Since the time of the ancient Greeks until only recently, geometry was studied and developed as a separate science. Today, we take for granted the many connections between mathematics and Euclidean geometry, but these connections were not so easily apparent to early thinkers. The quantity pi, by which the diameter of a circle is related to its circumference, is one of the few examples from antiquity in which some indication of the relationship of numbers to geometrical shapes was hinted at. But a more significant understanding of these relationships was not finally achieved until the seventeenth century.

In 1619, the French philosopher and mathematician René Descartes had a dream in which a fly flitted about the air of his bedroom. Descartes realized that the changing position of the fly in space could be described very accurately with respect to its distance from the walls, floor, and...