Proposition: Prop. 10.055: Root of Area contained by Rational Straight Line and Second Binomial

(Proposition 55 from Book 10 of Euclid's “Elements”)

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


Modern Formulation

If the rational straight line has unit length then this proposition states that the square root of a second binomial straight line is a first bimedial straight line: i.e., a second binomial straight line has a length \[\frac{\alpha}{\sqrt{1-\beta^{\,2}}}+\alpha,\] whose square root can be written \[\rho\,(\delta^{1/4}+\delta^{3/4}),\] where \[\rho=\sqrt{\frac{k(1+\beta)}{2(1-\beta)}}\quad\text{ and }\quad\delta=\frac{1-\beta}{1+\beta}.\] This is the length of a first bimedial straight line (see [Prop. 10.37]), 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