Content of type 'chapter'

Hero Metrica I.8 — Synthetic Incircle & Cyclic Quadrilateral Proof on (13, 14, 15) ( H E ): (x - 6)**2 + (y - 4)**2 - 16 = 0 ( M C ): (x - 7)**2 + (y + 4)**2 - 65 = 0 [ B C ]: y = 0 [ A B ]: 12*x - 5*y = 0 [ C A ]: -12*x - 9*y + 168 = 0 [ B E ]: -12*x + 18*y = 0 [ C G ]: -56*x/5 - 112*y/5 + 784/5 = 0 [ A H ]: 8*x + y - 52 = 0 [ H J ]: x - 6 = 0 [ T U ]: -16*sqrt(3)*x + 8*sqrt(3)*y + 64*sqrt(3) = 0 [ W X ]: x = 0 [ C L ]: 8*x - 14*y - 112 = 0 / H D /: |seg| = 4 / H E /: |seg| = 4 / H F /: |seg| = 4 / B G /: |seg| = 7 / A D /: |seg| = 7 / A F /: |seg| = 7 / B D /: |seg| = 6 / B E /: |seg| = 6 / C E /: |seg| = 8 / C F /: |seg| = 8 / C G /: |seg| = 21 B = (0, 0) C = (14, 0) A = (5, 12) H = (6, 4) E = (6, 0) D = (30/13, 72/13) F = (46/5, 32/5) G = (-7, 0) K = (4, 0) L = (0, -8) M = (7, -4) B C A H E D F G K L M

Metrica Book I (Preface & Props. 1–9): Triangles, Heron’s Formula & √720

Origins of Geometry in Egypt, Homogeneous Orthogonal Units, Rectangles & Triangles, Synthetic Incircle Proof of Δ = √(s(s−a)(s−b)(s−c)), and Iterative Square-Root Extraction — Schöne (1903), pp. 2–26 · Codex Constantinopolitanus Palatii Veteris 1, fols. 67r–72r

Hero Metrica I.18 Lemma — Pentagon Golden Section Identity (BA + AC)^2 = 5 AC^2 ( A B ): y**2 + (x + 1)**2 - 6 + 2*sqrt(5) = 0 [ A C ]: y = 0 [ B E ]: x = 0 [ A B ]: -x*sqrt(5 - 2*sqrt(5)) + y - sqrt(5 - 2*sqrt(5)) = 0 [ B D ]: x*sqrt(5 - 2*sqrt(5)) + y - sqrt(5 - 2*sqrt(5)) = 0 / A G /: |seg| = -1 + sqrt(5) / A B /: |seg| = sqrt(6 - 2*sqrt(5)) / B D /: |seg| = sqrt(6 - 2*sqrt(5)) / A C /: |seg| = 1 / C D /: |seg| = 1 A = (-1, 0) C = (0, 0) D = (1, 0) B = (0, sqrt(5 - 2*sqrt(5))) G = (-1 + sqrt(2)*(-sqrt(2)/2 + sqrt(10)/2), 0) A C D B G

Metrica Book I (Props. 10–39): Quadrilaterals, Regular Polygons, Circles & Conics

Trapezia & Irregular Quadrilaterals, the Regular Polygon Series (3-gon to 12-gon) & Golden Section Pentagon Lemma, Archimedes’ Plinthides & Circular Annulus, Circular & Parabolic Segments via the Method, Ellipse Area, and Curved Surfaces — Schöne (1903), pp. 26–90 · Codex Constantinopolitanus Palatii Veteris 1, fols. 72r–87r

Hero Metrica II.13 — Dionysodorus Torus Meridian Section (R = 14, r = 6) ( Z B ): y**2 + (x - 14)**2 - 36 = 0 [ A B ]: y = 0 [ D E ]: x - 14 = 0 [ I J ]: x = 0 [ L M ]: y - 6 = 0 [ O P ]: y + 6 = 0 / A Z /: |seg| = 14 / Z B /: |seg| = 6 A = (0, 0) C = (8, 0) B = (20, 0) Z = (14, 0) D = (14, 6) E = (14, -6) H1 = (0, 6) H2 = (0, -6) A C B Z D E H1 H2

Metrica Book II (Preface & Props. 1–20): Stereometry, Dionysodorus’s Torus, the Method & Platonic Solids

Eudoxus & Archimedes in Stereometry, Frusta & Hip-Roof Wedges, Spheres & Vaults, Dionysodorus’s Torus Theorem (9956 4/7), Archimedes’ Method Vaults, the Five Platonic Solids, and Hydrostatic Displacement — Schöne (1903), pp. 90–140 · Codex Constantinopolitanus Palatii Veteris 1, fols. 87r–99v

Hero Metrica III.1–2 — Vertex (5:3) & Parallel Mean-Proportional (3:1) Area Division ( M1 A ): (x - 5/2)**2 + (y - 6)**2 - 169/4 = 0 ( A R ): (x - 5)**2 + (y - 12)**2 - (15/4 + 3*sqrt(3))**2 - (9 - 5*sqrt(3)/4)**2 = 0 [ B C ]: y = 0 [ A B ]: 12*x - 5*y = 0 [ C A ]: -12*x - 9*y + 168 = 0 [ M N ]: y - 12 + 6*sqrt(3) = 0 / A V /: |seg| = 3*sqrt(281)/4 / P R /: |seg| = 13*sqrt(3)/4 / D E /: |seg| = 7*sqrt(3) B = (0, 0) C = (14, 0) A = (5, 12) V = (35/4, 0) P = (5/4, 3) R = (5/4 - 3*sqrt(3), 5*sqrt(3)/4 + 3) D = (5 - 5*sqrt(3)/2, 12 - 6*sqrt(3)) E = (5 + 9*sqrt(3)/2, 12 - 6*sqrt(3)) B C A V P R D E

Metrica Book III (Preface & Props. 1–23): Proportional Divisions of Figures, Justice & ∛100

Geometric Proportion & Distributive Justice (“Not By a Single Millet-Seed”), Dividing Triangles & Quadrilaterals, Euclid’s On Divisions, Exterior-Point Cuts, Solid Divisions, and Hero’s Cube-Root Algorithm — Schöne (1903), pp. 140–184 · Codex Constantinopolitanus Palatii Veteris 1, fols. 99v–107v

Hero Definitiones §58 — Universal Gnomon Realized on the Golden Rectangle in Q(phi) ( M C ): y**2 + (x - 1/2)**2 - 5/4 = 0 ( E B ): y**2 + (x - sqrt(5)/2 - 1/2)**2 - (-1/2 + sqrt(5)/2)**2 = 0 [ A B ]: y = 0 [ D E ]: x = 0 [ G H ]: x - 1 = 0 [ D C ]: y - 1 = 0 [ P Q ]: x - sqrt(5)/2 - 1/2 = 0 [ U V ]: y - sqrt(5)/2 + 1/2 = 0 / A E /: |seg| = 1/2 + sqrt(5)/2 / B E /: |seg| = -1/2 + sqrt(5)/2 A = (0, 0) B = (1, 0) D = (0, 1) C = (1, 1) M = (1/2, 0) E = (1/2 + sqrt(5)/2, 0) F = (1/2 + sqrt(5)/2, 1) G = (1/2 + sqrt(5)/2, -1/2 + sqrt(5)/2) H = (1, -1/2 + sqrt(5)/2) E1 = (1/2 - sqrt(5)/2, 0) A B D C M E F G H E1

Definitiones, Catoptrica & Dioptra: Genetic Axiomatics, Universal Gnomon, Shortest-Path Optics & Mountain Tunneling

Hero’s Genetic Definitions & Universal Gnomon (§58), Proclus on Hero’s Direct Proofs, the Variational Shortest-Path Proof of Reflection (Catoptrica Prop. 4), and Two-Portal Mountain Tunneling (Dioptra Ch. 15) — Hultsch (1864), pp. 7–23 · Schmidt (1900), pp. 324–372 · Schöne (1903), pp. 238–240