Unit II — Analytical Summary: Book II & the Critique of Theodosius

Deductive Architecture of Menelaus’s Sphaerica Book II (Propositions II.1–II.12)

1. Cevian Transversals Across Spherical Triangles (Props. II.1–II.9): Book II opens by constructing great-circle arcs cutting across an obtuse or acute spherical triangle at prescribed inclinations (Props. II.1–II.3), and deducing second-order monotonicity inequalities for equal arcs marked off along the base (Props. II.4–II.9): when base ΘΜ is subdivided by equal intervals ΘΚ = ΛΜ, the intermediate polar/cevian arcs satisfy strict convexity bounds (\(\widehat{\mathrm{M\Gamma}} + \widehat{\mathrm{\Gamma\Theta}} < \widehat{\mathrm{\Lambda\Gamma}} + \widehat{\mathrm{\Gamma K}}\) or the reverse depending on quadrant regime).

2. Superseding Theodosius’s *Sphaerica* Book III (Props. II.10–II.12): In Greek spherical astronomy (rising times of the zodiacal signs, ἀναφοραί), Theodosius (Sphaerica III.5, III.6, III.9) relied on intricate solid-geometry chord inequalities across multiple planes. On p. 75 ([§2.2]), Menelaus explicitly critiques these earlier proofs as lacking universality and firmness (“quorum quidem nonnulla vix satis firma videntur”), and collapses Theodosius III.5, III.6, and III.9 into Proposition II.10 ([§2.3]) in a single nine-line deduction from a single spherical triangle ΜΓΘ!

3. Bridge to Book III: Book II exhausts what pure qualitative second-difference inequalities on \(S^2\) can achieve: it proves which zodiacal arc rises faster or changes faster in declination, whereas Book III introduces the Menelaus Transversal Theorem (Prop. III.1) to compute the exact chord/sine magnitudes of those arcs.