CHAPTER XXII — Critical Commentary & Notes

On Its Thirteenth Most Worthy Effect

Historical and Philological Commentary

  1. The Golden Triangle (Euclid IV.10) and the Pentagon (Euclid IV.11–Euclid IV.12): In Euclid IV.10, Euclid constructs the $72^\circ\text{–}72^\circ\text{–}36^\circ$ isosceles triangle by dividing the leg $AB$ at $C$ via Euclid II.11 so that $AB \cdot BC = AC^2$, and placing $BD = AC$ as the base. In Euclid IV.11 he inscribes this golden triangle into a circle and bisects the two $72^\circ$ base angles to obtain the five vertices of the regular pentagon.

  2. Pacioli’s Insight into Euclid’s Axiomatic Method: Chapter XXII contains a remarkably lucid analysis of the architecture of Euclid’s Elements—one that directly anticipates modern historiography of Greek mathematics (and connects directly to the GEOMETOR G-Index dependency graph!). Pacioli explains why Euclid proves the golden section twice—first by geometric areas in Euclid II.11 ($AB \cdot BC = AC^2$) and again by proportions in Euclid VI.30 ($AB : AC = AC : BC$): because Euclid IV.10–Euclid IV.11 (the regular pentagon) stands in Book IV prior to Eudoxus’s general theory of proportions in Book V, Euclid could not legally invoke ratios in Book IV without violating his strict deductive rule of using only antecedenti (previously proved propositions) and never sequenti!

Constantin Winterberg’s Commentary (1889 Edition)

Note 9 (Cap. XXII): Cross-Reference to the Libellus de Quinque Corporibus Regularibus (Sachliche Anmerkungen #9, Cap. XXII, p. 211, l. 7 from top)

Compare on this pentagon construction the Tractatus de Quinque Corporibus Regularibus (Libellus, Tractate II, Case 20 ff.).

Anmerkung 9 (Cap. XXII, S. 211): Verweis auf den Tractat über die fünf regelmässigen Körper

Vgl. hierüber den Tractat über die fünf regelmässigen Körper (Libellus, Tract. II, Cas. 20 u. s. w.).

Philological and Folio Apparatus

  • 1509 Paganini Witness Folios: 0030 (f. 7v–f. 7v)

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/022_cap_XXII.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/022_cap_XXII.txt

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