内容 第10卷 · 285
命题 Propositio X.8
If two magnitudes have not to one another the ratio which a number has to a number, the magnitudes will be incommensurable.
若两量之比不等于两数之比，则两量不可公度。
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分步证明Step-by-step proof
1 / 4For let the two magnitudes A, B not have to one another the ratio which a number has to a number; I say that the magnitudes A, B are incommensurable.
设两量A、B之比不等于两数之比。
For, if they are commensurable, A will have to B the ratio which a number has to a number.
假设A、B可公度，则根据X.5，A与B之比等于两数之比。
[X. 5] But it has not; therefore the magnitudes A, B are incommensurable.
但假设中A与B之比不等于两数之比，矛盾。
因此A、B不可公度。
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