The identification of the point with the unit is referred to by Aristotle, _Phys._ Ε, 3. 227 a 27. [Sidenote: The numbers as magnitudes.] 146. It is clear that this way of regarding the point, the line, and the surface is closely bound up with the practice of representing numbers by dots arranged in symmetrical patterns, which we have seen reason for attributing to the Pythagoreans (§ 47). The science of geometry had already made considerable advances, but the old view of quantity as a sum of units had not been revised, and so a doctrine such as we have indicated was inevitable. This is the true answer to Zeller’s contention that to regard the Pythagorean numbers as spatial is to ignore the fact that the doctrine was originally arithmetical rather than geometrical. Our interpretation takes full account of that fact, and indeed makes the peculiarities of the whole system depend upon 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.”[792] Zeller holds that this is only an inference of Aristotle’s,[793] 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.[794] Footnote 792: Arist. _Met._ Μ, 6. 1080 b 18 sqq., 1083 b 8 sqq.; _de Caelo_, Γ, 1. 300 a 16 (R. P. 76 a). Footnote 793: Zeller, p. 381. Footnote 794: We learn from Plato, _Theaet._ 148 b 1, that Theaitetos called surds, what Euclid calls δυνάμει σύμμετρα, by the name of δυνάμεις, while rational square roots were called μήκη. Now in _Tim._ 31 c 4 we find a division of numbers into ὄγκοι and δυνάμεις, which seem to mean rational and irrational quantities. Cf. also the use of ὄγκοι in _Parm._ 164 d. 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. Probably, however, “Ekphantos” was a personage in a dialogue of Herakleides (Tannery, _Arch._ xi. pp. 263 sqq.), and Herakleides called the monads ἄναρμοι ὄγκοι (Galen, _Hist. Phil._ 18; _Dox._ p. 610).