Three Calculations of Pi Before 1800

This Demonstration shows three calculations of using the Gregory–Leibniz series with .
Then .
Let and be the subseries with positive and negative terms, respectively.


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According to [1, pp. 356–357], in 1699 Sharp used the Gregory series with to determine an approximation of to 72 decimal places (71 were correct). De Lagny, in 1706, went to 127 places (112 correct) using the same series.
Toward the end of the eighteenth century, F. X. von Zach saw a manuscript by an unknown author in the Radcliffe Library, Oxford, which gives the value of to 154 places (152 correct) [2].
In this Demonstration, we suppose that those mathematicians calculated values of terms to two or three decimals more than the reported results.
[1] W. W. Rouse Ball and H. S. M. Coxeter, Mathematical Recreations and Essays, 13th ed., New York: Dover Publications, 1987.
[2] B. Wardhaugh, "A ‘Lost’ Chapter in the Calculation of π: Baron Zach and MS Bodleian 949," Historia Mathematica, 42(3), 2015 pp. 343–351. doi:10.1016/j.hm.2015.01.002.
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