|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.||VIII.4|
|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.||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.|
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.
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.
The application of VIII.4 to find the least numbers continuously in the ratios c:d and e:f actually makes the proof more difficult. Here's a slightly shorter proof. Since
therefore, the ratio compounded from the ratios c:e and d:f of the sides is the ratio of the plane numbers a:b.
Next proposition: VIII.6
© 1996, 2002