James Burke did a very nice little explanation of Kepler’s calculations in one of the The Day the Universe Changed episodes. #5 in fact. Go to 31:00 here. Not a lot of math, but it gets the idea across.
Just a note here- if you are working with degrees and minutes, 90 degrees and 60 minutes, a 4-digit trig table should be adequate for calculations (90x60=5400). They had better than that by the time of Kepler.
I probably saw this programme when it came out. It does answer my questions quite well along with the passage cited by DPRK. I think the French guy must have had a more advanced method than the one described in the passage for calculating the distance of Uranus in such a short period of time.
By French guy, I assume you mean Laplace – of course he had a more advanced method than Kepler’s, because as a starting point he assumed all of Kepler’s laws and Newton’s laws of motion. Anyway, by the late 18th century they also had access to more and more powerful astronomical instruments and observations, so precision of seconds of arc was possible.
ETA as for calculating the orbit in a short amount of time, since you know the dynamics you can do it with a minimum of 3 celestial observations, and they already had many more than that for Uranus
Kepler, as we saw, used a geometric triangulation technique to get many points on the orbit of Mars, which he was able to fit on an ellipse. If you don’t care about a few percent error, you can even do this while assuming the orbit of the Earth is perfectly circular and that the Earth and Mars orbit in the same plane.
In practice, you’d do a first approximation based on assumptions like that, then you’d use the results of that to improve your model of the Earth’s orbit, then use the improved orbit of the Earth to make a new model for Mars, and so on. It converges fairly quickly.
There are also methods for solving for all of the parameters of all of the objects at once, but those are impractical without a computer.
It is fortunate, I think, that Kepler focused on Mars, and that Mars has a fairly eccentric orbit (it varies from 209 m km to 249 m km from the sun). If Mars’ orbit were more circular, like that of Venus (varies from 107 to 108 m km), he might never have discovered that the planets orbit in ellipses.
Kepler was also fortunate to have some really great data on Mars collected by Brahe.
I think that he focused on Mars precisely because it’s more eccentric. His first attempt would of course have been based on the assumption of circles. When he was trying to improve it, he’d naturally focus his attention on the cases where the circle model worked least well.