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It was months and years of difficult geometrical calculations from measurements made by Tycho Brahe, by his employee Johannes Kepler.

When Tycho was alive he would not let Kepler see the results of his work, but he let slip a few figures, tantalisingly, now and again. But after Tycho's death Kepler was clever and devious enough to get his hands on the results, which was fortunate because he had the ability to make use of them.

Kepler chose the orbit of Mars as a special subject, which was lucky because it has a relatively high eccentricity that shows up the difference between an ellipse and a circular orbit. After some false starts he realised that the figures fitted an elliptical orbit. Subsequently further observations demonstrated that elliptical orbits gave correct predictions of the positions of Mars and the other planets.

Kepler used the heliocentric principle, with the Sun at the centre. Although Kepler and Galileo were contemporary, Galileo never believed that the planets' orbits could be elliptical, and he stuck to the old systems of circles and epicycles of Ptolemy and Copernicus.

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