CHAPTER XVII — Critical Commentary & Notes

On the Eighth Effect, the Converse of the Preceding

Historical and Philological Commentary

1. Ptolemy’s *:cite:`Almagest <ptolemy:almagest:I.9>`* Book I, Ch. 9/10 (Calculation of the Table of Chords): Pacioli cites Gerard of Cremona’s Latin translation of Ptolemy’s :cite:`Almagest <ptolemy:almagest:I.9>` (Dictio prima, cap. 9 in manuscript/printed divisions corresponding to Heiberg Book I, Ch. 10, De magnitudine rectarum linearum quae in circulo ducuntur). To construct his trigonometric table of chords, Ptolemy divides the diameter into $120$ parts (so the radius / hexagon chord $\text{crd}(60^\circ) = 60$), and uses Euclid XIII.9 and Euclid XIII.10 simultaneously to compute the chord of $36^\circ$ (the side of the regular decagon, $\text{crd}(36^\circ) = \sqrt{5 \cdot 30^2} - 30 = \sqrt{4500} - 30 \approx 37;4,55$) and the chord of $72^\circ$ (the side of the regular pentagon).

Philological and Folio Apparatus

• 1509 Paganini Witness Folios: 0028 (f. 6v–f. 6v)

• Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/017_cap_XVII.txt

• Winterberg 1889 German Source: winterberg/03_divina_proportione_german/017_cap_XVII.txt

• Marginal Woodcut Diagrams: 1 figure(s) extracted from the 1509 Venice witness.