If an area is contained by a rational (straight line) and a first binomial (straight line) then the square root of the area is the irrational (straight line which is) called binomial. * For let the area $AC$ be contained by the rational (straight line) $AB$ and by the first binomial (straight line) $AD$. * I say that square root of area $AC$ is the irrational (straight line which is) called binomial.
If the rational straight line has unit length then this proposition states that the square root of a first binomial straight line is a binomial straight line: i.e., a first binomial straight line has a length \[\alpha+\alpha\,\sqrt{1-\beta^{\,2}},\] whose square root can be written \[\rho\,(1+\sqrt{\delta}),\] where \[\rho=\sqrt{\frac{\alpha\,(1+\beta)}2}\quad\text{ and }\quad\delta=\frac{1-\beta}{1+\beta.}\] This is the length of a binomial straight line (see [Prop. 10.36]), since $\rho, \delta,\alpha,\beta$ are all positive rational numbers.
Proofs: 1