# Q.E.D.

24 25 cIrclIng the cycloId a proof by dissection Start with a regular polygon under a line, mark one of its top corners, and start rolling it along the line. Every time the polygon comes to rest on the line, indicate the position of the marked corner by a dot. Stop when the coloured corner again touches the line and connect the dots by straight lines below, left. Dissecting the polygon, it quickly becomes clear that the area enclosed by the resulting curve is exactly three times the area of the polygon opposite top. Using a circle instead of a polygon, the resulting curve is a cycloid below, right, used with its relatives by the ancient Greeks to describe the orbits of the planets. Since a circle can be approximated by regular polygons, the area enclosed by the cycloid is also three times the area of the circle. The cycloid has many other important properties. For example, demonstrating the formidable power of Newtons and Leibnitzs new infinitesimal calculus, Johann Bernoulli proved in 1696 that the cycloid is the solution to the difficult classical problem of quickest descent. This means that if a particle slides along the cycloid from one of its ends to a second point, driven by gravity alone, it does so in less time than along any other curve connecting the two points. Puzzle out the two oneglance proofs lower, opposite