View source for Q.E.D.

{{Short description|Abbreviation at completion of a proof}}
{{other uses|QED (disambiguation){{!}}QED}}
'''Q.E.D.''' or '''QED''' is an [[initialism]] of the [[List of Latin phrases (full)|Latin phrase]] {{lang|la|'''quod erat demonstrandum'''}}, meaning "that which was to be demonstrated". Literally, it states "what was to be shown".<ref>{{Cite web|url=https://www.merriam-webster.com/dictionary/quod%20erat%20demonstrandum|title=Definition of QUOD ERAT DEMONSTRANDUM|website=www.merriam-webster.com|language=en|access-date=2017-09-03}}</ref> Traditionally, the abbreviation is placed at the end of [[Mathematical proof#Ending a proof|mathematical proofs]] and [[Philosophy|philosophical]] [[Logical argument|arguments]] in print publications, to indicate that the proof or the argument is complete.

== Etymology and early use ==
The phrase ''quod erat demonstrandum'' is a translation into [[List of Latin phrases|Latin]] from the [[Greek language|Greek]] {{lang|grc|ὅπερ ἔδει δεῖξαι}} ({{lang|grc-Latn|hoper edei deixai}}; abbreviated as ''ΟΕΔ''). The meaning of the Latin phrase is "that [thing] which was to be demonstrated" (with ''demonstrandum'' in the [[gerundive]]). 

The Greek phrase was used by many early Greek mathematicians, including [[Euclid]]<ref>[https://www.perseus.tufts.edu/hopper/text?doc=Euc.+2&fromdoc=Perseus%3Atext%3A1999.01.0085 Elements 2.5] by Euclid (ed. J. L. Heiberg), retrieved 16 July 2005</ref> and [[Archimedes]].

The Latin phrase is attested in a 1501 Euclid translation of [[Giorgio Valla]].<ref>{{cite web |last1=Valla |first1=Giorgio |title=Georgii Vallae Placentini viri clariss. De expetendis, et fugiendis rebus opus. 1 |url=https://www.digitale-sammlungen.de/en/view/bsb10148187?page=342,343&q=%22quod+erat+demonstrandum%22}}</ref> Its abbreviation ''q.e.d.'' is used once in 1598 by [[Johannes Praetorius]],<ref>{{cite web |last1=Praetorius |first1=Johannes |title=Ioannis Praetorii Ioachimici Problema, quod iubet ex Quatuor rectis lineis datis quadrilaterum fieri, quod sit in Circulo |url=https://www.digitale-sammlungen.de/en/view/bsb10989794?page=14,15&q=%22q.e.d.%22}}</ref> more in 1643 by [[Anton Deusing]],<ref>{{cite web |last1=Deusing |first1=Anton |title=Antonii Deusingii Med. ac Philos. De Vero Systemate Mundi Dissertatio Mathematica : Quâ Copernici Systema Mundi reformatur: Sublatis interim infinitis penè orbibus, quibus in Systemate Ptolemaico humana mens distrahitur |url=https://www.digitale-sammlungen.de/en/view/bsb10818398?page=154,155&}}</ref> extensively in 1655 by [[Isaac Barrow]] in the form ''Q.E.D.'',<ref>{{cite web |last1=Barrow |first1=Isaac |title=Elementa geometrie : libri XV. |url=https://www.digitale-sammlungen.de/en/view/bsb10235961?page=50,51&q=%22Q.E.D.%22}}</ref> and subsequently by many post-[[Renaissance]] mathematicians and philosophers.<ref>{{Cite web|url=http://jeff560.tripod.com/q.html|title=Earliest Known Uses of some of the Words of Mathematics (Q)|website=jeff560.tripod.com|access-date=2019-11-04}}</ref>

== Modern philosophy ==
[[File:Philippe van Lansberge 1604 QED.png|right|thumb|Philippe van Lansberge's 1604 [https://books.google.com/books?id=fg9KAAAAMAAJ&q=quod-erat-demonstrandum+date:0-1700 ''Triangulorum Geometriæ''] used {{lang|la|quod erat demonstrandum}} to conclude some proofs; others ended with phrases such as {{lang|la|sigillatim deinceps demonstrabitur}}, {{lang|la|magnitudo demonstranda est}}, and other variants.<ref>{{cite book|title=Triangulorum Geometriæ|author=Philippe van Lansberge|year=1604|pages=[https://archive.org/details/philippilansber00lansgoog/page/n8 1]–5|publisher=Apud Zachariam Roman|url=https://archive.org/details/philippilansber00lansgoog|quote=quod-erat-demonstrandum 0-1700.}}</ref>|418x418px]]

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

[[File:Spinoza Ethica Pars1 Prop1.jpg|thumb|400px|[[Baruch Spinoza|Spinoza]]'s original text of ''[[Ethics (Spinoza)|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 (Spinoza)|Ethics]]'' of [[Baruch Spinoza]], [[List of works published posthumously|published posthumously]] in 1677.<ref>{{Cite web|url=https://opentextbc.ca/modernphilosophy/chapter/baruch-spinoza-1632-1677/|title=Baruch Spinoza (1632–1677) – Modern Philosophy|website=opentextbc.ca|access-date=2019-11-04}}</ref> Written in Latin, it is considered by many to be Spinoza's ''[[Masterpiece|magnum opus]]''. The style and system of the book are, as Spinoza says, "demonstrated in [[Geometry|geometrical]] order", with [[axiom]]s and definitions followed by [[proposition (mathematics)|propositions]]. For Spinoza, this is a considerable improvement over [[René Descartes]]'s writing style in the ''[[Meditations on First Philosophy|Meditations]]'', which follows the form of a [[diary]].<ref>''The Chief Works of Benedict De Spinoza'', translated by R. H. M. Elwes, 1951. {{isbn|0-486-20250-X}}.</ref>

== Difference from Q.E.F. ==
There is another Latin phrase with a slightly different meaning, usually shortened similarly, but being less common in use. {{lang|la|Quod erat faciendum}}, originating from the Greek geometers' closing {{lang|grc|ὅπερ ἔδει ποιῆσαι}} ({{Transliteration|grc|hoper edei poiēsai}}), meaning "which had to be done".<ref>{{cite book | title=Disquisitiones Arithmeticae | isbn=9781493975600 | last1=Gauss | first1=Carl Friedrich | last2=Waterhouse | first2=William C. | date=7 February 2018 | publisher=Springer | url={{Google books|DyFLDwAAQBAJ|page=470|plainurl=yes}}}}</ref> 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.<ref>{{Cite web|url=http://mathworld.wolfram.com/QEF.html|title=Q.E.F.|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-11-04}}</ref> For example, Euclid's first proposition showing how to construct an [[equilateral triangle]], given one side, is concluded this way.<ref>{{Cite web|url=https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI1.html|title=Euclid's Elements, Book I, Proposition 1|website=mathcs.clarku.edu|access-date=2019-11-04}}</ref>

== Equivalent forms ==

[[File:Ihaveproveditmyself.jpg|thumb]]

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.

== Typographical forms used symbolically ==
{{Main|Tombstone (typography)}}
Due to the paramount importance of [[Mathematical proof|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 [[theorem]]s, [[lemma (mathematics)|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,<ref>This (generally accepted) claim was made in Halmos's autobiography, ''I Want to Be a Mathematician''.  The first usage of the solid black rectangle as an end-of-proof symbol appears to be in Halmos's ''Measure Theory'' (1950).  The intended meaning of the symbol is explicitly given in the preface.</ref> a practice which has become standard, although not universal. Halmos noted that he 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, so-called ''[[wikt:end mark|end marks]]''.<ref>{{Cite book|url=https://books.google.com/books?id=7VblBwAAQBAJ&q=tombstone&pg=PA3|title=I Want to Be a Mathematician: An Automathography|last=Halmos|first=Paul R.|year=1985|isbn=9781461210849|pages=403|publisher=Springer }}</ref><ref>{{cite web |last1=Felici |first1=James |title=The complete manual of typography : a guide to setting perfect type |url=https://archive.org/details/completemanualof0000feli/page/226 |publisher=Berkeley, CA : Peachpit Press |date=2003}}</ref> This symbol was later called the ''[[Tombstone (typography)|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 <math>\blacksquare</math> (filled square, \blacksquare) and sometimes, as a <math>\square</math> (hollow square, \square or \Box).<ref>See, for example, [[list of mathematical symbols]] for more.</ref> 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 (//, <math>//</math>) or four forward slashes (////, <math>////</math>).<ref>{{Cite book|title=Real and Complex Analysis|last=Rudin|first=Walter|publisher=McGraw-Hill|year=1987|isbn=0-07-100276-6}}</ref> In other cases, authors have elected to segregate proofs typographically—by displaying them as indented blocks.<ref>{{Cite book|title=Principles of Mathematical Analysis|last=Rudin|first=Walter|publisher=McGraw-Hill|year=1976|isbn=0-07-054235-X|location=New York}}</ref>

==In popular culture==
In [[Joseph Heller]]'s 1961 novel ''[[Catch-22]]'', [[Chaplain Tappman|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 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."<ref>{{cite book|url=https://books.google.com/books?id=X6k30_63q08C&pg=PA84|title=Catch-22|last=Heller|first=Joseph|access-date=15 July 2011|isbn=978-0-573-60685-4|year=1971|publisher=S. French }}</ref>

== See also ==
* [[List of Latin abbreviations]]
* [[A priori and a posteriori]]
* [[Bob's your uncle]]
* ''[[Ipso facto]]''
* [[Proof by contradiction#Notation|Q.E.A.]]
* {{slink|List of Latin phrases (E)|ergo}}

== References ==
{{Reflist|30em}}

==External links==
{{Wiktionary|quod erat demonstrandum|QED}}

*[http://jeff560.tripod.com/q.html Earliest Known Uses of Some of the Words of Mathematics (Q)]

[[Category:Initialisms]]
[[Category:Latin logical phrases]]
[[Category:Latin philosophical phrases]]
[[Category:Mathematical proofs]]
[[Category:Mathematical terminology]]