Book 10
55
And each of them is rational; therefore EL, that is, MR is rational, [X. 19] and MR is the rectangle MN, NO.
55
But, if two medial straight lines commensurable in square only and containing a rational rectangle be added together, the whole is irrational and is called a first bimedial straight line. [X. 37]
55
Therefore MO is a first bimedial straight line. Q. E. D. 39. Therefore BA, AG and BA, GE are pairs of rational straight lines commensurable in square only. The text has Therefore BA, AG, GE are rational straight lines commensurable in square only, which I have altered because it would naturally convey the impression that any two of the three straight lines are commensurable in square only, whereas AG, GE are commensurable in length (I. 18), and it is only the other two pairs which are commensurable in square only.
PROPOSITION 56.
56
If an area be contained by a rational straight line and the third binomial, the side of the area is the irrational straight line called a second bimedial.
56
For let the area ABCD be contained by the rational straight line AB and the third binomial AD divided into its terms at E, of which terms AE is the greater; I say that the side of the area AC is the irrational straight line called a second bimedial.
56
For let the same construction be made as before.
56
Now, since AD is a third binomial straight line, therefore AE, ED are rational straight lines commensurable in square only, the square on AE is greater than the square on ED by the square on a straight line commensurable with AE, and neither of the terms AE, ED is commensurable in length with AB. [X. Deff. II. 3]
56
Then, in manner similar to the foregoing, we shall prove that MO is the side of the area AC, and MN, NO are medial straight lines commensurable in square only; so that MO is bimedial.
56
It is next to be proved that it is also a second bimedial straight line.
