Since AG is incommensurable with EG, AH is also incommensurable with GK, that is, SN with NQ; [VI. 1, X. 11]
therefore MN, NO are incommensurable in square.
And, since AE is commensurable with AB, AK is rational; [X. 19]
and it is equal to the squares on MN, NO;
therefore the sum of the squares on MN, NO is also rational.
And, since DE is incommensurable in length with AB, that is, with EK,
while DE is commensurable with EF, therefore EF is incommensurable in length with EK. [X. 13]
Therefore EK, EF are rational straight lines commensurable in square only;
therefore LE, that is, MR, is medial. [X. 21]
And it is contained by MN, NO;
therefore the rectangle MN, NO is medial.
And the [sum] of the squares on MN, NO is rational, and MN, NO are incommensurable in square.
But, if two straight lines incommensurable in square and making the sum of the squares on them rational, but the rectangle contained by them medial, be added together, the whole is irrational and is called major. [X. 39]