Now since an unjust man is one who is unfair, and the unjust is the unequal, it is clear that corresponding to the unequal there is a mean, namely that which is equal;
for every action admitting of more and less admits of the equal also.
If then the unjust is the unequal, the just is the equal—a view that commends itself to all without proof; and since the equal is a mean, the just will be a sort of mean too.
Again, equality involves two terms at least. It accordingly follows not only （a） that the just is a mean and equal [and relative to something and just for certain persons * These words appear to be an interpolation. ], but also （b） that, as a mean, it implies certain extremes between which it lies, namely the more and the less; （c） that, as equal, it implies two shares that are equal; and （d） that, as just, it implies certain persons for whom it is just.
It follows therefore that justice involves at least four terms, namely, two persons for whom it is just and two shares which are just.
And there will be the same equality between the shares as between the persons, since the ratio between the shares will be equal to the ratio between the persons; for if the persons are not equal, they will not have equal shares; it is when equals possess or are allotted unequal shares, or persons not equal equal shares, that quarrels and complaints arise.
This is also clear from the principle of ‘assignment by desert.’ All are agreed that justice in distributions must be based on desert of some sort, although they do not all mean the same sort of desert; democrats make the criterion free birth; those of oligarchical sympathies wealth, or in other cases birth; upholders of aristocracy make it virtue.
Justice is therefore a sort of proportion; for proportion is not a property of numerical quantity only, but of quantity in general, proportion being equality of ratios, and involving four terms at least.
（That a discrete proportion * A ‘discrete proportion’ means one in which the two ratios are disconnected, being between different terms, whereas in a ‘continuous proportion’ they have one term in common. has four terms is plain, but so also has a continuous proportion, since it treats one term as two, and repeats it: for example, * Here the lecturer displayed a diagram. as the line representing term one is to the line representing term two, so is the line representing term two to the line representing term three; here the line representing term two is mentioned twice, so that if it be counted twice, there will be four proportionals.）
Thus the just also involves four terms at least, and the ratio between the first pair of terms is the same as that between the second pair. For the two lines representing the persons and shares are similarly divided * Here was another diagram （one would expect the sentence to run ‘Let two lines representing . . . have been similarly divided’）. Two segments, A and B, of one line represented two persons, two segments, C and D, of another their shares. It is shown that, if A:B::C:D, then A+C:B+D::A:B, i.e., if the shares are proportioned to the persons, their relative condition after receiving them will be the same as it was before. ;
then, as the first term is to the second, so is the third to the fourth; and hence, by alternation, as the first is to the third, so is the second to the fourth; and therefore also, as the first is to the second, so is the sum of the first and third to the sum of the second and fourth. Now this is the combination effected by a distribution of shares, and the combination is a just one, if persons and shares are added together in this way.
The principle of Distributive Justice, therefore, is the conjunction of the first term of a proportion with the third and of the second with the fourth; and the just in this sense is a mean between two extremes that are disproportionate, * i.e., A's just share lies between too large a share and too small a one, too large and too small here meaning more or less than is proportionate to A's claim. Cf. Bk. 2.6.4, third note, and 6.7. since the proportionate is a mean, and the just is the proportionate.
（This kind of proportion is termed by mathematicians geometrical proportion * We call this a proportion simply: cf. 4.3 and note. ; for a geometrical proportion is one in which the sum of the first and third terms will bear the same ratio to the sum of the second and fourth as one term of either pair bears to the other term.—
Distributive justice is not a continuous proportion, for its second and third terms, a person and a share, do not constitute a single term.） The just in this sense is therefore the proportionate, and the unjust is that which violates proportion. The unjust may therefore be either too much or too little; and this is what we find in fact, for when injustice is done, the doer has too much and the sufferer too little of the good in question;
though vice versa in the case of an evil, because a lesser evil in comparison with a greater counts as a good,
since the lesser of two evils is more desirable than the greater, but what is desirable is good, and the more desirable it is, the greater good it is.
This then is one kind of Justice.