CHAPTER LVI — Critical Commentary & Notes

On the Spherical Body and How It Is Formed (Plates XLI–XLII).

Historical & Mathematical Commentary

• Paganini 1509 Plates Referenced:

  • Plate XLI (0273.rst): Sperico Solido (Solid Sphere, labeled XL in the marginal roman numeral on Folio 0049).

  • Plate XLII (0275.rst): Sperico Vacuo (Hollow/Armillary Sphere with meridian and parallel ribs).

• Euclid’s Genetic Definition (Euclid XI.Def.14) vs. Theodosius’s Locus Definition: Pacioli contrasts the static locus definition of the sphere (attributed in medieval Latin manuals to Dionysius or Theodosius, Sphaerica I.Def.1) with Euclid’s kinematic definition (Euclid XI.Def.14) by the revolution of a semicircle around its fixed diameter—the exact kinematic method Euclid uses throughout Books XI–XIII to circumscribe a sphere around each of the five Platonic solids.

Philological and Folio Apparatus

• 1509 Paganini Witness Folios: 0049, 0050 (f. 17r–f. 17v)

• Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/056_cap_LVI.txt

• Winterberg 1889 German Source: winterberg/03_divina_proportione_german/056_cap_LVI.txt

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