# Construction of Components of First Bimedial

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## Theorem

In the words of Euclid:

*To find medial straight lines commensurable in square only which contain a rational rectangle.*

(*The Elements*: Book $\text{X}$: Proposition $27$)

## Proof

Let $\rho$ and $\rho \sqrt k$ be rational straight lines which are commensurable in square only.

Their mean proportional is $\rho \sqrt [4] k$ which is medial.

Let $x$ be such that:

- $\rho : \rho \sqrt k = \rho \sqrt [4] k : x$

which gives:

- $x = \rho k^{3/4}$

We have that:

- $\rho \frown \!\! - \rho \sqrt k$

where $\frown \!\! -$ denotes commensurability in square only.

Thus:

- $\rho \sqrt [4] k \frown \!\! - \rho k^{3/4}$

From Straight Line Commensurable with Medial Straight Line is Medial it follows that $\rho k^{3/4}$ is also medial.

$\blacksquare$

## Also see

## Historical Note

This proof is Proposition $27$ of Book $\text{X}$ of Euclid's *The Elements*.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 3*(2nd ed.) ... (previous) ... (next): Book $\text{X}$. Propositions