| Plane numbers have to one another the ratio compounded of the ratios of their sides. | ||
| Let A and B be plane numbers, and let the numbers C and D be the sides of A, and E and F the sides of B.
I say that A has to B the ratio compounded of the ratios of the sides. | ||
| The ratios being given which C has to E and D to F, take the least numbers G, H, and K that are continuously in the ratios C, E, D, and F, so that C is to E as G is to H, and D is to F as H is to K. Multiply D by E to make L. | VIII.4 | |
| Now, since D multiplied by C makes A, and multiplied by E makes L, therefore C is to E as A is to L. But C is to E as G is to H, therefore G is to H as A is to L. | VII.17 | |
| Again, since E multiplied by D makes L, and further multiplied by F makes B, therefore D is to F as L is to B. But D is to F as H is to K, therefore H is to K as L is to B. | VII.17 | |
| But it was also proved that, H as G is to H as A is to L, therefore, ex aequali, L as G is to K as A is 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. | ||
| Therefore, plane numbers have to one another the ratio compounded of the ratios of their sides. | ||
| Q.E.D. | ||
Book VIII Introduction - Proposition VIII.4 - Proposition VIII.6.