If a rational area be applied to a rational straight line, it produces as breadth a straight line rational and commensurable in length with the straight line to which it is applied.

For let the rational area AC be applied to AB, a straight line once more rational in any of the aforesaid ways, producing BC as breadth; I say that BC is rational and commensurable in length with BA. 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.

Therefore etc.