Book:Euclid/The Elements/Book XIII

From ProofWiki
Jump to navigation Jump to search

Euclid: The Elements: Book XIII

Published $c. 300 B.C.E$.


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
Lemma to 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
Lemma to 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
Porism to Proposition $16$: Construction of Regular Icosahedron within Given Sphere
Proposition $17$: Construction of Regular Dodecahedron within Given Sphere
Porism to Proposition $17$: Construction of Regular Dodecahedron within Given Sphere
Proposition $18$: Comparison of Sides of Five Platonic Figures
Endnote to Proposition $18$: Comparison of Sides of Five Platonic Figures
Lemma to Proposition $18$: Comparison of Sides of Five Platonic Figures


Sources