My father was fond of saying, "As every English schoolboy knows ...," right before invoking a result from linear algebra to prove something about projective planes. I thought of him and his remark at various times while looking through the interesting book presently under review.

The authors are all with the Open University in the United Kingdom, a university designed primarily for correspondence students. Its main educational tool is the correspondence text -- a magazine-like pamphlet that presents a single chapter of the topic at hand and contains five sections, comprising roughly fifty pages of well-written and well-presented material with a reasonable selection of problems. The present text grew out of the course *Introduction to Pure Mathematics*, which "has traditionally given geometry a central position," according to the introduction.

The book adopts the Klein approach, viewing a geometry as a set with an associated group of natural transformations. One then studies those properties left invariant by the transformation groups. A consequence is that, again from the introduction, "The book assumes basic knowledge of Group Theory and of Linear Algebra, as these are used throughout." Brief appendices on both topics are provided. (Very brief: eight pages total!)

The book contains nine chapters, numbered from zero. Chapter 0 is a short introduction that presents enough history to set the stage for Klein. Chapter 1 covers Conics in the real plane and also has a little to say about quadric surfaces in 3-space. The next three chapters form a unit: 2. Affine Geometry (in the real plane and including planar Euclidean geometry), 3. Projective Geometry: Lines (the real projective plane), and 4. Projective Geometry: Conics. The groups in question are matrix groups of small degree over the reals. The Möbius group on the complex projective line is the common point for the next three chapters: 5. Inversive Geometry, 6. Non-Euclidean Geometry, and 7. Spherical Geometry. The final Chapter 8 is entitled The Kleinian View of Geometry. It is followed by the appendices and over 100 pages of problem solutions.

There are many things to like about this book. It conveys the beauty and excitement of the subject, avoiding the dryness of many geometry texts. I was very pleased to see the classification of conics developed in Chapters 1 through 4. Conics are often treated in a cursory manner in calculus and then abandoned. Here Chapter 1 gives a careful discussion of the usual state of affairs, then Chapters 2 and 4 detail how the classification simplifies as the group of allowable transformations is enlarged. Required items, like the Theorems of Ceva and Menelaus, are proved not just because they must be (must they?) but also to illustrate and demonstrate the transformational arguments being introduced in Chapter 2. In the same spirit, Steiner's Porism is handled nicely in Chapter 5. As a demonstration of the power of projective duality, Pappus' Theorem and that of Brianchon are shown to be equivalent. Chapter 5 contains a nice section on the Riemann sphere and Chapter 6 one concerning non-Euclidean tessellations. The effort put in on the problems and answers is very impressive.

The language is clear, with little to fear from Anglicisms. (I had not previously known that my slope is the authors' gradient.) The standard Open University presentation is clean and pleasing to the eye, although the convention of having all marginal remarks (of which there are many and interesting) in the right margin probably works better in the original pamphlet form. Here the remarks on lefthand (even) pages get somewhat lost in the binding. The book is nearly free of misprints.

I did find some weaknesses in the book. It could have been more effectively edited. Its past as a sequence of pamphlets written by three different people is sometimes too evident. The style changes from chapter to chapter -- the early chapters are more synthetically and visually oriented while the later chapters are more analytic. A few topics get repeated several times, sometimes without reference to earlier occurrences or with changed notation. Chapter 8 on the Kleinian view exists only to pull together the other chapters. Much of its material would have been better inserted at the appropriate points in the earlier chapters. In particular, we should not have to wait until Chapter 8 to learn that the affine group of Chapter 1 sits naturally inside of the projective group of Chapter 2. Having supported the Klein approach and its unifying role in the introduction and Chapter 0, the authors should have emphasized it throughout the text. In Chapter 2, we find that the awkwardly defined parallel projections are special types of affine transformations, but which ones are they? An explicit matrix description would be enlightening. It would also be nice, in Chapter 3, to mention the connection between parallel projections and perspectivities with center on the ideal line at infinity.

With such a broad subject and so many possible approaches, it is natural for mathematical quibbles to arise. Having just done projective geometry, why not point out in Chapter 5 that the extended real plane is just the complex projective line, particularly when the transformation group under discussion is the Möbius group? For me, the Fundamental Theorem of Projective Geometry is the statement that every projectivity of a Desarguesian space is induced by a semilinear transformation. (For instance, see **Artin**, page 88, or **Baer**, page 44.) This beautiful joining of the synthetic and analytic worlds of geometry seems a perfect Kleinian result. The present book instead uses this terminology for the tamer, purely analytic result that, in the real projective plane, there is a unique linear projective transformation taking any ordered quadrilateral to another. (This is closer to what Baer calls the Second Fundamental Theorem of Projective Geometry on his page 68. See also **Artin**, page 98. **Veblen and Young**, pages 95-6, give a purely synthetic version in terms of the group generated by all perspectivities.) Another disagreement about terminology comes from the authors' use of the broad term non-Euclidean geometry to indicate only hyperbolic geometry. This "misnomer" is noted as such on page 322, but I think it should be avoided entirely.

I have only one strong mathematical objection to the book's treatment. On page 120 of Chapter 3, a perspective transformation of a projective plane is defined as a projective transformation induced by a linear transformation of 3-space taking some 2-flat (off the origin) isometrically (preserving distances) onto another. Such a definition is at best wrong-minded and at worst incorrect. As Veblen and Young state (their page 14), "It is evident that no properties that involve essentially the notion of measurement can have any place in projective geometry." In particular, the basic projective concept of perspective should not be approached via length preserving maps. In any case, the definition does not match the root idea of a perspectivity (described correctly on page 98). The discussion preceding the definition on page 120 vaguely associates a perspectivity centered at the origin with any projective transformation, with a special case motivating the definition. Following remarks then claim, without proof, that any perspectivity can be induced in the manner described. Even if this is true, it does not imply that the definition encapsulates perspectivities. For instance, the definition, as given, allows any 3 x 3 orthogonal matrix as a perspective transformation. Certainly perspectivities have natural definitions. See **Baer**, page 64, and **Veblen and Young**, page 72: analytically, perspectivities are induced by linear transformations that are scalar on a subspace of codimension 1; synthetically, they fix some projective line pointwise and, dually, some projective point (the center) linewise.

I wondered how I might use this book. In my university, as in many these days, there is a big gap in the geometry curriculum. There is an elementary course aimed mainly at secondary education students and to be taken soon after calculus, but then we jump to a beginning course in differential geometry. Although my father's English schoolboys might have sufficient background to handle a class taught out of this book, I do not think our education majors would, even those specializing in mathematics education. (Of course, today's English schoolboys might not have that background either.) Although the linear algebra and group theory needed are minimal, some understanding of their content, more than just definitions, is continually required for the full appreciation of this text. Instead I would encourage the use of the book as extra reference in an elementary geometry course or as the backbone for a capstone course, in which more linear algebra and group theory (say, crystallography) could easily and naturally be added as time allowed. The book's coherence and vision interestingly reveal its Kleinian approach to geometry as an elementary introduction to Lie groups in low dimension.

As already mentioned, the book is light and insightful enough to merit use in some form (despite the concerns voiced above). It raised in me a desire to teach elementary geometry again, something I have not felt or done for years.

## References

**E. Artin**, *Geometric Algebra*, Interscience Publishers, Inc., New York, 1957.

**R. Baer**, *Linear Algebra and Projective Geometry*, Academic Press, New York, 1952.

**O. Veblen and J.W. Young**, *Projective Geometry I*, Ginn and Company, Boston, 1938.

J. I. Hall is a Professor of Mathematics at Michigan State Univeristy