Epsilon (UK: /ɛpˈslən/,[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 εGREEK 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 ϵGREEK 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 ɛLATIN SMALL LETTER OPEN E and U+0190 ƐLATIN 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 sigma (Σ). The symbol , 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 as "e 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

Origin

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

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

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

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.

UncialUncial variantsCursive variantsMinusculeMinuscule with ligatures
inlineinlineinlineinlineinline

Uses

International Phonetic Alphabet

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

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

  • Greek Epsilon
Character information
PreviewΕεϵ϶
Unicode nameGREEK CAPITAL LETTER EPSILONGREEK SMALL LETTER EPSILONGREEK LUNATE EPSILON SYMBOLGREEK REVERSED LUNATE EPSILON SYMBOL
Encodingsdecimalhexdecimalhexdecimalhexdecimalhex
Unicode917U+0395949U+03B51013U+03F51014U+03F6
UTF-8206 149CE 95206 181CE B5207 181CF B5207 182CF B6
Numeric character referenceΕΕεεϵϵ϶϶
Named character referenceΕε, εϵ, ϵ, ϵ϶, ϶
DOS Greek132841569C
DOS Greek-2168A8222DE
Windows 1253197C5229E5
TeX\varepsilon\epsilon
  • Coptic Eie
Character information
Preview
Unicode nameCOPTIC CAPITAL LETTER EIECOPTIC SMALL LETTER EIE
Encodingsdecimalhexdecimalhex
Unicode11400U+2C8811401U+2C89
UTF-8226 178 136E2 B2 88226 178 137E2 B2 89
Numeric character referenceⲈⲈⲉⲉ
  • Latin Open E
Character information
PreviewƐɛ
Unicode nameLATIN CAPITAL LETTER
OPEN E
LATIN SMALL LETTER
OPEN E
LATIN SMALL LETTER
OPEN E WITH RETROFLEX HOOK
MODIFIER LETTER
SMALL OPEN E
Encodingsdecimalhexdecimalhexdecimalhexdecimalhex
Unicode400U+0190603U+025B7571U+1D937499U+1D4B
UTF-8198 144C6 90201 155C9 9B225 182 147E1 B6 93225 181 139E1 B5 8B
Numeric character referenceƐƐɛɛᶓᶓᵋᵋ
Character information
Previewɜɝ
Unicode nameLATIN 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
Encodingsdecimalhexdecimalhexdecimalhexdecimalhex
Unicode604U+025C605U+025D7572U+1D947583U+1D9F
UTF-8201 156C9 9C201 157C9 9D225 182 148E1 B6 94225 182 159E1 B6 9F
Numeric character referenceɜɜɝɝᶔᶔᶟᶟ
Character information
Previewʚɞ
Unicode nameLATIN SMALL LETTER
TURNED OPEN E
MODIFIER LETTER SMALL
TURNED OPEN E
LATIN SMALL LETTER
CLOSED OPEN E
LATIN SMALL LETTER
CLOSED REVERSED OPEN E
Encodingsdecimalhexdecimalhexdecimalhexdecimalhex
Unicode7432U+1D087500U+1D4C666U+029A606U+025E
UTF-8225 180 136E1 B4 88225 181 140E1 B5 8C202 154CA 9A201 158C9 9E
Numeric character referenceᴈᴈᵌᵌʚʚɞɞ
  • Mathematical Epsilon
Character information
Preview𝚬𝛆𝛦𝜀𝜠𝜺
Unicode nameMATHEMATICAL 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
Encodingsdecimalhexdecimalhexdecimalhexdecimalhexdecimalhexdecimalhex
Unicode120492U+1D6AC120518U+1D6C6120550U+1D6E6120576U+1D700120608U+1D720120634U+1D73A
UTF-8240 157 154 172F0 9D 9A AC240 157 155 134F0 9D 9B 86240 157 155 166F0 9D 9B A6240 157 156 128F0 9D 9C 80240 157 156 160F0 9D 9C A0240 157 156 186F0 9D 9C BA
UTF-1655349 57004D835 DEAC55349 57030D835 DEC655349 57062D835 DEE655349 57088D835 DF0055349 57120D835 DF2055349 57146D835 DF3A
Numeric character reference𝚬𝚬𝛆𝛆𝛦𝛦𝜀𝜀𝜠𝜠𝜺𝜺
Character information
Preview𝛜𝜖𝝐
Unicode nameMATHEMATICAL BOLD
EPSILON SYMBOL
MATHEMATICAL ITALIC
EPSILON SYMBOL
MATHEMATICAL BOLD ITALIC
EPSILON SYMBOL
Encodingsdecimalhexdecimalhexdecimalhex
Unicode120540U+1D6DC120598U+1D716120656U+1D750
UTF-8240 157 155 156F0 9D 9B 9C240 157 156 150F0 9D 9C 96240 157 157 144F0 9D 9D 90
UTF-1655349 57052D835 DEDC55349 57110D835 DF1655349 57168D835 DF50
Numeric character reference𝛜𝛜𝜖𝜖𝝐𝝐
Character information
Preview𝝚𝝴𝞔𝞮
Unicode nameMATHEMATICAL 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
Encodingsdecimalhexdecimalhexdecimalhexdecimalhex
Unicode120666U+1D75A120692U+1D774120724U+1D794120750U+1D7AE
UTF-8240 157 157 154F0 9D 9D 9A240 157 157 180F0 9D 9D B4240 157 158 148F0 9D 9E 94240 157 158 174F0 9D 9E AE
UTF-1655349 57178D835 DF5A55349 57204D835 DF7455349 57236D835 DF9455349 57262D835 DFAE
Numeric character reference𝝚𝝚𝝴𝝴𝞔𝞔𝞮𝞮
Character information
Preview𝞊𝟄
Unicode nameMATHEMATICAL SANS-SERIF
BOLD EPSILON SYMBOL
MATHEMATICAL SANS-SERIF
BOLD ITALIC EPSILON SYMBOL
Encodingsdecimalhexdecimalhex
Unicode120714U+1D78A120772U+1D7C4
UTF-8240 157 158 138F0 9D 9E 8A240 157 159 132F0 9D 9F 84
UTF-1655349 57226D835 DF8A55349 57284D835 DFC4
Numeric character reference𝞊𝞊𝟄𝟄

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

References

  1. ^ "epsilon". Oxford English Dictionary (3rd ed.). Oxford University Press. September 2005. (Subscription or UK public library 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