Metrica Book II (Preface & Props. 1–20): Stereometry, Dionysodorus’s Torus, the Method & Platonic Solids¶
Eudoxus & Archimedes in Stereometry, Frusta & Hip-Roof Wedges, Spheres & Vaults, Dionysodorus’s Torus Theorem (9956 4/7), Archimedes’ Method Vaults, the Five Platonic Solids, and Hydrostatic Displacement — Schöne (1903), pp. 90–140 · Codex Constantinopolitanus Palatii Veteris 1, fols. 87r–99v
Note
Edition Track (English): Original GEOMETOR scholarly translation and stereometric synthesis directly from Hermann Schöne's 1903 Teubner editio princeps (pp. 90–140), verified against 300 DPI archival page scans. Hover or click any highlighted geometric symbol (such as (ZB), DEH2H1, or AZ) to illuminate the corresponding element in the interactive meridian diagram. Switch tracks above for Greek (Teubner), Summary, or Notes.
Prooemium & Props. II.1–II.10: Eudoxus, Cones, Oblique Prisms, Wedges & Frusta¶
Book II, Prooemium (Schöne pp. 90–94; fols. 87r–88r). Having set forth in the First Book the mensuration of planar and curved surfaces, Hero opens the Second Book on solid bodies (στερεὰ σώματα) with a classic historical tribute: before Eudoxus of Cnidus, geometers did not possess rigorous proofs that every cylinder is triple the cone on the same base and of equal height ([XII.10]), nor that every prism is triple the pyramid; and after Eudoxus established those theorems by exhaustion, Archimedes surpassed all predecessors by proving that every sphere is two-thirds of its circumscribing cylinder (Sphere and Cylinder I.34).
Just as every planar area is referred to a unit square, every solid body is referred to an orthogonal unit cube (στερεὸς κύβος). Consequently, even when a cylinder, cone, or prism is oblique (κεκλιμένος), its solid content equals the product of its base area by its perpendicular altitude (kathetos) dropped orthogonally to the plane of the base—never by its inclined slant edge!
Propositions II.1–II.5 (Cones, Oblique Cylinders, Parallelepipeds, Prisms & Pyramids, Schöne pp. 94–108): Hero computes right and oblique cylinders (\(V = A_{\text{base}} \cdot h\)), cones (\(V = \frac{1}{3} A_{\text{base}} \cdot h\)), hexagonal and polygonal prisms, and pyramids on arbitrary triangular or polygonal bases, obtaining the perpendicular altitude \(h\) from the slant edge and circumradius (or base projection) via [I.47].
Propositions II.6–II.7, II.9–II.10 (Pyramidal & Conical Frusta — Hero’s Mean-Plus-Third-Difference Identity, Schöne pp. 108–120): Rather than computing a truncated pyramid or cone (frustum, κόλουρος) by extending it to its apex and subtracting two large pyramids, Hero transforms the classical multiplier \(\frac{a^2 + ab + b^2}{3}\) into his signature mean-square plus one-third half-difference-square identity:
\[\frac{a^2 + ab + b^2}{3} = \left(\frac{a + b}{2}\right)^2 + \frac{1}{3}\left(\frac{a - b}{2}\right)^2,\]so that for a square frustum of lower side \(a = 10\), upper side \(b = 2\), and perpendicular height \(h = 9\) (slant edge \(9\) or \(\sqrt{81 + 32}\)):
\[V_{\text{frustum}} = h \left[\left(\frac{a + b}{2}\right)^2 + \frac{1}{3}\left(\frac{a - b}{2}\right)^2\right],\]and similarly for a circular conical frustum of diameters \(D_1, D_2\):
\[V_{\text{cone-frustum}} = \frac{\pi}{4} h \left[\left(\frac{D_1 + D_2}{2}\right)^2 + \frac{1}{3}\left(\frac{D_1 - D_2}{2}\right)^2\right].\]Notice the engineering insight: the naive “average cross-section” rule \(h\left(\frac{a+b}{2}\right)^2\) underestimates the true volume by precisely \(\frac{1}{3} h \left(\frac{a-b}{2}\right)^2\)!
Proposition II.8 (The *Βωμίσκος* / Hip-Roof Wedge / Prismatoid, Schöne pp. 114–116): Generalizing to a solid bounded by two parallel dissimilar rectangles \((a_1 \times b_1)\) and \((a_2 \times b_2)\) at distance \(h\)—essential for temple altars (βωμίσκοι), stone lintels, and hipped roofs—Hero decomposes the solid into a central parallelepiped, two triangular prisms, and four corner pyramids, proving the exact universal prismatoid law:
\[V_{\text{bomiskos}} = h \left[\left(\frac{a_1 + a_2}{2}\right)\left(\frac{b_1 + b_2}{2}\right) + \frac{1}{3}\left(\frac{a_1 - a_2}{2}\right)\left(\frac{b_1 - b_2}{2}\right)\right].\]
Props. II.11–II.15: Spheres, Dionysodorus’s Torus (9956 4/7) & Archimedes’ Method Vaults¶
Propositions II.11–II.12 (Spheres, Spherical Segments & Vaulted Basins, Schöne pp. 120–126): By Archimedes' Sphere and Cylinder I.34, the sphere is \(\frac{2}{3}\) of the circumscribing cylinder (\(\frac{\pi}{4} d^2 \cdot d\)), giving \(V_{\text{sphere}} = \frac{\pi}{6} d^3 \approx \frac{11}{21} d^3\). For a spherical segment of base radius \(r_b\) and sagitta (height) \(h\) (Prop. II.12), Hero applies Sphere and Cylinder II.2 to derive the intrinsic formula requiring no separate lookup of the full sphere’s radius:
\[V_{\text{seg}} = \frac{1}{2}\pi r_b^2 h + \frac{1}{6}\pi h^3 = \frac{\pi}{6} h \left(3 r_b^2 + h^2\right).\]
Proposition II.13 — Dionysodorus’s Torus Theorem (*Περὶ τῆς σπείρας*, Schöne pp. 126–130; fols. 95v–96r):
Hero next preserves our sole surviving Greek witness to the lost treatise On the Torus (Περὶ τῆς σπείρας) by Dionysodorus of Caunus (fl. c. 200 BCE):
“Let it be required to measure a torus (:term:`speira`), having first set out its generation. Let there be in a plane a straight line AB and two points B, C (Hero’s \(\Gamma\) ) upon it. Let the circle BCDE be perpendicular to the underlying plane in which AB lies; and while point A remains fixed, let AB be carried round in the plane until it is restored to the same position, carrying round with it the circle BCDE remaining always perpendicular to the underlying plane. Then the circumference BCDE will generate a certain surface which they call spiric (toroidal)…”
(Diagram Mapping: A = fixed center of revolution \((0,0)\); C = inner rim \(\Gamma = (8,0)\); Z = generating circle center \((14,0)\); B = outer rim \((20,0)\); D, E = vertical diameter endpoints \((14, \pm 6)\); H1, H2 = endpoints \(H, \Theta\) on the vertical axis of revolution).
Dionysodorus’s Proportion: It was proved by Dionysodorus in On the Torus that as the generating circle (ZB) is to half the meridian parallelogram DEH2H1 (which has height \(\Delta E = 2r\) and width AZ \(= R\), so half its area is \(R r\)), so is the torus generated by (ZB) to the cylinder whose axis is \(H\Theta = 2r\) and whose base radius is AZ \(= R\) (volume \(\pi R^2 \cdot 2r = 2\pi R^2 r\)):
\[\frac{V_{\text{torus}}}{2\pi R^2 r} = \frac{\pi r^2}{R r} \qquad \Longrightarrow \qquad V_{\text{torus}} = 2\pi^2 R r^2.\]Hero’s Unrolled-Cylinder Centroid Demonstration: Hero immediately adds a breathtaking physical-geometric alternative («δυνατὸν δέ ἐστι καὶ ἄλλως μετρῆσαι») that directly anticipates Pappus’s Centroid Theorem (Synagoge VII): For \(AC = 8\) and \(BC = 12\) (so minor radius ZB \(r = 6\) and major radius AZ \(R = 14\)), the circle swept by center Z has diameter \(2R = 28\) and circumference \(2\pi R \approx \frac{22}{7} \times 28 = 88\). “When therefore the torus is unrolled and becomes like a cylinder” (ἁπλωθεῖσα ἄρα ἡ σπεῖρα καὶ γενομένη ὡς κύλινδρος), it has length \(L = 88\) and circular cross-section of diameter \(BC = 12\) (area \(\frac{11}{14} \times 144 = \frac{792}{7}\)), yielding the exact rational volume:
\[V_{\text{torus}} = 88 \times \frac{792}{7} = \frac{69{,}696}{7} = 9956\frac{4}{7}.\]
Propositions II.14–II.15 (Archimedes’ *Method* Solids: Cylindrical Ungula & Bicylinder Groin Vault, Schöne pp. 130–132; fols. 96v–97r): Immediately after Dionysodorus’s torus, Hero cites the two celebrated final propositions of Archimedes' Method (Ἐφοδικόν)—the very theorems whose Greek leaves were torn from or barely legible at the end of the Archimedes Palimpsest!
Prop. II.14 (Cylindrical Hoof / Ungula): When a plane cuts a right cylinder through a diameter of its base and meets the opposite generator at height \(h\)—useful for fluted niches and half-vaults—the cut-off wedge is exactly one-sixth of the circumscribing square prism (\(d \times d \times h\)):
\[V_{\text{ungula}} = \frac{1}{6} d^2 h = \frac{2}{3} r^2 h.\]Prop. II.15 (Intersection of Two Perpendicular Equal Cylinders — Groin Vault / Steinmetz Solid): When two equal right circular cylinders of diameter \(d\) intersect at right angles inside a cube of edge \(d\)—the monumental Roman cross-vault (καμάρα)—their common solid is two-thirds of the cube:
\[V_{\text{bicylinder}} = \frac{2}{3} d^3 = \frac{16}{3} r^3.\]
Props. II.16–II.20: The Five Platonic Solids & Irregular Hydrostatic Displacement¶
Propositions II.16–II.19 — Volumes of the Five Regular Solids (Schöne pp. 132–138; fols. 97r–99r):
Turning to the five cosmic regular polyhedra of Plato, Theaetetus, and Euclid ([XIII.13]–[XIII.17]), Hero decomposes the octahedron into two square pyramids and the icosahedron and dodecahedron into \(20\) triangular and \(12\) pentagonal pyramids meeting at the center of the circumscribed/inscribed sphere:
Prop. |
Regular Polyhedron |
Faces |
Exact Volume \(V(a)\) in \(\mathbb{Q}(\sqrt{2}, \sqrt{3}, \phi)\) |
|---|---|---|---|
Unit |
Cube (Κύβος) |
\(6\) squares |
\(a^3\) (foundational stereometric unit) |
II.16 |
Regular Tetrahedron (Πυραμίς) |
\(4\) equilateral \(\triangle\) |
\(\frac{\sqrt{2}}{12} a^3\) (altitude \(h = \sqrt{\frac{2}{3}} a\) ) |
II.17 |
Regular Octahedron (Ὀκτάεδρον) |
\(8\) equilateral \(\triangle\) |
\(\frac{\sqrt{2}}{3} a^3\) (two square pyramids of height \(\frac{a}{\sqrt{2}}\) ) |
II.18 |
Regular Icosahedron (Εἰκοσάεδρον) |
\(20\) equilateral \(\triangle\) |
\(\frac{5(3 + \sqrt{5})}{12} a^3 = \frac{5\phi^2}{6} a^3\) (inradius via \(\mathbb{Q}(\phi)\) ) |
II.19 |
Regular Dodecahedron (Δωδεκάεδρον) |
\(12\) regular pentagons |
\(\frac{15 + 7\sqrt{5}}{4} a^3 = \frac{5\phi^3}{2(2-\phi)} a^3\) (12 pentagonal pyramids) |
Notice the exact golden-ratio duality between II.18 and II.19: both depend directly upon the extreme-and-mean-ratio geometry established in Proposition I.18!
Proposition II.20 (Measuring Any Irregular Solid Body, Schöne pp. 138–140; fols. 99r–99v): Book II closes with an unforgettable synthesis of physics and geometry. How can one measure an irregular sculpted statue, rough boulder, or vessel (ἀκανόνιστον σῶμα) bounded by no geometric formula? Hero gives two exact operational methods:
Hydrostatic Immersion (*Archimedes’ Bath*): Place a watertight rectangular parallelepiped tank horizontally, fill it with water, submerge the irregular solid completely, and measure the vertical rise (or displaced overflow) of the rectangular prism of water; its length \(\times\) breadth \(\times\) depth gives the irregular solid’s exact volume in cubic units!
Plastic Clay/Wax Encasement: If the body is too monumental or immovable to immerse, pack wax or potter’s clay (κηρὸς ἢ πηλός) around it and plane the exterior flat with straightedges into a right-angled rectangular parallelepiped; measure that parallelepiped, strip off the clay and mold the clay alone into a smaller rectangular brick, and subtract the brick’s volume from the whole!