Then since, as A is to B, so is C to D,
and of A, C equimultiples G, H have been taken,
and of B, D other, chance, equimultiples L, 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, as C is to D, so is E to F,
and of C, E equimultiples H, K have been taken,
and of D, F other, chance, equimultiples M, 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, K are equimultiples of A, E, while L, N are other, chance, equimultiples of B, F;
therefore, as A is to B, so is E to F.
Complete text
Book I