Book 10
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]
71
Therefore the side of the area EI is a side of a rational plus a medial area, so that the side of the area AD is also a side of a rational plus a medial area.
71
Therefore etc. Q. E. D.
PROPOSITION 72.
72
If two medial areas incommensurable with one another be added together, the remaining two irrational straight lines arise, namely either a second bimedial or a side of the sum of two medial areas.
72
For let two medial areas AB, CD incommensurable with one another be added together; I say that the side of the area AD is either a second bimedial or a side of the sum of two medial areas.
72
For AB is either greater or less than CD.
72
First, if it so chance, let AB be greater than CD.
