Gonzalez Lodge, Commentary on Plato: Gorgias

δικαιότητα: an unusual word for justice, formed like κοσμιότης and ἰσότης. It was probably chosen here on account of the assonance. Similarly Prot. 331 b ταὐτόν ἐστι δικαιότης ὁσιότητι. Xenophon also uses the word.

καὶ ταῦτα: the regular Greek expression for “and that too.” See H. 612 a.

22 f.

ἡ γεωμετρική: adds a limitation to the ἰσότης, that absolute equality, which is impossible in any government. ἰσότης alone is purely arithmetical equality (a = b); ἰσότης γεωμετρική is ratio or proportion (a:b::c:d). Plato distinguishes between two kinds of ἰσότης in Leg. vi. 757 b, the one τὴν μέτρῳ ἴσην καὶ σταθμῷ καὶ ἀριθμῷ, the arithmetical; the second, τὴν ἀληθεστάτην καὶ ἀρίστην ἰσότητα, the geometric, which is not comprehensible by every man, and which is dependent upon the judgment and decision of the gods. Of this he says (ib.): τῷ μὲν μείζονι πλείω, τῷ δ᾽ ἐλάττονι σμικρότερα νέμει, μέτρια διδοῦσα πρὸς τὴν αὐτῶν φύσιν ἑκατέρῳ, καὶ δὴ καὶ τιμὰς μείζοσι μὲν πρὸς ἀρετὴν ἀεὶ μείζους, τοῖς δὲ τοὐναντίον ἔχουσιν ἀρετῆς τε καὶ παιδείας τὸ πρέπον ἑκατέροις ἀπονέμει κατὰ λόγον. On this ἰσότης, which is here opposed to πλεονεξία, is founded the administration of justice in the state (ib. τὸ πολιτικὸν τοῦτ᾽ αὐτὸ τὸ δίκαιον). Aris totle also (Nic. Eth. v. 6, 7, Bekk.) refers τὸ ἐν διανομῇ δίκαιον back to ἀναλογία, i.e. to ἰσότης λόγων, and adds, καλοῦσι δὲ τὴν τοιαύτην ἀναλογίαν γεωμετρικὴν οἱ μαθηματικοί. Isocrates expresses himself to the same effect in Areop. (vii.) 20 ff., where he says of Solon and Cleisthenes, μέγιστον αὐτοῖς συνεβάλετο πρὸς τὸ καλῶς οἰκεῖν τὴν πόλιν, ὅτι δυοῖν ἰσοτήτοιν νομιζομέναιν εἶναι, καὶ τῆς μὲν ταὐτὸν ἅπασιν ἀπονεμούσης, τῆς δὲ τὸ προσῆκον ἑκάστοις, οὐκ ἠγνόουν τὴν χρησιμωτέραν κτἑ. Thompson also quotes Plut. Mor. 719 b, according to which Lycurgus ‘expelled from Lacedaemon arithmetical equality, holding it to be democratic and levelling in principle, and introduced the geometric, as better suited to a temperate oligarchy and monarchy.’

γεωμετρίας γὰρ ἀμελεῖς: Plato considered mathematics as a necessary preliminary to philosophy. Hence the well-known μηδεὶς ἀγεωμέτρητος εἰσίτω μου τὴν στέγην upon his house.

εἶεν: see on 466 c. Cf. 472 d, 480 a.

ἢ ἐξελεγκτέος δὴ κτἑ.: the alternative appears as the obvious and necessary result of what precedes. The meaning and construction are similar to 482 b (ὡς οὐ κτἑ.). See on 467 a.