Summary — Metrica Book I (Props. 10–39)¶
Structural Overview of Quadrilaterals, Regular Polygons, Curvilinear Regions & Curved Surfaces
1. Deductive Architecture of Metrica I.10–I.39¶
Having solved the general triangle in Propositions I.1–I.9, Hero extends planar and surface mensuration across four progressively richer geometric domains:
Quadrilaterals & Trapezia (Props. I.10–I.16): Establishes why four side-lengths alone leave a quadrilateral underdetermined, reduces every trapezium (right, isosceles, acute, obtuse) to a parallelogram plus a differential scalene triangle of base \(|a - c|\) (solved by Prop. I.8), and decomposes general convex or dart-shaped (κοιλογώνιον) quadrilaterals along a given diagonal.
Regular Polygons from :math:`n = 3` to :math:`n = 12` (Props. I.17–I.25): Unifies Euclidean constructible polygons (\(3, 4, 5, 6, 8, 10, 12\)) with chord-table polygons (\(7, 9, 11\)), proving the Prop. I.18 Golden Section Lemma \((BA + A\Gamma)^2 = 5 A\Gamma^2\) over \(\mathbb{Q}(\phi)\) and supplying high-precision rational multipliers \(A_n / a^2\).
Circles, Annuli & Segments (Props. I.26–I.33): Combines Archimedes' Measurement of a Circle (\(\pi \approx \frac{22}{7}\)), lost Plinthides and Cylinders bounds, the concentric annulus formula, the \(4:1\) geometric series (Prop. I.27), intersecting-chords diameter recovery (Prop. I.30), and lower bounds on circular segments via inscribed parabolic exhaustion (Props. I.31–I.33).
Conic Regions & Curved Surfaces in 3D (Props. I.34–I.39): Measures the ellipse (\(\pi a b\), Prop. I.34), the parabolic segment (\(\frac{4}{3} \Delta_{\text{inscr}}\) citing Archimedes’ Method, Prop. I.35), and the curved surface areas of the right cylinder, right cone, sphere (\(4\pi r^2\)), and spherical cap (\(\pi c^2 = 2\pi R h\), Props. I.36–I.39).
2. Key Algebraic & Constructive Identities¶
Differential Triangle of a Trapezium (I.12–I.13): For parallel bases \(a > c\) and legs \(b, d\), the altitude \(h\) equals the altitude onto base \((a - c)\) of the triangle \((b, d, a - c)\):
\[A_{\text{trap}} = \frac{a + c}{a - c}\,\Delta(b, d, a - c).\]Pentagon Golden Section Lemma (I.18): In a \(36^\circ\text{-}54^\circ\text{-}90^\circ\) right triangle \(AB\Gamma\) (\(\angle \Gamma = 90^\circ, \angle A = 36^\circ\)):
\[(BA + A\Gamma)^2 = 5\,A\Gamma^2, \qquad \frac{BA}{A\Gamma} = \sqrt{5} - 1 = \frac{2}{\phi}, \qquad s(\angle A) = \sin^2(36^\circ) = \frac{5 - \sqrt{5}}{8}.\]Annulus (*Ἴτυς*, I.26): \(A_{\text{itys}} = \pi(R^2 - r^2) = \pi w (2r + w) \approx \frac{22}{7} w(D_2 + w)\).
Ellipse & Parabola (I.34–I.35): \(A_{\text{ellipse}} = \pi a b\) and \(A_{\text{parabola}} = \frac{4}{3} A_{\triangle}\).