Kitap 10
108
And they are applied to the rational straight line FG; therefore FH is rational and commensurable in length with FG, [X. 20] while FK is rational and incommensurable in length with FG; [X. 22] therefore FH is incommensurable in length with FK. [X. 13]
108
Therefore FH, FK are rational straight lines commensurable in square only; therefore KH is an apotome [X. 73], and KF the annex to it.
108
Now the square on HF is greater than the square on FK by the square on a straight line either commensurable with HF or not commensurable.
108
First, let the square on it be greater by the square on a straight line commensurable with it.
108
Now the whole HF is commensurable in length with the rational straight line FG set out; therefore KH is a first apotome. [X. Deff. III. 1]
108
But the side of the rectangle contained by a rational straight line and a first apotome is an apotome. [X. 91]
108
Therefore the side of LH, that is, of EC, is an apotome.
108
But, if the square on HF is greater than the square on FK by the square on a straight line incommensurable with HF, while the whole FH is commensurable in length with the rational straight line FG set out, KH is a fourth apotome. [X. Deff. III. 4]
108
But the side of the rectangle contained by a rational straight line and a fourth apotome is minor. [X. 94] Q. E. D.
PROPOSITION 109.
109
If from a medial area a rational area be subtracted, there arise two other irrational straight lines, either a first apotome of a medial straight line or a straight line which produces with a rational area a medial whole.
109
For from the medial area BC let the rational area BD be subtracted.
109
I say that the side of the remainder EC becomes one of two irrational straight lines, either a first apotome of a medial straight line or a straight line which produces with a rational area a medial whole.
109
For let a rational straight line FG be set out, and let the areas be similarly applied.
