§5.3ἐπεὶ δʼ ὅ τʼ ἄδικος ἄνισος καὶ τὸ ἄδικον ἄνισον, δῆλον ὅτι καὶ μέσον τι ἔστι τοῦ ἀνίσου.
Since the unjust man is unequal and the unjust thing is unequal, it is clear that there is also a certain mean of the unequal.
τοῦτο δʼ ἐστὶ τὸ ἴσον· ἐν ὁποίᾳ γὰρ πράξει ἔστι τὸ πλέον καὶ τὸ ἔλαττον, ἔστι καὶ τὸ ἴσον.
And this is the equal; for in whatever action there is the more and the less, there is also the equal.
εἰ οὖν τὸ ἄδικον ἄνισον, τὸ δίκαιον ἴσον· ὅπερ καὶ ἄνευ λόγου δοκεῖ πᾶσιν.
If, therefore, the unjust is unequal, the just is equal; which is what seems to everyone even without argument.
ἐπεὶ δὲ τὸ ἴσον μέσον, τὸ δίκαιον μέσον τι ἂν εἴη.
And since the equal is a mean, the just must be a mean.
ἔστι δὲ τὸ ἴσον ἐν ἐλαχίστοις δυσίν.
But the equal involves at least two terms.
ἀνάγκη τοίνυν τὸ δίκαιον μέσον τε καὶ ἴσον εἶναι καὶ πρός τι καὶ τισίν, καὶ ᾗ μὲν μέσον, τινῶν (ταῦτα δʼ ἐστὶ πλεῖον καὶ ἔλαττον), ᾗ δʼ ἴσον, δυοῖν, ᾗ δὲ δίκαιον, τισίν.
It is necessary, therefore, that the just be both a mean and equal, relative to something and for certain persons; and insofar as it is a mean, it must be between certain things (which are the more and the less), and insofar as it is equal, it must be between two things, and insofar as it is just, it must be for certain persons.
ἀνάγκη ἄρα τὸ δίκαιον ἐν ἐλαχίστοις εἶναι τέτταρσιν· οἷς τε γὰρ δίκαιον τυγχάνει ὄν, δύο ἐστί, καὶ ἐν οἷς, τὰ πράγματα, δύο.
Therefore, the just must involve at least four terms; for those for whom it happens to be just are two, and the things in which it is, the assets, are two.
καὶ ἡ αὐτὴ ἔσται ἰσότης, οἷς καὶ ἐν οἷς· ὡς γὰρ ἐκεῖνα ἔχει, τὰ ἐν οἷς, οὕτω κἀκεῖνα ἔχει·
And there will be the same equality between the persons and between the things; for as the latter, the things, are related, so also the former, the persons, are related.
εἰ γὰρ μὴ ἴσοι, οὐκ ἴσα ἕξουσιν, ἀλλʼ ἐντεῦθεν αἱ μάχαι καὶ τὰ ἐγκλήματα, ὅταν ἢ μὴ ἴσα ἴσοι ἢ μὴ ἴσοι ἴσα ἔχωσι καὶ νέμωνται.
For if they are not equal, they will not have equal shares; but from this arise battles and complaints, when either equals have and are distributed unequal shares, or unequals equal shares.
ἔτι ἐκ τοῦ κατʼ ἀξίαν τοῦτο δῆλον· τὸ γὰρ δίκαιον ἐν ταῖς νομαῖς ὁμολογοῦσι πάντες κατʼ ἀξίαν τινὰ δεῖν εἶναι, τὴν μέντοι ἀξίαν οὐ τὴν αὐτὴν λέγουσι πάντες, ἀλλʼ οἱ μὲν δημοκρατικοὶ ἐλευθερίαν, οἱ δʼ ὀλιγαρχικοὶ πλοῦτον, οἳ δʼ εὐγένειαν, οἱ δʼ ἀριστοκρατικοὶ ἀρετήν.
This is further clear from the principle of "according to merit"; for all agree that the just in distributions must be according to some merit, though all do not mean the same by merit: democratics say it is freedom, oligarchics wealth, others noble birth, and aristocratics virtue.
ἔστιν ἄρα τὸ δίκαιον ἀνάλογόν τι.
Therefore, the just is something proportional.
τὸ γὰρ ἀνάλογον οὐ μόνον ἐστὶ μοναδικοῦ ἀριθμοῦ ἴδιον, ἀλλʼ ὅλως ἀριθμοῦ· ἡ γὰρ ἀναλογία ἰσότης ἐστὶ λόγων, καὶ ἐν τέτταρσιν ἐλαχίστοις.
For proportion is a property not only of monadic number, but of number in general; for proportion is equality of ratios, and involves at least four terms.
ἡ μὲν οὖν διῃρημένη ὅτι ἐν τέτταρσι, δῆλον.
Now that discrete proportion involves four terms is clear.
ἀλλὰ καὶ ἡ συνεχής· τῷ γὰρ ἑνὶ ὡς δυσὶ χρῆται καὶ δὶς λέγει, οἷον ὡς ἡ τοῦ α πρὸς τὴν τοῦ β, οὕτως ἡ τοῦ β πρὸς τὴν τοῦ γ. δὶς οὖν ἡ τοῦ β εἴρηται· ὥστʼ ἐὰν ἡ τοῦ β τεθῇ δίς, τέτταρα ἔσται τὰ ἀνάλογα.
But continuous proportion does so too; for it uses the single middle term as two and mentions it twice, as for instance, "as the line A is to the line B, so is the line B to the line C." Since B is mentioned twice, if B is placed twice, the proportional terms will be four.
ἔστι δὲ καὶ τὸ δίκαιον ἐν τέτταρσιν ἐλαχίστοις, καὶ ὁ λόγος ὁ αὐτός· διῄρηται γὰρ ὁμοίως οἷς τε καὶ ἅ.
And the just also involves at least four terms, and the ratio is the same; for they are divided similarly, both the persons and the things.
ἔσται ἄρα ὡς ὁ α ὅρος πρὸς τὸν β, οὕτως ὁ γ πρὸς τὸν δ, καὶ ἐναλλὰξ ἄρα, ὡς ὁ α πρὸς τὸν γ, ὁ β πρὸς τὸν δ.
Therefore, as the term A is to B, so will C be to D, and hence alternately, as A is to C, B will be to D.
ὥστε καὶ τὸ ὅλον πρὸς τὸ ὅλον· ὅπερ ἡ νομὴ συνδυάζει, κἂν οὕτω συντεθῇ, δικαίως συνδυάζει.
So that the whole is related to the whole; which is what the distribution couples together, and if it is combined in this way, it couples them justly.
ἡ ἄρα τοῦ α ὅρου τῷ γ καὶ ἡ τοῦ β τῷ δ σύζευξις τὸ ἐν διανομῇ δίκαιόν ἐστι, καὶ μέσον τὸ δίκαιον τοῦτʼ ἐστί, τὸ δʼ ἄδικον τὸ παρὰ τὸ ἀνάλογον· τὸ γὰρ ἀνάλογον μέσον, τὸ δὲ δίκαιον ἀνάλογον.
Therefore, the coupling of the term A with C and of B with D is the just in distribution, and this just is a mean, and the unjust is what violates the proportion; for the proportional is a mean, and the just is proportional.
καλοῦσι δὲ τὴν τοιαύτην ἀναλογίαν γεωμετρικὴν οἱ μαθηματικοί· ἐν γὰρ τῇ γεωμετρικῇ συμβαίνει καὶ τὸ ὅλον πρὸς τὸ ὅλον ὅπερ ἑκάτερον πρὸς ἑκάτερον.
And mathematicians call this kind of proportion "geometrical"; for in geometrical proportion it happens also that the whole is related to the whole as each term is related to each.
ἔστι δʼ οὐ συνεχὴς αὕτη ἡ ἀναλογία· οὐ γὰρ γίνεται εἷς ἀριθμῷ ὅρος, ᾧ καὶ ὅ.
But this proportion is not continuous; for there does not come to be a single term in number for both the person and the thing.
τὸ μὲν οὖν δίκαιον τοῦτο, τὸ ἀνάλογον· τὸ δʼ ἄδικον τὸ παρὰ τὸ ἀνάλογον.
The just, then, is this proportional; and the unjust is what is contrary to the proportion.
γίνεται ἄρα τὸ μὲν πλέον τὸ δʼ ἔλαττον, ὅπερ καὶ ἐπὶ τῶν ἔργων συμβαίνει· ὁ μὲν γὰρ ἀδικῶν πλέον ἔχει, ὁ δʼ ἀδικούμενος ἔλαττον τοῦ ἀγαθοῦ.
Thus one share becomes more and the other less, which is also what happens in actual practice; for he who acts unjustly has more of the good, and he who is treated unjustly has less.
ἐπὶ δὲ τοῦ κακοῦ ἀνάπαλιν· ἐν ἀγαθοῦ γὰρ λόγῳ γίνεται τὸ ἔλαττον κακὸν πρὸς τὸ μεῖζον κακόν· ἔστι γὰρ τὸ ἔλαττον κακὸν μᾶλλον αἱρετὸν τοῦ μείζονος, τὸ δʼ αἱρετὸν ἀγαθόν, καὶ τὸ μᾶλλον μεῖζον.
But in the case of evil it is the reverse; for the lesser evil is reckoned as a good in comparison with the greater evil; for the lesser evil is more desirable than the greater, and what is desirable is a good, and what is more desirable is a greater good.
τὸ μὲν οὖν ἓν εἶδος τοῦ δικαίου τοῦτʼ ἐστίν.
This, then, is one species of the just.