Q.E.D. is an initialism of the Latin phrase quod erat demonstrandum, meaning "which is what had to be proven". The phrase is traditionally placed in its abbreviated form at the end of a mathematical proof or philosophical argument when what was specified in the enunciation—and in the setting-out—has been exactly restated as the conclusion of the demonstration. The abbreviation thus signals the completion of the proof.
Etymology and early use
The phrase quod erat demonstrandum is a translation into Latin from the Greek ὅπερ ἔδει δεῖξαι (hoper edei deixai; abbreviated as ΟΕΔ). Translating from the Latin into English yields, "what was to be demonstrated"; however, translating the Greek phrase ὅπερ ἔδει δεῖξαι produces a slightly different meaning. Since the verb "δείκνυμι" also means to show or to prove, a better translation from the Greek would read, "The very thing it was required to have shown." The phrase was used by many early Greek mathematicians, including Euclid and Archimedes.
In the European Renaissance, scholars often wrote in Latin, and phrases such as Q.E.D. were often used to conclude proofs.
Perhaps the most famous use of Q.E.D. in a philosophical argument is found in the Ethics of Baruch Spinoza, published posthumously in 1677. Written in Latin, it is considered by many to be Spinoza's magnum opus. The style and system of the book is, as Spinoza says, "demonstrated in geometrical order", with axioms and definitions followed by propositions. For Spinoza, this is a considerable improvement over René Descartes's writing style in the Meditations, which follows the form of a diary.
There is another Latin phrase with a slightly different meaning, and less common in usage. Quod erat faciendum, originating from the Greek geometers' closing ὅπερ ἔδει ποιῆσαι (hoper edei poiēsai), meaning "which had to be done". Euclid used this phrase to close propositions which were not proofs of theorems, but constructions. For example, Euclid's first proposition showing how to construct an equilateral triangle given one side is usually shortened to QEF.
Equivalents in other languages
Q.E.D. has acquired many translations in various languages, including:
|Arabic||هـ.ط.ث||وهو المطلوب إثباته|
|Armenian||Ի.Պ.Ա. (rarely abbreviated)||ինչը և պահանջվում էր ապացուցել|
|Bengali||অ. সি.||অতঃ সিদ্ধ|
|Bosnian||Š.T.D.||što je trebalo dokazati|
|Bulgarian||КТДД||Което трябваше да докажем
Което трябваше да се докаже
|Croatian||Š.T.D.||što je trebalo dokazati|
|Czech||C.B.D.||což bylo dokázati|
|wat moest bewezen worden
wat te bewijzen was
|Estonian||M.O.T.T.||mida oligi tarvis tõestada|
|Finnish||M.O.T.||mikä oli todistettava|
|French||C.Q.F.D.||ce qu'il fallait démontrer|
|Catalan||C.V.D.||com volíem demostrar|
|Georgian||რ.დ.გ||რისი დამტკიცებაც გვსურდა|
|German||w.z.b.w.||was zu beweisen war|
|Greek||Ο.Ε.Δ.||όπερ έδει δείξαι|
|Hebrew||.מ.ש.ל||מה שנדרש להוכיח|
|Hungarian||E.K.B. (rarely abbreviated)||Ezt kellett bizonyítani|
|Icelandic||Þ.S.S.Á.||Það sem sanna átti|
|Italian||C.V.D.||come volevasi dimostrare|
|Latvian||k.b.j.||kas bija jāpierāda|
|Polish||c.b.d.u.||co było do udowodnienia|
|Portuguese||C.Q.D.||como queríamos demonstrar|
|Romanian||c.c.t.d.||ceea ce trebuia demonstrat|
|Russian||ч.т.д.||что и требовалось доказать|
|Serbian||ш.т.д.||што је и требало да се докаже|
|Slovak||č.b.t.d.||čo bolo treba dokázať|
|Slovenian||k.e.d.||konec enega dokaza|
|Spanish||C.Q.D.||lo que se quería demostrar
como queda demostrado
|vilket skulle bevisas
vilket skulle visas
|Turkish||G.İ.B.||Gösterilmek istenen şey de buydu|
|що й слід було довести
що і треба було довести
|Vietnamese||đpcm.||Điều phải chứng minh|
There is no common formal English equivalent, though the end of a proof may be announced with a simple statement such as "this completes the proof", "as required", "hence proved", "ergo", or a similar locution. WWWWW or W5 - an abbreviation of "Which Was What Was Wanted" - has also been used. This is often considered to be more tongue-in-cheek than the usual Halmos symbol (see below) or Q.E.D.
When typesetting was done by a compositor with letterpress printing, complex typography such as mathematics and foreign languages were called "penalty copy" (the author paid a "penalty" to have them typeset, as it was harder than plain text). With the advent of systems such as LaTeX, mathematicians found their options more open, so there are several symbolic alternatives in use, either in the input, the output, or both. When creating TeX, Knuth provided the symbol ■ (solid black square), also called by mathematicians tombstone or Halmos symbol (after Paul Halmos, who pioneered its use as an equivalent of Q.E.D.). The tombstone is sometimes open: □ (hollow black square). Unicode explicitly provides the "End of proof" character U+220E (∎), but also offers ▮ (U+25AE, black vertical rectangle) and ‣ (U+2023, triangular bullet) as alternatives. Some authors have adopted variants of this notation with other symbols, such as two forward slashes (//), or simply some vertical white space, implying no further statements need to be made in the proof.
Modern humorous usage
In Joseph Heller's book Catch-22, the Chaplain, having been told to examine a forged letter allegedly signed by him (which he knew he didn't sign), verified that his name was in fact there. His investigator replied, "Then you wrote it. Q.E.D." The chaplain said he didn't write it and that it wasn't his handwriting, and the investigator replied, "Then you signed your name in somebody else's handwriting again."
In the 1978 sci-fi radio comedy, and later in the TV and novel adaptations of The Hitchhiker's Guide to the Galaxy, "Q.E.D." is referred to in the Guide's entry for the babel fish, when it is claimed that the babel fish is used as evidence for the non-existence of God.
- Euclid's Elements translated from Greek by Thomas L. Heath. 2003 Green Lion Press pg. xxiv
- Entry δείκνυμι at LSJ.
- Elements 2.5 by Euclid (ed. J. L. Heiberg), retrieved 16 July 2005
- Philippe van Lansberge (1604). Triangulorum Geometriæ. Apud Zachariam Roman. pp. 1–5.
- The Chief Works of Benedict De Spinoza, translated by R. H. M. Elwes, 1951. ISBN 0-486-20250-X.
- Donald E. Knuth, "Mathematical Typography", lecture to the ACM, 1975
- Heller, Joseph (1971). Catch-22. ISBN 9780573606854. Retrieved 15 July 2011.
- Adams, Douglas (2005). The Hitchhiker's Guide to the Galaxy. The Hitchhiker's Guide to the Galaxy (Film tie-in edition ed.). Basingstoke and Oxford: Pan Macmillan. pp. 62–64. ISBN 0-330-43798-4.
- Stephenson, Neal (1999). Cryptonomicon. New York, NY: Avon Books. ISBN 978-0-06-051280-4.
|Look up quod erat demonstrandum in Wiktionary, the free dictionary.|