Section II, Chapter Four: Inching Towards Universal Law
Astronomer Johannes Kepler’s (1571 - 1630) improvements to the Copernican model ultimately caused Newton to formulate his universal theory.
Note—this is a draft of Chapter Seven of a book I’m writing. The book is going to cover humanity’s deepest ideas from philosophy, physics, epistemology, and economics. Each chapter is meant to be short and digestible. Most of the chapters will explain just one or two ideas.
‘Chapter []’ indicates a future chapter that I’ve not yet written.
Section II, Chapter Four: Inching Towards Universal Law
In Chapter Five, we saw that Galileo improved upon Aristotle’s physics of motion by demonstrating that all objects accelerate towards the Earth at the same rate. And in Chapter Six, we explored Galileo’s principle of relativity, which implies that the laws of motion are the same in all reference frames. While Galileo’s concepts were critical preliminary steps towards the culmination that was Newton’s discovery of classical mechanics (see Chapter []), it was astronomer Johannes Kepler’s (1571 - 1630) improvements to the Copernican model that ultimately caused Newton to formulate his universal theory.
In conjecturing a heliocentric model of the solar system, Copernicus had resolved a number of theoretical issues with the Ptolemaic model (see Chapter Four). But Copernicus’ vision of planets orbiting the Sun in perfect circles was not quite right, and the dance of the planets told a far richer tale than Copernicus could have imagined without more granular and voluminous data.
The grueling work of gathering said data fell on the shoulders of Tycho Brahe (1546 - 1601), who had sought to track Mars’ orbit. Decades later, when Kepler pored through Brahe’s data, he noticed that most of the numbers corresponded to what one would expect if Mars’ orbit was indeed circular. But Kepler couldn’t ignore the glaring exceptions. As Livio quotes Kepler (brackets and ellipsis his): “If I had believed that we could ignore these eight minutes [of arc; about a quarter of the diameter of a full moon], I would have patched up my hypothesis…accordingly. Now, since it was not permissible to disregard, those eight minutes alone pointed the path to a complete reformation in astronomy.”
Kepler didn’t just show that Mars’ orbit was elliptical rather than circular. With Brahe’s data in one hand and his own calculations on that data in the other, Kepler formulated three laws of planetary motion:
Planets move in elliptical orbits with the Sun as one of the ellipse’s foci (every ellipse has two foci—points along its longer axis with particular mathematical properties that don’t concern us here).
Trace out the path of any planet from time t0 to t1. Then draw straight lines from the endpoints of that path to the Sun. No matter where the planet is on its trajectory, the area of the traced-out region is a constant for a given time interval. In other words, a planet sweeps out equal areas for equal times.
The square of a planet’s orbital period—the time it takes to complete one revolution around the Sun—is proportional to the cube of the longer radius of the ellipse that the planet traces out on its path.
To be sure, Kepler exerted a great deal of creativity and effort to produce his laws of planetary motion (he published the first two in 1609 and the third a decade later). Still, they are not quite as robust as the laws of modern physics: they lack an underlying explanation, and so we cannot say whether they apply to, for instance, planets of other solar systems. Said another way, Kepler’s laws are ‘merely’ mathematical expressions of regularities he matched to Brahe’s data. Expressions that describe regularities using precise mathematics without explanation are called phenomenological.
As Mlodniow writes, “In a sense, his laws were beautiful and concise descriptions of how the planets move through space, but in another sense they were empty observations, ad hoc statements that provided no insight about why such orbits should be followed.”
For decades, Kepler’s ad hoc improvements to the Copernican scheme languished in stasis, floating in the ether and untethered to robust explanation. Finally, in 1684, astronomer Edmond Halley met with architect and astronomer Christopher Wren and scientist Robert Hooke at the Royal Society of London to figure out the origins of Kepler’s phenomenological laws. Their proposed solution was “that Kepler’s laws would all follow if one assumed that the [S]un pulled each planet toward it with a force that grew weaker in proportion to the square of the planet’s distance, a mathematical form called an ‘inverse square law.’” For instance, if you triple the distance between a planet and the Sun, the attractive force between them decreases ninefold.
In the fall of the same year in which Halley and his colleagues made their conjecture, Newton sent Halley a nine-page treatise showing once and for all that, indeed, “all three of Kepler’s laws were…mathematical consequences of an inverse square law of attraction.”
In his proof, Newton relied on the idea that orbital motion is really the sum of two independent motions—a ‘tendency’ or ‘want’ to move in a straight line in the direction of its motion at a given instant, as well as a ‘tendency’ to fall in the direction of the Sun via an attractive force. These two are always at right angles to each other, and Newton used his own mathematical invention—calculus—to sum up the contributions of each of these two tendencies at each infinitesimal point along a planet’s trajectory.
As Mlodinow summarizes, “Orbital motion, in this view, is just the motion of some body that is continually deflected from its tangential path by the action of a force pulling it toward some center.”
Ecstatic that he’d been vindicated, Halley urged Newton to publish his treatise with the Royal Society. But Newton had caught his mouse and wasn’t quite finished playing with it: “Now I am upon this subject, I would gladly know the bottom of it before I publish my papers.”
Newton had unified free fall and orbital motion and explained Kepler’s laws, but there was still a yawning chasm between the physics of earth and sky. How did Galileo’s discoveries cohere with Newton’s recent accomplishments? What did the existence of an attractive force between the Sun and the planets imply about attractive forces between relatively miniscule objects like cannonballs and apples on the surface of the Earth? And for that matter, did these ‘tendencies’ of motion of celestial objects carry over to our everyday world? Finally, Newton was familiar with Galileo’s principle of relativity, which surely nudged him towards universalizing his nascent framework to all objects, all motion, all forces.
Three years later, in 1687, Halley got far more than what he’d bargained for. Newton published Principia Mathematica—and with it, humanity’s first scientific system of the world.
Thanks to Dennis Hackethal for early feedback.