CHAPTER LXVII — Critical Commentary & Notes

How It Is Manifestly Demonstrated That Every Lateral Pyramid Is One-Third (Subtripla) of Its Column, and Why Every Polygon of N Sides Resolves into N − 2 Triangles.

Historical & Mathematical Commentary

  • Paganini 1509 Marginal Diagrams: Folio 0057 (f. 21r) displays two woodcut diagrams (0057.g1.png and 0057.g2.png) showing the quadrilateral \(abcd\) divided by diagonal \(bd\) into 2 triangles (\(\triangle abd, \triangle bcd\)) and the pentagon \(abcde\) divided from vertex \(a\) by chords \(ac, ad\) into 3 triangles (\(\triangle abc, \triangle acd, \triangle ade\)).

  • Additivity of Ratios (Euclid V.12): Notice how Pacioli uses Euclid V.12 (if \(A_1 : B_1 = A_2 : B_2 = A_3 : B_3 = 3 : 1\), then \((A_1+A_2+A_3) : (B_1+B_2+B_3) = 3 : 1\)) to extend the \(3:1\) prism-to-pyramid volume theorem from the triangular prism (corpo seratile, Euclid XII.7) to any \(n\)-gonal prism decomposed into \(n-2\) triangular prisms.

Philological and Folio Apparatus

  • 1509 Paganini Witness Folios: 0056, 0057 (f. 20v–f. 21r)

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/067_cap_LXVII.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/067_cap_LXVII.txt

  • Marginal Woodcut Diagrams: 2 figure(s) extracted from the 1509 Venice witness.