Heiberg–Heath–GEOMETOR Concordance¶
Master cross-reference of Archimedes’ Opera Omnia across Johan Ludvig Heiberg’s critical Teubner editions (heiberg:1880, heiberg:1881), Sir Thomas Little Heath’s Cambridge editions (heath:1897, heath:1912), and the constructive & philosophical codices of the GEOMETOR Digital Scriptorium.
Seven-Unit Codex Architecture¶
Unit |
Treatise Unit |
Heiberg Teubner Locus |
Heath Cambridge Locus |
|---|---|---|---|
Unit 1 |
Measurement of a Circle (Κύκλου μέτρησις (Dimensio Circuli)) |
Vol. I (1880), pp. 256–271; 2nd ed. (1910), pp. 232–243 |
Heath (1897), pp. 91–98 |
Unit 2 |
On the Equilibrium of Planes, Books I & II (Ἐπιπέδων ἰσορροπιῶν αʹ βʹ (De Planorum Aequilibriis)) |
Vol. II (1881), pp. 142–241; 2nd ed. (1913), pp. 124–213 |
Heath (1897), Book I pp. 189–202; Book II pp. 203–220 |
Unit 3 |
Quadrature of the Parabola (Τετραγωνισμὸς παραβολῆς (Quadratura Parabolae)) |
Vol. II (1881), pp. 294–353; 2nd ed. (1913), pp. 262–315 |
Heath (1897), pp. 233–252 |
Unit 4 |
On the Sphere and Cylinder, On Spirals & On Conoids and Spheroids (Περὶ σφαίρας καὶ κυλίνδρου · Περὶ ἑλίκων · Περὶ κωνοειδέων καὶ σφαιροειδέων) |
Vol. I (1880), pp. 1–255, 272–499; Vol. II (1881), pp. 1–141 |
Heath (1897), pp. 1–90, 99–188 |
Unit 5 |
Book of Lemmas (Liber Assumptorum) (Liber Assumptorum (Kitāb al-Maʾkhūdhāt, tr. Thābit ibn Qurra)) |
Vol. II (1881), pp. 426–445; 2nd ed. (1913), pp. 510–525 |
Heath (1897), pp. 301–318 |
Unit 6 |
The Method of Mechanical Theorems Addressed to Eratosthenes (Περὶ τῶν μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένην ἔφοδος (Methodus)) |
Hermes 42 (1907), pp. 243–297; Opera Omnia Vol. II, 2nd ed. (1913), pp. 426–507 |
Heath (1912 Supplement), pp. 12–51 |
Unit 7 |
The Thirteen Semi-Regular Solids, Stomachion & The Sand-Reckoner (Τρισκαίδεκα ἡμικανονικὰ στερεά (Pappus V.19) · Στομάχιον · Ψαμμίτης (Arenarius)) |
Pappus ed. Hultsch I (1876), pp. 350–358; Heiberg II (1881), pp. 242–293; 2nd ed. (1913), pp. 216–259, 416–424 |
Heath (1897), pp. xxxvi, 221–232; Pappus V.19 Concordance |
Canonical Proposition & Ecosystem Concordance¶
Locus ID |
Treatise & Proposition |
Heiberg / Heath |
Theorem Summary |
GEOMETOR Ecosystem Links |
|---|---|---|---|---|
|
Heiberg I, 258–261 (2nd ed. I, 232–235); Heath 1897, pp. 91–93 |
Area of any circle equals a right-angled triangle with legs r and C |
euclid: XII.2, X.1; elements: Polygonal exhaustion & circle quadrature reduction; wildberger: Rational polygon bounding of circular area |
|
|
Heiberg I, 260–263 (2nd ed. I, 234–237); Heath 1897, p. 93 |
Ratio of circle area to square on diameter is approximately 11 : 14 |
euclid: XII.2; wildberger: Rational area approximation 11/14 |
|
|
Heiberg I, 262–271 (2nd ed. I, 236–243); Heath 1897, pp. 93–98 |
Circumference-to-diameter bounds: 3 10/71 < π < 3 1/7 via 96-gons and rational √3 bounds (265/153 < √3 < 1351/780) |
euclid: I.47, VI.3; elements: Bisection recurrence on cot(θ/2) = cot θ + csc θ; wildberger: Pell-type rational convergents to √3 (x² - 3y² = ±1, -2) |
|
|
Heiberg II, 142–145 (2nd ed. II, 124–125); Heath 1897, pp. 189–190 |
Seven axiomatic postulates of static balance, symmetry, superposition, and convexity of the center of gravity |
etc: Axiomatic foundation of homogeneous barycentric coordinates; hilbert: Axiomatization of statics parallel to geometric incidence/congruence |
|
|
Heiberg II, 152–159 (2nd ed. II, 132–139); Heath 1897, pp. 192–194 |
Law of the Lever: commensurable (I.6) and incommensurable (I.7) magnitudes balance at distances reciprocally proportional to weights |
euclid: V.Def.4, V.Def.5; pappus: Synagoge VIII (centers of gravity); etc: Weighted barycentric division (w₁ : w₂) |
|
|
Heiberg II, 162–169 (2nd ed. II, 140–147); Heath 1897, pp. 195–198 |
Center of gravity of any parallelogram lies at the intersection of its diagonals |
euclid: I.34; elements: Midpoint concurrency of parallelogram diagonals |
|
|
Heiberg II, 172–183 (2nd ed. II, 150–159); Heath 1897, pp. 198–201 |
Center of gravity of any triangle lies on each median (I.13) and at the concurrency of medians (I.14) — Kimberling X(2) |
etc: X(2) Centroid — barycentric (1 : 1 : 1), trilinear (bc : ca : ab); euclid: VI.2, VI.4; pappus: Synagoge VIII centroid theorems |
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|
Heiberg II, 182–187 (2nd ed. II, 158–163); Heath 1897, pp. 201–202 |
Center of gravity of a trapezium divides the segment joining midpoints of parallel sides in ratio (2b + a) : (2a + b) |
etc: Barycentric decomposition of quadrilaterals into triangles |
|
|
Heiberg II, 208–241 (2nd ed. II, 182–213); Heath 1897, pp. 208–220 |
Center of gravity of any parabolic segment divides its diameter in ratio 3 : 2 from the vertex (and frustum centroid) |
pappus: Pappus–Guldin centroid foundations |
|
|
Heiberg II, 294–297 (2nd ed. II, 262–265); Heath 1897, pp. 233–235 |
Mourning Conon of Samos, statement of Archimedes’ Lemma, and announcement of mechanical + geometric quadrature |
euclid: X.1, XII.2, XII.10; hilbert: Axiom V 1 (Archimedes’ Axiom) |
|
|
Heiberg II, 304–335 (2nd ed. II, 272–299); Heath 1897, pp. 238–246 |
Mechanical exhaustion via trapezium lever bounds proving parabolic segment = 4/3 of inscribed triangle |
etc: Triangle & trapezium centroids applied to conic quadrature |
|
|
Heiberg II, 336–353 (2nd ed. II, 300–315); Heath 1897, pp. 246–252 |
Pure geometric exhaustion via triangle tree (each tier = 1/4 previous) and exact finite identity A + B + … + Z + Z/3 = 4A/3 |
euclid: IX.35, X.1; wildberger: Exact algebraic quadrature of parabola without limits |
|
|
Heiberg I, 2–11 (2nd ed. I, 2–9); Heath 1897, pp. 1–4 |
Straight line as shortest path, curves/surfaces concave in the same direction (convexity), and Archimedes’ Lemma |
proclus: Friedlein p. 110 (straight line shortest path); hilbert: Grundlagen Axiom V 1 (Archimedes’ Axiom) |
|
|
On the Sphere and Cylinder I (Props. 33, 34 & Corollary (Tomb Theorem)) |
Heiberg I, 134–147 (2nd ed. I, 120–131); Heath 1897, pp. 39–44 |
Surface of sphere = 4 great circles; volume = 4 cones; circumscribed cylinder is 3/2 of sphere in both volume and total surface |
euclid: XII.10, XII.18; pacioli: Sacred invariant 3 : 2 of Archimedes’ Syracuse tomb |
|
Heiberg I, 172–181 (2nd ed. I, 152–161); Heath 1897, pp. 51–53 |
Surface of any spherical segment equals the circle whose radius is the chord from vertex of segment to base rim |
euclid: I.47, VI.8 |
|
|
Heiberg I, 194–223 (2nd ed. I, 170–195); Eutocius III, 130–176; Heath 1897, pp. 59–79 |
Volume of spherical segment (II.2) and cutting a sphere in a given ratio via cubic equation x²(a - x) = b c² preserved by Eutocius (II.4) |
euclid: XI.34, XII.10; pappus: Solid problems solvable by intersection of parabola and hyperbola |
|
|
Heiberg II, 62–83 (2nd ed. II, 56–75); Heath 1897, pp. 171–176 |
Polar subtangent to the Archimedean spiral r = aθ rectified to circumference 2πr of the first circle |
pappus: Synagoge IV.21–25 (kinematic classification of curves) |
|
|
Heiberg II, 94–101 (2nd ed. II, 84–91); Heath 1897, pp. 178–180 |
Area bounded by first turn of spiral and initial ray equals 1/3 of the first circle (via Σ k² bounds in Prop. 10) |
euclid: XII.2, XII.10; wildberger: Discrete sum of squares 3(1² + … + n²) = n(n+1)(2n+1)/2 |
|
|
Heiberg I, 378–499 (2nd ed. I, 342–445); Heath 1897, pp. 111–150 |
Ellipse area = πab (Prop. 4); paraboloid segment = 3/2 inscribed cone (Props. 21–22); hyperboloid & spheroid segments (Props. 25–32) |
euclid: XII.10, XII.12 |
|
|
Heiberg II, 428–431 (2nd ed. II, 510–513); Heath 1897, pp. 301–304 |
Homothetic collinearity on tangent circles (Prop. 1), semicircle tangent bisection (Prop. 2), and chord half-angle identity (Prop. 3) |
euclid: III.31, VI.2; pappus: Synagoge IV.14 |
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|
Heiberg II, 430–432 (2nd ed. II, 513–514); Heath 1897, pp. 304–305 |
Area of the Arbelos (shoemaker’s knife) equals the circle on the internal perpendicular tangent PN |
euclid: II.4, III.31, XII.2; elements: Exact circle-area cancellation (r₁+r₂)² - r₁² - r₂² = 2r₁r₂ |
|
|
Heiberg II, 432–435 (2nd ed. II, 514–516); Heath 1897, pp. 305–307 |
Archimedes’ Twin Circles in the partitioned Arbelos have equal diameter d = AC·CB / AB = r₁r₂/(r₁+r₂)·2 |
pappus: Synagoge IV (p. 230 Hultsch); wildberger: Harmonic radius invariant 1/r = 1/r₁ + 1/r₂ |
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|
Heiberg II, 435–438 (2nd ed. II, 516–519); Heath 1897, pp. 307–309 |
Circle inscribed in the Arbelos touching all three semicircles has diameter ratio λ : (1 + λ + λ²) (6/19 for 3:2) & Pappus chain c_n = n·d_n |
pappus: Synagoge IV.14–18 (‘ancient proposition’ on arbelos circle chain); wildberger: Descartes–Soddy rational curvature relation |
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|
Heiberg II, 439 (2nd ed. II, 519–520); Heath 1897, pp. 309–310 |
Angle Trisection by Neusis: inserting segment BC = r outside circle on secant through center O yields ∠AEB = (1/3)∠AOB |
euclid: I.5, I.32; pappus: Synagoge IV.36–44 (classification of plane, solid & neusis trisections) |
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|
Heiberg II, 439–443 (2nd ed. II, 520–523); Heath 1897, pp. 310–316 |
Orthogonal chords quadrance identity AO² + BO² + CO² + DO² = 4R² (Prop. 11) and polar orthocenter reciprocity (Prop. 12) |
wildberger: Pure quadrance invariant Q₁ + Q₂ + Q₃ + Q₄ = D²; pappus: Pole-polar reciprocity & Apollonius On Contacts |
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|
Heiberg II, 443–444 (2nd ed. II, 523–524); Heath 1897, pp. 316–317 |
Area of the Salinon (salt-cellar) bounded by four symmetric semicircles equals the circle on axial segment CF |
euclid: II.10, XII.2 |
|
|
Heiberg II, 444–445 (2nd ed. II, 524–525); Heath 1897, pp. 317–318 |
Regular pentagon & decagon chord-secant configuration cutting EC in extreme and mean ratio (golden section φ) with EM = radius |
euclid: XIII.9, XIII.10; divine: Golden section φ = (1 + √5)/2 in pentagon-decagon secants; pacioli: Divine proportion in regular pentagon/decagon |
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|
The Method of Mechanical Theorems (Letter to Eratosthenes & Lemmas 1–11) |
Heiberg Hermes 42, 243–247; II (2nd ed.), 426–434; Heath 1912, pp. 12–15 |
Distinction between mechanical heuristic indication (ἔφοδος) and synthetic proof; tribute to Democritus & Eudoxus on cone = 1/3 cylinder |
euclid: XII.7, XII.10; aristotle: Distinction between hoti (indication) and dioti (demonstrative proof) |
|
Heiberg II (2nd ed.), 434–446; Heath 1912, pp. 15–21 |
Mechanical slice-balancing of parabolic segment (Prop. 1: 4/3 ΔABC) and sphere + cone against cylinder (Prop. 2: V = (4/3)πr³ → S = 4πr²) |
etc: Triangle centroid W dividing median 2 : 1 (CK = 3KW) as fulcrum ratio |
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|
Heiberg II (2nd ed.), 446–484; Heath 1912, pp. 21–38 |
Mechanical discovery of volumes and centers of gravity of spheroids, paraboloids, hyperboloids, and spherical segments |
pappus: Synagoge VIII solid centroids |
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|
Heiberg II (2nd ed.), 484–507; Heath 1912, pp. 38–51 |
Cubature of curved solids equal to rectilinear polyhedra: cylindrical ungula = 1/6 prism (Props. 12–14) & bicylinder = 2/3 cube (Prop. 15) |
heron: Metrica citation of Archimedes’ Method on ungula and bicylinder |
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|
Thirteen Semi-Regular Solids (via Pappus Synagoge V.19) (Pappus V.19 (Solids I–XIII)) |
Heiberg Pappus ed. Hultsch I (1876), 350–358; Heiberg II (2nd ed.), 535–537; Heath 1897, p. xxxvi |
Complete census of the 13 convex vertex-transitive non-Platonic polyhedra (8, 14, 14, 14, 26, 26, 38, 32, 32, 32, 62, 62, 92 faces) |
euclid: XIII.13–18 (Five Platonic Solids); pappus: Synagoge V.19 face-species enumeration; pacioli: De Divina Proportione abscised & elevated golden polyhedra; divine: Icosidodecahedron, truncated icosahedron & snub dodecahedron over Q(φ) |
|
Stomachion (Loculus Archimedius) (Palimpsest Fragment & Arabic Dissection) |
Heiberg II (2nd ed., 1913), 416–424; Heath 1912 Intro, p. 7 |
Fourteen-tile rational areal dissection of the 12 × 12 square (536 distinct modulo-symmetry assemblies) |
wildberger: Rational lattice dissection & commensurable triangle areas |
|
Heiberg II, 242–293 (2nd ed. II, 216–259); Heath 1897, pp. 221–232 |
Aristarchus’s heliocentric cosmos, solar angular diameter bounds (90°/200 < d < 90°/164), myriad-octad exponents up to 10^(8·10^16), and < 10^63 (10^51 common) cosmic sand bound |
euclid: IX.8–9 (geometric progressions & law of exponents) |