Epsilon

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This article is about the Greek letter. For other uses, see Epsilon (disambiguation).
Epsilon uc lc.svg
Greek alphabet
Αα Alpha Νν Nu
Ββ Beta Ξξ Xi
Γγ Gamma Οο Omicron
Δδ Delta Ππ Pi
Εε Epsilon Ρρ Rho
Ζζ Zeta Σσς Sigma
Ηη Eta Ττ Tau
Θθ Theta Υυ Upsilon
Ιι Iota Φφ Phi
Κκ Kappa Χχ Chi
Λλ Lambda Ψψ Psi
Μμ Mu Ωω Omega
History
Use in other languages
Related topics

Epsilon (UK: /ɛpˈsaɪlən/,[1] US: /ˈɛpsɪlɒn/; 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 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.

The uppercase form of epsilon looks 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,[2][3] 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.[2] In TeX, \epsilon ( ) denotes the lunate form, while \varepsilon ( ) denotes the reversed-3 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, €.[4]

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).[5] 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 taken over from the Phoenician letter He (inline) 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 (inlineinline), 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.[6]

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.[7] 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 e-like sounds.

In Corinth, the normal function of Ε to denote /e/ and /ɛː/ was taken by a glyph resembling a pointed B (inline), while Ε was used only for long close /eː/.[8] The letter Beta, in turn, took the deviant shape inline.

In Sicyon, a variant glyph resembling an X (inline) was used in the same function as Corinthian inline.[9]

In Thespiai (Boeotia), a special letter form consisting of a vertical stem with a single rightward-pointing horizontal bar (inline) was used for what was probably a raised variant of /e/ in pre-vocalic environments.[10][11] 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 (Eucleidean) 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 (inline) 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.[12] 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 inverted-3 form became the basis for lower-case Epsilon in Greek typography during the modern era.

Uncial Uncial variants Cursive variants Minuscule Minuscule with ligatures
inline inline inline inline inline

Uses[edit]

International Phonetic Alphabet[edit]

Despite its pronunciation as mid, in the International Phonetic Alphabet, the Latin epsilon /ɛ/ represents open-mid 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, or the Latin lowercase epsilon ɛ (see above) is used in a variety of places:

Unicode[edit]

  • Greek Epsilon
Character information
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 decimal hex decimal hex decimal hex
Unicode 917 U+0395 949 U+03B5 1013 U+03F5 1014 U+03F6
UTF-8 206 149 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 Greek-2 168 A8 222 DE
Windows 1253 197 C5 229 E5
TeX \varepsilon \epsilon
  • Coptic Eie
Character information
Preview Ⲉ ⲉ
Unicode name COPTIC CAPITAL LETTER EIE COPTIC SMALL LETTER EIE
Encodings decimal hex decimal hex
Unicode 11400 U+2C88 11401 U+2C89
UTF-8 226 178 136 E2 B2 88 226 178 137 E2 B2 89
Numeric character reference Ⲉ Ⲉ ⲉ ⲉ
  • Latin Open E
Character information
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 decimal hex decimal hex decimal hex
Unicode 400 U+0190 603 U+025B 7571 U+1D93 7499 U+1D4B
UTF-8 198 144 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 information
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 decimal hex decimal hex decimal hex
Unicode 604 U+025C 605 U+025D 7572 U+1D94 7583 U+1D9F
UTF-8 201 156 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 information
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 decimal hex decimal hex decimal hex
Unicode 7432 U+1D08 7500 U+1D4C 666 U+029A 606 U+025E
UTF-8 225 180 136 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
Character information
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 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
UTF-8 240 157 154 172 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
UTF-16 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 information
Preview 𝛜 𝜖 𝝐
Unicode name MATHEMATICAL BOLD
EPSILON SYMBOL		 MATHEMATICAL ITALIC
EPSILON SYMBOL		 MATHEMATICAL BOLD ITALIC
EPSILON SYMBOL
Encodings decimal hex decimal hex decimal hex
Unicode 120540 U+1D6DC 120598 U+1D716 120656 U+1D750
UTF-8 240 157 155 156 F0 9D 9B 9C 240 157 156 150 F0 9D 9C 96 240 157 157 144 F0 9D 9D 90
UTF-16 55349 57052 D835 DEDC 55349 57110 D835 DF16 55349 57168 D835 DF50
Numeric character reference 𝛜 𝛜 𝜖 𝜖 𝝐 𝝐
Character information
Preview 𝝚 𝝴 𝞔 𝞮
Unicode name MATHEMATICAL SANS-SERIF
BOLD CAPITAL EPSILON		 MATHEMATICAL SANS-SERIF
BOLD SMALL EPSILON		 MATHEMATICAL SANS-SERIF
BOLD ITALIC CAPITAL EPSILON		 MATHEMATICAL SANS-SERIF
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
UTF-8 240 157 157 154 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
UTF-16 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
Character information
Preview 𝞊 𝟄
Unicode name MATHEMATICAL SANS-SERIF
BOLD EPSILON SYMBOL		 MATHEMATICAL SANS-SERIF
BOLD ITALIC EPSILON SYMBOL
Encodings decimal hex decimal hex
Unicode 120714 U+1D78A 120772 U+1D7C4
UTF-8 240 157 158 138 F0 9D 9E 8A 240 157 159 132 F0 9D 9F 84
UTF-16 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]

References[edit]

  1. ^ "epsilon". Oxford English Dictionary (3rd ed.). Oxford University Press. September 2005. (Subscription or participating institution membership required.)
  2. ^ a b Nick Nicholas: Letters Archived 2012-12-15 at Archive.today, 2003–2008. (Greek Unicode Issues)
  3. ^ 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.
  4. ^ "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.
  5. ^ Halmos, Paul R. (1960). Naive Set Theory. New York: Van Nostrand. pp. 5–6. ISBN 978-1614271314.
  6. ^ Jeffery, Lilian H. (1961). The local scripts of archaic Greece. Oxford: Clarendon. pp. 63–64.
  7. ^ Jeffery, Local scripts, p.24.
  8. ^ Jeffery, Local scripts, p.114.
  9. ^ Jeffery, Local scripts, p.138.
  10. ^ Nicholas, Nick (2005). "Proposal to add Greek epigraphical letters to the UCS" (PDF). Archived from the original (PDF) on 2006-05-05. Retrieved 2010-08-12.
  11. ^ Jeffery, Local scripts, p.89.
  12. ^ Thompson, Edward M. (1911). An introduction to Greek and Latin palaeography. Oxford: Clarendon. pp. 191–194.
  13. ^ Weisstein, Eric W. "Delta Function". mathworld.wolfram.com. Retrieved 2019-02-19.

Further reading[edit]

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