Commensurable magnitudes have to one another the ratio which a number has to a number.
可公度量之比等于某数比某数。
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Let A, B be commensurable magnitudes; I say that A has to B the ratio which a number has to a number. For, since A, B are commensurable, some magnitude will measure them.
设A、B为可公度量，则存在某量C可同时量尽A和B。
Let it measure them, and let it be C. And, as many times as C measures A, so many units let there be in D; and, as many times as C measures B, so many units let there be in E.
设C量尽A的次数为D，量尽B的次数为E，则单位量尽数D的次数等于C量尽A的次数，故C比A等于单位比D。
Since then C measures A according to the units in D, while the unit also measures D according to the units in it, therefore the unit measures the number D the same number of times as the magnitude C measures A; therefore, as C is to A, so is the unit to D; [VII. Def. 20] therefore, inversely, as A is to C, so is D to the unit. [cf. V. 7, Por.] Again, since C measures B according to the units in E, while the unit also measures E according to the units in it, therefore the unit measures E the same number of times as C measures B; therefore, as C is to B, so is the unit to E.
同理，C比B等于单位比E，故由反比得A比C等于D比单位。
But it was also proved that, as A is to C, so is D to the unit; therefore, ex aequali, as A is to B, so is the number D to E.
由等比定理，A比B等于数D比数E。