Scholarly Notes — Metrica Book III (Preface & Props. 1–23)¶
Commentary on Euclid’s On Divisions of Figures, Rational Quadrance Linearity, and Hero’s Cube-Root Formula
1. Metrica Book III & Euclid’s Lost On Divisions of Figures¶
Proclus (In Euclidem, Friedlein p. 69) records that Euclid composed a dedicated treatise On Divisions (Περὶ διαιρέσεων βιβλίον), lost in Greek but partially preserved in a 10th-century Arabic compendium by al-Sijzī (published by Woepcke in 1851; reconstructed by Archibald in 1915). Comparing Euclid’s On Divisions with Metrica Book III illuminates Hero’s genius:
Where Euclid gives pure existence and construction theorems (Data-style) for planar figures (triangles, trapezia, quadrilaterals, circles), Hero supplies both the rigorous Euclidean analysis/synthesis and explicit rational numerical algorithms on concrete survey triangles such as \((13, 14, 15)\).
Furthermore, Euclid’s On Divisions stops at two-dimensional figures; Props. III.20–III.23 extend proportional division into three-dimensional stereometry (pyramids, cones, frusta, and spheres), where straightedge-and-compass mean proportionals no longer suffice because triplicate ratios demand cube roots and conic intersections!
2. Rational Trigonometry View of Proposition III.2¶
Notice how Hero phrases the conclusion of his analysis in Proposition III.2 (Schöne 1903, p. 142):
«δοθεῖσα ἄρα καὶ ἡ ΑΔ δυνάμει» — “Therefore :math:`ADelta` is given in square (:term:`quadrance`).”
In Norman Wildberger's Rational Trigonometry (2005), area division by a parallel line is purely rational and linear in quadrance: for any rational division ratio \(m : n\), the cut quadrance \(Q(A, D) = \frac{m}{m+n} Q(A, B)\) belongs strictly to \(\mathbb{Q}\) (\(\frac{507}{4}\) on \((13, 14, 15)\) for \(3 : 1\)), and compass construction via the semicircle on \(AB\) ([VI.13]) constructs the exact Euclidean point \(D\) without numerical approximation. See the companion demonstration in Proportional Area Divisions & ∛100 (Metrica Book III).
3. Mathematical Structure of Hero’s Cube-Root Formula (Prop. III.20)¶
Hero’s cube-root extraction of \(\sqrt[3]{100}\) in Proposition III.20 (Schöne 1903, p. 178) attracted intense scrutiny from Curtze, Wertheim, Eneström, Thomas Heath (1921, Vol. II, pp. 341–343), and Dijksterhuis. Letting \(\ell^3 < N < u^3\) with \(u = \ell + 1\), \(e_\ell = N - \ell^3\), and \(e_u = u^3 - N\), Hero’s arithmetic steps evaluate:
Why is weighting \(e_\ell\) by \(u\) and \(e_u\) by \(\ell\) so vastly superior to naive linear interpolation \(\ell + \frac{e_\ell}{e_\ell + e_u}\)?
Naive linear interpolation between \(4^3 = 64\) and \(5^3 = 125\) gives \(4 + \frac{36}{61} \approx 4.5902\) (error \(-0.0514\), nearly forty times worse!).
Because \(f(x) = x^3\) is convex (\(f''(x) = 6x > 0\)), its slope \(3x^2\) is steeper near \(u = 5\) than near \(\ell = 4\). Weighting the lower deficit \(e_\ell\) by \(u\) and the upper excess \(e_u\) by \(\ell\) cancels the leading second-order curvature error of \(x^3\) across unit intervals \([\ell, \ell + 1]\)—producing \(\frac{65}{14} \approx 4.64286\) (error \(+0.00127\)) in a single rational fraction!