CHAPTER III — Critical Commentary & Notes

The Preface Being Ended, There Follows an Explanation of What Is Meant by This Name ‘Mathematical’

Historical and Philological Commentary

  1. The Lacuna for the Greek Word in the 1509 Printing: In the 1509 Paganini edition (Folio 0021r), the compositor left a blank space after derivato da for the Greek word $\mu\alpha\theta\eta\mu\alpha\tau\iota\kappa\acute{o}\varsigma$ (or $\mu\alpha\nu\theta\acute{\alpha}\nu\omega$ / $\mu\acute{\alpha}\theta\eta\mu\alpha$), which Pacioli regularly glosses in his Summa and his 1509 edition of Euclid as disciplinabile.

  2. Reforming the Quadrivium — Perspective as a Liberal Art: Chapter III is one of the defining documents of the Italian Renaissance elevation of the visual arts from artes mechanicae to artes liberales. Challenging Boethius’s canonical fourfold Quadrivium (Arithmetic, Music, Geometry, Astronomy), Pacioli argues on strictly parallel epistemological and mathematical grounds that Prospectiva (linear perspective and the science of visual rays, pioneered by Piero della Francesca, Leon Battista Alberti, and Leonardo da Vinci) stands on an equal footing with Music:

    • Sense Hierarchy: Sight (el vedere, the prima porta al intellecto) is nobler than hearing (l’udito).

    • Mathematical Object: Measure of the visual ray (linea visuale) and natural number vs. sonorous number (numero sonoro) and musical time (tempo de sue prolationi).

    • Proportional Scope: Arithmetic and geometric proportions of foreshortening vs. harmonic proportions.

  3. Second Ekphrasis of Leonardo’s *:term:`Last Supper <simulacro>`*: Pacioli specifies the exact dramatic moment depicted by Leonardo in the refectory of Santa Maria delle Grazie: Christ’s words Unus vestrum me traditurus est (Matthew 26:21), capturing the Apostles’ acti e gesti (“actions and gestures”, corresponding directly to Leonardo’s theory of moti mentali expressed through bodily movements).

Constantin Winterberg’s Commentary (1889 Edition)

  • Winterberg Footnote 1 (Cap. III, p. 179): Supply the verb exclude (scil. ausschliessen) regarding those who admit only athreefold division of the mathematical sciences. (Winterberg Fussnote 1 (Cap. III, S. 179): scil. ausschliessen.)

Philological and Folio Apparatus

  • 1509 Paganini Witness Folios: 0021, 0022 (f. 3r–f. 3v)

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/003_cap_III.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/003_cap_III.txt

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