Kepler, using very precise data provided by Tycho Brahe, recognized that an even greater simplification could come from abandoning the circular orbits of planets in favor of ellipses. He didn't reach this conclusion easily, and was never fully happy with it, apparently. He really tried to make Mars fit circles, but Brahe's data were too good.
Keplers Three Laws
Today, we remember Kepler's insight as 3 laws:
1. The orbits of the planets are ellipses, with the Sun at one focus.
This tells us that the motion is not uniform circular motion. Not
only is the shape of the orbit no longer a circle, but also the Sun
is not at the center. Remember the "eccentric" of Ptolemy?
This is now something that we expect, and only one position is allowed.
2. The line joining the planet to the Sun sweeps out equal areas in equal intervals of time.
This tells us that the planet sometimes moves quickly (when it is
closer to the Sun) and sometimes more slowly (when it is farther away).
3. The squares of the sidereal periods are proportional to the cubes of the semi-major axes.
What does this mean? It tells us that the planets are all obeying some common rules. It lets us figure out how long a new planet will take to go around the Sun if we know the size of its orbit. For example: The Earth takes 1 year to go around, and it is 1 astronomical unit from the Sun. So if there were a planet at 4 astronomical units, it would take 8 years to go around the Sun once: P squared = a cubed = 4x4x4 = 8x8 so P = 8.
Kepler was looking for patterns in nature, and he found three that
really mattered. These three rules helped Newton discover how gravity
works. Kepler also suspected there was some sort of force joining
the planets to the Sun, but it wasn't until Newton worked out the
details that we came to have a modern understanding of our solar system
and Earth's place in it.