Quadrature of the Parabola — Philological & Mathematical Notes

1. Title & Pre-Apollonian Conic Terminology

• As Heath (heath:1897, p. 233) notes, the manuscript title Τετραγωνισμὸς παραβολῆς cannot have been Archimedes' own original heading: throughout the body of the treatise (and in Conoids and Spheroids and The Method), Archimedes never writes παραβολή for the curve—a term coined shortly afterward by Apollonius of Perga from Pythagorean application of areas—but always ὀρθογωνίου κώνου τομή (section of a right-angled cone, τμῆμα ὀρθογωνίου κώνου τομῆς).

• Notice the three-stage epistemological progression across Archimedes’ corpus: 1. Heuristic Line-Slice Discovery: The Method, Prop. 1 (weighing chords MO one-by-one, discovered first). 2. Rigorous Mechanical Exhaustion: Quadrature, Props. 6–17 (replacing lines by finite-width trapezia compressed via Archimedes' Lemma). 3. Pure Synthetic Exhaustion: Quadrature, Props. 18–24 (eliminating mechanics altogether via the triangle tree).

2. Affine Invariance of the Ratio

Notice that Archimedes proves for every chord AC—not merely chords perpendicular to the axis of symmetry! In affine coordinates aligned with the diameter BV (parallel to the axis) and tangent BT (parallel to AC), the parabola has equation regardless of obliquity, and because oblique shear preserves area ratios, both (Prop. 21) and (Prop. 24) are affine invariants.

3. GEOMETOR Cross-Corpus Concordance

• Euclid (`geometor/euclid`) (geometor:euclid): Directly cited via archimedes:quadrature-of-parabola in Heath’s commentary on Elements X.1 (bisection exhaustion) and XII.2/XII.10.

• Hilbert (`geometor/hilbert`) (hilbert:1899): The Preface to Quadrature of the Parabola (alongside Sphere and Cylinder I, Assump. 5) is the locus classicus from which Axiom V 1 (Archimedes’ Axiom) enters modern axiomatic geometry and real analysis.