Book 8
4
Let the given ratios in least numbers be that of A to B, that of C to D, and that of E to F; thus it is required to find numbers in continued proportion which are the least that are in the ratio of A to B, in the ratio of C to D, and in the ratio of E to F.
4
Let G, the least number measured by B, C, be taken. [VII. 34]
4
And, as many times as B measures G, so many times also let A measure H, and, as many times as C measures G, so many times also let D measure K.
4
Now E either measures or does not measure K.
4
First, let it measure it.
4
And, as many times as E measures K, so many times let F measure L also.
4
Now, since A measures H the same number of times that B measures G, therefore, as A is to B, so is H to G. [VII. Def. 20, VII. 13]
4
For the same reason also, as C is to D, so is G to K, and further, as E is to F, so is K to L; therefore H, G, K, L are continuously proportional in the ratio of A to B, in the ratio of C to D, and in the ratio of E to F.
4
I say next that they are also the least that have this property.
4
For, if H, G, K, L are not the least numbers continuously proportional in the ratios of A to B, of C to D, and of E to F, let them be N, O, M, P.
4
Then since, as A is to B, so is N to O, while A, B are least, and the least numbers measure those which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent; therefore B measures O. [VII. 20]
