They who knew not Glasses had not so fair a pretence for the Divisibility ad infinitum
George Berkeley, Philosophical Commentaries
The central task of Chapter 5 was to examine the two arguments for the infinite f-divisibility of space that I have been categorizing as the 'classic' or 'traditional' arguments. These two arguments are by far the most popular in the Enlightenment literature and are also the most philosophically rewarding to study. However, for the sake of completeness in this appendix I briskly outline two other types of argument for infinite f-divisibility that are also present in the period literature. As we shall see, they can each be dealt with fairly quickly.
This first argument points out that, if space were only finitely f-divisible, then motion would have to be a series of discrete leaps as each f-indivisible part of the moving body jumps successively from one space atom to the next. This contradicts the idea that the motion of bodies in space is a continuous, smooth matter. One might also push this point and claim that 'motion' through staccato atomic leaps would really be no motion at all. Rather than actually moving, the body would simply be repeatedly disappearing and reappearing in an adjacent location. Moreover, if we then add the assumption that time is only finitely divisible (and this assumption is often made by those who argue for minima of space—since time seems relevantly analogous to space in terms of the arguments for its ultimate structure), it might then seem that all motion would have to proceed at an equal velocity: a constant advance of one space atom for each time atom. The conclusion we are supposed to draw is that these results are absurd, and thus that space must be infinitely f-divisible after all. Versions of this argument are reported by Charleton, and endorsed by Barrow and Keill. 44