Scholarly Notes — Metrica Book I (Props. 10–39)¶
Commentary on Golden-Section Polygon Algebra, Hipparchian Chord Tables, Archimedes’ Lost Plinthides, and the Method
1. Euclid Elements XIII.1 & the Pentagon Lemma in \(\mathbb{Q}(\phi)\)¶
In Proposition I.18 (Schöne 1903, p. 50), Hero compresses Euclid's Book XIII theory of the regular pentagon and decagon into a remarkably direct right-triangle lemma. By reflecting right triangle \(AB\Gamma\) (\(\angle A = 36^\circ\)) across \(B\Gamma\) to form isosceles triangle \(AB\Delta\) with apex angle \(108^\circ\) (\(\frac{6}{5}\) of a right angle), Hero identifies \(AB = B\Delta\) as the side of a regular pentagon and \(A\Delta = 2 A\Gamma\) as its diagonal.
By [XIII.8], \(A\Delta : AB = AB : (A\Delta - AB) = \phi : 1\). Invoking [XIII.1]—if \(A\Delta\) is cut in extreme and mean ratio with greater segment \(AB\), then adding half the whole (\(A\Gamma = \frac{1}{2} A\Delta\)) yields \((AB + A\Gamma)^2 = 5\,A\Gamma^2\)—Hero avoids nested fourth-degree radicals until the very end of the numerical evaluation! In Rational Trigonometry (Wildberger 2005), the spread at \(A\) (\(36^\circ\)) lives purely in \(\mathbb{Q}(\sqrt{5})\):
See the executable construction in Universal Golden Gnomon & Dioptra Mountain Tunnel and geometor.heron.constructions.construct_pentagon_lemma.
2. Pre-Ptolemaic Chord Tables in Propositions I.22 & I.24¶
Because the regular \(9\)-gon (\(40^\circ\) central angle, requiring angle trisection) and regular \(11\)-gon (\(32\frac{8}{11}^\circ\) central angle) cannot be constructed with unmarked straightedge and compass, Hero explicitly cites “the books on chords in a circle” (τὰ περὶ τῶν ἐν κύκλῳ εὐθειῶν). As Thomas Heath (1921, Vol. II, pp. 259, 326) and Otto Neugebauer observed, Hero’s ratios—specifically \(d / a_9 \approx \frac{17}{6}\) (equivalent to a chord of \(40^\circ\) near \(\frac{6}{17} d \approx 0.3529 d\) vs. exact \(\sin 20^\circ = 0.3420 d\)) and \(d / a_{11} \approx \frac{25}{7}\) (\(\frac{7}{25} d = 0.2800 d\) vs. exact \(\sin\frac{\pi}{11} \approx 0.28173 d\), accurate to \(0.6\%\)!)—preserve precious pre-Ptolemaic trigonometric data prior to Claudius Ptolemy's Almagest I.10, pointing back to Hipparchus of Nicaea.
3. Archimedes’ Lost Plinthides and Cylinders (Prop. I.26)¶
In Proposition I.26 (Schöne 1903, p. 66; fol. 81v), Hero preserves our sole surviving ancient testimony to Archimedes' work On Plinthides and Cylinders (Περὶ πλινθίδων καὶ κυλίνδρων). In the Codex Constantinopolitanus, the manuscript numerals read \(\frac{211875}{67441}\) (\(\approx 3.141635\)) and \(\frac{197888}{62351}\) (\(\approx 3.17377\), slightly corrupted in a single Greek digit; Paul Tannery and Hermann Schöne showed that restoring a single numeral yields tight four-decimal bounds straddling \(\pi = 3.14159265\dots\)). This proves that Archimedes’ famous \(3\frac{10}{71} < \pi < 3\frac{1}{7}\) in Measurement of a Circle was merely a simplified didactic bound, whereas Archimedes had computed \(\pi\) far more tightly in Plinthides and Cylinders.
4. Surfaces in Book I vs. Volumes in Book II¶
Why do the curved 3D surfaces of the cylinder, cone, sphere, and spherical cap appear at the end of Book I (Props. I.36–I.39) rather than in Book II? Hero adheres strictly to the dimensional ontology enunciated in his Book I Prooemium: Book I measures two-dimensional surfaces (ἐπιφάνειαι), whether planar or curved in three-dimensional space, because every surface—even a sphere’s—is measured in homogeneous unit squares (\(S_{\text{sphere}} = 4\pi r^2\)), whereas Book II is reserved exclusively for three-dimensional solid capacities measured in unit cubes.