James Adam, The Republic of Plato

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οὐκ ἂν ἐάσαις κτλ.: ‘you will not suffer to be mere irrational quantities, if they are to rule in the city and control the higher issues.’ ἄλογοι γραμμαί are irrational magnitudes (cf. Arist. περὶ ἀτόμων γραμμῶν 968^{b} 18), which Greek mathematicians treated “geometrically through a symbolism of irrational lines,” as in Euclid Bk. X (Gow Gk Math. p. 78). They are ἄλογοι or ἄρρητοι because “nicht aussprechbar” (Cantor Gesch. d. Math. p. 154 note), whereas rational lines are ῥηταί, ‘expressible’ (cf. Blass de Pl. Math. p. 18). In its application to Glauco's ‘children,’ ἄλογοι is active, and means of course μὴ λόγον ἔχοντες διδόναι (534 B). Has γραμμάς also any special application? Probably it has: otherwise the witticism seems unnecessarily far-fetched and frigid, even if we make every allowance for Plato's love of a mathematical jest (cf. Pol. 266 B), as well as for the interest which the subject of irrationals seems to have excited among the mathematicians of his day (see Theaet. 147 D ff. and Cantor l.c. pp. 182, 191, 203). Lucilius (II 20) has the line “vix vivo homini ac monogrammo” (“a dead-alive sketch of an anatomy” Tyrrell Lat. Poetry p. 175), and Cicero mocks at Epicurus' gods as “monogrammos” (N. D. II 59: cf. I 123 homunculi similem deum—liniamentis dumtaxat extremis, non habitu solido— praeditum etc., and other passages in Usener Epicurea p. 234). Perhaps Plato means to suggest that his “airy burgomasters,” as Milton calls them, would in such a case be only as it were mere silhouettes (“Schattenrisse” Bertram Bilderspr. Pl. p. 46) of rulers moving blindly to and fro in a sort of dreamland (cf. ὀνειροπολοῦντα 534 C and 533 C note). For other views see App. XVII.

τῶν μεγίστων. 525 B note

ἐρωτᾶν τε καὶ ἀποκρίνεσθαι κτλ. Plato concludes by emphasizing the most conspicuous and characteristic feature of the Socratic method: cf. Crat. 390 C.