James Adam, The Republic of Plato

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περὶ ποίων κτλ. On the derisive ποῖος see 522 D n. Mathematical units are in every case (ἕκαστον) equal each to each (πᾶν παντί), and destitute of parts; whereas sensible units (e.g. one horse, one cow etc.) are not equal to each other, and are divisible. In πᾶν παντί Plato copies the formal language of mathematics: cf. ἑκατέραν ἑκατέρᾳ and the like in Euclid passim. For the sense see Phil. 56 C ff., where these two kinds of number are made the basis of a distinction between philosophical or scientific and popular or unscientific ἀριθμητική. It should be carefully noted that a plurality of mathematical units is expressly recognised both here (ἴσον τε ἕκαστον πᾶν παντί κτλ.) and in Phil. l. c. (μονάδα μονάδος ἑκάστης τῶν μυρίων μηδεμίαν ἄλλην ἄλλης διαφέρουσαν). This entirely confirms what Aristotle tells us, viz. that Plato placed μαφηματικά between αἰσθητά and εἴδη, τῷ τὰ μὲν πόλλ᾽ ἄττα ὅμοια εἶναι, τὸ δὲ εἶδος αὐτὸ ἓν ἕκαστον μόνον (Met. A 6. 987^{b} 14 ff.). There are therefore three kinds of μονάδες in Plato's scheme—the Ideal μονάς, of which only one exists, the Mathematical and the Sensible, of each of which there are many. See on VI 510 D and App. I, where I have quoted further evidence on this subject, and endeavoured to explain the philosophical truth which is contained in the Platonic doctrine of mathematical numbers, magnitudes etc. as intermediates between the Ideas and sensibles.

ὧν κτλ. ὧν is for περὶ ὧν rather than <*> (as J. and C. hold): cf. VI 510 D οὐ περὶ τούτων διανοούμενοι, and (for the grammatical construction) III 402 A note διανοηθῆναι should be understood in the technical sense of VI 511 E.

τῷ ὄντι ἀναγκαῖον. Perhaps with a play on προσαναγκάζον (J. and C.): see on τῷ ὄντι VI 511 B.