CHAPTER III — Critical Commentary & Notes¶
On the Proportion of the Whole Well-Disposed Human Body to Its Head and Other Members According to Length and Breadth, and on the Perfect Numbers Six, Ten, and Sixteen.
Historical & Mathematical Commentary¶
Paganini 1509 Marginal Diagram: Folio 0069 (
f. 27r) includes the vertical graduated scale0069.g1.pngmarking the proportional divisions of the human body (\(\frac{1}{10}\), \(\frac{1}{8}\), \(\frac{1}{6}\), \(\frac{1}{4}\)).Euclid Euclid IX.36 and Algebraic *:term:`Censo de Censo <censo-de-censo>`*: Pacioli weaves together Vitruvius III.1.5–8 (on the numbers 10, 6, and 16), Euclid’s theorem on even perfect numbers (Euclid IX.36: \(2^{n-1}(2^n - 1)\) when \(2^n - 1\) is a Mersenne prime, e.g. \(2 \times 3 = 6\), \(4 \times 7 = 28\)), and Cossic algebra (censo primo \(= x^2 = 4\), :term:`censo de censo <censo-de-censo>` \(= x^4 = 16\)).
Constantin Winterberg’s Commentary (1889 Edition)¶
Winterberg Footnotes 1–5 (Arch. Cap. III, pp. 301–303): (1) Namely, equal to the distance from the chin to the base of the nostrils (vom Kinn bis zum Nasenstachel). (2) Namely, in breadth (scil. Breite). (3) The cubit (cubito)—from the elbow joint to the tip of the middle finger. (4) Square (Quadrat, $4^2 = 16$). (5) Biquadrate / fourth power (Biquadrat, $16^2 = 256$). (Winterberg Fussnoten 1–5 (Arch. Cap. III, S. 301–303): 1) D. h. wie der Abstand vom Kinn bis zum Nasenstachel. 2) scil. Breite. 3) d. h. vom Ellbogengelenk bis zur Mittelfingerspitze. 4) Quadrat. 5) Biquadrat.)
Philological and Folio Apparatus¶
1509 Paganini Witness Folios: 0068, 0069 (f. 26v–f. 27r)
Winterberg 1889 Italian Source:
winterberg/02_architectura_italian/003_cap_III.txtWinterberg 1889 German Source:
winterberg/04_architectura_german/003_cap_III.txtMarginal Woodcut Diagrams: 1 figure(s) extracted from the 1509 Venice witness.