Measurement of a Circle — Analytical Summary¶
Scope: Heiberg Vol. I, pp. 258–271 · Heath (1897), pp. 91–98 · Propositions 1–3
Architectural Argument¶
Although briefest among the extant works of Archimedes, Measurement of a Circle (Κύκλου μέτρησις) accomplishes two revolutions in Greek geometry:
Equivalence of Rectification and Quadrature (Prop. 1): Where Euclid in Elements XII. 2 (euclid:elements) proved only the proportionality of two circles to the squares on their diameters (), Archimedes establishes an exact areal identity between a curved boundary and a rectilinear figure: the circle equals the right-angled triangle whose legs are the radius and the circumference . Every question of circular area is thereby reduced to a single one-dimensional ratio .
Algorithmic Rational Interval Arithmetic (Prop. 3): Starting from the regular hexagon () and two extraordinary rational bounds for :
\[\frac{265}{153} < \sqrt{3} < \frac{1351}{780}, \qquad \left(265^2 - 3\cdot 153^2 = -2, \quad 1351^2 - 3\cdot 780^2 = +1\right),\]Archimedes iterates four half-angle recurrences ( sides) using only Eucl. VI. 3 (angle bisector) and Eucl. I. 47 (Pythagoras), systematically rounding every square root downward in the circumscribed recurrence and upward in the inscribed recurrence so that the inequalities remain 100% rigorous at every step.
Step-by-Step 96-Gon Bisection Ledger¶
Stage |
Polygon |
Circumscribed Cotangent / Secant Lower Bounds () |
Inscribed Cotangent / Cosecant Upper Bounds () |
|---|---|---|---|
Seed |
6-gon () |
, |
, |
Bisection 1 |
12-gon () |
, |
, |
Bisection 2 |
24-gon () |
, |
, |
Bisection 3 |
48-gon () |
, |
, |
Bisection 4 |
96-gon () |
, |