Sigma ( SIG-mə; uppercase Σ, lowercase σ, lowercase in word-final position ς; Greek: σίγμα) is the eighteenth letter of the Greek alphabet. When it is used at the end of a letter-case word (one that does not use all caps), the final form (ς) is used. In Ὀδυσσεύς (Odysseus), for example, the two lowercase sigmas (σ) in the center of the name are distinct from the word-final sigma (ς) at the end. In the system of Greek numerals, sigma has a value of 200. In general mathematics, Σ is used as an operator for summation. The Latin letter S derives from sigma while the Cyrillic letter Es (С) derives from a lunate form of this letter.
History The shape (Σς) and alphabetic position of sigma is derived from the Phoenician letter shin (𐤔). Sigma's original name may have been san, but due to the complicated early history of the Greek epichoric alphabets, san came to be identified as a separate letter in the Greek alphabet, represented as Ϻ. Herodotus reports that "san" was the name given by the Dorians to the same letter called "sigma" by the Ionians. According to one hypothesis, the name "sigma" may continue that of Phoenician samekh (𐤎), the letter continued through Greek xi, represented as Ξ. Alternatively, the name may have been a Greek innovation that simply meant 'hissing', from the root of σίζω (sízō, from Proto-Greek *sig-jō 'I hiss').
Lunate sigma
In handwritten Greek during the Hellenistic period (4th–3rd century BC), the epigraphic form of Σ was simplified into a C-like shape, which has also been found on coins from the 4th century BC onward. This became the universal standard form of sigma during late antiquity and the Middle Ages. Today, it is known as lunate sigma (uppercase Ϲ, lowercase ϲ), because of its crescent-like shape, and is still widely used in decorative typefaces in Greece, especially in religious and church contexts (i.e. the Christogram ΙϹ ΧϹ), as well as in some modern print editions of classical Greek texts. A dotted lunate sigma (sigma periestigmenon, Ͼ) was used by Aristarchus of Samothrace (220–143 BC) as an editorial sign indicating that the line thus marked is at an incorrect position. Similarly, a reversed sigma (antisigma, Ͻ), may mark a line that is out of place. A dotted antisigma (antisigma periestigmenon, Ͽ) may indicate a line after which rearrangements should be made, or to variant readings of uncertain priority. In Greek inscriptions from the late first century BC onwards, Ͻ was an abbreviation indicating that a man's father's name is the same as his own name, thus Dionysodoros Dionysodorou (Dionysodoros son of Dionysodoros) would be written Διονυσόδωρος Ͻ. In Unicode, the above variations of lunate sigma are encoded as U+03F9 Ϲ GREEK CAPITAL LUNATE SIGMA SYMBOL; U+03FD Ͻ GREEK CAPITAL REVERSED LUNATE SIGMA SYMBOL, U+03FE Ͼ GREEK CAPITAL DOTTED LUNATE SIGMA SYMBOL, and U+03FF Ͽ GREEK CAPITAL REVERSED DOTTED LUNATE SIGMA SYMBOL.
Derived alphabets Sigma was adopted in the Old Italic alphabets beginning in the 8th century BC. At that time a simplified three-stroke version, omitting the lowermost stroke, was already found in Western Greek alphabets, and was incorporated into classical Etruscan and Oscan, as well as in the earliest Latin epigraphy (early Latin S), such as the Duenos inscription. The alternation between three and four (and occasionally more than four) strokes was also adopted into the early runic alphabet (early form of the s-rune). Both the Anglo-Saxon runes and the Younger Futhark consistently use the simplified three-stroke version. The letter С of Cyrillic script and Ⲥ of the Coptic script originates in the lunate form of Sigma.
Uses
Language and linguistics In both Ancient and Modern Greek, the sigma represents the voiceless alveolar fricative [s]. In Modern Greek, this sound is voiced to the voiced alveolar fricative [z] when occurring before [m], [n], [v], [ð], or [ɣ]. The uppercase form of sigma (Σ) was re-borrowed into the Latin alphabet—more precisely, the International African Alphabet—to serve as the uppercase of modern esh (lowercase: ʃ). In phonology, σ is used to represent syllables. In linguistics, Σ represents the set of symbols that form an alphabet (see also computer science). In historical linguistics, Σ is used to represent a Common Brittonic consonant with a sound between [s] and [h]; perhaps an aspirated [ʃʰ].
Science and mathematics
Mathematics In general mathematics, lowercase σ is commonly used to represent unknown angles, additionally serving as a shorthand for "countably", whereas Σ is regularly used as the operator for summation, e.g.:
∑ k = 0 5 k = 0 + 1 + 2 + 3 + 4 + 5 = 15 {\displaystyle \sum _{k=0}^{5}k=0+1+2+3+4+5=15}
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