Chapter 3: Of what is just in ditribution and its rule of geometrical proportion.

The unjust man [in this limited sense of the word], we say, is unfair, and that which is unjust is unfair.

Now, it is plain that there must be a mean which lies between what is unfair on this side and on that. And this is that which is fair or equal; for any act that admits of a too much and a too little admits also of that which is fair.

If then that which is unjust be unfair, that which is just will be fair, which indeed is admitted by all without further proof.

But since that which is fair or equal is a mean between two extremes, it follows that what is just will be a mean.

But equality or fairness implies two terms at least.

It follows, then, that that which is just is both a mean quantity and also a fair amount relatively to something else and to certain persons—in other words, that, on the one hand, as a mean quantity it implies certain other quantities, i.e. a more and a less; and, on the other hand, as an equal or fair amount it involves two quantities, and as a just amount it involves certain persons.

That which is just, then, implies four terms at least: two persons to whom justice is done, and two things.

And there must be the same “equality” [i.e. the same ratio] between the persons and the things: as the things are to one another, so must the persons be. For if the persons be not equal, their shares will not be equal; and this is the source of disputes and accusations, when persons who are equal do not receive equal shares, or when persons who are not equal receive equal shares.

This is also plainly indicated by the common phrase “according to merit.” For in distribution all men allow that what is just must be according to merit or worth of some kind, but they do not all adopt the same standard of worth; in democratic states they take free birth as the standard, in oligarchic states they take wealth, in others noble birth, and in the true aristocratic state virtue or personal merit.

We see, then, that that which is just is in some sort proportionate. For not abstract numbers only, but all things that can be numbered, admit of proportion; proportion meaning equality of ratios, and requiring four terms at least.

That discrete proportion requires four terms is evident at once. Continuous proportion also requires four terms: for in it one term is employed as two and is repeated; for instance, a/b = b/c. The term b then is repeated; and so, counting b twice over, we find that the terms of the proportion are four in number.

That which is just, then, requires that there be four terms at least, and that the ratio between the two pairs be the same, i.e. that the persons stand to one another in the same ratio as the things.

Let us say, then, a/b = c/d, or alternando a/c = b/d.

The sums of these new pairs then will stand to one another in the original ratio [i.e. a + c/b + d = a/b or c/d ].

But these are the pairs which the distribution joins together; and if the things be assigned in this manner, the distribution is just.

This joining, then, of a to c and of b to d is that which is just in distribution; and that which is just in this sense is a mean quantity, while that which is unjust is that which is disproportionate; for that which is proportionate is a mean quantity, but that which is just is, as we said, proportionate.

This proportion is called by the mathematicians a geometrical proportion; for it is when four terms are in geometrical proportion that the sum [of the first and third] is to the sum [of the second and fourth] in the original ratio [of the first to the second or the third to the fourth].

But this proportion [as applied in justice] cannot be a continuous proportion; for one term cannot represent both a person and a thing.

That which is just, then, in this sense is that which is proportionate; but that which is unjust is that which is disproportionate. In the latter case one quantity becomes more or too much, the other less or too little. And this we see in practice; for he who wrongs another gets too much, and he who is wronged gets too little of the good in question: but of the evil conversely; for the lesser evil stands in the place of good when compared with the greater evil: for the lesser evil is more desirable than the greater, but that which is desirable is good, and that which is more desirable is a greater good.

This then is one form of that which is just.