内容 第9卷 · 270
命题 Propositio IX.29
If an odd number by multiplying an odd number make some number, the product will be odd.
若一个奇数乘以另一个奇数得某数，则其积为奇数。
奇数 A 乘奇数 B 得 C：C 由 A 个 B 相加，A、B 皆为奇数，故 C 为奇数个奇数之和，由 IX.23，C 为奇数。
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分步证明Step-by-step proof
1 / 4For let the odd number A by multiplying the odd number B make C; I say that C is odd.
设奇数A乘以奇数B得C，则C由与A中单位数相等个B相加而成。
For, since A by multiplying B has made C, therefore C is made up of as many numbers equal to B as there are units in A.
因A、B均为奇数，故C由奇数个奇数相加而成。
[VII. Def. 15] And each of the numbers A, B is odd; therefore C is made up of odd numbers the multitude of which is odd.
奇数个奇数之和为奇数，故C为奇数。
因此，奇数乘奇数之积为奇数。
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