CHAPTER XXXI — Critical Commentary & Notes¶
On the Rule and Way to Find the Sides of All Five Regular Bodies Placed in One and the Same Sphere, and Their Comparison with One Another.
Historical & Mathematical Commentary¶
Paganini 1509 Marginal Diagram: Folio 0036 (
f. 10v) contains the famous master diagram0036.g1.pngof Euclid XIII.18, displaying the semicircle with all five edges (\(ad\) = tetrahedron, \(fb\) = octahedron, \(db\) = cube, \(ah\) = icosahedron, \(dm\) = dodecahedron).Pacioli’s Observation on Campanus’s Ordering (“Not Without Mystery”): Pacioli astutely explains why Campanus of Novara placed the Cube (Campanus XIII.14) immediately after the Tetrahedron (XIII.13) and before the Octahedron (Campanus XIII.15): the single trisection point \(c\) (\(ac = 2\,cb\)) on diameter \(ab\) gives the tetrahedron edge \(ad\) as the longer chord and the cube edge \(db\) as the shorter chord in the very same right triangle \(adb\)!
Constantin Winterberg’s Commentary (1889 Edition)¶
Winterberg Footnotes 1–5 (Cap. XXXI, pp. 225–226): (1) Namely, $bd^2$. (2) That is, its square equals five times the square of the radius of the circle in which the pentagons of the icosahedron were described (compare Chapter XXIX, Euclid XIII.16). (3) Namely, the icosahedron. (4) Namely, of the sphere diameter (Euclid VI.8 Corollary, Euclid XIII.18). (5) Namely, the regular tetrahedron. (Winterberg Fussnoten 1–5 (Cap. XXXI, S. 225–226): 1) scil. bd². 2) D. h. sein Quadrat ist gleich dem fünffachen Quadrat des Radius desjenigen Kreises, in welchem die Pentagone beschrieben wurden (cfr. Cap. XXIX). 3) scil. Ikosaeder. 4) scil. vom Kugeldurchmesser. 5) scil. reguläre Tetraeder.)
Philological and Folio Apparatus¶
1509 Paganini Witness Folios: 0038, 0039, 0040 (f. 11v–f. 12v)
Winterberg 1889 Italian Source:
winterberg/01_divina_proportione_italian/031_cap_XXXI.txtWinterberg 1889 German Source:
winterberg/03_divina_proportione_german/031_cap_XXXI.txtMarginal Woodcut Diagrams: 0 figure(s) extracted from the 1509 Venice witness.