*Full Frontal Calculus* is an introductory text on single variable calculus, but with a novel approach of developing the calculus using infinitesimals rather than limits. The author states his reason for choosing infinitesimals early in the text. From Chapter 1, page 6, the author says

One does not take a course in Driver’s Education to learn the principles of the internal combustion engine, and one does not take freshman calculus to learn the subject’s deep logical underpinnings. One takes a course such as this to learn the calculus itself- to learn what it is, how to use it, and how to think in terms of it, for calculus is as much a way of thinking as it is a collection of computational tricks.

In setting the stage this way, the author addresses the student honestly, pointing out to the reader that they are not a professional mathematician and that

you need not, at this point in your academic career, concern yourself with the full details of calculus’s logical foundations, except to be reassured that these exist and are secure; you can study them (in either the limit-based version or the infinitesimal-based version) in the appropriate books or classes should you feel so inclined in the future.

With this viewpoint the author proceeds to develop the standard ideas and methods of calculus. Chapter 1 starts with an introduction to what the calculus is and gets to the heart of “infinitesimal thinking”. The author offers a very clear and intuitive explanation of what an infinitesimal is by an initial thought experiment: the problem of finding the probability of drawing a red ball from an urn containing infinitely many balls, only one of which is red. He explains how, by introducing infinitesimals, it makes sense to intuitively say the probability is “1 in infinity”. Having introduced the notion of infinitesimals, the author gets right to the matter at hand by stating that “Differential Calculus grows from a single idea: On an infinitesimal scale, curves are straight.” Using this initial framework and using the language of infinitesimals , the author develops the standard rules of differential calculus in subsequent chapters.

The rest of the chapters follow a sequence similar to what you might find in a standard Calculus textbook (Chapter 2 develops the standard differentiation rules; Chapter 3 covers the standard applications of differential calculus; and so on, up through Taylor Polynomials and Polar Coordinates). Of course, given the viewpoint of this text, these topics are all approached using the language of infinitesimals. (For example, the author discusses the definite integral from a framework of breaking the area into infinitely many rectangles that are infinitesimally thin and somehow summing their areas.) Enjoyably, there is also a “Chapter \( \pi \): Limits”, which brings in the more standard limit language and introduces the notion of the “limit form of the derivative” by viewing it as the ratio of infinitesimals.

All in all, the text does what the author intended, to introduce a student to differential and integral calculus, its meaning and uses, by the founding idea of Newton- infinitely small numbers, i.e. infinitesimals. As a result, the text is a different way of teaching the calculus that is, in a sense, informal from the traditional epsilon-delta approach—but succeeds in avoiding the “hand waving” often associated with teaching limits by appealing to the notion of infinitesimals. The text is supported by good examples and exercises throughout that follow this viewpoint, and the final section of the text includes solutions to selected problems. If used as a text for math majors, of course, the instructor will need to give a good deal of thought on how to transition from this approach to a traditional epsilon-delta treatment as encountered in a course such as Advanced Calculus.

Charles Traina is a Professor of Mathematics in the Dept. of Mathematics & Computer Science, of St. John’s College of Liberal Arts & Sciences of St. John’s University. Jamaica, N.Y. His research interests are Group Theory and Measure Theory.