# Book:Euclid/The Elements/Book XIII

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## Euclid:

## Euclid: *The Elements: Book XIII*

Published $\text {c. 300 B.C.E}$.

### Contents

Book $\text{XIII}$: The Five Platonic Solids

- Proposition $1$: Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio
- Proposition $2$: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio
- Proposition $3$: Area of Square on Lesser Segment of Straight Line cut in Extreme and Mean Ratio
- Proposition $4$: Area of Squares on Whole and Lesser Segment of Straight Line cut in Extreme and Mean Ratio
- Proposition $5$: Straight Line cut in Extreme and Mean Ratio plus its Greater Segment
- Proposition $6$: Segments of Rational Straight Line cut in Extreme and Mean Ratio are Apotome
- Proposition $7$: Equilateral Pentagon is Equiangular if Three Angles are Equal
- Proposition $8$: Straight Lines Subtending Two Consecutive Angles in Regular Pentagon cut in Extreme and Mean Ratio
- Proposition $9$: Sides Appended of Hexagon and Decagon inscribed in same Circle are cut in Extreme and Mean Ratio
- Proposition $10$: Square on Side of Regular Pentagon inscribed in Circle equals Squares on Sides of Hexagon and Decagon inscribed in same Circle
- Proposition $11$: Side of Regular Pentagon inscribed in Circle with Rational Diameter is Minor
- Proposition $12$: Square on Side of Equilateral Triangle inscribed in Circle is Triple Square on Radius of Circle
- Proposition $13$: Construction of Regular Tetrahedron within Given Sphere
- Proposition $14$: Construction of Regular Octahedron within Given Sphere
- Proposition $15$: Construction of Cube within Given Sphere
- Proposition $16$: Construction of Regular Icosahedron within Given Sphere
- Proposition $17$: Construction of Regular Dodecahedron within Given Sphere
- Proposition $18$: Comparison of Sides of Five Platonic Figures

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.4$: Euclid (flourished ca. $300$ B.C.)