Kitap 7
33
Therefore the numbers E, F, G measure the numbers A, B, C respectively according to the units in D. [VII. 16]
33
Therefore E, F, G measure A, B, C the same number of times; therefore E, F, G are in the same ratio with A, B, C. [VII. Def. 20]
33
I say next that they are the least that are in that ratio.
33
For, if E, F, G are not the least of those which have the same ratio with A, B, C, there will be numbers less than E, F, G which are in the same ratio with A, B, C.
33
Let them be H, K, L; therefore H measures A the same number of times that the numbers K, L measure the numbers B, C respectively.
33
Now, as many times as H measures A, so many units let there be in M; therefore the numbers K, L also measure the numbers B, C respectively according to the units in M.
33
And, since H measures A according to the units in M, therefore M also measures A according to the units in H. [VII. 16]
33
For the same reason M also measures the numbers B, C according to the units in the numbers K, L respectively;
33
Therefore M measures A, B, C.
33
Now, since H measures A according to the units in M, therefore H by multiplying M has made A. [VII. Def. 15]
33
For the same reason also E by multiplying D has made A.
33
Therefore the product of E, D is equal to the product of H, M.
33
Therefore, as E is to H, so is M to D. [VII. 19]
33
But E is greater than H; therefore M is also greater than D.
33
And it measures A, B, C: which is impossible, for by hypothesis D is the greatest common measure of A, B, C.
