During the 17th century the idea of an orbiting and rotating Earth became increasingly popular, but opponents of this view continued to point out that the theory had observable consequences that had never, in fact, been observed. Why, for instance, had astronomers failed to detect the annual parallax of the stars that "must" occur if Earth orbits the Sun? To address this problem, astronomers of the 17th and 18th centuries sought to measure the annual parallax of stars using telescopes. None of them succeeded. Annual stellar parallax was not successfully measured until 1838, when Friedrich Bessel detected the parallax of the star 61 Cygni. But the early failures to detect annual stellar parallax led to the discovery of a new (and entirely unexpected) phenomenon: the aberration of starlight. This paper recounts the story of the discovery of stellar aberration. It is accompanied by a set of activities and computer simulations that allow students to explore this fascinating historical episode and learn important lessons about the nature of science.