## Proposition 33

 If a number has its half odd, then it is even-times odd only. Let the number A have its half odd. I say that A is even-times odd only. Now that it is even-times odd is manifest, for the half of it, being odd, measures it an even number of times. VII.Def.9 I say next that it is also even-times odd only. If A is even-times even also, then it is measured by an even number according to an even number, so that the half of it is also measured by an even number though it is odd, which is absurd. VII.Def.8 Therefore A is even-times odd only. Therefore, if a number has its half odd, then it is even-times odd only. Q.E.D.
To say that a number is even-times odd only means that it is even-times odd, but it is not even-times even. As this proposition states, such numbers are the numbers which are twice odd numbers.

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