CHAPTER LXIX — Critical Commentary & Notes

On the Measure of All the Regular and Dependent Bodies, and the Panegyric to Duke Ludovico Maria Sforza and the 1498 Franciscan General Chapter in Milan.

Historical & Mathematical Commentary

  • Additivity and Subtractivity of Polyhedral Volumes: Mathematically, Cap. LXIX formulates the exact modern decomposition method for computing the volumes of Archimedean truncations (\(V_{\text{trunc}} = V_{\text{parent}} - \sum V_{\text{vertex pyramids}}\)) and Catalan/stellated cumulations (\(V_{\text{elev}} = V_{\text{core}} + \sum V_{\text{face pyramids}}\)).

  • Historical Witness to the Milanese Court in 1498:

    • The Ducal Chapel Choir (*dignissimi cantori*): Under Ludovico Sforza, the Cappella Musicale of the Castello Sforzesco (directed by Franchino Gaffurio and having included Josquin des Prez and Loyset Compère) was one of the premier musical institutions of Europe.

    • Santa Maria delle Grazie: Bramante’s monumental tribune (commissioned by Ludovico Sforza as the Sforza dynastic mausoleum) and Leonardo da Vinci's Last Supper in its refectory were both being completed in 1497–1498.

    • The Franciscan General Chapter of Pentecost 1498 in Milan: Presided over by the famous Conventual Minister General Francesco Sansone da Brescia (1414–1499, General 1475–1499, also a patron of learning) and attended by Cardinal Ippolito I d’Este (1479–1520, Archbishop of Milan and brother of Duchess Beatrice d’Este). This passage confirms that Pacioli was writing the final chapters of the Compendio in Milan between June and December 1498.

  • Comparison with Earlier Draft Track: The earlier divina_en_2/69.txt draft completely omitted the entire second half of Cap. LXIX (the panegyric to Ludovico Sforza, the Ducal Chapel choir, Santa Maria delle Grazie compared to Augustus’s Temple of Peace, and the eyewitness account of the 1498 Franciscan General Chapter in Milan with Francesco Sansone and Cardinal Ippolito d'Este). Everything is restored in full here.

Constantin Winterberg’s Commentary (1889 Edition)

Philological and Folio Apparatus

  • 1509 Paganini Witness Folios: 0058, 0059 (f. 21v–f. 22r)

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/069_cap_LXIX.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/069_cap_LXIX.txt

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