For, since A by multiplying B has made C,
therefore B measures C according to the units in A.
But the unit E also measures the number A according to the units in it;
therefore the unit E measures A the same number of times that B measures C.
Therefore, alternately, the unit E measures the number B the same number of times that A measures C. [VII. 15]
Again, since B by multiplying A has made D,
therefore A measures D according to the units in B.
But the unit E also measures B according to the units in it;
therefore the unit E measures the number B the same number of times that A measures D.
But the unit E measured the number B the same number of times that A measures C;
therefore A measures each of the numbers C, D the same number of times.
Therefore C is equal to D.
Q. E. D.
Complete text
Book I