Hero of Alexandria — Metrica, Definitiones, Catoptrica & Dioptra

Mensuration, Variational Optics, Geodesy, and Constructive Exactness in the GEOMETOR Digital Scriptorium

«Ἡ πρώτη γεωμετρία, καθάπερ ὁ παλαιὸς ἡμᾶς διδάσκει λόγος, περὶ τὰς ἐν Αἰγύπτῳ τῆς χώρας μετρήσεις καὶ διανομὰς ἠσχολεῖτο· διὰ γὰρ τὰς τοῦ Νείλου ἀναβάσεις τὰ ὅρια τῶν ἰδιοκτησιῶν ἠφανίζετο…»

“Original geometry, just as the ancient account teaches us, was occupied with the measurements and distributions of land in Egypt; for owing to the risings of the Nile, the boundaries of private holdings were obliterated…”

—Hero of Alexandria, Metrica I, Preface (ed. Hermann Schöne, Leipzig: B. G. Teubner, 1903, p. 2)

GEOMETOR • heron unites two complementary pillars of the GEOMETOR canon in a single scholarly and computational edition of Hero (Heron) of Alexandria (fl. c. 60 CE):

1. 4-Track Bilingual Book Codex (Hero of Alexandria — Metrica, Definitiones, Catoptrica & Dioptra): A folio-indexed Digital Scriptorium edition of Hero’s three-book masterwork *Metrica* (Μετρικά, rediscovered in 1896 by Richard Schöne in the 11th-century Codex Constantinopolitanus Palatii Veteris no. 1 and edited by Hermann Schöne, Teubner 1903), paired with foundational selections from *Definitiones* (ed. Hultsch 1864 / Heiberg 1912), *Catoptrica* (ed. Schmidt & Nix 1900), and *Dioptra* (ed. Schöne 1903). Every unit synchronizes four parallel tracks:

   • English: Original GEOMETOR scholarly translation directly from the scan-verified critical Greek editions, instrumented with interactive .euclid-graphic SVG diagrams and :g: / :eq: / :just: / :person: / :cite: / :term: markup.

   • Greek (Teubner): Polytonic Greek text transcribed via multimodal vision from 300 DPI archival scans of Schöne (1903), Schmidt (1900), and Hultsch (1864).

   • Summary: Proposition-by-proposition structural, algorithmic, and mathematical synopses.

   • Notes: Deep philosophical, historical, and Rational Trigonometry commentary cross-linked to executable symbolic demonstrations and sister GEOMETOR repositories (euclid, elements, proclus, pappus, wildberger, etc, divine).

2. 100% Exact Symbolic Python Constructions & Demos (Symbolic Constructions & Demos): Executable compass-and-straightedge proofs and mensuration engines in geometor.heron built on SymPy and geometor.model (Model(…, auto_intersect=False) with zero floating-point coordinates):

   • Heron’s Formula Synthetic Proof & Quadrea (Metrica I.8) — Hero’s synthetic incircle & cyclic-quadrilateral proof of \(\Delta = \sqrt{s(s-a)(s-b)(s-c)}\) (Metrica I.8) on \((13, 14, 15) \implies 84\) alongside Norman J. Wildberger's polynomial quadrea identity \(\mathcal{A} = 16\Delta^2\).

   • Iterative Square-Root Extraction on √720 (Metrica I.8) — Exact \(\mathbb{Q}\) rational sequence of Hero’s square-root extraction on \((7, 8, 9) \implies \sqrt{720}\) (\(x_0 = 27\), \(x_1 = \frac{161}{6}\), \(x_2 = \frac{51841}{1932}\), \(x_3, x_4\)).

   • Shortest-Path Law of Reflection (Catoptrica Prop. 4) — Hero’s Catoptrica Prop. 4 variational shortest-path proof of the Law of Reflection (\(S_{\text{inc}} = S_{\text{refl}}\) minimizes path length \(|\Gamma E| + |E\Delta|\)).

   • Proportional Area Divisions & ∛100 (Metrica Book III) — Metrica III.1–2 vertex (\(5:3\)) and parallel mean-proportional (\(3:1\), \(Q(A,D) = \frac{507}{4}\)) area divisions realizing geometric distributive justice.

   • Universal Golden Gnomon & Dioptra Mountain Tunnel — Hero’s universal gnomon (Definitiones §58) over \(\mathbb{Q}(\phi)\) and two-portal mountain tunneling by orthogonal right-triangle traversal (Dioptra Ch. 15).

Architecture of the Codex

Unit	Codex Chapter	Critical Source Pages	Interactive Constructions
I	Metrica Book I (Preface & Props. 1–9): Triangles, Heron’s Formula & √720	Schöne (1903), pp. 2–26	heron-i8-incircle.svg (Demo, √720)
II	Metrica Book I (Props. 10–39): Quadrilaterals, Regular Polygons, Circles & Conics	Schöne (1903), pp. 26–90	pentagon-golden-lemma.svg (\(\mathbb{Q}(\phi)\) Golden Lemma & Polygons 3–12)
III	Metrica Book II (Preface & Props. 1–20): Stereometry, Dionysodorus’s Torus, the Method & Platonic Solids	Schöne (1903), pp. 90–140	dionysodorus-torus-section.svg (Torus \(V = 2\pi^2 R r^2\), Vaults, 5 Solids)
IV	Metrica Book III (Preface & Props. 1–23): Proportional Divisions of Figures, Justice & ∛100	Schöne (1903), pp. 140–184	metrica-iii-area-division.svg (Demo)
V	Definitiones, Catoptrica & Dioptra: Genetic Axiomatics, Universal Gnomon, Shortest-Path Optics & Mountain Tunneling	Hultsch pp. 7–23 · Schmidt pp. 324–372 · Schöne pp. 238–240	heronian-golden-gnomon.svg, catoptrica-reflection.svg, dioptra-mountain-tunnel.svg (Optics, Gnomon & Dioptra)