ANoteontheGeometry andDescartes’s MathematicalWork

The Geometry is the only mathematical book written by Descartes. One can find somemathematical studies from his youth—constructions of proportional means and solutions of equations of third and fourth degrees—in the Regulae, the Excerpta Mathematica, and above all in the letters, mainly those to Beeckman, Mersenne, and Golius (Serfati2002, 56–72 and 95–104). All these subjects are treated in the Geometry. Note also the De Solidorum Elementis (1639?), an unpublished treatise on (convex) polyhedra that includes the equivalent of Euler’s formula (1751), namely F − E + V = 2 (cf. Descartes 1987). Descartes finished the Geometry at the very last moment, as the Meteorology was in print. It is divided into three Books but is difficult to read since it does not follow a coherent scheme, a paradox upon which Descartes insisted. I will distinguish three “threads of Ariadne” to lead us through the text: the acceptability of curves, the problem of Pappus, and the construction of roots “using curves”.