If two numbers by multiplying one another make certain numbers, the numbers so produced will be equal to one another.
若两数相乘得某两数，则所得两数相等。
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Let A, B be two numbers, and let A by multiplying B make C, and B by multiplying A make D; I say that C is equal to D. For, since A by multiplying B has made C, therefore B measures C according to the units in A.
设A、B为两数，A乘B得C，B乘A得D。
But the unit E also measures the number A according to the units in it; therefore the unit E measures A the same number of times that B measures C. Therefore, alternately, the unit E measures the number B the same number of times that A measures C.
因A乘B得C，故B依A中的单位数度量C。但单位E亦依其单位数度量A，故E度量A的次数等于B度量C的次数。
[VII. 15] Again, since B by multiplying A has made D, therefore A measures D according to the units in B. But the unit E also measures B according to the units in it; therefore the unit E measures the number B the same number of times that A measures D.
由交替比（VII.15），E度量B的次数等于A度量C的次数。
But the unit E measured the number B the same number of times that A measures C; therefore A measures each of the numbers C, D the same number of times.
又因B乘A得D，故A依B中的单位数度量D。但E亦依其单位数度量B，故E度量B的次数等于A度量D的次数。而E度量B的次数等于A度量C的次数，故A度量C和D的次数相同，因此C等于D。