Kitap 9
13
But, further, A has also by multiplying B made C; [IX. 11] therefore the product of A, B is equal to the product of F, G.
13
Therefore, proportionally, as A is to F, so is G to B. [VII. 19]
13
But A measures F; therefore G also measures B.
13
Let it measure it according to H.
13
Similarly then we can prove that H is not the same with A.
13
And, since G measures B according to H, therefore G by multiplying H has made B.
13
But further A has also by multiplying itself made B; [IX. 8] therefore the product of H, G is equal to the square on A.
13
Therefore, as H is to A, so is A to G. [VII. 19]
13
But A measures G;lt therefore H also measures A, which is prime, though it is not the same with it: which is absurd.
13
Therefore D the greatest will not be measured by any other number except A, B, C. Q. E. D.
PROPOSITION 14.
14
If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it.
14
For let the number A be the least that is measured by the prime numbers B, C, D; I say that A will not be measured by any other prime number except B, C, D.
14
For, if possible, let it be measured by the prime number E, and let E not be the same with any one of the numbers B, C, D.
14
Now, since E measures A, let it measure it according to F; therefore E by multiplying F has made A.
14
And A is measured by the prime numbers B, C, D.
14
But, if two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers; [VII. 30] therefore B, C, D will measure one of the numbers E, F.
14
Now they will not measure E; for E is prime and not the same with any one of the numbers B, C, D.
14
Therefore they will measure F, which is less than A: which is impossible, for A is by hypothesis the least number measured by B, C, D.
14
Therefore no prime number will measure A except B, C, D. Q. E. D.
PROPOSITION 15.
