 In letters of November 1643, shortly after the initial exchange concerning the union of mind and body, Descartes sets Elisabeth the classic geometrical problem of the three circles or Apollonius’s problem: to find a circle that touches each of three given circles on a plane. While Elisabeth’s solution is no longer available, Descartes’s comments indicate that Elisabeth had already mastered techniques of algebraic geometry. She is thought to have learned them from Johan Stampioen’s textbook. Elisabeth’s approach to the problem seems to have differed from Descartes’s own, and Descartes remarks on her solution having a symmetry and transparency in virtue of its using only a single variable that his lacked. Elisabeth’s recognized mathematical acumen is also evidenced by her involvement in the hiring of Frans van Schooten to the mathematical faculty at Leiden and John Pell’s effort to enlist her help in understanding Descartes’s Geometry.