CHAPTER LXI — Critical Commentary & Notes

On Five-Sided (Pentagonal) Lateral Columns and How the Species of Lateral Columns Proceed to Infinity (Plates XLIX–L).

Historical & Mathematical Commentary

  • Paganini 1509 Plates Referenced:

    • Plate XLIX (0289.rst, Pentagonal Prism, solid) and Plate L (0291.rst, Pentagonal Prism, skeletal).

  • Invocation of Euclid XIII.7: Pacioli deftly reminds the reader why in Chapter XXI (Euclid XIII.7) he proved that an equilateral pentagon requires three equal angles to force equiangularity (since a non-regular equilateral pentagon—such as a house-shaped pentagon—can have five equal sides and two right angles without being regular).

Constantin Winterberg’s Commentary (1889 Edition)

  • Winterberg Footnote 1 (Cap. LXI, p. 255): Rendering simili literally as ‘similar’ here would give a false sense; Pacioli means ‘uniform / homogeneous’ (gleichartig). (Winterberg Fussnote 1 (Cap. LXI, S. 255): simili = ähnlich, würde einen falschen Sinn geben.)

Philological and Folio Apparatus

  • 1509 Paganini Witness Folios: 0053 (f. 19r–f. 19r)

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/061_cap_LXI.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/061_cap_LXI.txt

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