CHAPTER II — Critical Commentary & Notes

Preface of the Reverend Father Magister Luca Pacioli of Borgo Sansepolcro, of the Order of Minorites and Professor of Sacred Theology, to the Compendium on the Divine Proportion drawn from the Mathematical Disciplines

Historical and Philological Commentary

1. Dantean and Petrarchan Echoes: Pacioli weaves vernacular literary classics directly into his Scholastic Latin-Italian prose:

   • el maistro de color ch’sano (“the Master of those who know”, Aristotle) is a direct quotation of Dante, Inferno IV.131.

   • l’uno pensier da l’altro scoppia (“one thought bursts forth from another”) quotes Dante, Inferno XXIII.10.

   • la gola, sonno e otiose piume (“gluttony, sleep, and idle feather-beds”) quotes the opening line of Petrarch’s famous sonnet La gola e ‘l sonno e l’oziose piume Hanno del mondo ogni vertù sbandita (Rerum vulgarium fragmenta VII).

2. Material Models of the Polyhedra: Pacioli explicitly states that he fashioned with his own hands (de mia propria mano materialmente) three-dimensional models of the regular and dependent polyhedra to accompany the treatise in the Ducal Library at Pavia/Milan. In May 1506, the Venetian Republic recorded Pacioli's presentation of three sets of sixty polyhedra fashioned in lightweight wood.

3. Roberto Valturio and the Urbino Frieze: Pacioli links classical military engineering (Archimedes at Syracuse, Caesar’s Rhine bridge, Vegetius, Frontinus) to Roberto Valturio’s De re militari (written c. 1455 for Sigismondo Pandolfo Malatesta of Rimini, printed at Verona in 1472) and the celebrated stone bas-reliefs (Formelle dell’Arte della Guerra, carved by Ambrogio Barocci after designs by Francesco di Giorgio Martini) that adorned the bench-frieze around the facade of Federico da Montefeltro’s Ducal Palace in Urbino.

4. Duns Scotus, Bartolus of Sassoferrato, and Euclid:

   • John Duns Scotus (Ordinatio / Opus Oxoniense II, dist. 2, p. 2, q. 5–6) used geometric arguments on the divisibility of the continuum from Euclid’s Elements to analyze angelic location (ubi angelicum) and motion.

   • Bartolus of Sassoferrato (1313–1357) composed his treatise De fluminibus seu Tyberiadis (1355) while watching the bends of the Tiber near Perugia; as Bartolus himself records in his prologue, he was assisted in the geometric division of alluvial islands and riverbeds by the Franciscan theologian and mathematician Guido da Perpignano / Guido de Perusio, citing Euclid’s Elements.

5. Pacioli’s Public Chair of Mathematics in Milan: Appointed by Ludovico Sforza in 1496, Pacioli notes at the end of Chapter II that he has just completed lecturing publicly through Books I–X of Euclid’s Elements (incorporating practical algebra and arithmetic alongside Campanus’s theoretical propositions) and is now devoting the remainder of his term to Books XI–XV and De Divina Proportione.

Constantin Winterberg’s Commentary (1889 Edition)

• Winterberg Footnote 1 (Cap. II, TOC p. 166 & p. 176): La Teberina (Tyberiadis)—a treatise by Bartolus of Sassoferrato on the alluvial deposits caused by the flooding of the Tiber. (Winterberg Fussnote 1 (Cap. II, Inh. S. 166): Ein Werk über die Tiberüberschwemmung.)

• Winterberg Footnote 2 (Cap. II, p. 176): Pacioli calls Euclid ‘the Megarian philosopher’ (Euclide Megarense)—throughout erroneously confusing the Socratic philosopher Euclid of Megara with the mathematician Euclid of Alexandria who lived in Egypt. (Winterberg Fussnote 2 (Cap. II, S. 176): Durchgehends fälschlich anstatt des in Aegypten lebenden Mathematikers gleichen Namens angegeben.)

Philological and Folio Apparatus

• 1509 Paganini Witness Folios: 0018, 0019, 0020, 0021 (f. 1v–f. 3r)

• Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/002_cap_II.txt

• Winterberg 1889 German Source: winterberg/03_divina_proportione_german/002_cap_II.txt

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