灯下 · 几何原本 · 第9卷命题 23 · 奇数个奇数之和为奇数

内容第9卷 · 264

命题 Propositio IX.23

If as many odd numbers as we please be added together, and their multitude be odd, the whole will also be odd.

若任意多个奇数相加，且它们的个数为奇数，则总和也是奇数。

fig-1

奇数个奇数 AB、BC、CD 相加，从 CD 中分出单位 DE，则 CE 为偶数，且 CA 为偶数（IX.22），故 AE 为偶数，加单位得 AD 为奇数。

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分步证明Step-by-step proof

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For let as many odd numbers as we please, AB, BC, CD, the multitude of which is odd, be added together; I say that the whole AD is also odd.
设奇数AB、BC、CD相加，其个数为奇数，总和为AD。
Let the unit DE be subtracted from CD; therefore the remainder CE is even.
从CD中减去单位DE，则余数CE为偶数。
[VII. Def. 7] But CA is also even; [IX. 22] therefore the whole AE is also even.
因CA为偶数，故AE为偶数。
[IX. 21] And DE is an unit.
而DE是单位，所以AD为奇数。

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