If two medial (straight lines), commensurable in square only, which contain a rational (area), are added together then the whole (straight line) is irrational - let it be called a first bimedial^{1} (straight line). * For let the two medial (straight lines), $AB$ and $BC$, commensurable in square only, (and) containing a rational (area), be laid down together. * I say that the whole (straight line), $AC$, is irrational.
Thus, a first bimedial straight line has a length expressible as \[\delta^{1/4}+\delta^{3/4},\]
for some positive rational number \(\delta\).
The first bimedial and the corresponding first apotome of a medial, whose length is expressible as
\[\delta^{1/4} - \delta^{3/4},\]
(see Prop. 10.74), are the positive roots of the quartic \[x^4-2\,\delta\sqrt{\delta} x^2+ \delta\left(1-\delta\right)^2 = 0.\]
Proofs: 1
Corollaries: 1
Proofs: 2 3 4 5 6 7 8 9
Propositions: 10 11 12 13 14 15
Literally, "first from two medials" (translator's note). ↩