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Efficiency of Some Machin-Like Formulas

This Demonstration shows the efficiency of some Machin-like formulas for computing π. Gregory numbers (defined by ) appear in the formulas.

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Machin-like formulas were used for several calculations of . Before 1706, Machin calculated to 100 places. In n 1789, Vega used Hutton's formula to calculate 140 places (126 correct). In 1841, Rutheford calculated to 208 places (152 correct). In 1844, Dase used Strassnitzky's formula to calculate to 205 places (200 correct). In 1847, Clausen went to 250 places (248 correct), using Machin's and Hutton's formula. In 1853 Rutheford extended his calculation to 440 places (all correct).
Reference
[1] W. W. Rouse Ball and H. S. M. Coxeter, Mathematical Recreations and Essays, 13th ed., New York: Dover Publications, 1987 pp. 356–357.
This Demonstration uses some code by Russ Johnson.
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