# 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. * For let the area $AC$ be contained by the rational (straight line) $AB$ and by the first binomial (straight line) $AD$. * I say that square root of area $AC$ 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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