| If an odd number measures an even number, then it also measures half of it. | ||
| Let the odd number A measure the even number B.
I say that it also measures the half of it. Since A measures B, let it measure it according to C. I say that C is not odd. | ||
| If possible, let it be so. Then, since A measures B according to C, therefore A multiplied by C makes B. Therefore B is made up of odd numbers the multitude of which is odd. Therefore B is odd, which is absurd, for by hypothesis it is even. Therefore C is not odd, therefore C is even. | IX.23 | |
| Thus A measures B an even number of times. For this reason then it also measures the half of it. | ||
| Therefore, if an odd number measures an even number, then it also measures half of it. | ||
| Q.E.D. | ||
Book IX Introduction - Proposition IX.29 - Proposition IX.31.