Read This!

The MAA Online book review column

Briefly Noted

December 2003

Perusal of the chapter headings in Calculus: the Elements would suggest that this is a fairly typical introductory calculus text. That is not the case, because the author has chosen to revive the notion of infinitesimals and to use it as a basis for the development of his material. Outside of a course on non-standard analysis, this is a very brave undertaking, for the simple reason that most students usually have only a fragmentary understanding of the real number system. Trying to cope with entities merely described as being "less than every positive real number and yet greater than zero" would likely compound this confusion.

To my mind, the book leaves many questions about infinitesimals unanswered. For example, if α is an infinitesimal, and x is a real number, what is the nature of the product xα? If there are infinitely small numbers, can there be infinitely large numbers? Can infinitesimals be added or multiplied? Do they conform to the algebra of real number system? etc, etc. Moreover, this opaqueness is transferred to the dependent concept of differentials, leading to unsupported assertions like "the ratio of dy to dx has a definite numerical value" and, immediately after that, "the quotient of a number by an infinitesimal cannot be a number." (p. 88).

The early chapters, which aspire to develop intuitive readiness for the more formal ideas that later emerge, caused me some headaches; I found the approach to be very wordy and the phraseology often clumsy and obscure. Consider, for example, the following sentence from p. 100: "... derivatives of complex functions which can be constructed out of the simpler ones are determined, by establishing how differentiation behaves with respect to the operations employed in construction." I could quote many instances of this sort.

In terms of the overall structure, the development of some of the main themes is fragmented and occasionally obscure. Take, for instance, the concept of limits, and techniques for dealing with them, which is central to understanding calculus. Yet treatment of this theme is dispersed over fifteen different locations in the book and the only formal definition of limit (the ε-δ-definition) is not put to use, and no alternative operational procedures for the determination of convergence emerge.

The book as a whole has several commendable features, such as the author's provision of a "philosophical" basis for the development of calculus and, with some reappraisal of his treatment of infinitesimals, it could prove to be useful preliminary text to a course on non-standard analysis. Chapter 6 on "growth functions" provides an alternative approach to the derivation of exponential functions and there are very many examples that illustrate the applicability of calculus to physics.

Although I wouldn't use this book with newcomers to calculus, it could prove useful as a source book from which selected readings could be made for use with later courses or other subjects (applied mathematics, physics etc). [Peter Ruane]

December 8, 2003: Michael Comenetz has sent us a response to this review. P. N. Ruane has replied to Comenetz's comments.


N. M. J. Woodhouse's comparatively short Special Relativity is a pleasure to read and therefore qualifies right off as a good source to use for learning about special relativity on your own. A lot of very nice material is touched on in its pages, presented in a natural sequence consonant with history, and is not improperly belabored. It's also rather informal in style. One gets the sense of breezing along pretty fast while, in actuality, a lot of material is being dealt with. So, an autodidactic reader had better be prepared to cover the book with marginal notes, to fill a thickish note-book with more carefully worked-out material, and to do most if not all of the exercises. The latter come in several flavors, by the way, including, by the author's own admission, quasi-standard "borrowed" problems (from standard older texts) and Oxford examination problems. It's meaty stuff.

The book does presuppose a certain amount of mathematical maturity in its audience, but not unduly so: a strong background in vector calculus is certainly called for and (very properly, I think) no apologies are made for matrices. Indeed, Woodhouse states in his preface to the book that "it is written for students who are more familiar with linear algebra than with electromagnetism." And this says a little about the reader or student for whom Woodhouse intends the book. Qua audience, the real question is specifically one of distinguishing between two ultimately different kinds of animals, the physicist and the mathematician who wants to learn some modern physics. Admittedly, this is a somewhat controversial business, but I propose that one tell-tale feature in this distinction has to do with the kind of prose one is comfortable with: mathematicians, I claim, hunger almost irrationally for theorems and proofs, very explicitly. Since Woodhouse's book has lots of theorems and proofs, it's clearly meant for mathematics students.

As I already indicated, the selection of topics in the book is very nice indeed, and is historically sound and will therefore reward the reader with an element of culture, to boot: he'll learn some history of modern physics. Classical mechanics and Maxwell give way to the properties of light, Einstein and Lorentz. The latter's famous transformations are dealt with in a general way so as to precipitate a marvellous treatment of Minkowski space, setting the stage for a discussion of the geometry of space-time. Then relative motion, relativistic collisions, and finally relativistic electrodynamics (!) are discussed, and the book ends with a chapter on tensors and isometries done very tersely but also very well.

I wish this book had been around when I was a student. [Michael Berg]


Mathematical biology is all the rage these days, but it has not always been so. When Vito Volterra, early in the 20th century, tried to model biological phenomena using differential equations, his work was controversial. Despite its unfortunate title, The Biology of Numbers is a useful window into this early period and a valuable source for those interested in Volterra's work and ideas. The book begins with a historical introduction to Volterra and his attempts to construct a mathematics of biological interactions and then collects much of his correspondence on the subject. It is not for the faint of heart: most of the letters are in French or Italian, and no translations are offered. For those willing (and able) to face the linguistic barriers, there is much to learn here.

Volterra's approach was deterministic, modeled on classical mechanics. His ambition was to find laws of population biology that were as realiable as the laws of physics. He even expressed some of his proposed laws as variational principles — a kind of biological Lagrangean mechanics. This approach was controversial from the beginning, and some of the issues sound very familiar. Should a mathematical model of biological interactions be deterministic or probabilistic? What counts as experimental evidence that the model is correct? Does a useful model need only to reproduce the behavior observed in nature, or should the "internal" features of the model reflect biological realities?

The letters to and from Volterra are organized by correspondent, in alphabetical order, from Marcel Brelot to William R. Thompson. This makes the discussion in each set of letters easier to follow, but makes a chronological tracing of Volterra's own thought much harder. Still, what emerges is quite fascinating. There is no question that this is an essential source for anyone wanting to understand the beginnings of mathematical biology. [Fernando Q. Gouvêa]

Publication Data

Calculus: The Elements, by Michael Comenetz. World Scientific, 2002. Paperback, 540pp, $38.00. ISBN 9810249047.

Special Relativity, by N. M. J. Woodhouse. Springer Undergraduate Mathematics Series. Springer, 2002. Paperback, 192pp., $34.95. ISBN 1852334266.

The Biology of Numbers: The Correspondence of Vito Volterra on Mathematical Biology, ed. by Giorgio Israel and Ana Millán Gasca. Science Networks, volume 26. Birkhäser, 2002. Hardcover, 405 pp., $89.95. ISBN 3-7643-6514-5.

Peter Ruane (ruane.p@blueyonder.co.uk) was Senior Lecturer in Mathematics Education at Anglia Polytechnic University, England. His research interests lie within the field of mathematics education and the history of geometry.

Michael Berg (mberg@lmu.edu) is professor of mathematics at Loyola Marymount University in Los Angeles, CA. His research interests are algebraic number theory and non-archimedian Fourier analysis.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College in Waterville, ME. He is interested in number theory, the history of mathematics, Christian theology, poetry, science fiction, comic books, politics, classics, and football (the real thing, not the American version).

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Read This! is the MAA Online book review column. Contributions are welcome; contact the editor if you'd like to be one of our reviewers. Books for review should be sent to the editor: Fernando Gouv&ecric;a, Dept. of Math&CS, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.

MAA Online is edited by Fernando Q. Gouvêa (fqgouvea@colby.edu).
Last modified: Thu Dec 11 12:34:24 Eastern Standard Time 2003