Book 10
114
For let an area, the rectangle AB, CD, be contained by the apotome AB and the binomial straight line CD, and let CE be the greater term of the latter; let the terms CE, ED of the binomial straight line be commensurable with the terms AF, FB of the apotome and in the same ratio; and let the side of the rectangle AB, CD be G; I say that G is rational.
114
For let a rational straight line H be set out, and to CD let there be applied a rectangle equal to the square on H and producing KL as breadth.
114
Therefore KL is an apotome.
114
Let its terms be KM, ML commensurable with the terms CE, ED of the binomial straight line and in the same ratio. [X. 112]
114
But CE, ED are also commensurable with AF, FB and in the same ratio; therefore, as AF is to FB, so is KM to ML.
114
Therefore, alternately, as AF is to KM, so is BF to LM; therefore also the remainder AB is to the remainder KL as AF is to KM. [V. 19]
114
But AF is commensurable with KM; [X. 12] therefore AB is also commensurable with KL. [X. 11]
114
And, as AB is to KL, so is the rectangle CD, AB to the rectangle CD, KL; [VI. 1] therefore the rectangle CD, AB is also commensurable with the rectangle CD, KL. [X. 11]
114
But the rectangle CD, KL is equal to the square on H; therefore the rectangle CD, AB is commensurable with the square on H.
114
But the square on G is equal to the rectangle CD, AB; therefore the square on G is commensurable with the square on H.
114
But the square on H is rational; therefore the square on G is also rational; therefore G is rational.
