If an area is contained by a rational (straight line) and a fourth binomial (straight line) then the square root of the area is the irrational (straight line which is) called major.
If the rational straight line has unit length then this proposition states that the square root of a fourth binomial straight line is a major straight line: i.e., a fourth binomial straight line has a length \[\alpha\,\left(1+\frac 1{\sqrt{1+\beta}}\right)\] whose square root can be written \[\rho\,\sqrt{\frac 12+\frac\delta{2\sqrt{1+\delta^{\,2}}}}+\rho\,\sqrt{\frac 12 -\frac\delta{2\sqrt{1+\delta^{\,2}}}},\] where \[\rho=\sqrt{\alpha}\quad\text{ and }\quad\delta^{\,2}=\beta.\]
This is the length of a major straight line (see [Prop. 10.39]), since $\rho, \delta,\alpha,\beta$ are all positive rational numbers.
Proofs: 1