Q.E.D. (also written QED) is an initialism of the Latin phrase quod erat demonstrandum, meaning "which is what had to be shown" or "thus it has been demonstrated." The phrase is traditionally placed in its abbreviated form at the end of a mathematical proof or philosophical argument when the original proposition 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 concluded this way. The phrase is usually shortened to QEF.
Equivalents in other languages
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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|
|Czech||C.B.D.||což bylo dokázati|
|Danish||H.S.B.||hvilket skulle bevises|
|wat moest bewezen worden
wat te bewijzen was
|Esperanto||K.E.P.||kio estis pruvenda|
|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 שצריך להוכיח|
|Hindi||इति सिद्धम||यही सिद्ध करना था|
|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|
|co było do udowodnienia
czego należało dowieść
co kończy dowód
|Portuguese||C.Q.D.||como queríamos demonstrar|
|Romanian||c.c.t.d.||ceea ce trebuia demonstrat|
|Russian||ч.т.д.||что и требовалось доказать|
|што је требало доказати
što je trebalo dokazati
|Slovak||č.b.t.d.||čo bolo treba dokázať|
|Slovenian||k.e.d.||konec enega dokaza|
|como queríamos demostrar
queda entonces 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.
Symbolic forms in typography
Due to the paramount importance of proofs in mathematics, mathematicians since the time of Euclid have developed conventions to demarcate the beginning and/or end of proofs. In English language texts, the formal statements of theorems, lemmas, and propositions are typically set in italics. The beginning of a proof usually follows immediately thereafter and is indicated by the word "Proof" in boldface or italics. On the other hand, several symbolic conventions exist to indicate the end of a proof. While some authors still use the classical abbreviation Q.E.D., this practice is increasingly viewed as archaic or even pretentious. Paul Halmos pioneered the use of a black square at the end of a proof as a Q.E.D sign, a practice which has become standard though not universal. Halmos claimed to have adopted this symbol from magazine typography in which simple geometric shapes had been used to indicate the end of an article. This symbol was later called the tombstone or Halmos symbol or even a halmos by mathematicians. The Halmos symbol is also often drawn on chalkboard to signal the end of a proof during a lecture, although this practice is not as common as its use in printed text.
The tombstone symbol appears in TeX as the character (filled square, \blacksquare) and (hollow square, \square). In the AMS Theorem Environment for LaTeX, the hollow square is the default end-of-proof symbol. Unicode explicitly provides the "End of proof" character U+220E (∎). Some authors use other Unicode symbols to note the end of a proof, including ▮ (U+25AE, black vertical rectangle) and ‣ (U+2023, triangular bullet). Other authors have adopted two forward slashes (//) or four forward slashes (////). In other cases, authors have elected to segregate proofs typographically by displaying them as indented blocks.
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 existence and non-existence of God.
Singer-songwriter Thomas Dolby's 1988 song "Airhead" includes the lyric, "Quod erat demonstrandum, baby," referring to the self-evident vacuousness of the eponymous subject; and in response, a female voice squeals, delightedly, "Oooh... you speak French!" 
- 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.
- Rudin, Walter (1987). Real and Complex Analysis. McGraw-Hill. ISBN 0-07-100276-6.
- Rudin, Walter (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. ISBN 007054235X.
- 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 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.
- "Airhead - Thomas Dolby". play.google.com. Retrieved 2016-09-15.
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