There has always been a mystery about how Kepler (1571-1630) discovered the laws of planetary motion: not until more than half a century later did Newton show how they can be explained within a gravitational framework. It is generally assumed that Kepler based his achievements on the full panoply of classical Greek geometry, and certainly he was very familiar with the advanced techniques developed by Archimedes and Apollonius, which he often referred to in his optical and stereometrical work. Yet I have found that for his astronomical success Kepler relied on nothing more complicated than the methods and results of Euclid. And since he did not know what sort of solution his enquiry would produce, my historical account of Kepler's discovery turns into a tale of detection – which began at the University of Tübingen in the early 1590s.