Book 5
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.
11
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 greatesti.e., the party that has too much. of the three the amount by which the mean is exceeded by him.
12
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.The mss. here insert the sentence that appears again at 5.9 init.
13
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;Literally where the law gives immunity, that is, does not give redress for inequality resulting from the contract. Should inequality result from a breach of the contract, this would of course be a case for the intervention of Corrective Justice in Voluntary Transactions (chap. 2 fin.).
