Book 10
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.
39
For let two straight lines AB, BC incommensurable in square, and fulfilling the given conditions [X. 33 ], be added together; I say that AC is irrational.
39
For, since the rectangle AB, BC is medial, twice the rectangle AB, BC is also medial. [X. 6 and 23, Por.]
39
But the sum of the squares on AB, BC is rational; therefore twice the rectangle AB, BC is incommensurable with the sum of the squares on AB, BC, so that the squares on AB, BC together with twice the rectangle AB, BC that is, the square on AC, is also incommensurable with the sum of the squares on AB, BC; [X. 16 ] therefore the square on AC is irrational, so that AC is also irrational. [X. Def. 4 ]
39
And let it be called major. Q. E. D.
PROPOSITION 40.
40
If two straight lines incommensurable in square which make the sum of the squares on them medial, but the rectangle contained by them rational, be added together, the whole straight line is irrational; and let it be called the side of a rational plus a medial area.
