内容 第8卷 · 221
命题 Propositio VIII.7
If there be as many numbers as we please in continued proportion, and the first measure the last, it will measure the second also.
若任意多个数成连比例，且第一个数能量尽最后一个数，则它也能量尽第二个数。
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分步证明Step-by-step proof
1 / 4Let there be as many numbers as we please, A, B, C, D, in continued proportion; and let A measure D; I say that A also measures B.
设A、B、C、D成连比例，且A量尽D。
For, if A does not measure B, neither will any other of the numbers measure any other.
假设A不量尽B，则根据第八卷命题6，这些数中任何一个都不能量尽另一个。
[VIII. 6] But A measures D.
但已知A量尽D，与假设矛盾。
因此A必量尽B。
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