Book 8
4
For the same reason C also measures O; therefore B, C measure O; therefore the least number measured by B, C will also measure O. [VII. 35]
4
But G is the least number measured by B, C; therefore G measures O, the greater the less: which is impossible.
4
Therefore there will be no numbers less than H, G, K, L which are continuously in the ratio of A to B, of C to D, and of E to F.
4
Next, let E not measure K.
4
Let M, the least number measured by E, K, be taken.
4
And, as many times as K measures M, so many times let H, G measure N, O respectively, and, as many times as E measures M, so many times let F measure P also.
4
Since H measures N the same number of times that G measures O, therefore, as H is to G, so is N to O. [VII. 13 and Def. 20]
4
But, as H is to G, so is A to B; therefore also, as A is to B, so is N to O.
4
For the same reason also, as C is to D, so is O to M.
4
Again, since E measures M the same number of times that F measures P, therefore, as E is to F, so is M to P; [VII. 13 and Def. 20] therefore N, O, M, P are continuously proportional in the ratios of A to B, of C to D, and of E to F.
4
I say next that they are also the least that are in the ratios A : B, C : D, E : F.
4
For, if not, there will be some numbers less than N, O, M, P continuously proportional in the ratios A : B, C : D, E : F.
4
Let them be Q, R, S, T.
