Historians generally agree that Galileo's new science of motion, in taking its inspiration from Archimedean mechanics, was an explicit break from the traditional causal and qualitative treatments of motion within Aristotelian natural philosophy. Relatively few, however, have noticed that an Aristotelian mechanical principle lurks at the core of the new science, a principle that Galileo explicitly adopted from an Aristotelian work and then deployed precisely because of the limitations of Archimedean statics. In this paper I shall sketch the history of this principle within the tradition of renaissance mechanics and show how it was almost universally taken as the fundamental principle of mechanics. I shall then show how Galileo first adopted it in the 1590s, modified it in 1600, applied it in 1612, and finally appealed to it in 1638 in a causal proof of the single postulate in his new science of motion. In this way I intend to show precisely how Galileo, taking up where his predecessors in Archimedean mechanics left off, extended their mechanics into a science of motion that explained the motion of bodies from a common evident physical principle.