Book 9
21
For let as many even numbers as we please, AB, BC, CD, DE, be added together; I say that the whole AE is even.
21
For, since each of the numbers AB, BC, CD, DE is even, it has a half part; [VII. Def. 6] so that the whole AE also has a half part.
21
But an even number is that which is divisible into two equal parts; [id.] therefore AE is even. Q. E. D.
PROPOSITION 22.
22
If as many odd numbers as we please be added together, and their multitude be even, the whole will be even.
22
For let as many odd numbers as we please, AB, BC, CD, DE, even in multitude, be added together; I say that the whole AE is even.
22
For, since each of the numbers AB, BC, CD, DE is odd, if an unit be subtracted from each, each of the remainders will be even; [VII. Def. 7] so that the sum of them will be even. [IX. 21]
22
But the multitude of the units is also even.
22
Therefore the whole AE is also even. [IX. 21] Q. E. D.
PROPOSITION 23.
23
If as many odd numbers as we please be added together, and their multitude be odd, the whole will also be odd.
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.
