Book 10
68
Therefore CF, FD are straight lines incommensurable in square which make, at the same time, the sum of the squares on them rational, but the rectangle contained by them medial; therefore the whole CD is the irrational straight line called major. [X. 39]
68
Therefore a straight line commensurable with the major straight line is major. Q. E. D.
PROPOSITION 69.
69
A straight line commensurable with the side of a rational plus a medial area is itself also the side of a rational plus a medial area.
69
Let AB be the side of a rational plus a medial area, and let CD be commensurable with AB; it is to be proved that CD is also the side of a rational plus a medial area.
69
Let AB be divided into its straight lines at E; therefore AE, EB are straight lines incommensurable in square which make the sum of the squares on them medial, but the rectangle contained by them rational. [X. 40]
69
Let the same construction be made as before.
69
We can then prove similarly that CF, FD are incommensurable in square, and the sum of the squares on AE, EB is commensurable with the sum of the squares on CF, FD, and the rectangle AE, EB with the rectangle CF, FD; so that the sum of the squares on CF, FD is also medial, and the rectangle CF, FD rational.
69
Therefore CD is the side of a rational plus a medial area. Q. E. D.
PROPOSITION 70.
70
A straight line commensurable with the side of the sum of two medial areas is the side of the sum of two medial areas.
