[32] ἐφήψατο, attigit, Cr. Frequent in Pindar.
[33] So Timon in the Silli, as quoted by Diog. Laert., VIII, vit. Pyth., c. 20.
[34] φυσιογονικὴν. The Barberine MS. has φυσιογνωμονικὴν, evidently inserted by some scribe who connected it with the absurd system of metoposcopy described in Book IV.
[35] κατὰ τὸ πλῆθος, multitudine, Cr.
[36] For definitions and examples of this term see Aristot., Metaphys., IV. c. 28.
[37] I cannot trace Hippolytus’ authority for attributing these neo-Pythagorean puerilities to Pythagoras himself. Diog. Laert., Aristotle and the rest represent him as saying only that the monad was the beginning of everything, and that from this and the undefined dyad numbers proceed. The general reader may be recommended to Mr. Alfred Williams Benn’s statement in The Philosophy of Greece (Lond., 1898), pp. 78 ff. that “the Greeks did not think of numbers as pure abstractions, but in the most literal sense as figures, that is to say, limited portions of space.”
[38] Macmahon thinks “number” and “monad” should here be transposed, as Pythagoras considered according to him the monad as “the highest generalization of number and a conception in abstraction.” Yet the monad was not the highest abstraction of current (Greek) philosophy. See Edwin Hatch, Influence of Greek Ideas upon the Christian Church (Hibbert Lectures), Lond., 1890, p. 255.
[39] δύναμις is here used like our own mathematical expression “power.” Why Hippolytus should associate it especially with the power of 2 does not appear. By Greek mathematicians it seems rather to be applied to the square root.
[40] κυβισθῇ, involvit, Cr. It cannot here mean “cubed.” Another mistake occurs in the same sentence, where it is said that the square multiplied by the cube is a cube. The sentence is fortunately repeated with the needful correction in Book IV, p. [116] infra. Macmahon gives the proper notation as (a2)2 = a4, (a2)3 = a6, (a3)3 = a9.
[41] μετενσωμάτωσις. The phrase which is here correctly used throughout, but which has somehow slipped into English as metempsychosis.