Editing Epsilon

{{Short description|Fifth letter of the Greek alphabet}}
{{Distinguish|Upsilon}} <!-- Letters with similar appearance are in the "See also" section below -->
{{about|the Greek letter|other uses|Epsilon (disambiguation)|and|E (disambiguation)}}
{{use dmy dates|date=May 2024}}
{{Greek Alphabet|letter=epsilon}}
{{Orthography notation}}
'''Epsilon''' ({{IPAc-en|US|ˈ|ɛ|p|s|ᵻ|l|ɒ|n|audio=LL-Q1860 (eng)-Flame, not lame-Epilson.wav}},<ref>{{cite encyclopedia |title=epsilon |encyclopedia=Longman Pronunciation Dictionary |first=John C. |last=Wells |author-link=John C. Wells |publisher=[[Longman]] |year=1990 |page=250 |location=[[Harlow]], England |isbn=0582053838}}</ref> {{IPAc-en|UK|ɛ|p|ˈ|s|aɪ|l|ə|n}};<ref>{{cite OED|epsilon}}</ref> uppercase '''{{lang|el|Ε}}''', lowercase '''{{lang|el|ε}}''' or  '''{{lang|el|ϵ}}'''; {{langx|el|έψιλον}}) is the fifth letter of the [[Greek alphabet]], corresponding phonetically to a [[mid<!-- not close-mid, see (1999) - Illustrations of the IPA: Modern Greek. --> front unrounded vowel]] {{IPA|el|e̞|IPA}} or {{IPA|el|ɛ̝|IPA}}. In the system of [[Greek numerals]] it also has the value five. It was derived from the [[Phoenician alphabet|Phoenician letter]] [[He (letter)|He]] <span style="background-color: white;">[[File:Phoenician he.svg|20px|He]]</span>. Letters that arose from epsilon include the Roman [[E]], [[Ë]] and [[Latin epsilon|Ɛ]], and Cyrillic [[Ye (Cyrillic)|Е]], [[Ye with grave|È]], [[Yo (Cyrillic)|Ё]], [[Ukrainian Ye|Є]] and [[E (Cyrillic)|Э]]. The name of the letter was originally {{lang|grc|εἶ}} ({{IPA|grc|êː|lang|link=yes}}), but it was later changed to {{wikt-lang|grc|ἒ ψιλόν}} ({{lang|Grc-Latn|e psilon}} 'simple e') in the Middle Ages to distinguish the letter from the [[digraph (orthography)|digraph]] {{lang|grc|αι}}, a former [[diphthong]] that had come to be pronounced the same as epsilon.

The [[uppercase]] form of epsilon is identical to Latin {{angbr|E}} but has its own [[code point]] in [[Unicode]]: {{unichar|0395|Greek capital letter epsilon}}. The [[lowercase]] version has two typographical variants, both inherited from [[history of the Greek alphabet|medieval Greek handwriting]]. One, the most common in modern typography and inherited from medieval [[minuscule]], looks like a reversed number "[[3]]" and is encoded {{unichar|03B5|Greek small letter epsilon}}. The other, also known as [[#Glyph variants|lunate]] or [[uncial script|uncial]] epsilon and inherited from earlier uncial writing,<ref name="nicholas1">Nick Nicholas: [http://www.tlg.uci.edu/~opoudjis/unicode/letters.html Letters] {{Webarchive|url=https://archive.today/20121215104041/http://www.tlg.uci.edu/~opoudjis/unicode/letters.html |date=2012-12-15 }}, 2003–2008. (''Greek Unicode Issues'')</ref><ref name="colwell">{{cite book|last=Colwell|first=Ernest C.|chapter=A chronology for the letters Ε, Η, Λ, Π in the Byzantine minuscule book hand|title=Studies in methodology in textual criticism of the New Testament|year=1969|place=Leiden|publisher=Brill|pages=127}}</ref> looks like a semicircle crossed by a horizontal bar: it  is encoded {{unichar|03F5|Greek lunate epsilon symbol}}. While in normal typography these are just alternative font variants, they may have different meanings as mathematical symbols: computer systems therefore offer distinct encodings for them.<ref name="nicholas1"/> In [[TeX]], <code>\epsilon</code> ( <math>\epsilon\!</math> ) denotes the lunate form, while <code>\varepsilon</code> ( <math>\varepsilon</math> ) denotes the [[epsilon number]]. Unicode versions 2.0.0 and onwards use {{char|ɛ}} as the lowercase Greek epsilon letter,<ref>{{cite book|chapter=Code Charts|chapter-url=https://www.unicode.org/versions/Unicode2.0.0/CodeCharts1.pdf|page=130|title=The Unicode Standard, Version 2.0|ISBN=0-201-48345-9}}</ref> but in version 1.0.0, {{char|ϵ}} was used.<ref>{{cite book|chapter=Code Charts|chapter-url=https://www.unicode.org/versions/Unicode1.0.0/CodeCharts1.pdf|page=130|title=The Unicode Standard, Version 1.0|volume=1|ISBN=0-201-56788-1}}</ref> The lunate or uncial epsilon provided inspiration for the [[euro sign]], {{char|€}}.<ref name="ec.europa.eu">{{cite web|url=https://ec.europa.eu/economy_finance/euro/cash/symbol/index_en.htm |title=European Commission – Economic and Financial Affairs – How to use the euro name and symbol |publisher=Ec.europa.eu |access-date=7 April 2010 |quote= Inspiration for the € symbol itself came from the Greek epsilon, {{char|ϵ}} – a reference to the cradle of European civilization – and the first letter of the word Europe, crossed by two parallel lines to 'certify' the stability of the euro.}}</ref>

There is also a '[[Latin epsilon]]', {{angbr|ɛ}} or "open e", which looks similar to the Greek lowercase epsilon. It is encoded in Unicode as {{unichar|025B |Latin small letter open e}} and {{unichar|0190|Latin capital letter open e}} and is used as an [[International Phonetic Alphabet|IPA]] phonetic symbol. This Latin uppercase epsilon, {{char|Ɛ}}, is not to be confused with the Greek uppercase {{char|Σ}} ([[sigma]])

The lunate epsilon, {{angbr|ϵ}}, is not to be confused with the [[set membership]] symbol {{char|∈}}.  The symbol <math>\in</math>, first used in set theory and logic by [[Giuseppe Peano]] and now used in mathematics in general for set membership ("belongs to"), evolved from the letter epsilon, since the symbol was originally used as an abbreviation for the Latin word {{lang|la|'''e'''st}}.  In addition, mathematicians often read the symbol {{char|∈}} as "element of", as in "1 is an element of the natural numbers" for <math>1\in\N</math>, for example.  As late as 1960, {{char|ɛ}} itself was used for set membership, while its negation "does not belong to" (now {{char|∉}}) was denoted by {{char|ε'}} (epsilon prime).<ref>{{Cite book|title=Naive Set Theory|last=Halmos|first=Paul R.|publisher=Van Nostrand|year=1960|isbn=978-1614271314|location=New York|pages=5–6}}</ref>  Only gradually did a fully separate, stylized symbol take the place of epsilon in this role.  In a related context, Peano also introduced the use of a backwards epsilon, {{char|϶}}, for the phrase "such that", although the abbreviation ''s.t.'' is occasionally used in place of {{char|϶}} in informal cardinals.

==History==

===Origin===
The letter {{angbr|Ε}} was adopted from the [[Phoenician alphabet|Phoenician]] letter [[He (letter)|He]] (<span style="background-color: white;">[[File:Phoenician he.svg|inline|x12px|alt=A letter that looks like a capital E with arms pointing left instead of right]]</span>) when Greeks first adopted alphabetic writing. In archaic Greek writing, its shape is often still identical to that of the Phoenician letter. Like other Greek letters, it could face either leftward or rightward (<span style="background-color: white;">[[File:Greek Epsilon left.svg|inline|x14px]][[File:Greek Epsilon archaic.svg|inline|x14px]]</span>), depending on the current writing direction, but, just as in Phoenician, the horizontal bars always faced in the direction of writing. Archaic writing often preserves the Phoenician form with a vertical stem extending slightly below the lowest horizontal bar. In the classical era, through the influence of more cursive writing styles, the shape was simplified to the current {{angbr|E}} glyph.<ref name="jeffery63">{{cite book |last=Jeffery |first=Lilian H. |title=The Local Scripts of Archaic Greece |publisher=Clarendon |year=1961 |place=Oxford |pages=63–64}}</ref>

===Sound value===
While the original pronunciation of the Phoenician letter ''He'' was {{IPA|[h]}}, the earliest Greek sound value of Ε was determined by the vowel occurring in the Phoenician letter name, which made it a natural choice for being reinterpreted from a consonant symbol to a vowel symbol denoting an {{IPA|[e]}} sound.<ref name="jeffery24">Jeffery, ''Local Scripts of Archaic Greece'', p. 24.</ref> Besides its classical Greek sound value, the short {{IPA|/e/}} phoneme, it could initially also be used for other {{IPA|[e]}}-like sounds. For instance, in early [[Attic Greek|Attic]] before {{Circa|500 BC}}, it was used also both for the long, [[open-mid vowel|open]] {{IPA|/ɛː/}}, and for the long [[close-mid vowel|close]] {{IPA|/eː/}}. In the former role, it was later replaced in the classic Greek alphabet by [[Eta]] ({{angbr|Η}}), which was taken over from eastern [[Ionic Greek|Ionic]] alphabets, while in the latter role it was replaced by the [[digraph (orthography)|digraph]] ⟨ΕΙ⟩.

===[[Archaic Greek alphabets#epichoric|Epichoric]] alphabets===
Some dialects used yet other ways of distinguishing between various e-like sounds.

In [[Ancient Corinth|Corinth]], the normal function of {{angbr|Ε}} to denote {{IPA|/e/}} and {{IPA|/ɛː/}} was taken by a glyph resembling a pointed B (<span style="background-color: white;">[[File:Greek Beta archaic.svg|inline|x14px]]</span>), while {{angbr|Ε}} was used only for long close {{IPA|/eː/}}.<ref name="jeffery114">Jeffery, ''Local Scripts of Archaic Greece'', p. 114.</ref> The letter [[Beta]], in turn, took the deviant shape <span style="background-color: white;">[[File:Greek Beta Corinth 1.svg|inline|x14px]]</span>.

In [[Sicyon]], a variant glyph resembling an {{angbr|X}} (<span style="background-color: white;">[[File:Greek Epsilon X-shaped.svg|inline|x14px]]</span>) was used in the same function as Corinthian <span style="background-color: white;">[[File:Greek Beta archaic.svg|inline|x14px]]</span>.<ref name="jeffery138">Jeffery, ''Local Scripts of Archaic Greece'', p. 138.</ref>

In [[Thespiae|Thespiai]] ([[Boeotia]]), a special letter form consisting of a vertical stem with a single rightward-pointing horizontal bar (<span style="background-color: white;">[[File:Greek Eta tack.svg|inline|x12px]]</span>) was used for what was probably a [[close vowel|raised]] variant of {{IPA|/e/}} in pre-vocalic environments.<ref name="nicholas2">{{cite web |last=Nicholas |first=Nick |title=Proposal to add Greek epigraphical letters to the UCS |url=http://www.tlg.uci.edu/~opoudjis/unicode/epigraphical.pdf |archive-url=https://web.archive.org/web/20060217000025/http://www.tlg.uci.edu/~opoudjis/unicode/epigraphical.pdf |url-status=dead |archive-date=February 17, 2006 |access-date=2010-08-12 |year=2005 }}</ref><ref name="jeffery89">Jeffery, ''Local Scripts of Archaic Greece'', p. 89.</ref> This tack glyph was used elsewhere also as a form of "[[Heta]]", i.e. for the sound {{IPA|/h/}}.

{{anchor|ϵ|Lunate epsilon|lunate epsilon}}<!--linked from [[Romanization of Greek]], [[ϵ]], [[Lunate epsilon]], &c.-->

===Glyph variants===
After the establishment of the canonical Ionian (Euclidean) [[Greek alphabet]], new glyph variants for Ε were introduced through handwriting. In the [[uncial script]] (used for literary [[papyrus]] manuscripts in late antiquity and then in early medieval [[vellum]] codices), the "[[lunate]]" shape (<span style="background-color: white;">[[File:Greek uncial Epsilon.svg|inline|x12px]]</span>) became predominant. In [[cursive]] handwriting, a large number of shorthand glyphs came to be used, where the cross-bar and the curved stroke were linked in various ways.<ref name="thompson">{{cite book |last=Thompson |first=Edward M. |title=An Introduction to Greek and Latin palaeography |publisher=Clarendon |year=1911 |place=Oxford |pages=191–194}}</ref> Some of them resembled a modern lowercase Latin "e", some a "6" with a connecting stroke to the next letter starting from the middle, and some a combination of two small "c"-like curves. Several of these shapes were later taken over into [[minuscule Greek|minuscule]] book hand. Of the various minuscule letter shapes, the inverted-3 form became the basis for lower-case Epsilon in Greek typography during the modern era.

{|class="wikitable"
!Uncial
!Uncial variants
!Cursive variants
!Minuscule
!Minuscule with ligatures
|-
|style="background-color: white !important;"|[[File:Greek uncial Epsilon.svg|inline|x20px]]
|style="background-color: white !important;"|[[File:Greek uncial variants Epsilon.svg|inline|x30px]]
|style="background-color: white !important;"|[[File:Greek cursive variants Epsilon.svg|inline|x30px]]
|style="background-color: white !important;"|[[File:Greek minuscule Epsilon.svg|inline|x40px]]
|style="background-color: white !important;"|[[File:Greek minuscule Epsilon with ligatures.svg|inline|x40px]]
|}

==Uses==

===International Phonetic Alphabet===
Despite its pronunciation as [[mid<!-- not close-mid, see Arvanti (1999) - Illustrations of the IPA: Modern Greek. --> front unrounded vowel|mid]], in the [[International Phonetic Alphabet]], the Latin epsilon {{IPAc-en|ɛ}} represents [[open-mid front unrounded vowel]], as in the English word ''pet'' {{IPAc-en|p|ɛ|t}}.

===Symbol===
The uppercase Epsilon is not commonly used outside of the Greek language because of its similarity to the [[Latin alphabet|Latin]] letter [[E]]. However, it is commonly used in [[structural mechanics]] with [[Young's Modulus]] equations for calculating tensile, compressive and areal [[Deformation (mechanics)|strain]].

The Greek lowercase epsilon {{code|ε}}, the lunate epsilon symbol {{code|ϵ}}, and the [[Latin epsilon|Latin lowercase epsilon]] {{code|ɛ}} (see above) are used in a variety of places:

* In [[engineering mechanics]], strain calculations ϵ = increase of length / original length. Usually this relates to extensometer testing of metallic materials.
* In [[mathematics]] 
** (In early [[calculus]] or [[nonstandard analysis]]) An infinitesimally small positive quantity is commonly denoted ε.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Epsilon |url=https://mathworld.wolfram.com/Epsilon.html |access-date=2025-01-30 |website=mathworld.wolfram.com |language=en |quote=In mathematics, a small positive infinitesimal quantity, usually denoted {{lang|el|ε}} or {{lang|el|ϵ}}, whose limit is usually taken as {{lang|el|ϵ}}->0.}}</ref> 
*** (In [[Mathematical analysis|analysis]]) By extension, a quantity thought of as "small", "negligible", or, especially, "arbitrarily small", is often denoted ε. For instance, quantities subject to a [[Limit (mathematics)|limit]] which takes them towards zero are often denoted ε; see [[(ε, δ)-definition of limit]].<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Limit |url=https://mathworld.wolfram.com/Limit.html |access-date=2025-01-30 |website=mathworld.wolfram.com |language=en}}</ref>
** [[David Hilbert|Hilbert]] introduced epsilon terms <math>\epsilon x.\phi</math> as an extension to [[first-order logic]]; see [[epsilon calculus]].
** it is used to represent the [[Levi-Civita symbol]].
** it is used to represent [[dual number]]s: <math>a+b \varepsilon</math>, with <math>\varepsilon^{2}=0</math> and <math>\varepsilon \neq 0</math>.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Dual Number |url=https://mathworld.wolfram.com/DualNumber.html |access-date=2025-01-30 |website=mathworld.wolfram.com |language=en}}</ref>
** it is sometimes used to denote the [[Heaviside step function]].<ref>{{Cite web|url=http://mathworld.wolfram.com/DeltaFunction.html|title=Delta Function|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com |access-date=2019-02-19}}</ref>
** in [[set theory]], the [[epsilon numbers (mathematics)|epsilon numbers]] are [[ordinal number]]s that satisfy the [[Fixed point (mathematics)|fixed point]] ε = ω<sup>ε</sup>. The first epsilon number, ε<sub>0</sub>, is the [[limit ordinal]] of the set {ω, ω<sup>ω</sup>, ω<sup>ω<sup>ω</sup></sup>, ...}.
** in numerical analysis and statistics it is used as the [[error term]]
** in [[group theory]] it is used as the [[idempotent]] group when e is in use as a variable name
* In [[computer science]]
** it often represents the [[empty string]], though different writers use a variety of other symbols for the empty string as well; usually the lower-case Greek letter [[lambda]] (λ).
** the [[machine epsilon]] indicates the upper bound on the relative error due to rounding in floating point arithmetic.<ref>{{Cite book |last=Überhuber |first=Christoph W. |title=Numerical Computation 1: Methods, Software, and Analysis |date=1997 |publisher=Springer |isbn=978-3-540-62058-7 |series=SpringerLink Bücher |location=Berlin, Heidelberg |pages=140 |quote=''eps'' frequently denotes his upper bound on the relative rounding error and is referred to as the ''machine epsilon''.}}</ref>
* In [[physics]], 
** it indicates the [[permittivity]] of a medium; with the subscript 0 (ε<sub>0</sub>) it is the [[permittivity of free space]].<ref>{{Cite web |title=Vacuum electric permittivity |url=https://physics.nist.gov/cgi-bin/cuu/Value?ep0%7Csearch_for=universal_in! |access-date=2025-02-10 |website=physics.nist.gov}}</ref>
** it can also indicate the [[Strain (materials science)|strain]] of a material (a ratio of extensions).<ref>{{Citation |last=Elert |first=Glenn |title=Special Symbols |date=2023 |work=The Physics Hypertextbook|quote= ε linear strain |url=https://physics.info/symbols/ |access-date=2025-02-01 |publisher=hypertextbook |language=en}}</ref>
** in [[quantum field theory]], it usually indicates the [[dimensional regularization]] parameter.<ref>{{cite book |last1=Peskin |first1=Michael E. |last2=Schroeder |first2=Daniel V. |title=An Introduction To Quantum Field Theory |date=4 May 2018 |publisher=CRC Press |isbn=978-0-429-97210-2 |edition=2nd |url=https://books.google.com/books?id=HUpaDwAAQBAJ |language=en}}</ref>
* In [[automata theory]], it shows a transition that involves no shifting of an input symbol.
* In [[astronomy]], 
** it stands for the fifth-brightest star in a [[constellation]] (see [[Bayer designation]]).
** Epsilon is the name for the most distant and most visible ring of [[Uranus]].
** In [[planetary science]], ε denotes the [[axial tilt]].<ref>{{Cite book |title=Cyclostratigraphy and astrochronology |date=2018 |publisher=Academic Press, an imprint of Elsevier |isbn=978-0-12-815098-6 |editor-last=Montenari |editor-first=Michael |edition=1st |series=Stratigraphy and Timescales |location=London San Diego, Calif. Cambridge, Mass. Oxford |pages=84 |quote=The Earth's orbital obliquity or axial tilt (ε) is the angle between the Earth's equatorial plane and its orbital plane,}}</ref>
* In [[chemistry]], it represents the [[molar extinction coefficient]] of a [[chromophore]].
* In [[economics]], ε refers to [[Elasticity (economics)|elasticity]].<ref>{{Cite book |last=Free |first=Rhona C. |title=21st century economics: a reference handbook |date=2010 |publisher=Sage |isbn=978-1-4129-6142-4 |location=Thousand Oaks (Calif.) |pages=93-94}}</ref>
* In [[statistics]], 
** it is used to refer to [[Errors and residuals in statistics|error terms]].
** it also can to refer to the degree of [[Mauchly's sphericity test|sphericity]] in [[Repeated measures design|repeated measures ANOVAs]].
* In [[agronomy]], it is used to represent the "[[photosynthetic]] efficiency" of a particular plant or crop.

==Unicode==

For accented Greek characters, see [[Greek diacritics#Computer encoding|Greek diacritics: Computer encoding]].

* {{unichar|0190|nlink=}}
* {{unichar|025B|html=}}
* {{unichar|025C|html=}}
* {{unichar|025D|html=}}
* {{unichar|025E|html=}}
* {{unichar|029A|html=}}
* {{unichar|0388|html=}}
* {{unichar|0395|html=}}
* {{unichar|03AD|html=}}
* {{unichar|03B5|html=}} ({{tt|\varepsilon}} in TeX)
* {{unichar|03F5|html=}} ({{tt|\epsilon}} in TeX)
* {{unichar|03F6|html=}}
* {{unichar|1D08|html=}}
* {{unichar|1D4B|html=}}
* {{unichar|1D4C|html=}}
* {{unichar|1D93|html=}}
* {{unichar|1D94|html=}}
* {{unichar|1D9F|html=}}
* {{unichar|2377|html=}}
* {{unichar|2C88|html=}}
* {{unichar|2C89|html=}}
* {{unichar|2CB6|html=}}
* {{unichar|2CB7|html=}}
* {{unichar|A7AB|html=}}
* {{unichar|1078F|html=}}
* {{unichar|1D6AC|html=}}{{efn|The {{sc|mathematical}} symbols are used only in math. Stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style.}}
* {{unichar|1D6C6|html=}}
* {{unichar|1D6DC|html=}}
* {{unichar|1D6E6|html=}}
* {{unichar|1D700|html=}}
* {{unichar|1D716|html=}}
* {{unichar|1D720|html=}}
* {{unichar|1D73A|html=}}
* {{unichar|1D750|html=}}
* {{unichar|1D75A|html=}}
* {{unichar|1D774|html=}}
* {{unichar|1D78A|html=}}
* {{unichar|1D794|html=}}
* {{unichar|1D7AE|html=}}
* {{unichar|1D7C4|html=}}
{{notelist}}

==Initial==
<gallery>
File:Lectionary 226 GA 0020b Epsilon.JPG|[[Initial]] epsilon in [[Lectionary 226]], folio 20 [[recto and verso|verso]]
File:Lectionary 226 (GA) Epsilon 0064b.JPG|folio 64 verso
File:Lectionary 226 GA 0125b Epsilon.JPG|folio 125 verso
</gallery>

==See also==
* [[Е]] and [[е]], the letter Ye  of the Cyrillic alphabet
* Є є, [[Ukrainian Ye]]
* Ԑ ԑ, [[Reversed Ze]]
* [[E (disambiguation)]]

==References==
{{reflist}}

== Further reading ==
{{Wiktionary|Ε|ɛ}}
*{{cite book|author-link=Paul Hoffman (science writer)|author-last1=Hoffman|author-first1=Paul|title=[[The Man Who Loved Only Numbers]]|publisher=Hyperion|year=1998|ISBN=0-7868-6362-5}}

[[Category:Greek letters]]
[[Category:Vowel letters]]