Before we get going, there is something which must be stated up front: this is a very British book. You should not be surprised by the appearance of words like "aeroplane" and "anticlockwise," but you should be surprised that flying out of London towards San Francisco takes you over Manchester and not the Channel. Also, since the first seven chapters are very conversational and visual in tone, while the eighth includes many of the proofs mentioned but omitted in the first seven, you should understand that when Higgins says his last chapter is "exceptional" he means that it's different from the others, not that he thinks it's noticeably better.
Within the last fifteen years, many books have been published which attempt to discuss mathematics in a manner which non-mathematicians can understand. While other disciplines have been pursuing this line of public relations for some time, mathematicians have only recently expended the effort to place their field of study in contexts which appeal to the general reader. Since this effort is so new, it's not surprising that few of these efforts have stood out from their kin. The books of this genre have begun to resemble each other, with much of their content overlapping from book to book. As more people take their turn at adding to the growing list of mathematics books for the mainstream reader, we should expect these books to become more focused, eschewing yet another mention of Fermat-Wiles or Ramanujan and the taxicab in favor of a detailed description of a smaller region of the field of mathematics. Science books have been doing this for a long time. We wouldn't expect a Stephen Jay Gould book to dwell on the work of a certain Swiss patent clerk, yet practically every popular mathematics book I've read tries to cover every branch of mathematics.
Mathematics for the Imagination takes a step toward more focused exposition. Higgins states in his preface that he aims to "...convey to my audience the history and development of mathematics throughout the ages and to explain some of its most interesting features." He focuses on geometry as the means by which to accomplish this, as it is the window through which our various forebears viewed the world and tried to explain its mysteries. It's a refreshing approach, as it allows him his second stated goal of avoiding as many formulas as possible, while providing a theme to unify his otherwise separable chapters. How well he succeeds is a function of the reader's patience with visual arguments and the author's rambling enthusiasm for his subject.
Higgins' writing bubbles over with information and unexpected connections between topics. For instance, one paragraph in the first chapter begins with a rhumb line spiral, moves into a biography of Thomas Harriot and his unsavory friends (Guy Fawkes and Christopher Marlowe among them), mentions the doomed Roanoke settlement in North Carolina, returns to a property of the spiral, and ends with Bernoulli's epitaph. Even if you're willing to get swept up in the narrative, it's a bit of a roller coaster ride.
In order to avoid formulas, Higgins pursues visual arguments as often as possible. This approach succeeds pretty well, and allows for several theorems and proofs that I've not come across in the popular literature. In a discussion of spherical geometry, Higgins demonstrates that non-congruent similar triangles are impossible because angular excess, the amount by which the sum of the angles of the triangle exceeds 180 degrees, is directly proportional to the area of the triangle. The explanation of Kepler's and Newton's work on planetary motion is done in a refreshing way, which then leads to the method currently being used to discover planets orbiting around distant stars. Heron's problem is not only included (an easily stated problem often overlooked by other authors), but it's put to good use several other times in the course of the book. Some explanations do bog down if you're not engrossed in the subject, but only one proof (that of Morley's Theorem in Chapter 8) is a bit convoluted, and any section that wears the reader down can be skipped, as the text is nicely episodic.
One of the surprising strengths of the book is the author's penchant for peppering his narrative with evidence which highlights the value, beauty, and uniqueness of mathematics. Mention is made of Lincoln and Darwin reading Euclid for fun. There's a nice differentiation made between how a mathematician and a chemist view the question of five-fold symmetry in quasicrystals. In discussing various proofs (all geometric) of the Pythagorean Theorem, he challenges the reader to come up with more, but admonishes that any proof needs to use the fact that the given triangle has a right angle, a point of subtlety our majors are often left to discover on their own. Indeed, this text offers the opportunity to work through some of the questions the ancients dealt with as they came to terms with the physical world.
As a text or supplement for a college course in mathematics for liberal arts or in the history of mathematics, this book is a little problematic. The wandering style is an obstacle for students trying to pull out the main themes. The expectations Higgins places on his readers are uneven. For instance, in the chapter on spherical geometry, the terms "similar" and "congruent" are used with only a sentence of explanation. However, when talking about Pythagorean geometry, the notion of "congruent" is introduced and discussed at great length. Most concepts are discussed in just enough detail for a casual reader to absorb the gist of the argument, but allow a more devoted one to explore the details at her leisure. However, some of the straight-edge and compass constructions are run through in laborious detail. It's unclear whether or not the enthusiasm Higgins so obviously feels and demonstrates in his writing would encourage or discourage a student who has this book as a required text. For a truly interested reader, however, one who wishes to broaden their mathematical experience or desires a more participatory introduction to the history of mathematics, this book fits the bill rather nicely.
Mathematics for the Imagination takes the admirable tack of focusing on geometry as a means to impart to a lay audience the development of mathematics. It's not a perfect book, but it contains enough less-traveled material, clear exposition, and unbridled enthusiasm that it stands out from the rest of crowd. It is certainly a worthwhile book for the interested reader, and, in the author's sense of the word, an exceptional one.
Steven Morics (Steven_Morics@redlands.edu) is Associate Professor of Mathematics at the University of Redlands in Southern California. He is currently enjoying a sabbatical in the much colder climate of Minnesota, pursuing interests and research in undergraduate mathematics education, mathematics and social choice, and bundling algorithms (procedures to stuff small children into snow pants and parkas before the school bus comes).