I shall not believe that you are entirely cured of mathematics so long as you maintain that these tiny bodies, about which we were disputing the other day, can be divided in infinitum.
Chevalier de Meré, in a letter to Pascal
If the interpretation advanced in Chapter 1 is correct, the fundamental challenge facing Enlightenment matter theory was the apparent conflict between the geometrization of nature on the one hand, and the actual parts metaphysic on the other. Geometry seemed to mandate matter's infinite divisibility; the actual parts doctrine to preclude it. Chapters 2 to 4 dealt with the one side of this conflict: the actual parts doctrine and its various corollaries (genuine and otherwise). In the current chapter I look at the other side of the conflict: the geometrization of nature and the case for infinite divisibility.
This chapter focuses on the two central arguments for infinite divisibility. Of the various arguments for infinite divisibility presented in early modern period, these two are the most philosophically interesting and also the most popular: they can justly be considered the classic arguments on this side of the debate. First there is (i) the argument from geometry, which purports to establish infinite divisibility by way of certain constructions in classical geometry. Although this was by far the most popular argument for infinite divisibility during the Enlightenment, it is given short shrift nowadays: it clearly presupposes that physical space and matter conform to the axioms of classical geometry, and thus begs the question. (The objection is certainly lethal. However, as we shall see, the more sophisticated early modern advocates of the argument from geometry were well aware of this possible petitio charge and attempted to answer it in various