Quadrature of the Parabola — Analytical Summary¶
Scope: Heiberg Vol. II, pp. 294–353 · Heath (1897), pp. 233–252 · Letter to Dositheus & Propositions 1–24
Two Complementary Demonstrations of ¶
Quadrature of the Parabola (archimedes:quadrature-of-parabola) presents the earliest exact area of a conic section in history, proved twice by independent engines:
Mechanical Exhaustion (Props. 1–17): Slices the segment into finite rectilinear trapezia (avoiding indivisibles of zero width, unlike The Method) and suspends them from a balance beam against the tangent triangle triangleQqT4triangleABC, proving .
Geometric Exhaustion (Props. 18–24): Recursively fills the two residual arches on AB, BC with secondary triangles triangleADB, triangleBEC whose combined area is always of the preceding stage, and closes the double reductio via the telescoping finite identity (Prop. 23) without ever invoking an actual infinite sum!
Structural Map of Propositions 1–24¶
Block |
Propositions |
Mathematical Role |
|---|---|---|
Preface |
Epistle to Dositheus |
Tribute to Conon; statement of Archimedes' Lemma & Eudoxean precedents |
Conic Lemmas |
Props. 1–5 |
Pre-Apollonian parabola symptomata & tangent-chord intercept ratio |
Statics Lemmas |
Props. 6–13 |
Lever equilibrium and upper/lower moment bounds for suspended triangles and trapezia |
Mechanical Proof |
Props. 14–17 |
Compression between circumscribed and inscribed trapezia |
Exhaustion Tree |
Props. 18–22 |
Binary parabolic triangle tree: each generation adds |
Telescoping Sum |
Props. 23–24 |
Exact finite remainder + double reductio ad absurdum |