Non-Euclidean Geometry

Author: H. S. M. Coxeter

Publisher: Cambridge University Press

ISBN: 9780883855225

Category: Mathematics

Page: 336

View: 5479

A reissue of Professor Coxeter's classic text on non-Euclidean geometry. It surveys real projective geometry, and elliptic geometry. After this the Euclidean and hyperbolic geometries are built up axiomatically as special cases. This is essential reading for anybody with an interest in geometry.

The Elements of Non-Euclidean Geometry

Author: D. M.Y. Sommerville

Publisher: Courier Corporation

ISBN: 0486154580

Category: Mathematics

Page: 288

View: 1596

Renowned for its lucid yet meticulous exposition, this classic allows students to follow the development of non-Euclidean geometry from a fundamental analysis of the concept of parallelism to more advanced topics. 1914 edition. Includes 133 figures.

A History of Non-Euclidean Geometry

Evolution of the Concept of a Geometric Space

Author: Boris A. Rosenfeld

Publisher: Springer Science & Business Media

ISBN: 1441986804

Category: Mathematics

Page: 471

View: 2497

The Russian edition of this book appeared in 1976 on the hundred-and-fiftieth anniversary of the historic day of February 23, 1826, when LobaeevskiI delivered his famous lecture on his discovery of non-Euclidean geometry. The importance of the discovery of non-Euclidean geometry goes far beyond the limits of geometry itself. It is safe to say that it was a turning point in the history of all mathematics. The scientific revolution of the seventeenth century marked the transition from "mathematics of constant magnitudes" to "mathematics of variable magnitudes. " During the seventies of the last century there occurred another scientific revolution. By that time mathematicians had become familiar with the ideas of non-Euclidean geometry and the algebraic ideas of group and field (all of which appeared at about the same time), and the (later) ideas of set theory. This gave rise to many geometries in addition to the Euclidean geometry previously regarded as the only conceivable possibility, to the arithmetics and algebras of many groups and fields in addition to the arith metic and algebra of real and complex numbers, and, finally, to new mathe matical systems, i. e. , sets furnished with various structures having no classical analogues. Thus in the 1870's there began a new mathematical era usually called, until the middle of the twentieth century, the era of modern mathe matics.

Introduction to Non-Euclidean Geometry

Author: Harold E. Wolfe

Publisher: Courier Corporation

ISBN: 0486320375

Category: Mathematics

Page: 272

View: 863

College-level text for elementary courses covers the fifth postulate, hyperbolic plane geometry and trigonometry, and elliptic plane geometry and trigonometry. Appendixes offer background on Euclidean geometry. Numerous exercises. 1945 edition.

Non-Euclidean Geometry

Author: Roberto Bonola

Publisher: Courier Corporation

ISBN: 048615503X

Category: Mathematics

Page: 448

View: 2386

Examines various attempts to prove Euclid's parallel postulate — by the Greeks, Arabs, and Renaissance mathematicians. It considers forerunners and founders such as Saccheri, Lambert, Legendre, W. Bolyai, Gauss, others. Includes 181 diagrams.

Non-Euclidean Geometries

János Bolyai Memorial Volume

Author: András Prékopa,Emil Molnár

Publisher: Springer Science & Business Media

ISBN: 0387295550

Category: Mathematics

Page: 506

View: 6141

"From nothing I have created a new different world," wrote János Bolyai to his father, Wolgang Bolyai, on November 3, 1823, to let him know his discovery of non-Euclidean geometry, as we call it today. The results of Bolyai and the co-discoverer, the Russian Lobachevskii, changed the course of mathematics, opened the way for modern physical theories of the twentieth century, and had an impact on the history of human culture. The papers in this volume, which commemorates the 200th anniversary of the birth of János Bolyai, were written by leading scientists of non-Euclidean geometry, its history, and its applications. Some of the papers present new discoveries about the life and works of János Bolyai and the history of non-Euclidean geometry, others deal with geometrical axiomatics; polyhedra; fractals; hyperbolic, Riemannian and discrete geometry; tilings; visualization; and applications in physics.

Introduction to Non-Euclidean Geometry


Publisher: Elsevier

ISBN: 1483295311

Category: Mathematics

Page: 274

View: 9674

An Introduction to Non-Euclidean Geometry covers some introductory topics related to non-Euclidian geometry, including hyperbolic and elliptic geometries. This book is organized into three parts encompassing eight chapters. The first part provides mathematical proofs of Euclid’s fifth postulate concerning the extent of a straight line and the theory of parallels. The second part describes some problems in hyperbolic geometry, such as cases of parallels with and without a common perpendicular. This part also deals with horocycles and triangle relations. The third part examines single and double elliptic geometries. This book will be of great value to mathematics, liberal arts, and philosophy major students.

Introductory Non-Euclidean Geometry

Author: Henry Parker Manning

Publisher: Courier Corporation

ISBN: 0486154645

Category: Mathematics

Page: 112

View: 4731

This fine and versatile introduction begins with the theorems common to Euclidean and non-Euclidean geometry, and then it addresses the specific differences that constitute elliptic and hyperbolic geometry. 1901 edition.

Non-Euclidean Geometry

Author: Stefan Kulczycki

Publisher: Courier Corporation

ISBN: 0486155013

Category: Mathematics

Page: 208

View: 3662

This accessible approach features stereometric and planimetric proofs, and elementary proofs employing only the simplest properties of the plane. A short history of geometry precedes the systematic exposition. 1961 edition.

A New Perspective on Relativity

An Odyssey in Non-Euclidean Geometries

Author: Bernard H. Lavenda

Publisher: World Scientific

ISBN: 9814340499

Category: Geometry, Non-Euclidean

Page: 668

View: 1449

Starting off from noneuclidean geometries, apart from the method of Einstein''s equations, this book derives and describes the phenomena of gravitation and diffraction. A historical account is presented, exposing the missing link in Einstein''s construction of the theory of general relativity: the uniformly rotating disc, together with his failure to realize, that the Beltrami metric of hyperbolic geometry with constant curvature describes exactly the uniform acceleration observed. This book also explores these questions: How does time bend? Why should gravity propagate at the speed of light? How does the expansion function of the universe relate to the absolute constant of the noneuclidean geometries? Why was the Sagnac effect ignored? Can Maxwell''s equations accommodate mass? Is there an inertia due solely to polarization? Can objects expand in elliptic geometry like they contract in hyperbolic geometry?

Euclidean and Non-euclidean Geometries

Author: Maria Helena Noronha

Publisher: N.A


Category: Mathematics

Page: 409

View: 634

Designed for undergraduate juniors and seniors, Noronha's (California State U., Northridge) clear, no-nonsense text provides a complete treatment of classical Euclidean geometry using axiomatic and analytic methods, with detailed proofs provided throughout. Non-Euclidean geometries are presented usi

The Foundations of Geometry and the Non-Euclidean Plane

Author: G.E. Martin

Publisher: Springer Science & Business Media

ISBN: 1461257255

Category: Mathematics

Page: 512

View: 3454

This book is a text for junior, senior, or first-year graduate courses traditionally titled Foundations of Geometry and/or Non Euclidean Geometry. The first 29 chapters are for a semester or year course on the foundations of geometry. The remaining chap ters may then be used for either a regular course or independent study courses. Another possibility, which is also especially suited for in-service teachers of high school geometry, is to survey the the fundamentals of absolute geometry (Chapters 1 -20) very quickly and begin earnest study with the theory of parallels and isometries (Chapters 21 -30). The text is self-contained, except that the elementary calculus is assumed for some parts of the material on advanced hyperbolic geometry (Chapters 31 -34). There are over 650 exercises, 30 of which are 10-part true-or-false questions. A rigorous ruler-and-protractor axiomatic development of the Euclidean and hyperbolic planes, including the classification of the isometries of these planes, is balanced by the discussion about this development. Models, such as Taxicab Geometry, are used exten sively to illustrate theory. Historical aspects and alternatives to the selected axioms are prominent. The classical axiom systems of Euclid and Hilbert are discussed, as are axiom systems for three and four-dimensional absolute geometry and Pieri's system based on rigid motions. The text is divided into three parts. The Introduction (Chapters 1 -4) is to be read as quickly as possible and then used for ref erence if necessary.

Non-Euclidean Geometry

Or, Three Moons in Mathesis

Author: Lillian Rosanoff Lieber

Publisher: N.A


Category: Geometry, Non-Euclidean

Page: 40

View: 6789

Non-Euclidean Geometry in the Theory of Automorphic Functions

Author: Jacques Hadamard

Publisher: American Mathematical Soc.

ISBN: 0821820303

Category: Mathematics

Page: 95

View: 6079

This is the English translation of a volume originally published only in Russian and now out of print. The book was written by Jacques Hadamard on the work of Poincare. Poincare's creation of a theory of automorphic functions in the early 1880s was one of the most significant mathematical achievements of the nineteenth century. It directly inspired the uniformization theorem, led to a class of functions adequate to solve all linear ordinary differential equations, and focused attention on a large new class of discrete groups. It was the first significant application of non-Euclidean geometry. The implications of these discoveries continue to be important to this day in numerous different areas of mathematics. Hadamard begins with hyperbolic geometry, which he compares with plane and spherical geometry. He discusses the corresponding isometry groups, introduces the idea of discrete subgroups, and shows that the corresponding quotient spaces are manifolds. In Chapter 2 he presents the appropriate automorphic functions, in particular, Fuchsian functions. He shows how to represent Fuchsian functions as quotients, and how Fuchsian functions invariant under the same group are related, and indicates how these functions can be used to solve differential equations. Chapter 4 is devoted to the outlines of the more complicated Kleinian case. Chapter 5 discusses algebraic functions and linear algebraic differential equations, and the last chapter sketches the theory of Fuchsian groups and geodesics. This unique exposition by Hadamard offers a fascinating and intuitive introduction to the subject of automorphic functions and illuminates its connection to differential equations, a connection not often found in other texts. This volume is one of an informal sequence of works within the History of Mathematics series. Volumes in this subset, ``Sources'', are classical mathematical works that served as cornerstones for modern mathematical thought.

János Bolyai, Non-Euclidean Geometry, and the Nature of Space

Author: Jeremy Gray,Professor of History of Mathematics Jeremy J Gray,János Bolyai

Publisher: MIT Press

ISBN: 9780262571746

Category: Mathematics

Page: 239

View: 7498

An account of the major work of Janos Bolyai, a nineteenth-century mathematician who set the stage for the field of non-Euclidean geometry. Janos Bolyai (1802-1860) was a mathematician who changed our fundamental ideas about space. As a teenager he started to explore a set of nettlesome geometrical problems, including Euclid's parallel postulate, and in 1832 he published a brilliant twenty-four-page paper that eventually shook the foundations of the 2000-year-old tradition of Euclidean geometry. Bolyai's "Appendix" (published as just that--an appendix to a much longer mathematical work by his father) set up a series of mathematical proposals whose implications would blossom into the new field of non-Euclidean geometry, providing essential intellectual background for ideas as varied as the theory of relativity and the work of Marcel Duchamp. In this short book, Jeremy Gray explains Bolyai's ideas and the historical context in which they emerged, were debated, and were eventually recognized as a central achievement in the Western intellectual tradition. Intended for nonspecialists, the book includes facsimiles of Bolyai's original paper and the 1898 English translation by G. B. Halstead, both reproduced from copies in the Burndy Library at MIT.

Euclidean and Non-Euclidean Geometries

Development and History

Author: Marvin J. Greenberg

Publisher: Macmillan

ISBN: 1429281332

Category: Mathematics

Page: 512

View: 940

This is the definitive presentation of the history, development and philosophical significance of non-Euclidean geometry as well as of the rigorous foundations for it and for elementary Euclidean geometry, essentially according to Hilbert. Appropriate for liberal arts students, prospective high school teachers, math. majors, and even bright high school students. The first eight chapters are mostly accessible to any educated reader; the last two chapters and the two appendices contain more advanced material, such as the classification of motions, hyperbolic trigonometry, hyperbolic constructions, classification of Hilbert planes and an introduction to Riemannian geometry.

The “Golden” Non-Euclidean Geometry

Hilbert's Fourth Problem, “Golden” Dynamical Systems, and the Fine-Structure Constant

Author: Alexey Stakhov,Samuil Aranson

Publisher: World Scientific

ISBN: 9814678317

Category: Mathematics

Page: 308

View: 3997

This unique book overturns our ideas about non-Euclidean geometry and the fine-structure constant, and attempts to solve long-standing mathematical problems. It describes a general theory of "recursive" hyperbolic functions based on the "Mathematics of Harmony," and the "golden," "silver," and other "metallic" proportions. Then, these theories are used to derive an original solution to Hilbert's Fourth Problem for hyperbolic and spherical geometries. On this journey, the book describes the "golden" qualitative theory of dynamical systems based on "metallic" proportions. Finally, it presents a solution to a Millennium Problem by developing the Fibonacci special theory of relativity as an original physical-mathematical solution for the fine-structure constant. It is intended for a wide audience who are interested in the history of mathematics, non-Euclidean geometry, Hilbert's mathematical problems, dynamical systems, and Millennium Problems. Contents:The Golden Ratio, Fibonacci Numbers, and the "Golden" Hyperbolic Fibonacci and Lucas FunctionsThe Mathematics of Harmony and General Theory of Recursive Hyperbolic FunctionsHyperbolic and Spherical Solutions of Hilbert's Fourth Problem: The Way to the Recursive Non-Euclidean GeometriesIntroduction to the "Golden" Qualitative Theory of Dynamical Systems Based on the Mathematics of HarmonyThe Basic Stages of the Mathematical Solution to the Fine-Structure Constant Problem as a Physical Millennium ProblemAppendix: From the "Golden" Geometry to the Multiverse Readership: Advanced undergraduate and graduate students in mathematics and theoretical physics, mathematicians and scientists of different specializations interested in history of mathematics and new mathematical ideas.