On his deathbed in 1879, Clark Maxwell wrote a referee’s report on a new paper by George FitzGerald, applying Maxwell’s theory to the reflection and refraction of electromagnetic waves. Maxwell was dead of cancer before George read it. In those last months Maxwell (who had designed the Cavendish Laboratory and was its first Professor) had also got about half way through revising a new edition of his book on electrodynamics. It did not contain “Maxwell’s equations” as we know them - instead (see the current Dover edition) we find 11 equations, some scalar, and some in quaternion form. Before he died, Maxwell, who wrote much light verse and poetry, had used a telephone and heard a primitive cylinder phonograph, but did not live to see the discovery of radio waves in 1888.
     This is a remarkable book - perhaps the best history of science text I’ve read, Koestler’s “Sleepwalkers” , Rhodes’ “Making of the atom bomb” and Kuhn’s “Structure of Scientific Revolutions” notwithstanding. It brings vividly to life the excitement surrounding the discovery of radio waves by Hertz, and the personalities involved with developing the relevant theory following Maxwell’s death - Fitzgerald, Lodge, Poynting, Larmor, Stokes, and Heaviside. But it is the mathematically  gifted Heaviside (1850-1925; cf generalized functions, Heaviside layer) who emerges as the unsung hero of this saga, despite the flair and fertile imagination of Fitzgerald (retarded potentials, “relativistic” contraction) and the crucial confidence given to their groping ideas by Herz’s experiment.  The book asks profound questions about how abstract new theories arise, and contains and discusses all the relevant equations in modern notation, including a discussion of the key question of whether fields or potentials are more fundamental.  Contrary to what we teach, we learn that Maxwell did not predict the existence of radio waves. The book is based on close analysis, much of it mathematical, of the hundreds of letters we still have between Fitzgerald, Heaviside and Lodge.
     Maxwell’s theory was based on Faraday’s idea of stresses and strains set up by charges in a surrounding invisible elastic medium (the ether) - the experimentalist Faraday is therefore the originator of field theory. It was Thomson (later Lord Kelvin) who introduced vortices into this medium to explain the Faraday rotation effect.
     Maxwell made these vortices the basis of his theory, which added “idler wheels” between them. These eventually produced  the displacement current, a verygreat discovery, which allows propagating solutions. He himself considered his finding that this medium supported waves which traveled with the velocity of light,  which could be predicted from static, known electromagnetic constants, to be his

 greatest discovery. (Yet, when he derived this famous result in Edinburgh, he had to wait, for family reasons, to the end of the summer in great anticipation before catching the steam-train up to London to get values of the constants needed to test his result!). But Maxwell probably never realized that an oscillating current would emit em waves, or dealt with any em waves other than light.

     Kelvin’s derivation of the “Telegraph equation” was also important. (It is now used , for example, to describe  neurons firing;  then it was used to explain delays on the new transatlantic submarine telegraph, where morse code did not travel, as expected, at the speed of light). This equation  later suggested a propagating wave solution of Maxwell’s equation. But Heaviside made a crucial  improvement to Kelvin’s result  by incorporating inductance, and so developed the condition for distortionless propagation - he described this work as the “Royal Road” to understanding electromagnetic wave propagation. Kelvin himself may never have read Maxwell’s book, and, having contributed so much to electrodynamics in the early stages, remained skeptical of   Maxwell’s equations to the end of his life in 1907.
     Heaviside, a working-class recluse who never really had a job, who never married,  and who started life as a telegraph boy, devoted his life to Maxwell’s work through his close friendship with the eminent Prof. Fitzgerald in Dublin. Archie Howie, who gave our Physics colloquium recently, recalled conversations with a colleague who had visited Heaviside in the nineteen twenties.  Heaviside was the first to derive “Maxwell’s equations” in their modern form (in 1884) - class prejudice and an early rejection from Phil. Mag. caused him to publish this (in 1885) and all his highly mathematical work in the telegrapher’s trade journal “The Electrician”. By 1900 his modern version of Maxwell’s equations had become widely accepted, and so, despite fierce opposition from the head of Britain’s telegraphy organization, Heaviside was eventually recognized at the end of his life with Fellowship in the Royal Society, and his papers  accepted in leading journals due to FitzGerald’s support. Heaviside’s book (“Electromagnetic Theory” 1912) became the Bible of the field, and the first modern EM text.  Oliver Lodge, the experimentalist, was also a strong proponent of Maxwell’s ideas, and through him we get a good feeling of

the excitement in the village of Cambridge when it was known that Maxwell’s book first appeared in the local bookshop. (Green’s book on String Theory had similar impact, but was far less technical - Cambridge was a much more specialized community in those days). But the “Woodstock” conference of the era was in at Bath, UK in 1888, when Hertz’s discovery was announced. We get a vivid picture of excited debate in the corridors, with Kelvin always the skeptic,  while the Young Turks defend their more abstract mathematical ideas, defending the propagation of waves in vacuum or the ether.
     Two points struck me forcibly on finishing this book. Although in his  last paper on electrodynamics, Maxwell ignored the mechanical scaffold and used purely Lagrangian methods, we are left to wonder if his theory could ever have  been developed without it. Fitzgerald described a simpler mechanical model for the ether than Maxwell’s - a two-dimensional array of wheels, each of which is connected to its four neighbors by rubber bands. Is it a coincidence that the  solution to the elasticity equations for this apparatus gives us Maxwell’s equations? We are led again to the question Frank Wilczek asks in the current issue of Physics Today, quoting Wigner : “ What is behind the unreasonable and miraculous accuracy of mathematics as a description of reality”. Wilczek goes on to cite the power of non-mathematical ideas in science (Darwin, the atomic theory in chemistry) and Feynman’s vision of a future expanded human intellect capable of understanding the qualitative content of equations. Maxwell evidently had highly developed skills of both kinds.
     Secondly, the theoretical difficulties they faced in 1880 were enormous. Remember that Maxwell worked only in the Coulomb gage of electrostatics, which required instantaneous propagation of potentials (but allowed fields finite time to propagate). It was FitzGerald, trusting his mechanical model, who came up with our modern “Lorentz gage” which allows both fields and potentials to propagate with the speed of light. Heaviside also had realized this earlier. This was one of  the hot topics, and the fundamental issue debated at Bath that summer. By focusing on the fields, and eliminating potentials , Heaviside was soon after able to derive the modern Maxwell’s equations. (He also introduced the constitutive equations) . But the hot topic of Bath 1888 is still with us - when we think of the Aharonov-Bohm experiment and the work of Wu and Yang, potentials have again become fundamental, in accordance with Maxwell’s original idea      
 
John Spence is Regents’ Professor of Physics at Arizona State University. For more information about Professor Spence and his research, please visit http://physics2.asu.edu/people/jspence