Kitap 9
29
And each of the numbers A, B is odd; therefore C is made up of odd numbers the multitude of which is odd.
29
Thus C is odd. [IX. 23] Q. E. D.
PROPOSITION 30.
30
If an odd number measure an even number, it will also measure the half of it.
30
For let the odd number A measure the even number B; I say that it will also measure the half of it.
30
For, since A measures B, let it measure it according to C; I say that C is not odd.
30
For, if possible, let it be so.
30
Then, since A measures B according to C, therefore A by multiplying C has made B.
30
Therefore B is made up of odd numbers the multitude of which is odd.
30
Therefore B is odd: [IX. 23] which is absurd, for by hypothesis it is even.
30
Therefore C is not odd; therefore C is even.
30
Thus A measures B an even number of times.
30
For this reason then it also measures the half of it. Q. E. D.
PROPOSITION 31.
31
If an odd number be prime to any number, it will also be prime to the double of it.
31
For let the odd number A be prime to any number B, and let C be double of B; I say that A is prime to C.
31
For, if they are not prime to one another, some number will measure them.
31
Let a number measure them, and let it be D.
31
Now A is odd; therefore D is also odd.
31
And since D which is odd measures C, and C is even, therefore [D] will measure the half of C also. [IX. 30]
31
But B is half of C; therefore D measures B.
31
But it also measures A; therefore D measures A, B which are prime to one another: which is impossible.
31
Therefore A cannot but be prime to C.
31
Therefore A, C are prime to one another. Q. E. D.
PROPOSITION 32.
32
Each of the numbers which are continually doubled beginning from a dyad is even-times even only.
32
For let as many numbers as we please, B, C, D, have been continually doubled beginning from the dyad A; I say that B, C, D are eventimes even only.
32
Now that each of the numbers B, C, D is even-times even is manifest; for it is doubled from a dyad.
32
I say that it is also even-times even only.
32
For let an unit be set out.
