^{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.