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.
Let them be Q, R, S, T.
Now since, as Q is to R, so is A to B,
while A, B are least,
and the least numbers measure those which have the same ratio with them the same number of times, the antecedent the antecedent and the consequent the consequent, [VII. 20]
therefore B measures R.
For the same reason C also measures R;
therefore B, C measure R.
Therefore the least number measured by B, C will also measure R. [VII. 35]
But G is the least number measured by B, C;
therefore G measures R.
And, as G is to R, so is K to S: [VII. 13]
therefore K also measures S.
But E also measures S;
therefore E, K measure S.
Therefore the least number measured by E, K will also measure S. [VII. 35]
But M is the least number measured by E, K;
therefore M measures S, the greater the less: which is impossible.
Therefore there will not be any numbers less than N, O, M, P continuously proportional in the ratios of A to B, of C to D, and of E to F;
therefore N, O, M, P are the least numbers continuously proportional in the ratios A : B, C : D, E : F.
Q. E. D.