Book 9
23
For let as many odd numbers as we please, AB, BC, CD, the multitude of which is odd, be added together; I say that the whole AD is also odd.
23
Let the unit DE be subtracted from CD; therefore the remainder CE is even. [VII. Def. 7]
23
But CA is also even; [IX. 22] therefore the whole AE is also even. [IX. 21]
23
And DE is an unit.
23
Therefore AD is odd. [VII. Def. 7] Q. E. D. 3. Literally let there be as many numbers as we please, of which let the multitude be odd. This form, natural in Greek, is awkward in English.
PROPOSITION 24.
24
If from an even number an even number be subtracted, the remainder will be even.
24
For from the even number AB let the even number BC be subtracted: I say that the remainder CA is even.
24
For, since AB is even, it has a half part. [VII. Def. 6]
24
For the same reason BC also has a half part; so that the remainder [CA also has a half part, and] AC is therefore even. Q. E. D.
PROPOSITION 25.
25
If from an even number an odd number be subtracted, the remainder will be odd.
25
For from the even number AB let the odd number BC be subtracted; I say that the remainder CA is odd.
25
For let the unit CD be subtracted from BC; therefore DB is even. [VII. Def. 7]
25
But AB is also even; therefore the remainder AD is also even. [IX. 24]
25
And CD is an unit; therefore CA is odd. [VII. Def. 7] Q. E. D.
PROPOSITION 26.
26
If from an odd number an odd number be subtracted, the remainder will be even.
26
For from the odd number AB let the odd number BC be subtracted; I say that the remainder CA is even.
