Unit IV — Analytical Summary: Harmonic Pencils, Concurrencies & Polarity

Deductive Architecture of Menelaus’s Sphaerica Book III (Propositions III.6–III.25)

1. Spherical Bisectors & Harmonic Division (*Props. III.6–III.8*, Halley pp. 91–95): From the Spherical Law of Sines (Prop. III.2), Menelaus derives both the internal and external spherical angle-bisector theorems (Prop. III.6). Combining orthogonal internal and external bisectors at the right angle of a right spherical triangle yields Proposition III.8: the exact spherical harmonic conjugate proportion \(\sin \widehat{\mathrm{\Gamma E}} : \sin \widehat{\mathrm{EA}} = \sin \widehat{\mathrm{\Gamma\Delta}} : \sin \widehat{\mathrm{\Delta A}}\) together with two sharp converses characterizing right angles and bisectors via harmonic cross-ratios—generalized in Pappus’s Synagoge VI/VII (ed. Hultsch, 1877) to full 4-ray cross-ratio pencil invariance.

2. Spherical Incenter & Orthocenter Concurrencies (*Props. III.9–III.10*, Halley pp. 95–97): - Prop. III.9 (Incenter): Equating two bisector sine ratios across a common cevian ΒΔΕ proves that the three spherical angle bisectors concur at the center Δ of the inscribed small circle. - Prop. III.10 (Orthocenter): Fifteen centuries before Giovanni Ceva (1678), Lazare Carnot (1806), and Jean-Victor Poncelet (1822), Menelaus constructs the orthic axis (trilinear polar) ΕΔΘ of the altitude intersection Ζ, applies Spherical Menelaus (Prop. III.1) three times to prove harmonic division (Α, Γ; Η, Θ), identifies Ζ as the incenter of the orthic spherical triangle ΔΕΗ via Prop. III.8 and Prop. III.9, and deduces ΒΗ ⊥ ΑΓ by the converse of Prop. III.8!

3. Astronomical Difference Proportions (*Props. III.11–III.25*, Halley pp. 97–112): Book III concludes with fine-grained arc-difference ratios along meridian and horizon circles, completing the mathematical engine of ancient and medieval spherical astronomy.