If a first magnitude have to a second the same ratio as a third to a fourth, and the third have to the fourth a greater ratio than a fifth has to a sixth, the first will also have to the second a greater ratio than the fifth to the sixth.
For let a first magnitude A have to a second B the same ratio as a third C has to a fourth D, and let the third C have to the fourth D a greater ratio than a fifth E has to a sixth F;
I say that the first A will also have to the second B a greater ratio than the fifth E to the sixth F.
For, since there are some equimultiples of C, E, and of D, F other, chance, equimultiples, such that the multiple of C is in excess of the multiple of D,
while the multiple of E is not in excess of the multiple of F, [V. Def. 7]
let them be taken, and let G, H be equimultiples of C, E, and K, L other, chance, equimultiples of D, F, so that G is in excess of K, but H is not in excess of L; and, whatever multiple G is of C, let M be also that multiple of A, and, whatever multiple K is of D, let N be also that multiple of B.
Now, since, as A is to B, so is C to D, and of A, C equimultiples M, G have been taken, and of B, D other, chance, equimultiples N, K,
therefore, if M is in excess of N, G is also in excess of K, if equal, equal, and if less, less. [V. Def. 5]
But G is in excess of K;
therefore M is also in excess of N.
But H is not in excess of L;
and M, H are equimultiples of A, E, and N, L other, chance, equimultiples of B, F;
[V. Def. 7]therefore A has to B a greater ratio than E has to F.
