I guess now they use a mix of observational data and supercomputer calculations, but how did they figure out the orbits of Uranus, Neptune and Pluto (no arguments about Pluto’s status please!) on the discovery of those?
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Kepler’s laws (which can pretty easily be derived from Newton’s law of gravity) tell that the orbit is an ellipse and planets move with certain speeds along their orbits. So, to describe the planet’s movement you only need to know two numbers: axis length (size) of the ellipse and its eccentricity (how stretched it is).
Do some measurements, gather a few points of data (you can describe pretty accurately at what angle the planet is seen from Earth) and then you have a system of equations with only two variables to find. Which can be solved with some trigonometry.
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