Abstract Alessandro Piccolomini and several other prominent 16th-century Aristotelians claimed that while a sphere touches a plane at a point in geometry, a material sphere touches a plane not at a point but over a small surface. These thinkers thereby called into question the reliability of geometric reasoning in physics. In this article I provide a detailed analysis of Galileo’s reply to such worries about geometry in the Second Day of his Dialogue Concerning the Two Chief World Systems. Because his infamous extrusion argument relies on the premise that a sphere touches a plane at a point, Galileo takes the opportunity to defend the argument and to attack geometry’s critics. According to the account I propose, Galileo argues for the applicability of geometry to physics by defending the legitimacy of geometric approximations and by advocating an expansive notion of geometric curve.