Book I — Generation of the Cone & Fundamental Symptoms

[Folio Heath 1896, Props. 1–27 [I. 1–60], pp. 1–52 (PDF 181–232)]

1. Definitions & Sections of the Oblique Cone [I. Defs. 1–8, I. 1–10]

[Folio Heath 1896, pp. 1–8 (PDF 181–188)]

Apollonius defines the conic surface with complete projective generality: if from a fixed apex A a straight line, indefinitely produced both ways, moves around the circumference of a circle BC not in the same plane with A, the double surface traced out consists of two vertically opposite cones meeting at A. The cone is right (ὀρθός) when the axis from A to the center of the base is perpendicular to the base, and oblique / scalene (σκαληνός) otherwise.

  • Axial Triangle & Parallel Circular Sections [I. 3–4]: Any plane through the apex A cuts the cone in a triangle; any plane parallel to the circular base BC cuts the cone in a circle.

  • Subcontrary (Antiparallel) Circular Sections [I. 5, I. 9]: If an oblique cone be cut through the axis at right angles to the base in the axial triangle ABC, and also cut by a second plane perpendicular to ABC cutting off toward the apex a triangle AKH similar but antiparallel (hypenantia) so that , the resulting section is likewise an exact circle (proved via [III.35] and [VI.4]). Every oblique circular cone thus carries two distinct conjugate families of circular sections.

2. Fundamental Symptoms via Application of Areas [Props. 1–4 / I. 11–14]

[Folio Heath 1896, pp. 8–14 (PDF 188–194)]

Let a cutting plane intersect the base in DE perpendicular to the base of the axial triangle ABC, cutting out on the cone a curve with diameter PM (starting at vertex P on generator AB). For any point Q on the curve with ordinate QV parallel to DE, the parallel circular section through V with diameter FG satisfies ([III.35]). Translating the product along the axial triangle yields the three immortal species:

I.11: Parabola (παραβολή) Exact Application: QV² = PL · PV = p x P V Q L p p x y I.12: Ellipse (ἔλλειψις) Defect: QV² = x(p − p x / d) P P′ V Q L R y² defect I.13: Hyperbola (ὑπερβολή) Excess: QV² = x(p + p x / d) P′ P V Q L R p x excess d

Plate 2 — Application of Areas (Book I.11–13 [Props. 1–3]). Left: parabolē (, exact application to orthia p). Center: elleipsis (, falling short by a similar rectangle). Right: hyperbolē (, exceeding by a similar rectangle).

Proposition 1 [I. 11] — The Parabola (παραβολή). If a cone be cut by a plane through its axis, and also by another plane cutting the base in a straight line perpendicular to the base of the axial triangle, and if further the diameter of the section be parallel to one of the sides of the axial triangle, then any straight line QV drawn from the section parallel to the common section of the cutting plane and the base to meet the diameter PM will be equal in square to the rectangle contained by PV and a certain fixed straight line PL (latus rectum), where :

\[QV^2 = PL \cdot PV \qquad \bigl(y^2 = p x\bigr).\]

Proposition 2 [I. 12] — The Hyperbola (ὑπερβολή). If the diameter of the section meet one side of the axial triangle and the other side produced beyond the vertex A at P, and if PL () be erected perpendicular to PP () with (for AFparallelPP), then QV2 equals the rectangle applied to PL with width PV, exceeding (ὑπερβάλλον, [VI.29]) by a figure similar and similarly situated to the rectangle :

\[QV^2 = PV \cdot PL + \frac{PL}{PP'} PV^2 \qquad \Longleftrightarrow \qquad QV^2 : PV \cdot VP' = p : d.\]

Proposition 3 [I. 13] — The Ellipse (ἔλλειψις). If the cutting plane meet both sides of the axial triangle on the same nape at P and P, then QV2 equals the rectangle applied to PL with width PV, falling short (ἐλλεῖπον, [VI.28]) by a figure similar and similarly situated to the rectangle :

\[QV^2 = PV \cdot PL - \frac{PL}{PP'} PV^2 \qquad \Longleftrightarrow \qquad QV^2 : PV \cdot VP' = p : d.\]
Proposition 4 [I. 14] — The Opposite Branches (αἱ ἀντικείμεναι). If the vertically opposite surfaces of the double cone be cut by a plane not passing through the vertex, the sections on the two napes have a common transverse diameter PP and equal parameters (latera recta) corresponding to it.

3. Tangents, Harmonic Division & Change of Diameter [Props. 12–27 / I. 33–60]

[Folio Heath 1896, pp. 22–52 (PDF 202–232)]

Apollonius establishes tangent lines without limits by proving that no straight line can fall between the tangent and the curve (extending [III.16]), and discovers that tangency is governed by harmonic division:

I.34–37: Harmonic Tangent Construction (A′, A ; V, T) = −1 & CV · CT = CA² A′V = 3a/2 VA = a/2 AT = a A′V : VA = 3 : 1 = A′T : TA (Harmonic Conjugate T of V wrt A′A) A′ C V A T P D tangent PT ∥ CD

Plate 3 — Tangent & Harmonic Division (Book I.34, 36–37 [Props. 13–14]). On diameter PP with center C, the ordinate foot V of contact point Q and the tangent meet T divide the vertices PP harmonically: , equivalently .

  • Parabola Tangent Bisection [Prop. 12 / I. 33, 35]: If QV be an ordinate to diameter PV of a parabola and TPPV be set off outside the vertex P, then TQ touches the parabola, and conversely every tangent satisfies (subtangent equals twice the abscissa).

  • Central Conic Harmonic Tangent [Props. 13–14 / I. 34, 36, 37]: For a hyperbola, ellipse, or circle with diameter PP and center C, TQ is tangent at Q if and only if T and the ordinate foot V divide PP harmonically:

    \[\frac{TP}{TP'} = \frac{PV}{VP'} \qquad \Longleftrightarrow \qquad CT \cdot CV = CP^2 \qquad \Longleftrightarrow \qquad CV \cdot VT = PV \cdot VP'.\]

  • Reference to Any New Conjugate Diameter [Props. 20–21 / I. 49–51]: Using pure Euclidean quadrangle-area identities ([I.43]), Apollonius proves that if any point Q on a conic be taken as a new vertex, the diameter through Q and the tangent at Q form a new oblique coordinate frame in which the exact same area-application equation holds with a transformed parameter .