Note 11 — Chapter VIII · PLATONIC COMPOSITIONS GENERALLY.

Themistius says (Orat. xxi. p. 245 D) that Plato sometimes lectured in the Peiræus, and that a crowd then collected to hear him, not merely from the city, but also from the country around: if he lectured De Bono, however, the ordinary hearers became tired and dispersed, leaving only τοὺς συνήθεις ὁμιλητάς.

It appears that Plato in his lectures delivered theories on the principles of geometry. He denied the reality of geometrical points — or at least admitted them only as hypotheses for geometrical reasoning. He maintained that what others called a point ought to be called “an indivisible line”. Xenokrates maintained the same doctrine after him. Aristotle controverts it (see Metaphys. A., 992, b. 20). Aristotle’s words citing Plato’s opinion (τούτῳ μὲν οὖν τῷ γένει καὶ διεμάχετο Πλάτων ὡς ὄντι γεωμετρικῷ δόγματι, ἀλλ’ ἐκάλει ἀρχὴν γραμμῆς· τοῦτο δὲ πολλάκις ἐτίθει τὰς ἀτόμους γραμμάς) must be referred to Plato’s oral lectures; no such opinion occurs in the dialogues. This is the opinion both of Bonitz and Schwegler in their comments on the passage: also of Trendelenburg, De Ideis et Numeris Platonis, p. 66. That geometry and arithmetic were matters of study and reflection both to Plato himself and to many of his pupils in the Academy, appears certain; and perhaps Plato may have had an interior circle of pupils, to which he applied the well-known exclusion — μηδεὶς ἀγεωμέτρητος εἰσίτω. But we cannot make out clearly what was Plato’s own proficiency, or what improvements he may have introduced, in geometry, nor what there is to justify the comparison made by Montucla between Plato and Descartes. In the narrative respecting the Delian problem — the duplication of the cube — Archytas, Menæchmus, and Eudoxus, appear as the inventors of solutions, Plato as the superior who prescribes and criticises (see the letter and epigram of Eratosthenes: Bernhardy, Eratosthenica, pp. 176-184). The three are said to have been blamed by Plato for substituting instrumental measurement in place of geometrical proof (Plutarch, Problem. Sympos. viii. 2, pp. 718, 719; Plutarch, Vit. Marcelli, c. 14). The geometrical construction of the Κόσμος, which Plato gives us in the Timæus, seems borrowed from the Pythagoreans, though applied probably in a way peculiar to himself (see Finger, De Primordiis Geometriæ ap. Græcos, p. 38, Heidelb. 1831).