It is amazing that Kepler determined his three laws by looking at data, without a calculator and using only pen and paper. It is conceivable how he proved his laws described the data after he had already conjectured them, but what I do not understand is how he guessed them in the first place.
I will focus in particular on Kepler's third law, which states that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of the orbit.
I assume that Kepler was working with data about the planets only, plus our own moon, and the sun. I make this assumption because I don't think Kepler had data about other moons, comets, or asteroids, which had not been observed by telescope yet. If this is true, knowing that Neptune, Uranus, and Pluto were not yet discovered when Kepler was alive, this means Kepler had less than 9 data points to work with.
My friend claims that it is totally concievably how Kepler guessed this relationship (although he provides no method of how Kepler might have done it), and also that Kepler's observations are "not that hard". As a challenge, I gave my friend a data table with one column labeled $x$, the other $y$, and 9 coordinates $(x,y)$ which fit the relationship $x^4=y^3$. I said "please find the relationship between $x$ and $y$", and as you might expect he failed to do so.
Please explain to me how in the world did Kepler guess this relationship working with so few data points. And if my assumption that the number of data points Kepler had at his disposal is small, is wrong, then I still think its quite difficult to guess this relationship without a calculator.