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Q.E.D. is an initialism of the Latin phrase quod erat demonstrandum, originating from the Greek analogous hóper ódei 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.[1] 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,[2] a better translation from the Greek would read, "what was required to be proved."[3] The phrase was used by many early Greek mathematicians, including Euclid[4] and Archimedes.

Modern philosophy

Philippe van Lansberge's 1604 Triangulorum Geometriæ used quod erat demonstrandum to conclude some proofs; others ended with phrases such as sigillatim deinceps demonstrabitur, magnitudo demonstranda est, and other variants.[5]

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.[6]


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:

LanguageAbbreviationsStands for...
CroatianŠ.T.D.što je trebalo pokazati
EstonianM.O.T.T.mida oligi tarvis tõestada
FinnishM.O.T.mikä on todistettu
FrenchC.Q.F.D.ce qu'il fallait démontrer
Germanw.z.b.w.was zu beweisen war
GreekΟ.Ε.Δ.όπερ έδει δείξαι
ItalianC.V.D.come volevasi dimostrare
Hebrew.מ.ש.למה שצריך להוכיח
Hindiइ.सि.इति सिद्धम्
Polishc.b.d.u.co było do udowodnienia
PortugueseC.Q.D.como queríamos demonstrar
Russianч.т.д.что и требовалось доказать

(although rarely used as an abbrevation)

Ezt kellett bizonyítani
queda entonces demostrado
queda esto demostrado
Vietnameseđpcm.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.

Electronic forms

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).[7] 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.[citation needed]

Modern humorous usage

Q.E.D. is sometimes jokingly claimed to abbreviate "quite easily done". Q.E.D. can also be used to ridicule the specious reasoning of another person by mockingly attaching it to the end of a poor argument, which was not in fact successfully demonstrated or presented.[citation needed]

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."[8]

Richard Feynman humorously called his magnum opus Quantum Electro Dynamics and abbreviated it as QED as a clever pun, underscoring this significant contribution to science.[citation needed]

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!"

In chapter six of The Hitchhiker's Guide to the Galaxy, by Douglas Adams, Q.E.D. is included in the following exchange:

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?"

See also


  1. ^ Euclid's Elements translated from Greek by Thomas L. Heath. 2003 Green Lion Press pg. xxiv
  2. ^ Entry δείκνυμι at LSJ.
  3. ^ Euclid's Elements translated from Greek by Thomas L. Heath. 2003 Green Lion Press pg. xxiv
  4. ^ Elements 2.5 by Euclid (ed. J. L. Heiberg), retrieved 16 July 2005
  5. ^ Philippe van Lansberge (1604). Triangulorum Geometriæ. Apud Zachariam Roman. pp. 1–5. http://books.google.com/books?id=fg9KAAAAMAAJ&pg=PT4&dq=quod-erat-demonstrandum+date:0-1700. 
  6. ^ The Chief Works of Benedict De Spinoza, translated by R. H. M. Elwes, 1951. ISBN 0-486-20250-X.
  7. ^ Donald E. Knuth, "Mathematical Typography", lecture to the ACM, 1975
  8. ^ Heller, Joseph. Catch-22. http://books.google.com/books?id=X6k30_63q08C&pg=PA84&lpg=PA84&dq=catch-22+q.e.d.+chaplain&onepage&f=false. Retrieved 15 July 2011. 

External links