If two commensurable magnitudes be added together, the whole will also be commensurable with each of them; and, if the whole be commensurable with one of them, the original magnitudes will also be commensurable.
若两可公度量相加，则整体与每个分量均可公度；若整体与其中一个分量可公度，则原两量也可公度。
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For let the two commensurable magnitudes AB, BC be added together; I say that the whole AC is also commensurable with each of the magnitudes AB, BC. For, since AB, BC are commensurable, some magnitude will measure them. Let it measure them, and let it be D.
设两可公度量AB、BC相加，整体为AC。因AB、BC可公度，存在某量D能量尽它们。
Since then D measures AB, BC, it will also measure the whole AC. But it measures AB, BC also; therefore D measures AB, BC, AC; therefore AC is commensurable with each of the magnitudes AB, BC.
D能量尽AB、BC，故亦能量尽整体AC。因此D能量尽AB、BC、AC，故AC与AB、BC均可公度。
[X. Def. 1] Next, let AC be commensurable with AB; I say that AB, BC are also commensurable. For, since AC, AB are commensurable, some magnitude will measure them. Let it measure them, and let it be D.
反之，设AC与AB可公度，则存在某量D能量尽AC、AB。
Since then D measures CA, AB, it will also measure the remainder BC. But it measures AB also; therefore D will measure AB, BC; therefore AB, BC are commensurable.
D能量尽CA、AB，故亦能量尽余量BC。又D能量尽AB，故D能量尽AB、BC，因此AB、BC可公度。