VII.16

If two numbers by multiplying one another make certain numbers, the numbers so produced will be equal to one another.

Proof

Let A, B be two numbers, and let A by multiplying B make C, and B by multiplying A make D; I say that C is equal to D.

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.