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.

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.