Then, since, as A is to B, so is D to E,
and of A, D equimultiples G, H have been taken,
and of B, E other, chance, equimultiples K, L,
therefore, as G is to K, so is H to L. [V. 4]
For the same reason also, as K is to M, so is L to N.
Since, then, there are three magnitudes G, K, M, and others H, L, N equal to them in multitude, which taken two and two together are in the same ratio,
therefore, ex aequali, if G is in excess of M, H is also in excess of N;
if equal, equal; and if less, less. [V. 20]
And G, H are equimultiples of A, D,
and M, N other, chance, equimultiples of C, F.
Therefore, as A is to C, so is D to F. [V. Def. 5]
Complete text
Book I