Epsilon Explained

. 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 Greek, Ancient (to 1453);: εἶ (pronounced as /grc/), but it was later changed to (e psilon 'simple e') in the Middle Ages to distinguish the letter from the digraph Greek, Ancient (to 1453);: 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: . 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 . 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 . 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 (

\epsilon

) denotes the lunate form, while \varepsilon (

\varepsilon

) denotes the reversed-3 form. Unicode versions 2.0.0 and onwards use as the lowercase Greek epsilon letter,[4] but in version 1.0.0, was used.[5] The lunate or uncial epsilon provided inspiration for the euro sign, .[6]

There is also a 'Latin epsilon', (ɛ) or "open e", which looks similar to the Greek lowercase epsilon. It is encoded in Unicode as and and is used as an IPA phonetic symbol. This Latin uppercase epsilon,, is not to be confused with the Greek uppercase (sigma)

The lunate epsilon, (ϵ), is not to be confused with the set membership symbol . 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"), evolved from the letter epsilon, since the symbol was originally used as an abbreviation for the Latin word Latin: '''e'''st. In addition, mathematicians often read the symbol as "element of", as in "1 is an element of the natural numbers" for

1\in\N

, for example. As late as 1960, itself was used for set membership, while its negation "does not belong to" (now) was denoted by (epsilon prime).[7] 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 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.[8]

Sound value

While the original pronunciation of the Phoenician letter He was pronounced as /[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 pronounced as /[e]/ sound.[9] Besides its classical Greek sound value, the short pronounced as //e// phoneme, it could initially also be used for other pronounced as /[e]/-like sounds. For instance, in early Attic before, it was used also both for the long, open pronounced as //ɛː//, and for the long close pronounced as //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 pronounced as //e// and pronounced as //ɛː// was taken by a glyph resembling a pointed B (

), while (Ε) was used only for long close pronounced as //eː//.[10] 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 .[11]

In Thespiai (Boeotia), a special letter form consisting of a vertical stem with a single rightward-pointing horizontal bar (

) was used for what was probably a raised variant of pronounced as //e// in pre-vocalic environments.[12] [13] This tack glyph was used elsewhere also as a form of "Heta", i.e. for the sound pronounced as //h//.

Glyph variants

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 cross-bar and the curved stroke were linked in various ways.[14] 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.

Uses

International Phonetic Alphabet

Despite its pronunciation as mid, in the International Phonetic Alphabet, the Latin epsilon represents open-mid front unrounded vowel, as in the English word pet .

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, the lunate epsilon symbol, and the Latin lowercase epsilon (see above) are used in a variety of places:

\epsilonx.\phi

as an extension to first-order logic; see epsilon calculus.

a+b\varepsilon

, with

\varepsilon2=0

and

\varepsilon0

.

Unicode

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.

See also

Further reading

The Man Who Loved Only Numbers. Hyperion, 1998. .

Notes and References

  1. Encyclopedia: epsilon . Longman Pronunciation Dictionary . John C. . Wells . John C. Wells . . 1990 . 250 . Harlow, England . 0582053838.
  2. Nick Nicholas: Letters, 2003–2008. (Greek Unicode Issues)
  3. Book: Colwell, Ernest C.. 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. 1969. Leiden. Brill. 127.
  4. Book: Code Charts. https://www.unicode.org/versions/Unicode2.0.0/CodeCharts1.pdf. 130. The Unicode Standard, Version 2.0. 0-201-48345-9.
  5. Book: Code Charts. https://www.unicode.org/versions/Unicode1.0.0/CodeCharts1.pdf. 130. The Unicode Standard, Version 1.0. 1. 0-201-56788-1.
  6. Web site: European Commission – Economic and Financial Affairs – How to use the euro name and symbol . Ec.europa.eu . 7 April 2010 . Inspiration for the € symbol itself came from the Greek epsilon, – 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..
  7. Book: Halmos, Paul R.. Naive Set Theory. Van Nostrand. 1960. 978-1614271314. New York. 5–6.
  8. Book: Jeffery, Lilian H.. The local scripts of archaic Greece. Oxford. Clarendon. 1961. 63–64.
  9. Jeffery, Local scripts, p. 24.
  10. Jeffery, Local scripts, p. 114.
  11. Jeffery, Local scripts, p. 138.
  12. Web site: Nicholas . Nick . Proposal to add Greek epigraphical letters to the UCS . https://web.archive.org/web/20060217000025/http://www.tlg.uci.edu/~opoudjis/unicode/epigraphical.pdf . dead . February 17, 2006 . 2010-08-12 . 2005 .
  13. Jeffery, Local scripts, p. 89.
  14. Book: Thompson, Edward M. . An introduction to Greek and Latin palaeography . Oxford . Clarendon . 1911 . 191–194.
  15. Web site: Delta Function. Weisstein. Eric W.. mathworld.wolfram.com . 2019-02-19.