Kitap 10
84
Similarly we can prove that EH is again an apotome and HN an annex to it.
84
Therefore to an apotome different rational straight lines are annexed which are commensurable with the wholes in square only: which was proved impossible. [X. 79]
84
Therefore no other straight line can be so annexed to AB.
84
Therefore to AB only one straight line can be annexed which is incommensurable in square with the whole and which with the whole makes the squares on them added together medial, twice the rectangle contained by them medial, and also the squares on them incommensurable with twice the rectangle contained by them. Q. E. D.
DEFINITIONS III.
1
1
Given a rational straight line and an apotome, if the square on the whole be greater than the square on the annex by the square on a straight line commensurable in length with the whole, and the whole be commensurable in length with the rational straight line set out, let the apotome be called a first apotome.
2
2
But if the annex be commensurable in length with the rational straight line set out, and the square on the whole be greater than that on the annex by the square on a straight line commensurable with the whole, let the apotome be called a second apotome.
3
3
But if neither be commensurable in length with the rational straight line set out, and the square on the whole be greater than the square on the annex by the square on a straight line commensurable with the whole, let the apotome be called a third apotome.
4
4
Again, if the square on the whole be greater than the square on the annex by the square on a straight line incommensurable with the whole, then, if the whole be commensurable in length with the rational straight line set out, let the apotome be called a fourth apotome;
5
5
if the annex be so commensurable, a fifth;
6
6
and, if neither, a sixth.
PROPOSITIONS 85—115.
PROPOSITION 85.
85
To find the first apotome.
85
Let a rational straight line A be set out, and let BG be commensurable in length with A; therefore BG is also rational.
