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数学 / 几何原本 / Proposition IX.7

第9卷命题 7 · 合数乘数得立体数

If a composite number by multiplying any number make some number, the product will be solid.

如果一个合数乘以任意数得到某个数,那么该乘积是一个立体数。

A A B B C C D D E E
fig-1

合数 A 与数 B 相乘得 C。D 为 A 的因子,E 记 D 量 A 的次数,故 A = D·E。因 A 乘 B = C 且 A = D·E,所以 C = D·E·B,即三数之积,故 C 为立体数。

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分步证明Step-by-step proof
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  1. For let the composite number A by multiplying any number B make C; I say that C is solid. For, since A is composite, it will be measured by some number.

    设合数A乘以任意数B得C,则C是立体数。

  2. [VII. Def. 13] Let it be measured by D; and, as many times as D measures A, so many units let there be in E.

    因为A是合数,所以它被某个数D量尽(VII.定义13)。

  3. Since then D measures A according to the units in E, therefore E by multiplying D has made A.

    设D量尽A的次数为E,则E乘以D得A(VII.定义15)。

  4. [VII. Def. 15] And, since A by multiplying B has made C, and A is the product of D, E, therefore the product of D, E by multiplying B has made C.

    由于A乘以B得C,且A是D、E之积,故D、E之积乘以B得C,因此C是立体数。