CHAPTER XLI — Critical Commentary & Notes¶
How the Exacedron or Cube Is Placed in the Ycocedron.
Historical & Mathematical Commentary¶
Elegance of Campanus’s Mediated Inscription: Rather than attempting a direct coordinate construction in the icosahedron, Campanus (and Pacioli) exploits the dual dodecahedron (XV.6) together with Euclid’s roof construction (Euclid XIII.17): since every regular dodecahedron contains 5 inscribed cubes (whose 12 edges are face diagonals of the 12 pentagons), inscribing the dual dodecahedron inside the icosahedron immediately places 8 of the icosahedron’s 20 face-centers at the corners of a regular cube!
Constantin Winterberg’s Commentary (1889 Edition)¶
Winterberg Footnote 1 (Cap. XLI, p. 231): Under the designation pyramide in Chapters XLI–XLVI, Pacioli always understands the regular triangular pyramid (the regular tetrahedron). (Winterberg Fussnote 1 (Cap. XLI, S. 231): Unter dieser Bezeichnung ist im Folgenden stets das dreiseitige reguläre Tetraeder verstanden.)
Philological and Folio Apparatus¶
1509 Paganini Witness Folios: 0042 (f. 13v–f. 13v)
Winterberg 1889 Italian Source:
winterberg/01_divina_proportione_italian/041_cap_XLI.txtWinterberg 1889 German Source:
winterberg/03_divina_proportione_german/041_cap_XLI.txtMarginal Woodcut Diagrams: 0 figure(s) extracted from the 1509 Venice witness.