Q.E.D. or QED is an initialism of the Latin phrase "quod erat demonstrandum", literally meaning "what was to be shown".[1] Traditionally, the abbreviation is placed at the end of a mathematical proof or philosophical argument in print publications to indicate that the proof or the argument is complete, and hence is used with the meaning "thus it has been demonstrated".[2]

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 phrase into English yields "what was to be demonstrated". However, translating the Greek phrase ὅπερ ἔδει δεῖξαι can produce a slightly different meaning. In particular, since the verb "δείκνυμι" also means to show or to prove,[3] a different translation from the Greek phrase would read "The very thing it was required to have shown."[4]

The Greek phrase was used by many early Greek mathematicians, including Euclid[5] and Archimedes. The translated Latin phrase (and its associated acronym) was subsequently used by many post-Renaissance mathematicians and philosophers, including Galileo, Spinoza, Isaac Barrow and Isaac Newton.[6]

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.[7]

During the European Renaissance, scholars often wrote in Latin, and phrases such as Q.E.D. were often used to conclude proofs.

Spinoza's original text of Ethics, Part 1, Q.E.D. is used at the end of Demonstratio of Propositio III on the right hand page

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.[8] Written in Latin, it is considered by many to be Spinoza's magnum opus. The style and system of the book are, 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.[9]

Difference from Q.E.F.

There is another Latin phrase with a slightly different meaning, usually shortened similarly, but being less common in use. Quod erat faciendum, originating from the Greek geometers' closing ὅπερ ἔδει ποιῆσαι (hoper edei poiēsai), meaning "which had to be done". Because of the difference in meaning, the two phrases should not be confused.

Euclid used the Greek original of Quod Erat Faciendum (Q.E.F.) to close propositions that were not proofs of theorems, but constructions of geometric objects.[10][2] For example, Euclid's first proposition showing how to construct an equilateral triangle, given one side, is concluded this way.[11]

Many times, mathematicians will only utilize faciendia as a result of the results of previous definitions or demonstradums. An idea of this is expressed within Topics (Aristotle), where he goes over the difference between a proposition and a problem. " For if it be put in this way, "'An animal that walks on two feet" is the definition of man, is it not?' or '"Animal" is the genus of man, is it not?' the result is a proposition: but if thus, 'Is "an animal that walks on two feet" a definition of man or no?' (or 'Is "animal" his genus or no?') the result is a problem." This is parallel to the idea of the difference between a Q.E.D. and a Q.E.F. A proposition (Q.E.D.) like this functions exactly the same way as it does for Euclid: the proposition is intended to prove a particular property, the problem (Q.E.F.) on the other hand requires multiple propositions in order to prove, or even construct an entirely new category. The problems are the dialectic's objective to solve. In a similar fashion, there are many different ways to construct a mathematical system to construct a triangle. There is only one triangle, however, and the triangle has definite properties. In this way, truth is sought within mathematics and philosophy in a congruous way. Euclid's Elements could be thought of as a document whose objective is to construct a dodecahedron and an icosahedron (Propositions 16 and 17 book XIII). Appollonius' On Conics Book I could be thought of as a document whose objective is to construct a pair of hyperbolas from two bisecting lines (Proposition 50 of book I). Propositions have historically been used in logic and mathematics to work towards solving a problem, and these fields both reflect that in their foundations through Euclid and Aristotle.

English equivalent

There is no common formal English equivalent, although the end of a proof may be announced with a simple statement such as "this completes the proof", "as required", "as desired", "as expected", "hence proved", "ergo", "so correct", or other similar locutions. WWWWW or W5 – an abbreviation of "Which Was What Was Wanted" – has been used similarly. Often this is considered to be more tongue-in-cheek than Q.E.D. or the Halmos tombstone symbol (see below).

Typographical forms used symbolically

Due to the paramount importance of proofs in mathematics, mathematicians since the time of Euclid have developed conventions to demarcate the beginning and end of proofs. In printed English language texts, the formal statements of theorems, lemmas, and propositions are set in italics by tradition. 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., it is relatively uncommon in modern mathematical texts. Paul Halmos pioneered the use of a solid black square at the end of a proof as a Q.E.D symbol, a practice which has become standard, although not universal. Halmos adopted this use of a symbol from magazine typography customs in which simple geometric shapes had been used to indicate the end of an article.[12] This symbol was later called the tombstone, the Halmos symbol, or even a halmos by mathematicians. Often the Halmos symbol is drawn on chalkboard to signal the end of a proof during a lecture, although this practice is not so common as its use in printed text.

The tombstone symbol appears in TeX as the character (filled square, \blacksquare) and sometimes, as a (hollow square, \square or \Box).[13] 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, a black vertical rectangle), and ‣ (U+2023, a triangular bullet). Other authors have adopted two forward slashes (//) or four forward slashes (////).[14] In other cases, authors have elected to segregate proofs typographically—by displaying them as indented blocks.[15]

Modern humorous use

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, to which the investigator replied, "Then you signed your name in somebody else's handwriting again."[16]

In the 1978 science-fiction radio comedy, and later in the television, novel, and film 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 – which serves the "mind-bogglingly" useful purpose of being able to translate any spoken language when inserted into a person's ear – is used as evidence for existence and non-existence of God. The exchange from the novel is as follows: "'I refuse to prove 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."[17]

In Neal Stephenson's 1999 novel Cryptonomicon, Q.E.D. is used as a punchline to several humorous anecdotes, in which characters go to great lengths to prove something non-mathematical.[18]

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!" [19]

See also


  1. "Definition of QUOD ERAT DEMONSTRANDUM". www.merriam-webster.com. Retrieved 2017-09-03.
  2. "The Definitive Glossary of Higher Mathematical Jargon — Q.E.D." Math Vault. 2019-08-01. Retrieved 2019-11-04.
  3. Entry δείκνυμι at LSJ.
  4. Euclid's Elements translated from Greek by Thomas L. Heath. 2003 Green Lion Press pg. xxiv
  5. Elements 2.5 by Euclid (ed. J. L. Heiberg), retrieved 16 July 2005
  6. "Earliest Known Uses of some of the Words of Mathematics (Q)". jeff560.tripod.com. Retrieved 2019-11-04.
  7. Philippe van Lansberge (1604). Triangulorum Geometriæ. Apud Zachariam Roman. pp. 1–5. quod-erat-demonstrandum 0-1700.
  8. "Baruch Spinoza (1632–1677) – Modern Philosophy". opentextbc.ca. Retrieved 2019-11-04.
  9. The Chief Works of Benedict De Spinoza, translated by R. H. M. Elwes, 1951. ISBN 0-486-20250-X.
  10. Weisstein, Eric W. "Q.E.F." mathworld.wolfram.com. Retrieved 2019-11-04.
  11. "Euclid's Elements, Book I, Proposition 1". mathcs.clarku.edu. Retrieved 2019-11-04.
  12. Halmos, Paul R. (1985). I Want to Be a Mathematician: An Automathography. p. 403. ISBN 9781461210849.
  13. See, for example, list of mathematical symbols for more.
  14. Rudin, Walter (1987). Real and Complex Analysis. McGraw-Hill. ISBN 0-07-100276-6.
  15. Rudin, Walter (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. ISBN 0-07-054235-X.
  16. Heller, Joseph (1971). Catch-22. ISBN 978-0-573-60685-4. Retrieved 15 July 2011.
  17. 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.
  18. Stephenson, Neal (1999). Cryptonomicon. New York, NY: Avon Books. ISBN 978-0-06-051280-4.
  19. "Airhead – Thomas Dolby". play.google.com. Retrieved 2016-09-15.
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