Find the curve down which a bead placed anywhere will fall to the bottom in the same amount of time. The solution is a
Cycloid, a fact first discovered and published by Huygens in *Horologium oscillatorium* (1673).
Huygens also constructed the first pendulum clock with a device to ensure that the
pendulum was isochronous by forcing the pendulum to swing in an arc of a Cycloid.

The parametric equations of the Cycloid are

(1) | |||

(2) |

To see that the Cycloid satisfies the tautochrone property, consider the derivatives

(3) | |||

(4) |

and

(5) |

Now

(6) |

(7) |

(8) |

so the time required to travel from the top of the Cycloid to the bottom is

(9) |

(10) |

(11) |

Now let

(12) | |||

(13) |

so

(14) |

**References**

Muterspaugh, J.; Driver, T.; and Dick, J. E. ``The Cycloid and Tautochronism.'' http://php.indiana.edu/~jedick/project/intro.html.

Muterspaugh, J.; Driver, T.; and Dick, J. E. ``P221 Tautochrone Problem.'' http://php.indiana.edu/~jedick/project/project.html.

Wagon, S. *Mathematica in Action.* New York: W. H. Freeman, pp. 54-60 and 384-385, 1991.

© 1996-9

1999-05-26