Proposition: Prop. 10.054: Root of Area contained by Rational Straight Line and First Binomial

Euclid's Formulation

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

Modern Formulation

If the rational straight line has unit length then this proposition states that the square root of a first binomial straight line is a binomial straight line: i.e., a first binomial straight line has a length \[\alpha+\alpha\,\sqrt{1-\beta^{\,2}},\] whose square root can be written \[\rho\,(1+\sqrt{\delta}),\] where \[\rho=\sqrt{\frac{\alpha\,(1+\beta)}2}\quad\text{ and }\quad\delta=\frac{1-\beta}{1+\beta.}\] This is the length of a binomial straight line (see [Prop. 10.36]), since $\rho, \delta,\alpha,\beta$ are all positive rational numbers.

Proofs: 1

Proofs: 1
Propositions: 2

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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