From Galois theory to algebraic number theory to the mathematics of astronomy and cosmology — you have to give Ian Stewart credit for versatility. Of course, Stewart deserves credit for other things as well. For decades now, in addition to his own mathematical contributions, Stewart has done far more than his share to make the subject accessible to others, both laypeople and mathematics students. In the book now under review, he discusses the mathematics of the universe, addressing — and sometimes critiquing — such issues as the big bang theory, black holes, the theory of multiverses and much more.

Like most of Stewart’s books, this one has much to recommend it. Its subject matter is very interesting, and I learned new things from it. Until reading this book, for example, I had assumed that the big bang theory was pretty much universally accepted as an explanation for the origin of the universe; I now know that recent developments require that the big bang theory be supplemented by other ideas (inflation, dark matter, dark energy) and that even here there is some room for doubt. Stewart explains the problems and offers alternative proposals.

Interspersed among all this, one finds occasional interesting nuggets of random trivia. I did not know, for example, that there are five references to comets in Shakespeare, three of which reflect the ancient belief that comets foretell disaster. There are also a couple of pages of nice color photographs, and a good five-page section called “Units and Jargon” in which a number of terms are defined.

However, notwithstanding these good points, I would not rank this book among Stewart’s best. The writing is at times dry, and sometimes reading this book was like reading an encyclopedia. For example:

Jupiter’s Trojans are closely associated with another fascinating group of asteroids, the Hilda family. These are in 3:2 resonance with Jupiter, and in a rotating frame they occupy a region shaped roughly like an equilateral triangle with vertices at L.4, L.5, and a point in Jupiter’s orbit that is diametrically opposite the planet. The Hildas ‘circulate’ slowly relative to the Trojans and Jupiter. Unlike most asteroids, they have eccentric orbits.

Additionally, although the book is clearly intended as a discourse on how mathematics gets used to address questions such as the ones mentioned above, the amount of mathematics that was actually described in the book turned out to be less than I expected. To a layperson, of course, “mathematics” is often equated with “equations”, and there are very few of these in the book. Even from a more sophisticated perception of what mathematics is, there seemed to be little in the way of actual mathematical explanations of ideas. (One notable exception is a nice discussion of non-Euclidean geometry and manifolds, and their role in relativity theory.) Mostly, the book just seemed to consist primarily of explanations of physics concepts, with allusions and references to, but not real discussions of, mathematics. Readers who read Stewart because of his ability to explain mathematical ideas might feel a bit let down.

Third, the book seems to suffer from an audience problem. Although it seems to be addressed to laypeople in science and mathematics, I think that such an audience might well find some of the discussions to be hard to understand. Even a more sophisticated reader might run into trouble; I had quite a time, for example, with Stewart’s discussion of Penrose diagrams.

Finally, Stewart is, on occasion, a little too free with factual assertions. He recounts, for example, the familiar story of how Gauss used surveying equipment to measure with surveying equipment the angle sum of a large triangle formed by three mountain peaks, and asserts as fact that Gauss did so to “find out the true shape of space.” However, this is the subject of some dispute; Buhler, in his *Gauss: A Biographical Study*, states that this account of Gauss’ motivation “is, as far as we know, a myth”. Also, in *Worlds out of Nothing*, Jeremy Gray refers to “the claim, much disputed in the literature, that Gauss surveyed three mountains in northern Germany with a view to deciding if space was Euclidean or non-Euclidean” and states that there is “no evidence at all” that this was in fact his motivation.

Because the end of the academic semester is fast approaching and I am in grading mode, I’ll give this book a B: a solid effort, though not a stellar one. Readers who have come to expect more from Ian Stewart might find themselves disappointed.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.