Book 10
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.
85
Let two square numbers DE, EF be set out, and let their difference FD not be square; therefore neither has ED to DF the ratio which a square number has to a square number.
85
Let it be contrived that, as ED is to DF, so is the square on BG to the square on GC; [X. 6, Por.] therefore the square on BG is commensurable with the square on GC. [X. 6]
85
But the square on BG is rational; therefore the square on GC is also rational; therefore GC is also rational.
85
And, since ED has not to DF the ratio which a square number has to a square number, therefore neither has the square on BG to the square on GC the ratio which a square number has to a square number; therefore BG is incommensurable in length with GC. [X. 9]
85
And both are rational; therefore BG, GC are rational straight lines commensurable in square only; therefore BC is an apotome. [X. 73]
85
I say next that it is also a first apotome.
85
For let the square on H be that by which the square on BG is greater than the square on GC.
