Starting with the story of Poincare's work on the phenomenon of chaos, this study traces the history of attempts to solve the problems of celestial mechanics first posed in Isaac Newton's Principia in 1686. In describing how mathematical rigour was brought to bear on one of our oldest fascinations - the motions of the heavens - they introduce the people whose ideas led to the field now called nonlinear dynamics. In presenting the modern theory of dynamic systems, the models underlying much of modern science are described pictorally, using the geometrical language invented by Poincare. More generally, the authors reflect on mathematical creativity and the roles that chance encounters, politics, and circumstance play in it.