CHAPTER XXVIII — Critical Commentary & Notes

How the Third Regular Body, Called the Octocedron of Eight Triangular Bases, Is Formed in the Sphere.

Historical & Mathematical Commentary

  • Paganini 1509 Marginal Diagram: Folio 0033 (f. 9r) provides diagram 0033.g2.png illustrating the bisected semicircle \(adb\) and the equatorial square \(efgh\) with its diagonals and perpendicular axis.

  • Three Perpendicular Axes: Notice Step 6: the octahedron is spanned by three mutually perpendicular diameters \(eg\), \(fh\), and \(lm\) intersecting at the center \(k\)—a structural property that Pacioli later calls “the glorious sign of the Holy Cross” in Cap. XLVI when explaining why the octahedron cannot be inscribed in the icosahedron.

Constantin Winterberg’s Commentary (1889 Edition)

  • Winterberg Footnote 1 (Cap. XXVIII, p. 218): Compare Chapter LXXI on the technical term cono (the apex of the pyramid). (Winterberg Fussnote 1 (Cap. XXVIII, S. 218): cfr. Cap. LXXI.)

Philological and Folio Apparatus

  • 1509 Paganini Witness Folios: 0033, 0034 (f. 9r–f. 9v)

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/028_cap_XXVIII.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/028_cap_XXVIII.txt

  • Marginal Woodcut Diagrams: 3 figure(s) extracted from the 1509 Venice witness.