Plane As The Nose On Descartes's Face

$Mathematicians$

René Descartes is perhaps most famous for his work in analytic geometry, and more exactly, for his invention of the Cartesian Plane. What most people don't know is that ``La géométrie" is actually written as a demonstration or application to geometry of Descartes's method of reasoning as stated in the Discourse (Katz, p436). The actually purpose is not to invent a method of analytic geometry. In fact, the Cartesian plane as known today to us is not found anywhere in ``La géométrie". Instead Descartes names two somewhat arbitrarily chosen line segments and uses them as reference for the rest of the line segments in the problem. The rest of the line segments are then stated in terms of the ``principal" line segments. This produces algebraic equations which can be solved by algebraic means. In other words, algebra is applied to geometry or geometry is translated into algebra.(Descartes himself states that it isn't necessary to draw a line as long one can express it by means of a single letter (Descartes, p298)). Another interesting point to note is that Descartes never states that the two principle lines are perpendicular. This idea then, of as the Cartesian plane, came much later in mathematics. Also, his choice of ``principal" line segments varies from problem to problem. He chooses the line segments he feels will best serve his purpose.

A last little bit of information is that Descartes is using his method to solve the four line problem of Apollonius. The problem is to find points from which lines are drawn to four given lines at given angles yield a ratio between the products of two of the lines with the other two (Katz, p438). According to Katz, the four line problem of Apollonius is one which also somewhat inspired Fermat's analytic geometry not just Descartes.