CHAPTER XXIX — Critical Commentary & Notes

How the Fourth Regular Body, Called the Ycocedron of Twenty Triangular Bases, Is Formed in the Sphere.

Historical & Mathematical Commentary

  • Paganini 1509 Marginal Diagram: Folio 0034 (f. 9v) includes diagram 0034.g1.png, showing the sphere diameter divided in \(4:1\) ratio (\(ac = 4\,cb\)) and the circle containing the pentagon and decagon.

  • Role of the Divine Proportion: This construction brings together nearly every theorem proved in Section 2: Euclid XIII.9 (sum of hexagon and decagon sides divided in extreme and mean ratio), Euclid XIII.10 (\(s_{5}^{2} = s_{6}^{2} + s_{10}^{2}\)), and Euclid XIII.11 (the side of the pentagon is a Minor, Euclid X.76).

  • Exact Formula: For a sphere of diameter \(D\), the radius of the circumcircle of the two parallel pentagonal cross-sections is \(R = \frac{D}{\sqrt{5}}\), and the edge of the icosahedron is \(a_{20} = R\sqrt{\frac{5 - \sqrt{5}}{2}} = D\sqrt{\frac{5 - \sqrt{5}}{10}}\). For \(D = 10\), \(a_{20} = \sqrt{50 - 10\sqrt{5}} = \sqrt{50 - \sqrt{500}}\).

Constantin Winterberg’s Commentary (1889 Edition)

Note 12 (Cap. XXIX): Construction of the Regular Icosahedron in the Sphere (Euclid XIII.16) (Sachliche Anmerkungen #12, Cap. XXIX, p. 220, l. 15 from top)

The edge $a$ of the regular icosahedron in terms of the circumradius $r$ of the sphere is:

\[a = \frac{4r}{\sqrt{10 + 2\sqrt{5}}}.\]

Passing a plane through the sphere diameter $ab = 2r$ and a polar cap edge $ad = a$ (which cuts the sphere in a semicircle $adb$), the projection $ac$ of $ad$ on the diameter $ab$ satisfies:

\[ac = \frac{ad^2}{ab} = \frac{a^2}{2r} = r\left(1 - \frac{1}{\sqrt{5}}\right), \qquad \text{whence } cm = r - ac = \frac{r}{\sqrt{5}},\]

and therefore the middle axial segment between the two parallel pentagon planes is $cc' = 2\,cm = \frac{2r}{\sqrt{5}} = \frac{d}{\sqrt{5}}$, where $d = 2r$ is the sphere diameter.

Now in the auxiliary semicircle of Chapter XXIX with $ac_{\text{aux}} = 2\,bc_{\text{aux}}$, we have $bd = \frac{d}{\sqrt{5}} = cc'$—that is, $bd$ equals the height of the ten-triangle antiprism belt (the segments called catheti in the text) erected perpendicular to the two pentagon planes at $c$ and $c'$. Moreover, the circumradius $r'$ of these two parallel pentagons (and of the inscribed decagon) with pentagon side $a$ is:

\[r' = a\sqrt{\frac{5 + \sqrt{5}}{10}} = \frac{d}{\sqrt{5}} = bd,\]

so the circumradius $r'$ of the pentagon circle equals both the line $bd$ of the text and the axial height $cc'$ of the perpendicular catheti, exactly as constructed in Euclid Euclid XIII.16!

Anmerkung 12 (Cap. XXIX, S. 220): Construction des regulären Ikosaeders in der Kugel (Euklid XIII.16)

Bekanntlich ist die Seite $a$ des regulären Ikosaeders, durch den Halbmesser $r$ der umschriebenen Kugel ausgedrückt:

\[a = \frac{4r}{\sqrt{10 + 2\sqrt{5}}}.\]

Denkt man sich nun durch die Axe $ab = 2r$ und eine Kante $ad$ des Ikosaeders eine Ebene gelegt, welche zugleich die Kugel im Halbkreise $adb$ schneidet, so ist nach bekannten Sätzen:

\[ac = \frac{ad^2}{ab} = \frac{a^2}{2r} = r\left(1 - \frac{1}{\sqrt{5}}\right),\]

woraus hervorgeht, dass $cm = r - ac = \frac{r}{\sqrt{5}}$, somit das Doppelte $cc' = \frac{2r}{\sqrt{5}} = \frac{d}{\sqrt{5}}$, wenn der Kugeldurchmesser mit $d = 2r$ bezeichnet wird. Nun ist dem Text zufolge $\frac{bd}{d} = \frac{1}{\sqrt{5}}$, sonach $bd = cc'$, d. h. gleich der Länge der im Text als Katheten bezeichneten Höhen, welche man in den Ecken der Fünfecke zu errichten hat. Der Radius $r'$ des diesen Fünfecken umschriebenen Kreises ist aber, wenn $a$ zugleich Fünfecksseite sein soll:

\[r' = a\sqrt{\frac{5 + \sqrt{5}}{10}} = \frac{d}{\sqrt{5}} = bd,\]

    1. es muss $r'$ genau gleich der Linie $bd$ des Textes und zugleich gleich den als Katheten bezeichneten Höhen sein, wie solches der dort angegebenen Construction zufolge auch wirklich der Fall ist.

  • Winterberg Footnotes 1–5 (Cap. XXIX, pp. 219–221): (1) When the total length of a line divided in extreme and mean ratio is rational, its segments are always irrational (apotomes), which gave rise to Euclid’s classification of the minor irrational line (Euclid X.76, Euclid XIII.16). (2) A rational diameter—one whose length itself or whose square is rational. (3) Catheti, i.e., perpendiculars; compare Chapter LXXI on the technical terms occurring here. (4) Namely, equal to the radius $bd$ of the circle. (5) In potentia, i.e., in their second power (square). (Winterberg Fussnoten 1–5 (Cap. XXIX, S. 219–221): 1) Wenn die ganze Länge der stetig zu theilenden Linie rational ist, so werden die Abschnitte stets irrational sein, was zu obiger heute nicht mehr gebräuchlicher Benennung Veranlassung gegeben. 2) scil. dessen Länge selbst oder deren Quadrat rational sei. 3) scil. Senkrechten; über die hier und im Folgenden vorkommenden Benennungen cfr. Cap. LXXI. 4) scil. dem Radius bd des Kreises gleich. 5) d. h. ihrer zweiten Potenz.)

Philological and Folio Apparatus

  • 1509 Paganini Witness Folios: 0034, 0035, 0036 (f. 9v–f. 10v)

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/029_cap_XXIX.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/029_cap_XXIX.txt

  • Marginal Woodcut Diagrams: 4 figure(s) extracted from the 1509 Venice witness.