GEOMETOR • menelaus

Menelaus of Alexandria (c. 70–140 CE) — Sphaerica (Sphaericorum Libri III) & Exact Symbolic Transversal Geometry

Digital Scriptorium 4-track Book Codex, scan-verified Latin & Greek primary witnesses, original English translation, cross-tradition scholarly concordance, and 100% exact SymPy / geometor.model constructions of the Planar & Spherical Menelaus Transversal Theorem, intrinsic spherical geometry, and projective Menelaus–Ceva harmonic duality.

A = (0, 0, 1) B = (3/5, 4/5, 0) C = (2/3, 2/3, 1/3) D = (9*sqrt(181)/181, 8*sqrt(181)/181, 6*sqrt(181)/181) E = (sqrt(3)/3, sqrt(3)/3, sqrt(3)/3) F = (3*sqrt(61)/61, 4*sqrt(61)/61, 6*sqrt(61)/61) A B C D E F
Menelaus's Sphaerica Book III, Proposition 1 (Theorema Menelai / Figura Catha): exact rational spherical triangle △ABC on S² crossed by transversal great circle DEF.

On the unit sphere S², great-circle transversal DF cuts the sides AB, BC, and CA of spherical triangle A, B, C at D, E, and F, satisfying \(\frac{\sin_{\pm}\widehat{AF}}{\sin_{\pm}\widehat{FB}} \cdot \frac{\sin_{\pm}\widehat{BD}}{\sin_{\pm}\widehat{DC}} \cdot \frac{\sin_{\pm}\widehat{CE}}{\sin_{\pm}\widehat{EA}} = -1\).

Primary Treatise (Codex)

Menelai Sphaericorum Libri III (c. 100 CE / Halley 1758)

Synchronized 4-Track Book Codex from Edmond Halley’s Oxford Edition (1758), Ptolemy’s Syntaxis Mathematica I.13 (ed. Heiberg 1898), and Björnbo’s Studien (1902).

Presents across Four Synchronized Units (English, Halley 1758 Latin & Heiberg 1898 Greek witnesses, Summary, and GEOMETOR Notes): Unit I (Book I: Intrinsic Spherical Triangles, Congruence, Excess \(A+B+C>\pi\), and AAA Congruence), Unit II (Book II: Spherical Arcs, Rising-Time Differences, and Superseding Theodosius), Unit III (Book III.1–5: The Planar & Spherical Menelaus Transversal Theorem, Chords of Double Arcs vs. Sines, and Spherical Law of Sines), and Unit IV (Book III.6–10: Spherical Angle Bisectors, Anharmonic Cross-Ratio Pencil Invariance, Orthocenter Concurrency, and Projective Legacy).

Exact Symbolic Demonstrations & Context