Q.E.D.

Q.E.D. or QED (sometimes italicized) is an initialism of the Latin phrase 'quod erat demonstrandum', literally meaning 'what was to be shown'. Traditionally, the abbreviation is placed at the end of a mathematical proof or philosophical argument to indicate that the proof or argument is complete, therefore used with the meaning 'thus it has been demonstrated'. Q.E.D. or QED (sometimes italicized) is an initialism of the Latin phrase 'quod erat demonstrandum', literally meaning 'what was to be shown'. Traditionally, the abbreviation is placed at the end of a mathematical proof or philosophical argument to indicate that the proof or argument is complete, therefore used with the meaning 'thus it has been demonstrated'. 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 Greek phrase was used by many early Greek mathematicians, including Euclid and Archimedes. During 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 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. 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 phrase, Quod Erat Faciendum (Q.E.F.), to close propositions that were not proofs of theorems, but constructions. For example, Euclid's first proposition showing how to construct an equilateral triangle, given one side, is concluded this way. Q.E.D. has acquired many translations in various languages, including: 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', 'hence proved', 'ergo', or by using a similar locution. 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 the usual Halmos symbol (see below) or Q.E.D.