Proposition: Prop. 10.057: Root of Area contained by Rational Straight Line and Fourth Binomial

Euclid's Formulation

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

Modern Formulation

If the rational straight line has unit length then this proposition states that the square root of a fourth binomial straight line is a major straight line: i.e., a fourth binomial straight line has a length \[\alpha\,\left(1+\frac 1{\sqrt{1+\beta}}\right)\] whose square root can be written \[\rho\,\sqrt{\frac 12+\frac\delta{2\sqrt{1+\delta^{\,2}}}}+\rho\,\sqrt{\frac 12 -\frac\delta{2\sqrt{1+\delta^{\,2}}}},\] where \[\rho=\sqrt{\alpha}\quad\text{ and }\quad\delta^{\,2}=\beta.\]

This is the length of a major straight line (see [Prop. 10.39]), 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