Eight minutes of arc

The best eyes in history

Copernicus changed the picture but not the data. This course puts the Danish astronomer Tycho Brahe on the same level as Copernicus in the story of the sun-centered system. That’s striking, because Tycho never accepted that the earth moves. His contribution was measurement.

Tycho built the finest observatory of his age and designed his own great instruments: quadrants, sextants, and armillary spheres, some several meters across. Night after night for twenty years, he measured the positions of the stars and planets to within a minute or two of arc, a thirtieth of a degree or better. One minute of arc is roughly the width of a quarter seen from the far end of a football field. Some historians call him the finest naked-eye observer who ever lived. His data were many times more precise than anything astronomers had before.

Tycho had his own model of the heavens, a compromise between the old and new systems. The earth stays still at the center. The sun and Moon circle the earth, and all the other planets circle the sun. It fit his observations as well as Copernicus’s system did, and it spared him the absurd-seeming idea that the huge, heavy earth was flying through space.

What the telescope showed

Galileo Galilei (1564–1642) is the bridge between ancient and modern physics, and his greatest gift was the experiment. In 1609 he heard of a Dutch invention for making distant things look near, built his own, improved it, and turned it on the sky. What he saw in the next few years struck at the heart of the classical cosmos:

• The Moon had mountains and valleys. It wasn’t a perfect, smooth sphere; it looked like another earth.
• Jupiter had four moons of its own circling it, not the earth. Not everything in the heavens revolves around us.
• Saturn looked oddly shaped, as if it had “ears.” The rings wouldn’t be recognized until the 1650s.
• Venus went through a full set of phases, from a thin crescent to nearly full, as the sun-centered system predicted. Ptolemy’s arrangement could not produce them.
• The sun had dark spots that came and went. Even the sun was blemished.

Each discovery was a blow to the idea of a perfect, unchanging heaven. The heavens began to look like the earth: rough, changing, and made of ordinary stuff. The news shook educated Europe, and eventually brought Galileo before the Inquisition.

Seventy tries

Johannes Kepler was lucky twice over. He was already convinced the sun sat at the center. And when Tycho died in 1601, Kepler got hold of his observations, over the objections of Tycho’s heirs. They were the best astronomical data in the world, and they would stay unmatched for generations.

He set out to calculate the orbit of Mars. What followed is a model case of how a law of physics is found. The data and the theory disagreed. Kepler trusted the data completely, so he changed the theory. He adjusted the size of the circle, its center, the planet’s speed, and calculated everything again by hand. He wrote that he had gone through the whole procedure at least seventy times.

His best circle came extremely close, but somewhere along the orbit it missed Tycho’s positions by eight minutes of arc, about a quarter of the width of the full Moon. Most astronomers would have blamed the observations. Kepler didn’t:

Since divine goodness has given us in Tycho Brahe a most diligent observer, from whose observations this error of eight minutes is revealed, … these eight minutes alone have led the way to the reform of all astronomy.

Johannes Kepler, New Astronomy, 1609

First he let the planet’s speed vary. Still no fit. Then he gave up the perfect circle itself and tried other shapes, until he reached the ellipse. The orbit fit. The result was three laws, the first two published in 1609 and the third in 1619:

1. Each planet moves in an ellipse, with the sun at one focus.
2. A line from the sun to the planet sweeps out equal areas in equal times. The planet moves fastest when it is closest to the sun.
3. The square of a planet’s orbital period is proportional to the cube of its average distance from the sun.

With these laws, the epicycles vanished. Each planet has one orbit, one shape, and one set of rules shared by every planet. The backward loops follow naturally, as Earth overtakes the outer planets. It was the simplicity and unity physicists love, and the classical cosmos could not survive it.

The snug cosmos breaks

Five beliefs sum up the classical cosmos. Gods, or at least a supreme intelligence, made the world. The earth is the motionless center of the universe. The heavens above the Moon are perfect. The universe runs according to purposes. And each element has its own nature. Picture it as a 世外桃源, a hidden paradise: a lovely, well-watered earth, the stars circling it, and the realm of the gods wrapped around it for protection. It was a small, snug, comfortable universe, where the earth and humans were special and everything had its purpose.

Its greatest achievement was to stop explaining nature as the will of the gods and to start asking what its laws are. But ancient physics was still observation plus speculation, and neither half was reliable. The speculation carried untested inheritances from myth, like the perfection of circles. The observation lacked measurement. Chinese tea culture gives an example. Tea connoisseurs described water coming to the boil by its bubbles: first “crab eyes,” then “fish eyes.” But is your crab eye the same as mine? Medieval Europe divided daylight into twelve hours, so a summer hour was longer than a winter hour. Without shared, precise measures, careful reasoning could still be wrong. From Galileo on, instruments and experiments took their proper place at the center of physics.

A universe too big to imagine

If the earth really orbits the sun, the nearby stars should seem to shift slightly as we move from one side of the orbit to the other, the way a nearby tree shifts against distant hills as you walk. Nobody could see any such shift. That had been an argument for a motionless earth. Now it meant something far stranger. The stars must be so distant that the shift is too small to see, and the universe unimaginably larger than the snug cosmos anyone had pictured. It might even be infinite.

The earth was no longer the center, no longer specially favored, no longer wrapped in protective spheres. It was one planet among others, floating in an enormous space. And if the earth wasn’t the center of the universe, the element earth had no natural place to fall toward. The core of Aristotle’s physics collapsed, and the rest of his cosmos came down with it. In this reading, it was humanity’s second great collapse of a worldview; the first was the passage from myth to the classical cosmos. The loss coincided with the Renaissance, with anger at the church and at a rigid social order. A new science was urgently needed to fill the vacuum. That is the subject of Course 4: Newton’s mechanical universe.