Conics

ΑΠΟΛΛΩΝΙΟΥ ΠΕΡΓΑΙΟΥ ΚΩΝΙΚΩΝ

APOLLONII PERGAEI CONICORUM LIBRI SEPTEM

Treatise on Conic Sections (c. 200 BCE)

English Edition in Modern Notation: Sir Thomas Little Heath (Cambridge University Press, 1896)

Critical Greek Text (Books I–IV): Johan Ludvig Heiberg (Leipzig: B. G. Teubner, 1891–1893)

Arabic-Latin Editio Princeps (Books V–VII): Thābit ibn Qurra (9th c.) & Edmond Halley (Oxford, 1710)

Conspectus Totius Operis

Composed at Alexandria and Pergamum around 200 BCE by Apollonius of Perga—acclaimed in antiquity as The Great Geometer—the Conics unified the geometry of second-degree curves by generating the parabola, ellipse, hyperbola, and opposite branches from a single arbitrary oblique circular double cone. Completing the Heath Cambridge Greek Trilogy alongside geometor/archimedes (1897) and geometor/euclid (1908), each unit of this 3-Track Digital Scriptorium Codex provides three synchronized reading tabs: English, Summary, and Notes.

• Unit 0 (Heath pp. vii–clxx): Preface, General Introduction & Apollonius’s Prefatory Letters — From Menaechmus, Aristaeus, Euclid, and Archimedes to Apollonius’s single oblique cone; the surviving Prefatory Letters to Eudemus and Attalus. (EN · Summary · Notes)

• Unit 1 (Book I · Props. 1–27 [I.1–60], pp. 1–52): Book I — Generation of the Cone & Fundamental Symptoms — Axial triangle, subcontrary circular sections, application of areas (, ), opposite branches, harmonic tangents, and change of conjugate diameter. (EN · Summary · Notes)

• Unit 2 (Book II · Props. 28–53 [II.1–53], pp. 53–94): Book II — Asymptotes, Conjugate Hyperbolas & Secant Intercepts — Construction of asymptotes, asymptotic hyperbola equation , bisected tangents, equal secant intercepts , conjugate hyperbolas, and finding principal axes. (EN · Summary · Notes)

• Unit 3 (Book III · Props. 54–76 [III.1–56], pp. 95–125): Book III — Chord Rectangles, Harmonic Poles & Polars, Foci & the 4-Line Locus — Generalization of Euclid III.35–36, harmonic pole-polar duality, central conic foci (), and the Three- and Four-Line Locus. (EN · Summary · Notes)

• Unit 4 (Book IV · Props. 77–83 [IV.1–57], pp. 126–138): Book IV — Number of Intersections & Tangencies of Conics — Converse polar tangents, proof that two distinct conics meet in at most four points, bitangent bounds, and pencils of conics. (EN · Summary · Notes)

• Unit 5 (Book V · Props. 84–113 [V.1–77], pp. 139–196): Book V — Normals, Minimum & Maximum Lines, Subnormals & the Evolute — Subnormals, diorismos of two, three, or four normals from an arbitrary point, determination of the evolute, and rectangular-hyperbola normal construction. (EN · Summary · Notes)

• Unit 6 (Books VI & VII · Props. 114–147 [VI.1–33, VII.1–51], pp. 197–254): Books VI & VII — Equal and Similar Conics & Conjugate Diameters — Congruence and similarity of conics, cutting a given section from a given right cone, and Apollonius’s Conjugate Diameter Theorems ( and ). (EN · Summary · Notes)

• Unit 7 (Pappus Book VII Appendix): Appendix — The Lost Analytic Treasury in Pappus Book VII — Problem of Apollonius (Tangencies CCC), Plane Loci (Circle of Apollonius & Isodynamic Points X(15), X(16)), Cutting Off of a Ratio, Determinate Section, and Inclinations (Neusis). (EN · Summary · Notes)

Critical Apparatus & Concordances

• Cross-Corpus Concordance & Symbolic Gap Audit — Complete mapping of the 387 classical propositions and 147 Heath propositions to euclid, elements, archimedes, pappus, etc, wildberger, Lean mathlib4, and sympy.geometry gaps.

• Bibliography & Critical Editions — Public Domain editions (Heath 1896, Heiberg 1891–1893, Halley 1710) and reception canon.

• Prosopography of the Conic Tradition — Chronological register of 29 geometers, patrons, translators, and editors.

• Technical Glossary of Apollonian Conics — Greek, Latin, and English lexicon of 18 core conic terms.