Book 5
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
In the mss. this sentence follows the next one.This 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 equalsIf 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. 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. Therefore 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.
