τὰς τέχνας κτἑ.: the branches pursued in the boys' schools, γραμματική, λογιστική, κιθαριστική,—thus about the same thought as ἐκ διδασκάλων ἀπαλλαγέντας (cf. 326 c).
ἄκοντας, ἄγοντες: note the paronomasia.
λογισμοὺς κτἑ.: here Protagoras refers unfavorably to those who teach these specialties as not being sophists in the best sense (λωβῶνται τοὺς νέους above); while, according to 316 d, such specialists have ever been true sophists; but the different purpose of the two arguments is obvious. The circle of the studies belonging to a complete education was enlarged just at this time; to γραμματική were added arithmetic, geometry, astronomy, and music; but as Callicles in Gorg. 485 a advises the pursuit of philosophy only ὅσον παιδείας χάριν, so the sophists warned against all that diverted from practical life, all mere theory. So Isocr. XV. 268 διατρῖψαι μὲν οὖν περὶ τὰς παιδείας ταύτας χρόνον τινὰ συμβουλεύσαιμ᾽ ἂν τοῖς νεωτέροις, μὴ μέντοι περιιδεῖν τὴν φύσιν τὴν ἑαυτῶν κατασκελετευθεῖσαν (not however to suffer their constitution to be impaired) ἐπὶ τούτοις. δεῖν δὲ (ἡγοῦμαι) τοὺς προὔργου τι ποιεῖν βουλομένους (those wishing to make some real progress) καὶ τῶν λόγων τοὺς ματαίους καὶ τῶν πράξεων τὰς μηδὲν πρὸς τὸν βίον φερούσας ἀναιρεῖν ἐξ ἁπασῶν τῶν διατριβῶν (to put away from their whole life vain words, and deeds which bring no profit).
The sophists professed to teach ἀρετή (cf. παρὰ τούτους τοὺς ὑπισχνουμένους ἀρετῆς διδασκάλους εἶναι Meno 91 b), but this virtue was only an external prudence in private and public affairs, cf. ταύτης τῆς ἀρετῆς, ᾗ οἱ ἄνθρωποι τάς τε οἰκίας καὶ τὰς πόλεις καλῶς διοικοῦσι καὶ τοὺς γονέας τοὺς αὑτῶν θεραπεύουσι, καὶ πολίτας καὶ ξένους ὑποδέξασθαί τε καὶ ἀποπέμψαι ἐπίστανται Meno 91 a.
