Luck was finally on Kepler's side. While he made a number
of mistakes in calculation and reasoning as he went along, they
always seemed to cancel each other out. At one point, he seemed
to have stumbled upon the right answer. His results almost matched
his predictions. They differed by only a very small error, eight
minutes of arc. No astronomer before Kepler would have paused at
such a figure – they would have blithely gone on and declared their
theory to be true. Kepler himself, at the time of his *Mysterium
Cosmographicum* had clung to his theory in the face of
conflicting data, deciding that the data must be wrong.

But Kepler had changed. Accuracy was now his watchword,
and eight minutes of arc error was unacceptable. So he threw out
his theory and went back to the drawing board.

In the end, it was the formulation of the first law that
gave him the most difficulty. Once Kepler was finally convinced
that the planetary orbits were oval-shaped, rather than circular,
he strove to find a mathematical formula that would describe the
shape of the ovals. But try as he might, he was unable to find
one. He worked on this problem for over a year, at one point complaining
to a friend that things would be so much easier if the ovals were
just ellipses. In fact they were, but Kepler was unable to see
it.

He worked and worked at the problem, finally coming up
with an equation that seemed to exactly describe the orbit. In
fact, it was the equation for an ellipse, but Kepler didn't recognize
it as such. While testing out his theory, he made a minor mistake
in the calculations and concluded that the equation must be incorrect.
Throwing up his hands in disgust, Kepler threw out the formula and
finally deciding to see what would happen if he treated the orbit
as if it was an ellipse. It wasn't until these calculations finally
led him to the same place he'd started that he realized he had
had the answer all along.