Kitap 10
38
Since then AB is incommensurable in length with BC, and, as AB is to BC, so is the square on AB to the rectangle AB, BC, therefore the square on AB is incommensurable with the rectangle AB, BC. [X. 11 ]
38
But the sum of the squares on AB, BC is commensurable with the square on AB, [X. 15 ] and twice the rectangle AB, BC is commensurable with the rectangle AB, BC. [X. 6 ]
38
Therefore the sum of the squares on AB, BC is incommensurable with twice the rectangle AB, BC. [X. 13 ]
38
But EH is equal to the squares on AB, BC, and HF is equal to twice the rectangle AB, BC.
38
Therefore EH is incommensurable with HF, so that DH is also incommensurable in length with HG. [VI. 1 , X. 11 ]
38
Therefore DH, HG are rational straight lines commensurable in square only; so that DG is irrational. [X. 36 ]
38
But DE is rational; and the rectangle contained by an irrational and a rational straight line is irrational; [cf. X. 20 ] therefore the area DF is irrational, and the side of the square equal to it is irrational. [X. Def. 4 ]
38
But AC is the side of the square equal to DF; therefore AC is irrational.
38
And let it be called a second bimedial straight line. Q. E. D.
PROPOSITION 39.
39
If two straight lines incommensurable in square which make the sum of the squares on them rational, but the rectangle contained by them medial, be added together, the whole straight line is irrational : and let it be called major.
