CHAPTER LXIV — Critical Commentary & Notes

On Pyramids and All Their Differences, and First on the Round Pyramid (Cone, Plates XLIII–XLIV).

Historical & Mathematical Commentary

  • Archimedean & Apollonian Classification of Cones (Euclid XI.Def.18): Before Apollonius of Perga generalized conic sections to slices of any circular cone, Menaechmus, Euclid, and Archimedes generated the three conic sections—the parabola (section of a right-angled cone), the ellipse (section of an acute-angled cone), and the hyperbola (section of an obtuse-angled cone)—by slicing the three species of cones defined in Euclid XI.Def.18 with a plane perpendicular to a generator! Pacioli gives the complete Euclidean proof via Euclid I.5, Euclid I.19, and Euclid I.32.

Constantin Winterberg’s Commentary (1889 Edition)

  • Winterberg Footnotes 1–2 (Cap. LXIV, p. 260): (1) The marginal cone diagram (Paganini Folio 0054, 0054.g1.png) is placed alongside for clarity. (2) Namely, directed (gerichtet sein) from the apex to the center of the circular base. (Winterberg Fussnoten 1–2 (Cap. LXIV, S. 260): 1) Des Verständnisses wegen ist die nebenstehende Figur hinzugefügt. 2) D. h. gerichtet sein.)

Philological and Folio Apparatus

  • 1509 Paganini Witness Folios: 0055 (f. 20r–f. 20r)

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/064_cap_LXIV.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/064_cap_LXIV.txt

  • Marginal Woodcut Diagrams: 2 figure(s) extracted from the 1509 Venice witness.