It is the thesis of both of these books that geometry, despite being removed from school and university mathematics syllabuses, is "still a subject of abiding beauty that provides tremendous intellectual satisfaction in return for effort" and "gives students a 'feel' for the reasons for studying various areas of mathematics (such as topology)I and services the needs of computer graphics, and so on". In short, geometry is due for a revival.
John Silvester's book, based on a first-year mathematics undergraduate course at King's College London, starts with a discussion of drawing, geometric construction, ruler-and-compass methods and angle bisection. This leads on to a discussion of plane geometry and isometries, triangles and triangle formulae. Later chapters deal with conics, affine and projective geometry, non-Euclidean geometry and algebraic geometry.
The approach is traditional - definitions are precise, propositions are posed, theorems and corollaries are proved - but it is not dull. Examples are well used to illustrate, and the author provides words of guidance and encouragement for the reader.
David Brannan et al 's book views geometry as "a space together with a set of transformations of that space". The authors explore various geometries: affine projective, inversive, non-Euclidean and spherical, making the last two geometries in particular very accessible.
The book arises from material developed for second and third-level Open University courses, but could form the basis of courses in geometry for mathematics undergraduates. It will also appeal to the general mathematical reader.
The book opens with a chapter on conics and quadric surfaces in Euclidean space. Chapters then focus on figures such as lines and conics in geometries other than Euclidean. The key to the book is the generation of various geometries from a space and particular groups of transformations.
This develops the subject in an attractive way, establishing classical results in affine, projective and inversive geometry.
Both texts require some group theory and linear algebra, although the necessary topics are covered in two short primers at the end of the Brannan text. The authors also assume knowledge of the arithmetic of complex numbers.
John Stone is principal lecturer in computing mathematics, Sheffield Hallam University.
Author - David A. Brannan, Matthew F. Esplen and Jeremy J. Gray
ISBN - 0 521 59193 7 and 59787 0
Publisher - Cambridge University Press
Price - £65.00 and £20.95
Pages - 497