Appendix IV: Winterberg’s Mathematical Commentary (1889)¶
Substantive Mathematical Notes (Sachliche Anmerkungen 1–17) by Constantin Winterberg (Vienna, 1889, pp. 338–352)
Constantin Winterberg appended seventeen substantive mathematical notes (Sachliche Anmerkungen, pp. 338–352) to his 1889 Vienna edition, reformulating Pacioli's numerical examples in modern algebraic notation, correcting Pacioli's claim in Cap. XLVI regarding the inscription of the octahedron in the icosahedron, and establishing a complete projective and polar-reciprocity theory of the twenty-one mutual inscriptions among the five Platonic solids.
Note 1 (Cap. VII): Equivalence of Pacioli’s Initial Characterization with Euclid’s Definition of Extreme and Mean Ratio¶
Corresponding Chapters: Cap. 7
Since the definition of continuous division (extreme and mean ratio) according to Euclid (Euclid VI.def.3, Euclid II.11) is stated at the close of Chapter VII—namely, when a given line $ac$ is divided at $b$ into two segments such that the square on the greater segment $ab^2$ equals the rectangle $ac \cdot bc$ contained by the whole line and the lesser segment—we demonstrate its identity with the numerical property stated at the beginning of Chapter VII (Euclid XIII.1):
Hypothesis: Let the line $ac$ be divided in extreme and mean ratio at $b$, so that $ab^2 = ac \cdot bc$.
Claim: $\left(ac + \frac{bc}{2}\right)^2 = 5\left(\frac{bc}{2}\right)^2$, or equivalently $\left(ab + \frac{ac}{2}\right)^2 = 5\left(\frac{ac}{2}\right)^2$.
Proof: By hypothesis, $ab^2 = ac(ac - ab)$, whence
\[ac^2 = ab^2 + ab \cdot ac \quad \text{and} \quad ab^2 + ab \cdot ac = ac^2.\]Adding $\left(\frac{ac}{2}\right)^2$ to both sides yields
\[\left(ab + \frac{ac}{2}\right)^2 = ab^2 + ab \cdot ac + \left(\frac{ac}{2}\right)^2 = ac^2 + \frac{ac^2}{4} = 5\left(\frac{ac}{2}\right)^2. \quad \text{Q.E.D.}\]
Note 2 (Cap. VII): Bisection of the Whole Line and Continuous Division of the Half¶
Corresponding Chapters: Cap. 7
This corollary (Euclid XIII.1 / Euclid XIII.2) follows immediately if we bisect the line $ab$ at $c$ ($ac = bc = \frac{1}{2}ab$) and divide $bc$ in extreme and mean ratio at $d$; considering the three parts of the line $ab$, the greater segment of the half-line together with the other half satisfies the fivefold square relation $ad^2 = 5\,ac^2$.
Note 3 (Cap. XII): Algebraic Proof of the Third Effect (Euclid XIII.3)¶
Corresponding Chapters: Cap. 12
Expressed algebraically for the line $ac$ divided in extreme and mean ratio at $b$ ($ab > bc$):
Hypothesis: $ab^2 = ac \cdot bc = (ab + bc)\,bc$.
Claim: $\left(bc + \frac{ab}{2}\right)^2 = 5\left(\frac{ab}{2}\right)^2$.
Proof: Since $ab^2 = bc^2 + ab \cdot bc$ by hypothesis, adding $\left(\frac{ab}{2}\right)^2$ to both sides yields
\[\left(bc + \frac{ab}{2}\right)^2 = bc^2 + ab \cdot bc + \left(\frac{ab}{2}\right)^2 = ab^2 + \frac{ab^2}{4} = 5\left(\frac{ab}{2}\right)^2.\]Substituting the numerical values of Chapter XII ($ac = 10$, $ab = \sqrt{125} - 5$, $bc = 15 - \sqrt{125}$) confirms the identity of both sides immediately.
Note 4 (Cap. XIII): Algebraic Proof of the Fourth Effect (Euclid XIII.5)¶
Corresponding Chapters: Cap. 13
In algebraic form, let the segment $bc$ be divided in extreme and mean ratio at $d$ ($cd > bd$), and let $ac = cd$ be adjoined to form the whole line $ad = ac + bc$:
Hypothesis: $cd^2 = bc \cdot bd$ and $ac = cd$.
Claim: $bc^2 = ad \cdot cd = ad \cdot ac$.
Proof: By hypothesis, $cd^2 = bc(bc - cd) = bc^2 - bc \cdot cd$, whence
\[bc^2 = cd^2 + bc \cdot cd = cd(cd + bc) = cd(ac + bc) = ad \cdot cd. \quad \text{Q.E.D.}\]
Note 5 (Cap. XIV): Algebraic Proof of the Fifth Effect (Euclid XIII.4)¶
Corresponding Chapters: Cap. 14
In general algebraic form, for the line $bc$ divided in extreme and mean ratio at $d$ ($cd > bd$):
Hypothesis: $cd^2 = bc \cdot bd$.
Claim: $bd^2 + bc^2 = 3\,cd^2$.
Proof: a) From Note 4, $bc^2 = bc \cdot cd + cd^2$. b) Expanding $bd^2 = (bc - cd)^2 = bc^2 - 2\,bc \cdot cd + cd^2$. Adding (a) and (b) together and substituting $bc^2 - bc \cdot cd = bc(bc - cd) = bc \cdot bd = cd^2$ gives:
\[bd^2 + bc^2 = 2\,cd^2 + bc(bc - cd) = 2\,cd^2 + bc \cdot bd = 3\,cd^2. \quad \text{Q.E.D.}\]
Note 6 (Cap. XVI): Seventh Effect — Hexagon Side (Radius) and Decagon Side (Euclid XIII.9)¶
Corresponding Chapters: Cap. 16
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.
Note 7 (Cap. XVIII): Ninth Effect — Diagonals of the Regular Pentagon (Euclid XIII.8)¶
Corresponding Chapters: Cap. 18
The proof follows immediately from the similarity of the isosceles triangles $\triangle bca \sim \triangle baf$ formed by the two intersecting chords $ac$ and $be$ (meeting at $f$) of the regular pentagon:
and since $ba = bc = cf$, we have $cf^2 = ca \cdot af$, so the chord $ca$ is cut in extreme and mean ratio at $f$.
Note 8 (Cap. XXI): Twelfth Effect — Ratio of the Cube Edge to the Icosahedron Edge in the Same Sphere¶
Corresponding Chapters: Cap. 21
Denoting by $s$ and $s'$ the edges of the cube and regular icosahedron inscribed in the same sphere, and letting the segment $ab$ be divided in extreme and mean ratio at $c$ ($ac > bc$), Pacioli's claim (following Campanus XIV.9) is:
Proof: Using the extreme and mean ratio relations (a) $ac^2 = ab \cdot bc$ and (b) $bc^2 = (ab - ac)^2 = ab^2 - 2\,ab \cdot ac + ac^2$, we have $ab^2 + bc^2 = 3\,ac^2 = 3\,ab \cdot bc$ (by Note 5) and $ab^2 + ac^2 = ab^2 + ab \cdot bc = ab(ab + bc)$. Hence the quotient under the radical simplifies to:
Solving $ac^2 = ab \cdot bc = ab(ab - ac)$ gives $bc = \frac{3 - \sqrt{5}}{2}\,ab$, and substituting this into the quotient yields:
The exact same value is obtained when we express $s$ and $s'$ in terms of the circumradius $r$ of the common sphere (Euclid XIII.15, Euclid XIII.16, Euclid XIII.18):
Note 9 (Cap. XXII): Cross-Reference to the Libellus de Quinque Corporibus Regularibus¶
Corresponding Chapters: Cap. 22
Compare on this pentagon construction the Tractatus de Quinque Corporibus Regularibus (Libellus, Tractate II, Case 20 ff.).
Note 10 (Cap. XXVI): Division of the Sphere Diameter by the Base of the Inscribed Tetrahedron (Euclid XIII.13)¶
Corresponding Chapters: Cap. 26
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:
Note 11 (Cap. XXVII): Division of the Sphere Diameter by the Inscribed Cube (Euclid XIII.15)¶
Corresponding Chapters: Cap. 27
The square of the diagonal of a square is double the square of its side ($ad^2 = 2\,bd^2$). Passing a plane through one of the three main space diagonals $ab$ of the cube and an edge $bd$, this plane intersects the circumscribed sphere in a great circle whose diameter is $ab$, while the two chords drawn from its endpoints to $d$ are the cube edge $bd$ and the face diagonal $ad$. By the right-triangle projection theorem (Euclid VI.8):
Since $ad^2 = 2\,bd^2$, dividing the two equations gives:
Note 12 (Cap. XXIX): Construction of the Regular Icosahedron in the Sphere (Euclid XIII.16)¶
Corresponding Chapters: Cap. 29
The edge $a$ of the regular icosahedron in terms of the circumradius $r$ of the sphere is:
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:
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:
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!
Note 13 (Cap. XXX): Verification of the Pentagon Diagonal in the Dodecahedron Construction (Euclid XIII.17)¶
Corresponding Chapters: Cap. 30
Retaining the lettering of the figure in Chapter XXX (where $ke$ is half the cube edge, divided in extreme and mean ratio at $l$ with $kl > le$, and $an = hn = 2\,kl$ is the cube edge): by the three-dimensional Pythagorean projection,
Since $kl^2 = ke \cdot le$, we can rewrite this as:
whence $ln = kl$, proving that the five vertices lie at equal radii and form an equilateral and equiangular pentagon (Euclid XIII.17).
Note 14 (Cap. XXX): Semicircle Proof of the Fifth Effect Used in the Dodecahedron Sphere Inscription¶
Corresponding Chapters: Cap. 30
This relation ($el^2 + ke^2 = 3\,kl^2$, already proved algebraically in Note 5) is seen most simply geometrically if we describe a semicircle on $ek$ as diameter, erect at $l$ a perpendicular meeting the semicircle at $m$, and draw the chords $em$ and $km$:
By hypothesis, $kl^2 = ke \cdot el$, and by the right triangle $emk$ in the semicircle, $em^2 = ke \cdot el$, so $em = kl$.
Moreover, $el^2 = em^2 - ml^2$ and $ke^2 = em^2 + mk^2$. Adding both gives:
\[el^2 + ke^2 = 2\,em^2 + (mk^2 - ml^2) = 2\,em^2 + kl^2 = 3\,em^2 = 3\,kl^2. \quad \text{Q.E.D.}\]
Note 15 (Cap. XLVI): Refutation of Pacioli’s Claim That the Octahedron Cannot Be Inscribed in the Icosahedron (Tartaglia & Isoperimetric Approximation to the Sphere)¶
Corresponding Chapters: Cap. 46
The assertion stated here by Pacioli (following Campanus XV.13) is manifestly false: in a completely analogous manner to how the octahedron is inscribed in the dodecahedron (Chapter XLIII, Euclid XV.9), the regular octahedron can also be inscribed in the regular icosahedron!
Construction of the Octahedron in the Icosahedron: One need only connect the midpoints of each pair of opposite parallel edges of the icosahedron. Among the 15 such segments (since the 30 edges of the icosahedron fall into 15 opposite parallel pairs), there are 5 sets of three mutually perpendicular axes meeting at the center of the icosahedron; the six endpoints of any such orthogonal triad of edge-midpoints form the 6 vertices of a regular octahedron inscribed in the icosahedron! (As Niccolò Tartaglia already demonstrated in his General Trattato di Numeri et Misure, refuting Campanus and Pacioli.)
Cosmological Bias vs. Isoperimetric Truth: The reason why Pacioli denies this elementary geometric fact stems from his desire to vindicate the Platonic cosmological hierarchy by assigning to the dodecahedron (as the symbol of the Quintessence / Heaven) the unique privilege of containing all four other regular solids, while restricting the icosahedron (Water) to only three. Pacioli's preference for the dodecahedron over the icosahedron would have been mathematically well-founded had he chosen as his criterion not how many other solids can be inscribed in it, but rather which regular solid inscribed in the same sphere approaches most closely to the sphere itself—that is, which solid encompasses the greatest surface area and the greatest volume! By this isoperimetric criterion (already proved by Hypsicles in Book XIV, Proposition 2, and by Pappus), the decision falls decisively in favor of the dodecahedron over the icosahedron.
Note 16 (Cap. XLVII–LII): Reciprocal (Dual) Polyhedra Generated by Pyramidal Elevation of Faces and Planar Truncation of Vertices¶
Corresponding Chapters: Cap. 47, Cap. 48, Cap. 49, Cap. 50, Cap. 51, Cap. 52
Regarding the generation of the regular bodies and their dependent derivatives, Pacioli distinguishes two primary classes: plane (convex) and truncated (*:term:`abscisi <abscisum>`*) bodies, each rendered both solid (*:term:`solidum <solidum>`*) and hollow/skeletal (*:term:`vacuum <vacuum>`*), and then modifies them further by elevating equilateral pyramids (*elevati*) outward upon their faces. (The inverse operation—excavating pyramids inward upon the same faces—is not mentioned.)
The star-like elevated polyhedra can likewise be inscribed in a sphere passing through all the pyramid apices. If we regard those pyramid apices as the vertices of a new convex polyhedron, we immediately obtain the reciprocal (dual) regular solid:
The elevated tetrahedron leads again to a regular tetrahedron (4 apices);
The elevated cube leads to the regular octahedron (6 apices);
The elevated octahedron (Stella Octangula) leads to the regular cube (8 apices);
The elevated icosahedron leads to the regular dodecahedron (20 apices);
The elevated dodecahedron leads to the regular icosahedron (12 apices).
More generally, the five regular solids correspond two by two as reciprocal (dual) pairs with respect to the numbers of their vertices ($V$), edges ($E$), and faces ($F$):
Tetrahedron $\leftrightarrow$ Tetrahedron (Self-Reciprocal): $E = 6$, $V = 4$, $F = 4$.
Cube $\leftrightarrow$ Octahedron: $E = 12$, with $V = 8, F = 6$ for the Cube and $V = 6, F = 8$ for the Octahedron.
Icosahedron $\leftrightarrow$ Dodecahedron: $E = 30$, with $V = 12, F = 20$ for the Icosahedron and $V = 20, F = 12$ for the Dodecahedron.
From any regular body, its reciprocal (dual) body arises in two dual ways:
By transforming faces into vertices (Elevation): Erecting pyramids upon the faces and taking their apices as the vertices of the new body (when the pyramid height is chosen so that the apices of the pyramids sharing a common vertex of the original solid become coplanar with that vertex, the new reciprocal solid circumscribes the sphere that circumscribed the original solid).
By transforming vertices into planes (Truncation): Slicing off the corners by planes until the cuts around each face meet at the centroid of that face, so that the original faces shrink to points (vertices) and the new body is circumscribed about the sphere that was inscribed in the original body.
Note 17 (Cap. XXXV–XLVI): Complete Projective Theory of the 21 Mutual Inscriptions of the Five Regular Solids via Polar Reciprocity in the Sphere¶
Corresponding Chapters: Cap. 35, Cap. 46
To understand the mutual relationships of the five regular solids systematically and resolve all possible questions of mutual inscription at a single stroke, we employ the projective theory of poles and polar planes with respect to a concentric sphere.
1. Direct vs. Reciprocal Correspondence and Polarity in the Sphere¶
Between the elements (vertices and bounding faces) of two polyhedra, we distinguish two kinds of correspondence:
Direct Correspondence: Each vertex of the inner solid corresponds to a vertex of the outer solid, or each face corresponds to a face.
Reciprocal Correspondence: Each vertex of the first solid corresponds to a bounding face of the second, or conversely each face of the first corresponds to a vertex of the second.
The simplest universal mechanism for establishing a mutual reciprocal correspondence is polar reciprocation with respect to a concentric sphere:
The pole of a plane cutting a sphere is the apex of the tangent cone touching the sphere along the circle of intersection; conversely, that plane is the polar plane of the point.
Taking a sphere concentric with any regular solid, the polar planes of all its vertices (or equivalently, the poles of all its faces) form its polar reciprocal regular solid:
Tetrahedron $\leftrightarrow$ Tetrahedron
Cube $\leftrightarrow$ Octahedron
Icosahedron $\leftrightarrow$ Dodecahedron
As the radius $R$ of the reciprocating sphere expands continuously from $0$ to $\infty$:
First Limit ($R = \rho$, the inradius of the original solid): The sphere touches the faces of the original solid from within; the poles of the faces are the centroids of the faces, so the reciprocal solid is inscribed in the original solid with its vertices at the face-centers.
Symmetric Dual Intersection ($R = \sqrt{r\rho}$, the geometric mean of the circumradius $r$ and inradius $\rho$): Both reciprocal solids are inscribed in the same outer sphere of radius $r$ and circumscribed about the same inner sphere of radius $\rho$!
Second Limit ($R = r$, the circumradius of the original solid): The sphere passes through the vertices of the original solid; the polar planes are the tangent planes at those vertices, so the original solid is now inscribed in its reciprocal solid!
2. Complete Classification of All 21 Possible Mutual Inscriptions¶
By means of polar reciprocation, all 20 mutual inscriptions between distinct regular solids (plus 1 self-inscription of the tetrahedron in the tetrahedron, making 21 cases in all) fall into three direct groups (a) and three polar-reciprocal groups (b), where an asterisk (*) marks the 12 cases treated by Euclid Book XV / Campanus / Pacioli in Chapters XXXV–XLVI:
(a) Direct Inscriptions (Vertices of Inner Solid) |
(b) Polar-Reciprocal Inscriptions (Faces of Outer Solid) |
|---|---|
Group I(a): Vertex on Vertex (Ecke auf Ecke) |
Group I(b): Face on Face (Fläche auf Fläche) |
1. Tetrahedron in Cube |
2. Octahedron in Tetrahedron |
3. Tetrahedron in Dodecahedron |
4. Icosahedron in Tetrahedron (omitted by Pacioli) |
5. Cube in Dodecahedron |
6. Icosahedron in Octahedron (omitted by Pacioli) |
Group II(a): Vertex on Face (Ecke auf Fläche) |
Group II(b): Face on Vertex (Fläche auf Ecke) |
7. Octahedron in Cube |
8. Cube in Octahedron |
9. Icosahedron in Dodecahedron |
10. Dodecahedron in Icosahedron |
11. Tetrahedron in Icosahedron |
12. Dodecahedron in Tetrahedron (omitted by Pacioli) |
13. Cube in Icosahedron |
14. Dodecahedron in Octahedron (omitted by Pacioli) |
Group III(a): Vertex on Edge (Ecke auf Kante) |
Group III(b): Edge on Face (Kante in Fläche) |
15. Octahedron in Dodecahedron |
16. Icosahedron in Cube (treated in Piero/Pacioli’s Libellus!) |
17. Octahedron in Icosahedron (denied by :person:`Pacioli <pacioli>` in Cap. XLVI; proved by :person:`Tartaglia <niccolo-tartaglia>` & Note 15) |
18. Dodecahedron in Cube (omitted by Pacioli) |
19. Tetrahedron in Octahedron |
20. Cube in Tetrahedron (omitted by Pacioli) |
21. Tetrahedron in Tetrahedron (vertices at face-centroids; omitted in Cap. XXXV–XLVI) |
(Self-reciprocal) |
3. Geometric Derivation of the Reciprocal Cases¶
Group I (1 $\leftrightarrow$ 2, 3 $\leftrightarrow$ 4, 5 $\leftrightarrow$ 6): Taking the polar reciprocals of two solids sharing common vertices on a circumscribed sphere converts the inner solid into the outer reciprocal solid and the outer solid into the inner reciprocal solid, with coincident faces replacing coincident vertices:
From 1 (Tetrahedron in Cube) arises 2 (Octahedron in Tetrahedron), where 4 of the 8 faces of the octahedron lie in the 4 faces of the tetrahedron and the 6 vertices of the octahedron lie at the 6 edge-midpoints of the tetrahedron.
From 3 (Tetrahedron in Dodecahedron) arises 4 (Icosahedron in Tetrahedron), where 4 faces of the icosahedron lie in the 4 faces of the circumscribing tetrahedron (obtained by extending the 4 icosahedron faces whose centroids form the inscribed tetrahedron of Case 11 until they meet!).
From 5 (Cube in Dodecahedron) arises 6 (Icosahedron in Octahedron), where 8 faces of the icosahedron lie in the 8 faces of the octahedron and all 12 vertices of the icosahedron lie on the 12 edges of the octahedron (dividing them in the golden ratio!).
Group II (7 $\leftrightarrow$ 8, 9 $\leftrightarrow$ 10, 11 $\leftrightarrow$ 12, 13 $\leftrightarrow$ 14):
7 $\leftrightarrow$ 8 (Octahedron $\leftrightarrow$ Cube) and 9 $\leftrightarrow$ 10 (Icosahedron $\leftrightarrow$ Dodecahedron) are dual pairs directly.
From 11 (Tetrahedron in Icosahedron), reciprocating with respect to the sphere touching the icosahedron and circumscribing the dodecahedron yields 12 (Dodecahedron in Tetrahedron): passing tangent planes through the 4 common vertices of the dodecahedron and its inscribed tetrahedron (Case 3) forms the circumscribing tetrahedron touching 4 vertices of the dodecahedron.
From 13 (Cube in Icosahedron) arises 14 (Dodecahedron in Octahedron), where the centers of the 8 octahedron faces contain the 8 vertices of the dodecahedron that form an inscribed cube.
Group III (15 $\leftrightarrow$ 16, 17 $\leftrightarrow$ 18, 19 $\leftrightarrow$ 20): Because the two solids do not share a common circumsphere when vertices fall on edges, we reciprocate with respect to any concentric sphere (e.g., the sphere circumscribing the outer solid), transforming vertices on edges into faces containing edges:
From 15 (Octahedron in Dodecahedron) arises 16 (Icosahedron in Cube), where 6 opposite parallel edges of the icosahedron lie in the 6 square faces of the cube along their midlines in three mutually perpendicular directions, so that all 12 vertices of the icosahedron rest in pairs on the 6 faces of the cube! Remarkably, although omitted here in the Compendio, this exact inscription is explicitly treated in Piero della Francesca / Pacioli's companion Libellus de Quinque Corporibus Regularibus!
From 17 (Octahedron in Icosahedron) arises 18 (Dodecahedron in Cube), where 6 opposite parallel edges of the dodecahedron lie along the midlines of the 6 faces of the cube (while the 8 remaining vertices of the dodecahedron, corresponding to an inscribed cube, lie free inside).
From 19 (Tetrahedron in Octahedron) arises 20 (Cube in Tetrahedron), where the 4 faces of the tetrahedron contain 4 edges of the cube (two opposite top edges of the cube and the two perpendicular bottom edges).