Proposition: Prop. 10.059: Root of Area contained by Rational Straight Line and Sixth Binomial

(Proposition 59 from Book 10 of Euclid's “Elements”)

If an area is contained by a rational (straight line) and a sixth binomial (straight line) then the square root of the area is the irrational (straight line which is) called the square root of (the sum of) two medial (areas).

For let the area $ABCD$ be contained by the rational (straight line) $AB$ and the sixth binomial (straight line) $AD$, which has been divided into its (component) terms at $E$, such that $AE$ is the greater term.
So, I say that the square root of $AC$ is the square root of (the sum of) two medial (areas).

fig054e

Modern Formulation

If the rational straight line has unit length then this proposition states that the square root of a sixth binomial straight line is the square root of the sum of two medial area: i.e., a sixth binomial straight line has a length \[\sqrt{\alpha}+\sqrt{\beta}\] whose square root can be written \[\alpha^{1/4}\left(\sqrt{\frac 12+\frac{\delta}{2\sqrt{1+\delta^2}}}+\sqrt{\frac 12-\frac{\delta}{2\sqrt{1+\delta^2}}}\right),\] where \[\delta^2=\frac{\alpha-\beta}\beta.\] This is the length of the square root of the sum of two medial area (see [Prop. 10.41]).