Scholae dictae sunt, non ab otio ac vacatione omni, sed quod, ceteris rebus omissis, vacare liberalibus studiis pueri debent; ut etiam ludi appellantur, in quibus minime luditur, ne tristi aliquo nomine fug<iunt pueri suo fungi mu>nere.
(Festus, De Verborum Significatu p. 470 Lindsay)

Schools are called scholae (lit. spare time), not because one enjoys leisure or freedom from all work there, but because children should set all else aside and invest their ‘free time’ in the liberal studies. Likewise they are called ludi (lit. games), even though there’s no ‘playing’ (ludere) whatsoever in them, lest a less cheerful name would discourage children from doing their duty. (tr. David Bauwens)

Militem: Aelius a mollitia κατὰ ἀντίφρασιν dictum putat, eo, quod nihil molle, sed potius asperum quid gerat; sic ludum dicimus, in quo minime luditur.
(Paulus Diaconus, Epitoma Festi p. 109 Lindsay)

Soldier: Aelius thinks the world for soldier (miles) is derived by antiphrasis from mollitia (weakness), because they do nothing that is weak, on the contrary: they do what is rough. Likewise we say ludus (school), even though there’s no ‘playing’ (ludere) there. (tr. David Bauwens)

Lucus quia umbra opacus parum luceat et ludus quia sit longissime a lusu et Ditis quia minime dives.
(Aelius Stilo fr. 59 Funaioli)

A grove is called lucus because on account of all the shadows there’s hardly any light (lux) there; a school is called ludus because it’s very far from any kind of play (lusus); and we call Pluto Dis or Ditis because he’s hardly rich (dives). (tr. David Bauwens)