fact, and indeed makes the peculiarities of the whole system depend on it. Aristotle is very decided as to the Pythagorean points having magnitude. "They construct the whole world out of numbers," he tells us, "but they suppose the units have magnitude. As to how the first unit with magnitude arose, they appear to be at a loss."[1] Zeller holds that this is only an inference of Aristotle's,[2] and he is probably right in this sense, that the Pythagoreans never felt the need of saying in so many words that points had magnitude. It does seem probable, however, that they called them ὄγκοι.[3]
Zeller, moreover, allows, and indeed insists, that in the Pythagorean cosmology the numbers were spatial, but he raises difficulties about the other parts of the system. There are other things, such as the Soul and Justice and Opportunity, which are said to be numbers, and which cannot be regarded as constructed of points, lines, and surfaces.[4] Now it appears to me that this is just the meaning of a passage in which Aristotle criticises the Pythagoreans. They held, he says, that in one part of the world Opinion prevailed, while a little above it or below it were to be found Injustice or Separation or Mixture, each of which was, according to them, a number. But in the very same regions of the heavens were to be found things having magnitude which were also numbers. How can this be, since justice has no magnitude?[5] This means surely that the Pythagoreans
- ↑ Arist. Met. M, 6. 1080 b 18 sqq., 1083 b 8 sqq.; De caelo, Γ, 1. 300 a 16 (R. P. 76 a).
- ↑ Zeller, p. 381.
- ↑ Zeno in his fourth argument about motion, which, we shall see (§ 163), was directed against the Pythagoreans, used ὄγκοι for points. Aetios, i. 3, 19 (R. P. 76 b), says that Ekphantos of Syracuse was the first of the Pythagoreans to say that their units were corporeal. Cf. also the use of ὄγκοι in Plato, Parm. 164 d, and Galen, Hist. Phil. 18 (Dox. p. 610), Ἡρακλείδης δὲ ὁ Ποντικὸς καὶ Ἀσκληπιάδης ὁ βιθυνὸς ἀνάρμους ὄγκους τὰς ἀρχὰς ὑποτίθενται τῶν ὅλων.
- ↑ Zeller, p. 381.
- ↑ Arist. Met. A, 8. 990 a 22 (R. P. 81 e). I read and interpret thus "For, seeing that, according to them, Opinion and Opportunity are in a given part of the world, and a little above or below them Injustice and Separation and Mixture,—in proof of which they allege that each of these is a number,—and seeing that it is also the case (reading συμβαίνῃ with
