It's amazing that Kepler derived his three laws emperically and then Newton rederived them from his own laws of motion. Its conceivable how Kepler derived the first and third laws, but the second law? How would Kepler have measured the area of the arcs swept out in equal time intervals just by looking at the data and why would he have the motivation to do this in the first place?
Kepler analyzed an enormous amount of data provided by Tycho Brahe. The data provided by Tycho Brahe must have represented a monumental effort.
In my opinion, it must have been more grueling for Brahe to collect all the raw data than it was for Kepler to analyze the data. The amount of number crunching Brahe did was staggering.
If you could plot all the data to scale on a sheet of paper and actually see the ellipses and the time lapses between locations, I don't think it was all that much of a leap to see that the longer, narrower slices of pie could possibly equal the areas of the shorter, fatter slices of pie. To me, that's pretty intuitive.
Motivations for finding answers to huge questions of the day? There are at least several: Fame, glory, wealth, prestige, the thrill of being the only person on Earth to know a truth for a short period of time before you publish your results, etc.
1$\begingroup$ No I don't mean motivation like that. I mean motivation for thiking about areas. Because its not so easy to draw an ellipse which evolves over time: you can graph a few points and write the point in time at each point, and the best thing you could do today is make an animation which traces the ellipse over time. Even using an animation, the areas relationship is not obvious. I would like some more detail please. $\endgroup$ Jan 24, 2015 at 4:07
$\begingroup$ @Joshua Benabou - I would say that the amount of data Tycho Brahe provided was so voluminous, that precise ellipses could have drawn by Kepler. After that, I think Kepler's intuition would have revealed the equal area possibility and he would have begun measuring. The "equal area in equal time" concept seems very intuitive to me. $\endgroup$ Jan 24, 2015 at 4:20
1$\begingroup$ Its not easy to notice that from data though, because you would need to draw the ellipse, and at each point on the ellipse, know the corresponding point in time. Just looking at time coordinates like that does not make the area thing obvious at all. $\endgroup$ Jan 24, 2015 at 5:11
$\begingroup$ Take this with grain of salt, but here is what I think probably happened. The orbits are almost circles, so the constant rate of area sweeping was most probably quite obvious and uninteresting first. The really difficult part was to realize that the discrepancies between description based on circular or else shaped models (ovals) of the orbit of Mars and the actual motion of the planet could be resolved by assuming the orbits are ellipses. $\endgroup$ Jan 24, 2015 at 5:32
$\begingroup$ I heard in some document this took Kepler some time to realize. He didn't think of ellipses as probable for quite some time. After he realized them, it was probably quite easy to check that if the radius is drawn from focus instead of the center, the rate of area sweeping was still constant. $\endgroup$ Jan 24, 2015 at 5:33