The Brachistochrone Problem

The [b]Brachistochrone problem[/b] is one of the most famous problems in the history of mathematics and was posed by the Swiss mathematician John Bernoulli in 1696 as a challenge ''to the most acute mathematicians of the entire world''. The problem can be stated as follows: Given two points on a plane at different heights, what is the shape of the wire down which a bead will slide (without friction) under the influence of gravity so as to pass from the upper point to the lower point in the shortest amount of time?

This applet will approximate the time of descent along three different paths: straight line (this is the shortest path but not the quickest), circular (this was the path investigated by Galileo) and the cycloid (the Brachistochrone). Drag the points C,D, etc. to obtain a polygonal path that approximates the curve.