 Consequently, after the publication of Commandino’s translation of the Collection, early modern mathematicians gave added attention to the questions of whether and why these curves should be used in geometrical problem-solving. The spiral and quadratrix were prominent in such discussions, because, as noted above, they could be used to address some of the more famous outstanding geometrical problems, namely, angle trisection and squaring the circle.  [2]  For instance, in his second and expanded (1589) edition of Euclid’s Elements (which was first published in 1574) as well as in his Geometria practica (1604), Christoph Clavius discusses the status of the quadratrix. Accepting the objections to the description of the quadratrix detailed by Pappus in the Collection, Clavius supplies what he deems a “truly geometrical” construction of the curve that would legitimize its use in geometrical problem-solving, and in solving the problem of squaring the circle in particular. His construction is a pointwise one: We begin with a quadrant of a circle (as in Pappus’s description) but rather than relying on the intersection of uniformly moving segments to describe the curve, Clavius proceeds by first identifying the points of intersection between segments that bisect the quadrant and segments that bisect the arc of the quadrant. 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 (arbitrarily many) intersecting points, which are evenly spaced along the sought after curve. Therefore, to construct the quadratrix according to Clavius’s method, we still go beyond basic straightedge and compass constructions (connecting the points in this case cannot be done by straightedge, as in the case of bisection), but one need not consider the simultaneous motions of lines as Pappus’s construction requires. (See Bos 2001, 161–162 for Clavius’s construction of the quadratrix and compare with Pappus’s construction on Bos 2001, 40–42. For Descartes’ assessment of Clavius’s pointwise construction see  section 3.3  below.)