A moving Earth, a flattened Earth

A pendulum in the Pantheon

In the spring of 1851 Parisians were invited to a strange show. Under the dome of the Panthéon, the great domed building on the Left Bank, hung a heavy ball of about 28 kilograms (62 pounds), a brass shell filled with lead, on a steel wire 67 meters (220 feet) long. A ring of sand lay on the floor beneath it, and a small pin on the ball nicked the sand at every pass. The ball swung back and forth, slowly, once every sixteen seconds or so. The newspaper that announced the exhibition put a question to its readers: “Have you seen the earth turn?”

The man behind it was Léon Foucault, a young French physicist who had tried the idea first in his cellar with a two-meter wire and a five-kilogram weight. His reasoning was simple. A pendulum is free to swing in any direction, and once it is swinging it has no reason to change the plane in which it swings. If the floor turns under it, the plane of the swing must seem to turn the other way. Within about half an hour the line of nicks in the sand had indeed begun to creep around. At the latitude of Paris the swing appears to turn about eleven degrees an hour, clockwise, a full circle in about 32 hours. At the North Pole it would take a day; at the equator it would not turn at all.

Visitors were not seeing a trick. They were watching the earth rotate, for the first time, on a human scale. Yet people had argued about whether the earth moves for more than two thousand years. This lesson follows that argument, and a second one that began as soon as the first was settled: if the earth is a sphere and it spins, what shape is it, exactly?

Why the earth stood still

Almost everyone who thought about it believed that the earth was at rest, and for good reasons. Aristotle argued that earth, the heavy element, moves naturally toward the center of the universe, so the earth lies at the center and has nowhere to go. He also noted what any of us can check: a stone thrown straight up comes down in the same place. Aristotle mentions that some Pythagoreans thought otherwise. In their picture the center held a fire, not the sun, and the earth, together with a hidden counter-earth, circled it. He rejected the idea as bending the facts to fit a theory.

In the third century BC Aristarchus of Samos went further: according to Archimedes, he put the sun and the stars at rest and made the earth circle the sun. His book is lost, but Archimedes records the answer he gave to the obvious objection that the stars would then appear to shift as the earth moves: the sphere of stars is so large that the earth’s whole orbit is like a point compared with it. That was correct, and it was ahead of anything anyone could measure.

The best statement of the case against a moving earth is in Ptolemy’s Almagest (Book I, chapter 7), written around 150 CE. If the earth turned once a day, Ptolemy reasoned, it would be the fastest of all motions; clouds and anything thrown or flying would seem to be left behind to the west; if the air went along with the earth, nothing in it would ever seem to outrun the air. He even granted that a turning earth made the heavens simpler. But the world we live in, he said, would then look quite unlike the world we see. These were reasonable objections, and nobody could answer them until the idea of inertia was found.

Copernicus’s On the Revolutions of the Heavenly Spheres was printed in Nuremberg in 1543, the year he died. (Course 3, The cosmos before Newton, tells that story; here we follow only what it did to the earth.) Copernicus gave the earth not one motion but three: a daily rotation, a yearly circuit of the sun, and a third, slow turning of its axis that kept the axis pointing the same way through the year. He still used circles and small circles on circles, and the earth became the third planet from the sun. The book left two great objections standing. Nobody could see the stars shift as the earth moved, and nobody could say why we feel nothing of the speed.

The second objection fell first. In his Dialogue Concerning the Two Chief World Systems (1632), Galileo asked readers to shut themselves in a cabin below decks on a large ship with some flies, butterflies, a bowl of fish and a dripping bottle. Nothing inside would change whether the ship stood still or sailed smoothly along. The argument had older roots (Copernicus had a version), and Galileo presented it as reasoning, not as an experiment. In October 1640 the French philosopher Pierre Gassendi did try the experiment on a galley at Marseille, dropping a stone from the mast, and found that it landed at the foot of the mast whether the ship was moving or not. In 1687 Newton made the principle a law: a body keeps its state of rest or of steady motion in a straight line unless a force acts on it.

That is why we do not feel the earth’s motion. At the equator the ground moves at 465 meters a second (about 1,040 miles an hour), and around the sun we travel at nearly 30 kilometers a second, but we feel acceleration, not steady speed. The sideways acceleration of the earth’s spin is only about a third of one percent of gravity.

The earth becomes a planet

If the earth moves, it is a body among bodies. Galileo drew that conclusion in Sidereus Nuncius (March 1610). On a thin crescent moon the dark part is faintly lit, the “ashen light.” Galileo explained it as sunlight reflected from the earth: the moon is lit by the earth just as we are lit by the moon. Leonardo da Vinci had given the same explanation a hundred years before, in a private notebook. Galileo put it in print, and promised to show that the earth moves and shines brighter than the moon. Seen from the moon, the earth would be a planet.

The idea that people could go and look was not new. In the second century CE the Greek writer Lucian sent a ship to the moon in his comic True History, and in his Icaromenippus a man with an eagle’s and a vulture’s wings sees an earth far smaller than the moon looks. In 1516 Ludovico Ariosto’s hero rides a chariot to the moon in Orlando Furioso. What changed after Copernicus was the seriousness of the question. Kepler’s Somnium (written about 1608, published in 1634 after his death) imagines the sky of people living on the moon, in which the earth is their moon, showing phases. Fiction after Kepler began to ask what a journey would take and what one would see. The earth was a place you might see from outside.

Proof that the earth moves around the sun took a long time. In 1725 the English astronomer James Bradley was looking for the apparent shift of a star, called parallax, that the earth’s motion should cause. He found a yearly shift, but of the wrong kind: the star moved on the wrong schedule. By 1728 he had the answer: starlight arrives at a moving earth the way rain falls into a moving car, slanted by our speed. This is the aberration of light, and its size, about 20.5 seconds of arc, matches the ratio of the earth’s speed to the speed of light. Bradley announced it in January 1729. It was the first direct evidence of the earth’s orbital motion.

Parallax itself, the thing Bradley had been looking for, was found in 1838. The German astronomer Friedrich Bessel measured the faint star 61 Cygni against its neighbors with a new instrument and found a shift of about 0.31 seconds of arc, which put the star about ten light-years away. (Modern measurements give 0.286 seconds of arc and 11.4 light-years.) That is the angle a U.S. quarter takes up when seen from about ten miles away. Two others had results at about the same time: Thomas Henderson for Alpha Centauri (reported in 1839) and Friedrich Struve for Vega (tentatively in 1837, more firmly in 1839). Copernicus had said the stars must be very far away. Now there was a number.

The numbers are humbling. Make a model in which the distance from the earth to the sun is one meter. The sun is then a marble about a centimeter across, the earth a speck as thick as a hair, and Pluto 40 meters away. The nearest star, Proxima Centauri, would be about 270 kilometers (170 miles) away. Measuring the stars had given the earth an address, and it was not a prominent one.

This course reads a human lesson into that change. Before Copernicus, it suggests, the earth was the unique stage of the world; afterward it was one planet among planets, and people began to ask whether they belonged on it or might leave it. There is something to this, though it is only a reading: the same discovery also made the earth a place, a body with a shape, a size, and a history, which could be studied as a whole.

The onion and the olive

A sphere that spins is not quite a sphere. Newton saw this in the Principia (1687). Every part of the earth is pulled toward the center by gravity, but at the equator the spin throws things outward, and the equator should bulge. Newton worked out how much, by asking what shape a rotating ball of fluid must have so that water in channels from the pole and from the equator would balance, and he found that the earth’s axes should be in the ratio 230 to 229: the earth is about 1/230 flatter at the poles than a sphere. The Dutch scientist Christiaan Huygens, who was no follower of Newton, reached 1/578 by a different assumption in 1690.

There was a measurement already pointing the same way. In 1672–73 the French astronomer Jean Richer took a pendulum clock to Cayenne, in French Guiana, five degrees north of the equator. It lost about two and a half minutes a day, and to keep Paris time he had to shorten the pendulum by about 2.8 millimeters. Newton used the result: the pendulum swings slower where gravity is weaker, and gravity is weaker at the equator, partly because of the spin and partly because the equator is farther from the center.

Yet the shape remained disputed, because another set of measurements said the opposite. Measure the length of one degree of latitude, that is, how far you must travel north for the pole star to rise one degree. On a sphere it is the same everywhere. On a flattened earth the surface is flatter near the poles, so a degree is longer there. On an earth stretched toward the poles, like an egg or an olive, it is shorter. In 1718 Jacques Cassini and colleagues finished measuring the arc of the meridian across France, from Dunkirk to the Mediterranean. They found the northern degrees slightly shorter than the southern: 56,960 toises against 57,097 (a toise is about 1.95 meters). That meant an earth stretched at the poles. The Cassini family held that view, and many followers of Descartes agreed with them; Mairan, a leading Cartesian, argued for it in 1720.

The usual telling, and the one this course started from, is a duel between two theories. Descartes imagined space filled with swirling vortices that carried the planets and pressed things toward the center; his followers, in this telling, expected the swirl to squeeze the earth into an olive. Newton, who attacked the vortices, expected an onion. The history is messier (see “Another view”), but the stake was real: if the earth were an olive, Newton’s gravity was in trouble.

The French Academy of Sciences decided to settle it by measuring. It sent one expedition to the equator and another to the far north. The equatorial expedition left La Rochelle on 16 May 1735 for what is now Ecuador: Louis Godin, Pierre Bouguer, and Charles-Marie de La Condamine. They measured an arc of about three degrees in the Andes near Quito, working at altitude for years. The fieldwork ended in 1743, eight years after they set out. They found a degree of about 56,750 toises, or 110.6 kilometers. The northern expedition, led by Pierre Louis Maupertuis with Alexis Clairaut and the Swedish astronomer Anders Celsius, reached the Gulf of Bothnia in 1736. They measured a 14-kilometer baseline on the frozen Torne River, triangulated north into Lapland, and came home in 1737 (Maupertuis survived a shipwreck on the way). They found a degree of about 57,438 toises, or 111.9 kilometers: longer. The Paris degree was 57,060. The degree grows toward the pole. The earth is an onion.

Voltaire, who was in London at the time of Newton’s death and helped carry Newtonian ideas to France, had already summed up the quarrel in his Letters on the English (published in French in 1734): in Paris, he wrote, you picture the earth as shaped like a melon; in London it is flattened on both sides. When Maupertuis returned, Voltaire is said to have congratulated him on having flattened the poles and the Cassinis. The argument did not end overnight. Jacques Cassini attacked the Lapland work, and Celsius defended it. A second French survey of the Paris meridian in 1739–40 found the degrees growing northward after all, in line with Lapland. Newtonian gravity was not widely accepted in France until the 1740s, but by the middle of the century the flattened earth had won.

Modern satellite measurements give the earth’s flattening as 1 part in 298.257. The equatorial radius is 6,378.137 kilometers and the polar radius 6,356.752, a difference of 21.4 kilometers. That is not a lot in a planet about 12,700 kilometers across, but it is more than twice the height of Everest above sea level. Newton’s figure for the bulge, about 28 kilometers, was too large because he assumed the earth had the same density throughout; a dense core makes the flattening smaller. And the earth is not even a perfect ellipsoid: sea level departs from the smooth shape by up to about 100 meters, higher near Iceland and lower south of India. The shape of the earth is a measurement that keeps being refined.