Book IV — Number of Intersections & Tangencies of Conics

[Folio Heath 1896, Props. 77–83 [IV. 1–57], pp. 126–138 (PDF 306–318)]

In the Preface to Book IV (addressed to Attalus after the death of Eudemus), Apollonius states the exact purpose of the fourth Book: in how many points at most can sections of cones meet one another, or meet the circumference of a circle, whether by simple intersection or by tangency—and in particular how a double-branch hyperbola meets another conic, resolving the polemic between Conon of Samos and Nicoteles of Cyrene.

1. Converse Harmonic Polarity [IV. 1–23]

[Folio Heath 1896, p. 126 (PDF 306)]

Propositions IV.1–23 supply the exact converse of Book III’s harmonic polar theorems (III.30–40): if from an external point T secants meet a conic in RR and harmonic conjugates M of T with respect to RR are marked on each secant, those harmonic conjugates lie along the straight polar chord of contact joining the points of tangency from T—giving a pure straightedge-only construction of tangents!

2. Non-Coincidence of Arcs & the Four-Point Intersection Bound [Props. 77–83 / IV. 24–57]

[Folio Heath 1896, pp. 126–138 (PDF 306–318)]

Plate 8 — At Most Four Intersections & the Four-Point Conic Pencil (Book IV.24–57 [Props. 77–83]). Two distinct conics (here an ellipse, a circle, and a two-branch hyperbola) can intersect in at most four points ABCD; their diagonal triangle EFG is self-polar with respect to every conic in the pencil.

Proposition 77 [IV. 24] — Identity Principle for Conic Arcs. No two conics can intersect in such a way that part of one of them is common to both, while the rest is not. (Proved via chord-bisection diameters: two curves sharing an arc qQPQ must share the same diameter PV and half-chords everywhere.)

Proposition 78 [IV. 25, 38, 43] — Maximum of Four Intersection Points. Two conics (including a circle or a double-branch hyperbola) cannot intersect in more than four points.

Proof Method via Harmonic Polarity: Suppose two conics met in five points ABCDE taken in order along a convex arc. Join AB and CD meeting at T. On both conics simultaneously, the harmonic conjugate of T with respect to AB is a single point M1, and with respect to CD is a single point M2. Hence both conics share the exact same polar line M1M2 of pole T, forcing any fifth common point E to yield contradictory harmonic conjugates on a secant through T!

Propositions 79–83 [IV. 26–57] — Tangency Multiplicity Counting. - Single Tangency (`Prop. 79 [IV. 26, 41–42, 46]`): Two conics touching at one point cannot intersect in more than two other points (). - Double Tangency / Bitangency (`Prop. 80 [IV. 27–30, 49–54]`): Two conics touching one another at two distinct points cannot intersect at any other point (). - Opposite-Branch Separations (`Props. 81–83 [IV. 35–37, 55–57]`): Complete enumeration of how single-branch conics and opposite hyperbolic branches distribute up to four intersections across napes.