For, if they are not prime to one another, some number will measure them.
Let some number measure them, and let it be C.
And, as many times as C measures A, so many units let there be in D,
and, as many times as C measures B, so many units let there be in E.
Since C measures A according to the units in D,
therefore C by multiplying D has made A. [VII. Def. 15]
For the same reason also C by multiplying E has made B.
Thus the number C by multiplying the two numbers D, E has made A, B;
therefore, as D is to E, so is A to B; [VII. 17]
therefore D, E are in the same ratio with A, B, being less than they: which is impossible.
Therefore no number will measure the numbers A, B.
Therefore A, B are prime to one another.
Q. E. D.
Complete text
Book I