Eight arcminutes of Mars break the perfect circle

Kepler arrived at Tycho Brahe's establishment outside Prague in 1600, assigned the planet nobody could model. He reportedly bet a colleague he would finish its orbit in eight days. It took roughly five years of hand arithmetic; logarithms were not published until 1614. He later said he had worked the Mars calculations through some seventy times.

Kepler's vicarious hypothesis used a circular orbit, an equant and the full Ptolemaic toolkit, tuned until it reproduced Tycho's observed longitudes of Mars to within two arcminutes. He then tested it against the latitudes, which fix how far away Mars is, not only which direction it lies in. The fit collapsed: the Sun–Mars distances the model implied were out by a few per cent. Correcting the geometry to fit the distances threw the longitudes off by eight arcminutes.

Eight arcminutes is about a quarter of the apparent width of the Moon, roughly eight times what an unaided eye can resolve, and Ptolemaic astronomy had lived with errors several times larger. Kepler would not accept it: 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."

He called what followed a war on Mars. He tried an egg-shaped oval and found it wrong in the opposite direction, undershooting by almost exactly as much as the circle had overshot. He recognised 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. Mars has the most eccentric orbit of the planets a pre-telescopic observer could measure well — Venus's is very nearly circular — so its departure from a circle was the largest available, and Tycho's record was the only data precise enough to show the departure was real.