Epsilon (uppercase Ε, lowercase ε or lunate ϵ; Greek: έψιλον) is the fifth letter of the Greek alphabet, corresponding phonetically to a mid front unrounded vowel /e/. In the system of Greek numerals it also has the value five. It was derived from the Phoenician letter He . Letters that arose from epsilon include the Roman E, Ë and Ɛ, and Cyrillic Е, È, Ё, Є and Э.
The name of the letter was originally εἶ (Ancient Greek: [êː]), but the name was changed to ἒ ψιλόν (e psilon "simple e") in the Middle Ages to distinguish the letter from the digraph αι, a former diphthong that had come to be pronounced the same as epsilon.
In essence, the uppercase form of epsilon looks identical to Latin E. The lowercase version has two typographical variants, both inherited from medieval Greek handwriting. One, the most common in modern typography and inherited from medieval minuscule, looks like a reversed "3". The other, also known as lunate or uncial epsilon and inherited from earlier uncial writing,^{[1]}^{[2]} looks like a semicircle crossed by a horizontal bar. 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.^{[1]} In Unicode, the character U+03F5 "Greek lunate epsilon symbol" (ϵ) is provided specifically for the lunate form. In TeX, \epsilon
($\epsilon \!$) denotes the lunate form, while \varepsilon
($\varepsilon \!$) denotes the reversed3 form.
There is also a Latin epsilon or "open e", which looks similar to the Greek lowercase epsilon. It is encoded in Unicode as U+025B ("Latin smallletter open e", ɛ) and U+0190 ("Latin capitalletter open e", Ɛ) and is used as an IPA phonetic symbol. The lunate or uncial epsilon has also provided inspiration for the euro sign (€).
The lunate epsilon (ϵ) is not to be confused with the set membership symbol (∈); nor should the Latin uppercase epsilon (Ɛ) be confused with the Greek uppercase sigma (Σ). The symbol $\in$, first used in set theory and logic by Giuseppe Peano and now used in mathematics in general for set membership ("belongs to") did, however, evolve from the letter epsilon, since the symbol was originally used as an abbreviation for the Latin word "est". In addition, mathematicians often read the symbol $\in$ as "element of", as in "1 is an element of the natural numbers" for $1\in \mathbb {N}$, for example. As late as 1960, $\epsilon$ itself was used for set membership, while its negation "does not belong to" (now $\notin$) was denoted by $\epsilon '$ (epsilon prime).^{[3]} 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, $\backepsilon$, for the phrase "such that", although the abbreviation "s.t." is occasionally used in place of $\backepsilon$ in informal cardinals
History[edit]
The letter Ε was taken over from the Phoenician letter He () 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 (), 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 E glyph.^{[4]}
Sound value[edit]
While the original pronunciation of the Phoenician letter He was [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 [e] sound.^{[5]} Besides its classical Greek sound value, the short /e/ phoneme, it could initially also be used for other [e]like sounds. For instance, in early Attic before c.500 B.C., it was used also both for the long, open /ɛː/, and for the long close /eː/. In the former role, it was later replaced in the classic Greek alphabet by Eta (Η), which was taken over from eastern Ionic alphabets, while in the latter role it was replaced by the digraph spelling ΕΙ.
Epichoric alphabets[edit]
Some dialects used yet other ways of distinguishing between various elike sounds.
In Corinth, the normal function of Ε to denote /e/ and /ɛː/ was taken by a glyph resembling a pointed B (), while Ε was used only for long close /eː/.^{[6]} The letter Beta, in turn, took the deviant shape .
In Sicyon, a variant glyph resembling an X () was used in the same function as Corinthian .^{[7]}
In Thespiai (Boeotia), a special letter form consisting of a vertical stem with a single rightwardpointing horizontal bar () was used for what was probably a raised variant of /e/ in prevocalic environments.^{[8]}^{[9]} This tack glyph was used elsewhere also as a form of "Heta", i.e. for the sound /h/.
Glyph variants[edit]
After the establishment of the canonical classic Greek alphabet^{[clarify]}, 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 () became predominant. In cursive handwriting, a large number of shorthand glyphs came to be used, where the crossbar and the curved stroke were linked in various ways.^{[10]} 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 book hand. Of the various minuscule letter shapes, the inverted3 form became the basis for lowercase Epsilon in Greek typography during the modern era.
Uncial

Uncial variants

Cursive variants

Minuscule

Minuscule with ligatures






International Phonetic Alphabet[edit]
Despite its pronunciation as mid, in the International Phonetic Alphabet, the Latin epsilon represents openmid front unrounded vowel, as in the English word pet .
The uppercase Epsilon is not commonly used outside of the Greek language because of its similarity to the Latin letter E however, it is commonly used in structural mechanics with Young's Modulus equations for calculating tensile, compressive and Areal strain.
The Greek lowercase epsilon ε, the lunate epsilon symbol ϵ, or the Latin lowercase epsilon ɛ (see above) is 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 (particularly calculus), an arbitrarily small positive quantity is commonly denoted ε; see (ε, δ)definition of limit.
 In reference to this, the late mathematician Paul Erdős also used the term "epsilons" to refer to children (Hoffman 1998, p. 4).
 In mathematics, Hilbert introduced epsilon terms $\epsilon x.\phi$ as an extension to first order logic; see epsilon calculus.
 In mathematics, it is used to represent the LeviCivita symbol.
 In mathematics, it is used to represent dual numbers: a + bε, with ε^{2} = 0 and ε ≠ 0.
 In mathematics, it is sometimes used to denote the Heaviside step function.^{[11]}
 In set theory, the epsilon numbers are ordinal numbers that satisfy the fixed point ε = ω^{ε}. The first epsilon number, ε_{0}, is the limit ordinal of the set {ω, ω^{ω}, ω^{ωω}, ...}.
 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 lowercase Greek letter lambda (λ).
 In computer science, the machine epsilon indicates the upper bound on the relative error due to rounding in floating point arithmetic.
 In physics, it indicates the permittivity of a medium; with the subscript 0 (ϵ_{0}) it is the permittivity of free space.
 In physics, it can also indicate the strain of a material (a ratio of extensions).
 In automata theory, it shows a transition that involves no shifting of an input symbol.
 In astronomy, it stands for the fifthbrightest star in a constellation (see Bayer designation).
 In astronomy, Epsilon is the name for Uranus' most distant and most visible ring.
 In planetary science, ε denotes the axial tilt.
 In chemistry, it represents the molar extinction coefficient of a chromophore.
 In economics, ε refers to elasticity.
 In statistics, it is used to refer to error terms.
 In statistics, it also can to refer to the degree of sphericity in repeated measures ANOVAs.
 In agronomy, it is used to represent the "photosynthetic efficiency" of a particular plant or crop.
Unicode[edit]
Character 
Ε 
ε 
ϵ 
϶

Unicode name 
GREEK CAPITAL LETTER EPSILON 
GREEK SMALL LETTER EPSILON 
GREEK LUNATE EPSILON SYMBOL 
GREEK REVERSED LUNATE EPSILON SYMBOL

Encodings 
decimal 
hex 
decimal 
hex 
decimal 
hex 
decimal 
hex

Unicode 
917 
U+0395 
949 
U+03B5 
1013 
U+03F5 
1014 
U+03F6

UTF8 
206 149 
CE 95 
206 181 
CE B5 
207 181 
CF B5 
207 182 
CF B6

Numeric character reference 
Ε 
Ε 
ε 
ε 
ϵ 
ϵ 
϶ 
϶

Named character reference 
Ε 
ε 


DOS Greek 
132 
84 
156 
9C 




DOS Greek2 
168 
A8 
222 
DE 




Windows 1253 
197 
C5 
229 
E5 




TeX 

\varepsilon 
\epsilon 

Character 
Ⲉ 
ⲉ

Unicode name 
COPTIC CAPITAL LETTER EIE 
COPTIC SMALL LETTER EIE

Encodings 
decimal 
hex 
decimal 
hex

Unicode 
11400 
U+2C88 
11401 
U+2C89

UTF8 
226 178 136 
E2 B2 88 
226 178 137 
E2 B2 89

Numeric character reference 
Ⲉ 
Ⲉ 
ⲉ 
ⲉ

Character 
Ɛ 
ɛ 
ᶓ 
ᵋ

Unicode name 
LATIN CAPITAL LETTER OPEN E 
LATIN SMALL LETTER OPEN E 
LATIN SMALL LETTER OPEN E WITH RETROFLEX HOOK 
MODIFIER LETTER SMALL OPEN E

Encodings 
decimal 
hex 
decimal 
hex 
decimal 
hex 
decimal 
hex

Unicode 
400 
U+0190 
603 
U+025B 
7571 
U+1D93 
7499 
U+1D4B

UTF8 
198 144 
C6 90 
201 155 
C9 9B 
225 182 147 
E1 B6 93 
225 181 139 
E1 B5 8B

Numeric character reference 
Ɛ 
Ɛ 
ɛ 
ɛ 
ᶓ 
ᶓ 
ᵋ 
ᵋ

Character 
ɜ 
ɝ 
ᶔ 
ᶟ

Unicode name 
LATIN SMALL LETTER REVERSED OPEN E 
LATIN SMALL LETTER REVERSED OPEN E WITH HOOK 
LATIN SMALL LETTER REVERSED OPEN E WITH RETROFLEX HOOK 
MODIFIER LETTER SMALL REVERSED OPEN E

Encodings 
decimal 
hex 
decimal 
hex 
decimal 
hex 
decimal 
hex

Unicode 
604 
U+025C 
605 
U+025D 
7572 
U+1D94 
7583 
U+1D9F

UTF8 
201 156 
C9 9C 
201 157 
C9 9D 
225 182 148 
E1 B6 94 
225 182 159 
E1 B6 9F

Numeric character reference 
ɜ 
ɜ 
ɝ 
ɝ 
ᶔ 
ᶔ 
ᶟ 
ᶟ

Character 
ᴈ 
ᵌ 
ʚ 
ɞ

Unicode name 
LATIN SMALL LETTER TURNED OPEN E 
MODIFIER LETTER SMALL TURNED OPEN E 
LATIN SMALL LETTER CLOSED OPEN E 
LATIN SMALL LETTER CLOSED REVERSED OPEN E

Encodings 
decimal 
hex 
decimal 
hex 
decimal 
hex 
decimal 
hex

Unicode 
7432 
U+1D08 
7500 
U+1D4C 
666 
U+029A 
606 
U+025E

UTF8 
225 180 136 
E1 B4 88 
225 181 140 
E1 B5 8C 
202 154 
CA 9A 
201 158 
C9 9E

Numeric character reference 
ᴈ 
ᴈ 
ᵌ 
ᵌ 
ʚ 
ʚ 
ɞ 
ɞ

Character 
𝚬 
𝛆 
𝛦 
𝜀 
𝜠 
𝜺

Unicode name 
MATHEMATICAL BOLD CAPITAL EPSILON 
MATHEMATICAL BOLD SMALL EPSILON 
MATHEMATICAL ITALIC CAPITAL EPSILON 
MATHEMATICAL ITALIC SMALL EPSILON 
MATHEMATICAL BOLD ITALIC CAPITAL EPSILON 
MATHEMATICAL BOLD ITALIC SMALL EPSILON

Encodings 
decimal 
hex 
decimal 
hex 
decimal 
hex 
decimal 
hex 
decimal 
hex 
decimal 
hex

Unicode 
120492 
U+1D6AC 
120518 
U+1D6C6 
120550 
U+1D6E6 
120576 
U+1D700 
120608 
U+1D720 
120634 
U+1D73A

UTF8 
240 157 154 172 
F0 9D 9A AC 
240 157 155 134 
F0 9D 9B 86 
240 157 155 166 
F0 9D 9B A6 
240 157 156 128 
F0 9D 9C 80 
240 157 156 160 
F0 9D 9C A0 
240 157 156 186 
F0 9D 9C BA

UTF16 
55349 57004 
D835 DEAC 
55349 57030 
D835 DEC6 
55349 57062 
D835 DEE6 
55349 57088 
D835 DF00 
55349 57120 
D835 DF20 
55349 57146 
D835 DF3A

Numeric character reference 
𝚬 
𝚬 
𝛆 
𝛆 
𝛦 
𝛦 
𝜀 
𝜀 
𝜠 
𝜠 
𝜺 
𝜺

Character 
𝝚 
𝝴 
𝞔 
𝞮

Unicode name 
MATHEMATICAL SANSSERIF BOLD CAPITAL EPSILON 
MATHEMATICAL SANSSERIF BOLD SMALL EPSILON 
MATHEMATICAL SANSSERIF BOLD ITALIC CAPITAL EPSILON 
MATHEMATICAL SANSSERIF BOLD ITALIC SMALL EPSILON

Encodings 
decimal 
hex 
decimal 
hex 
decimal 
hex 
decimal 
hex

Unicode 
120666 
U+1D75A 
120692 
U+1D774 
120724 
U+1D794 
120750 
U+1D7AE

UTF8 
240 157 157 154 
F0 9D 9D 9A 
240 157 157 180 
F0 9D 9D B4 
240 157 158 148 
F0 9D 9E 94 
240 157 158 174 
F0 9D 9E AE

UTF16 
55349 57178 
D835 DF5A 
55349 57204 
D835 DF74 
55349 57236 
D835 DF94 
55349 57262 
D835 DFAE

Numeric character reference 
𝝚 
𝝚 
𝝴 
𝝴 
𝞔 
𝞔 
𝞮 
𝞮

These characters are used only as mathematical symbols. Stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style.
Initial[edit]
References[edit]
 ^ ^{a} ^{b} Nick Nicholas: Letters, 2003–2008. (Greek Unicode Issues)
 ^ Colwell, Ernest C. (1969). "A chronology for the letters Ε, Η, Λ, Π in the Byzantine minuscule book hand". Studies in methodology in textual criticism of the New Testament. Leiden: Brill. p. 127.
 ^ Halmos, Paul R. (1960). Naive Set Theory. New York: Van Nostrand. pp. 5–6. ISBN 9781614271314.
 ^ Jeffery, Lilian H. (1961). The local scripts of archaic Greece. Oxford: Clarendon. pp. 63–64.
 ^ Jeffery, Local scripts, p.24.
 ^ Jeffery, Local scripts, p.114.
 ^ Jeffery, Local scripts, p.138.
 ^ Nicholas, Nick (2005). "Proposal to add Greek epigraphical letters to the UCS" (PDF). Archived from the original (PDF) on 20060505. Retrieved 20100812.
 ^ Jeffery, Local scripts, p.89.
 ^ Thompson, Edward M. (1911). An introduction to Greek and Latin palaeography. Oxford: Clarendon. pp. 191–194.
 ^ Weisstein, Eric W. "Delta Function". mathworld.wolfram.com. Retrieved 20190219.
Further reading[edit]

Look up Ε or ɛ in Wiktionary, the free dictionary. 