Brachistochrone Problem

Here is one of the most famous problems in the history of mathematics. It can be described in the following way: ***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?***



In **1638**, **Galileo** studied the problem in his famous work **Two new sciences**. His solution to the problem was to first find a **straight line** from point A to the point on a **vertical line** which it would reach the quickest. He calculated the time needed for the point to move from A to B in a straight line, then he showed that the point would reach B **more quickly** if it travelled along the two line segments AC followed by CB where C is a point on an **arc of a circle**.