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 persons¹], 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 proportion² has four terms is plain, but so also has a continuous proportion, since it treats one term as two, and repeats it: 1131bfor example,³ 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 divided⁴; 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,⁵ since the proportionate is a mean, and the just is the proportionate. (This kind of proportion is termed by mathematicians geometrical proportion⁶; 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.

The remaining kind is Corrective Justice, which operates in private transactions, both voluntary and involuntary. This justice is of a different sort from the preceding. For justice in distributing common property always conforms with the proportion we have described (since when a distribution is made from the common stock, it will follow the same ratio as that between the amounts which the several persons have contributed to the common stock); and the injustice opposed to justice of this a kind is a violation of this proportion. But the just in private transactions, although it is the equal in a sense (and the unjust the unequal), 1132ais not the equal according to geometrical but according to arithmetical proportion.⁷ For it makes no difference⁸ whether a good man has defrauded a bad man or a bad one a good one, nor whether it is a good or a bad man that has committed adultery; the law looks only at the nature of damage, treating the parties as equal, and merely asking whether one has done and the other suffered injustice, whether one inflicted and the other has sustained damage. Hence the unjust being here the unequal, the judge endeavors to equalize it: inasmuch as when one man has received and the other has inflicted a blow, or one has killed and the other been killed, the line⁹ representing the suffering and doing of the deed is divided into unequal parts, but the judge endeavors to make them equal by the penalty or loss¹⁰ he imposes, taking away the gain. (For the term “gain” is used in a general way to apply to such cases, even though it is not strictly appropriate to some of them, for example to a person who strikes another, nor is “loss” appropriate to the victim in this case; but at all events the results are called “loss” and “gain” respectively when the amount of the damage sustained comes to be estimated.) Thus, while the equal is a mean between more and less, gain and loss are at once both more and less in contrary ways, more good and less evil being gain and more evil and less good loss; and as the equal, which we pronounce to be just, is, as we said, a mean between them, it follows that Justice in Rectification¹¹ will be the mean between loss and gain. This is why when disputes occur men have recourse to a judge. To go to a judge is to go to justice, for the ideal judge is so to speak justice personified. Also, men require a judge to be a middle term or medium—indeed in some places judges are called mediators—, for they think that if they get the mean they will get what is just. Thus the just is a sort of mean, inasmuch as the judge is a medium between the litigants. Now the judge restores equality: if we represent the matter by a line divided into two unequal parts, he takes away from the greater segment that portion by which it exceeds one-half of the whole line, and adds it to the lesser segment. When the whole has been divided into two halves, people then say that they “have their own,” having got what is equal. ¹²This is indeed the origin of the word dikaion (just): it means dicha (in half), as if one were to pronounce it dichaion; and a dikast (judge) is a dichast (halver). The equal is a mean by way of arithmetical proportion between the greater and the less. For when of two equals¹³ a part is taken from the one and added to the other, the latter will exceed the former by twice that part, since if it had been taken from the one but not added to the other, the latter would exceed the former by once the part in question only. 1132bTherefore the latter will exceed the mean by once the part, and the mean will exceed the former, from which the part was taken, by once that part. This process then will enable us to ascertain what we ought to take away from the party that has too much and what to add to the one that has too little: we must add to the one that has too little the amount whereby the mean between them exceeds him, and take away from the greatest¹⁴ of the three the amount by which the mean is exceeded by him. Let the lines AA, BB, CC be equal to one another; let the segment AE be taken away from the line AA, and let the segment CD be added to the line CC, so that the whole line DCC exceeds the line EA by CD+CF; then DCC will exceed BB by CD.¹⁵ The terms “loss” and “gain” in these cases are borrowed from the operations of voluntary exchange. There, to have more than one’s own is called gaining, and to have less than one had at the outset is called losing, as for instance in buying and selling, and all other transactions sanctioned by law;¹⁶ while if the result of the transaction is neither an increase nor a decrease, but exactly what the parties had of themselves, they say they “have their own” and have neither lost nor gained. Hence Justice in Involuntary Transactions is a mean between gain and loss in a sense: it is to have after the transaction an amount equal to the amount one had before it.