Euclidean and Non-Euclidean Geometries

Development and History

Author: Marvin J. Greenberg

Publisher: Macmillan

ISBN: 9780716724469

Category: Mathematics

Page: 483

View: 2397

This classic text provides overview of both classic and hyperbolic geometries, placing the work of key mathematicians/ philosophers in historical context. Coverage includes geometric transformations, models of the hyperbolic planes, and pseudospheres.
Posted in Mathematics

Euclidean and Non-Euclidean Geometries

Development and History

Author: Marvin J. Greenberg

Publisher: Macmillan Higher Education

ISBN: 1429281332

Category: Mathematics

Page: 512

View: 7669

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.
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Euclidean and Non-Euclidean Geometry International Student Edition

An Analytic Approach

Author: Patrick J. Ryan

Publisher: Cambridge University Press

ISBN: 0521127076

Category: Mathematics

Page: 232

View: 2025

This book gives a rigorous treatment of the fundamentals of plane geometry: Euclidean, spherical, elliptical and hyperbolic.
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Non-Euclidean Geometry

Author: H. S. M. Coxeter

Publisher: Cambridge University Press

ISBN: 9780883855225

Category: Mathematics

Page: 336

View: 7439

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.
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Ideas of Space

Euclidean, Non-Euclidean, and Relativistic

Author: Jeremy Gray

Publisher: Oxford University Press

ISBN: 0198539355

Category: Mathematics

Page: 242

View: 2702

Ideas of Space is a lively and readable account of the development of Euclidean, non-Euclidean, and relativistic ideas of the shape of the universe. For this new edition the author has updated much of the material and added a chapter on the emerging story of the Arabic contribution.
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Non-Euclidean Geometry

A Critical and Historical Study of Its Development

Author: Roberto Bonola

Publisher: Courier Dover Publications

ISBN: N.A

Category: Mathematics

Page: 389

View: 4666

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.
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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: 2405

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.
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Geometry: Euclid and Beyond

Author: Robin Hartshorne

Publisher: Springer Science & Business Media

ISBN: 0387226761

Category: Mathematics

Page: 528

View: 7806

This book offers a unique opportunity to understand the essence of one of the great thinkers of western civilization. A guided reading of Euclid's Elements leads to a critical discussion and rigorous modern treatment of Euclid's geometry and its more recent descendants, with complete proofs. Topics include the introduction of coordinates, the theory of area, history of the parallel postulate, the various non-Euclidean geometries, and the regular and semi-regular polyhedra.
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Geometry Through History

Euclidean, Hyperbolic, and Projective Geometries

Author: Meighan I. Dillon

Publisher: Springer

ISBN: 3319741357

Category: Mathematics

Page: 350

View: 1782

Presented as an engaging discourse, this textbook invites readers to delve into the historical origins and uses of geometry. The narrative traces the influence of Euclid’s system of geometry, as developed in his classic text The Elements, through the Arabic period, the modern era in the West, and up to twentieth century mathematics. Axioms and proof methods used by mathematicians from those periods are explored alongside the problems in Euclidean geometry that lead to their work. Students cultivate skills applicable to much of modern mathematics through sections that integrate concepts like projective and hyperbolic geometry with representative proof-based exercises. For its sophisticated account of ancient to modern geometries, this text assumes only a year of college mathematics as it builds towards its conclusion with algebraic curves and quaternions. Euclid’s work has affected geometry for thousands of years, so this text has something to offer to anyone who wants to broaden their appreciation for the field.
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Introduction To Non-Euclidean Geometry

Author: Harold E. Wolfe

Publisher: Read Books Ltd

ISBN: 1446547302

Category: Science

Page: 260

View: 1736

Many of the earliest books, particularly those dating back to the 1900s and before, are now extremely scarce and increasingly expensive. We are republishing these classic works in affordable, high quality, modern editions, using the original text and artwork.
Posted in Science

Modern Geometries

Non-Euclidean, Projective, and Discrete

Author: Michael Henle

Publisher: Pearson College Division

ISBN: 9780130323132

Category: Mathematics

Page: 389

View: 1145

Engaging, accessible, and extensively illustrated, this brief, but solid introduction to modern geometry describes geometry as it is understood and used by contemporary mathematicians and theoretical scientists. Basically non-Euclidean in approach, it relates geometry to familiar ideas from analytic geometry, staying firmly in the Cartesian plane. It uses the principle geometric concept of congruence or geometric transformation--introducing and using the Erlanger Program explicitly throughout. It features significant modern applications of geometry--e.g., the geometry of relativity, symmetry, art and crystallography, finite geometry and computation. Covers a full range of topics from plane geometry, projective geometry, solid geometry, discrete geometry, and axiom systems. For anyone interested in an introduction to geometry used by contemporary mathematicians and theoretical scientists.
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Sources of Hyperbolic Geometry

Author: John Stillwell

Publisher: American Mathematical Soc.

ISBN: 9780821809228

Category: Mathematics

Page: 153

View: 9868

This book presents, for the first time in English, the papers of Beltrami, Klein, and Poincare that brought hyperbolic geometry into the mainstream of mathematics. A recognition of Beltrami comparable to that given the pioneering works of Bolyai and Lobachevsky seems long overdue--not only because Beltrami rescued hyperbolic geometry from oblivion by proving it to be logically consistent, but because he gave it a concrete meaning (a model) that made hyperbolic geometry part of ordinary mathematics. The models subsequently discovered by Klein and Poincare brought hyperbolic geometry even further down to earth and paved the way for the current explosion of activity in low-dimensional geometry and topology. By placing the works of these three mathematicians side by side and providing commentaries, this book gives the student, historian, or professional geometer a bird's-eye view of one of the great episodes in mathematics. The unified setting and historical context reveal the insights of Beltrami, Klein, and Poincare in their full brilliance.
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Introduction to Hyperbolic Geometry

Author: Arlan Ramsay,Robert D. Richtmyer

Publisher: Springer Science & Business Media

ISBN: 1475755856

Category: Mathematics

Page: 289

View: 8027

This book is an introduction to hyperbolic and differential geometry that provides material in the early chapters that can serve as a textbook for a standard upper division course on hyperbolic geometry. For that material, the students need to be familiar with calculus and linear algebra and willing to accept one advanced theorem from analysis without proof. The book goes well beyond the standard course in later chapters, and there is enough material for an honors course, or for supplementary reading. Indeed, parts of the book have been used for both kinds of courses. Even some of what is in the early chapters would surely not be nec essary for a standard course. For example, detailed proofs are given of the Jordan Curve Theorem for Polygons and of the decomposability of poly gons into triangles, These proofs are included for the sake of completeness, but the results themselves are so believable that most students should skip the proofs on a first reading. The axioms used are modern in character and more "user friendly" than the traditional ones. The familiar real number system is used as an in gredient rather than appearing as a result of the axioms. However, it should not be thought that the geometric treatment is in terms of models: this is an axiomatic approach that is just more convenient than the traditional ones.
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A Radical Approach to Real Analysis

Author: David M. Bressoud

Publisher: MAA

ISBN: 9780883857472

Category: Mathematics

Page: 323

View: 9259

Second edition of this introduction to real analysis, rooted in the historical issues that shaped its development.
Posted in Mathematics

The Foundations of Geometry

Author: David Hilbert

Publisher: N.A

ISBN: N.A

Category: Geometry

Page: 143

View: 5700

Posted in Geometry

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: 7402

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.
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Euclidean Geometry and Transformations

Author: Clayton W. Dodge

Publisher: Courier Corporation

ISBN: 0486138429

Category: Mathematics

Page: 304

View: 7286

This introduction to Euclidean geometry emphasizes transformations, particularly isometries and similarities. Suitable for undergraduate courses, it includes numerous examples, many with detailed answers. 1972 edition.
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A New Perspective on Relativity

An Odyssey in Non-Euclidean Geometries

Author: Bernard H. Lavenda

Publisher: World Scientific

ISBN: 9814340480

Category: Science

Page: 668

View: 2768

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?
Posted in Science