Front Page Titles (by Subject) 4.: Of that which is just in correction, and its rule of arithmetical proportion. - The Nicomachean Ethics
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4.: Of that which is just in correction, and its rule of arithmetical proportion. - Aristotle, The Nicomachean Ethics 
The Nichomachean Ethics of Aristotle, trans. F.H. Peters, M.A. 5th edition (London: Kegan Paul, Trench, Truebner & Co., 1893).
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Of that which is just in correction, and its rule of arithmetical proportion.
It remains to treat of the other form, viz. that which is just in the way of redress, the sphere of which is private transactions, whether voluntary or involuntary.
This differs in kind from the former.
For that which is just in the distribution of a common stock of good things is always in accordance with the proportion above specified (even when it is a common fund that has to be divided, the sums which the several participants take must bear the same ratio to one another as the sums they have put in), and that which is unjust in the corresponding sense is that which violates this proportion.
But that which is just in private transactions* is indeed fair or equal in some sort, and that which is unjust is unfair or unequal; but the proportion to be observed here is not a geometrical proportion as above, but an arithmetical one.
For it makes no difference whether a good man defrauds a bad one, or a bad man a good one, nor whether a man who commits an adultery be a good or a bad man; the law looks only to the difference created by the injury, treating the parties themselves as equal, and only asking whether the one has done, and the other suffered, injury or damage.
That which is unjust, then, is here something unequal [or unfair] which the judge tries to make equal [or fair]. For even when one party is struck and the other strikes, or one kills and the other is killed, that which is suffered and that which is done may be said to be unequally or unfairly divided; the judge then tries to restore equality by the penalty or loss which he inflicts upon the offender, subtracting it from his gain.
For in such cases, though the terms are not always quite appropriate, we generally talk of the doer’s “gain” (e.g. the striker’s) and the sufferer’s “loss;” but when the suffering has been assessed by the court, what the doer gets is called “loss” or penalty, and what the sufferer gets is called “gain.”
What is fair or equal, then, is a mean between more or too much and less or too little; but gain and loss are both more or too much and less or too little in opposite ways, i.e. gain is more or too much good and less or too little evil, and loss the opposite of this.
And in the mean between them, as we found, lies that which is equal or fair, which we say is just.
That which is just in the way of redress, then, is the mean between loss and gain.
When disputes arise, therefore, men appeal to the judge:* and an appeal to the judge is an appeal to that which is just; for the judge is intended to be as it were a living embodiment of that which is just; and men require of a judge that he shall be moderate [or observe the mean], and sometimes even call judges “mediators” (μεσιδίους), signifying that if they get the mean they will get that which is just.
That which is just, then, must be a sort of mean, if the judge be a “mediator.”
But the judge restores equality; it is as if he found a line divided into two unequal parts, and were to cut off from the greater that by which it exceeds the half, and to add this to the less.
But when the whole is equally divided, the parties are said to have their own, each now receiving an equal or fair amount.
But the equal or fair amount is here the arithmetic mean between the more or too much and the less or too little. And so it is called δίκαιον (just) because there is equal division (δίχα); δίκαιον being in fact equivalent to δίχαιον, and δικαστής (judge) to διχαστής.
If you cut off a part from one of two equal lines and add it to the other, the second is now greater than the first by two such parts (for if you had only cut off the part from the first without adding it to the second, the second would have been greater by only one such part); the second exceeds the mean by one such part, and the mean also exceeds the first by one.
Thus we can tell how much to take away from him who has more or too much, and how much to add to him who has less or too little: to the latter’s portion must be added that by which it falls short of the mean, and from the former’s portion must be taken away that by which it exceeds the mean.
To illustrate this, let AA′, BB′, CC′ be three equal lines:—
From AA′ let AE be cut off; and let CD [equal to AE] be added to CC′; then the whole DCC′ exceeds EA′ by CD and CZ [equal to AE or CD], and exceeds BB′ by CD.
And this* holds good not only in geometry, but in the arts also; they could not exist unless that which is worked upon received an impression corresponding in kind and quantity and quality to the exertions of the artist.
But these terms, “loss” and “gain,” are borrowed from voluntary exchange. For in voluntary exchange having more than your own is called gaining, and having less than you started with is called losing (in buying and selling, I mean, and in the other transactions in which the law allows free play); but when the result to each is neither more nor less but the very same amount with which he started, then they say that they have their own, and are neither losers nor gainers. That which is just, then, is a mean between a gain and a loss, which are both contrary to the intention,* and consists in having after the transaction the equivalent of that which you had before it.
[* ]In the way of redress, as given by the law-courts: later on (cap. 5) he gives as an after-thought the kind of justice which ought to regulate buying and selling, etc. See note on p. 152.
[* ]The δικασταί at Athens combined the functions of judge and jury.
[* ]The point to be illustrated is, that in these private transactions what one man gains is equal to what the other loses, so that the penalty that will restore the balance can be exactly measured. Of this principle (on which the possibility of justice does in fact depend) Aristotle first gives a simple geometrical illustration, and then says that the same law holds in all that man does: what is suffered by the patient (whether person, as in medicine, or thing, as in sculpture or agriculture) is the same as what is done by the agent. This paragraph occurs again in the next chapter (5, 9): but it can hardly have come into this place by accident; we rather see the author’s thought growing as he writes. I follow Trendelenburg (who omits the passage here) in inserting ὅ before ἐποίει, but not in omitting τὸ before πάσχον.
[* ]For the aim of trade is neither profit nor loss, but fair exchange, i.e. exchange (on the principle laid down in ch. 5) which leaves the position of the parties as the state fixed it (by distributive justice, ch. 3). But when in the private transactions of man with man this position is disturbed, i.e. whenever either unintentionally, by accident or negligence, or intentionally, by force or fraud, one has bettered his position at the expense of another, corrective justice steps in to redress the balance. I read αὐτὰ δἰ αὐτω̂ν and accept Stewart’s interpretation of these words, and in part Jackson’s interpretation of τω̂ν παρὰ τὸ ἑκούσιον, but cannot entirely agree with either as to the sense of the whole passage.