- Taschenbuch: 464 Seiten
- Verlag: Dover Publications Inc.; Auflage: Revised ed. (1. Dezember 1988)
- Sprache: Englisch
- ISBN-10: 0486658120
- ISBN-13: 978-0486658124
- Größe und/oder Gewicht: 16,6 x 2,2 x 23,4 cm
- Durchschnittliche Kundenbewertung: 1 Kundenrezension
- Amazon Bestseller-Rang: Nr. 155.300 in Fremdsprachige Bücher (Siehe Top 100 in Fremdsprachige Bücher)
Geometry: A Comprehensive Course (Dover Books on Mathematics) (Englisch) Taschenbuch – 1. Dezember 1988
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Throughout the text the author emphasizes the use of basic algebraic techniques as an aid to finding clear and simple proofs. In more than one case a result is proved several times, each proof illustrating a different technique. In the first chapter, the utility of the vector approach is highlighted by using vector arithmetic, inner-products and exterior algebra to prove several classic theorems of plane geometry. In later chapters elementary group theory, Moebius transformations and linear algebra are used extensively in the discussions of the mappings of the Euclidean plane, of the mappings of the inversive plane, and of projective geometry respectively.
Basically, this is a good, detailed undergraduate introduction to geometry. It's perhaps a little less entertaining than Coxeter's introductory geometry books but it has a much friendlier price-tag.
There are some good written introductory books. It depends on your goals. Books like Foundations of Geometry (Dover Books on Mathematics) (has no solutions to the exercises that I know) by Wylie. The more I read this book, the more and more I am coming to like Geometry and appreciate the subject. Another nice first book to study alongside Wylie's is Euclidean Geometry and Transformations (Dover Books on Mathematics) by Dodge which has hints and answers, but seems more advanced than Wylie. Wylie's is an axiomatic approach, and I'm liking Wylie's book much more than Dodge's, but Dodge seems more of an analytic and algebraic approach to Euclid. Wylie's other book Introduction to Projective Geometry (Dover Books on Mathematics) (has answers in back to odd exercises I believe) is analytic and algebraic and is not like the "Foundations" book in writing and clarity. It is an advanced text and you will need some mathematical maturity and experience I think to study it even though it has "Introduction" in the Title. It's nice to "shift gears" and go back and forth between the two approaches, axiomatic and analytic/algebraic, since they are complementary. Also Modern Geometries by Smart or Geometry by Brannan. Both these books are pretty good but more expensive. You can get used ones, even the Dover books, for a good price or an older edition. Afterward, try Geometry Revisited (Mathematical Association of America Textbooks) (has hints and answers), or Pedoe's as a second book. The books from Dover seem inexpensive whereas Smart's and Brannan's books are $50.00 to $200.00 new in price, that is one reason I like Dover books and they are well known scholarly books. However, some are better written and pedagogically friendlier than others. You can preview most of them on Google or at Dover to see if you like the Author's style and writing/communication skills. Pode's at my first encounter, seemed lacking in clear communication, especially the proofs. Wylie's proofs are more detailed and clear, but you will still need to give your rapt attention to follow the proofs and use pencil and paper at times. Many of the exercises are writing proofs of the Theorems in the text, which could explain why there are no answers. Brannan's book has worked-out solutions to most of the Problems in the chapters but none for the end of chapter exercises. Smart's has answers too. One thing I do like about Pedoe's book is that it has an introductory chapter to algebraic geometry. Perhaps this book would be good for some as a second or third book.
Coxeter's books Introduction to Geometry (Wiley Classics Library) and Geometry Revisited (Mathematical Association of America Textbooks) are very difficult in many ways and hard to comprehend at times. Being a Russian textbook, its level of "Introductory" and "Elementary" is not the same as in the USA or some other countries. However, "Introduction" is a classic Geometry book as is the "Revisited" book for advanced study. I would start with the "Revisited" book before his "Intro" book. They are advanced books in my view and may be a good second or third book. "Revisited" may be good for a Secondary Mathematics Teacher looking for an advanced treatment of their level of Geometry. Yet the books cited above do just as well and are more student friendly. Coxeter's books are written for those who plan to go on in advanced study of mathematics or competitions, especially the "Revisited" book which seems to be all theory and proof, quite rigorous though it claims to be elementary. So it is limited in its scope in my view, with much "holding of the breath" needed and time to work out the extremely challenging exercises at times. Not something really for someone who wants to be practical, concrete, and remain at an advanced secondary school level.
Of course, starting from the very beginning, the study of Euclid's Elements is good because the books cited above assume the reader knows Euclid's Elements. There are some good books on this as well. Dover has all thirteen books in three volumes. But there are some recent books available that are acclaimed: Euclid's Elements of Geometry has helpful notes and is a bilingual Greek/English text. Euclid's Elements which is a reasonably priced book and highly acclaimed. Wylie's "Foundations" book cited above, is an axiomatic treatment of Euclid and non-Euclidean geometry, a nice first book. Hilbert's Foundations of Geometry book is a classic, but I am not sure if it would be a good first book.
Do some shopping: there are many geometry books, even advanced ones to sample. Pick one that fits your needs and goals that you can learn from and that is not just pompous.
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."