Kitap 10
3
and if neither of the terms be commensurable in length with the rational straight line set out, let the whole be called a third binomial.
4
4
Again, if the square on the greater term be greater than the square on the lesser by the square on a straight line incommensurable in length with the greater, then, if the greater term be commensurable in length with the rational straight line set out, let the whole be called a fourth binomial;
5
5
if the lesser, a fifth binomial;
6
6
and if neither, a sixth binomial.
PROPOSITIONS 48—84.
PROPOSITION 48.
48
To find the first binomial straight line.
48
Let two numbers AC, CB be set out such that the sum of them AB has to BC the ratio which a square number has to a square number, but has not to CA the ratio which a square number has to a square number; [Lemma I after X. 28] let any rational straight line D be set out, and let EF be commensurable in length with D.
48
Therefore EF is also rational.
48
Let it be contrived that, as the number BA is to AC, so is the square on EF to the square on FG. [X. 6, Por.]
48
But AB has to AC the ratio which a number has to a number; therefore the square on EF also has to the square on FG the ratio which a number has to a number, so that the square on EF is commensurable with the square on FG. [X. 6]
48
And EF is rational; therefore FG is also rational.
