Book 7
15
And, since the units BG, GH, HC are equal to one another, and the numbers EK, KL, LF are also equal to one another, while the multitude of the units BG, GH, HC is equal to the multitude of the numbers EK, KL, LF, therefore, as the unit BG is to the number EK, so will the unit GH be to the number KL, and the unit HC to the number LF.
15
Therefore also, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents; [VII. 12] therefore, as the unit BG is to the number EK, so is BC to EF.
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.
