 Descartes’ reading of the ancients aside, what’s important for understanding his own peculiar interpretation of geometrical curves is the distinction he draws between the “accuracy of construction” of a curve, which he renders an issue for mechanics, and the “exactness of reasoning,” which he deems as the sole requirement for accepting a curve as legitimately geometrical. In making this claim, Descartes is carving out a unique place for his notion of geometrical curves: He abandons the “accuracy of construction” criterion that Clavius adopted in his early works to render a curve acceptable in geometrical problem-solving and also the claim forwarded by Viète that instrumentally-constructed curves were not to be considered geometrical (see  section 1.1  above). As Descartes’ presentation implies, both these sorts of criteria confuse issues of mechanics with the “exactness of reasoning” that is the sole concern of geometry. Thus, as Book Two continues, Descartes reiterates that to determine the geometrical status of a curve we must lay our focus on issues of exact and clear reasoning and, specifically, on the question of whether a curve can be constructed by exact and clear motions. After presenting the postulate that “two or more lines can be moved, one upon the other, determining by their intersection other curves,” Descartes explains,