Book 10
18
But, if any straight line be commensurable in square with a given rational straight line, then, if it is also commensurable in length with it, it is called in this case also rational and commensurable with it both in length and in square; but, if again any straight line, being commensurable in square with a given rational straight line, be incommensurable in length with it, it is called in this case also rational but commensurable in square only.]
PROPOSITION 19.
19
The rectangle contained by rational straight lines commensurable in length is rational.
19
For let the rectangle AC be contained by the rational straight lines AB, BC commensurable in length; I say that AC is rational.
19
For on AB let the square AD be described; therefore AD is rational. [X. Def. 4]
19
And, since AB is commensurable in length with BC, while AB is equal to BD, therefore BD is commensurable in length with BC.
19
And, as BD is to BC, so is DA to AC. [VI. 1]
19
Therefore DA is commensurable with AC. [X. 11]
19
But DA is rational; therefore AC is also rational. [X. Def. 4]
19
Therefore etc.
PROPOSITION 20.
20
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.
