Construction of Polyhedron in Outer of Concentric Spheres/Porism

Theorem

In the words of Euclid:

But if in another sphere also a polyhedral solid be inscribed similar to the solid in the sphere $BCDE$, the polyhedral solid in the sphere $BCDE$ has to the polyhedral solid in the other sphere the ratio triplicate of that which the diameter of the sphere $BCDE$ has to the diameter of the other sphere.

(The Elements: Book $\text{XII}$: Proposition $17$ : Porism)

Proof

We have that the polyhedron as constructed in Proposition $17$ of Book $\text{XII} $: Construction of Polyhedron in Outer of Concentric Spheres is divided into pyramids which are similar in number and arrangement.

But from Porism to Proposition $8$ of Book $\text{XII} $: Volumes of Similar Tetrahedra are in Triplicate Ratio of Corresponding Sides:

similar pyramids are to one another in the triplicate ratio of their corresponding sides.

Therefore the pyramid whose base is $KBPS$ and whose apex is $A$ has to the similarly arranged pyramid in the other sphere the triplicate ratio of their corresponding sides: that is, $AB$ to the radius of the other sphere.

The same applies to all the other pyramids.

It follows from Proposition $12$ of Book $\text{V} $: Sum of Components of Equal Ratios:

the sum total of all the pyramids that form the whole polyhedron in the one sphere has to the sum total of all the pyramids that form the whole polyhedron in the other sphere the triplicate ratio of the radii of the spheres.

That is, the triplicate ratio of the diameters of the spheres.

$\blacksquare$

Historical Note

This proof is Proposition $17$ of Book $\text{XII}$ 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 $\text{XII}$. Propositions