The Britannica Guide to Geometry by Publishing, Britannica Educational -- Read -- Imperial Library of Trantor

Index

Cover Page Title Page Copyright Page Contents Introduction Chapter 1: History of Geometry

Ancient Geometry: Practical and Empirical

Finding the Right Angle Locating the Inaccessible Estimating the Wealth

Ancient Geometry: Abstract and Applied

The Three Classical Problems

Doubling the Cube Trisecting the Angle Squaring the Circle

Idealization and Proof The Euclidean Synthesis Gnomonics and the Cone Astronomy and Trigonometry

Calculation Epistemology

Ancient Geometry: Cosmological and Metaphysical

Pythagorean Numbers and Platonic Solids Measuring the Earth and Heavens

The Post-Classical Period

Passage Through Islam Europe Rediscovers the Classics Linear Perspective

Transformation

French Circles

Projective Geometry Cartesian Geometry

Geometrical Calculus The World System

Relaxation and Rigour

Projection Again Non-Euclidean Geometries A Grand Synthesis The Real World

Chapter 2: Branches of Geometry

Euclidean Geometry

Fundamentals Plane Geometry

Congruence of Triangles Similarity of Triangles Areas Pythagorean Theorem Circles Regular Polygons Conic Sections and Geometric Art

Solid Geometry

Volume Regular Solids

Conic Section

Analytic Definition Greek Origins Post-Greek Applications

Analytic Geometry

Elementary Analytic Geometry Analytic Geometry of Three and More Dimensions

Vector Analysis Projections

Algebraic Geometry

Projective Geometry

Parallel Lines and the Projection of Infinity Projective Invariants Projective Conic Sections

Differential Geometry

Curvature of Curves Curvature of Surfaces Shortest Paths on a Surface

Non-Euclidean Geometry

Comparison of Euclidean, Spherical, and Hyperbolic Geometries Spherical Geometry Hyperbolic Geometry

Topology

Basic Concepts of General Topology

Simply Connected Topological Equivalence Homeomorphism Topological Structure

Algebraic Topology

Fundamental Group Knot Theory

Graph Theory

Chapter 3: Biographies

Ancient Greek and Islamic Geometers

Apollonius of Perga Archimedes

His Life His Works His Influence

Archytas of Tarentum Conon of Samos Eratosthenes of Cyrene Euclid

Life Sources and Contents of the Elements Euclid’s Axioms and Euclid’s Common Notions Renditions of the Elements Other Writings Assessment

Eudoxus of Cnidus

Life Mathematician Assessment

Heron of Alexandria Hippias of Elis Hippocrates of Chios Menaechmus Omar Khayyam Pappus of Alexandria Pythagoras Thales of Miletus Theaetetus

Pre-Modern (Pre-1800) Geometers

Bonaventura Cavalieri Giovanni Ceva Girard Desargues René Descartes Leonhard Euler Gaspard Monge, count de Péluse Gilles Personne de Roberval Simon Stevin

Modern Geometers

Lars Valerian Ahlfors Pavel Sergeevich Aleksandrov James W. Alexander II Sir Michael Francis Atiyah Eugenio Beltrami Enrico Betti János Bolyai Charles-Julien Brianchon Luitzen Egbertus Jan Brouwer Michel Chasles Shiing-shen Chern William Kingdon Clifford Pierre René Deligne Simon Kirwan Donaldson Vladimir Gershonovich Drinfeld Alexandre Grothendieck David Hilbert Gaston Maurice Julia Felix Klein Niels Fabian Helge von Koch Kodaira Kunihiko Nikolay Ivanovich Lobachevsky Benoit Mandelbrot John Willard Milnor Hermann Minkowski August Ferdinand Möbius Mori Shigefumi David Bryant Mumford Sergey Petrovich Novikov Grigori Perelman Henri Poincaré Jean-Victor Poncelet Bernhard Riemann Jean-Pierre Serre Wacław Sierpiński Stephen Smale Karl Georg Christian von Staudt Jakob Steiner René Frédéric Thom William Paul Thurston Oswald Veblen Vladimir Voevodsky André Weil Wendelin Werner Shing-Tung Yau

Appendix of Geometric Terms and Concepts

Algebraic Surface Angle Trisection: Archimedes’ Method Angle Trisection: Quadratrix of Hippias Axiom Axiomatic Method Brachistochrone Bridge of Asses Brouwer’s Fixed Point Theorem Catenary Ceva’s Theorem Circle Compactness Cone Coordinate Systems Cross Ratio Curve Cycloid Cylinder Desargues’s Theorem Dimension Duality The Elements Since the Middle Ages Ellipse Ellipsoid Envelope Euclid’s Windmill Euclidean Space Method of Exhaustion Fractal Golden Ratio Graph Harmonic Construction Hausdorff Space Hilbert Space Hippocrates’ Quadrature of the Lune Hyperbola Hyperboloid Incommensurables Isometric Drawing Königsberg Bridge Problem Line Measuring the Earth, Classical and Arabic Measuring the Earth, Modern Metric Space Parabola Paraboloid Parallel Postulate Pencil Pi Platonic Solid Polygon Projection Pythagorean Theorem Space-Time Spiral Square Thales’ Rectangle Topological Space

Glossary Bibliography Index

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