Book 10
85
Now since. as ED is to FD, so is the square on BG to the square on GC, therefore also, convertendo, [v. 19, Por.] as DE is to EF, so is the square on GB to the square on H.
85
But DE has to EF the ratio which a square number has to a square number, for each is square; therefore the square on GB also has to the square on H the ratio which a square number has to a square number; therefore BG is commensurable in length with H. [X. 9]
85
And the square on BG is greater than the square on GC by the square on a straight line commensurable in length with BG.
85
And the whole BG is commensurable in length with the rational straight line A set out.
85
Therefore BC is a first apotome. [X. Deff. III. 1]
85
Therefore the first apotome BC has been found. (Being) that which it was required to find.
PROPOSITION 86.
86
To find the second apotome.
86
Let a rational straight line A be set out, and GC commensurable in length with A; therefore GC is rational.
86
Let two square numbers DE, EF be set out, and let their difference DF not be square.
86
Now let it be contrived that, as FD is to DE, so is the square on CG to the square on GB. [X. 6, Por.]
86
Therefore the square on CG is commensurable with the square on GB. [X. 6]
86
But the square on CG is rational; therefore the square on GB is also rational; therefore BG is rational.
86
And, since the square on GC has not to the square on GB the ratio which a square number has to a square number, CG is incommensurable in length with GB. [X. 9]
