Scholarly Notes — Metrica Book II (Preface & Props. 1–20)¶
Commentary on Dionysodorus of Caunus, Centroid Unrolling, the Archimedes Palimpsest, and the Bomiskos
1. Dionysodorus’s On the Torus & the Prehistory of Pappus’s Centroid Theorem¶
Proposition II.13 (Schöne 1903, pp. 126–130; fols. 95v–96r) is one of the jewels recovered when Richard Schöne and Hermann Schöne brought the Codex Constantinopolitanus to light in 1896–1903. Prior to 1896, Dionysodorus of Caunus was known primarily through Eutocius's commentary on Archimedes' Sphere and Cylinder II.4 for cutting a sphere in a given ratio via the intersection of a parabola and a rectangular hyperbola.
Hero preserves both Dionysodorus’s rigorous exhaustion proportion—comparing the torus to the cylinder of radius \(R\) and height \(2r\) in the ratio \(\pi r^2 : R r\)—and his own intuitive centroid reformulation: unrolling (ἁπλωθεῖσα) the torus into a right circular cylinder whose length equals the circumference \(2\pi R\) traced by the center of gravity \(Z\) of the generating meridian circle! Two and a half centuries later, Pappus of Alexandria generalized this exact principle to arbitrary planar domains in Synagoge Book VII (the Pappus–Guldinus Centroid Theorem). Notice also that because \(\pi\) enters squared in \(V = 2\pi^2 R r^2\), Hero evaluates \(\pi^2 \approx \left(\frac{22}{7}\right)^2 = \frac{484}{49}\), producing the rational signature \(\frac{69696}{7} = 9956\frac{4}{7}\) (vs. exact \(1008\pi^2 \approx 9948.56\)).
2. Metrica II.14–II.15 & the 1906 Discovery of Archimedes’ Method¶
History records a dramatic convergence between Metrica and the Archimedes Palimpsest. In 1903, Hermann Schöne published Hero’s explicit citations in Metrica I.35, II.14, and II.15 to Archimedes' hitherto unknown treatise The Method (Ἐφοδικόν), revealing that Archimedes had proved both the cylindrical ungula (\(V = \frac{1}{6}d^2 h\)) and the bicylinder groin vault (\(V = \frac{2}{3}d^3\)). Just three years later, in 1906, Johan Ludvig Heiberg traveled to Constantinople (to the Metochion of the Holy Sepulchre, steps away from the Old Seraglio where the Metrica manuscript lay!) and deciphered the Archimedes Palimpsest, finding the Method of Mechanical Theorems Addressed to Eratosthenes! Crucially, because the final folios of the Archimedes Palimpsest containing the classical exhaustion proof of Proposition 15 (the bicylinder) were lost, Hero’s Metrica II.15 remains an indispensable witness confirming that Archimedes completed and circulated the theorem for architectural vaults (καμάραι).
3. Why Hero Rewrote the Frustum Formula (Props. II.6–II.10)¶
Egyptian Moscow Mathematical Papyrus Problem 14 (c. 1850 BCE) already computed the square frustum via \(\frac{h}{3}(a^2 + ab + b^2)\). Why did Hero systematically rewrite this throughout Book II as \(h\left[\left(\frac{a+b}{2}\right)^2 + \frac{1}{3}\left(\frac{a-b}{2}\right)^2\right]\)?
Direct Error Correction of Surveyor Practice: Everyday Roman and Egyptian masons frequently averaged top and bottom dimensions \(\bar{a} = \frac{a+b}{2}\) and multiplied \(\bar{a}^2 h\). Hero’s decomposition isolates \(\bar{a}^2 h\) as the leading term and exhibits \(\frac{1}{3}\left(\frac{a-b}{2}\right)^2 h\) as the exact positive convexity correction.
Seamless Generalization to Dissimilar Rectangles (Prop. II.8): Whereas \(a^2 + ab + b^2\) has no obvious symmetric extension when length and breadth taper at different rates (\(a_1 \ne a_2, b_1 \ne b_2\)), Hero’s mean-plus-third-difference form generalizes immediately by replacing squares with products: \(\bar{a}\bar{b} + \frac{1}{3}\Delta a\,\Delta b\), which is equivalent to Simpson’s prismatoid rule \(\frac{h}{6}(A_{\text{top}} + 4 A_{\text{mid}} + A_{\text{bot}})\)!