James Clerk Maxwell is known primarily for the four equations that bear his name. That is unfortunate. As impressive as that achievement was, it represents only a small part of his life’s work and only one product of his genius. Maxwell’s *Treatise on Electricity and Magnetism *shows us more. It is a major contribution that is indicative of his breadth and foresight, and offers us a good deal more than a textbook on electromagnetic theory. Maxwell uses that work to show how the treatment of electricity and magnetism can suggest a new approach to mathematical physics at large.

Maxwell’s *Treatise *was originally intended to meet two distinct and rather conflicting goals. One was to put the science of electricity and magnetism on a firm experimental and theoretical foundation. The second was to produce a textbook for the Cavendish Laboratory program and the mathematical tripos examinations at Cambridge University. That combination unfortunately manages to obscure the unity of Maxwell’s line of thought. The current book is a study of the *Treatise*, a commentary and an appreciation designed to extract the main ideas and recover Maxwell’s original themes.

Howard Fisher, author of the commentary, proceeds by identifying sections of the *Treatise* that are most relevant to what he calls the “central argument”. He then provides commentary, elaboration and elucidation of Maxwell’s arguments. There are a few challenges along the way. Maxwell shows a certain fondness for the quaternions, a fondness not shared by many modern readers, so Fisher provides vector equivalents in his commentary. Maxwell’s proofs tend to be terse, but the author’s commentary draws them out in detail and brings the notation into more modern form. The book’s format puts Maxwell’s original text on the main portion of each page with the author’s commentary below a conspicuous separator line.

Maxwell implicitly presents his subject as something founded on the physical and visual, preferably described in prose, and supported by equations and symbolic reasoning. He expresses some disdain for those (“professed mathematicians”) incapable of taking ideas seriously unless they are expressed in symbolic form. For him the essence of mathematical reasoning is something separate from the purely formal and symbolic.

The parts of the *Treatise* that Fisher selects begin with Preliminary Material, which includes discussions of physical quantities and their measurements, as well as mathematical topics like line and surface integrals and even Stokes’ theorem and its proof. These are followed by Part I: Electrostatics; Part 2: Electrokinematics; Part III: Magnetism; and Part IV: Electromagnetism. In each of the parts some (and occasionally many) chapters of the original text are omitted by Fisher. Generally these are more specific examples or sections more aligned with tripos preparation material.

The “Central Argument” of the title is not explicitly identified, but appears to refer to Maxwell’s progression from basic electrical and magnetic phenomena (much of this depending in the work of Faraday) to the interpretation of them as manifestations of continuous electric and magnetic fields and then culminating with a theory of light as wave motion within those fields.

Fisher also regards the “new science of connected systems” that Maxwell describes in Chapter V of Part IV as a particularly important part of the work. Maxwell argues in favor of Lagrange’s method for representing complex systems only in terms of the variables that are observable. He also advocates generalizing beyond conventional mechanical concepts such as force and mass, concepts of limited value for electricity and magnetism.

Not much more is demanded of the reader than some acquaintance with multivariable calculus, some basic physics, and the willingness to dig in. Maxwell’s work is not easy reading, but Fisher’s commentary makes it much more accessible and his insights are valuable.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.