Q.E.D. orQED is aninitialism of theLatin phrasequod erat demonstrandum, meaning "that which was to be demonstrated". Literally, it states "what was to be shown".[1] Traditionally, the abbreviation is placed at the end ofmathematical proofs andphilosophicalarguments in print publications, to indicate that the proof or the argument is complete.
The phrasequod erat demonstrandum is a translation intoLatin from theGreekὅπερ ἔδει δεῖξαι (hoper edei deixai; abbreviated asΟΕΔ). The meaning of the Latin phrase is "that [thing] which was to be demonstrated" (withdemonstrandum in thegerundive). However, translating the Greek phraseὅπερ ἔδει δεῖξαι can produce a slightly different meaning. In particular, since the verb"δείκνυμι" means bothto show orto prove,[2] a different translation from the Greek phrase would read "The very thing it was required to have shown."[3]
The Greek phrase was used by many early Greek mathematicians, includingEuclid[4] andArchimedes.
The Latin phrase is attested in a 1501 Euclid translation ofGiorgio Valla.[5] Its abbreviationq.e.d. is used once in 1598 byJohannes Praetorius,[6] more in 1643 byAnton Deusing,[7] extensively in 1655 byIsaac Barrow in the formQ.E.D.,[8] and subsequently by many post-Renaissance mathematicians and philosophers.[9]
During the European Renaissance, scholars often wrote in Latin, and phrases such asQ.E.D. were often used to conclude proofs.
Perhaps the most famous use ofQ.E.D. in a philosophical argument is found in theEthics ofBaruch Spinoza,published posthumously in 1677.[11] Written in Latin, it is considered by many to be Spinoza'smagnum opus. The style and system of the book are, as Spinoza says, "demonstrated ingeometrical order", withaxioms and definitions followed bypropositions. For Spinoza, this is a considerable improvement overRené Descartes's writing style in theMeditations, which follows the form of adiary.[12]
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' closingὅπερ ἔδει ποιῆσαι (hoper edei poiēsai), meaning "which had to be done".[13] 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.[14] For example, Euclid's first proposition showing how to construct anequilateral triangle, given one side, is concluded this way.[15]
There is no common formal English equivalent, although the end of a proof may be announced with a simple statement such as "thus it is proved", "this completes the proof", "as required", "as desired", "as expected", "hence proved", "ergo", "so correct", or other similar phrases.
Due to the paramount importance ofproofs in mathematics, mathematicians since the time ofEuclid have developed conventions to demarcate the beginning and end of proofs. In printed English language texts, the formal statements oftheorems,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 claims to have pioneered the use of a solid black square (or rectangle) at the end of a proof as a Q.E.D. symbol,[16] a practice which has become standard, although not universal. Halmos noted that he adopted this use of a symbol from magazinetypography customs in which simple geometric shapes had been used to indicate the end of an article, so-calledend marks.[17][18] This symbol was later called thetombstone, theHalmos symbol, or even ahalmos 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 inTeX as the character (filled square, \blacksquare) and sometimes, as a (hollow square, \square or \Box).[19] In the AMS Theorem Environment forLaTeX, 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 (////,).[20] In other cases, authors have elected to segregate proofs typographically—by displaying them as indented blocks.[21]
InJoseph Heller's 1961 novelCatch-22,the Chaplain, having been told to examine a forged letter allegedly signed by him (which he knew he didn't sign), verified that hisname was in fact there. His investigator replied, "Then you wrote it. Q.E.D." The chaplain said he did not write it and that it was not his handwriting, to which the investigator replied, "Then you signed your name in somebody else's handwriting again."[22]
quod-erat-demonstrandum 0-1700.
