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Spectrum
Non-Euclidean Geometry
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 text should be of use to anybody with an interest in geometry.
- GenresMathematicsGeometry
353 pages, Paperback
First published May 10, 2014
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About the author
H.S.M. Coxeter
26Ìýbooks14ÌýfollowersHarold Scott MacDonald "Donald" Coxeter (1907-2003), CC, FRS, FRSC was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
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June 7, 2015
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 years elapsed between the publication of the fifth and sixth editions. Fortunately, like so many things in the world, trends in mathematics are cyclic and one can hope that the geometric cycle is on the rise. We in mathematics owe so much to geometry. It is generally conceded that much of the origins of mathematics is due to the simple necessity of maintaining accurate plots in settlements.
The only book from the ancient history of mathematics that all mathematicians have heard of is the “Elements� by Euclid. It is one of the most read books of all time, arguably the only book without a religious theme still in widespread use over 2000 years after the publication of the first edition. The geometry taught in high schools today is with only minor modifications found in the Euclidean classic.
There are other reasons why geometry should occupy a special place in our hearts. Most of the principles of the axiomatic method, the concept of the theorem and many of the techniques used in proofs were born and nurtured in the cradle of geometry. For many centuries, it was nearly an act of faith that all of geometry was Euclidean. That annoying fifth postulate seemed so out of place and yet it could not be made to go away. Many tried to remove it, but finally the Holmsean dictum of “once you have eliminated the impossible, what is left, no matter how improbable, must be true,� had to be admitted. There were in fact three geometries, all of which are of equal validity. The other two, elliptic and hyperbolic, are the main topics of this wonderful book.
Coxeter is arguably the best geometer of the twentieth century but there can be no argument that he is the best explainer of geometry of that century. While fifty years is a mere spasm compared to the time since Euclid, it is certainly possible that students will be reading Coxeter far into the future with the same appreciation that we have when we read Euclid. His explanations of the non-Euclidean geometries is so clear that one cannot help but absorb the essentials. In so many ways, Euclidean geometry is but the middle way between the other two geometries, a point well made and in great detail by Coxeter.
Geometry is a jewel that was born on the banks of the Nile river and we should treasure and respect it as the seed from which so much of our basic reasoning sprouted. For this reason you should buy this book and keep a copy on your shelf.
Published in Smarandache Notions Journal, reprinted with permission and this review also appears on Amazon.
The only book from the ancient history of mathematics that all mathematicians have heard of is the “Elements� by Euclid. It is one of the most read books of all time, arguably the only book without a religious theme still in widespread use over 2000 years after the publication of the first edition. The geometry taught in high schools today is with only minor modifications found in the Euclidean classic.
There are other reasons why geometry should occupy a special place in our hearts. Most of the principles of the axiomatic method, the concept of the theorem and many of the techniques used in proofs were born and nurtured in the cradle of geometry. For many centuries, it was nearly an act of faith that all of geometry was Euclidean. That annoying fifth postulate seemed so out of place and yet it could not be made to go away. Many tried to remove it, but finally the Holmsean dictum of “once you have eliminated the impossible, what is left, no matter how improbable, must be true,� had to be admitted. There were in fact three geometries, all of which are of equal validity. The other two, elliptic and hyperbolic, are the main topics of this wonderful book.
Coxeter is arguably the best geometer of the twentieth century but there can be no argument that he is the best explainer of geometry of that century. While fifty years is a mere spasm compared to the time since Euclid, it is certainly possible that students will be reading Coxeter far into the future with the same appreciation that we have when we read Euclid. His explanations of the non-Euclidean geometries is so clear that one cannot help but absorb the essentials. In so many ways, Euclidean geometry is but the middle way between the other two geometries, a point well made and in great detail by Coxeter.
Geometry is a jewel that was born on the banks of the Nile river and we should treasure and respect it as the seed from which so much of our basic reasoning sprouted. For this reason you should buy this book and keep a copy on your shelf.
Published in Smarandache Notions Journal, reprinted with permission and this review also appears on Amazon.
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