Today, Kepler is perhaps best known for his three laws of planetary motion. Two of those laws were first introduced in his seminal work of 1609, Astronomia Nova, or the New Astronomy. Kepler's first law states that the planets travel around the sun in elliptical orbits, with the sun positioned at one of the ellipse's foci.
This was an almost heretical idea, even more so that Copernicus's new system. For over two thousand years, astronomers, philosophers, and theologians had believed that the planets traveled with uniform motion around circular orbits. In fact, part of Copernicus's intention in the creation of his system was to preserve circular motion. Who was Kepler to go against the wisdom of millennia?
Kepler's new law finally made sense of the astronomical data. His second law, which he actually discovered first, contributed to the demolition of the ancient assumptions. It stated that the planets swept out equal areas of their orbits in equal times. He was forced to dispose of the idea of circular planetary orbits, and had to reject the ancient belief that the planets traveled their orbits with a consistent speed. Instead, he tweaked the notion of uniform motion. Kepler discovered that the planets' speeds varied as they circled the sun – they went faster when they were at a point on their orbit closer to the sun than they did when they were farther away from it. But the area of the elliptical orbit that was covered in a certain amount of time always remained the same.
Kepler's first two laws were important for a number of reasons. They made sense of the universe's structure – astronomers could finally throw out the epicycles and the equant, and construct a simplified version of the Copernican universe. The epicycles had never been intended to model the actual motion of the planets; they were only there to preserve the appearance of uniform circular motion. Now that there was no need for such preservation, astronomy could for the first time describe the physical reality of the universe. Kepler also reiterated his belief that a force emanating from the sun causes the motion of the six planets. He was the first astronomer to fully address the cause of celestial motion, rather than the mere mathematical description of it.
The book itself offers an interesting insight into Kepler's mind, as he records the path he took to get to the two laws – mistakes and all. And Kepler made quite a few mistakes.
Kepler often commented later that had he not been assigned to work on the shape of Mars's orbit, he would never have figured out the planetary orbits. Only Mars's orbit was irregular enough to offer the necessary data. Kepler called it an act of "Divine Providence" that the problem had fallen into his lap. The Mars orbit represented Kepler's greatest challenge yet. In the dedication of the Astronomia Nova, he refers to it as "the mighty victor over human inquisitiveness, who made a mockery of all the stratagems of astronomers."
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.