Metrica Book I (Preface & Props. 1–9) — Analytical Summary¶
Scope: Schöne (1903), pp. 2–26 · Cod. Constantinopolitanus 1, fols. 67r–72r
Architectural Progression¶
Unit I opens Hero of Alexandria's Metrica by establishing the epistemological and metrological foundations of Greek mensuration and ascending in nine propositions from the unit square to the universal side-only area law for any triangle:
Philosophical Prooemium (Schöne pp. 2–6): - Recounts the origin of geometry in annual Nile flood resurveying (matching the Eudemian history preserved in Proclus, In Euclidem 64.16–65.7). - Formulates the Axiom of Homogeneous Orthogonal Comparison: lines are measured by linear units (cubit, foot, palm, dactyl), surfaces (embadon) by the orthogonal unit square, and solids by the orthogonal unit cube. - Announces the triadic design of the Metrica: Book I (plane & curved surfaces), Book II (solid bodies), Book III (proportional divisions).
Altitude-Based Triangle Mensuration (Props. I.1–I.7, Schöne pp. 6–18): - I.1–I.3: Oblong rectangle (\(5 \times 3 = 15\)), right triangle (\(3, 4, 5 \implies 6\)), isosceles triangle (\(10, 10, 12 \implies 48\)). - I.4–I.6: Uses [II.13] (acute, \((13, 14, 15) \implies B\Delta = 5, A\Delta = 12, \Delta = 84\)) and [II.12] (obtuse, \((13, 11, 20) \implies B\Delta = 5, A\Delta = 12, \Delta = 66\)) to locate the foot of the altitude and compute area. - I.7: Eliminates the acute/obtuse branch test via a unified algebraic projection formula.
Heron’s Formula & Iterative Root Extraction (Props. I.8–I.9, Schöne pp. 18–26): - I.8a (Algorithm & √720): Evaluates \((7, 8, 9) \implies s = 12 \implies 12 \cdot 5 \cdot 4 \cdot 3 = 720\), and introduces the quadratic Newton–Heron iteration \(x_{k+1} = \frac{1}{2}(x_k + N/x_k)\) from \(x_0 = 27\) (\(27^2 = 729\)) to \(x_1 = 26\frac{5}{6} = \frac{161}{6}\) (error \(x_1^2 - 720 = \frac{1}{36}\)) and \(x_2 = \frac{51841}{1932}\). - I.8b (Synthetic Proof & (13, 14, 15)): Proves \(\Delta^2 = s(s-a)(s-b)(s-c)\) synthetically using only the incircle ([IV.4]), equal tangents ([III.17]), cyclic quadrilateral \(\Gamma HB\Lambda\) ([III.22]), similar right triangles ([VI.4]), and the right-triangle altitude theorem ([VI.8]), culminating in \((13, 14, 15) \implies \sqrt{7056} = 84\). - I.9: Extends triangle mensuration to sides given only in square (irrational lengths).
Proposition Synopsis Table¶
Locus |
Schöne Pages |
Canonical Paradigm |
Key Euclidean / Algebraic Principle |
|---|---|---|---|
Prooemium |
pp. 2–6 |
Nile resurveying; unit square & cube |
[I.def.22]; homogeneous orthogonal units |
Props. I.1–I.3 |
pp. 6–10 |
\(5\times 3\); \((3,4,5)\); \((10,10,12)\) |
[I.34], [I.41], [I.47] |
Props. I.4–I.7 |
pp. 10–18 |
\((13,14,15) \to 84\); \((13,11,20) \to 66\) |
[II.12], [II.13] projection laws |
Prop. I.8 |
pp. 18–24 |
\((7,8,9) \to \sqrt{720}\); \((13,14,15) \to 84\) |
[IV.4], [III.22], [VI.8]; \(x_{k+1}=\frac{1}{2}(x_k + N/x_k)\) |
Prop. I.9 |
pp. 24–26 |
Irrational side quadrances |
Iterative rational root extraction |