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Geometry Revisited

Mathematical Association of America
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In 1961 a book appeared with the widely embracing title Introduction to Geometry. Its author was H. S. M. Coxeter who, in the preface, said that ‘For the past thirty or forty years, most Americans have somehow lost interest in geometry. The present book constitutes an attempt to revitalize this sadly neglected subject’.

It’s hard to say what effect that book has had in terms of the author’s overall aim; but I know that it was highly valued by many mathematical educators of that era. Unfortunately, it seemed to exert (at most) transient influence over the development of geometry in school or university mathematics. This, I feel, was because it isn’t a textbook as such, but rather, in the spirit of Birkhoff and MacLane, more akin to a ‘survey of modern geometry’. Moreover, it was regarded as too challenging for direct student use, and it is still hard to see how one could base an introductory course on it.

Six years later, however, there appeared this book, co-authored with S.L. Greitzer. Its purpose was the same as Coxeter’s book of 1961, but the contents and structure are entirely different, and much more likely to achieve the stated aim.

Of course, it may seem strange to be reviewing a book that was published over forty years ago but, having been unable to locate any previous review of this excellent book, it is surely a case of ‘better late than never’.

The title and the contents give a clue to the book’s main purpose. That is, to reawaken interest in geometry amongst those who have some prior knowledge of the subject, however rusty that may be. Consequently, as it says on the back cover, the early chapters take the reader by easy stages from very simple ideas into the core of the subject. For this purpose, the six chapters are sequenced as follows:

  1. Points and lines connected with a triangle
  2. Properties of circles.
  3. Collinearity and concurrence
  4. Transformations
  5. Inversive geometry
  6. Projective geometry

The chapter headings, in themselves, give little clue as to the style of the book. It is written with great clarity; it is well-illustrated and charmingly interspersed with interesting historical observations. In the process, the ideas and influence of over one hundred geometers are discussed — including lesser knowns ones such as Oene Bottema, Napoleon Bonaparte and A.Peaucellier.

Among many beautiful and surprising theorems included in this book are those by Ceva, Menelaus, Morley, Brianchon, Steiner-Lehmus and Feuerbach. The proofs in themselves are always interesting, and often non-standard.

But, apart from triangle theorems, many fascinating aspects of circles, quadrilaterals and conics are revealed, and the book explains the importance of the transformational viewpoint, beginning with the isometries and leading nicely into inversion and projective transformations.

Overall, I found the chapter that introduces inversive geometry particularly enjoyable; it includes a metric for inversive distance that relates very nicely to Steiner’s porism. As for the book’s final chapter, the approach to projective geometry is synthetic and perhaps, to quote an English saying, ‘not everyone’s cup of tea’.

Lastly, the exercises in the book are of equal interest as the main text itself. They extend the ideas previously introduced and encourage investigational participation.

Having read this book from cover-to-cover and worked through every problem, I can declare it to totally error-free. It deserves never to be out of print and could readily form the basis of an introductory geometry course in any undergraduate programme.

Peter Ruane, like many others whose pedagogical aspirations are thwarted by the constraints of curricular intertia, wistfully regrets that he never had opportunity to teach a course based upon a book like this one.

Date Received: 
Sunday, November 9, 2008
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H. S. M. Coxeter and S. L. Greitzer
Anneli Lax New Mathematical Library 19
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P. N. Ruane
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Tuesday, December 2, 2008