27. ῥυθμούς τε καὶ ἁρμονίας. ῥυθμοί times or rhythms (cf. Rep. III. 399E ff.), ἁρμονίαι scales (ibid. 398Dff.). ῥυθμός and ἁρμονία between them make up μουσική in the narrower sense: see Symp. 187A ff., where music is defined as περὶ ἁρμονίαν καὶ ῥυθμὸν ἐρωτικῶν ἐπιστήμη—περὶ ἁρμονίαν since it reconciles ὀξύ and βαρύ, περὶ ῥυθμόν since it reconciles ταχύ and βραδύ.
28. οἰκειοῦσθαι ταῖς ψυχαῖς τῶν παίδων. Sauppe quotes Rep. III. 401D κυριωτάτη ἐν μουσικῇ τροφή, ὅτι μάλιστα καταδύεται εἰς τὸ ἐντὸς τῆς ψυχῆς ὄ τε ῥυθμὸς καὶ ἁρμονία καὶ ἐρρωμενέστατα ἅπτεται αὐτῆς.
29. εὐρυθμότεροι καὶ εὐαρμοστότεροι. See Rep. III. 400C ff., where, after it is shown that τὸ εὔρυθμον and τὸ εὐάρμοστον imply εὐλογία, Plato continues (400D) εὐλογία ἄρα καὶ εὐαπμοστία καὶ εὐσχημοσύνη καὶ εὐπυθμία εὐηθείᾳ ἀκολουθεῖ, οὐχ ἣν ἄνοιαν οὖσαν ὑροκοπιζόμενοι καλοῦμεν ὡς εὐήθειαν, ἀλλὰ τὴν ὡς ἀληθῶς εὖ τε καὶ καλῶς τὸ ἦθος κατεσκευασμένην διάνοιαν.
33. εἰς παιδοτρίβου: Protagoras passes to γυμναστική, the second great division of Greek education: Rep. II. 376E.
34. ὑπηρετῶσι τῇ διανοίᾳ. Plato asserts that the true object of γυμναστική is not to cultivate the body, but to educate the soul to the proper mean between hardness and softness: Rep. III. 410C ff. On the soul as the mistress of the body see Phaedo, ch. 43.