Q.E.D. is an initialism of the Latin phrase quod erat demonstrandum, which translates as "which was 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] Aristotle (APo.90b34) 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]

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, Yossarian 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."[7]

Richard Feynman humourously deliberately called his magnum opus Quantum Electro Dynamics and abbreviated it as QED as a clever pun, underscoring this monumental contribution to science.


There is another Latin phrase with a slightly different meaning, and less common in usage. Quod erat faciendum is translated as "what was to have been done". This is usually shortened to QEF. The expression quod erat faciendum is a translation of the Greek geometers' closing ὅπερ ἔδει ποιῆσαι (hoper edei poiēsai). 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.

Equivalents in other languages

Q.E.D. has acquired many translations in various foreign languages, including:

Language Abbreviations Stands for...
French C.Q.F.D. ce qu'il fallait démontrer
Georgian რ.დ.გ.(r.d.g.) რისი დამტკიცებაც გვინდოდა(risi damtkic'ebac' gvindoda)
German W.Z.B.W. was zu beweisen war
Italian C.V.D. come volevasi dimostrare
Portuguese C.Q.D. como queríamos demonstrar
Russian ч.т.д. что и требовалось доказать
Spanish Q.E.D.
queda entonces demostrado
queda esto demostrado

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", or a similar locution. Likewise, in Hungarian, the abbreviation of the Latin phrase is usually used.

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).[8] 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.[citation needed]

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. ^ 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. 
  8. ^ Donald E. Knuth, "Mathematical Typography", lecture to the ACM, 1975

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