CHAPTER XXVI — Critical Commentary & Notes

How the First Regular Body, Called the Tetracedron or Pyramid of Four Triangular Bases, Is Formed and Placed in the Sphere.

Historical & Mathematical Commentary

  • Paganini 1509 Marginal Diagram: Folio 0032 (f. 8v) includes diagram 0032.g1.png, showing the semicircle \(adb\) cut at \(c\) (\(ac = 2\,cb\)) alongside the base circle and triangle of the tetrahedron.

  • Mathematical Identity: For a sphere of diameter \(D\), the edge \(a_{4}\) of the inscribed regular tetrahedron satisfies \(D^{2} : a_{4}^{2} = 3 : 2\), or \(a_{4} = D\sqrt{\frac{2}{3}}\).

  • Comparison with Earlier Draft Track: The earlier draft omitted Steps 5–7 (how the perpendicular \(hk = ac\) from the center of the circumscribed circle of radius \(cd\) proves \(ke = da\) via Euclid XIII.12 and Euclid XI.4).

Constantin Winterberg’s Commentary (1889 Edition)

Note 10 (Cap. XXVI): Division of the Sphere Diameter by the Base of the Inscribed Tetrahedron (Euclid XIII.13) (Sachliche Anmerkungen #10, Cap. XXVI, p. 216, l. 12 from top)

In every regular tetrahedron, the four altitudes intersect at the center of the circumscribed sphere (of radius $r$) in the ratio $1 : 3$. Denoting the smaller segment (from the sphere center to the base plane) by $x = \frac{r}{3}$, and extending the altitude $ac = r + x$ beyond the base plane to meet the sphere at $b$ (so that $ab = 2r$ is the sphere diameter and $bc = r - x$), the ratio of the extension $bc$ to the tetrahedron altitude $ac$ is:

\[\frac{bc}{ac} = \frac{r - x}{r + x} = \frac{r - \frac{r}{3}}{r + \frac{r}{3}} = \frac{\frac{2}{3}r}{\frac{4}{3}r} = \frac{1}{2}, \qquad \text{whence } ab : ac = 3 : 2 \text{ and } ac : bc = 2 : 1.\]

Anmerkung 10 (Cap. XXVI, S. 216): Theilung des Kugeldurchmessers beim regulären Tetraeder (Euklid XIII.13)

Bekanntlich verhalten sich in jedem regulären Tetraeder die Abschnitte der Höhen, welche sich im Centrum der umschriebenen Kugel vom Radius $r$ schneiden, zu einander wie $1 : 3$. Bezeichnet man nun den kleinern Abschnitt mit $x$ und denkt sich ferner eine der Höhen über die Grundfläche hinaus bis zum Durchschnitt mit der Kugel verlängert, so ergibt sich das Verhältniss dieser Verlängerung $bc$ zu der Höhe $ac$ des Tetraeders:

\[\frac{bc}{ac} = \frac{r - x}{r + x}, \qquad \text{folglich, da } x = \frac{r}{3} \text{ ist:} \quad \frac{bc}{ac} = \frac{1}{2}.\]

Alles Uebrige ergibt sodann der Text von selber.

  • Winterberg Footnotes 1–3 (Cap. XXVI, pp. 215–216): (1) Instead of the term ‘edge’ (Kante), Pacioli frequently uses the synonymous designation ‘side’ (lato / Seite). (2) Namely, the diameter of the circumscribed sphere (scil. der umbeschriebenen). (3) Compare Chapter LXXI on the technical term corausto (the slant edge of the pyramid). (Winterberg Fussnoten 1–3 (Cap. XXVI, S. 215–216): 1) Statt des Ausdrucks „Kante“ kehrt häufig die dasselbe besagende Bezeichnung „Seite“ wieder. 2) scil. der umbeschriebenen. 3) cfr. Cap. LXXI.)

Philological and Folio Apparatus

  • 1509 Paganini Witness Folios: 0032 (f. 8v–f. 8v)

  • Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/026_cap_XXVI.txt

  • Winterberg 1889 German Source: winterberg/03_divina_proportione_german/026_cap_XXVI.txt

  • Marginal Woodcut Diagrams: 1 figure(s) extracted from the 1509 Venice witness.