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Customer Reviews (18)

Excellent examples
Finally, an understandable upper-level math text! Greenberg walks you step-by-step through the proofs of the foundation ideas for each chapter (with selected exceptions left as examples. However, these proceed predictably from other examples that do have full proofs). There are numerous exercises at the end of each chapter, typically professors will choose a handful that they find interesting or amusing and assign those. No answers are at the back of the book. The problems require proofs and I have yet to see a text that provides answers to proofs problems in the back. Anyhow, many problems have multiple approaches. I actually found this book easier to understand than the professor.

an excellent and really untertaining book
There are already 16 reviews of this excellent and exciting book so i will only add that some people complained about the great number of results of the core text the reader is asked to search proofs as exercises. For a mathematically inclined reader this is not such a big trouble because most of these exercises have extended indications which math-inclined people can easily transform in a complete and sound proof. For myself i had almost no trouble with them (there is one exception with the section on axioms of beetwenness in chapter 3, i took the strategy of admitting the propositions of this section i could not prove as complementary axioms (these propositions are visualy obvious and easy to accept) so i proceeded further for the really interesting matters).
So the difficulty is only for people who did not have a mathematical training as college junior. Even in this casethey can learn a lot about the nature and purpose of pure mathematics and, if they are persistent and enduring, how to read and write mathematical proofs.
As a Frenchman i wonder why such a good book has not been translated in French, it really deserves it because books in French on geometry are so often unexciting and boring.

Quintessential Work on Non-Euclidean Geometry
I had the pleasure of reading and studying the Second Edition of this text while in college.This course with this text was my favorite course during all of my undergraduate math courses.

Being a fan of the subject, I was eager to see the new Fourth Edition of the text.The Fourth Edition is quite expanded from earlier editions, going past the wonderful main story of the Parallel Postulate - told better by Greenberg than any other author, IMHO - and diving into the different non-Euclidean geometries that "open one's eyes" by setting aside the "obvious axiom of a unique parallel". The last chapters are greatly enhanced, with a superb presentation of the issue of straightedge and compass constructions in the Hyperbolic plane.

This presentation of Non-Euclidean geometry is more serious than the "popularized" books on advanced mathematical topics.If you're looking for a "light, fun" reading of this topic, this is not the book for you.

I feel that the real power of the story of the maturing of intellectual thought, so brilliantly portrayed in the story of the Parallel Postulate, must be experienced, through the effort (and often hard work) of actually **doing** geometry, rather than just reading lightly about it. If you want to dive in and actual experience geometry (and the consequent rewards), then this is the book for you.The explanations are magnificent, the problems are wonderful (and, at times, very challenging), all culminating in the "wow!" of modifying the Euclidean way of thinking to a new and beautiful alternate geometrical universe.

As other reviewers have noted, this text reads like a great novel - a drama involving geometry.If PBS/Nova ever make a "What does Parallel mean anyway?" show, this text will be the basis for that show.

I believe this Fourth Edition can be considered the quintessential text on this topic, on which all future discussion of the topics can be based, including both the introductory materials, as well as moving to the forefront of research on many topics in Hyperbolic geometry.

For a university course, weaker students will find this text quite challenging, and possibly too hard.For average students, this text will provide sufficient challenge and interest, and ample areas in the text that will not overwhelm.For advanced students, this text will certainly challenge in many different directions and interests, both in the later chapter discussions, and various problems throughout.

Greenberg's writing is meticulous - you will never find an error, a comma out of place, nor a sentence that is not perfect.

Euclidean and Non-Euclidean Geometries, Fourth Edition, by Marvin Jay Greenberg
The Fourth Edition of M.J. Greenberg's textbook is a wonderful addition to the geometry textbook literature. No praise could be higher than to say that it is even better--indeed, a good deal better--than the highly regarded earlier editions. There are important revisions to each of the chapters and appendices, some of them extensive. As Greenberg aptly notes: "this book is a resource for a wide variety of students, from the naive to the sophisticated, from the non-mathematical-but-educated to the mathematical wizards."In this reviewer's opinion, Greenberg's fourth edition along with the Robin Hartshorne's mathematically more technical Geometry: Euclid and Beyond (2000)--a text to which Greenberg repeatedly makes reference--are far and away the most informed, up-to-date, and historically and philosophically sensitive geometry texts on the market today. No one with an interest in the foundations of geometry can afford to be without copies of these two great works.

A very good book about Geometry
This is a very good book about Euclidean and Non-Euclidean Geometries.
Well written, this book introduces to the lector in the historical context of the development of the Geometry.
I enjoyed very much.

Why is it so cheap, now (April, 2008) ?Because, this is the 3rd edition and exists a new 4th edition since September 28, 2007. ... Read more

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This book gives a rigorous treatment of the fundamentals of plane geometry: Euclidean, spherical, elliptical and hyperbolic. The primary purpose is to acquaint the reader with the classical results of plane Euclidean and nonEuclidean geometry, congruence theorems, concurrence theorems, classification of isometries, angle addition and trigonometrical formulae. However, the book not only provides students with facts about and an understanding of the structure of the classical geometries, but also with an arsenal of computational techniques for geometrical investigations. The aim is to link classical and modern geometry to prepare students for further study and research in group theory, Lie groups, differential geometry, topology, and mathematical physics. The book is intended primarily for undergraduate mathematics students who have acquired the ability to formulate mathematical propositions precisely and to construct and understand mathematical arguments. Some familiarity with linear algebra and basic mathematical functions is assumed, though all the necessary background material is included in the appendices. ... Read more

Customer Reviews (2)

Too Advanced for most
This is so rigorous it is only for the advanced mathematician. I was looking for something much more accessible. I'll have to keep looking.

Great math book
This book about euclidean and non-euclidean geometry is great! A must for researh or math class! ... Read more

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This book develops a self-contained treatment of classical Euclidean geometry through both axiomatic and analytic methods. Concise and well organized, it prompts readers to prove a theorem yet provides them with a framework for doing so.Chapter topics cover neutral geometry, Euclidean plane geometry, geometric transformations, Euclidean 3-space, Euclidean n-space; perimeter, area and volume; spherical geometry; hyperbolic geometry; models for plane geometries; and the hyperbolic metric.Amazon.com Review
What's so sacred about parallel lines? Students and general readers who want a solid grounding in the fundamentals of space would do well to let M. Helena Noronha's Euclidean and Non-Euclidean Geometries be their guide. Noronha, professor of mathematics at California State University, Northridge, breaks geometry down to its essentials and shows students how Riemann, Lobachevsky, and the rest built their own by re-evaluating the parallel postulate. Each chapter devotes itself to rigorous study of one topic: neutral geometry, Euclidean 3-space, hyperbolic geometry, and more reveal themselves to the reader through the author's clear analyses and proofs. Problem sets help the student become comfortable with techniques and reach the conclusions through their own work, gaining a visceral understanding impossible through passive reading. Little mathematical background is needed beyond a bit of set theory, calculus, and a willingness to persevere. --Rob Lightner ... Read more

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Introduction to NON-EUCLIDEAN GEOMETRY by HAROLD E. WOLFE . PREFACE:This book has been written in an attempt to provide a satisfactory textbook to be used as a basis for elementary courses in Non-Euclid ean Geometry. The need for such a volume, definitely intended for classroom use and containing substantial lists of exercises, has been evident for some time. It is hoped that this one will meet the re quirements of those instructors who have been teaching the subject tegularly, and also that its appearance will encourage others to institute such courses. x The benefits and amenities of a formal study of Non-Euclidean Geometry are generally recognized. Not only is the subject matter itself valuable and intensely fascinating, well worth the time of any student of mathematics, but there is probably no elementary course which exhibits so clearly the nature and significance of geometry and, indeed, of mathematics in general. However, a mere cursory acquaintance with the subject will not do. One must follow its development at least a little way to see how things come out, and try his hand at demonstrating propositions under circumstances such that intuition no longer serves as a guide. For teachers and prospective teachers of geometry in the secondary schools the study of Non-Euclidean Geometry is invaluable. With out it there is strong likelihood that they will not understand the real nature of the subject they are teaching and the import of its applications to the interpretation of physical space. Among the first books on Non-Euclidean Geometry to appear in English was one, scarcely more than a pamphlet, written in 1880 by G. Chrystal. Even at that early date the value of this study for those preparing to teach was recognized. In the preface to this little brochure, Chrystal expressed his desire to bring pangeometrical speculations under the notice of those engaged in the teaching of geometry He wrote It will not be supposed that I advocate the introduction of pan geometry as a school subject it is for the teacher that I advocate such a study. It is a great mistake to suppose that it is sufficient for the teacher of an elementary subject to be just ahead of his pupils. No one can be a good elementary teacher who cannot handle his subject with the grasp of a master. Geometrical insight and wealth of geometrical ideas, either natural or acquired, are essential to a good teacher of geometry and I know of no better way of cultivat ing them than by studying pan geometry. Within recent years the number of American colleges and uni versities which offer courses in advanced Euclidean Geometry has increased rapidly. There is evidence that the quality of the teaching of geometry in our secondary schools has, accordingly, greatly improved. But advanced study in Euclidean Geometry is not the only requisite for the good teaching of Euclid. The study of Non-Euclidean Geometry takes its place beside it as an indispensable part of the training of a well-prepared teacher of high school geometry. This book has been prepared primarily for students who have completed a course in calculus. However, although some mathe matical maturity will be found helpful, much of it can be read profitably and with understanding by one who has completed a secondary school course in Euclidean Geometry. He need only omit Chapters V and VI, which make use of trigonometry and calcu lus, and the latter part of Chapter VII. In Chapters II and III, the historical background of the subject has been treated quite fully. It has been said that no subject, when separated from its history, loses more than mathematics. This is particularly true of Non-Euclidean Geometry... ... Read more

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This is a reissue of Professor Coxeter's classic text on non-Euclidean geometry. It begins with a historical introductory chapter, and then devotes three chapters to surveying real projective geometry, and three to elliptic geometry. After this the Euclidean and hyperbolic geometries are built up axiomatically as special cases of a more general 'descriptive geometry'. This is essential reading for anybody with an interest in geometry. ... Read more

Customer Reviews (1)

The beauty of geometry is captured
Originally published in 1942, this book has lost none of its power in the last half century.It is a commentary on the recent demise of geometry in many curricula that 33 yearselapsed between the publication of the fifthand sixth editions. Fortunately, like so manythings in the world, trendsin mathematics are cyclic, and one can hope that the geometriccycle is onthe rise. We in mathematics owe so much to geometry. It is generally conceded that much of the origins of mathematics is due to the simplenecessity ofmaintaining accurate plots in settlements. The only book fromthe ancient history of mathematics that all mathematicians have heard of isthe Elements by Euclid. It is one of the most read books of all time,arguably the only book without a religious theme still inwidespread useover 2000 years after the publication of the first edition. The geometry taught in high schools today is with only minor modifications found in theEuclidean classic.
There are other reasons why geometry should occupya special place in our hearts. Most of the principles ofthe axiomaticmethod, the concept of the theorem and many of the techniques used inproofs were born and nurtured in the cradle of geometry. For manycenturies, it was nearly anact of faith that all of geometry wasEuclidean. That annoying fifth postulate seemed so out ofplace and yet itcould not be made to go away. Many tried to remove it, but finally theHolmseandictum of ,"once you have eliminated the impossible, what isleft, not matter how improbable,must be true", had to be admitted. Therewere in fact three geometries, all of which are of equalvalidity. Theother two, elliptic and hyperbolic, are the main topics of this wonderfulbook.
Coxeter is arguably the best geometer of this century but therecan be no argument that he is thebest explainer of geometry of thiscentury. While fifty years is a mere spasm compared to thetime sinceEuclid, it is certainly possible that students will be reading Coxeter farinto the futurewith the same appreciation that we have when wereadEuclid. His explanations of thenon-Euclidean geometries is so clear thatone cannot help but absorb the essentials. In so manyways, Euclideangeometry is but the middle way between the two other geometries. A pointwellmade and in great detail by Coxeter.
Geometry is a jewel thatwas born on the banks of the Nile river and we should treasure and respectit as the seed from which so much of our basic reasoning processessprouted. For thisreason, you should buy this book and keep a copy onyour shelf.

Published in Smarandache Notions Journal, reprintedwith permission. ... Read more

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The geometry of the hyperbolic plane has been an active and fascinating field of mathematical inquiry for most of the past two centuries. This second edition of Hyperbolic Geometry has been thoroughly rewritten and updated. Chapter 4 focuses on planar models of hyperbolic plane that arise from complex analysis and looks at the connections between planar hyperbolic geometry and complex analysis.

Customer Reviews (3)

Excellent book
This is an excellent introduction to hyperbolic geometry. It assumes knowledge of euclidean geometry, trigonometry, basic complex analysis, basic abstract algebra, and basic point set topology. That material is very well presented, and the exercises shed more light on what is being discussed. Plus, solutions to all the exercises are at the end of the book.

Very good introduction
I used this text along with Tristan Needham's "Visual Complex Analysis" to get a full dose of the geometric beauty inherent in studying complex variables. I found it to be a nice complement to the second year course in geometry at Cambridge University. Anderson does a wonderful job of working out in detail lots of examples so that you can get the algorithmic practice of solving problems. However this is not merely a cookbook. Rather, core elements of the theory are presented from the ground up, with plenty of time spent on understanding the group structure of Mobius transformations in various settings. Disc and upper-half plane models are treated as well as more general models. I recommend you buy both this book and Needham's if you want to appreciate the world of complex numbers.

great book
this is a really great introduction to hyperbolic geometry.especially if you want to study gammas acting on the upper half plane.it starts at a much lower level then any other text. ... Read more

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Euclidean plane geometry is one of the oldest and most beautiful topics in mathematics. Instead of carefully building geometries from axiom sets, this book uses a wealth of methods to solve problems in Euclidean geometry. Many of these methods arose where existing techniques proved inadequate. In several cases, the new ideas used in solving specific problems later developed into independent areas of mathematics. This book is primarily a geometry textbook, but studying geometry in this way will also develop students' appreciation of the subject and of mathematics as a whole. For instance, despite the fact that the analytic method has been part of mathematics for four centuries, it is rarely a tool a student considers using when faced with a geometry problem. Methods for Euclidean Geometry explores the application of a broad range of mathematical topics to the solution of Euclidean problems. ... Read more

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Stahl's Second Edition continues to provide students with the elementary and constructive development of modern geometry that brings them closer to current geometric research. At the same time, repeated use is made of high school geometry, algebra, trigonometry, and calculus, thus reinforcing the students' understanding of these disciplines as well as enhancing their perception of mathematics as a unified endeavor. This distinct approach makes these advanced geometry principles accessible to undergraduates and graduates alike. ... Read more

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The focus of this book is geometric properties of general sets and measures in Euclidean spaces. Applications of this theory include fractal-type objects, such as strange attractors for dynamical systems, and those fractals used as models in the sciences. The author provides a firm and unified foundation for the subject and develops all the main tools used in its study, such as covering theorems, Hausdorff measures and their relations to Riesz capacities and Fourier transforms. The last third of the book is devoted to the Besicovitch-Federer theory of rectifiable sets, which form in a sense the largest class of subsets of Euclidean space possessing many of the properties of smooth surfaces. ... Read more

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Ideal for mathematics majors and prospective secondary school teachers, Euclidean and Transformational Geometry provides a complete and solid presentation of Euclidean geometry with an emphasis on how to solve challenging problems.The author examines various strategies and heuristics for approaching proofs and discusses the process students should follow to determine how to proceed from one step to the next, through numerous problem solving techniques.A large collection of problems, varying in level of difficulty, are integrated throughout the text, and suggested hints for the more challenging problems appear in the instructor's solutions manual for use at instructor's discretion. ... Read more

Customer Reviews (4)

Euclidean and Transformational Geometry
"Euclidean and Transformational Geometry: A Deductive Inquiry" is a text that every mathematics teacher should have a copy of.Not only does it provide a comprehensive coverage of geometry, but it reads like a historical journey through the development of mathematics. The order of the topics is so logical that the reader cannot help but leap from theorem to theorem and take wonderful vacations along the way into islands of interesting problems.

This book's major strength is its clever combination of challenge, clarity, and instruction to teach ideas. The concepts are so clearly presented that students can easily learn them and the skillfully done illustrations augment the book's clarity. Each step of instruction is included and labeled so that the student will not miss some crucial step in their thinking. Multiple paths to the solution of a problem are presented so that the student learns alternative ways of thinking about that concept. This variation and the easy to understand style of writing makes this book interesting and an intellectually stimulating read.

As a high school teacher I find this book to be the richest single resource I have. Most of my students are directed to its well-worn pages multiple times throughout their time with me. I send them to read further about something that interests them, or to read an alternate explanation of a concept that they are struggling with, or simply to find some interesting and challenging problems to work on at home. I believe it is my duty to provide each student with problems that challenge them and are attainable at their level. Dr. Libeskind's book provides me with enough material for all of them, even the brightest of my students, from basic Algebra to Calculus. The problems are not necessarily hard, though some are, but more importantly they're interesting. Many students come early to my class or stay after simply to talk with me, or each other about problems they are working on from this book.

I highly recommend "Euclidean and Transformational Geometry" to all math instructors at the middle school, high school and college levels. Not only has reading it and doing the problems myself greatly enhanced my own understanding of geometry, it has made the subject become beautifully alive for me and the students I share it with. It is a book I cannot imagine being without.

Memorable Geometry
Fifteen years ago I was fortunate to be a student in Professor Libeskind's geometry course at the University of Oregon.The problems that he presented to us, and the way that he emphasized multiple approaches and deep understandings, helped shape my career as a high school mathematics teacher more than any other singular experience.I have now been teaching for thirteen years, and geometry is one of my favorite courses to teach.

My experience in Professor Libeskind's class was unforgettable.I wanted to share some of the problems with my honors students, and was overjoyed to discover that the wonderful collection of problems from that course have now been published as a book!I have yet to find a person who has opened this book and is not immediately interested in finding a solution to the famous Treasure Island Problem.The presentation, clarified through ample diagrams, immediately draws one into the world of exploring.Professor Libeskind writes in a style that invites students of all levels, encourages success, and provides support and assistance to those who need it.He patiently allows the student to make the connections first, but his clear explanations that reveal multiple connections between topics ensure that the student will fully understand the depth of the content.

The development of proof is one of the greatest strengths of this book.High school geometry texts typically begin with a list of theorems and then expect students to construct proofs.As a result, many students look at a proof and have no idea how to proceed.In this book, Professor Libeskind guides students through the process of constructing a proof as an extension of an investigation.I strive to teach my students proof through a similar process, and I am eagerly anticipating Professor Libeskind's high school geometry text.

The wide range of topics introduced and the connections developed between them help to foster a passion for mathematics.One of my favorite aspects of the book is the experience of using Geometer's Sketchpad to investigate problems that formerly led me to use reams of paper.Although the same exercises can be completed without Geometer's Sketchpad, it definitely adds to the experience and helps to keep students motivated.I have not found another geometry book that covers as much material and involves the student in learning as much as this one.

Fantastic Resource
As a middle school math teacher, I am constantly struggling to find textbooks that I can use with my advanced students.There is a dearth of high-quality math textbooks aimed at advanced middle school and high school students, and I usually have to create my own lesson plans using bits and pieces from a number of sources.Though Dr. Libeskind's Euclidean and Transformational Geometry is intended to be a college-level text, I have found it a gold mine for for my advanced middle school classes.The format of the text very closely matches the way I present concepts to my students: each section begins with an introduction to new concepts and vocabulary, followed by simple diagrams and illustrations, and then a theorem and its proof.The exercises are designed to tie back into the main text discussion, and each section builds upon concepts that earlier sections have introduced.

Textbooks that I have worked with in the past have rarely given the same level of focus to writing and understanding proofs that this one does.My experience suggests that while many students find thinking about theorems in terms of proofs foreign at first, they quickly acclimate to the process if proofs are readily available for every property that they encounter.Dr. Libeskind's book is fantastic in this respect because he provides proofs for every mathematical relationship that the text proposes.The book is written to actively encourage students to get into the proof mindset so that they can deconstruct problems, and it is the only textbook that I have come across that does this effectively.

I am also impressed with Dr. Libeskind's ability to collect the diverse topics that his book covers and arrange them in a simple and logical way.This book is straightforward and to the point; no time is wasted on extraneous diagrams, pictures, or problems.The content is clear and concise, but unlike many other textbooks it also encourages the reader to think through problems himself rather than simply providing statements.Even the page layout, which separates ancillary discussions and exercises from the main text using color-coded side-bars and subsections, has been designed with simplicity and ease-of-use in mind.

Though my school district mandates the use of particular textbooks for middle school and high school math, I would like to have several copies of this book in my classroom for students to use as a resource.I have found it vastly superior to many other textbooks that cover the same subject, and though it is aimed at college-level courses, it has been very useful in my lesson planning.

Review from Karen
I highly recommend this book by Professor Libeskind, because it has many outstanding features which make it is vastly superior to the typical college geometry textbook.This geometry textbook is a unique and extraordinary resource for students.It is an elegant book, beautifully written and illustrated, that will introduce students to the process of mathematics - that is to say, to explore interesting problems, discover for themselves possible solutions, and to verify a solution through the process of writing a proof.This book will be treasured by students long after they have completed their coursework and will surely be a continuing resource for students who enter the teaching profession.As a high school math teacher, I can also recommend this book for advanced high school geometry students.

With its beautiful illustrations, uncluttered diagrams, and clear writing style, this book will pique interest by offering students intriguing problems to consider.The book incorporates several features that will develop student appreciation for mathematics, including historical notes about mathematicians that give thoughtful glimpses into the personal lives of those who have contributed to the development of mathematics.The multicultural nature of the discipline of mathematics is clearly described in these notes, and in reading these notes, students will gain a deep respect for the contributions other times and cultures to present-day mathematics.

The book contains a wide range of problems designed to challenge students at every level of understanding.The author's clear belief is that all college students can engage in mathematics at a meaningful level, even beginning students.This textbook is written to develop an in-depth understanding of geometry, and also contains material that will challenge advanced students.Traditional geometric constructions with compass and straight-edge are approached as the outcome of exploration and discovery, rather than as mere techniques.Computer geometry software activities are also included in the text (i.e., Geometer Sketchpad).Sections on recursive formulas for evaluating ð, trigonometric functions, isometries, extremal problems, and complex numbers provide options for providing more complex material for advanced students.
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Originally published in 1909.This volume from the Cornell University Library's print collections was scanned on an APT BookScan and converted to JPG 2000 format by Kirtas Technologies.All titles scanned cover to cover and pages may include marks notations and other marginalia present in the original volume. ... Read more

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This introduction to the geometry of lines and conics in the Euclidean plane is example-based and self-contained, assuming only a basic grounding in linear algebra. Including numerous illustrations and several hundred worked examples and exercises, the book is ideal for use as a course text for undergraduates in mathematics, or for postgraduates in the engineering and physical sciences. ... Read more

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Geometry of lines and planes in the Euclidean plane
The content of this book is not what I expected from the title. My thoughts were that it would be a book of traditional geometry, based on the Euclidean set of axioms. Instead, the book covers the geometry of lines and conics in the Euclidean plane.
It begins with the representation of points and lines as vectors and how length and distance are computed in the Euclidean plane. From this, the equations of the three standard categories of conics, as well as all of the associated figures such as the asymptotes are examined. Understanding the material requires knowledge of the basics of linear algebra, in particular how to work with matrices and determinants.
The presentation is well done, based on a large number of worked examples and many figures. If your interest is in learning the formulaic representations of conics in 2-space, then this book is right for you. However, I do consider the title misleading, the book is not about geometry as we usually consider it in the Euclidean sense. It deals with an application of geometry as applied to a specific class of figures and equations.

Published in Journal of Recreational Mathematics, reprinted with permission.
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Examines various attempts to prove Euclid’s parallel postulate—by the Greeks, Arabs and Renaissance mathematicians. Ranging through the 17th, 18th, and 19th centuries, it considers forerunners and founders such as Saccheri, Lambert, Legendre, W. Bolyai, Gauss, Schweikart, Taurinus, J. Bolyai and Lobachewsky.
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A very old classic
In Einstein's day this might have been a very good read! It is very well written. It is like reading a Spanish concurrent to Russell. A little reading finds it is a translation of a 1912 text. With general Relativity being a product of the understanding of the velocity based non Euclidean geometry of Lorentz who based his work on Poincare who based his work on Klein who based his work on... you see that history is important in an axiomaticdevelopment like this has been! But for a modern student of geometry, this book is much like buying a copy of Euclid's book on geometry: a reference that might help with understanding, but is so far out of date that it can be very little help in current problem! ... Read more

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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. Halsted, both reproduced from copies in the Burndy Library at MIT. ... Read more

Customer Reviews (1)

Readable history, difficult paper
I find Bolyai's paper quite hopeless to read; it's a strange choice for semi-popular publication. Gray's introduction is very pleasant and interesting and full of historical background, but his commentary on Bolyai's actual paper is quite short and not always clear. He does comment extensively on Bolyai's squaring of the circle but this construction is too complicated to be very enjoyable. This result, and Bolyai's entire approach, depends on hyperbolic trigonometric formulae. He saw that such formulae should exist by finding a correspondence between a hyperbolic plane and a surface in hyperbolic space whose geometry is Euclidean (F-surface, horosphere). Today we may interpret this in terms of the half-space model. As our hyperbolic plane we can take a hemisphere centred at the origin and as the horosphere we can take a plane z=c. Lines on the hemisphere are of course intersections with planes perpendicular to the x-y-plane, and lines on the horosphere are Euclidean lines. Under vertical projection of one onto the other lines go to lines and angles are preserved. So a Euclidean right-angle triangle with one vertex at the z-axis correspond to a hyperbolic right-angle triangle with one vertex at the z-axis. And by rotating about the z-axis we see that the ratio of circumferences of the circles generated by the other two vertices is the same on both surfaces. This relates side lengths and thus gives a way of transferring Euclidean trigonometry to the hyperbolic plane. But in hyperbolic geometry the circumference does not grow linearly with the radius (but rather as the hyperbolic sine of the radius, as Bolyai shows later using the angle of parallelism formula), so Euclidean trigonometry does not transfer literally. ... Read more

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Develops a simple non-Euclidean geometry and explores some of its practical applications through graphs, research problems, and exercises. Includes selected answers. ... Read more

Customer Reviews (5)

Excellent for high school teachers and students
I use the ideas in this book in my mathematics teaching in high school. Students learn to think of the world as Euclidean through most of their instruction; Taxicab Geoemetry gives teachers a very straghtforward way to introduce non-Eucliean Geometry. Admittedly, this book is not thorough, and it is very open ended (which I consider to be positive). Nevertheless, for its intended audience it is outstanding.

Disappointing
Very simplistic treatment, with some results left for the reader to work through exercises.The chapters are almost non-existent, with all the book being mainly exercises.

Excellent for what it is
Before purchasing this book, realize what it is.This is a book about non-euclidean geometry.Specifically, a specialized form of non-euclidian geometry affectionately referred to as taxi-cab geometry.This is not atable top book, but is a book for mathemeticians and those interested inmathematics.Others need not apply (regardless of how interesting thetopic is).This is an excellent introduction to non-euclidean geometrybecause it strips away common misconceptions about the nature ofnon-euclidean geometries.This text is excellent for grade school childrenand those who would like to branch into more advanced non-euclideangeometries like hyperbolic.

This is a book for a math student only.
I thought that this book would be about geometry of exotic (but real)places in outer space (such as a black hole, for example).Instead, itturned out to be a lethally boring book, full of math problems, that wasLESS interesting than my geometry book in high school!

A simple, real-world example of non-Euclidean geometry
Some years ago, I was employed by a company that built mapping software. One of the projects I worked on was the design of features that allowed for the computation of the shortest path from one position to another followingonly roads. This form of travel is similar to the taxicab geometry in thatmovement is restricted to lines. The only difference is that roads can beplaced at any location or angle whereas the lines in taxicab geometry areequidistant and perpendicular. Think of it as the geometry of graph paper.As I constructed the program, I was struck by how so much of classicalEuclidean geometry does not apply. Yet, the geometry is generally easier tounderstand because it is almost always how we move from place to place.
The phrase non-Euclidean geometry generally conjures up thoughts ofcurved space and Riemannian geometry. However, in this delightfully simplebook, a natural non-Euclidean geometry is developed in great detail. Verylittle mathematics is needed to understand the geometry, if you can markand understand the points on a grid of graph paper, nearly all of thetopics will make sense. A large number of problems are included at the endof each chapter and solutions to many appear in an appendix.
Theproblems cover topics such as finding the point(s) of minimum distancebetween two or more points and what the taxicab analogues of circles andellipses are. Determining the point of minimum distance between three ormore points is a hard problem in standard geometry, but fairly simple inthe taxicab geometry.
Geometry is the godfather of abstractmathematics, being first used to codify the parceling of land and theconstruction of cities. In this book, you will learn how to minimizefunctions based on the restrictions of traveling through cities, a taskwith many applications in the world. ... Read more

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Customer Reviews (1)

A Classic on Euclidean geometry
Recently Dover has reissued two classics on Euclidean geometry, College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle (Dover Books on Mathematics) and this book. Both books were originally issued in the first half of the 20th century and both were aimed at a college level audience. Both of them also have a considerable amount of so called triangle geometry. As triangle geometry has seen a large upsurge the last years, especially during the last two decennia,there is certainly a need for an English book that gives an overview of the subject including the recent results. These books are useful in this respect but as they are both from the first half of the 20th century, they are out of date. Until a modern treatment of the subject will be available, these two books and the resources on the www will have to do. Altshiller Courts' book has a great set of exercises that can be used as a training ground for geometric problem solving. The problems in Johnsons' book mostly ask for proofs of theorems that are ommited in the text (that's why I give 4 stars). Another drawback of Johnsons' book is that there is no attention paid to geometric constructions. If you are interested in the subject, buy both, its certainly value for money.

The book assumes that you are familiar with simple geometrical concepts like congruence of triangles, parallelograms, circles and the most elementary theorems and constructions as can be found in Kiselev's book Kiselev's Geometry / Book I. Planimetry.

The table of contents:

I Introduction
Prerequisites
Points at infinity
Notation
Directed angles

II Similar Figures
Homotheticfigures
Centers of similitude of two circles
Similar figures in general

III Coaxal circles and inversions
The radical axis
Coaxal circles
Inversions

IV Triangles and Polygons
Ratios in the triangle
Quadrangles and quadrilaterals
The theorem of Ptolemy
Triangle and quadrangle theorems
Polygon theorems and exercises
Theorems concerning areas

V Geometry of Circles
The power theorem of Casey
Circles of antisimilitude
Poles and polars
Stereographic projection

VI Tangent Circles
Circles tangent to two circles
Steiner chains; the arbelos
The problem of Apollonius
Four circles touching a circle

VII The theorem of Miquel
The Miquel theorem
Pedal triangles and circles; Simson lines

VII Theorems of Ceva and Menelaos
Theorems of Ceva and Menelaos; applications
Isogonal conjugates

IX Three Notable Points
Fundamental properties of orthocenter and circumcenter
The orthocentric system
Properties of the median point
The polar circle

X Inscribed and Escribed Circles
Fundamental properties
Algebraic formulas; principle of transformation

XI The nine point circle
Properties of the nine point circle
The theorem of Feuerbach
Further properties of Simson lines

XII Symmedian Point and Other Notable Points
Symmedians and the symmedian point
The isogonic centres
Nagel point, Spieker circle, Fuhrmann circle

XIII Triangles in Perspective
The theorem of Desargues
The theorems of Pascal and Brianchon

XIV Pedal Triangles and Circles
Pedal triangles and circles of a quadrangle
Fontené's theorems; the theorem of Feuerbach
The orthopole

XV Shorter Topics
Statical theorems: center of gravity, resultant of vectors
The cyclic quadrangle and its orthocenters
The theorem of Morley
Circles of Droz-Farny
Miscellaneous exercises

XVI The Brocard Configuration
The Brocard points and their properties
The Tucker circles
The Brocard triangles and the Brocard circle
Steiner point and Tarry point
Related triangles

XVII Equibrocardal Triangles
The Neuberg circles
Vertical projection of triangles
Circles of Appolonius and isodynamic points
The circles of Schoute
Generalizations of Brocard geometry

XVIII Three Similar Figures
Similar figures on the sides of a triangle
Three similar figures in general

Index ... Read more

Product Description
Exposition of fourth dimension, concepts of relativity as Flatland characters continue adventures. Popular, easily followed yet accurate, profound. Topics include curved space time as a higher dimension, special relativity and shape of space-time. Accessible to layman but also of interest to specialist. 141 illustrations.
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Customer Reviews (22)

geometry, relativity and the fourth dimemsion
too abstract. Didn't touch on relativity until the 4th chapter and had trouble following the book til then.

Well Written Introduction to the Fourth Dimension
A great mathematical read! Fascinating diagrams. Begins with accessible concepts for all who love geometry. Gets into spacetime later in the book. Considers some philosophical/spiritual elements too, but mostly geared toward math and physics. A classic read. Highly recommend!

Good intro to related topics to Special Relativity
I found this work to be quite readable and something I can introduce to people with less math background.However, reading the book raises more questions than answers in my mind regarding the history of mathematics.For example, non-Euclidean geometry has been around for a long time and is the basis for ancient systems of navigation.

Similarly, for those who have studied the history of astrology (and its initimate relationship in the ancient world to navigation and agriculture), a great number of things (for example, the divisions of the houses) are all based on spherical geometry and many go back nearly two thousand years.For anyone who has ever known that the earth was a sphere, many of these problems were largely taken for granted.The only real problem with disproving Euclid's 5th postulate has been defining parallel lines on a sphere.I am not entirely sure that Rucker answers this in looking at the flattened sphere because the sphere could be rotated to make any two lines parallel.

Otherwise, I think this is a decent beginner book relating to the subjects in question.It is a useful work and I would generally highly recommend it as an introduction.

Instructive, and interesting
I found the book to be both educational, in that I learned great deal about geomtery and the history of diemsions from this book, as well as being fun to read. Both interesting and intellectually stimulating--I find this combination rare. I recommend ths book to anyone interested in the field.

With few exceptions, it is a readable, stepwise explanation of how the universe is structured
To understand relativity, it is necessary to understand geometry, specifically how a straight line can be curved. For nearly everyone, any attempt to understand four-dimensional space begins with understanding how a three-dimensional creature would appear to a two-dimensional one. One of the earliest and still the greatest of all introductions to going up a dimension is "Flatland" by Edwin A. Abbott. Quite naturally and sensibly, Rucker starts with Abbott's rendition of the properties of Flatland.
Rucker then moves on to the idea of curved space, where the shortest distance between two points is a "straight line", which is curved by the properties of the space. The space that we occupy is curved by the presence of matter, as Einstein claimed in his relativity theories. Furthermore, movement causes shrinkage in the direction of the movement and the slowing of time, which causes time to become just another dimension of space. As counterintuitive as this may appear, Einstein's relativity theory has been verified over and over again to a large number of significant figures.
One of the best things about this book is that Rucker has included problems at the end of each chapter. These problems reinforce the concepts of the chapter; it is unfortunate that no solutions were included.
In this book, Rucker steps the reader through all of the background material necessary to understand relativity and four-dimensional space. With few exceptions, the accounts are understandable to anyone with an understanding of college algebra.
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