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.
For let the two commensurable magnitudes / A B /, / B C / be added together; I say that the whole / A C / is also commensurable with each of the magnitudes / A B /, / B C /.
For, since / A B /, / B C / are commensurable, some magnitude will measure them. Let it measure them, and let it be D.
Since then D measures / A B /, / B C /, it will also measure the whole / A C /.
But it measures / A B /, / B C / also;
therefore D measures / A B /, / B C /, / A C /; therefore / A C / is commensurable with each of the magnitudes / A B /, / B C /. [G.372]
Next, let / A C / be commensurable with / A B /; I say that / A B /, / B C / are also commensurable.
For, since / A C /, / A B / are commensurable, some magnitude will measure them. Let it measure them, and let it be D.
Since then D measures / A C /, / A B /, it will also measure the remainder / B C /.
But it measures / A B / also;
therefore D will measure / A B /, / B C /; therefore / A B /, / B C / are commensurable. [G.372]
