VIII.5

Plane numbers have to one another the ratio compounded of the ratios of their sides.

Proof

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, as C is to E, so is G to H, and, as D is to F, so is H to K. VIII. 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

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.