CHAPTER XI — Critical Commentary & Notes

On Its Second Essential Effect

Historical and Philological Commentary

  1. Converse of the First Effect (Euclid XIII.2): In Euclid’s Elements Book XIII, Proposition 2 states: “If the square on a straight line be five times the square on a segment of it, then when the double of the said segment is cut in extreme and mean ratio, the greater segment is the remaining part of the original straight line.” Pacioli formulates this directly as the converse of Chapter VII/X: if $(a + c)^2 = 5c^2$ where $a + b = 2c$, then $c = \frac{a+b}{2}$ and $a$ is the major golden segment of $a+b$.

Philological and Folio Apparatus

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

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/011_cap_XI.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/011_cap_XI.txt

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