Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.

But, as H is to G, so is A to B;
therefore also, as A is to B, so is N to O.

For the same reason also,

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.

I say next that they are also the least that are in the ratios A : B, C : D, E : F.

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;