If an unit measure any number, and another number measure any other number the same number of times, alternately also, the unit will measure the third number the same number of times that the second measures the fourth.
若单位量尽某数，另一数量尽另一数相同次数，则交替地，单位量尽第三数的次数等于第二数量尽第四数的次数。
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For let the unit A measure any number BC, and let another number D measure any other number EF the same number of times; I say that, alternately also, the unit A measures the number D the same number of times that BC measures EF. For, since the unit A measures the number BC the same number of times that D measures EF, therefore, as many units as there are in BC, so many numbers equal to D are there in EF also.
设单位A量尽数BC，另一数D量尽数EF，且次数相同。
Let BC be divided into the units in it, BG, GH, HC, and EF into the numbers EK, KL, LF equal to D. Thus the multitude of BG, GH, HC will be equal to the multitude of EK, KL, LF.
将BC分为单位BG、GH、HC，将EF分为与D相等的数EK、KL、LF，则BG、GH、HC的个数等于EK、KL、LF的个数。
And, since the units BG, GH, HC are equal to one another, and the numbers EK, KL, LF are also equal to one another, while the multitude of the units BG, GH, HC is equal to the multitude of the numbers EK, KL, LF, therefore, as the unit BG is to the number EK, so will the unit GH be to the number KL, and the unit HC to the number LF. Therefore also, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents; [VII. 12] therefore, as the unit BG is to the number EK, so is BC to EF.
由于单位BG、GH、HC彼此相等，数EK、KL、LF也彼此相等，且个数相等，故单位BG比数EK等于单位GH比数KL，也等于单位HC比数LF。
But the unit BG is equal to the unit A, and the number EK to the number D. Therefore, as the unit A is to the number D, so is BC to EF.
根据VII.12，所有前项之和比所有后项之和等于一个前项比一个后项，因此单位BG比数EK等于BC比EF。但单位BG等于单位A，数EK等于数D，故单位A比数D等于BC比EF。