If a number by multiplying two numbers make certain numbers, the numbers so produced will have the same ratio as the numbers multiplied.

For let the number A by multiplying the two numbers B, C make D, E; I say that, as B is to C, so is D to E.

For, since A by multiplying B has made D, therefore B measures D according to the units in A.

But the unit F also measures the number A according to the units in it; therefore the unit F measures the number A the same number of times that B measures D.

Therefore, as the unit F is to the number A, so is B to D. [VII. Def. 20]

For the same reason, as the unit F is to the number A, so also is C to E; therefore also, as B is to D, so is C to E.

Therefore, alternately, as B is to C, so is D to E. [VII. 13] Q. E. D.