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. [8]

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. [9]

1This 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. [10] For when of two equals2 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.

1 In the mss. this sentence follows the next one.

2 If a=b, then （b+n）-（a-n）=2n, and （b+n）-a=N, and （b+n）-（b+n）+（a-n）/2=n=（b+n）+（a-n）/2-（a-n）. Aristotle, of course, represented the quantities by lines, not algebraically.

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