Kitap 10
18
But the sum of BF, DC is commensurable in length with DC; [X. 6] therefore BC is also incommensurable in length with DC, [X. 13] so that, separando, BD is also incommensurable in length with DC. [X. 16]
18
Therefore etc.
18
[Lemma. Since it has been proved that straight lines commensurable in length are always commensurable in square also, while those commensurable in square are not always commensurable in length also, but can of course be either commensurable or incommensurable in length, it is manifest that, if any straight line be commensurable in length with a given rational straight line, it is called rational and commensurable with the other not only in length but in square also, since straight lines commensurable in length are always commensurable in square also.
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.
