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Mathematica is a general-purpose program to perform mathematical computations that was created by Wolfram Research Inc. The system is designed for use in both educational and professional settings. The basic system can do symbolic, graphical, and numerical computations and includes a comprehensive programming language that extends its capabilities. Some versions of the system can create “mathematical notebooks” that document mathematical calculations. These notebooks can include text and graphics along with Mathematica input and output. Mathematica has aroused much interest in the mathematical and scientific communities, and reviews of the system have appeared in both the popular and the scientific presses [1–4]. It has even been described as “the first system to tackle all of mathematics.” A more sober appraisal would describe it as a new computer algebra system, similar to MACSYMA, MAPLE, SMP, muMath, DERIVE, or REDUCE, that has a moderate amount of symbolic power, extensive graphical capabilities, and some numerical capabilities. The system has a nice user interface and a well-designed computer language with a powerful pattern-matching capability. All of this is combined in a package that is relatively easy to use. Hyperbole aside, Mathematica is a well-designed piece of software, but it does not tackle all of mathematics, and its mathematical capabilities are not significantly different from those of other computer algebra systems. Some initial impressions of Mathematica's capabilities are given by Barwise [1], Simon [2], and Wayner [4]. This review will concentrate on the book Mathematica: a system for doing mathematics by computer rather than on the Mathematica system. A press release from the publisher describes the book as “accessible to anyone who wants to use Mathematica—from high school students and those with no prior experience with computers, to advanced engineers, mathematicians and scientists—the book describes the basic Mathematica system and details its use on all computers which run the program. . . . ” In other words, the book is primarily a manual on how to use the Mathematica system. However, since it contains some involved examples of programs written in the Mathematica language and some discussion about various styles of mathematical programming, it can also be described as a programming language book on how to represent mathematical algorithms and mathematical knowledge in the Mathematica language. Although the publishers feel the audience ranges from high school students to professionals, a more realistic lower bound for much of the material is college sophomores or juniors with strong backgrounds in applied mathematics and computer science. Some sections will be accessible only to advanced students or professionals. Indeed, the first sample session with the system (page 3) includes a command to find a root of the Bessel function J 0( x) and another one to evaluate the Riemann zeta function at a complex point. Wolfram describes the Mathematica system as consisting of two parts, a “kernel” that contains the mathematical knowledge in the system and does the calculations, and a “front end” that manages the interaction with the user. The book's primary concern is the kernel. Wolfram considers the “mathematical notebook” part of the front end and describes it only briefly. I found this disappointing since this fascinating feature attracted me to the system and separates Mathematica from other mathematics computation systems. The book contains five chapters (numbered 0 to 4) and two large appendices. Chapter 0, “A Tour of Mathematica,” is a brief overview of Mathematica's capabilities. Chapter 1, “A Practical Introduction to Mathematica,” contains a detailed tutorial on the basic numerical, symbolic, and graphical commands of the system. The tutorial includes numerous examples that illustrate the various commands. The chapter concludes with some material on interfacing Mathematica with C and FORTRAN programs, the computer's operating system, and the T 2EX formatting system. Chapter 2, “The Structure of Mathematica,” provides a detailed overview of the Mathematica programming language and other miscellaneous aspects of the system. Chapter 3, “Advanced Mathematics in Mathematica,” describes the semantics of the system commands in greater detail. Chapter 4, “Mathematica as a Computing Language,” describes various approaches to mathematical programming, including procedural programming, functional programming, object-oriented programming, logic programming, and rule-based programming. The author believes that rule-based programming, using mathematical transformation rules and the pattern-matching capability of the system, often provides the most efficient way to write mathematics programs and represent mathematical knowledge on a computer. This chapter compares the Mathematica language to BASIC, FORTRAN, C, Pascal, APL, LISP, and PostScript but, unfortunately, not to other computer algebra languages that have capabilities similar to those of Mathematica. Appendix A provides some examples of symbolic and graphical programs; Appendix B is a reference guide to the system. This book is perhaps the best documentation currently available for a mathematics system. It is generally well written and well laid out. The author describes most (but not all) of the examples clearly, and it is generally easy to locate material in the book. Commands are conveniently summarized in tables (highlighted with gray backgrounds) which appear throughout the book. Keep in mind that the book is more of a manual than a textbook. The author has included no exercises, and the presentation places more emphasis on describing the details of the system than on describing the algorithms that make the system work. Although the author occasionally alludes to fundamental concepts in computer algebra, such as the Risch integration algorithm and Gro¨bner bases, and gives a Mathematica program for Gosper's summation algorithm, he provides no references for those who wish to explore the material in greater detail. The book gives the impression that software systems like Mathematica have great potential to alter the practice of mathematics. In fact, the cover of the book describes Mathematica as “powerful enough to tackle frontier problems in science and mathematics.” Unfortunately, the author gives the reader little guidance about which problems are most appropriate for the software. He states that future books will describe the use of Mathematica in specific application areas. The book can be used for a course in symbolic mathematical programming or as a supplemental reference manual for other courses that might use the Mathematica software.