 We notice in Descartes’ remarks concerning geometry in particular that the “entirely new science” he proposes will provide an exhaustive classification for problem-solving, where each of his three classes is determined by the curves needed for solution. This suggests an important overlap between Descartes’ three classes of geometrical problems and Pappus’s three classes, which, recall, were separated based on the types of curves required for solution: Planar problems are solvable by straightedge and compass, solid problems by conics, and line-like problems by more complicated curves that have an “inconstant and changeable origin.” However, there is also a significant difference between their classifications insofar as Descartes strongly suggests that those problems that require “imaginary” curves for their solution do not have a legitimately geometrical solution. Namely, just as some problems of arithmetic “can be imagined but not be solved,” so too in geometry, there is a class of problems that require curves that are “certainly only imaginary,” i.e., curves generated by “diverse motions,” and thus that are not geometrical in a proper sense. In this respect, Descartes is moving from Pappus’s descriptive classification to a normative one that separates geometrical curves from non-geometrical curves, and thereby distinguishes problems that have a legitimate geometrical solution from those that do not. Just as importantly, we see in Descartes’ letter his attempt to expand the scope of legitimate geometrical constructions beyond straightedge and compass by appealing to the motions needed to construct a curve. Specifically, as we see in the passage above, Descartes relies on the “single motions” of his “new types of compasses, which [he says] are no less exact and geometrical…than the common ones used to draw circles” in order to mark out a new class of problems that have legitimate geometrical solutions.