Geometry: A Comprehensive Course (Dover Books on Mathematics) by Dan Pedoe (PDF)

Description

“A lucid and masterly survey.” — Mathematics GazetteProfessor Pedoe is widely known as a fine teacher and a fine geometer. His abilities in both areas are clearly evident in this self-contained, well-written, and lucid introduction to the scope and methods of elementary geometry. It covers the geometry usually included in undergraduate courses in mathematics, except for the theory of convex sets. Based on a course given by the author for several years at the University of Minnesota, the main purpose of the book is to increase geometrical, and therefore mathematical, understanding and to help students enjoy geometry.Among the topics discussed: the use of vectors and their products in work on Desargues’ and Pappus’ theorem and the nine-point circle; circles and coaxal systems; the representation of circles by points in three dimensions; mappings of the Euclidean plane, similitudes, isometries, mappings of the inversive plane, and Moebius transformations; projective geometry of the plane, space, and n dimensions; the projective generation of conics and quadrics; Moebius tetrahedra; the tetrahedral complex; the twisted cubic curve; the cubic surface; oriented circles; and introduction to algebraic geometry.In addition, three appendices deal with Euclidean definitions, postulates, and propositions; the Grassmann-Pluecker coordinates of lines in S3, and the group of circular transformations. Among the outstanding features of this book are its many worked examples and over 500 exercises to test geometrical understanding.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐An interesting book covering multiple topics not usually contained in the same book, this goes quite well logically from Euclidean Geometry to natural generalizations in affine and projective geometry. To finish it off, we have one of the more classical non-euclidean geometries through a discussion (nowhere near complete, though) of hyperbolic geometry. It is an easily accessible and interesting read for a prospective math education major. It is of little interest, however, to any other audience.My biggest issue, though clearly seen if one were to use the table of contents, is that this is not comprehensive at either the college or high school levels. This assumes a STRONG background in both co-ordinate geometry and synthetic geometry (involving proofs). It also assumes familiarity with linear algebra, complex numbers and trigonometry. Moreover, the material moves quickly and leaves some of its developments as exercises as opposed to actually fully developing and discussing the material.The material on hyperbolic geometry is woefully small (does not cover upper half plane model and no hyperbolic trigonometry) and incomplete as to make one wonder why it should be included until one realizes the price is low enough to be happy with so much other material, and the projective and affine geometry mentioned in this book isn’t very general at all. A developed discussion of elliptic and spherical geometry is also missing. While a great introduction, the book doesn’t go anywhere near in depth enough, and even wide enough, to be comprehensive.Another issue I have with this book is that its coverage of Euclidean geometry is rather boring and covers few of the more classical and widely used theorems in math contests. I would recommend Johnson’s Advanced Euclidean Geometry as a follow-up to the high school geometry course instead. As for the college geometry course, I recommend Sossinsky’s Geometries, which requires, as most college geometry textbooks these days requires, a good understanding of ABSTRACT algebra (emphasis as this is not simply abstract in a general sense but a whole field elevated above high school algebra with its own terminology and structures to work with).Pedoe seems to be halfway between going towards a traditional classical algebraic geometry approach and the use of analytic methods, which is usually frowned upon by the “elite” and “purists” as being impure, ugly, and lacking that special quality touted as “mathematical maturity.” This, unfortunately, leads to a bit of a confused text with some ideas left hanging, subtleties left uncovered, and technical details unnecessarily emphasized.The title should be something like “Geometry: An Advanced Course” or “College Geometry.”

⭐An interesting selection of topics in geometry. I loved the use of complex numbers in developing the theory of isometries and similarities in the Euclidean plane. The representation of circles as points in 3-space yields interesting insights into coaxal systems. The development of projective geometry, in n dimensions and then specializing to the projective plane and projective 3-space, with conics in the former and quadric surfaces in the latter, is an interesting blend of the synthetic and the analytic (or “algebraic” as Pedoe calls it). I especially appreciated the discussion of 3-space and quadrics, since so many treatments limit themselves to conics in the plane.This book is not for the novice. In particular, if this is the reader’s first exposure to projective geometry, I think he or she could wind up quite bewildered by it all. A better place to start is Coxeter’s “Projective Geometry”, which is mainly synthetic, and gets the principal results without bouncing between different dimensions, algebraic vs. synthetic, etc.The exercises are excellent! What a wonderful choice of interesting and enlightening results. No dreary “working out the details of the theory” here!

⭐This is a comprehensive book in Geometry, not a fundamentals book. There are two distinguishable differences between a fundamentals-type book vs. a comprehensive-type book, in that a comprehensive book gives a survey of, or a smorgasbord, of topics that are inter-relatable. Fundamentals typically go more in depth than a comprehensive book.If you are looking for a book that assumes you understand some higher topics in mathematics (i.e. college sophomore-level maths), then by all means, you are the assumed audience. This is not meant as a replacement to Euclid’s “The Elements” or the book would have entirely been a lot thicker in size and probably more in cost. This is not a “Master Math: Geometry” type book.I bought this book because of the section on projective geometry, but I also wanted a book to reference other topics as well. I’m using this book as a supplement rather than an actual course.

⭐I passed through Geometry in high school without learning much if anything. My thought process at the time was “I’m not headed to college so why do I need this?” Well, I did go to college years later and lack of this knowledge did impact the me. I was limited in the courses I could take because of a lack of math skills. Now I’m over 40 years older and trying to cover gaps in my skill sets of which math is definitely short coming. To move forward to calculous I definitely need to improve in the areas of algebra and geometry. This book is exceptional in covering the geometry shortfall and I highly recommend it to anyone in need of a geometry start or refresher.

⭐I would rate the material that’s in the book higher than three stars. I’m giving it three stars because of material that is missing. This says “comprehensive” and so I thought it would include at least the basics of Euclid’s Elements. The assumptions and postulates are stated but there’s absolutely nothing relating to the proofs as presented in Euclid’s original nor any insight into the concept of axiomatization. I bought this book hoping to gain a deeper understanding of why Euclid made the axioms he made, you know straight edge and compass but no ruler. And how he went to great lengths to prove things that would be obvious had he introduced the concept of “area”. But there’s basically zilch in this book on that front. So I don’t see how this could be called a “comprehensive” course.

⭐Need an answer key for this book to see if your learning the material!

⭐was reported as used, came brand-new.So far, I love this book.

⭐Excellent book.

⭐Excellent

⭐Everything was fine, thank you!

⭐Great book that covers a faschinating topic very well. Great for those who got the basic skills of high school mathematics and geometry.

⭐WHy buy a book if you are not going to see what it offers? This book offers a view of geometry that some traditionalists might not want to see but few geometers nowadays would want to see themselves bound by the traditions of the past! An excellent book by a master of his trade that deserves to be read by anybody who calls/thinks himself a mathematician.

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