Book 10
58
And, since DE is commensurable in length with AB, that is, with EK, while DE is commensurable with EF, therefore EF is also commensurable with EK. [X. 12]
58
And EK is rational; therefore EL, that is, MR, that is, the rectangle MN, NO, is also rational. [X. 19]
58
Therefore MN, NO are straight lines incommensurable in square which make the sum of the squares on them medial, but the rectangle contained by them rational.
58
Therefore MO is the side of a rational plus a medial area [X. 40] and is the side of the area AC. Q. E. D.
PROPOSITION 59.
59
If an area be contained by a rational straight line and the sixth binomial, the side of the area is the irrational straight line called the side of the sum of two medial areas.
59
For let the area ABCD be contained by the rational straight line AB and the sixth binomial AD, divided into its terms at E, so that AE is the greater term; I say that the side of AC is the side of the sum of two medial areas.
59
Let the same construction be made as before shown.
59
It is then manifest that MO is the side of AC, and that MN is incommensurable in square with NO.
59
Now, since EA is incommensurable in length with AB, therefore EA, AB are rational straight lines commensurable in square only; therefore AK, that is, the sum of the squares on MN, NO, is medial. [X. 21]
59
Again, since ED is incommensurable in length with AB, therefore FE is also incommensurable with EK; [X. 13] therefore FE, EK are rational straight lines commensurable in square only; therefore EL, that is, MR, that is, the rectangle MN, NO, is medial. [X. 21]
59
And, since AE is incommensurable with EF, AK is also incommensurable with EL. [VI. 1, X. 11]
59
But AK is the sum of the squares on MN, NO, and EL is the rectangle MN, NO; therefore the sum of the squares on MN, NO is incommensurable with the rectangle MN, NO.
59
And each of them is medial, and MN, NO are incommensurable in square.
59
Therefore MO is the side of the sum of two medial areas [X. 41], and is the side of AC. Q. E. D.
