Eight arcminutes of Mars break the perfect circle
A stubborn eight-arcminute gap between Tycho's Mars positions and every possible circular orbit forced Kepler to the ellipse, and rebuilt astronomy around it.
Kepler did not choose Mars. He was handed it. Arriving at Tycho Brahe's establishment outside Prague in 1600, he was assigned the planet nobody could model, and reportedly bet a colleague he would have its orbit finished inside eight days. It took him roughly five years of arithmetic done entirely by hand — logarithms would not be published until 1614 — and he later admitted to having worked the Mars calculations through some seventy times.
His first success is the part he threw away. Kepler built what he called the vicarious hypothesis: a thoroughly old-fashioned model, circular orbit, equant point, the full Ptolemaic toolkit, tuned until it reproduced Tycho's observed longitudes of Mars to within two arcminutes. By the standard of every astronomer who had ever lived, that was a solved planet. Then he tested the same model against the planet's latitudes, which pin down how far away Mars actually is and not merely which direction it lies in. The fit collapsed — the Sun–Mars distances the model implied were out by tens of per cent. Correcting the geometry so the distances came out right threw the longitudes off by eight arcminutes.
Eight arcminutes is about a quarter of the apparent width of the Moon, and sits near the edge of what an unaided eye can register at all. Ptolemaic astronomy had lived comfortably with errors several times larger. Kepler refused to, and his reason was not aesthetic — it was that he knew exactly how good the data were. Tycho did not make eight-arcminute errors. "These eight minutes alone," he wrote in chapter 19, "will have led the way to the reformation of all of astronomy."
What followed, in his own word, was a war on Mars. He tried an egg-shaped oval and found it wrong in the opposite direction from the circle, overshooting by almost exactly as much as the circle had undershot. Somewhere in that symmetry the answer was sitting in plain sight, and he missed it for months before recognising the ellipse in 1605. Astronomia Nova, printed in 1609, sets out the first two laws: a planet traces an ellipse with the Sun at one focus, and the line from planet to Sun sweeps equal areas in equal times.
The circle had been the assumed shape of celestial motion since Plato, and it was Mars that broke it. Mars has the most eccentric orbit of the planets a pre-telescopic observer could measure well, so its departure from a circle was the largest available — and Tycho's record was the only dataset precise enough to prove the departure was real rather than sloppiness. Had Kepler been assigned Venus, whose orbit is very nearly circular, he could have finished his career believing in circles and been unable to tell the difference.