In this paper, I examine the manner in which Descartes defends his Vortex Hypothesis in Part III of the Principles of Philosophy, and expand on Ernan McMullin’s characterization of the methodology that Descartes uses to support his planetary system. McMullin illuminates the connection between the deductive method of Part III and the method Descartes uses in earlier portions of the Principles, and he brings needed light to the role that imaginative constructions play in Descartes’s explanations of the phenomena. I develop (...) McMullin’s reading by bringing further attention to the constraints that Descartes places on the imagination in Part III. I focus in particular on the way in which Descartes uses metaphysical truths concerning God’s nature to support his general description of the planetary system, and on the way he relies on a mathematical standard of intelligibility to defend his proposals about the configuration of matter. Attending to the role of metaphysics and mathematics in Part III shows that Descartes’s arguments for the explanatory power of the Vortex Hypothesis are more effective than McMullin suggests. The reading I forward also offers important perspective on how Descartes’s hypotheses in Part III can be seen as both metaphysically and mathematically well-grounded. (shrink)