CHAPTER XXXIII — Critical Commentary & Notes¶
On the Proportion of the Surfaces and Solid Volumes of All the Regular Bodies to One Another.
Historical & Mathematical Commentary¶
Dual Pairs Sharing Inspheres: Campanus’s additions to Book XIV highlight one of the deepest symmetries in solid geometry: when all five Platonic solids are inscribed in the same sphere of radius \(R = \frac{1}{2}D\), dual solids share the exact same insphere radius \(r_{\text{in}}\) and the exact same face circumradius \(\rho\):
Cube (6 faces) & Octahedron (8 faces): \(r_{\text{in}} = \frac{R}{\sqrt{3}}\), \(\rho = R\sqrt{\frac{2}{3}}\).
Dodecahedron (12 faces) & Icosahedron (20 faces): \(r_{\text{in}} = R\sqrt{\frac{5+2\sqrt{5}}{15}}\), \(\rho = R\sqrt{\frac{10-2\sqrt{5}}{15}}\).
Tetrahedron (self-dual, 4 faces): \(r_{\text{in}} = \frac{1}{3}R\), \(\rho = R\frac{2\sqrt{2}}{3}\).
Because \(V = \frac{1}{3} r_{\text{in}} \cdot A\), any two dual regular solids inscribed in the same sphere have \(V_{1} : V_{2} = A_{1} : A_{2}\).
Constantin Winterberg’s Commentary (1889 Edition)¶
Winterberg Footnote 1 (Cap. XXXIII, p. 226): Pacioli's traditional Boethian proportion names (sesquitertia = $4:3$, etc.) are retained here for brevity. (Winterberg Fussnote 1 (Cap. XXXIII, S. 226): Der Ausdruck ist hier der Kürze wegen beibehalten = 4:3.)
Philological and Folio Apparatus¶
1509 Paganini Witness Folios: 0040, 0041 (f. 12v–f. 13r)
Winterberg 1889 Italian Source:
winterberg/01_divina_proportione_italian/033_cap_XXXIII.txtWinterberg 1889 German Source:
winterberg/03_divina_proportione_german/033_cap_XXXIII.txtMarginal Woodcut Diagrams: 0 figure(s) extracted from the 1509 Venice witness.