At the Centre of the Shadow
Science doesn't always work the way you think it should
How do you prove how nature works?
I want to try to answer this question by reference to a legendary historical episode from nineteenth-century French physics. Interpreted simplistically, it provides a rather romantic but wholly mythical view of the scientific method at work. Interpreted carefully, it holds an object-lesson for those seeking a better understanding of the nature of the scientific enterprise.
Imagine you are a physicist. How do you go about your job? We might suppose that you draw on your experiences of the natural world, gained through observation or experiment, or both. In your attempt to explain what you find, you conceive an idea, from which you derive a conceptual structure which connects what you find with your vision of nature’s detailed, inner workings. You might build this structure using models or ‘toy’ systems that have the virtue of conceptual and mathematical simplicity: perfectly elastic point-particles, or waves flowing in an elastic medium, or vortices, fields, or other abstract representations, analogies, or metaphors.
You translate this construction into the scientific language through which we attempt to describe physical reality—mathematics. Congratulations. You have formulated a theory, a genuine conjecture which, the philosopher Karl Popper explained, is ‘a highly informative [guess] about the world which although not verifiable (i.e. capable of being shown to be true) can be submitted to severe crucial tests’.
Your theory does an admirable job of predicting what is already known, but this is not enough. You use the mathematics to poke and squeeze and stretch and push and shove the theory to reveal something new and unusual about itself, perhaps something completely unanticipated. The result is a singular, novel prediction, perhaps for something nobody has yet experienced or even thought to look for.
You then experiment with nature or make some new observations on it. If the new evidence supports your conjecture you can pause to congratulate yourself before launching a search for further evidence. If the new evidence refutes your conjecture, you can abandon it and start the search for another. You might be disappointed for a time, but you are not downhearted. ‘Every serious test of a theory is an attempt to refute it’. A theory that is in principle refutable is, by Popper’s definition, a scientific theory. You can at least console yourself with the knowledge that what you are doing is science.
We acknowledge that nature can sometimes be rather coy: it doesn’t always want to give up its secrets too easily, or too cheaply. For every conjecture that succeeds there are many doomed to fail. Those that look great on paper, those that are so conceptually or mathematically beautiful or elegant that they ‘must be true’, don’t often survive the brutal assault from experiment and observation. They turn out to be completely false and, once again, we shake our heads and remind ourselves that we must deal with nature as it is, not nature as we might wish it to be.
As our theoretical descriptions of reality have matured in the last hundred years or so, we’ve also had to learn to be extraordinarily patient. In ‘foundational’ physics—the basic physics of space, time, matter, light, energy, and the universe—our conjectures and their mathematical formulations have now run far ahead of our capacity to put them to the test. It is not unusual to have to wait fifty years or more for experimental science to catch up with speculation and hypothesis.
But, every now and again, science does something rather spectacular. By the force of intellectual will, happy circumstance, or just sheer good fortune, everything comes together. When this happens, the experience can be exhilarating. Jaws drop. Skin tingles. A scientific discovery is simply one of the most extraordinary experiences a human being can enjoy. And although this experience is historically often unique to an individual, it is nevertheless something that can be shared, at least in part, again and again, even two hundred years after it happened.
What then? Surely, rationality prevails, and we take another step on the long road to scientific progress. The discovery demonstrates the superiority of one conjecture over another, and scientists—being rational people—happily embrace it and abandon the failed theory they have cherished for much of their professional lives.
Hmm. Let’s see.
The Prize
We turn to Paris, 1818. Three years earlier Napoleon I had been defeated at the Battle of Waterloo by the armies of the Seventh Coalition. He had been deposed as head of the French state and exiled by the British for a second time, this time to the distant island of Saint Helena in the South Atlantic. His return to power had lasted just 111 days. The Bourbon constitutional monarch, Louis XVIII, was restored to the French throne, also for a second time.
The mathematician and astronomer Pierre-Simon Laplace had once helped the young Bonaparte when he was a student at the Ecole Militaire. As Napoleon’s star rose, Laplace had courted his patronage, not least because Napoleon understood the importance of science in post-revolutionary France.
Their relationship was significant. Following his successful coup in November 1799, Napoleon had appointed Laplace as his Minister of the Interior. Alas, Laplace was not suited to the task. Though ‘a mathematician of first rank’, he made for a mediocre administrator. ‘[H]e sought quibbles and saw problems everywhere, bringing the spirit of infinitesimals into the realm of governance’. No doubt Napoleon’s assessment was informed by Laplace’s apparent disloyalty during his return from exile in the ‘Hundred Days’. With Napoleon’s first defeat and abdication in 1814, Laplace had (to some onlookers rather too quickly), shifted his allegiance to the restored king. He had been made a marquis in 1817, placing him in the hierarchy of French noblemen above a baron or count, below a duke or prince.
Many considered Laplace to be France’s Isaac Newton, or at least a worthy successor, committed to the mechanical conception of nature that Newton had helped to establish in the late seventeenth and early eighteenth centuries, the foundation of what we call classical physics. He held a position of great influence in the prestigious Academy of Sciences in Paris. With the chemist Claude Louis Berthollet, he had founded the influential Society of Arcueil, a small but elite group of scientists that would gather informally during summer weekends at the country homes of Berthollet or Laplace in the Paris suburb. As our story begins, he had resolved to use his influence to chart a final victory for his particular neo-Newtonian version of physics.
Newton’s reign was unquestioned in almost every aspect of our understanding of the physical world, from the motions of material objects on Earth to the grand spectacle of the cosmos. But there was one aspect of his description of nature in which Newton was perceived by some to be fallible. His corpuscular, or ‘emission’ theory of light had dominated physics since the publication of Opticks in 1704. In France, Laplace, Jean Baptiste Biot, and Étienne-Louis Malus had built on these foundations to develop a theory of light based on ‘rays’ composed of streams of particles that interact with matter through short-range forces.
But this particulate or emission interpretation of light had suffered some robust challenges, primarily from Dutch mechanical philosopher Christiaan Huygen’s alternative wave theory and particularly from reports of recent experiments on light diffraction and interference by the English physician Thomas Young, and others.
These interference phenomena produce patterns of alternating bright and dark ‘fringes’, readily explained using a wave theory of light in which overlapping wave peaks combine to produce bigger peaks, troughs combine to produce deeper troughs (giving bright fringes), and peaks and troughs combine to cancel each other out (giving dark fringes). Interference posed stubborn conundrums for an emission theory constructed from particles.
In a France battered by years of political turmoil and revolution, the ‘émissionaires’ sensed rebellion. They needed to find a way to demonstrate the superiority of the emission theory, confound the wave theory, and suppress the uprising. In a move worthy of Napoleon himself, in 1817 Laplace and Biot arranged for the Academy of Sciences to sponsor a prize for an original work that best explained the phenomenon of diffraction in terms of the emission theory. The prize was announced on 17 March 1817, with a closing date for submission of 1 August 1818.
The Memoir
But the émissionaires hadn’t reckoned with Dominique François Jean Arago, an accomplished experimenter and scientific adventurer, and a man of legendary exploits. Arago had been elected to the Academy of Sciences in 1809 (with Laplace’s support) and was also a member of the Society of Arcueil. He tended to favour the wave theory, or was at least sufficiently open-minded to consider its possibility.
The prize announcement caught Arago by surprise, but he swiftly fought a rear-guard action, ensuring a place for himself on the committee that would judge the prize submissions. He was unable to change the wording of the prize announcement, however, which required candidates to explain the fringe patterns ‘by mathematical inductions, [to deduce] the motions of the rays in their passage near bodies’. A literal interpretation of the prize question thus in principle excluded submissions based on the wave theory, which had no use for ‘rays’.
Both Arago and André-Marie Ampère nevertheless pressed Arago’s young protégé, Augustin Jean Fresnel, to submit a memoir in competition for the prize. Fresnel had worked as a civil engineer, but had greeted Napoleon’s return from exile in 1814 as an affront to civilisation and had offered his services to the royalist resistance. He was suspended from his position and put under surveillance. With Arago’s help, during this period of enforced leisure he was able to turn his attention fully to the theoretical and experimental study of light. He resumed his engineering career in 1815 following the second restoration.
Arago had been greatly impressed with Fresnel’s work in optics, the branch of physics that involves the study of light, and felt confident that he could win the prize and strike a blow in support of the wave theory.
Fresnel made his submission, titled ‘Mémoire sur la Diffraction de la Lumière’ on 29 July 1818. This was essentially a synthesis of theoretical principles based on the wave theory, involving some innovative mathematics combined with some new, exquisitely detailed experimental observations.
The Theorem
The prize committee included Laplace, Biot, Arago, Joseph Louis Gay-Lussac and Siméon Denis Poisson. Laplace and especially Biot were committed émissionaires. Gay-Lussac, most notable today for his work in clarifying the molecular formula of water (H2O), could be counted on to be impartial. Although he had some reservations, the accomplished mathematician Poisson was inclined towards the emission theory. The committee was therefore stacked in favour of the émissionaires and against Fresnel.
Poisson was intrigued, and somewhat perturbed, by Fresnel’s use in his analysis of the calculus of definite integrals. And, being more mathematically astute, he noticed something in the formulation that Fresnel had overlooked. Fresnel had used his theory to predict the pattern of fringes created by light diffraction in the shadow cast from an object with a straight edge. But his approach was quite general, completely independent of the type of object casting the shadow and causing the diffraction.
Poisson now applied it to a situation in which a distant source of light is blocked by a small circular disk. We can only speculate that he must have smiled to himself when he did the calculations, for they suggested a seemingly quite absurd result. Poisson called it a ‘singular theorem’. According to the wave theory as formulated by Fresnel, the circular disk would cast a shadow—this much was obvious—but it further predicted that at the centre of the shadow should be a spot as bright as if the disk did not exist.
It was clear to Poisson that something had gone wrong. The fact that the wave theory predicted this absurd result must surely be taken to indicate that it provides an inadequate description of light. Poisson wrote to Fresnel to advise him of this problem.
The Experiment
It seemed that the conjecture based on waves, when cast into Fresnel’s mathematical language, could be pushed and stretched to a point at which it broke faith with nature. But neither Arago nor Fresnel were so easily persuaded. There was no escaping the conclusion—if Fresnel’s general description was to be taken at face value then Poisson’s singular theorem was undoubtedly correct. The notion that a bright spot should be found at the centre of a dark shadow seemed ridiculous but, Arago now asked himself, what if it is actually true?
Arago set himself to the task of finding out. He constructed a circular screen from a metal disk measuring just two millimetres in diameter, mounted on a parallel-sided glass plate using a little sealing wax. When illuminated with a distant source of light, the disk cast a circular shadow, as expected.
At the centre of the shadow Arago found a bright spot. This is today called Poisson’s spot or Arago’s spot, and can be revealed quite readily in the laboratory using modern laser light sources.
The rest of the story surely writes itself. The émissionaires had no choice but to accept this compelling experimental evidence in favour of the wave theory. Fresnel had predicted the precise details of the interference fringes for the case of a straight edge, already demonstrating the superiority of the wave theory over its rival. And the wave theory had now yielded a stunning novel prediction for the case of diffraction from a circular disk which, though wholly counter-intuitive, had been shown to be true. There was no accounting for this result using the emission theory. Being rational people, committed to the scientific method, the émissionaires bowed to the evidence and abandoned the emission theory.
Fresnel, the heroic underdog in this story, prevailed. His memoir was duly awarded the prize and the emission theory was consigned to the waste bin of science history. The discovery of Poisson’s spot entered scientific folklore as a singular example of the power of an experimental test of a novel prediction. (Except, as Arago himself later discovered, this was not so novel. The bright spot had been observed in previous experiments, though without the context of the wave theory the experimenters had not known what to make of it.)
The Object-lesson
But this is not quite how it worked out. Fresnel did win the prize, but he had virtually no competition. There was only one other submission, whose author is not recorded, which the committee judged inadequate. In Arago’s report of the committee’s deliberations, published in May 1819, Fresnel is commended only for his precision experiments and his novel use of mathematics. Arago’s summary is written in the language of ‘rays’ and makes no mention of the wave theory which underpins Fresnel’s work. The discovery of Poisson’s spot is given relatively short shrift, in just two sentences, in which the phenomenon is also discussed in terms of rays. An appendix published after the main report refers to another ‘singular’ case, of diffraction through a small circular hole for which Fresnel calculated the equally counter-intuitive result that there should appear a dark spot at the centre of the bright image projected on a far screen. Fresnel had gone on to show that this dark spot, the ‘negative’ of Poisson’s spot, can be observed experimentally.
Despite what many subsequent historical reconstructions might have concluded, Fresnel’s memoir, and the singular discovery of Poisson’s spot, did not immediately spell the end of the emission theory.
Why not? Perhaps, in drafting his report, Arago had sought to placate the powerful émissionaires, who had remained stubbornly unpersuaded by Fresnel’s mathematics and in denial of the new evidence. Or perhaps Arago himself suffered doubts common to all physicists for whom the Newtonian ‘way of thinking’ had become second-nature. Fresnel’s work was based on Huygen’s principle—every point on a wavefront serves as a secondary source of spherical waves which combine to produce a further wavefront. In this way the light wave was thought to move through the ‘ether’, a tenuous medium imagined to fill all space. In correspondence with Arago, even Young questioned the way Fresnel had made use of Huygen’s principle. Those hoping to learn more from Fresnel’s prize memoir had to exercise patience: it was not published until 1826. Fresnel’s health was poor for most of his life but it deteriorated in the early 1820s and he died in 1827 at the age of 39.
Fresnel and Arago had continued to collaborate on aspects of light polarisation. This could be accommodated within the wave theory framework, but it threw up more puzzles that were not quite so easy to explain away. In particular, there was no real accounting for why light waves appeared be transverse waves—oscillating up and down only at right-angles to the direction in which the light moves.
Despite what might appear to be overwhelming evidence (with the hindsight afforded by our modern-day perspective), the simple truth is that those in positions of authority in French physics were not overwhelmed by these results. The physicists were loath to give up the concept of the light ray as a physical thing in favour of the ray as a mathematical construct of the wave theory. And why discard a theory judged more than adequate for more than a hundred years when there might still be the possibility of using it to explain diffraction and interference?
In August 1818, the power and influence of the French émissionaires was strong but it was also waning. Laplace was 70 (he died in 1827). Biot continued to argue with Fresnel and Arago—sometimes bitterly—particularly on the topic of polarisation, and insisted that far from overturning the emission theory, Fresnel’s experiments provided a striking confirmation of it. Poisson remained unsatisfied with Fresnel’s mathematics and debated the issues with him for the next four years. Skirmishes continued into the early 1820s.
That the situation was confused is reflected in French physics textbooks of the time. In his discussion of diffraction in 1824, François Sulpice Beudant points out the challenges posed by Fresnel’s experiments, but nevertheless chooses to present only the emission theory because ‘it is easier to grasp’. In contrast, in his 1823 textbook, Jean Claude Eugène Péclet embraces Fresnel’s wave theory although, like many physicists of the period, he struggled with Fresnel’s account of polarisation.
At this point in the story we’re entitled to ask how the wave theory eventually came to dominate. The answer lies not in France, but in England and Ireland. The wave theory was eagerly adopted by a group of young Cambridge University mathematicians led by their Lucasian Professor, George Biddell Airy. John Herschel (son of William Herschel, who had discovered the planet Uranus in 1781), also embraced the wave theory in a rather confused and confusing encyclopaedia article published in 1827.
On 28 June 1833, the third meeting in Cambridge of the newly-formed British Association for the Advancement of Science witnessed a tense showdown between the new ‘wave men’—Airy, Herschel, William Whewell, Humphrey Lloyd, and the Royal Astronomer of Ireland, William Rowan Hamilton—and the British emission theorists David Brewster, Richard Potter, and John Barton. Hamilton led the charge, not with the discovery of Poisson’s spot but with his own discovery that the wave theory predicted something called internal and external conical refraction. These phenomena had been demonstrated experimentally by Lloyd.
Brewster counter-challenged with the patterns of dark ‘lines’ in light passed through molecular gases, showing that the gases selectively absorbed certain colours and not others (or all)—surely a problem for the wave theory. But the challenge was disarmed by Herschel. It probably didn’t much matter. The ‘Cambridge faction and its Dublin allies’ were in the ascendant, on their way to becoming a new scientific elite. Brewster was forced to retreat to a ‘beleaguered defence … sniping at Whewell from the columns of the Edinburgh Review’. From 1834 the wave theory had become the new orthodoxy.
The wave theory became dominant because it ‘exemplified the new mathematical physics [the Cambridge group] sought to promote’. Of course this was all about the superiority of the wave theory as a description of light, but it was also about a theory that was more amenable to abstract mathematical analysis which fed the ambitions of young mathematicians who would come to dominate the science of optics in Britain. It is said that history is written by the victors, and in Whewell’s History of the Inductive Sciences (1837) he writes: ‘When we look at the emission-theory of light, we see exactly what we may consider as the natural course of things in the career of a false theory’.
Nature would have the last laugh. The ‘ether’ required to support light waves could not be found and, in 1905, Albert Einstein eliminated the need for it by proposing his ‘light-quantum hypothesis’. Light, he suggested, consists of quanta (which are particle-like), whose energy is governed by the frequency of the light (a wave-like property). Einstein’s light-quanta would later be called photons. This is not, after all, about either-or, particle or wave. In some curious way that nobody really understands, quantum theory is about both. Although clearly superior to both the emission and wave theories as a description of light, almost 120 years later we are still struggling to come to terms with what quantum theory is trying to tell us.
This episode and its object-lesson should serve to remind us that science is first and foremost a human activity and, when it comes to the content of their science, scientists do not always behave as rationally as simplistic historical reconstructions would have us believe. Although most, when challenged, will strenuously deny it, scientists arguably do behave perfectly rationally ‘in the round’, when considering the content of their science in the context of the beliefs about how nature works that they have developed throughout their careers, and in the context of their position and authority within the scientific community.
A friend of mine and I were talking about the course of science, and he made the perfect quip: "Science proceeds despite scientists."