If an area is contained by a rational (straight line) and a second binomial (straight line) then the square root of the area is the irrational (straight line which is) called first bimedial. * For let the area $ABCD$ be contained by the rational (straight line) $AB$ and by the second binomial (straight line) $AD$. * I say that the square root of area $AC$ is a first bimedial (straight line).\
If the rational straight line has unit length then this proposition states that the square root of a second binomial straight line is a first bimedial straight line: i.e., a second binomial straight line has a length \[\frac{\alpha}{\sqrt{1-\beta^{\,2}}}+\alpha,\] whose square root can be written \[\rho\,(\delta^{1/4}+\delta^{3/4}),\] where \[\rho=\sqrt{\frac{k(1+\beta)}{2(1-\beta)}}\quad\text{ and }\quad\delta=\frac{1-\beta}{1+\beta}.\] This is the length of a first bimedial straight line (see [Prop. 10.37]), since $\rho, \delta,\alpha,\beta$ are all positive rational numbers.
Proofs: 1