Well, the unjust man we have said is unequal, and the abstract “Unjust” unequal: further, it is plain that there is some mean of the unequal, that is to say, the equal or exact half (because in whatever action there is the greater and the less there is also the equal, i.e. the exact half). If then the Unjust is unequal the Just is equal, which all must allow without further proof: and as the equal is a mean the Just must be also a mean. Now the equal implies two terms at least: it follows then that the Just is both a mean and equal, and these to certain persons; and, in so far as it is a mean, between certain things (that is, the greater and the less), and, so far as it is equal, between two, and in so far as it is just it is so to certain persons. The Just then must imply four terms at least, for those[9] to which it is just are two, and the terms representing the things are two. And there will be the same equality between the terms representing the persons, as between those representing the things: because as the latter are to one another so are the former: for if the persons are not equal they must not have equal shares; in fact this is the very source of all the quarrelling and wrangling in the world, when either they who are equal have and get awarded to them things not equal, or being not equal those things which are equal. Again, the necessity of this equality of ratios is shown by the common phrase “according to rate,” for all agree that the Just in distributions ought to be according to some rate: but what that rate is to be, all do not agree; the democrats are for freedom, oligarchs for wealth, others for nobleness of birth, and the aristocratic party for virtue.