CHAPTER XXXVIII — Critical Commentary & Notes

How the Exacedron or Cube Is Formed in the Octocedron.

Historical & Mathematical Commentary

• Paganini 1509 Marginal Diagram: Folio 0039 (f. 12r) contains diagram 0039.g1.png showing the octahedron and the cube whose 8 vertices sit at the centroids of the 8 triangular faces.

• Exact Ratio: If the octahedron has edge \(s_{\text{octa}}\), its equatorial square has diagonal \(s_{\text{octa}}\sqrt{2}\). The centroids of two opposite faces lie at \(\frac{1}{3}\) of the way from the equatorial midpoints toward the poles, so the face diagonal of the inscribed cube is \(\frac{2}{3}s_{\text{octa}}\), and the edge of the inscribed cube is \(s_{\text{cube}} = \frac{\sqrt{2}}{3}s_{\text{octa}}\), giving \(s_{\text{octa}}^{2} : s_{\text{cube}}^{2} = 9 : 2\).

Philological and Folio Apparatus

• 1509 Paganini Witness Folios: 0042 (f. 13v–f. 13v)

• Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/038_cap_XXXVIII.txt

• Winterberg 1889 German Source: winterberg/03_divina_proportione_german/038_cap_XXXVIII.txt

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