Book 9
20
Now A, B, C measure DE; therefore G also will measure DE.
20
But it also measures EF.
20
Therefore G, being a number, will measure the remainder, the unit DF: which is absurd.
20
Therefore G is not the same with any one of the numbers A, B, C.
20
And by hypothesis it is prime.
20
Therefore the prime numbers A, B, C, G have been found which are more than the assigned multitude of A, B, C. Q. E. D.
PROPOSITION 21.
21
If as many even numbers as we please be added together, the whole is even.
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.
