Customer Reviews

Circles: A Mathematical View (Spectrum)

5 star:		(1)
4 star:		(1)
3 star:		(0)
2 star:		(0)
1 star:		(0)

 

 

This is the only book I've seen concentrating solely on the geometry of the circle. So it discusses topics not usually covered in general, lower-level (high school or elementary college) texts: inversion, centers of similitude, coaxal systems, others. (There's no point in telling you what's in this book, since you can "Look Inside" and see for yourself.) This is intermediate in its level of difficulty, and certainly requires a familiarity of more than basic geometry and algebra. There is nothing of the elementary properties and theorems of the circle involving: arcs, radii, diameter, chords, secants, tangents, inscribed geometric figures, etc. This knowledge of all this is all presumed. It seems to be oriented to a person very comfortable with the material being presented using mathematical symbols and terms. This is why I gave it 4 stars, rather than 5.

Four chapters:
Chapter 1 discusses some more esoteric properties of circles (see the "Look Inside!").
Chapter 2 "adopt(s) a completely different point of view from that pursued in the first chapter." (page 26) I couldn't figure out how to summarize it for you.
Chapter 3 discusses the circle's part in "the Poincare model of hyperbolic non-Euclidean geometry." (page 44)
Chapter 4 talks about the circle's property of enclosing the greatest area with a given perimeter.

This text, problems, and solutions of this "new" edition are verbatim the same as the previous 1979 (corrected) edition. Even the typeset and layout are identical. Pedoe (1910-1998) has added three items to this "new" edition.
1) A 27-page "Chapter 0, designed to introduce readers the special vocabulary of circle concepts" (from back cover of new book). Even this "introduction" (the first six pages) would never be considered elementary.
2) There is a 17-page appendix about the life of Karl Feuerbach, and his work on the 9-point circle.
3) This edition has added an index.

I hope this helps.

Although it is the simplest of all nonlinear geometric forms, the circle is far from trivial. It is indeed a pleasure that The Mathematical Association of America chose to reprint an update of this classic first printed in 1957. Geometry teaching has been in retreat for many years in the US and that has been a sad (and very bad) thing. It is also puzzling as so many people say that the reason why they cannot do mathematics (i.e. algebra) is that they need to see something in order to understand it. Furthermore, the first mathematical education that most children receive contains the differentiation of shapes and their different properties.
Circles and lines as used in geometry are abstractions that are easily grasped, much simpler to many than the abstract generalizations of algebra. One can only hope that this book signals a rebirth in interest in geometry education. Without question, it can be used as a text for that education and would help parent a rebirth. To remedy this modern affliction and make the material available to the current readership, a chapter zero was included. This new chapter is used to introduce the background concepts and terminology that could be assumed when is was first published.
No one can truly appreciate the intellectual achievements of the ancients as summarized by Euclid without doing some of the problems. There is also a stark beauty to a form of mathematics where the tools are a compass, straightedge and a mind. Particularly in the age of calculators and computers. All of the basic, ancient results concerning circles are covered as well as some very recent ones. The theorems are well presented and compete without being overdone. In keeping with the ancient traditions, pencil, paper, compass and straightedge are the only tools used. A short collection of solved exercises is also included.
Like the books of Euclid, this work will grow old but never dated. It was destined to be a classic when is was first printed and remains so today.

Published in Smarandache Notions Journal, reprinted with permission.