 In his 26 March 1619 letter to Beeckman, Descartes does not elaborate on the “new types of compasses” to which he refers; he simply reports to Beeckman in the early portions of the letter that he has, in a short time, “discovered four conspicuous and entirely new demonstrations with the help of my compasses” (AT X, 154). Fortunately, more details about these compasses and Descartes’ demonstrations are included in Cogitationes privatae, or Private Reflections (ca. 1619–1620), a text in which Descartes applies three different “new compasses” (often referred to by commentators as “proportional compasses”) to the problems of (1) dividing a given angle into any number of equal parts, (2) constructing the roots of three types of cubic equations, and (3) describing a conic section. In the first two cases, as Descartes treats the angular section and mean proportional problems, the compasses on which he relies are used to generate a curve that will solve the problem at hand.