Now, since D by multiplying C has made A, and by multiplying E has made L,
therefore, as C is to E, so is A to L. [VII. 17]
But, as C is to E, so is G to H;
therefore also, as G is to H, so is A to L.
Again, since E by multiplying D has made L,
and further by multiplying F has made B,
therefore, as D is to F, so is L to B. [VII. 17]
But, as D is to F, so is H to K;
therefore also, as H is to K, so is L to B.
But it was also proved that, as G is to H, so is A to L;
therefore, ex aequali, as G is to K, so is A to B. [VII. 14]
But G has to K the ratio compounded of the ratios of the sides;
therefore A also has to B the ratio compounded of the ratios of the sides.
Q. E. D.
Complete text
Book I