Summary — Book IV (Propositions 77–83 [IV. 1–57])¶
1. Intersection & Tangency Multiplicity Table¶
Apollonius’s 57 propositions in Book IV establish the exact intersection-multiplicity conservation law for two distinct plane quadrics:
Contact Configuration |
Classical Props. |
Maximum Distinct Common Points |
|---|---|---|
Shared continuous arc |
IV.24 [Prop. 77] |
Impossible unless the two conics are identical |
Transverse intersections only |
IV.25, 38, 43 [Prop. 78] |
At most 4 points () |
Tangent at 1 point |
IV.26, 41, 46 [Prop. 79] |
At most 2 further intersections () |
Tangent at 2 points (bitangent) |
IV.27–30, 49–54 [Prop. 80] |
0 further intersections () |
Opposite-branch distributions |
IV.35–57 [Props. 81–83] |
Total across both branches never exceeds 4 |