11453

Machin's Computation of Pi

Around 1706, John Machin found the arc tangent formula .
He expanded it using Gregory's series to compute to over 100 digits. The convergence of this series is much more rapid than that for the simple Gregory–Leibniz series: .
Let and be the series for and . Manchin's formula is then .
Split and into four series according to the signs of their coefficients:
,
,
,
.
Machin used values truncated to 103 decimals, which means that only terms greater than are calculated.
To make sure that all terms of and with indices are less than , it is necessary that . For , this means .

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Let , be an arc of radius 1, and be perpendicular to , , , and . Construct points and on the arc , and and on the straight line , so that and .
Then and , .
So , .
See [1, pp. 8–9].
References
[1] V. F. Rickey. "Machin's Formula for Computing Pi." (Apr 28, 2016) fredrickey.info/talks/10-03-13-HPM-DC-Machin/10-03-13-HPM_DC_Machin.pdf.
[2] MacTutor. "John Machin." (Apr 28, 2016) www-history.mcs.st-andrews.ac.uk/Biographies/Machin.html.
[3] Wikipedia. "John Machin." (Apr 28, 2016) en.wikipedia.org/wiki/John_Machin.
[4] W. W. Rouse Ball, H. S. M. Coxeter, Mathematical Recreations and Essays, 13th ed., New York: Dover Publications, 1987 p. 356.
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