CHAPTER IV — Critical Commentary & Notes

On Those Things Which the Reader Must Observe for the Understanding of This Treatise

Historical and Philological Commentary

  1. Citation Convention (`la prima del primo`): Pacioli codifies the standard medieval and Renaissance citation formula for Euclid’s Elements (in the fifteen-book Campanus of Novara edition, where Book XIV is Hypsicles’ treatise on the dodecahedron and icosahedron, and Book XV is the late-antique treatise on inscribing the five Platonic solids in one another). Note that propositional numbering in the Campanus Latin edition occasionally differs by one or two from the modern Heiberg Greek text (notably in Books VI, X, XIII, and XIV); in our critical commentary we cross-reference both the Campanus number cited by Pacioli and the standard Heiberg/GEOMETOR G-Index identifier.

  2. Algebraic Abbreviations (*Cosa, Censo, Cubo*): Pacioli connects the symbolic syncopation of Renaissance Italian abacus algebra (R. = radice, p. = più, m. = meno, co. = cosa, ce. = censo, cu. = cubo) to the professional notation systems of apothecaries, goldsmiths, astrologers, and merchants.

  3. Geometric-Arithmetical Isomorphism: Pacioli explicitly equates the arithmetical operations of multiplicatione / producto ($a \cdot b$) and quadrato ($a^2$) with their Euclidean Book II geometric counterparts: rectangolo (the rectangle contained by two segments) and potentia (the square on a segment, rendering the Greek $\delta\acute{\upsilon}\nu\alpha\mu\iota\varsigma$, as in the Pythagorean Theorem, Euclid I.47, l’ultima del primo).

Constantin Winterberg’s Commentary (1889 Edition)

Philological and Folio Apparatus

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

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/004_cap_IV.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/004_cap_IV.txt

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