Unit I — Analytical Summary: Intrinsic Spherical Foundations

Deductive Architecture of Menelaus’s Sphaerica Book I (Propositions I.1–I.38)

1. The Intrinsic Axiomatic Shift (Defs. I–VI): Unlike Theodosius, who treated circles on a sphere solely via 3D Euclidean planes and chords, Menelaus defines the spherical triangle intrinsically on \(S^2\) by minor great-circle arcs (\(< 180^\circ\)) and dihedral inclinations.

2. Neutral vs. Positively Curved Theorems (Props. I.1–I.10): Menelaus establishes spherical angle copying (I.1), isosceles base angles and converse (I.2–I.3), SAS congruence (I.4), the triangle inequality (I.5), and SSS / Hinge inequalities (I.7–I.9) without invoking parallel lines. Proposition I.10 introduces the decisive curvature threshold: the relation between an exterior angle and a remote interior angle depends on whether the two adjacent sides sum to less than, equal to, or greater than a semicircle.

3. Spherical Angle-Sum Excess (Prop. I.11): By completing sides ΑΒ and ΑΓ to the antipodal vertex Δ (forming a spherical lune of two semicircles) and applying I.10, Menelaus proves directly that \(\angle_{\text{ext}} < \angle \mathrm{A} + \angle \mathrm{B}\) and hence \(\angle \mathrm{A} + \angle \mathrm{B} + \angle \mathrm{\Gamma} > 2\text{ right angles}\)—the historic first rigorous demonstration of positive spherical excess.

4. AAA Spherical Congruence (Prop. I.17) & Cevian Inequalities (I.18–I.38): Without needing polar triangles (developed explicitly in the Renaissance by Viète and Snell), Menelaus constructs a vertical SAS reflection at vertex Β, produces ΙΘ to Κ on ΑΓ, and applies Proposition I.10 twice to equate two semicircle sums (\(\widehat{\mathrm{K\Theta}} + \widehat{\mathrm{K\Gamma}} = \widehat{\mathrm{IK}} + \widehat{\mathrm{KA}}\)), immediately forcing \(\widehat{\mathrm{\Theta I}} = \widehat{\mathrm{A\Gamma}}\)! The remainder of Book I (Props. I.18–I.38) investigates medians, angle bisectors, and segments cut off on spherical arcs.