Kitap 10
95
Therefore the remainder LN is the irrational straight line called that which produces with a rational area a medial whole; [X. 77] and it is the side of the area AB.
95
Therefore the side of the area AB is a straight line which produces with a rational area a medial whole. Q. E. D.
PROPOSITION 96.
96
If an area be contained by a rational straight line and a sixth apotome, the side of the area is a straight line which produces with a medial area a medial whole.
96
For let the area AB be contained by the rational straight line AC and the sixth apotome AD; I say that the side of the area AB is a straight line which produces with a medial area a medial whole.
96
For let DG be the annex to AD; therefore AG, GD are rational straight lines commensurable in square only, neither of them is commensurable in length with the rational straight line AC set out, and the square on the whole AG is greater than the square on the annex DG by the square on a straight line incommensurable in length with AG. [X. Deff. III. 6]
96
Since then the square on AG is greater than the square on GD by the square on a straight line incommensurable in length with AG, therefore, if there be applied to AG a parallelogram equal to the fourth part of the square on DG and deficient by a square figure, it will divide it into incommensurable parts. [X. 18]
96
Let then DG be bisected at E, let there be applied to AG a parallelogram equal to the square on EG and deficient by a square figure, and let it be the rectangle AF, FG; therefore AF is incommensurable in length with FG.
96
But, as AF is to FG, so is AI to FK. [VI. 1] therefore AI is incommensurable with FK. [X. 11]
96
And, since AG, AC are rational straight lines commensurable in square only, AK is medial. [X. 21]
96
Again, since AC, DG are rational straight lines and incommensurable in length, DK is also medial. [X. 21]
96
Now, since AG, GD are commensurable in square only, therefore AG is incommensurable in length with GD.
