successively adding ones. This includes all the numbers that can be expressed by means of our ordinary numerals, and since such numbers can be made greater and greater, without ever reaching an unsurpassable maximum, it is easy to suppose that there are no other numbers. But this supposition, natural as it is, is mistaken.
Whether the Pythagoreans themselves believed space and time to be composed of indivisible points and instants is a debatable question. It would seem that the distinction between space and matter had not yet been clearly made, and that therefore, when an atomistic view is expressed, it is difficult to decide whether particles of matter or points of space are intended. There is an interesting passage in Aristotle's Physics, where he says:"The Pythagoreans all maintained the existence of the void, and said that it enters into the heaven itself from the boundless breath, inasmuch as the heaven breathes in the void also; and the void differentiates natures, as
- 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, το πόσον, and the other to the how much, το 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 (την σφαιρικήν) contemplates continued quantity so far as it is of a self-motive nature. (Proclus, ed. Friedlein, p. 35. As to the distinction between το πηλίκον, continuous, and το πόσον, discrete quantity, see Iambl., in Nicomachi Geraseni Arithmeticam introductionem, ed. Tennulius, p. 148.)" Cf. p. 48.
- Referred to by Burnet, op. cit., p. 120.
- iv., 6. 213b, 22; H. Ritter and L. Preller, Historia Philosophiæ Græcæ, 8th ed., Gotha, 1898, p. 75 (this work will be referred to in future as " R. P.").