Bibliography & Primary Sources

Critical register of public-domain Greek/Latin editions of Hero of Alexandria, ancient mathematical treatises cited in the Metrica, Catoptrica, and Dioptra, and companion GEOMETOR works. Every :cite: reference resolves directly below.

Critical Editions of Hero of Alexandria

Friedrich Hultsch (ed.) — Heronis Alexandrini Geometricorum et Stereometricorum Reliquiae (Berlin: Weidmann, pp. 1–41 (Definitiones)) [1864]

Key: hultsch:1864 Public-domain Greek critical edition of Hero’s Definitiones and the Heronian metrological corpus. Provenance: https://archive.org/details/heronisalexandr00hultgoog

Hermann Schöne (ed.) — Heronis Alexandrini Opera quae supersunt omnia, Vol. III: Rationes dimetiendi et Commentatio dioptrica (Leipzig: B. G. Teubner, pp. 2–315) [1903]

Key: schoene:1903 Editio princeps of Hero’s Metrica (Books I–III) from Codex Constantinopolitanus Palatii Veteris 1, together with the critical Greek text of the Dioptra. Provenance: https://archive.org/details/heronisalexandri03hero

Johan Ludvig Heiberg (ed.) — Heronis Alexandrini Opera quae supersunt omnia, Vol. IV: Heronis Definitiones cum variis collectionibus (Leipzig: B. G. Teubner, §§1–138) [1912]

Key: heiberg:1912 Teubner standard section numbering of Hero’s Definitiones collated with Hultsch (1864).

Wilhelm Schmidt & Ludwig Nix (eds.) — Heronis Alexandrini Opera quae supersunt omnia, Vol. II, Fasc. 1: Mechanica et Catoptrica (Leipzig: B. G. Teubner, pp. 301–373) [1900]

Key: schmidt:1900 Critical edition of Hero’s Catoptrica (William of Moerbeke’s 1269 Latin text and Olympiodorus’s verbatim Greek shortest-path fragment, pp. 368–372). Provenance: https://archive.org/details/heronisalexandr02schogoog

Ancient Mathematical & Philosophical Authorities

Apollonius of Perga — Conics (Κωνικά) (Book I (oblique cone & elliptic section)) [c. 200 BCE]

Key: apollonius:conics Underpins Metrica I.34 (ellipse as oblique section of a right circular cylinder) and II.1–4 (oblique cones and cylinders).

Archimedes — Measurement of a Circle (Κύκλου μέτρησις) (Props. 1–3 (ed. Heiberg Vol. I, pp. 258–271)) [c. 240 BCE]

Key: archimedes:circle Cited explicitly in Metrica I.26 for Circle = (1/2) * circumference * radius and the rational bound π ≈ 22/7 (giving Circle : d² ≈ 11 : 14).

Archimedes — On Conoids and Spheroids (Περὶ κωνοειδέων καὶ σφαιροειδέων) (Prop. 4 (ed. Heiberg Vol. I)) [c. 220 BCE]

Key: archimedes:conoids Cited in Metrica I.34 for the exact area of an ellipse (πab, the circle whose diameter is the geometric mean √(2a·2b) of the axes).

Archimedes — The Method of Mechanical Theorems (Περὶ τῶν μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένην ἔφοδος / Ἐφοδικόν) (Props. 12–15 (ed. Heiberg Vol. II, 1913)) [c. 215 BCE]

Key: archimedes:method Cited by name (ἐν τῷ Ἐφοδικῷ) in Metrica I.32, II.14, and II.15 for the cylindrical ungula (hoof = 1/6 circumscribing square prism) and the bicylinder groin vault (2/3 circumscribing cube). Hero’s Metrica was the sole surviving witness to these theorems until Heiberg discovered the Archimedes Palimpsest in 1906.

Archimedes — Quadrature of the Parabola (Τετραγωνισμὸς παραβολῆς) (Prop. 24 (ed. Heiberg Vol. II)) [c. 240 BCE]

Key: archimedes:parabola Cited in Metrica I.32 & I.35 for the exact 4/3 ratio of any parabolic segment to the inscribed triangle of equal base and height, and used heuristically in I.31–33 to bound circular segments.

Archimedes — On Plinths and Cylinders (Περὶ πλινθίδων καὶ κυλίνδρων — lost) (Cited in Metrica I.26 (Schöne p. 66)) [c. 230 BCE]

Key: archimedes:plinthides Lost Archimedean treatise cited solely by Hero in Metrica I.26 as containing finer rational bounds for the circumference-to-diameter ratio before simplifying to 22 : 7 for practical mensuration.

Archimedes — On the Sphere and Cylinder (Περὶ σφαίρας καὶ κυλίνδρου) (Books I–II (I.33–34, I.42–43, II.2; ed. Heiberg Vol. I)) [c. 225 BCE]

Key: archimedes:sphere-cylinder Cited in Metrica I.37–39 (spherical surface = 4 great circles; spherical segment surface = circle of radius equal to polar chord) and Metrica II.11–12 & III.23 (sphere = 2/3 circumscribing cylinder; spherical segment volume; cutting a sphere in a given ratio).

Dionysodorus — On the Torus (Περὶ τῆς σπείρας — lost) (Cited in Metrica II.13 (Schöne pp. 126–130)) [c. 200 BCE]

Key: dionysodorus:speira Lost Hellenistic treatise whose central theorem—expressing the volume of a torus (σπεῖρα) as 2π²Rr²—survives exclusively through Hero’s Metrica II.13.

Euclid — Data (Δεδομένα) (Props. 1–94 (ed. Menge)) [c. 300 BCE]

Key: euclid:data Provides the deductive ‘given’ (δοθέν) analysis language used in every proposition of Hero’s Metrica before the numerical synthesis (συντεθήσεται δὴ οὕτως).

Euclid — On Divisions of Figures (Περὶ διαιρέσεων βιβλίον) (Preserved in Arabic epitome of al-Sijzī & Fibonacci’s Practica geometriae) [c. 300 BCE]

Key: euclid:divisions Classical precursor to Book III of Hero’s Metrica on dividing plane figures into given area ratios; Hero generalizes the theory to arbitrary quadrilaterals, polygons, concentric circles, pyramids, cones, frusta, and spheres.

Euclid — Elements (Στοιχεῖα) (Books I–XIII (ed. Heiberg / trans. Heath, geometor/euclid & geometor/elements)) [c. 300 BCE]

Key: euclid:elements Axiomatic foundation of Hero’s synthetic proofs: Book II gnomons and obtuse/acute projections (II.12–13), Book III cyclic quadrilaterals (III.22), Book IV incircle (IV.4), Book VI similarity and areas (VI.4, VI.8, VI.19), Book XII exhaustion, and Book XIII golden-section regular polyhedra (XIII.8–18). Provenance: https://geometor.github.io/euclid

Pappus of Alexandria — Synagoge (Mathematical Collection) (Book VII.42 (Centroid Theorem) & Book VIII (Mechanics) (geometor/pappus)) [c. 340 CE]

Key: pappus:synagoge Companion GEOMETOR treatise; generalizes Hero’s Metrica II.13 ‘unrolled torus’ argument to arbitrary solids of revolution and preserves fragments of Hero’s Mechanics. Provenance: https://geometor.github.io/pappus

Proclus Diadochus — Commentary on the First Book of Euclid’s Elements (Friedlein 1873, pp. 196, 305, 323–326, 340–346, 429 (geometor/proclus)) [c. 470 CE]

Key: proclus:in-euclidem Companion GEOMETOR treatise preserving five explicit testimonies to ‘Heron the Mechanician’ on postulate economy, direct proof without reductio, and alternative Euclidean constructions. Provenance: https://geometor.github.io/proclus

Companion GEOMETOR & Modern Critical Scholarship

Clark Kimberling — Encyclopedia of Triangle Centers (ETC) (X(1) Incenter (geometor/etc)) [1998–present]

Key: kimberling:etc Companion GEOMETOR registry of triangle centers; X(1) (the incenter H of Hero’s Metrica I.8 diagram) has trilinears 1 : 1 : 1 and barycentrics a : b : c. Provenance: https://geometor.github.io/etc

Norman J. Wildberger — Divine Proportions: Rational Trigonometry to Universal Geometry (Wild Egg Books, Ch. 7 (Archimedes’ Function & Quadrea, geometor/wildberger W.07)) [2005]

Key: wildberger:2005 Companion GEOMETOR formalization expressing 16Δ² purely polynomially in side quadrances Q₁, Q₂, Q₃ as A = (Q₁+Q₂+Q₃)² − 2(Q₁²+Q₂²+Q₃²). Provenance: https://geometor.github.io/wildberger

Sir Thomas Little Heath — A History of Greek Mathematics, Vol. II: From Aristarchus to Diophantus (Oxford: Clarendon Press, Ch. XVIII (‘Heron of Alexandria’), pp. 298–354) [1921]

Key: heath:1921 Classic public-domain English mathematical analysis of the chronology of Hero, the Constantinople manuscript of the Metrica, Prop. I.8, square- and cube-root iteration, Catoptrica, and Dioptra.