CHAPTER XLVII

How the Sphere Is Placed and Inscribed in Each of the Said Five Regular Bodies.

To conclude the fifteenth book (Campanus XV.13), having shown how all five regular bodies are inscribed in the sphere and how they are inscribed in one another, we now demonstrate the converse: how a sphere is inscribed inside each of the five regular bodies so that it touches the center of every one of its faces.

Inscribing a Sphere in Any of the Five Regular Solids (Campanus XV.13)

In any given one of the five regular bodies (tetracedron, cubo, octocedron, ycocedron, or duodecedron), to inscribe a sphere that touches all the plane bases (faces) of that body at their centers.

Construction and Proof ([III.14], [XI.11], [III.16], [XI.Def.14])

  1. Every one of the five regular bodies can be inscribed in a circumscribing sphere ([XIII.13]–[XIII.17]), whose center \(a\) is the common center of the regular body.

  2. From this center \(a\) of the circumscribing sphere, draw perpendiculars ([XI.11]) to all the plane bases (faces) of the regular body—namely 4 perpendiculars in the tetracedron, 6 in the cube, 8 in the octocedron, 20 in the ycocedron, and 12 in the duodecedron (see the marginal diagram 0042.g1.png on Folio 0042).

  3. Each plane face of the regular body cuts the circumscribing sphere in a circle that circumscribes that face. Because all the faces of any single regular body are equal and equilateral polygons, all their circumscribing circles are equal circles in the circumscribing sphere.

  4. By the fourteenth of the third, equal circles (and equal chords) in the same sphere lie at equal perpendicular distances from the center of the sphere; and the perpendicular from the center of the sphere to the plane of each circle falls upon the exact center of that circle (and therefore the center of that regular face).

  5. Consequently, all the perpendiculars \(ab, ac, ad, \dots\) drawn from the center \(a\) of the body to the centers \(b, c, d, \dots\) of its faces are mutually equal!

  6. With center \(a\) and radius equal to one of these perpendiculars \(ab\), describe a semicircle in a plane passing through \(ab\) and revolve it around its diameter ([XI.Def.14]).

  7. As the semicircle revolves, its circumference passes through the center \(b, c, d, \dots\) of every face of the regular body; and because each face plane is perpendicular to the radius \(ab, ac, \dots\) at its extremity, by the sixteenth of the third the inscribed sphere touches (continge) every face at its center without cutting through any face!

And recall from Chapter XXXII and Chapter XXXIII that when all five regular bodies are inscribed in one and the same outer sphere, the inner sphere inscribed in the duodecedron is equal to the inner sphere inscribed in the ycocedron, and the inner sphere inscribed in the cube is equal to the inner sphere inscribed in the octocedron!