Read This! The MAA Online book review column Ruler and the Round: Classic Problems in Geometric Constructions by Nicholas D. Kazarinoff Reviewed by P. N. Ruane

1. Ground rules for ruler and compass constructions leading to some historical discussion of the problems of angle trisection, cube duplication and squaring the circle etc. It then places the notion of constructible geometric figures in the context of analytic geometry, thereby providing algebraic classification of constructible numbers. The concept of fields of real numbers then leads to a discussion of roots of algebraic equations.

2. The second part treats non-constructible regular polygons and introduces the algebra associated with them. This begins with consideration of irreducibility and factorisation, unique factorisation of quadratic integers, finite dimensional vector spaces, algebraic fields and non-constructible regular polygons.

An appraisal of this book must, of course, be given in relation to the intended readership, which is defined in the preface as consisting of three possible groups:

'...high school students who have studied plane geometry for at least one term, high school seniors and college students...'

In fact, if the book were to be used at undergraduate level, it should be after students have taken introductory courses in algebra and geometry. In this way, it could form the basis of a very useful consolidating-module linking the two areas of mathematics by revealing geometric applications of basic algebraic ideas (fields, vector spaces etc).

On the other hand, if high school teachers were inclined to introduce students to the geometry of angle trisection, cube doubling and polygonal constructions, they would do well to consider the way that such material is presented elsewhere. For example, Courant and Robbins [1], give a very accessible treatment, which would not overwhelm students with a plethora of new algebraic concepts.

Apart from the second chapter, which aspires to lay down 'ground rules' for ruler and compass constructions, the book is clearly written and is mathematically sound. The content is developed in such a way that geometric ideas and algebraic techniques are always kept in close contact. Furthermore, conclusive discussion of the three of the four classic problems is achieved. However, a proof that circles are not 'squarable' with straightedge and compass is not given, but it is nicely explained how this may be done by use of the quadratrix.

This book is highly recommended; it presents mathematics in its most attractive form and within an interesting historical context. To teach a course based upon its contents could be a most rewarding experience.

Reference:

[1] What is Mathematics? by R. Courant and H. Robbins (Oxford University Press, 2^nd Edition)

Publication data: Ruler and the Round: Classic Problems in Geometric Constructions, by Nicholas D. Kazarinoff. Dover, 2003. Paperback, 138pp., $7.95. ISBN 0-486-42515-0

Peter Ruane (ruane.p@blueyonder.co.uk) was Senior Lecturer in Mathematics Education at Anglia Polytechnic University, England. His research interests lie within the field of mathematics education and the history of geometry.

Posted October 6, 2003

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