CHAPTER XVI — Critical Commentary & Notes¶
On Its Seventh Inestimable Effect
Historical and Philological Commentary¶
Hexagon and Decagon Inscribed in the Same Circle (Euclid XIII.9): By Euclid IV.15.p, the side $s_6$ of the regular hexagon inscribed in a circle of radius $R$ is equal to the radius itself ($s_6 = R$, so the diameter is $2R = 2(\sqrt{125}-5) = \sqrt{500}-10$). Proposition 9 of Book XIII proves that when $s_6$ and the side $s_{10}$ of the regular decagon inscribed in the same circle are placed in a straight line, the whole line $s_6 + s_{10}$ is cut in extreme and mean ratio, and $s_6$ is the greater segment: $$(s_6 + s_{10}) : s_6 = s_6 : s_{10} = \phi$$
Constantin Winterberg’s Commentary (1889 Edition)¶
Note 6 (Cap. XVI): Seventh Effect — Hexagon Side (Radius) and Decagon Side (Euclid XIII.9) (Sachliche Anmerkungen #6, Cap. XVI, p. 205, l. 11 from bottom)
This follows immediately from Note 4 (Euclid XIII.5), once one observes that the side of the regular decagon inscribed in a circle is the greater segment of the radius (the hexagon side) divided in extreme and mean ratio.
Anmerkung 6 (Cap. XVI, S. 205): Siebenter Effect — Sechsecksseite und Zehnecksseite (Euklid XIII.9)
Dies folgt unmittelbar aus Anmerkung 4, wenn man berücksichtigt, dass die Zehnecksseite der grössere Abschnitt des stetig getheilten Radius (der Sechsecksseite) ist.
Philological and Folio Apparatus¶
1509 Paganini Witness Folios: 0028 (f. 6v–f. 6v)
Winterberg 1889 Italian Source:
winterberg/01_divina_proportione_italian/016_cap_XVI.txtWinterberg 1889 German Source:
winterberg/03_divina_proportione_german/016_cap_XVI.txtMarginal Woodcut Diagrams: 0 figure(s) extracted from the 1509 Venice witness.