CHAPTER XXV — Critical Commentary & Notes

How It Is Not Possible for There to Be More Than Five Regular Bodies in Nature, and Why.

Historical & Mathematical Commentary

  • Source in Euclid and Campanus: This famous classification proof comes from the concluding Scholium to Euclid XIII.18 (integrated into the proposition text in Campanus of Novara’s Latin edition as XIII.18 corollary), resting on Euclid XI.21 (the sum of plane angles of any convex solid angle is less than \(4R\)) and Euclid I.32 (the interior angle sum of an \(n\)-gon is \((n-2)\cdot 2R\)).

  • Comparison with Earlier Draft Track: In divina_en_2/25.txt, the angle calculations in fractions of a right angle (\(\frac{2}{3}R\), \(2\frac{2}{3}R\), \(3\frac{1}{3}R\), \(1\frac{1}{5}R\), \(3\frac{3}{5}R\), \(1\frac{1}{3}R\)) were condensed into a brief table. Here the complete rhetorical and deductive structure of Pacioli's 1509 text (0031.rst–0032.rst, f. 8r–f. 8v) is translated in full.

Constantin Winterberg’s Commentary (1889 Edition)

  • Winterberg Footnote 1 (Cap. XXV, p. 214): Namely, each regular solid can be accurately inscribed in the others (scil. der eine lässt sich genau in den andern einbeschreiben). (Winterberg Fussnote 1 (Cap. XXV, S. 214): scil. der eine lässt sich genau in den andern einbeschreiben.)

Philological and Folio Apparatus

  • 1509 Paganini Witness Folios: 0031, 0032 (f. 8r–f. 8v)

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/025_cap_XXV.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/025_cap_XXV.txt

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