Cartesian co-ordinates

Descartes’ analytical geometry is often loosely defined as the application of algebra to geometry. In fact this was old hat, having been explored by many mathematicians from Archimedes onwards. His actual contribution was to develop a framework and techniques for describing geometric curves in algebraic terms.

Descartes’ great innovation was to identify a point in a plane by specifying its distances (say x and y) from two fixed lines drawn at right angles in the plane. Consider an equation f(x,y)=0 which may be satisfied by an infinite number of values x and y. These determine the co-ordinates of an infinite number of points which form a curve. The equation therefore expresses geometrical properties of the curve true at every point on it. Properties can then be worked out and expressed algebraically. (Phew….)

This development opened up a realm of possibilities as many geometric problems could be translated directly into equivalent algebraic terms for solution. For example, Descartes pointed out that by specifying two curves in the plane you could identify where they cross by finding the roots of the resulting equations.

René then went on to examine the interesting geometrical properties of curves using his new system. His most famous curve is the ‘folium of Descartes’ shown here. It may not look like much but for mathematicians it’s terribly exciting...honest.