Metrica Book I (Preface & Props. 1–9) — GEOMETOR Commentary & Notes

GEOMETOR Mathematical & Historical Commentary

  1. The 1896 Constantinople Rediscovery & Archimedian Attribution of I.8: For over a millennium, Hero's Metrica was known only through fragmentary citations in Eutocius (In Archimedis Dim. Circ.), until Richard Schöne and Paul Tannery identified the complete three-book treatise in 1896 in the 11th-century Codex Constantinopolitanus Palatii Veteris no. 1 (fols. 67r–107v), edited by Hermann Schöne in 1903. Medieval Arabic geometer al-Bīrūnī reports that the triangle area theorem \(\Delta = \sqrt{s(s-a)(s-b)(s-c)}\) originated with Archimedes, while Metrica I.8 preserves its oldest extant Greek synthetic proof—duplicated verbatim in Hero’s Dioptra Ch. 30.

  2. Fourth-Degree Magnitude (*Dynamodynamis*) Inside Greek Geometry: Classical Euclidean geometry restricts magnitudes to three homogeneous spatial dimensions (line, square, solid). At Schöne p. 20, lines 22–25, Hero boldly writes: “the rectangle \(\Gamma\Theta \cdot EH\) is the square root of \(\Gamma\Theta^2 \cdot EH^2\) “—multiplying a square by a square (a fourth-degree dynamodynamis, named explicitly at Metrica I.17, Schöne p. 48) inside a synthetic proof! This conceptual leap bridges classical geometric algebra (Elements Book II) and Diophantine symbolic algebra.

  3. Archimedes’ Function / Wildberger’s Quadrea (:math:`mathcal{A} = 16Delta^2`): In Norman J. Wildberger's Rational Trigonometry (formalized in geometor/wildberger W.07), expanding \(16 s(s-a)(s-b)(s-c)\) in the side quadrances \(Q_a = a^2, Q_b = b^2, Q_c = c^2\) eliminates every square root:

    \[\mathcal{A}(Q_a, Q_b, Q_c) = (Q_a + Q_b + Q_c)^2 - 2(Q_a^2 + Q_b^2 + Q_c^2) = 16 s(s-a)(s-b)(s-c) = 16\Delta^2.\]

    On Hero’s two benchmark triangles:

    • For \((7, 8, 9)\), \((Q_a, Q_b, Q_c) = (49, 64, 81) \implies \mathcal{A} = 194^2 - 2(12738) = 11520 = 16 \times 720\).

    • For \((13, 14, 15)\), \((Q_a, Q_b, Q_c) = (196, 225, 169) \implies \mathcal{A} = 590^2 - 2(117562) = 112896 = 16 \times 7056 = (4 \times 84)^2\).

  4. Quadratic Convergence of Hero’s Root Algorithm: Hero’s rule \(x_{k+1} = \frac{1}{2}\!\left(x_k + \frac{N}{x_k}\right)\) satisfies the exact algebraic error recurrence:

    \[x_{k+1}^2 - N = \left(\frac{x_k^2 - N}{2 x_k}\right)^2 > 0.\]

    Every iterate \(k \ge 1\) is therefore a strict rational upper bound whose quadratic excess squares at every step: from \(x_0^2 - 720 = +9\) (\(x_0 = 27\)), step 1 gives \(\frac{9^2}{4 \cdot 729} = \frac{1}{36}\) (\(x_1 = \frac{161}{6}\)), and step 2 gives \(\frac{1}{4 \cdot 36 \cdot 25921} = \frac{1}{3{,}732{,}624}\) (\(x_2 = \frac{51841}{1932}\)), accurate to 8 significant decimal digits in two hand divisions!

Interactive GEOMETOR Demonstrations & Sister Concordance