[338] τίνες ξύμφωνοι ἀριθμοὶ, &c.
[339] Η καὶ διαλεκτικὸν καλεῖς τὸν λόγον ἐκάστου λαμβάνοντα τῆς οὐσίας; (§ 14).
[340] ὥσπερ θριγγὸς τοῖς μαθήμασιν ἡ διαλεκτικὴ ἦμιν ἐπάνω κεῖσθαι. (§ 14).]
[341] Pol. vi. § 19.
[342] He adds, "This oraton, this visible world, I will not say has any connexion with ouranon, heaven, that I may not be accused of playing upon words."
[343] It is plain that Plato, by Hypotheses, in this place, means the usual foundations of Arithmetic and Geometry; namely, Definitions and Postulates. He says that "the arithmeticians and geometers take as hypotheses (hυποθεμενοι) odd and even, and the three kinds of angles (right, acute, and obtuse); and figures, (as a triangle, a square,) and the like." I say his "hypotheses" are the Definitions and Postulates, not the Axioms: for the Axioms of Arithmetic and Geometry belong to the Higher Faculty, which ascends to First Principles. But this Faculty operates rather in using these axioms than in enunciating them. It knows them implicitly rather than expresses them explicitly.
[344] διάνοιαν άλλ' οὐ νοῦν.
[345] The Diagram, as here described, would be this:
| Intelligible World. | Visible World. | ||
| Intuition. | Conception. | Things. | Images. |
Plato supposes the whole, and each of the two parts, to be divided in the same ratio, in order that the analogy of the division in each case may be represented.