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