CHAPTER XX — Critical Commentary & Notes

On Its Eleventh Most Excellent Effect

Historical and Philological Commentary

1. Direct Calculation of the Decagon from the Radius ($R = 10$): Whereas Chapters XVI and XVII took $s_6 + s_{10} = 10$ (so that the radius $R = s_6 = \sqrt{125} - 5$), Chapter XX takes the hexagon side (the circle’s radius $R$) itself as $10$ (so the circle’s diameter is $20$). Since $(s_6 + s_{10}) : s_6 = s_6 : s_{10}$ (Euclid XIII.9), $s_6$ is divided in extreme and mean ratio when a segment equal to $s_{10}$ is marked off from it: thus the side of the inscribed decagon is simply the major golden segment of the radius ($s_{10} = \sqrt{125} - 5$). Note that Pacioli uses circumscripto dal medesimo cerchio in the sense of “surrounded/encompassed by the same circle” (i.e., inscribed within the circle).

Philological and Folio Apparatus

• 1509 Paganini Witness Folios: 0029 (f. 7r–f. 7r)

• Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/020_cap_XX.txt

• Winterberg 1889 German Source: winterberg/03_divina_proportione_german/020_cap_XX.txt

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