Successive application of Archimedes' Recurrence Formula gives the Archimedes algorithm, which can be used to
provide successive approximations to (Pi). The algorithm is also called the Borchardt-Pfaff
Algorithm. Archimedes obtained the first rigorous approximation of by Circumscribing and Inscribing -gons on a Circle. From Archimedes'
Recurrence Formula, the Circumferences and of the circumscribed and
inscribed Polygons are

(1) | |||

(2) |

where

(3) |

(4) | |||

(5) |

where . The first iteration of Archimedes' Recurrence Formula then gives

(6) | |||

(7) |

Additional iterations do not have simple closed forms, but the numerical approximations for , 1, 2, 3, 4 (corresponding to 6-, 12-, 24-, 48-, and 96-gons) are

(8) |

(9) |

(10) |

(11) |

(12) |

By taking (a 96-gon) and using strict inequalities to convert irrational bounds to rational bounds at each step,
Archimedes obtained the slightly looser result

(13) |

**References**

Miel, G. ``Of Calculations Past and Present: The Archimedean Algorithm.'' *Amer. Math. Monthly* **90**, 17-35, 1983.

Phillips, G. M. ``Archimedes in the Complex Plane.'' *Amer. Math. Monthly* **91**, 108-114, 1984.

© 1996-9

1999-05-25