Book 10
71
Therefore the side of EI is binomial; so that the side of AD is also binomial.
71
Next, let the square on EH be greater than the square on HK by the square on a straight line incommensurable with EH.
71
Now the greater straight line EH is commensurable in length with the rational straight line EF set out; therefore EK is a fourth binomial. [X. Deff. II. 4]
71
But EF is rational; and, if an area be contained by a rational straight line and the fourth binomial, the side of the area is the irrational straight line called major. [X. 57]
71
Therefore the side of the area EI is major; so that the side of the area AD is also major.
71
Next, let AB be less than CD; therefore EG is also less than HI, so that EH is also less than HK.
71
Now the square on HK is greater than the square on EH either by the square on a straight line commensurable with HK or by the square on a straight line incommensurable with it.
71
First, let the square on it be greater by the square on a straight line commensurable in length with itself.
71
Now the lesser straight line EH is commensurable in length with the rational straight line EF set out; therefore EK is a second binomial. [X. Deff. II. 2]
71
But EF is rational, and, if an area be contained by a rational straight line and the second binomial, the side of the square equal to it is a first bimedial; [X. 55] therefore the side of the area EI is a first bimedial, so that the side of AD is also a first bimedial.
71
Next, let the square on HK be greater than the square on HE by the square on a straight line incommensurable with HK.
71
Now the lesser straight line EH is commensurable with the rational straight line EF set out; therefore EK is a fifth binomial. [X. Deff. II. 5]
71
But EF is rational; and, if an area be contained by a rational straight line and the fifth binomial, the side of the square equal to the area is a side of a rational plus a medial area. [X. 58]
