Unit I — Scholarly & GEOMETOR Symbolic Commentary

Philological Apparatus & Exact SymPy Verification on \(S^2(\mathbb{Q})\)

1. Philological Note on Costard’s Preface & Direct Proofs

In Costard's Praefatio (1758, p. iv), the Bodleian Huntington Hebrew witness is traced explicitly to Isḥāq ibn Ḥunayn’s ninth-century Baghdad translation and Jacob ben Machir’s Montpellier version. As Axel Anthon Björnbo (1902, §5) and Thomas Little Heath (1921, Vol. II pp. 262–265) emphasize, Menelaus systematically avoids reductio ad absurdum throughout Book I—matching Proclus’s testimony (Friedlein p. 345) regarding Menelaus’s direct proof of Euclid Elements I.25.

2. Exact Rational Certification of Spherical Excess (geometor.menelaus.spherical)

In spherical_excess_report(), for any three rational vertices \(\mathbf{A}, \mathbf{B}, \mathbf{C} \in S^2(\mathbb{Q})\) lying in an open hemisphere, the Euler–Eriksson formula expresses the tangent of half the spherical excess \(E = A + B + C - \pi\) as an exact rational number with zero square roots or inverse cosines:

\[\tan\!\left(\frac{E}{2}\right) = \frac{|\det(\mathbf{A}, \mathbf{B}, \mathbf{C})|}{1 + \mathbf{A}\cdot\mathbf{B} + \mathbf{B}\cdot\mathbf{C} + \mathbf{C}\cdot\mathbf{A}} \in \mathbb{Q}_{> 0}.\]

For our canonical rational triangle \(A = (0, 0, 1)\), \(B = (3/5, 4/5, 0)\), \(C = (2/3, 2/3, 1/3)\):

  • \(|\det(\mathbf{A}, \mathbf{B}, \mathbf{C})| = \frac{2}{15}\),

  • \(1 + \mathbf{A}\cdot\mathbf{B} + \mathbf{B}\cdot\mathbf{C} + \mathbf{C}\cdot\mathbf{A} = 1 + 0 + \frac{14}{15} + \frac{1}{3} = \frac{34}{15}\),

  • whence \(\tan(E/2) = \frac{2/15}{34/15} = \frac{1}{17} > 0\), certifying \(A + B + C > \pi\) (Sphaerica Prop. I.11) purely in \(\mathbb{Q}\)!

3. Polar Triangle Duality vs. Menelaus’s Intrinsic Proof of Prop. I.17

Modern textbooks prove AAA spherical congruence (Sphaerica Prop. I.17) by passing to the polar triangle \(\triangle A^*B^*C^*\) (polar_triangle()), where \(\cos a^* = -\cos A\) reduces AAA on \(\triangle ABC\) to SSS (Prop. I.8) on \(\triangle A^*B^*C^*\). Remarkably, Menelaus’s own proof on pp. 18–19 ([§1.5]) achieves the exact same result within the primal sphere using only vertical SAS copying and two semicircle applications of Proposition I.10—demonstrating that positive curvature \(K = +1\) fixes an absolute unit of length (the quadrant / semicircle) and eliminates non-congruent homothetic similarity (contrast geometor/hilbert on the independence of Euclid’s Parallel Postulate).