Concordantia Menelai (Cross-Tradition Concordance)

Halley (1758) · Maurolico (1558) · Björnbo (1902) · Ptolemy Almagest I.13 (1898) · Euclid · GEOMETOR

This master concordance aligns the principal definitions, lemmas, and propositions of Menelaus of Alexandria’s Sphaerica across Edmond Halley’s Oxford Latin edition (1758), Francesco Maurolico’s Messina editio princeps (1558), Axel Anthon Björnbo’s critical analysis (1902), Claudius Ptolemy’s Greek witnesses (Heiberg 1898), Euclid’s Elements, and sister GEOMETOR repositories.

Master Concordance of Menelaus’s Sphaerica & Transversal Legacy

Codex Unit	Halley (1758)	Maurolico / Björnbo	Greek Witness	Euclidean Parallel / Contrast	GEOMETOR Constructive & Ecosystem Links
Unit I	Praefatio [pp. i–viii]; Lib. I Def. I–VI [p. 1]	Maur. I Def. 1–6; Björnbo §3–§5	Pappus Synagoge VI (τρίπλευρον)	Elements I Def. 15–23; XI Def. 6, 18	SphericalTriangle; geometor/euclid
Unit I	Lib. I Prop. 1 [p. 2]	Maur. I.1; Björnbo I.1	Theodosius Sphaerica I–II	Elements I.23 (copy rectilinear angle)	elements:G.39; GreatCircle
Unit I	Lib. I Prop. 4, 5, 8 [pp. 5–8]	Maur. I.4–5, 8; Björnbo I.4–5, 8	Proclus In Eucl. p. 345 (direct I.25)	Elements I.4 (SAS), I.20 / XI.20, I.8 (SSS), I.24–25	geometor/proclus; intrinsic SAS/SSS congruence
Unit I	Lib. I Prop. 11 [p. 11]	Maur. I.11; Björnbo I.11	Antipodal lune ΑΒΔΓ	Overturns Elements I.16 & I.32 (\(A+B+C > \pi\))	spherical_excess_report(); geometor/hilbert
Unit I	Lib. I Prop. 17 [pp. 18–19]	Maur. I.18; Björnbo I.17	Non-existence of non-congruent similar spherical triangles	Contrasts Elements VI.4 (planar AAA similarity)	polar_triangle() (AAA congruence)
Unit II	Lib. II Prop. 1, 6–12 [pp. 48–79]	Maur. II.1–19; Björnbo II.1–12	Pappus Synagoge VI; Theodosius III.5–9	Euclid Phaenomena (rising-time arcs)	geometor/pappus (Book VI astronomical tradition)
Unit III	Lib. III Lemma I [p. 83]	Maur. III Praem.; Björnbo §9	Almagest I.13 (Heiberg pp. 68–70)	Elements I.31, VI.2, VI.4, VI.23	construct_menelaus_planar(); geometor/altshiller-court Ch. VI
Unit III	Lib. III Prop. 1 & Scholion [pp. 80–86]	Maur. III.1 (Figura Catha); Björnbo III.1	Almagest I.13 (Heiberg pp. 74–76)	Central projection from sphere center onto planar transversal	verify_spherical_menelaus(), gnomonic_projection_menelaus()
Unit III	Lib. III Prop. 2–5 [pp. 86–90]	Maur. III.2–6; Björnbo III.2–5	Applied in Almagest I.14–16 & II	Spherical Rule of Four Quantities & Law of Tangents	spherical_excess_report() (sine_law_common_ratio)
Unit IV	Lib. III Prop. 6–8 [pp. 91–94]	Maur. III.7–9; Björnbo III.6–8	Anticipates Pappus VII.129 cross-ratio invariance on \(S^2\)!	Elements VI.3 & external angle-bisector ratio	construct_external_bisector_axis(), verify_spherical_cross_ratio_invariance(); geometor/pappus
Unit IV	Lib. III Prop. 9–10 [pp. 95–97]	Maur. III.10–11; Björnbo III.9–10	Complete spherical quadrilateral & altitude concurrency	Elements IV.4 (incenter) & orthic axis	spherical_orthocenter_menelaus(), construct_trilinear_polar(); geometor/etc X(1), X(4), X(7)