If an odd number be prime to any number, it will also be prime to the double of it.
如果一个奇数与某个数互素，那么它也与该数的两倍互素。
奇数 A 与 B 互素，C = 2B。假设 A 与 C 有公因子 D，则 D 必为奇且量 C/2 = B，从而 D 量互素 A、B，矛盾，故 A 与 C 互素。
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For let the odd number A be prime to any number B, and let C be double of B; I say that A is prime to C. For, if they are not prime to one another, some number will measure them.
设奇数A与数B互素，且C是B的两倍。
Let a number measure them, and let it be D. Now A is odd; therefore D is also odd.
假设A与C不互素，则存在某数D同时度量A和C。
And since D which is odd measures C, and C is even, therefore [D] will measure the half of C also. [IX. 30] But B is half of C; therefore D measures B.
由于A是奇数，D也是奇数；又D度量偶数C，故D也度量C的一半，即B。
But it also measures A; therefore D measures A, B which are prime to one another: which is impossible. Therefore A cannot but be prime to C.
但D也度量A，因此D度量互素的A和B，矛盾。所以A与C互素。