Note 24 — LECTURE VI THE PROBLEM OF INFINITY CONSIDERED HISTORICALLY

There is some reason to think that the Pythagoreans distinguished between discrete and continuous quantity. G. J. Allman, in his Greek Geometry from Thales to Euclid, says (p. 23): “The Pythagoreans made a fourfold division of mathematical science, attributing one of its parts to the how many, τὸ πόσον, and the other to the how much, τὸ πηλίκον; and they assigned to each of these parts a twofold division. For they said that discrete quantity, or the how many, either subsists by itself or must be considered with relation to some other; but that continued quantity, or the how much, is either stable or in motion. Hence they affirmed that arithmetic contemplates that discrete quantity which subsists by itself, but music that which is related to another; and that geometry considers continued quantity so far as it is immovable; but astronomy (τὴν σφαιρικήν) contemplates continued quantity so far as it is of a self-motive nature. (Proclus, ed. Friedlein, p. 35. As to the distinction between τὸ πηλίκον, continuous, and τὸ πόσον, discrete quantity, see Iambl., in Nicomachi Geraseni Arithmeticam introductionem, ed. Tennulius, p. 148.)” Cf. p. 48.