“Science is written in the language of mathematics,” proclaimed Galileo in 1623. And over the past few centuries science has become ever more mathematical. Nowadays, mathematics seems to hold total hegemony, particularly in the fields of quantum physics and relativity – the teaching of modern physics seems to involve deriving an endless series of equations.
But though it is an important tool, mathematical analysis is not the only way of approaching scientific enquiry. Scientists also need to develop concepts on which to build the mathematics and carry out experiments to test and demonstrate their ideas. And they also need to translate the equations back into physical concepts and verbal explanations to make them comprehensible. These other aspects have long been undervalued – in both the teaching and practice of physics – and this has damaged and is continuing to damage our understanding of the world around us.
Nowhere is this better exemplified than in the science of rotation and spin, which might at first glance appear to be a shining example of the triumph of mathematics. In his 1687 magnum opus
Principia, Isaac Newton laid out the mathematical workings of our solar system: he showed how the laws of motion and gravity explain how the planets orbit around the sun, and how the spin of the earth causes it to bulge, drives the tides and makes its tilted axis slowly wobble. Over the next hundred years, Newton’s analysis was extended and translated into modern mathematical language. All the problems of cosmology appeared to have been solved, the first of many occasions when scientists have mistakenly thought they had uncovered all the secrets of the universe.
Yet Newton’s triumph was only made possible by his more down-to-earth contemporary Robert Hooke. It was Hooke who made the conceptual leap that an object moving in a circle is travelling at a constant speed but is also accelerating at right angles towards the centre of the circle. He also went on to show experimentally how a universal gravity could provide the force that causes the planets to orbit around the sun and the moon around Earth. He hung a large ball, representing Earth, from the ceiling and a small ball, representing the moon, from the large ball, before pulling them away from vertical and setting them moving. The tension in the ropes, representing gravity, provided the inward force that kept them travelling around in a circle.
Unfortunately, Newton, who came to dominate world science, had little time for such conceptual and experimental approaches, insisting that equations were the only way to describe physical reality. His influence impeded further conceptual advances in mechanics and consequently progress in cosmology. For instance, it delayed our understanding of how the solar system was created.
The accepted model – the nebular hypothesis – was put forward in the 18th century by such luminaries as the philosopher Immanuel Kant and the mathematician Pierre-Simon Laplace. The hypothesis proposed that the solar system formed from a spinning ball of dust and gas. Gravity flattened the ball into a disc before the attraction between the particles pulled them together into planets and moons, all orbiting in the same plane and in the same direction.
All seemed well until the 1850s when engineers such as William Rankine finally developed a new mechanical concept – the conservation of angular momentum – 150 years after the conservation of linear momentum had been accepted. This new concept revealed a potential flaw in the nebular hypothesis that had remained hidden in Newton’s equations. To have shrunk to its size and to spin so slowly, the sun must have lost almost all its angular momentum, something that seemed to break this new law of nature.
It was only 40 years ago that a convincing explanation was proposed about how the sun lost its angular momentum. The charged particles shot out by the sun in the solar wind are channelled within magnetic fields before being flung out slowing the spin of the material that remained and allowing gravity to draw it inwards. It was only two years ago that this explanation was finally verified by the Parker Solar Probe, which found that the solar particles were channelled up to 32 million kilometres outwards before being released. And only in October 2023 did the James Webb Space Telescope reveal the same process occurring in the newly forming solar system of the star HH212.
The overreliance on mathematics also delayed our understanding of how the spin of Earth makes it habitable. By the end of the 18th century, Laplace had derived equations describing how Earth’s spin deflects bodies of water moving over its surface. However, even he failed to observe that it would also affect solid objects and gases, so his work was ignored by the early meteorologists.
This only changed in 1851, when the French physicist Jean Foucault produced a free-hanging pendulum that demonstrated Laplace’s forces in action. The forces diverted the bob to the right during each sweep so that its plane of swing gradually rotated, like a Spirograph drawing. Not only did this prove the spin of Earth to a sceptical public, but it showed schoolteacher William Ferrel that Laplace’s forces would also deflect air masses moving around Earth’s surface. This would explain how global air currents are deflected east and west to form the three convection cells that cover each hemisphere and create the world’s climate zones, and how they divert winds into rotating weather systems, creating depressions, hurricanes and anticyclones. Modern meteorology was born.
In 1835, the French engineer Gaspard-Gustave de Coriolis produced more general equations describing the forces on bodies moving within a rotating reference frame. However, since these were in a paper examining the efficiency of water wheels, his work was largely ignored by scientists. Instead, it was a simple experiment that enabled geophysicists to understand how Earth’s spin diverts fluid movements in its interior and produces its magnetic field.
In 1911, the British physicist G. I. Taylor investigated how beakers of water behave when they are set spinning. The water quickly spins with the beaker and its surface rises in a parabola until the extra pressure counters the centrifugal force on the water. What’s interesting is how the water behaves when it is disturbed. Its movement changes the centrifugal force on it, as Coriolis’s equations predicted, so that when heated from below, it moves not in huge convection currents but up and down in narrow rotating columns. This discovery led the geophysicists Walter Elsasser and Edward Bullard to realise that the same forces would deflect convection currents in Earth’s metal outer core that are driven by radioactive decay. They are diverted into north-to-south columns of rotating metal that act like self-excited dynamos, producing the magnetic field that shields Earth from charged particles. A simple laboratory demonstration had illuminated events in Earth’s core that had been hidden in Coriolis’s equations.
Today, perhaps the most damaging failure to translate the mathematics of spin into easy-to-grasp concepts is in the fields of biomechanics and sports science. Our bodies are complex systems of rotating joints, but despite the sophistication of modern motion analysis software, few researchers realise that accelerating our joints can produce torques that actively accelerate our limbs. Biomechanics researchers are only starting to realise that accelerating our bodies upwards at the start of each step swings our arms and legs when we walk, and that a sling action straightens them at the end of each step.
In the same way, when we throw things, we use a multi-stage sling action; rotating our shoulders accelerates first our upper arm, then our forearm and finally our hands. And the reason we can wield heavy sledgehammers and swing wooden clubs to smash golf balls down the fairway is that their handles act as further sling elements; they accelerate forwards due to the centrifugal forces on them without us having to flex our wrists. Failing to articulate these simple mechanical concepts has made biomechanics ill-equipped to communicate with and help physiotherapists, sports coaches and roboticists.
And there is still confusion about the simplest aspects of rotation among physicists. Even Richard Feynman, for instance, was unable to explain the so-called Dzhanibekov effect – why spinning wing nuts on the International Space Station flip every few seconds. This was despite the fact that the mathematician Leonhard Euler had shown this should happen almost 300 years ago. The same is also true of more down-to-earth events: how children power playground swings and how cats land on their feet, for example.
The truth is that the basics of physics, despite involving simple mathematics, are harder to grasp than we tend to think. It took me two years, for instance, to master just the science of spin and rotation for my latest book. We need to spend more time thinking about, visualising and demonstrating basic physical concepts. If we do, we could produce a generation of physicists who can communicate better with everyone else and discover more about the world around us. The answers are probably already there, hidden in the equations.
