Pentagon and polygons

For several centuries a good education was considered incomplete without the study of Euclidean geometry - which made Euclid's Elements , composed in about 300BC, one of the most frequently printed books in history. But the Elements was not an unqualified success as a textbook. In 1871 an organisation that set itself the task of seeking alternatives to Euclid was formed. But supporters of Euclid were powerful and it was not until the early 20th century that a greater range of textbooks became available.

It is not hard to sympathise with 19th-century mathematicians who declared that Euclid had put them off geometry for life when the text they had used had become formalised into "a model of soul-destroying systematisation". Had they had access to Geometry Civilised , how different their view of geometry would have been. For J. L. Heilbron sees geometry not simply as a formal system residing between the covers of a textbook but existing everywhere in the world around him. He delights in its aesthetic qualities, the systematic reasoning it provokes, its integration of the pictorial and the verbal, and its broad cultural diversity. Stonehenge, Islamic mosaics, Vedic altars, European cathedrals, Monticello, the Pentagon - all these and more contribute to his story.

Heilbron begins by setting geometry in a cultural context. He provides an overview of different geometries from differing times and places - Greek, Babylonian, Egyptian, Chinese, Indian and

Islamic - and the account is enriched by discussions of geometry as science and as technique, and of geometry and gender. In his second chapter he looks at the idea and necessity of proof. While in no way diminishing the notion of formal proof, Heilbron happily embraces a pragmatic view - when an adequate demonstration suffices to win the argument, the formality can be eschewed. Appropriately the third chapter is all about triangles. Plane trigonometry is introduced, and some of the earthy origins of geometry and its application to practical problems begin to emerge. The practicality of understanding the properties of triangles is demonstrated through examples from sources ranging from Egyptian papyri to the writings of Galileo.

In the next chapter the

Pythagorean theorem takes centre stage. Several different proofs of the theorem are discussed, including the earliest known due to the Chinese. Irrational numbers appear and there is a further venture into trigonometry.

In chapter five the behaviour of circles, both with and without attendant plane figures, is thoroughly explored, and regular

polygons are constructed. The two objects are then elegantly brought together in the context of the design of gothic windows. Procedures for estimating the value of pi are also discussed. The book concludes with a chapter devoted to applying some of the geometrical principles previously developed to a selection of more challenging problems such as Huygens's improved method for computing pi, the design of Lavoisier's great burning machine, and further examples associated with Gothic tracery.

Little is assumed with regard to mathematical knowledge, basic definitions and techniques are explained, and the text is studded with useful tables and calculations. And while Euclid's presence is over-arching, specific references to the Elements are sensibly reduced through the inclusion of tables listing the Euclidean elements and propositions used in each chapter. The illustrations are excellent and there are diagrams aplenty. Integrated within the text is a wide range of exercises and solutions derived from a variety of times and cultures.

This is a handsome book, well researched and entertainingly written. It shows how powerfully a historically informed account can communicate. If you thought Euclidean geometry was something only your great-grandparents did - try it, you will be surprised.

June Barrow-Green is research fellow in the history of mathematics, Open University.