CHAPTER X — Critical Commentary & Notes

Sequel to the First Proposed Effect

Historical and Philological Commentary

  1. *Restorare e rempire* (*al-jabr*): Notice Pacioli’s technical use of the verb restorare (“to restore”): adding $+5$ cancels the subtracted $-5$ in $\sqrt{125} - 5$, which is the literal meaning of the Arabic al-jabr (restauratio, restoring a deficient or subtracted quantity).

  2. Correlation with Euclid Euclid XIII.1: Algebraically, since $a = \sqrt{5\left(\frac{a+b}{2}\right)^2} - \frac{a+b}{2}$, adding $\frac{a+b}{2}$ immediately isolates $\sqrt{5}\left(\frac{a+b}{2}\right)$, whose square is $5\left(\frac{a+b}{2}\right)^2$. Thus Pacioli illuminates why Euclid begins Book XIII with Euclid XIII.1: it is the geometric equivalent of completing the square on the golden ratio quadratic equation.

Philological and Folio Apparatus

  • 1509 Paganini Witness Folios: 0026 (f. 5v–f. 5v)

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/010_cap_X.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/010_cap_X.txt

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