If four magnitudes be proportional, and the first be commensurable with the second, the third will also be commensurable with the fourth; and, if the first be incommensurable with the second, the third will also be incommensurable with the fourth.
若四个量成比例，且第一个与第二个可公度，则第三个与第四个也可公度；若第一个与第二个不可公度，则第三个与第四个也不可公度。
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Let A, B, C, D be four magnitudes in proportion, so that, as A is to B, so is C to D, and let A be commensurable with B; I say that C will also be commensurable with D. For, since A is commensurable with B, therefore A has to B the ratio which a number has to a number.
设A、B、C、D四个量成比例，即A比B等于C比D，且A与B可公度。
[X. 5] And, as A is to B, so is C to D; therefore C also has to D the ratio which a number has to a number; therefore C is commensurable with D.
因A与B可公度，故A与B的比等于某两数之比（X.5）。
[X. 6] Next, let A be incommensurable with B; I say that C will also be incommensurable with D. For, since A is incommensurable with B, therefore A has not to B the ratio which a number has to a number.
由比例关系，C与D的比也等于该两数之比，故C与D可公度（X.6）。
[X. 7] And, as A is to B, so is C to D; therefore neither has C to D the ratio which a number has to a number; therefore C is incommensurable with D.
若A与B不可公度，则A与B的比不等于任何两数之比（X.7），从而C与D的比也不等于任何两数之比，故C与D不可公度（X.8）。