CHAPTER LXVIII — Critical Commentary & Notes

How Short or Truncated Pyramids (Pyramidi Corte ouer Scapezze) Are Measured.

Historical & Mathematical Commentary

• Subtraction of the *Pyramidella* vs. Heron’s Mean-Base Formula: In the :cite:`Summa de Arithmetica <pacioli:summa>` (Part II, Tractatus Geometriae, Dist. VI), Pacioli gives both this geometric completion-and-subtraction method (\(V = \frac{1}{3}B_1 h_{\text{total}} - \frac{1}{3}B_2 h_{\text{top}}\)) and the equivalent algebraic formula \(V = \frac{1}{3}h(B_1 + \sqrt{B_1 B_2} + B_2)\). Notice how the completion-and-subtraction method prepares the exact conceptual model for measuring truncated (*:term:`abscisi <abscisum>`*) polyhedra in the very next chapter (Cap. LXIX)!

Philological and Folio Apparatus

• 1509 Paganini Witness Folios: 0057, 0058 (f. 21r–f. 21v)

• Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/068_cap_LXVIII.txt

• Winterberg 1889 German Source: winterberg/03_divina_proportione_german/068_cap_LXVIII.txt

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