It is now commonplace for textbooks on ordinary and partial differential equations to include a chapter on numerical methods. This book constitutes a more radical departure from traditional approaches to these subjects. The use of computational tools is an essential adjunct to the text, leading to a recasting of the basic material covered. It remains to be seen whether such an idiosyncratic strategy will prove either popular or effective.

This work is the first of a projected three-volume set devoted respectively to single ordinary differential or difference equations, to systems of ordinary differential or difference equations, and to linear partial differential equations. Only the first volume is available, so only a preliminary assessment of the concept can be attempted. The authors have developed a package of 12 interactive graphics programs for the Macintosh computer, called MacMath; the programs are referred to repeatedly in both the text and the exercises, so access to them would be mandatory for a student taking a course based on the text. The authors give little indication of the capabilities of these programs, so one is left to draw inferences based on examples. For some reason, the authors do not indicate how a student or instructor could gain access to these programs. I accidentally discovered in a recent brochure from the publisher that the software and a user manual are, or soon will be, available [1] for a price 25 percent higher than that of the text.

The book contains five chapters. The first chapter deals with qualitative methods. This classical starting point for the study of differential equations is revived by the use of automatic graphics software to construct direction fields, approximate solutions, and the like. The treatment is more extensive than what is now customary, and it is integrated into the subsequent discussion. The second chapter presents elementary analytical methods for solving selected first-order ordinary differential equation initial value problems, and the third covers three elementary numerical methods. The discussion of numerical errors is more extensive than in most such presentations, though it is essentially by example. The fourth chapter concerns itself with questions of the existence and uniqueness of solutions, but not in the customary vein. The approach is a classical one revisited and recast, proving existence and uniqueness as consequences of the spirit (though not the language) of the “stability plus consistency equals convergence” paradigm applied to the elementary numerical methods of the previous chapter, supplemented by results from the qualitative theory of chapter 1. The fifth chapter investigates the currently fashionable topic of iterated maps, thought of as discrete dynamical systems: the more natural language of difference equations is eschewed. Graphical illustrations generated by the MacMath package play a key role, but the connection to the rest of the book remains tenuous.

The book is generally well written but provides little entry to the literature. I am sure students would enjoy playing with the computer. I remain skeptical that the amount learned is commensurate with the time and effort expended, and I suspect that instructors will regret the lack of coverage of material traditionally included in such a course. This may be an unfair comment on the concept since the real payoff may come in the second volume and cannot be evaluated at this point: hints to this effect are sprinkled throughout the text. Since the first volume is apparently intended for the novice, my guess would be that few students or instructors will reach the third volume.