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.
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.
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.
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.
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.
The ratio compounded from two given ratios a : b and b : c is just the ratio a : c. But if the middle term b is not shared by the two given ratios, then equal ratios must be found that do have a shared middle term.
To find the ratio compounded from two given ratios a : b and c : d, first find e, f, and g so that e : f = a : b and f : g = c : d. Then, the ratio compounded from the ratios a : b and c : d will be the same as the ratio compounded from the ratios e : f and f : g, namely e : g. For numbers, this construction was done in the previous proposition VIII.4.
therefore, the ratio of the plane numbers cd : ef will do.
Euclid’s proof is complicated because (1) his symbolic notation is limited, and (2) the application of VIII.4 to find the least numbers continuously in the ratios c : d and e : f makes the proof more difficult.
Since a = cd, therefore c : e = a : de, and so g : h = a : de. Since b = ef, therefore d : f = de : b, and so h : k = de : b. From the two proportions g : h = a : de and h : k = de : b therefore, ex aequali, g : k = a : b. Thus, ratio the plane numbers is the ratio compounded of the ratios of their sides.