内容 第7卷 · 210
命题 Propositio VII.35
If two numbers measure any number, the least number measured by them will also measure the same.
如果两个数都量尽某个数，那么它们所量尽的最小数也能量尽该数。
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分步证明Step-by-step proof
1 / 4For let the two numbers A, B measure any number CD, and let E be the least that they measure; I say that E also measures CD.
设两数A、B量尽某数CD，且E是它们量尽的最小数。
For, if E does not measure CD, let E, measuring DF, leave CF less than itself.
假设E不量尽CD，则让E量尽DF，余下CF小于E。
Now, since A, B measure E, and E measures DF, therefore A, B will also measure DF.
由于A、B量尽E，且E量尽DF，所以A、B也量尽DF。
But they also measure the whole CD; therefore they will also measure the remainder CF which is less than E: which is impossible.
但A、B也量尽整个CD，因此它们也量尽余数CF，而CF小于E，这是不可能的。
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