CHAPTER LVII — Critical Commentary & Notes

How All Five Regular Bodies Are Placed in the Sphere, How Stonemasons May Carve Them from a Spherical Stone, and the Anecdote of Count Girolamo Riario and Melozzo da Forlì in Rome.

Historical & Mathematical Commentary

  • Count Girolamo Riario, Melozzo da Forlì, and Pacioli in Rome (c. 1477–1480 or 1484–1488): This vivid autobiographical narrative is one of the richest historical documents of Renaissance workshop culture:

    • Count Girolamo Riario (1443–1488), Lord of Imola and Forlì and Captain General of the Church under his uncle Pope Sixtus IV, began constructing his Roman palace near Sant’Apollinare (incorporated into Palazzo Altemps) in the late 1470s / early 1480s.

    • Melozzo da Forlì (1438–1494), the celebrated pictor papalis and master of sotto in su perspective (who also came from Riario’s city of Forlì and knew Piero della Francesca), joined Pacioli in setting the geometric trap for Riario’s boastful master stonemason.

  • The Tau (:math:`mathbf{T}`) Optical Illusion: Pacioli's remark on the two equal rods placed in the shape of a T (doi verghette equali in forma de tau, cioè così T, inanze ali occhi suoi, ora vna ora l’altra più longa giudicaua, Folio 0050–0051) is the earliest documented scientific description of the Vertical-Horizontal (T) Optical Illusion in Western history (nearly four centuries before Fick, 1851, and Oppel, 1855)!

  • Comparison with Earlier Draft Track: The earlier divina_en_2/57.txt draft cut out the entire second half of this chapter—omitting the three carpenter jokes, the Tau optical illusion, the entire story of Count Girolamo Riario, Melozzo da Forlì, and the Roman stonemason’s wager and gift of a cloak, and Cicero’s story of Hiero and Simonides! Here the complete text of Folios 0050.rst–0051.rst (f. 17v–f. 18r) is translated in full.

Constantin Winterberg’s Commentary (1889 Edition)

  • Winterberg Footnote 1 (Cap. LVII, p. 249): Point $d$, which is missing in the 1509 woodcut diagram (Folio 0050, 0050.g1.png), must be conceived in the initial plane from which the rotation of the semicircle begins. (Winterberg Fussnote 1 (Cap. LVII, S. 249): Punkt d, welcher in der Zeichnung fehlt, muss in der Ebene gedacht werden, von wo die Drehung beginnt.)

Philological and Folio Apparatus

  • 1509 Paganini Witness Folios: 0050, 0051 (f. 17v–f. 18r)

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/057_cap_LVII.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/057_cap_LVII.txt

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