Let A be to B as C is to D, and let C be to D as E is to F.
I say that A is to B as E is to 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 of A and C equimultiples G and H have been taken, and of B and D other, arbitrary, equimultiples L and M, therefore, if G is in excess of L, H is also in excess of M; if equal, equal; and if less, less.
Again, since C is to D as E is to F, and of C and E equimultiples H and K have been taken, and of D and F other, arbitrary, equimultiples M and N, therefore, if H is in excess of M, K is also in excess of N; if equal, equal; and if less, less.
But we saw that, if H was in excess of M, G was also in excess of L; if equal, equal; and if less, less, so that, in addition, if G is in excess of L, K is also in excess of N; if equal, equal; and if less, less.
And G and K are equimultiples of A and E, while L and N are other, arbitrary, equimultiples of B and F, therefore A is to B as E is to F.
Therefore, ratios which are the same with the same ratio are also the same with one another.
The magnitudes may be of three different kinds with A and B of one kind, C and D of a second kind, and E and F of a third kind.
This proposition is used very frequently whenever ratios are used.