Brachistochrone Problem -- from Wolfram MathWorld

Find the shape of the curve down which a bead sliding from rest and accelerated by gravity will slip (without friction) from one point to another in the least time. The term derives from the Greek (brachistos) "the shortest" and (chronos) "time, delay." The brachistochrone problem was one of the earliest problems posed in the calculus of variations. Newton was challenged to solve the problem in 16