Kitap 9
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.
32
Since then as many numbers as we please beginning from an unit are in continued proportion, and the number A after the unit is prime, therefore D, the greatest of the numbers A, B, C, D, will not be measured by any other number except A, B, C. [IX. 13]
32
And each of the numbers A, B, C is even; therefore D is even-times even only. [VII. Def. 8]
32
Similarly we can prove that each of the numbers B, C is even-times even only. Q. E. D.
PROPOSITION 33.
33
If a number have its half odd, it is even-times odd only.
33
For let the number A have its half odd; I say that A is even-times odd only.
33
Now that it is even-times odd is manifest; for the half of it, being odd, measures it an even number of times. [VII. Def. 9]
33
I say next that it is also even-times odd only.
