CHAPTER VIII — Critical Commentary & Notes¶
How a Quantity Divided According to the Proportion Having a Mean and Two Extremes Is to Be Understood
Historical and Philological Commentary¶
Pacioli’s Canonical Numerical Exemplar ($a + b = 10$): Throughout Chapters VIII–XXII (and later in the polyhedral measurements), Pacioli adopts $10$ as the standard rational whole divided in extreme and mean ratio. Setting the greater part $a = x$ (una cosa) and the lesser part $b = 10 - x$, the defining equation $10(10 - x) = x^2$ gives the quadratic equation $x^2 + 10x = 100$ (censo e cose equale a numero). By al-Khwarizmi’s rule in Algebra et Almucabala: $$a = \sqrt{\left(\frac{10}{2}\right)^2 + 100} - \frac{10}{2} = \sqrt{125} - 5$$ $$b = 10 - (\sqrt{125} - 5) = 15 - \sqrt{125}$$
Arithmetic of Apotomes (*Residui*, Euclid X.73): Pacioli writes radicals in pure single-radical form so that $10 \cdot \sqrt{125} = \sqrt{100 \cdot 125} = \sqrt{12500}$. Thus:
Product of extremes: $10 \cdot (15 - \sqrt{125}) = 150 - \sqrt{12500}$.
Square of the mean: $(\sqrt{125} - 5)^2 = 125 + 25 - 2 \cdot 5\sqrt{125} = 150 - \sqrt{12500}$.
Notice also that in line 24 of the 1509 print (and Winterberg's transcription), 15 m. R. 125 was twice misprinted as o. m. R. 125; our translation restores the mathematically exact 15 - sqrt{125}.
Campanus X.79 = Heiberg Euclid X.73: In Euclid’s Book X, an apotome ($\dot{\alpha}\pi\omicron\tau\omicron\mu\acute{\eta}$, rendered residuum by Gerard of Cremona and Campanus) is defined in Euclid X.73 (which is Proposition 79 in Campanus’s numbering because Campanus counted several lemmas as numbered propositions), followed by the six species of apotomes in X.79–84 (Campanus X.85–90). Specifically, $\sqrt{125} - 5$ is a fourth apotome (residuum quartum), and $15 - \sqrt{125}$ is a first apotome (residuum primum).
Philological and Folio Apparatus¶
1509 Paganini Witness Folios: 0025 (f. 5r–f. 5r)
Winterberg 1889 Italian Source:
winterberg/01_divina_proportione_italian/008_cap_VIII.txtWinterberg 1889 German Source:
winterberg/03_divina_proportione_german/008_cap_VIII.txtMarginal Woodcut Diagrams: 0 figure(s) extracted from the 1509 Venice witness.