Proposition: Prop. 10.056: Root of Area contained by Rational Straight Line and Third Binomial

(Proposition 56 from Book 10 of Euclid's “Elements”)

If an area is contained by a rational (straight line) and a third binomial (straight line) then the square root of the area is the irrational (straight line which is) called second bimedial.

Modern Formulation

If the rational straight line has unit length then this proposition states that the square root of a third binomial straight line is a second bimedial straight line: i.e., a third binomial straight line has a length \[\sqrt{\alpha}\,\left(1+\sqrt{1-\beta^{\,2}}\right),\] whose square root can be written \[\rho\,\left(\alpha^{1/4}+\frac{\sqrt \delta}{\alpha^{1/4}}\right),\] where \[\rho=\sqrt{\frac {1+\beta}2}\quad\text{ and }\quad\delta=\alpha\,\frac{1-\beta}{1+\beta}.\]

This is the length of a second bimedial straight line (see [Prop. 10.38]), since $\rho, \delta,\alpha,\beta$ are all positive rational numbers.

Proofs: 1

Proofs: 1
Propositions: 2

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