For on AB let the square AD be described;
therefore AD is rational. [X. Def. 4]
But AC is also rational;
therefore DA is commensurable with AC.
And, as DA is to AC, so is DB to BC. [VI. 1]
Therefore DB is also commensurable with BC; [X. 11]
and DB is equal to BA;
therefore AB is also commensurable with BC.
But AB is rational;
therefore BC is also rational and commensurable in length with AB.