Ratio of Volumes of Regular Dodecahedron and Regular Icosahedron in Same Sphere/Lemma
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Lemma to Ratio of Volumes of Regular Dodecahedron and Regular Icosahedron in Same Sphere
In the words of Hypsicles of Alexandria:
- If two straight lines be cut in extreme and mean ratio, the segments of both are in one and the same ratio.
(The Elements: Book $\text{XIV}$: Proposition $8$ : Lemma)
Proof
Let $AB$ be cut at $C$ in extreme and mean ratio where $AC$ is the greater segment.
Let $DE$ be cut at $F$ in extreme and mean ratio where $DF$ is the greater segment.
It is to be demonstrated that:
- $AB : AC = DE : DF$
Thus:
\(\ds AB \cdot BC\) | \(=\) | \(\ds AC^2\) | ||||||||||||
\(\ds DE \cdot EF\) | \(=\) | \(\ds DF^2\) | ||||||||||||
\(\ds \therefore \ \ \) | \(\ds AB \cdot BC : AC^2\) | \(=\) | \(\ds DE \cdot EF : DF^2\) | |||||||||||
\(\ds \therefore \ \ \) | \(\ds 4 \cdot AB \cdot BC : AC^2\) | \(=\) | \(\ds 4 \cdot DE \cdot EF : DF^2\) | |||||||||||
\(\ds \therefore \ \ \) | \(\ds \left({4 \cdot AB \cdot BC + AC^2}\right) : AC^2\) | \(=\) | \(\ds \left({4 \cdot DE \cdot EF + DE^2}\right) : DF^2\) | Proposition $18$ of Book $\text{V} $: Magnitudes Proportional Separated are Proportional Compounded | ||||||||||
\(\ds \therefore \ \ \) | \(\ds \left({AB + BC}\right)^2 : AC^2\) | \(=\) | \(\ds \left({DE + EF}\right)^2 : DF^2\) | Proposition $5$ of Book $\text{II} $: Difference of Two Squares | ||||||||||
\(\ds \therefore \ \ \) | \(\ds \left({AB + BC}\right) : AC\) | \(=\) | \(\ds \left({DE + EF}\right) : DF\) | |||||||||||
\(\ds \therefore \ \ \) | \(\ds \left({AB + BC + AC}\right) : AC\) | \(=\) | \(\ds \left({DE + EF + DF}\right) : DF\) | Proposition $18$ of Book $\text{V} $: Magnitudes Proportional Separated are Proportional Compounded | ||||||||||
\(\ds \therefore \ \ \) | \(\ds 2 \cdot AB : AC\) | \(=\) | \(\ds 2 \cdot DE : DF\) | |||||||||||
\(\ds \therefore \ \ \) | \(\ds AB : AC\) | \(=\) | \(\ds DE : DF\) |
$\blacksquare$
Historical Note
This proof is Proposition $8$ of Book $\text{XIV}$ of Euclid's The Elements.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): The So-Called Book $\text{XIV}$, by Hypsicles