Kitap 9
36
For, if possible, let some number P measure FG, and let P not be the same with any of the numbers A, B, C, D, E, HK, L, M.
36
And, as many times as P measures FG, so many units let there be in Q; therefore Q by multiplying P has made FG.
36
But, further, E has also by multiplying D made FG; therefore, as E is to Q, so is P to D. [VII. 19]
36
And, since A, B, C, D are continuously proportional beginning from an unit, therefore D will not be measured by any other number except A, B, C. [IX. 13]
36
And, by hypothesis, P is not the same with any of the numbers A, B, C; therefore P will not measure D.
36
But, as P is to D, so is E to Q; therefore neither does E measure Q. [VII. Def. 20]
36
And E is prime; and any prime number is prime to any number which it does not measure. [VII. 29]
36
Therefore E, Q are prime to one another.
36
But primes are also least, [VII. 21] and the least numbers measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent; [VII. 20] and, as E is to Q, so is P to D; therefore E measures P the same number of times that Q measures D.
36
But D is not measured by any other number except A, B, C; therefore Q is the same with one of the numbers A, B, C.
36
Let it be the same with B.
36
And, however many B, C, D are in multitude, let so many E, HK, L be taken beginning from E.
36
Now E, HK, L are in the same ratio with B, C, D; therefore, ex aequali, as B is to D, so is E to L. [VII. 14]
