Book 10
110
And, since BC is incommensurable with BD, that is, GH with GK, HF is also incommensurable with FK; [VI. 1, X. 11] therefore FH, FK are rational straight lines commensurable in square only; therefore KH is an apotome. [X. 73]
110
If then the square on FH is greater than the square on FK by the square on a straight line commensurable with FH, while neither of the straight lines FH, FK is commensurable in length with the rational straight line FG set out, KH is a third apotome. [X. Deff. III. 3]
110
But KL is rational, and the rectangle contained by a rational straight line and a third apotome is irrational, and the side of it is irrational, and is called a second apotome of a medial straight line; [X. 93] so that the side of LH, that is, of EC, is a second apotome of a medial straight line.
110
But, if the square on FH is greater than the square on FK by the square on a straight line incommensurable with FH, while neither of the straight lines HF, FK is commensurable in length with FG, KH is a sixth apotome. [X. Deff. III. 6]
110
But the side of the rectangle contained by a rational straight line and a sixth apotome is a straight line which produces with a medial area a medial whole. [X. 96]
110
Therefore the side of LH, that is, of EC, is a straight line which produces with a medial area a medial whole. Q. E. D.
PROPOSITION 111.
111
The apotome is not the same with the binomial straight line.
111
Let AB be an apotome; I say that AB is not the same with the binomial straight line.
111
For, if possible, let it be so; let a rational straight line DC be set out, and to CD let there be applied the rectangle CE equal to the square on AB and producing DE as breadth.
111
Then, since AB is an apotome, DE is a first apotome. [X. 97]
