Proposition 12

Let any number of magnitudes A, B, C, D, E, and F be proportional, so that A is to B as C is to D, and as E is to F.

I say that A is to B as the sum of A, C, and E is to the sum of B, D, and F.

Take equimultiples G, H, and K of A, C, and E, and take other, arbitrary, equimultiples L, M, and N of B, D, and F.

Then since A is to B as C is to D, and as E is to F, and of A, C, and E equimultiples G, H, and K have been taken, and of B, D, and F other, arbitrary, equimultiples L, M, and N, therefore, if G is in excess of L, then H is also in excess of M, and K of N; if equal, equal; and if less, less. So that, in addition, if G is in excess of L, then the sum of G, H, and K is in excess of the sum of L, M, and N; if equal, equal; and if less, less.

Now G and the sum of G, H, and K are equimultiples of A and the sum of A, C, and E, since, if any number of magnitudes are each the same multiple the same number of other magnitudes, then the sum is that multiple of the sum.

For the same reason L and the sum of L, M, and N are also equimultiples of B and the sum of B, D, and F, therefore A is to B as the sum of A, C, and E is to the sum of B, D, and F.

Therefore, if any number of magnitudes are proportional, then one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents.