Book 7
15
But the unit BG is equal to the unit A, and the number EK to the number D.
15
Therefore, as the unit A is to the number D, so is BC to EF.
15
Therefore the unit A measures the number D the same number of times that BC measures EF. Q. E. D.
PROPOSITION 16.
16
If two numbers by multiplying one another make certain numbers, the numbers so produced will be equal to one another.
16
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.
16
For, since A by multiplying B has made C, therefore B measures C according to the units in A.
16
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.
16
Therefore, alternately, the unit E measures the number B the same number of times that A measures C. [VII. 15]
16
Again, since B by multiplying A has made D, therefore A measures D according to the units in B.
16
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.
16
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.
16
Therefore C is equal to D. Q. E. D. The Greek has οἰ γενόμενοι ἐξ αὐτῶν, the (numbers) produced from them. By from them Euclid means from the original numbers, though this is not very clear even in the Greek. I think ambiguity is best avoided by leaving out the words.
