Unit III — Analytical Summary: Gnomonic Reduction & Spherical Trigonometry

Deductive Chain of Menelaus’s Sphaerica Book III (Propositions III.1–III.5)

  1. Planar Menelaus Lemma (*Lemma I*, Halley p. 83; *Almagest* I.13, Heiberg p. 69): Drawing a single auxiliary parallel (ΑΗ ∥ ΒΕ or ΕΗ ∥ ΓΔ) converts a complete quadrilateral (a triangle ΑΒΔ cut by a transversal ΓΖΕ) into two pairs of Euclid VI.4 similar triangles, compounding two simple ratios through an intermediate segment ΖΔ to give both the katà sýnthesin (composition) and katà diaíresin (division) planar product of three ratios equal to unity.

  2. Chord-Sine Projection (*Lemma II*, Halley p. 83; *Almagest* I.13, Heiberg pp. 70–72): Any radius ΔΒ from the center of a circle meets a chord ΑΓ at Ε and its arc at Β, dividing the chord in the exact ratio of the perpendiculars (sines / half-chords of double arcs) dropped from Α and Γ onto diameter ΔΒ: \(\mathrm{AE} : \mathrm{E\Gamma} = \sin \widehat{\mathrm{AB}} : \sin \widehat{\mathrm{B\Gamma}} = \operatorname{crd}(2\widehat{\mathrm{AB}}) : \operatorname{crd}(2\widehat{\mathrm{B\Gamma}})\).

  3. 3D Plane-Intersection / Gnomonic Reduction (*Prop. III.1*, Halley pp. 80–82; *Almagest* I.13, Heiberg pp. 74–75): On the sphere centered at Β, the plane of the transversal great circle ΑΘΕ cuts the plane of the chordal triangle ΝΛΜ along a single straight line ΣΔΚ (by Euclid Elements XI.3, or parallel to diameter ΕΒΚ by XI.9 in the semicircle-sum case). Applying Lemma I to planar triangle ΝΛΜ cut by ΣΔΚ and translating all three chordal ratios into arc-sine ratios via Lemma II yields Spherical Menelaus in one stroke!

  4. Immediate Trigonometric Consequences (*Props. III.2–III.5*): Specializing two arcs of the six-arc Menelaus configuration to quadrants gives the right-angled spherical triangle canon (Prop. III.1 Coroll. 2, p. 86), the Spherical Law of Sines (Prop. III.2, pp. 86–87—formulated intrinsically by Abū Naṣr Manṣūr as the Rule of Sine freeing astronomy from full six-arc quadrilaterals), the Rule of Four Quantities & Tangents (Prop. III.3, p. 88), and sum-and-difference tangent identities (Props. III.4–III.5, pp. 88–90).