Q.E.D. is an initialism of the Latin phrase quod erat demonstrandum, originating from the Greek analogous hóper édei deîxai (ὅπερ ἔδει δεῖξαι), meaning "which had to be demonstrated". 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. An example of Q.E.D. signifying the successful completion of a geometry proof is available in Chapter 2 of the Wikibooks Geometry text.
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 shows how to construct an equilateral triangle given one side. It is usually shortened to QEF.
Equivalents in other languages
Q.E.D. has acquired many translations in various languages, including:
|Arabic||و.هـ.م||وهو المطلوب إثباته|
|Armenian||Ի.Պ.Ա.(rarely used as an abbreviation)||ինչը և պահանջվում էր ապացուցել|
|Bengali||অ. সি.||অতঃ সিদ্ধ|
|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/což se mělo dokázat|
|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|
|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|
|Hungarian||E.K.B. (rarely used as an abbreviation)||Ezt kellett bizonyítani|
|Icelandic||Þ.S.S.Á.||Það sem sanna átti|
|Italian||C.V.D.||come volevasi dimostrare|
|Norwegian||Q.E.D.||hvilket skulle demonstreres
hvilket skulle bevises
|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||Q.E.D.||lo que se quería demostrar|
|vilket skulle bevisas
vilket skulle visas
|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
Q.E.D. is sometimes jokingly claimed to abbreviate "quite easily done".
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, but the investigator's faulty logic caused him to point out, "Then you signed your name in somebody else's handwriting again."
Thomas Dolby, in his 1988 song "Airhead", imagines a conversation with the titular young woman and says "quod erat demonstrandum, baby", to which she squeals the eager reply "ohhh, you speak French!"
- The argument goes something like this: "I refuse to prove that I exist," says God, "for proof denies faith, and without faith I am nothing."
- "But," says Man, "the Babel fish is a dead giveaway, isn't it? It could not have evolved by chance. It proves you exist, and so therefore, by your own arguments, you don't. QED."
- "Oh dear," says God, "I hadn't thought of that," and promptly vanishes in a puff of logic.
Another potential English transliteration is in the movie "Ice Princess". Michelle Trachtenberg's lead character Casey is a physics nerd who writes a paper on the physics of figure skating concluding she has shown passion for the topic, "QED," to which her best friend replies, "What has been Quite Easily Demonstrated?"
Dannay and Lee, authors of the famous Ellery Queen mystery novels, often had their protagonist refer to his solving of the murder case as "Queen's Elementary Deduction".
- 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.
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