Epsilon
Greek alphabet  



History  
Use in other languages  
Related topics  
Epsilon (/ˈɛpsɪlɒn/,^{[1]} UK also /ɛpˈsaɪlən/;^{[2]} 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 IPA: [e̞] or IPA: [ɛ̝]. 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 it was later 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.
The uppercase form of epsilon is identical to Latin E but has its own code point in Unicode: U+0395 Ε GREEK CAPITAL LETTER EPSILON. 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 number "3" and is encoded U+03B5 ε GREEK SMALL LETTER EPSILON. The other, also known as lunate or uncial epsilon and inherited from earlier uncial writing,^{[3]}^{[4]} looks like a semicircle crossed by a horizontal bar: it is encoded U+03F5 ϵ 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.^{[3]} In TeX, \epsilon
( ) denotes the lunate form, while \varepsilon
( ) denotes the reversed3 form.
There is also a 'Latin epsilon', ɛ or "open e", which looks similar to the Greek lowercase epsilon. It is encoded in Unicode as U+025B ɛ LATIN SMALL LETTER OPEN E and U+0190 Ɛ LATIN CAPITAL LETTER OPEN E and is used as an IPA phonetic symbol. The lunate or uncial epsilon provided inspiration for the euro sign, €.^{[5]}
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 Σ (sigma). The symbol , 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 est. In addition, mathematicians often read the symbol ∈ as "element of", as in "1 is an element of the natural numbers" for , for example. As late as 1960, ε itself was used for set membership, while its negation "does not belong to" (now ∉) was denoted by ε' (epsilon prime).^{[6]} 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, ϶, for the phrase "such that", although the abbreviation s.t. is occasionally used in place of ϶ in informal cardinals.
History[edit]
Origin[edit]
The letter Ε was adopted 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.^{[7]}
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.^{[8]} 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 BC, 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/.^{[9]} 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 .^{[10]}
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.^{[11]}^{[12]} 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 classical 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 () 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.^{[13]} 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 

Uses[edit]
International Phonetic Alphabet[edit]
Despite its pronunciation as mid, in the International Phonetic Alphabet, the Latin epsilon /ɛ/ represents openmid front unrounded vowel, as in the English word pet /pɛt/.
Symbol[edit]
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
, and the Latin lowercase epsilon 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
 (particularly calculus), an infinitesimally small positive quantity is commonly denoted ε; see (ε, δ)definition of limit.
 Hilbert introduced epsilon terms as an extension to firstorder logic; see epsilon calculus.
 it is used to represent the LeviCivita symbol.
 it is used to represent dual numbers: , with and .
 it is sometimes used to denote the Heaviside step function.^{[14]}
 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 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 lowercase Greek letter lambda (λ).
 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.
 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).
 Epsilon is the name for the most distant and most visible ring of Uranus.
 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.
 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]
 Greek Epsilon
Preview  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  dec  hex  dec  hex  dec  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, ε  ϵ, ϵ, ϵ  ϶, ϶  
DOS Greek  132  84  156  9C  
DOS Greek2  168  A8  222  DE  
Windows 1253  197  C5  229  E5  
TeX  \varepsilon  \epsilon 
 Coptic Eie
Preview  Ⲉ  ⲉ  

Unicode name  COPTIC CAPITAL LETTER EIE  COPTIC SMALL LETTER EIE  
Encodings  decimal  hex  dec  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 
Preview  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  dec  hex  dec  hex  dec  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 
Preview  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  dec  hex  dec  hex  dec  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 
Preview  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  dec  hex  dec  hex  dec  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 
 Mathematical Epsilon
Preview  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  dec  hex  dec  hex  dec  hex  dec  hex  dec  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 
Preview  𝛜  𝜖  𝝐  

Unicode name  MATHEMATICAL BOLD EPSILON SYMBOL 
MATHEMATICAL ITALIC EPSILON SYMBOL 
MATHEMATICAL BOLD ITALIC EPSILON SYMBOL  
Encodings  decimal  hex  dec  hex  dec  hex 
Unicode  120540  U+1D6DC  120598  U+1D716  120656  U+1D750 
UTF8  240 157 155 156  F0 9D 9B 9C  240 157 156 150  F0 9D 9C 96  240 157 157 144  F0 9D 9D 90 
UTF16  55349 57052  D835 DEDC  55349 57110  D835 DF16  55349 57168  D835 DF50 
Numeric character reference  𝛜 
𝛜 
𝜖 
𝜖 
𝝐 
𝝐 
Preview  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  dec  hex  dec  hex  dec  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 
Preview  𝞊  𝟄  

Unicode name  MATHEMATICAL SANSSERIF BOLD EPSILON SYMBOL 
MATHEMATICAL SANSSERIF BOLD ITALIC EPSILON SYMBOL  
Encodings  decimal  hex  dec  hex 
Unicode  120714  U+1D78A  120772  U+1D7C4 
UTF8  240 157 158 138  F0 9D 9E 8A  240 157 159 132  F0 9D 9F 84 
UTF16  55349 57226  D835 DF8A  55349 57284  D835 DFC4 
Numeric character reference  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]
Initial epsilon in Lectionary 226, folio 20 verso
References[edit]
 ^ Wells, John C. (1990). "epsilon". Longman Pronunciation Dictionary. Harlow, England: Longman. p. 250. ISBN 0582053838.
 ^ "epsilon". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)
 ^ ^{a} ^{b} Nick Nicholas: Letters Archived 20121215 at archive.today, 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.
 ^ "European Commission – Economic and Financial Affairs – How to use the euro name and symbol". Ec.europa.eu. Retrieved 7 April 2010.
Inspiration for the € symbol itself came from the Greek epsilon, 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.
 ^ 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]
 Hoffman, Paul; The Man Who Loved Only Numbers. Hyperion, 1998. ISBN 0786863625.