Mathematics and science instructors are all too frequently faced with the task of convincing skeptical students that their disciplines are profoundly *human* endeavors full of the same measures of excitement, frustration, courage, and controversy that are found in the history or arts classroom. Two pedagogical approaches commonly used to attack this misconception are seminars for freshman science majors and courses on the history and philosophy of science for general audiences. In such venues, the primary focus is (or should be) on an understanding of how science progresses and how it is done, rather than on the seemingly sterile “end products” with which students must grapple in typical survey courses. For instructors teaching such courses and searching for good material for student discussion and contemplation, Robyn Arianrhod’s *Einstein’s Heroes: Imagining the World Through the Language of Mathematics* is an ideal starting point.

Arianrhod’s narrative deftly interweaves the progress of James Clerk Maxwell’s development of the theory of electromagnetism (leading to the derivation of Maxwell’s “wonderful” equations, sure to appear on a student T-shirt at a mathematical meeting near you) with brief yet cogent summaries of various stages in the development of “unified” physical theories and the mathematical language needed to explain them to the wider world. Albert Einstein was sufficiently impressed by the economy and elegance of Maxwell’s achievement that he displayed the British scientist’s portrait in his laboratory, and the quality of Einstein’s much-praised exegeses of the general and special theories of relativity owed a great deal to the clarity of thought and expression evinced by Maxwell in his most significant works. But Maxwell in turn drew inspiration from “heroes” of his own — hence the plural in this book’s title.

In developing his theory, Maxwell sought to do for electromagnetic phenomena what Isaac Newton had so famously done for gravitation: to fit experimental results onto a grand theoretical framework describing *how* such results took place. Ironically, it was Newton’s theory itself — which had worked so well for describing planetary motions and static electric and magnetic phenomena (as in Coulomb’s Law) that 19th-century scientists felt fully justified in extending it to electromagnetic induction — that loomed as the largest obstacle to an acceptance of Maxwell’s new ideas. Michael Faraday’s notion that fields consisting of “lines of force”, rather than the Newtonian principle of “action-at-a-distance”, could better serve to explain the behavior of electromagnetic forces provided the key insight, but Faraday lacked the mathematical tools to construct a coherent theory. Despite his own limitations as a “nuts-and-bolts” mathematician, Maxwell used the more powerful “weapons” of the calculus of vector fields — and his own considerable rhetorical skill — to reconcile the “Newtonian” and “Faradayan” approaches in a quartet of elegant equations that Arianrhod likens to “a four-line poem”.

The aforementioned “poetic parallel” is anything but accidental. The overarching theme of *Einstein’s Heroes* is that mathematics is a universal human language — complete with its own rules of grammar, structure, and form — that has come to serve as the primary means by which scientists describe the physical universe. Arianrhod admirably sustains this note whether she is discussing vector operations or the “meaning” of the terms in Newton’s Second Law of Motion. (In the hands of a less gifted author, this might have lent the narrative a tone of pedantry or condescension, but Arianrhod has a knack for explaining things simply, as opposed to simplistically.)

In addition to its descriptive power, Arianrhod argues, mathematics is at its core a language of *imagination*. Just as the use of the proper language can make imaginary and unseen literary personalities and worlds come alive, so too the right sort of mathematical language can permit one to move beyond over-reliance on comforting but misleading physical analogies and to uncover *real* physical entities whose existence may not have been previously suspected, but which can be derived directly “from the equations.” In this manner, Maxwell used his equations to predict the existence of radio waves and to hypothesize that light itself is an electromagnetic phenomenon. In turn, Maxwell inspired Einstein to apply the field idea to gravitation in his general theory of relativity. Telecommunications, black holes, and the idea of an expanding universe are just some of the consequences of these extensions discussed by Arianrhod.

What makes this book ideal for a general student audience is its excellent evocation of the progress of scientific ideas and the interplay of scientific personalities. Arianrhod includes enough personal anecdotes and historical scene-setting to clearly establish that science does not transpire in a “vacuum” but travels, so to speak, through a “field” of cultural cross-currents that may affect its progress. As a result, she demonstrates, scientific progress is as much about philosophical changes (“paradigm shifts”) as it is about technical advancement. Science also progresses through the medium of human interaction, and Maxwell’s work with lifelong friend P. G. Tait to hammer out the details of vector calculations is described in humorous detail by Arianrhod. The narrative even includes a “tragic hero” of sorts: Sir William Thomson, Lord Kelvin, a legitimately great scientist who did not accept Maxwell’s theory and (as he himself admitted late in life) fell fatally “behind the curve” as a result.

The book’s time scale ranges from the ancient Greeks’ development of deductive geometry to advances in late-20th-century physics, and the narrative treats all subjects in between with the same combination of straightforward narrative and (when required) concrete, even homely, analogy. Open-minded students will come away from reading *Einstein’s Heroes* with a newfound appreciation of the role of scientific inquiry and insight in the larger human enterprise — and, hopefully, will acquire a hankering to “take the ‘field’” themselves.

Christopher E. Barat (f-barat@mail.vjc.edu) is an associate professor of mathematics at Villa Julie College. He received a B.S. in Mathematics from the University of Notre Dame and Sc.M. and Ph.D. degrees in Applied Mathematics from Brown University. He has previously taught at Randolph-Macon College, J. Sargeant Reynolds Community College, and Virginia State University.