If a number neither be one of those which are continually doubled from a dyad, nor have its half odd, it is both eventimes even and even-times odd.
如果一个数既不是从2开始连续加倍得到的数之一，也没有其半数为奇数，那么它既是偶倍偶数又是奇倍偶数。
数 A 既不属于从 2 加倍数列，半也不为奇数；连续二分必到达奇数因子，故 A 既是偶倍偶数也是偶倍奇数。
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For let the number A neither be one of those doubled from a dyad, nor have its half odd; I say that A is both even-times even and even-times odd. Now that A is even-times even is manifest; for it has not its half odd.
设数A既不是从2开始连续加倍得到的数之一，也没有其半数为奇数。
[VII. Def. 8] I say next that it is also even-times odd. For, if we bisect A, then bisect its half, and do this continually, we shall come upon some odd number which will measure A according to an even number.
首先，A是偶倍偶数，因为它的半数不是奇数（根据定义VII.8）。
For, if not, we shall come upon a dyad, and A will be among those which are doubled from a dyad: which is contrary to the hypothesis. Thus A is even-times odd.
其次，若连续平分A及其一半，最终会得到一个奇数，该奇数以偶数倍量A；否则会得到2，从而A属于从2开始连续加倍的数，与假设矛盾。
But it was also proved even-times even.
因此A是奇倍偶数，且已证是偶倍偶数。