CHAPTER V — Critical Commentary & Notes

On the Fitting Title of the Present Treatise

Historical and Philological Commentary

  1. Coining the Name *:term:`Divina Proportione <divina-proportione>`*: Prior to Pacioli’s treatise, Euclid’s division of a line in extreme and mean ratio (Euclid VI.def.3, $\ddot{\alpha}\kappa\rho\omicron\varsigma \kappa\alpha\grave{\iota} \mu\acute{\epsilon}\sigma\omicron\varsigma \lambda\acute{\omicron}\gamma\omicron\varsigma$) was known in medieval Latin exclusively by Gerard of Cremona’s and Campanus of Novara’s rendering proportio habens medium et duo extrema (“proportion having a mean and two extremes”). Chapter V is the historic locus where Pacioli christens the ratio $\phi = \frac{1+\sqrt{5}}{2}$ the :term:`*Divine Proportion <divina-proportione>`* (:term:`Divina Proportione <divina-proportione>`), grounding the name in five Scholastic-Platonic analogies:

    1. Unicity: Unlike multiplex or superparticular ratios ($2:1, 3:1, 3:2, 4:3, \dots$), which form infinite families of species, extreme and mean ratio is unique (una sola e non più).

    2. Trinitarian Structure: Continuous proportion requires a minimum of three terms ($a+b : a :: a : b$), mirroring the una substantia in tribus personis of Trinitarian theology.

    3. Apophatic Ineffability: The incommensurability of $\phi$ (Euclid XIII.6) is compared to negative (apophatic) theology: just as God transcends finite verbal definition, $\phi$ transcends rational numerical expression (occulta e secreta).

    4. Self-Similarity and Invariance: Echoing the Neoplatonic and Augustinian maxim that God is totus in toto et totus in qualibet parte, $\phi$ remains invariant across all scales and self-replicates upon subdivision or addition (Euclid XIII.4, Euclid XIV.2).

    5. Cosmological Formative Principle (Plato’s *:cite:`Timaeus <plato:timaeus:53c-56c>`*): Combining Aristotle’s :cite:`De Caelo <aristotle:de-caelo:III.8>` (the celestial :term:`quinta essentia <quinta-essentia>`) with Plato’s :cite:`Timaeus <plato:timaeus:53c-56c>` 53c–56c (Tetrahedron = Fire, Cube = Earth, Octahedron = Air, Icosahedron = Water, Dodecahedron = the Cosmos/Heaven), Pacioli argues that because the regular pentagon and dodecahedron—and the mutual proportions of all five Platonic solids inscribed in a single sphere (Euclid XIII.13–Euclid XIII.18)—depend essentially on $\phi$, the Divine Proportion is the formal geometric principle of cosmic order.

Constantin Winterberg’s Commentary (1889 Edition)

  • Winterberg Footnote 1 (Cap. V, p. 181): By the product $ab \cdot bc$ (‘contained under the whole and lesser part’), Euclid and Pacioli mean the rectangle (Rechteck), in contrast to the product multiplied by the cosine of the angle of inclination, which corresponds to the area of an oblique parallelogram. (Winterberg Fussnote 1 (Cap. V, S. 181): D. i. Rechteck im Gegensatze zu dem mit dem cos. des Neigungswinkels zu multiplicirenden Producte, welches dem Inhalt eines schiefwinkeligen Parallelogramms entspricht.)

Philological and Folio Apparatus

  • 1509 Paganini Witness Folios: 0022, 0023 (f. 3v–f. 4r)

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/005_cap_V.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/005_cap_V.txt

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