CHAPTER XIX — Critical Commentary & Notes

On Its Tenth Supreme Effect

Historical and Philological Commentary

  1. Universal Similarity of Golden-Divided Lines (Campanus Lemma to Euclid XIV.2): In Hypsicles’ Book XIV of the Elements, Proposition 2 proves that the same circle circumscribes both the pentagon of the dodecahedron and the triangle of the icosahedron inscribed in the same sphere. In his commentary on XIV.2, Campanus of Novara inserted and proved a fundamental lemma (often cited in medieval and Renaissance mathematics simply as part of XIV.2): if two straight lines are cut in extreme and mean ratio, they are cut similarly ($AB : AC = DE : DF$ and $AC : CB = DF : FE$).

  2. The Six Operations of Book V: Pacioli enumerates the six canonical Scholastic operations on proportions from Book V of Euclid’s Elements:

    • Permutata (alternando, Euclid V.16): $a : b = c : d \implies a : c = b : d$

    • Conversa (invertendo, Euclid V.7.p): $a : b = c : d \implies b : a = d : c$

    • Congionta (componendo, Euclid V.18): $a : b = c : d \implies (a+b) : b = (c+d) : d$

    • Disgionta (dividendo, Euclid V.17): $a : b = c : d \implies (a-b) : b = (c-d) : d$

    • Eversa (convertendo, Euclid V.19.p): $a : b = c : d \implies a : (a-b) = c : (c-d)$

    • Equa proportionalita (ex aequali, Euclid V.22).

Philological and Folio Apparatus

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

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/019_cap_XIX.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/019_cap_XIX.txt

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