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XVII.—The ς′ of Diophantus
Published online by Cambridge University Press: 06 July 2012
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The Arithmetical, or, as we should say, Algebraic, Problems of Diophantus involve a single unknown quantity, symbolised by ς′ or ς°′. The number for which this symbol stands is described as πλῆθος μονάδων ἄλογον, and is spoken of as ὁ ἀόριστος ἀριθμός, the undefined number, or simply as ὁ ἀριθμός, the number, par excellence, of the problem in question.
By all commentators, with but one exception so far as I know, the ς′ of Diophantus is taken as an abbreviation of ἀριθμός, and I find the same statement made unquestioningly in the standard Palæography of Gardthausen. The mathematicians, while expressing no doubt as to the fact, show, however, in some cases, a clear enough perception of the difficulty of accounting for such an abbreviation, and are not all agreed as to its more precise origin. Nesselmann, Cantor, and most others, treat the ς as the final letter of ἀριθμός: Heath, who devotes many pages to a discussion of the subject, puts forward, as an alternative and preferable hypothesis, a suggestion that the ς may be a corrupt cursive abbreviation of the two first letters αρ of the same word.
- Type
- Research Article
- Information
- Earth and Environmental Science Transactions of The Royal Society of Edinburgh , Volume 38 , Issue 3 , 1897 , pp. 607 - 609
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- Copyright © Royal Society of Edinburgh 1897
References
page 607 note * I am not quite content with Mr Heath's rendering (Dioph. of Alex., p. 57) of πλῆθος μονάδων ἄλογον as “a number of units of which no account is given, or undefined,” but I rather think that ἄλογος is here used in a more technical sense. The especial significance of ἄλογος is irrational, usually in the sense of “surd,” i.e., a number having no determinable “ratio”: I take it here to refer, in a similar way, to a number whose “ratio” is not yet determined. The equivalent phrase ὁ ἀόριστος ἀριθμός is precisely comparable to the Aristotelian ἀόριστος δνάς. ἡ ἀόριστος δνάς is, in our nomenclature, 2n or 2x; and the statement (Arist, ., Met., xii. 7, 1081aGoogle Scholar) that ὁ ἀριθμός ἐστιν ἐκ τοῦ ἑνὸς καὶ τῆς δνάδος ὁ ἀριθμός ἐστιν ἐκ τοῦ ἑνὸς καὶ τῆς δνάδος ἀοςρίστον, simply means that any number can be expressed either as 2n or 2n + 1. In like manner ὁ ἀόριστος ἀριθμός is n or x.
page 607 note † The one writer who declines to commit himself as to the derivation of the ς from ἀριθμός is Gow (Hist, of Gk. Mathem., p. 108, footnote); the symbol, he says, “may be Egyptian, or Indian, or Babylonian, and may reveal an entirely unknown chapter in the history of mathematics.” He is inclined to look for the origin of the ς (and also of the Diophantine Ϯ) in some pictorial hieroglyph.
page 608 note * Treatise on Algebra, London, 8vo, 2 vols., 1820Google Scholar; vol. i. pt. i. p. 225. This writer's interesting and sympathetic account of the Diophantine analysis is not alluded to by Heath.
page 608 note † Hist of Math., p. 286.