Claudius Ptolemy

Chord Trigonometry, Cyclic Quadrilaterals, Spherical Menelaus Calculus, and Stereographic Projection

Οἶδ’ ὅτι θνητὸς ἔφυν καὶ ἐφήμερος· ἀλλ’ ὅταν ἄστρων μαστίζω πυκινὰς ἀμφιδρόμους ἕλικας, οὐκέτ’ ἐπιψαύω γαίης ποσίν, ἀλλὰ παρ’ αὐτῷ Ζηνὶ διοτρεφέος πίμπλαμαι ἀμβροσίης.

I know that I am mortal and a creature of a day; yet when I trace the close-set wheeling spirals of the stars, my feet no longer touch the earth, but beside Zeus himself I take my fill of god-nurturing ambrosia.

—Epigram prefixed to Claudius Ptolemy's Syntaxis Mathematica (Heiberg 1898, p. 4)

Overview

GEOMETOR Ptolemy (geometor-ptolemy) unites a scholarly Digital Scriptorium Book Codex with 100% exact symbolic SymPy / ``geometor.model`` constructions and proofs for the two foundational geometric treatises of Claudius Ptolemy (c. 100–170 CE):

  1. *Almagest* (*Μαθηματικὴ Σύνταξις*, Book I, Chapters 1–14): Edited from J. L. Heiberg's Teubner Greek edition (Syntaxis Mathematica, Vol. I.1, 1898), establishing cosmic sphericity, the sexagesimal chord circle ($R = 60^p$, $D = 120^p$), Ptolemy’s single-circle construction of the regular decagon and pentagon over $\mathbb{Q}(\phi)$, Ptolemy’s Theorem on Cyclic Quadrilaterals ($AC \cdot BD = AB \cdot CD + BC \cdot DA$), the difference/sum/half-arc chord calculus, Ptolemy’s Inequality Lemma ($\operatorname{crd} 1^\circ \approx 1;2,50 \implies \pi \approx 3;8,30 = \frac{377}{120}$), and Planar & Spherical Menelaus Transversal Trigonometry.

  2. *Planisphaerium* (Sections 1–4): Edited from J. L. Heiberg's Teubner Latin edition (Opera Astronomica Minora, Vol. II, 1907) of Hermann of Carinthia's 1143 translation, establishing stereographic projection of the celestial sphere from the South Pole onto the equatorial plane, conformal circle-preservation via Apollonius's subcontrary cone section (Conics I.5), circular inversion of opposite parallels ($r_{\text{Cap}} \cdot r_{\text{Can}} = R_{\text{eq}}^2$), and the geometry of the planispheric astrolabe.

Architecture of the Repository

  • Digital Scriptorium Codex (Almagest I & Planisphaerium): Four multi-track units (:layout: book) pairing original English translations with verbatim polytonic Greek (1898) and scientific Latin (1907) primary texts, executive geometric summaries, historical commentary, interactive .euclid-graphic SVGs, and linked apparatus registries (Concordance, Bibliography, Persons, Glossary).

  • Exact Symbolic Demonstrations (Symbolic Demonstrations): Pure-symbolic Python constructions (Model(…, auto_intersect=False)) free of floating-point coordinates, verifying every chord, cyclic quadrilateral, Menelaus transversal ratio, and stereographic circle over $\mathbb{Q}$, $\mathbb{Q}(\phi)$, and exact radical towers.

  • Python API Reference (Modules): Documented modules in geometor.ptolemy (pentagon_chords, cyclic_quadrilateral, menelaus_sphaerica, stereographic).