|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.|
Next proposition: IX.31