For, if possible, let A measure C.
And, however many A, B, C are, let as many numbers F, G, H, the least of those which have the same ratio with A, B, C, be taken. [VII. 33]
Now, since F, G, H are in the same ratio with A, B, C, and the multitude of the numbers A, B, C is equal to the multitude of the numbers F, G, H,
therefore, ex aequali, as A is to C, so is F to H. [VII. 14]
And since, as A is to B, so is F to G,
while A does not measure B,
therefore neither does F measure G; [VII. Def. 20]
therefore F is not an unit,
for the unit measures any number.
Now F, H are prime to one another. [VIII. 3]
And, as F is to H, so is A to C;
therefore neither does A measure C.
Similarly we can prove that neither will any other measure any other.
Q. E. D.
Complete text
Book I