Book 10
47
And they are rational; therefore EH, HN are rational straight lines commensurable in square only; therefore EN is a binomial straight line divided at H. [X. 36 ]
47
Similarly we can prove that it is also divided at M.
47
And EH is not the same with MN; therefore a binomial has been divided at different points: which is absurd. [X. 42 ]
47
Therefore a side of the sum of two medial areas is not divided at different points; therefore it is divided at one point only.
DEFINITIONS II.
1
1
Given a rational straight line and a binomial, divided into its terms, such that the square on the greater term is greater than the square on the lesser by the square on a straight line commensurable 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 first binomial straight line;
2
2
but if the lesser term be commensurable in length with the rational straight line set out, let the whole be called a second binomial;
3
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.
