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τὸ τοῦ Ὁμήρου κτλ. I formerly proposed to omit πεπονθέναι, but a precise parallel is furnished by Symp. 198 C ἀτεχνῶς τὸ τοῦ Ὁμήρου ἐπεπόνθη. ἄν makes πεπονθέναι equivalent to the future perfect—a more vigorous form of expression than the future (or aorist with ἄν) would be. The quotation (which is from Od. XI 489, cf. supra III 386 C) “has a curious felicity, being the words of Achilles in expressing his detestation of the world of shades (lit. shadows) in comparison with the world of human life” (Bosanquet). It is better, I think, to connect ἄλλῳ with θητευέμεν (Ameis on Od. l.c.) than with παρά (as Schneider does). καὶ ὁτιοῦν κτλ. ἄν (which Richards would alter to δή) is as suitable here as before, since ἂν πεπονθέναι depends in both cases directly on δοκεῖς. The confusion of ἄν and δή is not so frequent as some have thought: see on V 450 C. δοξάζειν=‘opine’ is technical: for the cave is an allegory of τὸ δοξαστόν (514 A note).
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