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Descartes and the Fly

Who invented the coordinate plane? Sam knows a math legend that answers this question:

 René Descartes was a French man who lived in the 1600s. When he was a child, he was often sick, so the teachers at his boarding school let him stay in bed until noon. He went on staying in bed until noon for almost all his life. While in bed, Descartes thought about math and philosophy. One day, Descartes noticed a fly crawling around on the ceiling. He watched the fly for a long time. He wanted to know how to tell someone else where the fly was. Finally he realized that he could describe the position of the fly by its distance from the walls of the room. When he got out of bed, Descartes wrote down what he had discovered. Then he tried describing the positions of points, the same way he described the position of the fly. Descartes had invented the coordinate plane! In fact, the coordinate plane is sometimes called the Cartesian plane, in his honor.

Sam likes this story, because it is about flies. Sam spends lots of time trying to find flies, just like Descartes does in the story. But is the story true? Or is it just a legend?

The story of the coordinate plane turns out to be a long story, with many parts. It starts long before Descartes, in Ancient Greece.