Box 9: Cadmus and Harmonia

Algebra and Geometry had developed and flourished largely apart from one another.  Today we have grown so used to their marriage that it is difficult to realize what a huge step it was getting these two together, a huge step taken independently by two 17th century French mathematical giants Descartes and Fermat:  the development of co-ordinate or analytical geometry.  

Mini Biography - Rene Descartes

Pierre de Fermat: Biography of a Great Thinker

These two mathematicians came at this development from two opposite perspectives:  Fermat started with an algebraic equation and then described the geometric figure that you get from it.  Descartes started with the geometric figure and showed how its properties could be described by an algebraic equation.

Many streams of thought flowed in together to lead to this development.  Omar Khayyam, the great Persian mathematician developed a geometric approach to solving the general cubic equation (the algebraic equation where x is raised to the third power).  Musical notation is essentially a graph of pitch and time.

Fermat & Descartes: Origins of Coordinate Geometry Part 1

Fermat & Descartes: Origins of Coordinate Geometry Part 2

 There is an amusing story of how Descartes came up with the idea of the co-ordinate grid for analytic geometry:

From the Math Education Page:  Cartesian Coordinate Objectives

How Descartes Created the Coordinate Plane

The conic sections studied by the ancient Greeks all have their graph equations:

Try these equations in a graphing calculator.  There are many free graphing calculators available online.  Here is an excellent full featured one to try, provided by Desmos.  If you click on "Edit Graph" (in the lower right corner) you can experiment with the ellipse and see the effect of altering various parameters: