 Recall that in addition to his emphasis on the “precise and exact” motions that can be used to describe legitimately geometrical curves, Descartes also claims that these curves “can be conceived of as described by a continuous motion or by several successive motions.” As such, we would reasonably expect that the geometrical construction of these curves should not proceed pointwise in the manner of Book One, where Descartes constructed the Pappus curves by solving the equations to which the problem had been reduced. However, when Descartes treats the five-line Pappus Problem in Book Two, he in fact offers a pointwise construction of the Pappus curve. He then remarks that the pointwise construction of this “geometric” Pappus curve is importantly different from the pointwise construction of non-geometrical, “mechanical” curves: