CHAPTER LVIII — Critical Commentary & Notes

On Oblong Bodies, That Is, Bodies Longer or Higher Than They Are Wide: And First on the Round Column (Cylinder, Plates XLIII–XLIV).

Historical & Mathematical Commentary

  • Paganini 1509 Marginal Diagram & Plates:

    • Folio 0052 (f. 18v) displays diagram 0052.g1.png showing the axial rectangle \(dcef\) of the cylinder with its axis \(ab\) and diagonal \(de\) intersecting at the center \(g\).

    • Plate XLIII (0277.rst) and Plate XLIV (0279.rst): Note that in the 1509 plates, Plates XLIII–XLIV show the round cone (Cono Solido / Vacuo) while the marginal text numerals shifted slightly between prisms and cones; Cap. LXX explains how to match the Roman numerals in the margins.

  • Comparison with Earlier Draft Track: The earlier divina_en_2/58.txt draft omitted the entire Euclidean proof (Euclid I.15, Euclid I.26, Euclid III.31) that the center \(g\) bisects both the axis \(ab\) and the diagonal \(de\) so that every right cylinder is inscribable in a sphere.

Philological and Folio Apparatus

  • 1509 Paganini Witness Folios: 0051, 0052 (f. 18r–f. 18v)

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/058_cap_LVIII.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/058_cap_LVIII.txt

  • Marginal Woodcut Diagrams: 1 figure(s) extracted from the 1509 Venice witness.