Geometry

Summary

The only mathematical work written by Descartes, La géométrie, was published by the Leiden printer Jan Maire in 1637 as the last of three tracts appended to the Discourse on Method. While the Dioptrics and the Meteors were said to use and exemplify Descartes’ philosophical method, the Geometry was actually held to demonstrate it (AT I 478, 621; CSMK 77–78).

Before 1636, there is no mention in Descartes’ writings of a publication corresponding to the Geometry. Nonetheless, the published work does take up ideas found already in the Rules and contemporary correspondence such as the numerical expression of powers and the construction of mean proportionals by the intersection of a circle and a parabola. Anticipating claims he would later make for the Geometry itself, he already announces to Isaac Beeckman in March 1619 his plan to formulate an “entirely new science” by which all problems that can be posed concerning all kinds of quantity, continuous or discrete, can be generally solved (AT X 156). In effect, his aim then, as later in the Geometry, was to show that after him nothing would remain to be discovered in mathematics (AT I 340, CSMK 51; AT II 361–62; AT X 157).

Descartes’ catalyst for writing the Geometry was the Pappus problem that Jacob Golius sent to him in 1631. The ancient solution of this problem was unknown to the seventeenth century, and Descartes recognized the importance of the interplay of algebraic equations and geometry in solving it. The solution that he discovered in 1632 was accorded a prominent place in the Geometry and served to illustrate the power of his geometrical calculus (AT I 323, 244, 478; CSMK 78).