Book 10
47
And since, by hypothesis, the sum of the squares on AC, CB is medial, therefore EG is also medial.
47
And it is applied to the rational straight line EF; therefore HE is rational and incommensurable in length with EF. [X. 22 ]
47
For the same reason HN is also rational and incommensurable in length with EF.
47
And, since the sum of the squares on AC, CB is incommensurable with twice the rectangle AC, CB, therefore EG is also incommensurable with GN, so that EH is also incommensurable with HN. [VI. 1 , X. 11 ]
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
