CHAPTER LXVI — Critical Commentary & Notes

On the Way and Rule to Know How to Measure Every Complete Pyramid, Round or Lateral.

Historical & Mathematical Commentary

  • Epistemological Clarity on :math:`pi approx frac{22}{7}` vs. :math:`V_{text{cone}} = frac{1}{3}V_{text{cylinder}}`: Pacioli makes a remarkably sharp mathematical distinction on Folio 0056 (f. 20v): while the numerical area of the circular base (\(\frac{11}{14}d^{2}\)) is an Archimedean approximation (per la ignorantia dela quadratura del cerchio), the proportional theorem that the cone is exactly one-third (\(\frac{1}{3}\)) of the cylinder (Euclid XII.10, proved by Eudoxus’s method of exhaustion) is exact without the slightest error!

Constantin Winterberg’s Commentary (1889 Edition)

  • Winterberg Footnotes 1–2 (Cap. LXVI, p. 263): (1) That is, according to the numerical value of its volume (dem Zahlenwerthe ihres Inhalts nach), since $\pi$ is irrational, even though geometrically the cone is exactly $\frac{1}{3}$ of the cylinder (Euclid XII.10). (2) Namely, given numerical data (scil. Daten). (Winterberg Fussnoten 1–2 (Cap. LXVI, S. 263): 1) d. i. dem Zahlenwerthe ihres Inhalts nach. 2) scil. Daten.)

Philological and Folio Apparatus

  • 1509 Paganini Witness Folios: 0056 (f. 20v–f. 20v)

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/066_cap_LXVI.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/066_cap_LXVI.txt

  • Marginal Woodcut Diagrams: 4 figure(s) extracted from the 1509 Venice witness.