Measurement of a Circle — Philological & Mathematical Notes

1. Textual State & Proposition Order

  • Fragmentary Byzantine Recension: As Heiberg (heiberg:1880) and Heath (heath:1897, pp. xxiv, 93) observe, our surviving Greek text of Κύκλου μέτρησις bears marks of a late editor or epitomator (connected with the sixth-century school of Isidore of Miletus): the Syracusan Doric dialect has been stripped in favor of Attic/Koine forms (κύκλος, περιφέρεια, διάμετρος), and Proposition 2 () stands illogically before Proposition 3, upon whose bound it depends, while lacking Archimedes’ characteristic inequality phrasing (“approximately” or upper/lower bounds).

  • Testimony of Heron: Heron of Alexandria (heron:metrica, I.25–26) cites Archimedes’ Measurement of a Circle for and mentions another treatise (On Plinthides and Cylinders) in which Archimedes gave still closer fractional bounds, paralleled by Apollonius's Okytokion attested by Eutocius (eutocius:in-archimedem).

2. Origin of the Pell Approximants and

Why does Archimedes state without explanation that ?

  • As analyzed in heath:1897 (Introduction, Chapter IV, pp. lxxvii–lxxxiv), both fractions belong to the convergents of the simple continued fraction of :

    \[1, \quad \frac{2}{1}, \quad \frac{5}{3}, \quad \frac{7}{4}, \quad \frac{19}{11}, \quad \frac{26}{15}, \quad \frac{71}{41}, \quad \frac{97}{56}, \quad \mathbf{\frac{265}{153}}, \quad \frac{362}{209}, \quad \frac{989}{571}, \quad \mathbf{\frac{1351}{780}}.\]

  • Equivalently, iterating Heron’s / Babylonians’ reciprocal-averaging bounds starting from via and produces , then (satisfying the Pell-type equation ), and one step further (satisfying Pell’s equation , accurate to six decimal places: vs. ).

  • Eutocius of Ascalon (eutocius:in-archimedem, Heiberg Vol. III, pp. 228–261) verifies every squared mixed fraction in Greek alphabetic numerals step-by-step.

3. GEOMETOR Cross-Corpus Concordance

  • Euclid & Constructive Elements (geometor:euclid): Directly extends euclid XII.2 (exhaustion by inscribed bisected squares) while adding circumscribed polygons whose perimeters require the explicit convexity assumptions of Unit 4 (Sphere and Cylinder I, Assump. 1–2).

  • Exact Symbolic Field Towers (geometor/model & geometor/fundamentals): Whereas the exact algebraic coordinates of the 96-gon live in the nested quadratic tower of degree 16, Archimedes projects each layer back onto via certified rational interval enclosures—anticipating rigorous interval arithmetic by twenty-one centuries.

  • Rational Trigonometry (geometor:wildberger): Notice that Archimedes’ half-angle recurrence operates entirely on rational spread/cotangent ratios without transcendental angle measure.