Kitap 9
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.
15
If three numbers in continued proportion be the least of those which have the same ratio with them, any two whatever added together will be prime to the remaining number.
15
Let A, B, C, three numbers in continued proportion, be the least of those which have the same ratio with them; I say that any two of the numbers A, B, C whatever added together are prime to the remaining number, namely A, B to C; B, C to A; and further A, C to B.
15
For let two numbers DE, EF, the least of those which have the same ratio with A, B, C, be taken. [VIII. 2]
