CHAPTER LXV — Critical Commentary & Notes

On Lateral (Polygonal) Pyramids and Their Varieties: Right, Oblique, and Truncated (Plates LI–LIX).

Historical & Mathematical Commentary

  • Paganini 1509 Marginal Diagrams & Plates:

    • Folio 0055 (f. 20r) and Folio 0056 (f. 20v) contain 10 marginal woodcut diagrams (0055.g1.png–0055.g3.png and 0056.g1.png–0056.g7.png) depicting the triangular, quadrangular, and pentagonal pyramids and their truncated frusta (tronca tonda A, corta triangola B, tagliata quadrangola C).

    • Plates LI–LIX (0293.rst–0309.rst) show Leonardo's full-page perspective drawings of the triangular, quadrangular, and pentagonal pyramids and the truncated pyramid.

Constantin Winterberg’s Commentary (1889 Edition)

  • Winterberg Footnote 1 (Cap. LXV, p. 262): By pyramide corta non equale, Pacioli means a right frustum of a pyramid (gerade abgestumpfte Pyramide) whose top and bottom bases are unequal parallel similar polygons. (Winterberg Fussnote 1 (Cap. LXV, S. 262): Gerade Pyramide.)

Philological and Folio Apparatus

  • 1509 Paganini Witness Folios: 0055, 0056 (f. 20r–f. 20v)

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/065_cap_LXV.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/065_cap_LXV.txt

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