Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.

PROPOSITION 5.

Let A, B be plane numbers, and let the numbers C, D be the sides of A, and E, F of B;
I say that A has to B the ratio compounded of the ratios of the sides.

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,

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]

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,

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