The lonesome grave lies in a tiny crumbling stone church beside Loch Ken in southwest Scotland. You might think some otherwise undistinguished local lord lies here – until you see the name on the granite headstone: James Clerk Maxwell.

This is the final resting place of a giant of physics, the man who discovered that light is a wave created by the mutual push of magnetic and electric impulses. His discovery 150 years ago opened the door to the modern era of wireless communication.

Ask physicists to rank their heroes, and Maxwell is in the top three, standing a shade below Newton and Einstein. But when it comes to being celebrated by the public, somehow Maxwell got left behind. Einstein’s image is well known and Newton’s pilgrims regularly flock to his tomb at Westminster. But few of us would recognise Maxwell’s face or know of the forgotten grave in the crumbling kirk.

It is a pity, because Maxwell was one of the most likeable men in the annals of science. How can you not like a man who sends a heartfelt letter of condolence on the death of a friend’s dog? A man who patiently nursed his dying father, and later his wife, and who regularly gave up his time to volunteer at the new “Working Men’s Colleges” for tradesmen? It seems that everyone who knew him thought of him as kind and generous, albeit a little eccentric. He was “one of the best men who ever lived”, according to his childhood friend and biographer, Lewis Campbell.

Born in Edinburgh, Maxwell grew up on his family estate at Glenlair, not far from the church where he is buried. He became laird of Glenlair when he was only 24, on the death of his father with whom he shared a close bond – he was an only child and his mother died when he was eight.

His interest in physics was a natural extension of his fascination with how things work, and his love of nature. Maxwell was also a gifted mathematician. At 14 he developed a new method of constructing some unusual geometric curves; his father brought it to the attention of the Edinburgh Royal Society which declared “the simplicity and elegance of the method” worthy of publication in their proceedings. In his early 20s, Maxwell used Newton’s laws to show mathematically that Saturn’s rings are not solid, as they appear through a telescope, but are made of many smaller bodies. His paper won him a prize from Cambridge and more than a century later in the 1980s, Voyager proved him right.

Maxwell also loved language: he was a poet and a lucid science writer. His feeling for mathematics and for language came together in his unique approach to the most important scientific problem of his time: understanding electromagnetism.

**We don’t often think** about the importance of language in making scientific theories. Yet language clearly shapes our perception of the world. Place English speakers with their single word for snow amid the Inuit of Alaska, and they’ll be at a loss to describe details of the landscape. The Inuit by contrast have dozens of words to describe snow in all its forms.

Language can also express a prejudice or lock in an attitude. For instance, do we describe a wilderness as beautiful or threatening?

If everyday language can be so subjective, then what about scientific language? Science’s guiding principle is objectivity: scientists must be able to agree upon the results of a given experiment, regardless of their personal beliefs. At the same time science stands at the cusp between reality and language – between what is really out there, and what we are able to describe. Maxwell’s genius lay in recognising that problem and painstakingly searching for the right mathematical language to minimise it. But we’ll get to that shortly.

First, let’s set the scene. What did physics look like when young Maxwell entered the picture?

Most auspiciously he was born in 1831, the same year that the self-educated English physicist Michael Faraday made an astounding discovery. Faraday slid a magnet through a coiled wire, and presto, an electric current began flowing through it – no batteries required. Ten years earlier, Danish physicist, Hans Øersted had discovered the opposite: by switching on an electric current, he found a nearby magnetic compass needle jumped, as if the changing electric current were itself a magnet.

This mysterious interaction between electricity and magnetism was called “electromagnetism”.

No-one knew how this force was actually transmitted between wires and magnets, but most physicists assumed it acted instantaneously, without any intermediary mechanism – the same way gravity, magnetism and static electricity seemed to act. Apples immediately fall to the ground. Iron nails placed near a magnet are immediately pulled towards it, and two electric charges immediately attract or repel each other.

This kind of remote, instantaneous process was dubbed “action at a distance”. But by giving it such a name, physicists were playing with words: it was just an intuitive idea yet it seemed to be confirmed by mathematics – particularly Newton’s inverse square law (which says the force of gravity between two objects decreases according to the square of the distance between them). Using Newton’s laws you could work out the paths of planets without worrying about *how* the force of gravity travelled from the Sun to the Earth, for instance. So if the maths worked without the need to consider how gravity moved through space, then maybe gravity really did act at a distance?

This is how most of Newton’s disciples saw it – although Newton himself did not. As it happened, action at a distance was a headache for Newton: in the late 1600s many scholars dismissed his whole theory because they couldn’t imagine how gravity could possibly act instantaneously across a vast chasm of nothingness, no intermediary required.

Nevertheless, Newton’s laws worked. As well as describing planetary motion, they accurately predicted phenomena such as the date of the return of Halley’s comet and the existence of Neptune (deduced from distortions in the orbit of Uranus). In time, action at a distance was accepted as self-evident. This idea became even more entrenched when, in 1785 – nearly a century after Newton formulated his theory – French physicist Charles- Augustin de Coulomb showed that the electric force between two charged particles obeyed the same kind of inverse square law as gravity.

With the discovery of electromagnetism in the 19th century, the picture became much more complicated. Most physicists – the so-called Newtonians – assumed action at a distance still applied.

But Faraday dissented. The co-discoverer of electromagnetism thought the electromagnetic force must be communicated step by step through space, just as a breeze blowing through a farmer’s field moves every stalk in turn. Indeed he used the term “field” to describe the space around magnets and currents, and he imagined the field contained lines of force radiating from the electric and magnetic sources. He didn’t believe gravity acted remotely either, and knew that Newton had never thought so: just because the maths seemed to imply action at a distance, that didn’t mean the concept was real. “Newton was no Newtonian,” Faraday once quipped.

But the 19th-century Newtonians held sway. Their maths worked brilliantly for gravity, as well as for static electric and magnetic forces; they saw no reason not to apply it to electromagnetism too. As George Airy, Britain’s Astronomer Royal summed it up: “I declare that I can hardly imagine anyone … to hesitate an instant in the choice between the simple and precise [Newtonian] action, on the one hand, and anything so vague as lines of force on the other …”

* * *

**This was the intellectual** backdrop against which Maxwell, newly graduated from Cambridge, began developing a theory to explain electromagnetism. He carefully considered both sides of the debate. But something about Faraday’s lines of force resonated with him. As a two-year-old at Glenlair, he had been amazed that when he pulled a rope in one room, a bell rang in another, as if by magic. Then he discovered the holes in the walls where the bell-wires came through, and he dragged his father through the house, enthusiastically pointing them out.

Now, all these years later, he recalled those bell-wires. They deepened his conviction that just as a bell needed a wire, so electromagnetic effects must act through the agency of some sort of field.

Faraday’s lack of formal education meant he hadn’t been able to present his field concept in mathematical language. This is why Airy called the idea vague, and why few mainstream physicists paid it much attention. Maxwell believed that if he could find the right mathematical language to describe Faraday’s meticulous measurements of the forces surrounding electromagnetic objects, then perhaps the Newtonians might reconsider their objections.

For Maxwell, language held the key to unlocking the true nature of electromagnetism. He felt physicists’ choice of language was partly influenced by their style of thinking – by whether they tended to think primarily in mathematical terms or with the help of concrete images. Each style had its advantages and disadvantages, which Maxwell summed up later in a speech to the British Association for the Advancement of Science. The “natural mathematicians” are “quick to appreciate the significance of mathematical relationships”, he said. But the problem in physics was that such a thinker was often “indifferent” as to whether or not “quantities actually exist in nature which fulfil this relationship”. (Maxwell might also have been thinking here about those who used mathematics to justify a “magical” concept such as action at a distance.)

On the other hand, some scientists need concrete imagery to flesh out their equations. Such thinkers, Maxwell said, “are not content unless they can project their whole physical energies into the scene which they conjure up. They learn at what rate the planets rush through space, and they experience a delightful feeling of exhilaration. They calculate the forces with which the heavenly bodies pull at one another, and they feel their own muscles straining with the effort. To such men, [concepts such as] momentum, energy, and mass are not mere abstract expressions of the results of scientific enquiry. They are words of power, which stir their souls like memories of childhood.”

Maxwell employed both types of thinking.

Maxwell was clear: analogies were useful as scaffolds in erecting a theory, but should not be taken for reality.

**To explore Faraday’s field** idea, Maxwell began by searching for analogies, “words of power” that conjure up concrete physical images. In his first electrical paper, he showed how Faraday’s imaginary lines of force around magnets and electric charges could be modelled using the analogy of streamlines, such as you see in eddies in a flowing river. The mathematics of fluid flow had been pioneered in the 18th century, and Maxwell adapted these equations so they fitted the meticulous data Faraday had collected measuring the forces in the space around magnets and current-carrying wires.

Maxwell sent a copy of his streamlines paper to Faraday. Now 65, and depressed at the mainstream rejection of his idea, Faraday was overjoyed to hear from this unknown 25-year-old. He wrote to Maxwell that he had “never communicated with one of your mode and habit of thinking”. He also asked if Maxwell could express his work in “common language” as well as in mathematical “hieroglyphics” so that he could understand it?

Maxwell spent the next five years developing mathematical descriptions of various mechanical models that helped him imagine how a field could transmit changing electromagnetic forces. Yet he knew that such models did not qualify as a true theory of electromagnetism. It was about language again: by imagining Faraday’s field to be like a fluid – or like heat, or acting via mechanical cogs and flywheels – he was making assumptions for which he had no actual evidence.

Maxwell was clear in his writing about this: models and analogies are useful as scaffolds in erecting a theory, but they should not be mistaken for reality. So, with the mathematical insights he’d gained from his models now in hand, he dismantled his scaffolds, and started erecting his theory from scratch. His goal was to build a mathematical theory using only established physical principles, and the data gleaned from the papers of electromagnetic experimentalists such as Faraday. It would take him three more years.

Finally, by 1865, Maxwell was able to describe all that was known about electromagnetism in a set of “partial differential equations”. That in itself was a remarkable feat. Then he combined his equations and carried out one more mathematical operation.

And something extraordinary happened …

He found himself looking at the mathematical description of a transverse wave – the sort that travels along a plucked string.

With growing excitement, Maxwell realised his purely electromagnetic wave had exactly the same signature as a light wave – the same form, the same speed. The mathematical coincidence was too delicious to ignore. In his 1865 paper, he announced with understated triumph: “We have strong reason to conclude that light itself (including radiant heat, and other radiation if any) is electromagnetic …”

**In one fell swoop**, Maxwell’s equations seemed to resolve two of the major conundrums in physics: how electromagnetism is transmitted through space and the nature of light.

As Einstein said later of this discovery: “Imagine Maxwell’s feelings … at this thrilling moment! To few men in the world has such an experience been vouchsafed.”

If Maxwell was right, then Faraday was vindicated: electromagnetism did not act instantaneously at a distance, but through a field. And fluctuations in this field were propagated as waves. Just as a wave rippling along a string can vibrate with a range of frequencies, so too electromagnetic waves had different frequencies, some of which we perceive as light.

The rippling light wave was created by the mutual nudging of electric and magnetic fields. It was not so different to the way a Mexican wave travels across a stadium: one row of fans stand up and sit down, and trigger the next row to do the same. But in this case, the stadium is populated by electric and magnetic fields, each nudging the other on.

Maxwell hadn’t assumed anything about the nature of light.

**But did the waves** that Maxwell described mathematically actually exist? To prove it, someone would have to generate an electromagnetic wave from electrical and magnetic impulses. German physicist Heinrich Hertz took up the challenge.

He rigged up an electric circuit that sent sparks jumping back and forth between a spark gap made of two brass knobs. According to Maxwell the changing electric current would generate an electromagnetic wave.

To detect it, Hertz set up a receiver a few metres away – a loop of wire with a spark gap but no source of electricity. Then he started generating sparks from his oscillator. Lo and behold, across the room, another series of sparks began oscillating in the receiver!

To prove that these sparks had been generated from energy carried by an electromagnetic wave, Hertz aimed his oscillator at a metal screen. If the oscillator really did produce waves, then the screen would reflect the incoming waves, and a reflected wave would combine with an incoming one to form a series of “nodes” where the two waves cancel each other out.

This is what happens when the ripples from two pebbles dropped in a pond combine. By moving his apparatus, Hertz did indeed detect the neutral spots that must be the nodes; measuring the distance between adjacent nodes, he found the wavelength of his radiation. The year was 1887, more than two decades after Maxwell’s theory was published. Hertz had produced the first deliberately engineered radio waves. They had the same speed as light but a different frequency and wavelength, so they are part of what is now called the electromagnetic spectrum.

Sadly, Maxwell did not live to see Hertz’s confirmation of his theory. Nor did he live to see the widespread consequences of his mathematical field analysis, so that today we speak not only of electromagnetic fields, but also of gravitational and quantum fields.

A hundred and fifty years ago, though, Maxwell’s theory was so controversial that many of his peers refused to teach it. Critics such as Maxwell’s friend, Lord Kelvin, thought mathematics should describe tangible facts, analogies and models. They did not think that mathematical language itself might reveal new knowledge about the physical world.

Today, Einstein’s E=mc^{2} is the best known example of the power of mathematical language to reveal hidden truths. Einstein hadn’t set out to prove that energy and matter were essentially the same – but there it was in his equation!

Similarly, Maxwell hadn’t assumed anything about the nature of light – but there, hidden in his electromagnetic equations were mathematical waves travelling at the speed of light.

They were not ordinary waves travelling through a medium such as water or air, but purely mathematical waves of changing electric and magnetic intensity. In abandoning his earlier concrete models, Maxwell instinctively seemed to know that in the unseen realms – the “hidden, dimmer regions where thought weds fact”, as he put it – the closest we may come to perceiving physical reality is to imagine it mathematically.

The shy poetic Maxwell may not have achieved the celebrity status of Einstein, but in 2015 – the Year of Light – he deserves to be celebrated. You could do worse than don one of the T-shirts occasionally worn by students at university physics departments. They read: “And God said … Maxwell’s equations … and there was light.”

### Robyn Arianrhod

Robyn Arianrhod is a senior adjunct research fellow at the School of Mathematical Sciences at Monash University. Her research fields are general relativity and the history of mathematical science.

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