# 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. * For let the area $ABCD$ be contained by the rational (straight line) $AB$ and by the third binomial (straight line) $AD$, which has been divided into its (component) terms at $E$, of which $AE$ is the greater. * I say that the square root of area $AC$ 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

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