CHAPTER XXIV — Critical Commentary & Notes

On the Tenth Effect of Our Divine Proportion, Which Introduces the Formation of the Five Regular Bodies.

Historical & Mathematical Commentary

  • Role as Bridge to Section 3: Cap. XXIV concludes the chain of planar theorems from Book XIII (Euclid XIII.1–Euclid XIII.12) by pairing Euclid XIII.10 (\(s_{5}^{2} = s_{6}^{2} + s_{10}^{2}\)) with Euclid XIII.9 (\(s_{6} : s_{10} = (s_{6}+s_{10}) : s_{6}\)). This Pythagorean triad of regular polygon sides (\(s_5, s_6, s_{10}\)) is the exact geometric engine Euclid uses in Euclid XIII.16 to construct the icosahedron and in Euclid XIII.17 to construct the dodecahedron.

  • Comparison with Earlier Draft Track: The earlier divina_en_2 draft reduced this chapter to a three-line summary without the corollary on Euclid XIII.9 or the numerical illustration. The present translation restores the full 1509 Paganini text (0031.rst, f. 8r).

Philological and Folio Apparatus

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

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/024_cap_XXIV.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/024_cap_XXIV.txt

  • Marginal Woodcut Diagrams: 0 figure(s) extracted from the 1509 Venice witness.