If two straight lines (which are) incommensurable in square, making the sum of the squares on them medial, and the (rectangle contained) by them medial, and, moreover, incommensurable with the sum of the squares on them, are added together then the whole straight line is irrational - let it be called the square root of (the sum of) two medial (areas). * For let the two straight lines, $AB$ and $BC$, incommensurable in square, (and) fulfilling the prescribed (conditions), be laid down together [Prop. 10.35]. * I say that $AC$ is irrational.
Thus, the square root of (the sum of) two medial (areas) has a length expressible as \[\beta^{1/4}\left(\sqrt{\frac 12+\frac{\alpha}{2\sqrt{1+\alpha^2}}} +\sqrt{\frac 12-\frac{\alpha}{2\sqrt{1+\alpha^2}}}\right)\] for some positive rational numbers \(\alpha,\beta\).
This and the corresponding irrational with a minus sign^{1}, whose length is expressible as \[\beta^{1/4}\left(\sqrt{\frac 12+\frac{\alpha}{2\sqrt{1+\alpha^2}}} -\sqrt{\frac 12-\frac{\alpha}{2\sqrt{1+\alpha^2}}}\right)\] (see [Prop. 10.78]), are the positive roots of the quartic \[x^4-2\sqrt{\beta}x^2+ \frac{\beta\alpha^2}{1+\alpha^2}= 0.\]
Proofs: 1
Proofs: 1 2 3 4
Propositions: 5 6 7
Euclid calls it "that which makes with a medial (area) a medial whole" (see [Prop. 10.78]). ↩