| If as many odd numbers as we please are added together, and their multitude is odd, then the sum is also odd. | ||
| Let as many odd numbers as we please, AB, BC, and CD, the multitude of which is odd, be added together.
I say that the sum AD is also odd. |
||
| Subtract the unit DE from CD, therefore the remainder CE is even. | VII.Def.7 | |
| But CA is also even, therefore the sum AE is also even. | IX.22
IX.21 | |
| And DE is a unit. Therefore AD is odd. | VII.Def.7 | |
| Therefore, if as many odd numbers as we please are added together, and their multitude is odd, then the sum is also odd. | ||
| Q.E.D. | ||
Book IX Introduction - Proposition IX.22 - Proposition IX.24.