For let two numbers F, G, the least that are in the ratio of A, C, D, B, be taken, three numbers H, K, L with the same property,
and others more by one continually, until their multitude is equal to the multitude of A, C, D, B. [VIII. 2]
Let them be taken, and let them be M, N, O, P.
It is now manifest that F by multiplying itself has made H and by multiplying H has made M,
while G by multiplying itself has made L and by multiplying L has made P. [VIII. 2, Por.]
And, since M, N, O, P are the least of those which have the same ratio with F, G,
and A, C, D, B are also the least of those which have the same ratio with F, G, [VIII. 1]
while the multitude of the numbers M, N, O, P is equal to the multitude of the numbers A, C, D, B,
therefore M, N, O, P are equal to A, C, D, B respectively;
therefore M is equal to A, and P to B.
Now, since F by multiplying itself has made H,
therefore F measures H according to the units in F.
But the unit E also measures F according to the units in it;
therefore the unit E measures the number F the same number of times as F measures H.
Therefore, as the unit E is to the number F, so is F to H. [VII. Def. 20]
Again, since F by multiplying H has made M,
therefore H measures M according to the units in F.
But the unit E also measures the number F according to the units in it;
therefore the unit E measures the number F the same number of times as H measures M.
Therefore, as the unit E is to the number F, so is H to M.
But it was also proved that, as the unit E is to the number F, so is F to H;
therefore also, as the unit E is to the number F, so is F to H, and H to M.
But M is equal to A;
therefore, as the unit E is to the number F, so is F to H, and H to A.
For the same reason also, as the unit E is to the number G, so is G to L and L to B.
Therefore, as many numbers as have fallen between A, B in continued proportion,
so many numbers also have fallen between each of the numbers A, B and the unit E in continued proportion.
Q. E. D.
Complete text
Book I