# Proposition: Prop. 10.058: Root of Area contained by Rational Straight Line and Fifth Binomial

### Euclid's Formulation

If an area is contained by a rational (straight line) and a fifth binomial (straight line) then the square root of the area is the irrational (straight line which is) called the square root of a rational plus a medial (area) . * For let the area $AC$ be contained by the rational (straight line) $AB$ and the fifth binomial (straight line) $AD$, which has been divided into its (component) terms at $E$, such that $AE$ is the greater term. * So I say that the square root of area $AC$ is the irrational (straight line which is) called the square root of a rational plus a medial (area) . ### Modern Formulation

If the rational straight line has unit length then this proposition states that the square root of a fifth binomial straight line is the square root of a rational plus a medial area: i.e., a fifth binomial straight line has a length $\alpha\,(\sqrt{1+\beta}+1)$ whose square root can be written $\rho\,\sqrt{\frac{\sqrt{1+\delta^{2}}+\delta}{2\,(1+\delta^{2})}}+\rho\,\sqrt{\frac{\sqrt{1+\delta^{2}}-\delta}{2\,(1+\delta^{2})}},$ where $\rho=\sqrt{\alpha\,(1+\delta^{2})}\quad\text{ and }\quad\delta^{2}=\beta.$ This is the length of the square root of a rational plus a medial area (see [Prop. 10.40]), since $\rho, \delta,\alpha,\beta$ are all positive rational numbers.

Proofs: 1

Proofs: 1
Propositions: 2

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