Summary — Metrica Book II (Preface & Props. 1–20)

Structural Overview of Stereometry, Toroidal Revolution, Archimedean Vaults & Regular Polyhedra

1. Deductive Architecture of Metrica Book II

Book II advances from two-dimensional surfaces to three-dimensional solid capacities measured in unit cubes:

  1. Historical & Dimensional Foundation (Prooemium): Credits Eudoxus (cone = \(\frac{1}{3}\) cylinder; pyramid = \(\frac{1}{3}\) prism) and Archimedes (sphere = \(\frac{2}{3}\) cylinder), and establishes Cavalieri-like invariance for oblique cylinders and prisms under shear (\(V = A_{\text{base}} \cdot h_{\perp}\)).

  2. Polyhedral & Conical Architecture (Props. II.1–II.10): Measures right and oblique cones, cylinders, prisms, pyramids, pyramidal/conical frusta via the mean-plus-third-difference identity, and the rectangular hip-roof wedge (*βωμίσκος*, Prop. II.8).

  3. Spheres, Torus & *Method* Vaults (Props. II.11–II.15): Measures the sphere, spherical segment/basin, Dionysodorus's torus (Prop. II.13, \(V = 2\pi^2 R r^2\)), and Archimedes’ Method cylindrical ungula (\(\frac{1}{6}d^2 h\)) and bicylinder groin vault (\(\frac{2}{3}d^3\), Props. II.14–II.15).

  4. Cosmic Polyhedra & Physical Displacement (Props. II.16–II.20): Derives the volumes of the Five Platonic Solids (Props. II.16–II.19) and solves arbitrary irregular solids via hydrostatic displacement or clay/wax orthogonal encasement (Prop. II.20).

2. Canonical Stereometric Formulas

  • Frustum Identity (II.6–II.10):

    \[\frac{a^2 + ab + b^2}{3} = \left(\frac{a + b}{2}\right)^2 + \frac{1}{3}\left(\frac{a - b}{2}\right)^2.\]

  • Dionysodorus’s Torus & Hero’s Centroid Cylinder (II.13):

    \[V_{\text{torus}} = (2\pi R)(\pi r^2) = 2\pi^2 R r^2, \qquad V(14, 6)\big|_{\pi = 22/7} = 9956\frac{4}{7}.\]

  • Archimedean *Method* Solids (II.14–II.15):

    \[V_{\text{ungula}} = \frac{1}{6} d^2 h, \qquad V_{\text{bicylinder}} = \frac{2}{3} d^3.\]