CHAPTER LIV — Critical Commentary & Notes

On the Body of Seventy-Two Bases (Septuaginta Duarum Basium), Its Origin, and Its Architectural Use in Cupolas and Vaults (Plates XXXIX–XL).

Historical & Mathematical Commentary

  • Paganini 1509 Plates Referenced:

    • Plate XXXIX (0269.rst): Septuagintaduarum Basium Planum Solidum (72-base faceted sphere, solid).

    • Plate XL (0271.rst): Septuagintaduarum Basium Vacuum (72-base faceted sphere, hollow/coffered).

  • Euclid Book XII, Proposition 17 (Campanus XII.14): Euclid’s construction in Euclid XII.17 inscribes a polyhedron of quadrangular (trapezoidal) zones and polar triangles between two concentric spheres by subdividing the equator and meridians into \(4n\) equal parts—the exact classical source for Renaissance perspective drawings of the mazzocchio (by Paolo Uccello, Piero della Francesca in De Prospectiva Pingendi, and Leonardo da Vinci) and for the coffered hemisphere of the Pantheon (La Rotonda) and Bramante’s Santa Maria presso San Satiro and Santa Maria delle Grazie in Milan.

  • Comparison with Earlier Draft Track: The earlier divina_en_2/54.txt draft omitted Pacioli's discussion of the Pantheon (La Rotonda), Santa Maria delle Grazie, San Satiro, Ludovico Sforza’s devices (la scopetta and la cetta), and Pythagoras’s Angulus Iustitiae. All are restored in full here (0046.rst–0047.rst, f. 15v–f. 16r).

Constantin Winterberg’s Commentary (1889 Edition)

  • Winterberg Footnotes 1–3 (Cap. LIV, pp. 244–245): (1) Namely, generated from the truncated hexahedron (cuboctahedron), so that the truncation operation is repeated twice on the ordinary cube. (2) Pacioli's remark on elevating the 72-base body requires modification, since the faces of the 72-base body are no longer equilateral polygons and therefore cannot support equilateral triangular pyramids of equal edge length. (3) Namely, the 72-faced body (72-Flächner). (Winterberg Fussnoten 1–3 (Cap. LIV, S. 244–245): 1) scil. aus dem abgeschnittenen Hexaeder; die Operation des Abschneidens wiederholt sich also zweimal beim gewöhnlichen Kubus. 2) Dies scheint insofern wenigstens nicht zutreffend und einer Modification zu bedürfen, als die Längen der Seitenkanten ihrer Pyramiden nicht mehr den bisherigen Anforderungen entsprechen können, indem ihre Grundflächen selbst keine gleichseitigen Figuren sind. 3) scil. 72-Flächner.)

Philological and Folio Apparatus

  • 1509 Paganini Witness Folios: 0047, 0048 (f. 16r–f. 16v)

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/054_cap_LIV.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/054_cap_LIV.txt

  • Marginal Woodcut Diagrams: 0 figure(s) extracted from the 1509 Venice witness.