Euclid:Proposition/XIV/8/Summary

Proposition

In the words of Hypsicles of Alexandria:

If $AB$ be any straight line divided at $C$ in extreme and mean ratio, $AC$ being the greater segment, and if we have a cube, a dodecahedron and an icosahedron inscribed in one and the same sphere, then:

$(1) \quad \left({\text{side of cube} }\right) : \left({\text{side of icosahedron} }\right) = \sqrt {AB^2 + AC^2} : \sqrt {AB^2 + BC^2}$;

$(2) \quad \left({\text{surface of dodecahedron} }\right) : \left({\text{surface of icosahedron} }\right)$

$ = \left({\text{side of cube} }\right) : \left({\text{side of icosahedron} }\right)$;

$(3) \quad \left({\text{content of dodecahedron} }\right) : \left({\text{content of icosahedron} }\right)$

$ = \left({\text{surface of dodecahedron} }\right) : \left({\text{surface of icosahedron} }\right)$;

and $(4) \quad \left({\text{content of dodecahedron} }\right) : \left({\text{content of icosahedron} }\right)$

$ = \sqrt {AB^2 + AC^2} : \sqrt {AB^2 + BC^2}$

(The Elements: Book $\text{XIV}$: Proposition $8$ : Summary)

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous): The Contents of the So-Called Book $\text{XIV}$. By Hypsicles