Book 5
10
Thus the just also involves four terms at least, and the ratio between the first pair of terms is the same as that between the second pair. For the two lines representing the persons and shares are similarly dividedHere was another diagram (one would expect the sentence to run Let two lines representing . . . have been similarly divided). Two segments, A and B, of one line represented two persons, two segments, C and D, of another their shares. It is shown that, if A:B::C:D, then A+C:B+D::A:B, i.e., if the shares are proportioned to the persons, their relative condition after receiving them will be the same as it was before.;
11
then, as the first term is to the second, so is the third to the fourth; and hence, by alternation, as the first is to the third, so is the second to the fourth; and therefore also, as the first is to the second, so is the sum of the first and third to the sum of the second and fourth. Now this is the combination effected by a distribution of shares, and the combination is a just one, if persons and shares are added together in this way.
