For, the ratios being given which C has to E and D to F, let the least numbers G, H, K that are continuously in the ratios C : E, D : F be taken, so that, as C is to E, so is G to H, and, as D is to F, so is H to K. [[VIII. 4](elem.8.4)]

And let D by multiplying E make L.

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](elem.7.17)]

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](elem.7.17)]

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](elem.7.14)]

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.
1, 5, 29, 31. compounded of the ratios of their sides. As in [VI. 23](elem.6.23), the Greek has the less exact phrase, compounded of their sides.