James Adam, The Republic of Plato

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ἀναμεμιγμένων . ἀναμεμιγμένοι would be more usual, but the genitive lays more stress on the participial clause: cf. Thuc. III 13. 6 βοηθησάντων δὲ ὑμῶν προθύμως πόλιν τε προσλήψεσθε κτλ., and other examples quoted in Kühner Gr. Gr. II p. 666. See also infra on 459 C. Here, too, it should be noted that the addition of a parenthetical οἶμαι helps to render ἀναμεμιγμένων independent of ἔσονται. The genitive absolute in ἰδίᾳ δὲ—κεκτημένου may also, as Jackson suggests, have influenced Plato's choice of construction in this clause. Plato perhaps thought of Sparta when he wrote the present sentence: cf. Plut. Lyc. 15. 1 ἦν μὲν οὖν καὶ ταῦτα παρορμητικὰ πρὸς γάμους: λέγω δὲ τὰς πομπὰς τῶν παρθένων καὶ τὰς ἀποδύσεις καὶ τοὺς ἀγῶνας ἐν ὄψει τῶν νέων, ἀγομένων οὐ γεωμετρικαῖς, ἀλλ᾽ ἐρωτικαῖς, ὥς φησιν ὁ Πλάτων, ἀνάγκαις.

γεωμετρικαῖς γε: sc. ἀναγκαῖα, with which the dative goes, as in Soph. 252 D ταῖς μεγίσταις ἀνάγκαις ἀδύνατον (cited by J. and C.). We have here one of the earliest assertions of the famous doctrine which has played so large and important a part in the history of philosophy—the doctrine of the so-called ‘necessity’ of mathematical reasoning. See for instance Mill's Logic Book II c. 5. In the rest of this sentence Schneider suspects that Glauco is paraphrasing some passage of poetry. τὸν πολὺν λεών certainly sounds tragic.