If two straight lines (which are) incommensurable in square, making the sum of the squares on them medial, and the (rectangle contained) by them rational, are added together then the whole straight line is irrational - let it be called the square root of a rational plus a medial (area) . * For let the two straight lines, $AB$ and $BC$, incommensurable in square, (and) fulfilling the prescribed (conditions), be laid down together [Prop. 10.34]. * I say that $AC$ is irrational.
Thus, the square root of a rational plus a medial (area) has a length expressible as \[\sqrt{\frac{\sqrt{1+\rho^2}+\rho}{2\,(1+\rho^2)}} +\sqrt{\frac{\sqrt{1+\rho^2}-\rho}{2\,(1+\rho^2)}},\]
for some positive rational number \(\rho\).
This and the corresponding irrational with a minus sign^{1} whose length is expressible as
\[\sqrt{\frac{\sqrt{1+\rho^2}+\rho}{2\,(1+\rho^2)}} -\sqrt{\frac{\sqrt{1+\rho^2}-\rho}{2\,(1+\rho^2)}},\]
(see Prop. 10.77), are the positive roots of the quartic \[x^4-\frac 2{\sqrt{1+\rho^2}}x^2+ \frac{\rho^2}{(1+\rho^2)^2} = 0.\]
Proofs: 1
Proofs: 1 2 3 4
Propositions: 5 6 7
which Euclid calls "that which makes with a rational (area) a medial whole" (see Prop. 10.77). ↩