内容 第7卷 · 193
命题 Propositio VII.18
If two numbers by multiplying any number make certain numbers, the numbers so produced will have the same ratio as the multipliers.
若两数分别乘以同一数，则所得两积之比等于原两数之比。
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分步证明Step-by-step proof
1 / 4For let two numbers A, B by multiplying any number C make D, E; I say that, as A is to B, so is D to E.
设两数A、B乘以同一数C，分别得D、E。
For, since A by multiplying C has made D, therefore also C by multiplying A has made D.
因A乘C得D，由乘法交换律，C乘A也得D。
[VII. 16] For the same reason also C by multiplying B has made E.
同理，C乘B得E。
Therefore the number C by multiplying the two numbers A, B has made D, E.
故数C乘A、B得D、E，因此D比E等于A比B。
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