If two medial (straight lines), commensurable in square only, which contain a medial (area) , are added together then the whole (straight line) is irrational - let it be called a second bimedial (straight line).^{1}
The second bimedial is a straight line whose length is expressible as \[\alpha^{1/4}+\frac{\sqrt{\beta}}{\alpha^{1/4}},\]
where $\alpha$ and $\beta$ are positive rational numbers.
The second bimedial and the second apotome of a medial (see Prop. 10.75), whose length is expressible as
\[\alpha^{1/4}-\frac{\sqrt{\beta}}{\alpha^{1/4}},\]
are the positive roots of the quartic \[x^4-2\,\left(\frac{\left(\alpha+\beta\right)}{\sqrt{\alpha}}\right)\,x^2+ \left(\frac{\left(\alpha-\beta\right)^2}{\alpha}\right) = 0.\]
Proofs: 1
Corollaries: 1
Proofs: 2 3 4 5 6 7
Propositions: 8 9 10 11 12
Literally, "second from two medials" (translator's note). ↩