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Q.E.D. is an abbreviation of the Latin phrase "quod erat demonstrandum" (literally, "that which was to be demonstrated"). The phrase is written in its abbreviated form at the end of a mathematical proof or philosophical argument, to signify that the last statement deduced was the one to be demonstrated, so the proof is complete. This page lists direct English translations of common Latin phrases, such as veni vidi vici and et cetera. ... QED can mean several different things: Q.E.D. Latin Quod erat demonstrandum, used at the end of mathematical proofs The QED project intended to construct a formalized database of all mathematical knowledge The QED text editor program Quantum electrodynamics, a field of physics Quantum Effect Devices, a maker of... Wikipedia does not have an article with this exact name. ... Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 151 languages. ... This page lists direct English translations of common Latin phrases, such as veni vidi vici and et cetera. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ... For other uses, see Philosophy (disambiguation). ... In logic, an argument is a set of statements, consisting of a number of premises, a number of inferences, and a conclusion, which is said to have the following property: if the premises are true, then the conclusion must be true or highly likely to be true. ...

Etymology and early use

The phrase is a translation into Latin of the original Greek ὅπερ ἔδει δειξαι. (hoper edei deixai) which was used by many early mathematicians including Euclid^[1] and Archimedes. These mathematicians, in particular Euclid, are credited with founding axiomatic mathematics with its emphasis on establishing truths by logical deduction (rather than experimentation or assertion); their use of this phrase symbolizes this emphasis, as well as marking this important step in the development of mathematical philosophy. For other uses, see Euclid (disambiguation). ... For other uses, see Archimedes (disambiguation). ...

Modern philosophy

Philippe van Lansberge's 1604 Triangulorum Geometræ used "quod erat demonstrandum" to conclude some proofs; others ended with phrases such as "figillatim deinceps demunstrabitur," "magnitudo demonstranda est," and other variants.

In the European renaissance, mathematical books were typically written in Latin, and phrases such as "quod erat demonstrandum" were often used to conclude proofs. This article is about the European Renaissance of the 14th-17th centuries. ... For other uses, see Latin (disambiguation). ...

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 his magnum opus. Ethics is a philosophical book written by Baruch Spinoza. ... Baruch de Spinoza (‎, Portuguese: , Basque: , Latin: ) (November 24, 1632 – February 21, 1677) was a Dutch philosopher of Portuguese Jewish origin. ... Posthumous means after death. ... 1677 (MDCLXXVII) was a common year starting on Friday of the Gregorian calendar (or a common year starting on Monday of the 10-day slower Julian calendar). ... For other uses, see Latin (disambiguation). ... Magnum opus (sometimes Opus magnum, plural magna opera), from the Latin meaning great work,[1] refers to the best, most popular, or most renowned achievement of an author, artist, or composer, and most commonly one who has contributed a very large amount of material. ...

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.^[2] For other uses, see Geometry (disambiguation). ... For the algebra software named Axiom, see Axiom computer algebra system. ... This article is about the word proposition as it is used in logic, philosophy, and linguistics. ... Descartes redirects here. ... The title page of the Meditations Meditations on First Philosophy (subtitled In which the existence of God and the real distinction of mind and body, are demonstrated) is a philosophical treatise written by René Descartes first published in Latin in 1641 . ... == c programming[[a--203. ...

Current usage

Currently, it has become so symbolic of irrefutable logic that "Q.E.D." is occasionally used in non-mathematical contexts as well to intensify assertions; in this context it has little connection with rigorous deduction, however, and is more tongue-in-cheek. Sarcasm is the making of remarks intended to mock the person referred to (who is normally the person addressed), a situation or thing. ...

Q.E.F.

There is another Latin phrase with a slightly different meaning, and less common in usage. Quod erat faciendum is translated as "which was to be done." This is usually shortened to Q.E.F.. As with Q.E.D., Q.E.F. is a translation of the Greek geometers' closing ὅπερ ἔδει ποιῆσαι (hoper edei poiēsai). Euclid used this phrase to close propositions which were not precisely "proofs", but rather exemplar constructions. The distinction between Q.E.D. and Q.E.F. is roughly equivalent to the distinction between a proof and an illustration of the proof. For other uses, see Euclid (disambiguation). ...

Q.E.D. in languages other than Latin

As mathematics has come to be written in languages other than Latin, Q.E.D. has acquired many translations; in French and German (two of the main languages of Western mathematics) it is respectively C.Q.F.D, for "ce qu'il fallait démontrer" (or sometimes "ce qui finit la démonstration"), and W.Z.B.W, for "was zu beweisen war". There does not appear to be a common formal English equivalent, though the end of a proof may be announced with a simple statement such as "this completes the proof" or a similar locution. The equivalent in Ancient Greek was "ὅπερ ἔδει δεῖξαι" and could be translated as "which had to be proved". In modern greek texts sometimes the "ο.ε.δ." initials are used at the end of a mathematical proof. Beginning of Homers Odyssey The Ancient Greek language is the historical stage of the Greek language[1] as it existed during the Archaic (9th–6th centuries BC) and Classical (5th–4th centuries BC) periods in Ancient Greece. ...

Electronic forms

With computers frequently being used to "write" proofs (see LaTeX), there are several symbolic alternatives in use. The most popular symbol is ■ (solid black square), also called tombstone or Halmos symbol (after Paul Halmos, who pioneered its use). 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. Professor Paul Sally at the University of Chicago is known for ending proofs with a pirate face symbol.^[3] This article is about the typesetting system. ... The tombstone, or halmos, symbol — (Unicode U+220E) — is used in mathematics to denote the end of a proof. ... Paul Halmos Paul Richard Halmos (March 3, 1916 — October 2, 2006) was a Hungarian-born American mathematician who wrote on probability theory, statistics, operator theory, ergodic theory, functional analysis (in particular, Hilbert spaces), and mathematical logic. ... The Unicode Standard, Version 5. ... In drawing, painting and sculpture white space is often referred to as negative space. ... Paul Sally is a professor of mathematics at the University of Chicago, where he is the director of undergraduate mathematics instruction. ...

Notes

1. ^ Elements 2.5 by Euclid (ed. J. L. Heiberg), retrieved 16 July 2005
2. ^ The Chief Works of Benedict De Spinoza, translated by R. H. M. Elwes, 1951 (available online - links found on the Wikipedia page Ethics (book)).
3. ^ Sally, Paul. "Foundations of Mathematical Analysis." Textbook for Math 199. University of Chicago Department of Mathematics. Chicago, IL.

is the 197th day of the year (198th in leap years) in the Gregorian calendar. ... Year 2005 (MMV) was a common year starting on Saturday (link displays full calendar) of the Gregorian calendar. ... Ethics is a philosophical book written by Baruch Spinoza. ...

External links

• Earliest Known Uses of Some of the Words of Mathematics (Q)
• Robin Hartshorne note on QED