Summary — Book I (Propositions 1–27 [I. 1–60])

1. Deductive Chain of Book I

  1. The Cone (`I.1–10`): Oblique double cone, axial triangle, parallel circular sections (I.4), and antiparallel subcontrary sections (I.5, 9).

  2. Fundamental Symptoms (`Props. 1–4 [I.11–14]`): - Parabola (`I.11`): (exact application). - Hyperbola (`I.12`): (application with excess; two-vertex ratio ). - Ellipse (`I.13`): (application with defect; two-vertex ratio ). - Opposite Branches (`I.14`): Equal parameter on both napes.

  3. Conjugate Diameters (`Props. 5–6 [I.15–16]`): Existence of center C bisecting PP and secondary conjugate diameter DD of length satisfying .

  4. Secants & Tangents (`Props. 7–15 [I.17–38]`): - Parabola subtangent law: . - Central conic harmonic tangent law: and .

  5. Transformation of Oblique Coordinates (`Props. 16–21 [I.42–51]`): Area-equivalence of trapezoids between diameter, tangent, and ordinate transfers the symptom from the original cone-derived diameter to any diameter QQ on the conic.

  6. Synthetic Construction of Conics (`Props. 22–27 [I.52–60]`): Given in a plane a diameter PP, ordinate angle , and parameter , Apollonius constructs in 3D a right circular cone whose plane section reproduces the prescribed parabola, hyperbola, conjugate hyperbolas, or ellipse—proving existence and completeness of planar quadratic curves.