[Footnote 30: By these are meant the planets, including therein the Sun and Moon. Cf. Sextus Empiricus, _Adversus Astrologos_, p. 343 (Cod.) _passim_.] [Footnote 31: τὰ ὅλα = entities which must needs differ from one another in kind. The phrase is thus used by Plato, Aristotle and all the neo-Platonic writers.] [Footnote 32: ἐφήψατο, _attigit_, Cr. Frequent in Pindar.] [Footnote 33: So Timon in the _Silli_, as quoted by Diog. Laert., VIII, _vit. Pyth._, c. 20.] [Footnote 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.] [Footnote 35: κατὰ τὸ πλῆθος, _multitudine_, Cr.] [Footnote 36: For definitions and examples of this term see Aristot., _Metaphys._, IV. c. 28.] [Footnote 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.”] [Footnote 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.] [Footnote 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.] [Footnote 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.] [Footnote 41: μετενσωμάτωσις. The phrase which is here correctly used throughout, but which has somehow slipped into English as metempsychosis.] [Footnote 42: So Diog. Laert., VIII, _vit. Pyth._, c. 4.] [Footnote 43: Diodorus of Eretria is not otherwise known, Aristoxenus is mentioned by Cicero, _Quæst. Tusculan._, I, 18, as a writer on music.]