 This distinction between the pointwise construction of “geometric” and “mechanical” curves serves two rather important purposes in the program of La Géométrie: (1) Descartes can establish that the Pappus curves he has pointwise constructed are in fact “geometrical” and thereby complete his synthesis (or demonstration) of the general Pappus Problem, and (2) he can maintain a boundary between intelligible “geometric” curves and unintelligible “mechanical” curves. Without a clear indication of why the pointwise construction of a Pappus curve is “geometric,” Descartes would have to allow “mechanical” curves such as the spiral and quadratrix into the domain of geometrical curves, since these curves can also be pointwise constructed. Recall for instance Clavius’s pointwise construction of the quadratrix. According to Clavius’s description, we begin with a quadrant of a circle and then identify the points of intersection between segments that bisect the quadrant and segments that bisect the arc of the quadrant (see  figure 9).  That is, we identify the several intersecting points of segments which are constructible by straightedge and compass, and then, to generate the quadratrix, we connect the intersecting points, which are evenly spaced along the sought after curve. Why is such a pointwise construction not “geometric”? Because, according to Descartes, if we proceed as Clavius does, “not any point of the required curve may be found at pleasure.” Specifically, given the restrictions of Euclidean construction, we are only able to divide the given arc into \(2^n\) parts. As such, what Descartes suggests is that it is not possible to divide the arc any way we please, and we cannot therefore locate any arbitrary point along the curve by use of pointwise construction. In the case of the “geometric” curves, however, we can find any arbitrary point on the curve by appeal to the equations corresponding to the problem; or borrowing Bos’s terminology, Descartes is claiming that “geometric” curves, and the Pappus curves in particular, can be generated by “generic” pointwise constructions.