For, if CA, AB are not incommensurable, some magnitude will measure them.
Let it measure them, if possible, and let it be D.
Since then D measures CA, AB, therefore it will also measure the remainder BC.
But it measures AB also;
therefore D measures AB, BC.
Therefore AB, BC are commensurable;
but they were also, by hypothesis, incommensurable: which is impossible.
Therefore no magnitude will measure CA, AB;
therefore CA, AB are incommensurable. [X. Def. 1]
Similarly we can prove that AC, CB are also incommensurable.
Therefore AC is incommensurable with each of the magnitudes AB, BC.