Proposition: Prop. 10.038: Second Bimedial is Irrational

(Proposition 38 from Book 10 of Euclid's “Elements”)

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


Modern Formulation

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


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

Thank you to the contributors under CC BY-SA 4.0!



Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

  1. Prime.mover and others: "Pr∞fWiki",, 2016


  1. Literally, "second from two medials" (translator's note).