第9卷命题 33 · 半为奇数则仅为偶倍奇数

For let the number A have its half odd; I say that A is even-times odd only.

设数A的一半是奇数，则A是偶倍奇数，因为其半数为奇数，以偶数次度量A。

Now that it is even-times odd is manifest; for the half of it, being odd, measures it an even number of times.

假设A同时也是偶倍偶数，则A可被一个偶数按偶数次度量。

[VII. Def. 9] I say next that it is also even-times odd only.

因此A的一半也将被一个偶数度量，但该半数是奇数。

For, if A is even-times even also, it will be measured by an even number according to an even number; [VII. Def. 8] so that the half of it will also be measured by an even number though it is odd: which is absurd.

这与奇数不能被偶数整除的事实矛盾，故A仅为偶倍奇数。