The Calculus of Explanations

Note: I’m going to start publishing these sort of pop-science history posts more regularly over at thecalculusofexplanations.substack.com – if you like this kind of thing, please follow me there!

I recently visited Paris for the second time. Unfortunately I was too early to witness the fix for The lie at the heart of the Eiffel Tower. However there is of course a lot more to see in Paris, especially if you’re interested in the history of science.

In particular I made sure to visit the Panthéon, which I highly recommend both for its historical and scientific interest. I think it’s worth pausing here to clarify the difference between (at least) three different buildings in Europe that share remarkably similar names – because I can’t be the only one that’s gotten them confused.

The Parthenon refers to the remains of a temple in Athens dedicated to Athena, from the Greek parthénos meaning ‘virgin, unmarried woman’.

The Pantheon is a temple ‘to all the gods’ (Pantheion) in Rome, used as a Catholic church since the 7th century.

The Panthéon is a grand domed monument in Paris, located in the Latin Quarter on the Montagne Sainte-Geneviève.

The Panthéon has been many things. Due to a confusing and turbulent history, (it was originally built as a church of St Genevieve) it’s a tribute to the Catholic Church, the Monarchy, the revolutionary figures of the 18th century, and a mausoleum holding the remains of some of the great men and women of the republic.

To be honest the reason I wanted to visit was it was also the site of Foucault’s pendulum experiment – a replica of which slowly swings back and forth in the exact centre of the building – a 28kg spherical mass hanging from the triple-domed roof by a 67m cable.

You can see a video I took below. I promise I haven’t slowed it down – the period of the pendulum (time it takes to complete one oscillation) is proportional to the square root of its length, and its a very long pendulum!

Léon Foucault was an engineer who is famous for the elegance and genius of his experiments, among which was the first serious attempt to measure the speed of light.

The experiment he’s most famous for however is the pendulum that bears his name, which was the first definitive proof that the Earth turns on its axis.

The idea is stunning in its simplicity – a heavy mass swinging in one direction has a lot of inertia, and thus tends to remain moving in that direction.

As the Earth (and thus the church) spins beneath the pendulum, it appears from our perspective that the pendulum is changing the direction of its oscillation – a phenomenon called “precession”.

The genius of Foucault was to design a hinge mechanism that would allow the pendulum to not only swing, but to smoothly rotate (or rather, to allow the church to rotate as the pendulum’s motion stays fixed).

At the North Pole this precession would take 24 hours as the Earth spins below the pendulum. As you change latitude it takes longer, with the effect disappearing as you get to the equator.

At the latitude of Paris the precession takes about 32 hours, and you can set your watch by it – there are markings around the pendulum that allow you to tell the time using the precession, and the angle noticeably changed during our visit.

Underneath the Panthéon is the crypt, which holds the remains of many important revolutionary figures, as well as more modern heroes that resisted the Nazis during WW2. There are also famous mathematicians and physicists such as Lagrange, Painlevé, Langevin and Carnot.

There is only one statue in the crypt, which is that of the famous philosopher Voltaire.

Voltaire is a household name, and for good reason. A key figure of the Enlightment, a philosopher, writer and satirist, he is famous for his witty quips and criticism of the Church.

“Those who can make you believe absurdities can make you commit atrocities.”

One of the most famous quotes attributed to Voltaire is actually a misattribution

“I disapprove of what you say, but I will defend to the death your right to say it.”

which is apparently from writer Evelyn Beatrice Hall, of course writing under a pseudonym, in a work summarising Voltaire’s views.

But we aren’t here to talk about Voltaire, who is already well known.

Far less well known is his long term romantic partner, the mathematician and physicist Émilie du Châtelet.

Gabrielle Émilie Le Tonnelier de Breteuil was born in Paris in 1706 into an aristocratic family. By all accounts, Émilie received a better than average education as a child, but of course as a young woman the imperative given to her took the form of an arranged marriage followed quickly by the birth of three children.

After an agreement with her husband that they would maintain a household for their children but otherwise live separately, Émilie reentered social, and equally importantly, intellectual life.

By 28 Émilie was studying with Alexis Clairaut, an excellent mathematician who, later on in life, used Newton’s work on gravity to solve a particular case of the 3-body problem – this gives us a hint about the types of contributions Émilie would end up making as well.

She took an avid interest in public debates, going as far as dressing like a man in order to enter the salons where scientific ideas were discussed, after being rejected due to her sex.

It was likely in one of these intellectual hot houses that she met Voltaire – recently returned from England, where he had become an admirer of Isaac Newton.

The relationship blossomed, with Émilie hosting Voltaire at her husbands estate, where they built huge libraries, installed research laboratories and conducted experiments and research in physics.

At some point they both separately entered a competition to explain the nature of fire, where apparently they disagreed – Voltaire thought that fire must have mass (ha, what an idiot), where of course the hero of our story, Émilie correctly guessed it does not.

In fact, Voltaire did acknowledge Émilie as his equal or perhaps even his superior intellectually, proclaiming that she was

“A great man whose only fault was being a woman.”

During this period of intense collaboration, Voltaire introduced Émilie to Newton’s masterpiece, the Philosophiæ Naturalis Principia Mathematica, a work of such shocking scope and depth that it redefined science and mathematics forever.

At this time the European continent still subscribed to the ideas of René Descartes – that nature was mechanical, that space was filled with a fluid called “aether” and the planets were carried around the sun by twisting vortices in this fluid.

In the Principia Newton puts forth another view: that space was truly empty, there was no need for the aether, that the Sun and planets had a “gravity” that pulled them together, a force that was proportional to the inverse of the distance between two bodies squared. That this force required no fluid, no medium through which to transmit its effect, it simply followed from all observable data and experiment.

That the laws of motion derived from this basic relationship explains the movement of projectiles and planets alike, the curves of comets and the motion of the moon, the flow of tides and the oscillations of the axis of the Earth. A single principal unifying terrestrial and celestial mechanics. To explain all this however, Newton required a new tool, one that he had to develop himself. That tool we now know as the Calculus.

However genius the Principia’s methods and results, and however popular it may have been in England, there were some issues with it gaining wide acceptance in the rest of Europe.

It was written in Latin, which few people read. There were only a few hundred copies printed, with astronomer Edmond Halley shouldering the cost of the enterprise – the Royal Society paid him in volumes of their first publication “A History of Fishes”.

And finally, Newton’s results were, counterintuitively, expressed largely using classical geometry rather than the calculus that he himself was developing, still very much in its nascent form.

Enter Émilie du Châtelet, who spent years translating the Principia into French. But her work went much further than simple translation. She added hundreds of explanatory notes, including a few corrections of Newton’s mistakes. Read that again. Do you know the size of the ovaries required to CORRECT ISAAC NEWTON!?

But her real stroke of genius was to realise that calculus, the tool that Newton had developed to describe the dance of the celestial spheres, was itself an incredibly valuable thing – not just a tool but an enormously important framework for developing new mathematics. Much of her work translating the Principia involved extending and clarifying the explanations using a more modern form of calculus than Newton conceived. Her work is still the standard translation in French to this day.

In addition to this weighty achievement, Émilie participated vigorously in the vis viva (“living force”) debate – an argument over exactly which quantities were conserved by a body in motion. Whether momentum, the product of mass and velocity was the only important quantity, or whether something even more fundamental was at play.

Émilie carefully examined experiments, especially those of Willem s’ Gravesande, in which balls dropped into soft clay produced craters whose depth increased with the square of impact speed.

She championed the idea that the product of mass and velocity squared was just as important as momentum, one of the foundational ideas of all later physics. Today we place a factor of one half in front of this product, and refer to this concept as kinetic energy.

Émilie du Châtelet was a translator, a synthesiser, a collaborator and an educator – but she also had her own deep and original mathematical insights.

She bridged philosophy and science, arguing that physics should be underpinned by a rigorous mathematical foundation.

She brought Newtonian physics to France, and shed a little bit more light on the concepts of force, motion and energy. She transformed him into something the rest of Europe could read, understand and build upon, forming the essential basis of our modern industrial and technological world.

She also championed the education of women and transformed the Château de Cirey into one of Europe’s leading centres of scientific scholarship.

She stands a legitimate French icon and should be just as well known as Descartes, Rousseau and yes, even Voltaire.

Émilie died in 1749 from childbirth, aged only 42.

Émilie completed the final revisions to her translation of the Principia while heavily pregnant. She died shortly after giving birth, aware that childbirth was dangerous at her age and reportedly working with great urgency to finish the manuscript.