Customer Reviews (3)

Extraordinary Scholarship
This book conclusively reveals that fourth dimension theories and related spiritual quests constitute, at the very least, a key pillar in the development of modern art, if not its foundation. The book gives the lie to strictly formalist interpretations of the historic impetus for modern art. Perhaps more importantly, it highlights that vast amounts of art history omits the intended metaphysical content of early modern painting. If the record were set straight sooner, one wonders whether the trajectory of art would have led to contemporary art that is more substantive and enriching than it so often is. Artists such as Brice Marden, Agnes Martin, Astrid Colomar, Anne Truitt and others whose work is infused with spiritual depth would be seen as those whose work is most linked to, rather than most divorced from, the artistic heritage of modernism. With even out-of-print paperback copies of this ground-breaking book selling for well over $100, a new printing would be as welcomed as it is deserved.

thought provoking
One should not practice so-called fine arts without reading this book!!! One should be able to fill the b l a n k in his/her understanding of the contemporary relationship between arts & science! Worthy of yourvaluable time!

excellent
i highly recommend this book to anyone interested in turn of the century art and/or concepts of higher space.very insightful on a subject once widespread and now obscure, i.e. the fourth dimension. ... Read more

Product Description
In his classic work of geometry, Euclid focused on the properties of flat surfaces. In the age of exploration, mapmakers such as Mercator had to concern themselves with the properties of spherical surfaces. The study of curved surfaces, or non-Euclidean geometry, flowered in the late nineteenth century, as mathematicians such as Riemann increasingly questioned Euclid's parallel postulate, and by relaxing this constraint derived a wealth of new results. These seemingly abstract properties found immediate application in physics upon Einstein's introduction of the general theory of relativity.

In this book, Eisenhart succinctly surveys the key concepts of Riemannian geometry, addressing mathematicians and theoretical physicists alike. ... Read more

Customer Reviews (1)

The best classical-style exposition of Riemannian Geometry.
I bought the Russian translation of this book in 1954 and found that this is the best source of the Riemannian geometry, not only for a beginner (as I was at that time), but also for every specialist. Some items fullydiscussed there by L.P. Eisenhart were even rediscovered decades later ---and published another time as new results. This book is, of course, writtenin the old good traditional style, one will not find here, e.g., Cartan'sforms, but it is really an everlasting treasure. Look also for theContinuous Groups of Transformations by the same author. ... Read more

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Lawrence Edwards researched and taught projective geometry for more than 40 years. Here, he presents a clear and artistic understanding of the intriguing qualities of this geometry. Illustrated with over 200 instructive diagrams and exercises, this book will reveal the secrets of space to those who work through them. It is also a valuable resource for Steiner-Waldorf teachers. ... Read more

Customer Reviews (1)

A Different Kind of Math Book
I was first introduced to Projective Geometry by Lawrence Edwards in an upper level undergraduate Projective Geometry course, and I believe that this book is not like any other math text book that I ever had in my undergraduate career.This may sound strange, but I felt as if this book read more like a novel.Its chapters often include some very philosophical (almost poetic) statements, and new concepts are introduced via constructions. I believe you could call this a synthetic approach to the subject. This made learning and re-leaning projective geometry easier. After I finished the course I would occasionally pick the book up the book, grab a pencil and paper, or (better yet) use The Geometer's Sketchpad and continue on with the book.Yet later on, I preferred having a more direct and concise book (such as the one by H.S.M. Coxeter) text to refer back to.
Edwards states in his Foreword that much of the material beyond chapter 13 doesn't appear in this same form anywhere else in English.One of the subjects discussed in chapter 13 and beyond is Von Staudt's theory of the imaginary.I actually did a research project on this topic, and I must agree with Edwards.Many of the projective geometry books that I have read only go far enough to briefly mention that elliptic involutions determine imaginary points.However there are a couple books out there that devote their last chapter to each author's own development of the imaginary. Otherwise, the information comes from scattered articles or foreign language (mostly German) texts.There is one book that I found, J.L.S.Hatton's The Theory of The Imaginary in Geometry: Together with the Trigonometry of the Imaginary that is wholly devoted to the theory of the imaginary.However, Hatton's work is (in my opinion) very dense, and it also includes analytical approaches as well.
Furthermore, if you briefly look over the table of contents you will probably find other topics that are not typically presented in other projective geometry books (such as path curves), but the only topics I really searched for dealt with the imaginary.
I truly learned a great deal from this book, having read it both for school work and for fun. However, like I mentioned above, I found it sometimes useful to have another book on the subject to refer to at times.Also some of the terms that Edwards uses (e.g. "breathing involution") may actually be his own creation (I'm not sure), and this made cross-referencing the book with other projective geometry texts a little harder.Lastly, I should mention that when I had this book in class my teacher pointed out errors in it, but there were only one or two of them.
The Theory Of The Imaginary In Geometry: Together With The Trigonometry Of The Imaginary (1920)Projective GeometryThe Geometer's Sketchpad: Dynamic Geometry Software for Exploring Mathematics ... Read more

Product Description
This book presents, for the first time in English, the papers of Beltrami, Klein, and Poincaré 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 Poincaré 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 Poincaré in their full brilliance. ... Read more

Customer Reviews (2)

This could be the beginning of a beautiful seminar course
Hyperbolic geometry is mathematics at its best: deep classical roots; stunning intrinsic beauty and conceptual simplicity; diverse and profound applications. In this source book we see how three great masters worked to understand this new and exciting geometry.

First, Beltrami's two 1868 papers. The geodesic geometry of surfaces of constant negative curvature such as the pseudosphere capture much of the essence of hyperbolic geometry. However, one does not find the actual hyperbolic plane lying around in three-space. But Beltrami has a way of mapping a surface of constant curvature into the Euclidean plane such that geodesics go to lines. From this point of view the previously intractable step--how to go from a hyperbolic surface to the hyperbolic plane--suggests itself immediately, and we obtain the projective disc model. Now, one way of looking at this construction is to say that it consists of putting a constant-curvature metric on a disc. This point of view is sufficiently abstract to work in n dimensions, as Beltrami shows in his second paper. As a bonus he exploits two other constant-curvature metrics to obtain the other two fundamental models of hyperbolic geometry: the conformal disc model and the half plane model. (Especially for the second paper one is very grateful for Stillwell's introductions.)

Next, Felix Klein. Instead of differential geometry, Klein approches the subject from the point of view of projective geometry. Indeed, Beltrami's projective disc metric begs to be interpreted in terms of projective geometry: the distance between two points in the circle is easily expressed in terms of the cross-ratio of these two points and the two colinear points on the circle. Similarly, projective geometry subsumes spherical and Euclidean geometry as well.

Lastly, there are three little texts by Poincaré, from a third viewpoint: complex function theory. The isometries of Beltrami's half plane model are readily described in terms of linear fractional transformations (in fact, the harmony is even more marked in three dimensions, as Poincaré soon realises). But we can also go "backwards", i.e. we can deduce Beltrami's metric from the isometry group. This proves to be a very rewarding shortcut indeed, since we can employ the built-in geometry of complex function theory.

Learn from those who discovered it
The intellectual power of the human race is never more in evidence than when discoveries are made that are counter to common sense. Examples are quantum mechanics, relativity and non-Euclidean geometry. While the study of such topics is fascinating in the way it forces you to suspend disbelief until the absurd becomes knowledge, almost as interesting is how the topic was discovered. All forms of non-Euclidean geometry were derived from futile attempts to prove Euclid's parallel postulate. The thought processes as the inevitable was conceded and the consequences determined are solid lessons in how scientific and mathematical progress is made.
In this book, we hear from those instrumental in developing the consequences of hyperbolic geometry. The book consists of translations of original papers by E. Beltrami, F. Klein and H. Poincare. In reading them, you are allowed to be there at the creation, learning firsthand how a revolution in mathematics was made. I found the papers to be fascinating, learning many aspects of hyperbolic geometry that I did not know before.
Mathematical progress is commonly measured by nonlinear sticks. The papers of this book not only show you how hyperbolic geometry was developed, but many of the consequences. It is ideal for a short course in non-Euclidean geometry. ... Read more

Customer Reviews (1)

complete but dense
Forders' book is as far as I know the only book in English that is written with the purpose of presenting the foundations of euclidean geometry to be used in the context of studying geometry. Most of the books on foundations of geometry study the relationships between geometric and algebraic objects, for instance the fact that the Papos Pascal theorem in the plane is equivalent to the commutative law in the coordinate field. Another thing studied in those books is the relationships between axiom systems of the different euclidean and non euclidean geometries. The books are certainly not meant to develop geometry.

Forders' book is a real geometry book. The only drawback is that the book is not easy to read. It is a difficult and dense book. Every sentence for instance has its own number and the method of numbering seems to have some meaning. Foundations of euclidean geometry must be a life threatening problem for you if you were to decide on using this book. But lets' not be to negative, after two chapters one gets used to the mannerism (I even started to like it). There are some nice rewards: a description of the concept of the cross, a generalisation of the angle concept that allows some awkward proofs that use "cases" to be simplified.

There are no problems in the book (hm, maybe the book is its own problem).

If you are not a trained mathematician skip the first chapter, it is not really needed in the next chapters though it contains a proof that the archimedean axiom follows from the continuity axiom.

It should not be difficult to find a copy, for instance on Abebooks. ... Read more

Customer Reviews (1)

geometry review
great book, designed as a general overview, the proofs are very extensive and detailed which is great as a preparatory if you are majoring in mathematics. The solid geometry is scattered and distributed in different sections thats my gripe. I give this 4 stars. ... Read more

Product Description
The purpose of this book is to demonstrate that complex numbers and geometry can be blended together beautifully. This results in easy proofs and natural generalizations of many theorems in plane geometry, such as the Napoleon theorem, the Ptolemy-Euler theorem, the Simson theorem, and the Morley theorem. The book is self-contained - no background in complex numbers is assumed - and can be covered at a leisurely pace in a one-semester course. Many of the chapters can be read independently. Over 100 exercises are included. The book would be suitable as a text for a geometry course, or for a problem solving seminar, or as enrichment for the student who wants to know more. ... Read more

Customer Reviews (1)

New ways to do old things
Proofs in "pure" geometry are easy to understand but difficult to conceive. When presented with the opportunity to do such proofs, many people suffer from a brain cramp similar to that experienced by writers.One of the most common phrases heard when I was teaching is similar to thefollowing, "I can follow the proof once it is done, but how do you think oftrying those steps?" In this book, the author performs a marriage ofcomplex numbers and geometry that can sometimes serve to point the aspiringgeometer in the proper direction.
Contrary to their name, complexnumbers are easy to understand and manipulate. Only basic knowledge ofalgebra is essential. In this case, the author uses all of Chapter One tointroduce the fundamental ideas of their use. After that, things getexciting. Applying this knowledge to geometry, we see new ways to do old,sometimes very old things. In many cases, the approach is general, in thatit is easy to see how such ideas can be used to attack other problems.Large numbers of exercises are included at the end of each chapter.
Worthy of inclusion in any library, this author shows that it is alwayspossible to develop new ways to solve old problems.

Published inJournal of Recreational Mathematics, reprinted with permission. ... Read more

Product Description
This is an OCR edition without illustrations or index. It may have numerous typos or missing text. However, purchasers can download a free scanned copy of the original rare book from GeneralBooksClub.com. You can also preview excerpts from the book there. Purchasers are also entitled to a free trial membership in the General Books Club where they can select from more than a million books without charge. Original Published by: University press in 1897 in 226 pages; Subjects: Geometry; Mathematics / General; Mathematics / Geometry / General; Mathematics / History & Philosophy; Mathematics / Logic; Philosophy / General; Philosophy / Logic; Philosophy / History & Surveys / Modern; ... Read more

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This text for advanced undergraduates emphasizes the logical connections of the subject. The derivations of formulas from the axioms do not make use of models of the hyperbolic plane until the axioms are shown to be categorical; the differential geometry of surfaces is developed far enough to establish its connections to the hyperbolic plane; and the axioms and proofs use the properties of the real number system to avoid the tedium of a completely synthetic approach. The development includes properties of the isometry group of the hyperbolic plane, tilings, and applications to special relativity. Elementary techniques from complex analysis, matrix theory, and group theory are used, and some mathematical sophistication on the part of students is thus required, but a formal course in these topics is not a prerequisite. ... Read more

Product Description
Although it arose from purely theoretical considerations of the underlying axioms of geometry, the work of Einstein and Dirac has demonstrated that hyperbolic geometry is a fundamental aspect of modern physics. In this book, the rich geometry of the hyperbolic plane is studied in detail, leading to the focal point of the book, Poincare's polygon theorem and the relationship between hyperbolic geometries and discrete groups of isometries. Hyperbolic 3-space is also discussed, and the directions that current research in this field is taking are sketched. This will be an excellent introduction to hyperbolic geometry for students new to the subject, and for experts in other fields. ... Read more

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Customer Reviews (5)

An excellent book with just one minor thing missing
It's a beautiful book, interesting and lovely. I enjoy mathematical problems and I enjoy studying Japanese culture. This book satisfies both my interests!

There is just one minor thing that's missing - the Japanese names are written entirely in Romanji. When I first started reading the book I kept wondering what is "sangaku" in Kanji. Most of my Japanese friends have not even heard of it. After much searching, I discovered the kanji for sangaku is ZZŠz, or mathematics tablets. The same romanji, "sangaku", could also have been these two kanji, ZZŠw, study of mathematics.

I hope the next edition of this book would include kanjis or at list an index of the kanji, it makes life easier for students of Japanese.

Bottom line is, this is an excellent book, and anyone interested in Japanese culture would be interested in this book.

Like pirahnas on a hapless animal
I have been using a number of the Sangakus from Sacred Geometry in my High School Pre-Calculus classes to get things rolling at the start of the class. The kids are loving them! Watchingthe kids last class get the problem and go to work on it reminded me of watching piranha's go after a hapless animal-- maybe a bit less graphic. The problems are just great-- they really hook the kids, really get them trying stuff, and they do a fantastic job of building up and connecting their skills. Of course I am having a great time with them too!

Further, the book is just a pleasure to read. Everything about it-- prose, graphics, mathematics, quality of production-- is just top notch.

Excellent book
The last (for the moment) title of Fukagawa&Rothman is really excellent. Not only the printing is superb, but the mathematical content is also outstanding. Strongly recomended to every lover of geometry...

Another Brilliant One
I am always interested in what Tony Rothman has to say. He is the real deal, teaches physics at Princeton, Harvard, etc., who comes up with revolutionary insights you just can't find anywhere else. SACRED MATHEMATICS is a revelation and a tremendous challenge, another brilliant one in this writer's repertoire.

I began my Rothman studies after reading INSTANT PHYSICS, which pretty much brought me up to speed in what had always intrigued yet baffled me. Then I was amazed with his majestic DOUBT AND CERTAINTY followed by the jaw-dropping, myth-busting EVERYTHING'S RELATIVE. I couldn't get enough so I started backtracking and discovered the Pulitzer Prize nominated A PHYSICIST ON MADISON AVENUE and SCIENCE A LA MODE, where he maybe first established his continual theme of treating science with the skeptical irreverence it often deserves. In between, I discovered articles in SCIENTIFIC AMERICAN, DISCOVER, ISAAC ASIMOV'S SCIENCE FICTION MAGAZINE and THE NEW REPUBLIC, not to mention some weighty scientific papers and reports. Finally, I found his science fiction novel, THE WORLD IS ROUND, with which the movie industry might finally have the tools to do justice.

Tony Rothman is a great and gifted writer and SACRED MATHEMATICS is a beautifully illustrated book of art, religion, history and geometry. I see from his web site that a novel about The Great Seige of Malta is next. I anxiously anticipate that and hope that both APOCHRYPHA and the plays there mentioned will soon be published.

I strongly recommend SACRED MATHEMATICS and, in fact, everything written by Tony Rothman to anyone, who in a world too often full of nonsense and lies, cherishes instead reality and truth. Rothman's voice is beautiful and unique.

Beautiful Mathematics
For anyone who truly loves mathematics, this book is a must have.
Simply put, the book tells the story of sangaku, geometry problems which were painted in color on wooden tablets and displayed at Buddhist temples and Shinto shrines throughout Japan. Most of the sangaku were composed by people from all walks of life-priests, farmers, children women, samurai, etc.-between 1600 and 1900. Approximately 900 of the old tablets have survived and even today one is occasionally found at an abandoned temple/shrine. Tony Rothman has assisted Mr. Fukagawa Hidetoshi, a retired Japanese high school teacher, who is one of the world's foremost experts in sangaku, in producing a beautiful book. Various chapters discuss Japan and temple geometry, the Chinese foundation of mathematics, Japanese mathematics and mathematicians of the Edo period. In addition, the book contains over 200 sangaku problems ranging from very elementary to extremely difficult. The book also contains extensive excerpts from the diary of Yamaguchi Kanzan, a Japanese mathematician, who treked through Japan during the 1800s collecting sangaku problems. Finally, there are chapters on East and West, Japanese attempts to handle differentiation and integration, and inversion. The book contains numerous diagrams which accompany the problems and there are 16 color plates. In summary, this book captures a beautiful form of vanished mathematics which was artistic/religious in nature. Mr. Fukagawa Hidetoshi and Mr. Rothman are to be congratulated for producing a superb book which tells the story of this vanished mathematical/religious art form. Buy your copy today. This book contains enough history, mathematics, art, and religion to keep one's intellect perplexed for years. ... Read more

Product Description
This remarkable book has endured as a true masterpiece of mathematical exposition. There are few mathematics books that are still so widely read and continue to have so much to offer--after more than half a century! The book is overflowing with mathematical ideas, which are always explained clearly and elegantly, and above all, with penetrating insight. It is a joy to read, both for beginners and experienced mathematicians.

"Hilbert and Cohn-Vossen" is full of interesting facts, many of which you wish you had known before, or had wondered where they could be found. The book begins with examples of the simplest curves and surfaces, including thread constructions of certain quadrics and other surfaces. The chapter on regular systems of points leads to the crystallographic groups and the regular polyhedra in $\mathbb{R}^3$. In this chapter, they also discuss plane lattices. By considering unit lattices, and throwing in a small amount of number theory when necessary, they effortlessly derive Leibniz's series: $\pi/4 = 1 - 1/3 + 1/5 - 1/7 + - \ldots$. In the section on lattices in three and more dimensions, the authors consider sphere-packing problems, including the famous Kepler problem.

One of the most remarkable chapters is "Projective Configurations". In a short introductory section, Hilbert and Cohn-Vossen give perhaps the most concise and lucid description of why a general geometer would care about projective geometry and why such an ostensibly plain setup is truly rich in structure and ideas. Here, we see regular polyhedra again, from a different perspective. One of the high points of the chapter is the discussion of Schlafli's Double-Six, which leads to the description of the 27 lines on the general smooth cubic surface. As is true throughout the book, the magnificent drawings in this chapter immeasurably help the reader.

A particularly intriguing section in the chapter on differential geometry is Eleven Properties of the Sphere. Which eleven properties of such a ubiquitous mathematical object caught their discerning eye and why? Many mathematicians are familiar with the plaster models of surfaces found in many mathematics departments. The book includes pictures of some of the models that are found in the Göttingen collection. Furthermore, the mysterious lines that mark these surfaces are finally explained!

The chapter on kinematics includes a nice discussion of linkages and the geometry of configurations of points and rods that are connected and, perhaps, constrained in some way. This topic in geometry has become increasingly important in recent times, especially in applications to robotics. This is another example of a simple situation that leads to a rich geometry.

It would be hard to overestimate the continuing influence Hilbert-Cohn-Vossen's book has had on mathematicians of this century. It surely belongs in the "pantheon" of great mathematics books. ... Read more

Customer Reviews (8)

l'imagination au pouvoir
Ce livre est magnifique, on sent le génie de Hilbert dans ce livre. Parfois un peu difficile à suivre, il est bon de complété sa lecture avec d'autres livres.
Il donne une introduction à a peu près toutes les parties de la géométrie et va parfois très loin dans le sujet.
Un plaisir...

A classic on Geometry
A pearl! Anyone interested in Geometry shouldn't miss the lucid presentation of the great Hilbert.

Many beautiful things
This is a marvellous book. I will illustrate by one sample from each chapter (except chapter 1 on "the simplest curves and surfaces" which is the least exciting chapter).

Chapter 2 on "regular system of points" contains a beautiful derivation of Leibnitz' series pi/4=1-1/3+1/5-1/7+... If we draw a large circle centred at the origin then of course a good measure of its area is the number of integer points it contains. Now, for any such point, x^2+y^2 is an integer less than r^2. So the number of such points can be obtained by going through all integers less than r^2 and counting how many times it can be written as the sum of two squares. But this is a classical problem in number theory and the solution is known. So this number theoretic result essentially tells us the area of a large circle, so it implies an expression for pi, namely Leibnitz' series.

Chapter 3 is on projective geometry. We go through many projective configurations that are not seen very often today, but still the classics are the best, such as Desargues' theorem. If we have a triangular pyramid and cut it with two planes to get two triangles then the three points of intersection of the extensions of corresponding sides will or course be on a line (the intersection of the two planes), which is the three-dimensional Desargues' theorem. But by projecting the triangles onto one of the walls of the pyramid we get two projectively related plane triangles and the theorem holds for them also. All we have to do to prove the plane Desargues' theorem is to prove that all such configurations can be obtained in his way (i.e. that one can always erect an appropriate pyramid based on two projectively related plane triangles) which is practically obvious.

Chapter 4 is on differential geometry. The fundamental concept of differential geometry is curvature, which is a number that indicates how curved a surface is at a given point. It may be defined as follows. We draw a little circle around the point on the surface and consider all the normals to the surface at these points. Take these normals and put them with their origin at the center of a sphere; then they will sweep out a section of the surface of the sphere. The curvature is the ratio of the area enclosed on the surface and that on the sphere as the circle is taken infinitesimally small. This quantity is seen to be invariant under bending by triangulating the surface; then the the circles are polygons with fixed angles and the theorem follows from the fact that the area of a spherical triangle is determined by its angles (proof omitted here; see any Stillwell geometry book for Harriot's beautiful proof (a.k.a. "Euler's proof")). Now, there are two fundamentally different types of points. Either the surface bends in the same direction in every direction, as on a sphere, or it bends in different directions like a saddle. In the first case the boundary on the sphere traced out by the normals has the same orientation as the boundary on the surface; in the second case the orientation is reversed. So, using signed area, the second type of points have negative curvature. A typical surface will have areas of positive curvature and areas of negative curvature and in between there will be lines of zero curvature. An absolutely wonderful, although perhaps not entirely successful, application of this concept is Klein's Apollo Belvidere hypothesis that the curves of zero curvature on a human face determine beauty.

Chapter 5 on kinematics contains a determination of the curve that "we may observe ... every day in cups and tin cans when the light shines on them", i.e. the coffee cup caustic. With the sun at x=-infinity, the radius that makes an angle theta with the x-axis will point to a point where the angle of reflection is also theta. Consider a concentric circle of half the radius, and another circle with the other half of the radius as its diameter. The arc cut out of the inner circle by the radius and the x-axis is equal to the arc cut out of the outer circle by the radius and the reflected ray (arc with central angle theta in the big circle = arc with central angle 2*theta in the small cirlce). The shape of the caustic follows by rolling the outer circle on the inner. The reflected light rays are tangent to this curve since they are perpendicular to the line connecting the generating point with the center of motion (intersection of the two circles).

From chapter 6 on topology one nice result is that any continuous mapping of a disc onto itself has a fixed point. For suppose it did not. Then any point in the circle can be connected with its image by an arrow. Now consider the point on the boundary. The arrow direction varies continuously as we walk once around the circle, and it end up where it started so it must have made an integer number of revolutions. But there is also a tangent at each point, and the tangent of course make one revolution as we walk once around. The arrows always point to some point in the disc so they could never point in a direction parallel to the tangent so the arrows in fact have to make one revolution also (they would have to be parallel to the tangent for a moment to overtake it, and if they stood still they would be parallel to the tangent "at six o'clock" so to speak). But if we consider the same situation for a concentric circle inside the disc then it too must have arrows making one revolution because the number of revolutions can not make jumps since the new circle is obtained by continuous shrinking of the circumference circle. But as we shrink this circle to infinitesimal radius then all its arrows point in the same direction, so they don't make one revolution, so we have a contradiction. One sees similarly that a continuous mapping of the sphere onto itself also has a fixed point. Since the projective plane is the sphere with diametrically opposite points identified this proves that any projective transformation has a fixed point.

Don't expect to find it "easy."
I agree that this book, co-authored by the co-greatest mathematician of the first quarter of the twentieth century, is a masterpiece to be treasured and kept in print, as other reviewers have stated.

However: The Preface states: "This book was written to bring about a greater enjoyment of mathematics, by making it easier for the reader to penetrate to the essence of mathematics without having to weight himself down under a laborious course of studies."

All I can say is that if you read this and find it "easy," then you have terrific mathematical talent! Yes, the drawings and the intuitive descriptions are helpful, but much of the book is so obscure that I have been told that one of the world's leading geometers is working on an annotated edition explaining what the authors were talking about. On topics which I had already studied elsewhere, I found the presentation illuminating.

I still recommend this book.

Beautiful, Rewarding, and Deep.
I have some 47 books in the geometry section of my shelves.If I had to discard 40 of these, Geometry and the Imagination would be among the 7 remaining.

Geometry is the study of relationships between shapes, and this book helps you see how shapes fit together.Ultimately, you must make the connections in your mind using your mind's eye.The illustrations and text help you make these connections.This is a book that requires effort and delivers rewards. ... Read more

Product Description
Clear explanations, an uncluttered and appealing layout, and examples and exercises featuring a variety of real-life applications have made this text popular among students year after year. This latest edition of Swokowski and Cole's ALGEBRA AND TRIGONOMETRY WITH ANALYTIC GEOMETRY retains these features. The problems have been consistently praised for being at just the right level for precalculus students like you. The book also provides calculator examples, including specific keystrokes that show you how to use various graphing calculators to solve problems more quickly. Perhaps most important-this book effectively prepares you for further courses in mathematics. ... Read more

Customer Reviews (3)

Excelent!!
Very happy with this seller. Book arrived faster thsn expected! Would with not doubt purchase again from seller

Fast delivery!!
It was in good condition and fast shipping. Even though I bought the wrong book, they understand and gave me a full refund. I would recommend them some one else.

Holy cow, I'm a buyer not a writer!!
This is the second time we ordered the book.The first time the D--- computer or the Amazon.com site kept defaulting to a different zip code--the second four digits.Ii happened so often that I thought it had been altered and went along with it.Obviously the PO ignored everything else in the address and it couldn't be delivered. The Amazon site tried again on the second order to insert that last four digits of the zip-code.It was like telling Hal, "Hal I can't let you do that."After 30 + years I do know my zip code.Amazon get your act in order!! It happened the second time and after numerous attempts I got the %#$@&&* site to recognize the correct last four digits. I guess 1984 didinfiltrateand immerseour society with thought control.

But back to the second attempt.The book was delivered as promised and on time.It was the book we desired, desperately needed, and it was the merchandise as advertised.In all respects the second seller did all that he/she could to be an honest, efficient, and complaint merchant.We were pleased with the transaction!! ... Read more

Product Description
The word barycentric is derived from the Greek word barys (heavy), and refers to center of gravity. Barycentric calculus is a method of treating geometry by considering a point as the center of gravity of certain other points to which weights are ascribed. Hence, in particular, barycentric calculus provides excellent insight into triangle centers. This unique book on barycentric calculus in Euclidean and hyperbolic geometry provides an introduction to the fascinating and beautiful subject of novel triangle centers in hyperbolic geometry along with analogies they share with familiar triangle centers in Euclidean geometry. As such, the book uncovers magnificent unifying notions that Euclidean and hyperbolic triangle centers share.

In his earlier books the author adopted Cartesian coordinates, trigonometry and vector algebra for use in hyperbolic geometry that is fully analogous to the common use of Cartesian coordinates, trigonometry and vector algebra in Euclidean geometry. As a result, powerful tools that are commonly available in Euclidean geometry became available in hyperbolic geometry as well, enabling one to explore hyperbolic geometry in novel ways. In particular, this new book establishes hyperbolic barycentric coordinates that are used to determine various hyperbolic triangle centers just as Euclidean barycentric coordinates are commonly used to determine various Euclidean triangle centers.

The hunt for Euclidean triangle centers is an old tradition in Euclidean geometry, resulting in a repertoire of more than three thousand triangle centers that are known by their barycentric coordinate representations. The aim of this book is to initiate a fully analogous hunt for hyperbolic triangle centers that will broaden the repertoire of hyperbolic triangle centers provided here. ... Read more

Product Description

This book follows Klein’s proposal of studying geometry by looking at the symmetries (or rigid motions) of the space in question. In this way the classical geometries are studied: Euclidean, affine, elliptic, projective and hyperbolic. For simplicity the focus is on the two-dimensional case, which is already rich enough, though some aspects of the 3 or $n$-dimensional geometries are included. Once plane geometry is well understood, it is much easier to go into higher dimensions.

The book appeals to, and develops, the geometric intuition of the reader. Some basic notions of algebra and analysis are also used to get better understandings of various concepts and results.

Customer Reviews (1)

Introduction to Classical Geometries.
This book is translated from Spanish, and therefore cheap.Occasionally you hit a word they forgot to translate, but it does not get in the way of reading.Written pretty tersely, so if you are not comfortable with taking classes open to graduates, you will have a hard time reading this book.On the other hand, if you are, you will find this book very concise, and not overburdened with meaningless examples. ... Read more

Product Description
An important new perspective on AFFINE AND PROJECTIVE GEOMETRY

This innovative book treats math majors and math education students to a fresh look at affine and projective geometry from algebraic, synthetic, and lattice theoretic points of view.

Affine and Projective Geometry comes complete with ninety illustrations, and numerous examples and exercises, covering material for two semesters of upper-level undergraduate mathematics. The first part of the book deals with the correlation between synthetic geometry and linear algebra. In the second part, geometry is used to introduce lattice theory, and the book culminates with the fundamental theorem of projective geometry.

While emphasizing affine geometry and its basis in Euclidean concepts, the book:

• Builds an appreciation of the geometric nature of linear algebra
• Expands students' understanding of abstract algebra with its nontraditional, geometry-driven approach
• Demonstrates how one branch of mathematics can be used to prove theorems in another
• Provides opportunities for further investigation of mathematics by various means, including historical references at the ends of chapters

Throughout, the text explores geometry's correlation to algebra in ways that are meant to foster inquiry and develop mathematical insights whether or not one has a background in algebra. The insight offered is particularly important for prospective secondary teachers who must major in the subject they teach to fulfill the licensing requirements of many states. Affine and Projective Geometry's broad scope and its communicative tone make it an ideal choice for all students and professionals who would like to further their understanding of things mathematical. ... Read more