

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    $\begingroup$ I'll stick with the classical $\{(x,y) : y \geq f(x)\}$. $\endgroup$ Sep 29, 2015 at 10:01
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    $\begingroup$ @FedericoPoloni: however, this one is not proper :-) $\endgroup$
    – M.G.
    Sep 29, 2015 at 10:41
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    $\begingroup$ When I defended my PhD, a renown mathematician and authority in my field showed up. This surprised me slightly because, though I of course knew her, I had never interacted with her in any way. After the defense, she came to see me and asked if we could talk about "some of the beautiful ideas in the manuscript", which surprised me much more. Turns out she wanted to talk about the epigraph. $\endgroup$
    – Olivier
    Sep 29, 2015 at 11:31
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    $\begingroup$ I think it would be helpful if you made clear in the question what exactly you are looking for/what an epigraph is. Some answers seem just like quotes, and we had such a question already mathoverflow.net/questions/7155/famous-mathematical-quotes $\endgroup$
    – user9072
    Sep 29, 2015 at 13:37
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    $\begingroup$ Isn't this the epitome of opinion-based questions? $\endgroup$ Sep 29, 2015 at 15:50

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I'm fond of the one from Volume 1 of Zariski and Samuel's Commutative Algebra (Springer-Verlag, New York, 1958):

Le juge: Accusé, vous tâcherez d'être bref.

L'accusé: Je tâcherai d'être clair.

        —G. Courteline

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Dennis Gaitsgory begins his "Outline of the proof of the geometric Langlands conjecture for $GL_2$" (available here) with the following interesting quote in German:

“In jedem Minus steckt ein Plus. Vielleicht habe ich so etwas gesagt, aber man braucht das doch nicht allzu wörtlich zu nehmen.” - Robert Musil, Der Mann ohne Eigenschaften.

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    $\begingroup$ Can you explain what is hidden behind this epigraph? $\endgroup$ Oct 3, 2015 at 2:07
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    $\begingroup$ "There's a plus in every minus. Maybe I did say something like that but one doesn't need to take it so literally." $\endgroup$ Oct 3, 2015 at 16:57
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Go back to An-Fang, the Peace Square at An-Fang, the beginning place at An-Fang, where all things start (...) An-Fang was near a city, the only living city with a pre-atomic name (...) the headquarters of the people programmer was at An-Fang, and there the mistake happened : a ruby trembled. Two tourmaline nets failed to rectify the laser beam. A diamond noted the error. both the error and the correction went into the general computer. -Cordwainer Smith, The Dead Lady Of Clown Town, 1964

Epigraph to Jean-Yves Girard, "Locus Solum" http://iml.univ-mrs.fr/~girard/0.pdf

There was no turning back after I read that one.

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Recently stumbled on a great one: "The infinite wedge representation and the reciprocity law for algebraic curves" by Arbarello, De Concini and Kac (in "Theta functions—Bowdoin, 1987", PSPM 49, Part I, pp. 171—190) has this epigraph

"Только то и держится на гвозде,
Что не делится без остатка на два."
Joseph Brodsky, Roman Elegies

Literal translation - "Only that keeps hanging on a nail, which is not divisible without remainder by two".

There also is author's own translation but it is (at least for me) extremely cryptic without the surrounding context. Let me still reproduce it here, though.

For a nail holding something one would divide by two—­
were it not for remainders—there is no gentler quarry.

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