Cross-Corpus Concordance & Symbolic Gap Audit

This concordance maps the 387 classical Greek and Arabic propositions of Apollonius’s Conics (Books I–VII) and T. L. Heath’s 147 consolidated propositions (1896) across the Digital Scriptorium Codex units, Euclid’s Elements, Archimedes, Pappus’s Synagoge Book VII, Kepler, Kimberling’s Encyclopedia of Triangle Centers (etc), Wildberger’s Rational Trigonometry (wildberger), Lean mathlib4, and current symbolic conic gaps in sympy.geometry.

1. Classical vs. Heath Proposition Synopsis (Books I–VII)

Book

Classical Props.

Heath (1896) Range

Heath Pages

Primary Textual Witness

Book I

60

Props. 1–27 (plus Cone Lemmas I.1–10)

pp. 1–52

Greek (Heiberg Vol. I, 1891; Vat. gr. 206)

Book II

53

Props. 28–53

pp. 53–94

Greek (Heiberg Vol. I, 1891; Vat. gr. 206)

Book III

56

Props. 54–76

pp. 95–125

Greek (Heiberg Vol. I, 1891; Vat. gr. 206)

Book IV

57

Props. 77–83

pp. 126–138

Greek (Heiberg Vol. II, 1893; Vat. gr. 206)

Book V

77

Props. 84–113

pp. 139–196

Arabic of Thābit ibn Qurra / Latin of Halley (1710; Bodleian Marsh 667)

Book VI

33

Props. 114–122

pp. 197–220

Arabic of Thābit ibn Qurra / Latin of Halley (1710; Bodleian Marsh 667)

Book VII

51

Props. 123–147

pp. 221–254

Arabic of Thābit ibn Qurra / Latin of Halley (1710; Bodleian Marsh 667)

Total

387

Propositions 1–147

pp. 1–254

Books I–IV Greek; Books V–VII Arabic/Latin

2. Locus-by-Locus Cross-Corpus Concordance

Books I, II, IV, V, VI, VII Prefaces — Intro Part II, Ch. I

  • Codex Unit: Preface, Introduction & Prefatory Letters (Heath pp. lxix–lxxv · PDF 77–83)

  • Geometric Plate: —

  • Euclid Foundations: VI.27, VI.28, VI.29, XII.10

  • GEOMETOR & Formal Links: euclid, archimedes, pappus

  • Doctrinal Summary: Epistolary prefaces to Eudemus and Attalus dividing Books I–IV (elementary foundation, ἀγωγὴ στοιχειώδης) from Books V–VIII (special extensions, περιουσιαστικώτερα).

I. 1–10 — The Cone (Introductory Props.)

  • Codex Unit: Book I — Generation of the Cone & Fundamental Symptoms (Heath pp. 1–8 · PDF 181–188)

  • Geometric Plate: fig_oblique_cone_sections.svg

  • Euclid Foundations: III.35, VI.4, XI.Def.18

  • GEOMETOR & Formal Links: euclid, elements, pappus

  • Doctrinal Summary: Generation of the arbitrary oblique scalene circular double cone, axial triangle, parallel circular sections (I.4), and antiparallel subcontrary circular sections (I.5, I.9).

I. 11 — Proposition 1

  • Codex Unit: Book I — Generation of the Cone & Fundamental Symptoms (Heath pp. 8–9 · PDF 188–189)

  • Geometric Plate: fig_application_of_areas.svg

  • Euclid Foundations: I.44, II.14, VI.4, VI.27

  • GEOMETOR & Formal Links: euclid, elements, archimedes, wildberger

  • Doctrinal Summary: Fundamental symptom of the Parabola (παραβολή): cutting plane parallel to a generator of the axial triangle yields QV² = PL · PV (y² = px), exact application of area.

I. 12 — Proposition 2

  • Codex Unit: Book I — Generation of the Cone & Fundamental Symptoms (Heath pp. 9–11 · PDF 189–191)

  • Geometric Plate: fig_application_of_areas.svg

  • Euclid Foundations: II.6, VI.4, VI.29

  • GEOMETOR & Formal Links: euclid, elements, wildberger

  • Doctrinal Summary: Fundamental symptom of the Hyperbola (ὑπερβολή): cutting plane meeting both napes yields QV² = PV · PL + (PL/PP’) PV², application exceeding by a figure similar to d × p.

I. 13 — Proposition 3

  • Codex Unit: Book I — Generation of the Cone & Fundamental Symptoms (Heath pp. 11–13 · PDF 191–193)

  • Geometric Plate: fig_application_of_areas.svg

  • Euclid Foundations: II.5, VI.4, VI.28

  • GEOMETOR & Formal Links: euclid, elements, archimedes, wildberger

  • Doctrinal Summary: Fundamental symptom of the Ellipse (ἔλλειψις): cutting plane meeting all generators of one nape yields QV² = PV · PL − (PL/PP’) PV², application falling short by a figure similar to d × p.

I. 14, 15, 16 — Propositions 4–6

  • Codex Unit: Book I — Generation of the Cone & Fundamental Symptoms (Heath pp. 13–17 · PDF 193–197)

  • Geometric Plate: fig_oblique_cone_sections.svg

  • Euclid Foundations: VI.4, VI.29

  • GEOMETOR & Formal Links: euclid, pappus

  • Doctrinal Summary: Opposite branches (αἱ ἀντικείμεναι) cut from the double cone have equal latus rectum and form a single two-branched curve with center bisecting PP’ and secondary conjugate diameter.

I. 33–38 — Propositions 12–15

  • Codex Unit: Book I — Generation of the Cone & Fundamental Symptoms (Heath pp. 25–30 · PDF 205–210)

  • Geometric Plate: fig_tangent_harmonic_division.svg

  • Euclid Foundations: II.5, II.6, III.16, VI.17

  • GEOMETOR & Formal Links: euclid, pappus, mathlib4

  • Doctrinal Summary: Tangent construction and harmonic division: TP = PV for the parabola (I.33, 35) and TP : TP’ = PV : VP’ (equivalently CT · CV = CP²) for central conics (I.34, 36–37).

I. 49–51 — Propositions 20–21

  • Codex Unit: Book I — Generation of the Cone & Fundamental Symptoms (Heath pp. 36–41 · PDF 216–221)

  • Geometric Plate: fig_tangent_harmonic_division.svg

  • Euclid Foundations: I.43, VI.4

  • GEOMETOR & Formal Links: euclid, wildberger

  • Doctrinal Summary: Invariance of the conic symptom under transformation from the original cone-derived diameter to any new diameter QQ’ and tangent at Q.

II. 1–14 — Propositions 28–35

  • Codex Unit: Book II — Asymptotes, Conjugate Hyperbolas & Secant Intercepts (Heath pp. 53–62 · PDF 233–242)

  • Geometric Plate: fig_hyperbola_asymptotes_secant.svg

  • Euclid Foundations: II.5, II.6, X.1

  • GEOMETOR & Formal Links: euclid, archimedes, descartes

  • Doctrinal Summary: Construction of asymptotes (PL² = ¼ p·d), proof of non-intersection, bisected tangent segment LL’ at P (II.3, 9), equal secant intercepts RQ = Q’R’ (II.8, 16), asymptotic equation xy = const (II.12), and indefinite approach (II.14).

II. 15–23 — Propositions 28, 36–39

  • Codex Unit: Book II — Asymptotes, Conjugate Hyperbolas & Secant Intercepts (Heath pp. 53, 62–64 · PDF 233, 242–244)

  • Geometric Plate: fig_hyperbola_asymptotes_secant.svg

  • Euclid Foundations: II.6, VI.4

  • GEOMETOR & Formal Links: euclid, wildberger

  • Doctrinal Summary: Opposite branches and conjugate hyperbolas share common asymptotes; conjugate diameters of original hyperbola swap transverse and secondary roles on the conjugate hyperbola.

II. 44–53 — Propositions 47–53

  • Codex Unit: Book II — Asymptotes, Conjugate Hyperbolas & Secant Intercepts (Heath pp. 72–83 · PDF 252–263)

  • Geometric Plate: fig_tangent_harmonic_division.svg

  • Euclid Foundations: III.17, III.31

  • GEOMETOR & Formal Links: euclid, elements, pappus

  • Doctrinal Summary: Finding a diameter, center, and principal orthogonal axes of a given conic; constructing tangents from an external point by harmonic division and compass-and-straightedge.

III. 16–29 — Propositions 57–61

  • Codex Unit: Book III — Chord Rectangles, Poles & Polars, Foci & the 4-Line Locus (Heath pp. 97–101 · PDF 277–281)

  • Geometric Plate: fig_three_four_line_locus.svg

  • Euclid Foundations: III.35, III.36, III.37

  • GEOMETOR & Formal Links: euclid, pappus, newton

  • Doctrinal Summary: Ratio of rectangles under segments of intersecting chords PO·OQ : RO·OS equals the ratio of squares on parallel tangents—generalizing Euclid III.35–36 (power of a point) from circles to all conics.

III. 30–40 — Propositions 62–66

  • Codex Unit: Book III — Chord Rectangles, Poles & Polars, Foci & the 4-Line Locus (Heath pp. 102–108 · PDF 282–288)

  • Geometric Plate: fig_harmonic_pole_polar.svg

  • Euclid Foundations: VI.2, VI.4

  • GEOMETOR & Formal Links: pappus, mathlib4, wildberger

  • Doctrinal Summary: Harmonic properties of poles and polars: any secant through pole T is divided harmonically by the conic at R, R’ and the chord of contact (polar) Qq at M: TR : TR’ = MR : MR’.

III. 45–52 — Propositions 69–73

  • Codex Unit: Book III — Chord Rectangles, Poles & Polars, Foci & the 4-Line Locus (Heath pp. 113–118 · PDF 293–298)

  • Geometric Plate: fig_central_conic_foci.svg

  • Euclid Foundations: I.47, II.5, II.6, III.21, III.31

  • GEOMETOR & Formal Links: euclid, pappus, kepler, wildberger

  • Doctrinal Summary: Focal properties of central conics (‘points arising out of the application’, AS·SA’ = b²): optical reflection equality (III.48 [Prop. 71]), pedal auxiliary circle (III.49 [Prop. 72]), and constant sum/difference SP ± S’P = AA’ = 2a (III.51–52 [Prop. 73]).

III. 53–56 — Propositions 74–76

  • Codex Unit: Book III — Chord Rectangles, Poles & Polars, Foci & the 4-Line Locus (Heath pp. 119–125 · PDF 299–305)

  • Geometric Plate: fig_three_four_line_locus.svg

  • Euclid Foundations: VI.16, VI.23

  • GEOMETOR & Formal Links: pappus, descartes

  • Doctrinal Summary: Projective product of tangent intercepts and synthetic key to the Three-Line and Four-Line Locus of Euclid, Apollonius, Pappus (Book VII), and Descartes (La Géométrie, 1637).

IV. 1–23 — Introductory Note to Book IV

  • Codex Unit: Book IV — Number of Intersections & Tangencies of Conics (Heath pp. 126 · PDF 306)

  • Geometric Plate: fig_harmonic_pole_polar.svg

  • Euclid Foundations: III.17, VI.4

  • GEOMETOR & Formal Links: pappus, mathlib4

  • Doctrinal Summary: Converse harmonic polar construction of tangents from an external point across single-branch and opposite-branch hyperbolas.

IV. 24–57 — Propositions 77–83

  • Codex Unit: Book IV — Number of Intersections & Tangencies of Conics (Heath pp. 126–138 · PDF 306–318)

  • Geometric Plate: fig_four_point_conic_pencil.svg

  • Euclid Foundations: III.10, III.13

  • GEOMETOR & Formal Links: euclid, pappus

  • Doctrinal Summary: Two distinct conics cannot share an arc (IV.24 [Prop. 77]), intersect in at most four points (IV.25, 38, 43, 55 [Props. 78–83]), and if tangent at one point meet in at most two others—the ancient foundation of Bézout’s theorem (2 × 2 = 4) and five-point conic uniqueness.

V. 1–34 — Propositions 84–94

  • Codex Unit: Book V — Normals, Minimum & Maximum Lines, Subnormals & the Evolute (Heath pp. 139–159 · PDF 319–339)

  • Geometric Plate: fig_normals_evolute.svg

  • Euclid Foundations: I.19, III.7, III.8, III.16

  • GEOMETOR & Formal Links: euclid, archimedes

  • Doctrinal Summary: Minimum and maximum straight lines from axis points to a conic; constant parabola subnormal NG = ½ p_a (V.13, 27) and central-conic subnormal proportion NG : CN = p_a : AA’ (V.14–15, 28–29); perpendicularity of minimum/maximum lines to the tangent (normals).

V. 51–63 — Propositions 99–106

  • Codex Unit: Book V — Normals, Minimum & Maximum Lines, Subnormals & the Evolute (Heath pp. 168–186 · PDF 348–366)

  • Geometric Plate: fig_normals_evolute.svg

  • Euclid Foundations: II.5, V.8, VI.28

  • GEOMETOR & Formal Links: archimedes, descartes, wildberger

  • Doctrinal Summary: Exact diorismos of the number of normals through an arbitrary point O (2, 3, or 4 normals), determining point-by-point the evolute envelope of every conic and constructing normal feet via an auxiliary rectangular hyperbola.

VI. 1–33 — Definitions 1–6 & Propositions 114–122

  • Codex Unit: Books VI & VII — Equal and Similar Conics & Conjugate Diameters (Heath pp. 197–220 · PDF 377–400)

  • Geometric Plate: fig_oblique_cone_sections.svg

  • Euclid Foundations: VI.Def.1, VI.20, XI.24

  • GEOMETOR & Formal Links: euclid, elements

  • Doctrinal Summary: Congruence and similarity of conic sections and segments: all parabolas are similar (VI.11 [Prop. 115]); central conics are similar iff their figures d × p are similar (VI.12–13 [Prop. 116]); embedding any given conic inside a prescribed right cone (VI.28–30 [Props. 120–122]).

VII. 12, 13, 29, 30, 31 — Propositions 129 & 136

  • Codex Unit: Books VI & VII — Equal and Similar Conics & Conjugate Diameters (Heath pp. 228–236 · PDF 408–416)

  • Geometric Plate: fig_conjugate_diameters_theorems.svg

  • Euclid Foundations: I.35, I.47, II.5, II.6, VI.23

  • GEOMETOR & Formal Links: euclid, wildberger

  • Doctrinal Summary: Apollonius’s First and Second Conjugate Diameter Theorems: (1) PP’² ± DD’² = AA’² ± BB’² (a’² ± b’² = a² ± b²); (2) area of circumscribed conjugate parallelogram LL’MM’ equals rectangle of axes AA’ · BB’ (a’b’ sin ω = ab).

VII. 33–51 — Propositions 137–147

  • Codex Unit: Books VI & VII — Equal and Similar Conics & Conjugate Diameters (Heath pp. 236–254 · PDF 416–434)

  • Geometric Plate: fig_conjugate_diameters_theorems.svg

  • Euclid Foundations: V.8, V.10, VI.27

  • GEOMETOR & Formal Links: euclid, ibn-al-haytham, halley

  • Doctrinal Summary: Extrema and monotonic variations of d’ ± d’’, d’·d’’, p’, and d’² ± p’² across conjugate diameters, forged explicitly as diorismos machinery for the lost Book VIII.

Pappus Synagoge VII (Περὶ ἐπαφῶν) — Intro Part II, Ch. I & Viète (1600)

  • Codex Unit: Appendix — The Lost Analytic Treasury in Pappus Book VII (Heath pp. lxviii · PDF 76)

  • Geometric Plate: fig_problem_of_apollonius_ccc.svg

  • Euclid Foundations: III.17, III.36, IV.4, IV.5

  • GEOMETOR & Formal Links: pappus, elements, etc

  • Doctrinal Summary: Apollonius’s On Tangencies: the ten contact problems (PPP through CCC, up to 8 tangent circles), reconstructed by Viète’s Apollonius Gallus (1600) and connected to ETC X(181) (Apollonius Point).

Pappus Synagoge VII (Τόποι ἐπίπεδοι II) — Circle of Apollonius & Analytic Treasury

  • Codex Unit: Appendix — The Lost Analytic Treasury in Pappus Book VII (Heath pp. lxviii, cxxv–cxxvi · PDF 76, 133–134)

  • Geometric Plate: fig_apollonius_circle_locus.svg

  • Euclid Foundations: VI.3, VI.A

  • GEOMETOR & Formal Links: pappus, elements, etc, wildberger

  • Doctrinal Summary: Apollonius’s Plane Loci Book II: the locus PA : PB = k (k ≠ 1) is the Circle of Apollonius on harmonic diameter H_in H_ex; the three Apollonius circles of a triangle form a coaxal pencil intersecting at the Isodynamic Points X(15) and X(16), alongside Cutting Off of a Ratio, Determinate Section, and Inclinations.

3. Symbolic Conic Capabilities & sympy.geometry Gap Audit

A central architectural purpose of geometor-apollonius is specifying the exact algebraic and projective operations required for a future constructive conic engine beyond the axis-aligned limitations of sympy.geometry:

General planar conic Ax² + Bxy + Cy² + Dx + Ey + F = 0 & oblique-diameter frame (I.11–14, I.49–60)

  • Current ``sympy.geometry`` Status: Partial / Axis-Aligned Only (sympy.geometry.ellipse.Ellipse and parabola.Parabola lack general rotated/projective hyperbola and two-branch oblique-diameter conic kernel)

  • GEOMETOR Constructive Target: geometor.model exact algebraic/projective Conic element over Q(sqrt(d)) and Q(phi)

Harmonic pole-polar bijection, chord of contact, and self-polar triangle (III.30–40 [Props. 62–66])

  • Current ``sympy.geometry`` Status: Missing in sympy.geometry (no polar_line(point) or pole_point(line) on conics; formalized in Lean mathlib4 PolePolar.lean)

  • GEOMETOR Constructive Target: geometor.pappus & geometor.model symmetric 3×3 matrix polarity P ↦ C·P

Four-line locus conic pencil C₁ + λ C₂ = 0, five-point conic constructor, and exact 4-point intersection solver (III.53–56 [Props. 74–76] & IV.24–57 [Props. 77–83])

  • Current ``sympy.geometry`` Status: Missing in sympy.geometry (no conic_through_five_points or 4-point pencil resolver)

  • GEOMETOR Constructive Target: geometor.pappus Braikenridge–Maclaurin / determinant 6×6 exact symbolic constructor

Exact 4-normal cubic/quartic foot resolver from arbitrary point O and rational/algebraic evolute envelope (V.51–63 [Props. 99–109])

  • Current ``sympy.geometry`` Status: Missing in sympy.geometry (Ellipse.normal_lines uses numeric/Groebner fallbacks that stall on symbolic parameters; no evolute method)

  • GEOMETOR Constructive Target: Apollonius rectangular-hyperbola intersection constructor (V.62–63 [Prop. 106])

Exact symbolic solver for all 8 tangent circles to three given circles (Problem of Apollonius CCC) (Lost Tangencies (Pappus VII) / Viète (1600))

  • Current ``sympy.geometry`` Status: Missing in sympy.geometry (Circle supports only basic tangent lines from a point)

  • GEOMETOR Constructive Target: Gergonne radical-center / pole-polar 8-solution algebraic solver in geometor.elements / geometor.etc