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) |
(3) |
(4) | |||
(5) |
(6) | |||
(7) |
(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 Eric W. Weisstein