A portrait of Emilie du Chatelet attributed to Maurice-Quentin de La Tour. – DeAgostini / Getty Images

A spirited debate raged in 18th century Europe about what was driving the movements of the planets. In England, Sir Isaac Newton and his followers said it was gravity: the same invisible force that propels a falling apple also commands the planets in their marvellously ordered paths.

On the other side of the Channel, many Continentals favoured René Descartes’ theory of a swirling cosmic “ether” that, like a celestial tornado, swept up the planets in its wake.

This disagreement is more than an historical curiosity – it went to the heart of what it takes for a proposition to qualify as a truly scientific theory.

An unlikely pair of champions helped win the victory for Newton in Continental Europe: France’s best known and most controversial playwright, Voltaire, and his lover, mathematician Émilie du Châtelet. Her scientific work includes what is still the definitive French translation of Newton’s Principia. Yet after her death she was all but forgotten. If she was remembered at all, her achievements were often belittled, lost in the shadow of the “great men” in her life. But modern-day historians have rediscovered Émilie, and her story is inspiring new generations of women mathematicians, myself included.

When Emilie moved to Cirey to join her lover, tongues wagged salaciously.

Born in Paris in 1706, she is surely the most glamorous female mathematician in history. Tall and aristocratic, passionate in both her intellectual and amorous pursuits, she was larger than life. Too large for most people at the time: too ambitious, too intellectual, too emotional and too sexually liberated. Too much of a feminist, too: she pulled no punches when writing of her struggle to educate herself in higher mathematics and physics (because girls were denied access to good schools, let alone universities): “If I were king,” she wrote, “I would reform an abuse which effectively cuts back half of humanity. I would have women participate in all human rights, and above all, those of the mind.”

At 26, she captivated Voltaire, who was seduced by her brains as well as her beauty. He was already notorious as an upstart commoner with a wicked wit. Émilie, by contrast, was born to the aristocratic life; her father had been chief of protocol at Louis XIV’s court at Versailles. She’d been married off at 18 to the Marquis du Châtelet, with whom she soon had three children. Having done their duty for the Châtelet line, she and her husband then lived relatively separate lives – a common situation in aristocratic families. Less common was the remarkable friendship that developed between husband and wife, so that ultimately the marquis supported not only Émilie’s unusual ambition, but also her passionate relationship with Voltaire. Taking a lover was the norm at that time of arranged marriages, but Émilie and Voltaire scandalised polite society when they set up house together: extra-marital love affairs were supposed to be discreet dalliances, not alternative marriages. Curiously, their domestic arrangement – and their role as Newtonian revolutionaries – were as interconnected as the mysteries of the cosmos they set out to explain.

Newton’s proposal that gravity held the planets in place was based on observations and mathematics. Descartes’ theory of a swirling cosmic ether was an imaginative conjecture. – Bob Venables / Illustration Ltd

Voltaire’s attraction to Newton sprang from the playwright’s exasperation with France’s conservatives and elites – something he made clear in his satirical writing. By the time he met Émilie in 1733, his propensity for upsetting powerful people had already landed him in the Bastille for 11 months – and in the late 1720s he had been forced into exile for a couple of years. A fortunate exile, as it turned out, because he’d gone to England, where he met some of Newton’s leading disciples – Newton himself was well into his 80s by then.

London had been abuzz with Newton, and when the great man died in 1727, Voltaire attended his funeral in Westminster Abbey. Such official veneration of a scientist was unknown in Voltaire’s France, and it impressed him no end. So much so that he wrote a series of essays about the English: their constitutional monarchy, relative religious tolerance, rational Newtonian science, and new breed of empiricist philosophers, notably Newton’s friend and disciple John Locke.

Voltaire published these essays in England. In early 1734 he told a friend he was holding off publishing the longer French version – Lettres Philosophiques – for fear of the clergy of the French court. The French edition included an unfavourable critique of French mathematician Blaise Pascal’s religious writings, and a defence of Locke’s assertion that thought might arise via a material mechanism – an idea that had led nervous theologians on both sides of the Channel to assume Locke was saying there was no such thing as an immortal soul.

To tangle with religious dogma was dangerous. But Voltaire’s support of Locke’s and Newton’s ideas also challenged French national pride. One of his essays criticised the “Cartesians” who dominated Paris’s Academy of Sciences. These men – followers of 17th century philosopher René Descartes – had great difficulty with Newton’s theory of planetary motion. How could the Sun’s gravity reach across millions of kilometres of empty space to influence the planets? They thought it smacked of pseudo-science – like astrology or alchemy. This is ironical, in hindsight, because today we regard Descartes’s theory as pseudo-scientific, with its swirling vortices of invisible ether dragging the planets in their orbits. No one knew what this ether was made of, or why it swirled like a tornado. Voltaire pointed out the hypocrisy of believing in magical ethereal whirlpools while rejecting gravitational attraction. His essay shows that for many 17th century theorists, the rules of what constitutes a truly scientific theory had not yet gelled.

Mathematics was crucial to Newton’s approach. Not that Voltaire was on top of the mathematical subtleties that showed just how superior Newton’s theory was – he would need Émilie’s help for that. But such help would have to wait because in April 1734, Voltaire’s French publisher released Lettres in France without his permission. An arrest warrant was issued and Voltaire went into hiding. Émilie raged to her friends that France’s treatment of its greatest writer was unjust. Her appeals to the authorities, as well as those of her husband and other aristocratic friends, bore fruit. Voltaire was allowed to return to France, where he lived under a kind of house arrest at the Châtelets’ run-down château at Cirey, in Champagne.

When Émilie moved to Cirey to join her lover, tongues wagged salaciously, even hatefully, because she had dared to flout the rules of propriety. She and Voltaire set about turning Cirey into an informal academy where they studied, wrote, discussed philosophy and hosted free-thinking intellectuals. It was an idyllic arrangement, although sometimes Voltaire felt he wasn’t working hard enough on his poetry and plays. “Too often,” he said, “the supper, Newton and Émilie carry me away”. He was referring to their preparations for a serious popularisation of Newton’s ideas, to be called Elements of Newton’s Philosophy.

Portrait of Voltaire, the controversial Enlightenment playwright who took up Newton’s cause after living in London during his exile from France. – Leemage / Corbis

Émilie would take this project even further in a 180-page “commentary” she appended to her translation of Newton’s Principia. This included a relatively accessible reader’s guide to the main arguments in Newton’s gravitational theory of planetary motion. It also described applications of Newton’s theory by her eminent mathematical friends and sometime tutors, Alexis Clairaut and the dashing Pierre-Louis Moreau de Maupertuis, as well as an update on Newton’s gravitational theory of the tides by their colleague, Swiss mathematician Daniel Bernoulli. Émilie’s appendix also included her own reworking of some of the Principia’s key proofs in the language of calculus. Newton (and independently German mathematician-philosopher Gottfried Leibniz) had invented calculus – the maths that describes and predicts how things change, such as the position of a falling apple or a planet in the sky. But apparently Newton felt calculus was too new to convince people of the validity of his radical gravitational theory. Instead he established most of his arguments with ingenious but idiosyncratic geometrical proofs – the kind of logical, rigorous approach perfected by the ancient Greeks. Émilie re-wrote some of these proofs using the cutting-edge dy/dx calculus notation that had been developed by Leibniz.

Emilie’s brilliance lay in her ability to understand the subtleties of both Newton’s and Liebniz’s philosophy.

Émilie’s fame among European intellectuals came not from her translation of the Principia but from an earlier work of popular science – called Institutions de Physique (Fundamentals of Physics) – in which she bravely attempted to integrate the work of Newton and Leibniz. Scientific opinion at that time tended to favour either the Englishman or the German. It was not only about nationalism, it was also a debate about what constitutes a theory of nature. Newton focused on providing testable explanations for what we can observe in the Universe, while Leibniz emphasised philosophical questions about the nature of existence. Émilie’s brilliance lay in her ability to understand the subtleties of both Newton’s theory and Leibniz’s philosophy.

Voltaire, on the other hand, was entranced with Newton and didn’t bother much with Leibniz – he and Émilie remained in feisty disagreement on the matter. In his novella, Candide, he would lampoon Leibniz’s “best possible world” philosophy – Leibniz’s attempt to reconcile God’s goodness with the suffering and evil in the world.

By the mid-1740s, however, it was Emilie’s work on the Principia that was closest to her heart – although translating 500 pages of Latin and intricate geometry, and checking and re-checking her calculus proofs, was arduous. “I have never made such a sacrifice for reason as I have by staying here and finishing this book. It is an awful job, for which one needs a head and a constitution of iron,” she lamented. Nevertheless, both she and Voltaire had been seduced by Newton’s logic. In showing how profoundly the human mind can penetrate the mysteries of nature, Newton gave his disciples hope that reason would triumph over superstition, ushering in a rational, secular approach not only to “natural philosophy”, but also to politics and ethics.

Emilie du Chatelet integrated the work of Newton and Leibniz in her work, Fundamentals of Physics. – ViaLibri

In particular, Émilie and Voltaire realised that Newton had created the blueprint for modern theoretical physics. He did this by keeping religion and philosophy separate from what we can actually observe, and from what we can infer from those observations. The Cartesians, on the other hand, belonged to the past – an era when scientific theorists were primarily philosophers. The natural philosopher’s job was not so much to measure and quantify as to be metaphysical – to look “beyond” physical observations to the ultimate cause or nature of a phenomenon. Take the ether whirlpool hypothesis: it was an attempt to imagine what might be causing the planets to move through the sky, and it was consistent with the “self-evident” notion that forces must make direct contact with objects if they are to move them. But there was no evidence for this ethereal substance. Nor did the ether theory predict anything about the actual movements of the planets. In fact, Newton showed a mathematical vortex was incompatible with German astronomer Johannes Kepler’s planetary observations.

Newton, on the other hand, began with Kepler’s analyses. (Kepler in turn had spent years sifting through Danish astronomer Tycho Brahe’s observations of the planets’ positions in the sky at different times of the year.) Kepler was able to fit a mathematical curve to each planet’s orbit and he realised they form ellipses around the Sun. He also found relationships between the size of an orbit and the time it took for the planet to orbit the Sun. Kepler’s laws provided an accurate mathematical description of these orbits.

But Newton’s goal was to develop a theory about why the planets moved at all.

Newton used Kepler’s analysis to show that the force needed to move a planet on an elliptical path around the Sun must obey an “inverse-square law”. That is, as the distance from the Sun increases, the force becomes weaker; for instance, if the distance increases two-fold, the force is only a quarter of what it was before – the inverse of two squared. So planets further from the Sun experience less force and have wider, slower orbits. Newton also showed that moons with circular orbits, and comets whose orbits were elliptical, parabolic or hyperbolic, were also governed by this law.

But Newton’s genius did not stop there: he realised this celestial inverse square force was the same force that makes apples fall on Earth. In other words, planets and moons were falling around their parent body. Galileo had explored the nature of the acceleration due to gravity by rolling metal balls down a plank, publishing his results in 1638, 49 years before the publication of the Principia. To show that this same gravitational force was acting on the Moon, Newton calculated the Moon’s circular acceleration (based on its speed and distance from Earth) and found that it was about 1/3,600th of the acceleration of a falling body here on Earth. The Moon is about 60 times further from the centre of the Earth than we are on its surface, so the inverse-square law fitted!

Of course, Newton’s chain of reasoning was much more complex than this. What is important, from a modern perspective, is that his theory could be used to make testable predictions. Newton lived to witness one of its earliest confirmations – the total solar eclipse of 1715 that caused darkness to fall across England, northern Europe and northern Asia. It was a thrilling public occasion, but an even more spectacular confirmation came 32 years after his death: the return of Halley’s comet in 1759.

In the 1730s and 40s Émilie and Voltaire helped to articulate and popularise Newton’s extraordinary achievement – a paradigm shift in our understanding of the Universe. In Elements of Newton’s Philosophy, Voltaire said the Cartesian emphasis on metaphysical causes was “the surest way of losing our way. Instead, [like Newton] let us follow step by step what actually happens in nature: like voyagers who have arrived at the mouth of a river, we must travel up the river before imagining where its source is located.”

Not even Newton’s great contemporaries Leibniz and Christiaan Huygens had understood Newton’s paradigm: they agreed the theory of gravity was a mathematical tour de force that accorded remarkably well with the physical evidence of planetary motion. But they were not convinced gravity could act across the vastness of the Universe. Since Newton had given no idea of how this might happen, they rejected his theory as a return to mysticism.

Newton had, in fact, tried unsuccessfully to find a mechanism by which gravity acted, but he refused to include untested speculations in his rigorous Principia. He left such a discovery to posterity (to Einstein, so far), saying: “It is enough that gravity really exists and acts according to the laws we have set forth, and is sufficient to explain all the motions of the heavenly bodies and of our sea [the tides].” Enough indeed: today we know that for most applications within the Solar System, Newton’s theory is accurate to one part in ten million.

Emilie and Voltaire helped to articulate and popularise Newton’s theories.

But not even Newton got everything right – especially when it came to the nature of light and heat. Back in 1738, this was such an open question that the Paris Academy of Sciences made it the topic of its annual essay competition, which Voltaire planned to enter. He and Émilie had an impressive laboratory with a large reflecting telescope, high-quality prisms, lenses and accurate measuring scales. They were fascinated by Newton’s optical experiments – including those that proved white light is made up of the spectrum of colours. But Newton did not have a theory about the fundamental composition of light, although he suggested it was made of tiny particles.

Voltaire assumed that heat, too, was made of particles. With Émilie’s help he heated huge amounts of metal in the forge at Cirey, weighing the metal before and after heating to see if he could detect an increase in mass – the mass of the additional heat particles. After months he had no consistent results, and Émilie began to believe heat had no weight. Voltaire was so passionately Newtonian that he wouldn’t listen to her arguments – which included the possibility that the extraneous charcoal coating the burned metal would weigh more than the heat, so that the experiment could never work.

Émilie also believed, contrary to Newton, that light had no weight. Her conclusion was based on an ingenious thought experiment. She calculated that even if a light particle weighed less than a trillionth as much as a cannon ball it would feel like a cannon ball when it hit our eyes because it travelled so fast!

She had other innovative ideas about light and heat, too – for instance, that the different colours of light would have different amounts of energy and different temperatures, a conjecture that would be confirmed half a century later. Émilie did not chase up her ideas on light and heat with experiments. But she expressed them in a solo entry for the Paris Academy’s essay competition. It was submitted anonymously and in secret. She didn’t want to hurt Voltaire’s feelings by publicly disagreeing with him – and didn’t want to expose herself to ridicule, as a woman daring to enter male territory. The only person she trusted with her secret was her husband! (He joined the Cirey household during rare respites from his military service – and his mistresses.)

As it turned out, Voltaire was proud of Émilie’s essay, and thought she should have won the competition. And she thought he should have done: she complained to a friend that the Academicians were too Cartesian to be impartial. But not so partial that they didn’t agree that each of the papers from Cirey was sufficiently interesting for publication, along with the winners, in the Academy’s proceedings. And so Émilie became the first woman to have a scientific paper published in this prestigious journal.

Emilie’s commentary and translation of Newton’s Principia was published a decade after her death. – Camille Souget Librairie

There is no happy ending to this story. In 1749 – having penetrated the scientific establishment in a way few women had ever done – Émilie died as only a woman can die, after giving birth (to the child of her new lover, the Marquis de Saint Lambert). She was 42. Society gossips believed she’d got what she deserved for living so outrageously, so freely. Voltaire stayed with her until the end. Although they were no longer lovers, he’d remained “a tiny planet in her vortex, hobbling along in her orbit”, as he wrote in a letter to a friend.

Émilie had hoped her work on Newton would live forever. But soon after her death, her scientific reputation faded, too. Voltaire lost interest in science, and her Principia languished in a drawer – until Clairaut ushered it into print. He’d checked her calculus proofs in the months before she died, and he’d refined Newton and Halley’s calculations to obtain the accurate prediction of the return of Halley’s comet in 1759. What better celebration than to publish Émilie’s book in the same year! She would have been delighted: she’d known the comet held a key to securing Newton’s reputation.

And I am delighted to be able to celebrate her achievements here, and to honour the sacrifice she made “for reason”. She was an inspiration to me in my own journey into higher mathematics, and she is continuing to inspire women – and men – because of her mathematical achievements against such odds, and her courage in living life to the full.

What makes a great theory today?

Forget the apple. It was the maths that led Newton to his theory of gravity. – Photos.com / Getty Images

Technology has opened the window to a Universe that seems even more mysterious than it did three centuries ago. We are dazzled by dark energy, dark matter, and quantum-generated multiverses consisting not only of our own “best possible world” – but every possible world. What happened to Newton’s method: proceed from a set of physical principles and mathematical laws to a beautiful theory? Some of these new ideas seem to take us back to Descartes’s ether.

But Newton would probably have had no trouble with our weird modern theories. Like his theory of gravity, modern theories make bold leaps that the Cartesians would never have accepted. We still don’t really know what gravity is, so what would the Cartesians have made of such counter-intuitive ideas as mass-energy equivalence, or curved space-time or antimatter? Although these ideas are weird, they have arisen from theories based, like Newton’s, on experiment and mathematics – in contrast to the ether.

A great theory does not have to be – indeed cannot be – a perfect fit to the physical world. Newton’s gravitational theory was long ago subsumed into Einstein’s general theory of relativity, which itself will no doubt need updating as we learn more about the Universe. Yet both theories successfully predicted new and unexpected physics, and they remain incredibly accurate: Newton’s theory fits observed planetary motion to better than 0.0001%, while Einstein’s theory has passed all experimental tests so far. This is why they will always be what Roger Penrose called, in The Emperor’s New Mind, “superb” theories. (Others in this category include Maxwell’s theory of electromagnetism, quantum electrodynamics, and quantum mechanics.)

During the process of finding new theories, or adapting old ones so they are a better fit with more accurate data, all sorts of wild conjectures emerge and heated debates take place. Using Penrose’s terminology, these emerging theories can be categorised as “useful”– such as the Big Bang standard model – or “tentative” – such as the cyclic model of the Universe, or string theory that unites gravity with the other fundamental forces. A theory will become “superb” only when it has made predictions that are experimentally confirmed at a high level of accuracy and with a wide range of applications or explanations.