Cross-Corpus Geometric Concordance

Bidirectional mapping linking Claudius Ptolemy's propositions in Almagest Book I and Planisphaerium to sister repositories across the GEOMETOR canon and to executable symbolic modules in geometor.ptolemy.

Ptolemy Locus	Geometric Proposition / Construction	Companion GEOMETOR Canon	Symbolic Module (geometor.ptolemy)
Almagest I.1 (Heiberg I, pp. 4–7)	Tripartite division of theoretical philosophy (physics, mathematics, theology)	aristotle (Metaphysics E.1, 1026a18–19); proclus (Prologue I)	—
Almagest I.10a (Heiberg I, pp. 32–35)	Single-circle construction of regular decagon ($s_{10} = R/\phi$) & pentagon ($s_5 = \sqrt{R^2 + s_{10}^2}$)	euclid / elements (II.6, VI.Def.3, XIII.9, XIII.10); divine ($\mathbb{Q}(\phi)$)	construct_pentagon_decagon()
Almagest I.10b (Heiberg I, pp. 36–37)	Ptolemy’s Theorem on cyclic quadrilaterals ($AC \cdot BD = AB \cdot CD + BC \cdot DA$)	euclid (III.21, VI.4, VI.16, II.1); casey (Sequel VI.6); altshiller-court	construct_ptolemy_quadrilateral()
Almagest I.10b (Heiberg I, pp. 38–42)	Chord Difference, Half-Arc Lemma, Chord Sum, & Irreducibility of Angle Trisection	euclid / elements (I.47, III.31, VI.8)	chord_difference(), chord_half_arc()
Almagest I.10b & VI.7 (Heiberg I, pp. 43–46, 513)	Ptolemy’s Inequality Lemma ($\frac{\operatorname{crd}\alpha}{\operatorname{crd}\beta} < \frac{\alpha}{\beta}$), $\operatorname{crd}(1^\circ) \approx 1;2,50$, $\pi \approx 3;8,30 = \frac{377}{120}$	archimedes (Dimensio Circuli Prop. 3); euclid (VI.3, VI.33)	verify_ptolemy_pi()
Almagest I.13a (Heiberg I, pp. 68–70)	Planar Menelaus Sector Lemma (kata synthesin & kata diairesin via $EH \parallel \Gamma\Delta$)	menelaus; pappus (Synagoge VII); euclid (I.31, VI.4)	construct_planar_menelaus()
Almagest I.13b–I.14 (Heiberg I, pp. 70–78)	3D Spherical Menelaus Theorem ($\Theta, K, \Lambda$ collinear) & Solar Declination Law	menelaus (Sphaerica III.1); euclid (VI.1, XI.3)	verify_spherical_menelaus(), compute_solar_declination()
Planisphaerium §§1–3 (Heiberg II, pp. 227–232)	Stereographic projection of equator, tropics, oblique ecliptic ($\angle mdt = 90^\circ$), & horizon	apollonius (Conics I.5 subcontrary section); euclid (III.31)	construct_astrolabe_planisphere()
Planisphaerium §4 (Heiberg II, pp. 233–234)	Circular inversion of opposite parallels ($r_{\text{Cap}} \cdot r_{\text{Can}} = R_{\text{eq}}^2$; $92;8,15 \times 39;4,19 \approx 60^2$)	casey; altshiller-court (Ch. IX Inversion)	prove_parallel_inversion_symbolically()