A chronology of π

Pre computer calculations of π

 Mathematician Date Places Comments Notes 1 Rhind papyrus 2000 BC 1 3.16045 (= $4(\large\frac{8}{9}\normalsize )^{2}$) Click for note 1 2 Archimedes 250 BC 3 3.1418 (average of the bounds) Click for note 2 3 Vitruvius 20 BC 1 3.125 (= $\large\frac{25}{8}\normalsize$) Click for note 3 4 Chang Hong 130 1 3.1622 (= √10) Click for note 4 5 Ptolemy 150 3 3.14166 Click for note 5 6 Wang Fan 250 1 3.155555 (= $\large\frac{142}{45}\normalsize$) Click for note 6 7 Liu Hui 263 5 3.14159 Click for note 7 8, Zu Chongzhi 480 7 3.141592920 (= $\large\frac{355}{113}\normalsize$) Click for note 8 9 Aryabhata 499 4 3.1416 (= $\large\frac{62832}{20000}\normalsize$) Click for note 9 10 Brahmagupta 640 1 3.1622 (= √10) Click for note 10 11 Al-Khwarizmi 800 4 3.1416 Click for note 11 12 Fibonacci 1220 3 3.141818 Click for note 12 13 Madhava 1400 11 3.14159265359 Click for note 13 14 Al-Kashi 1430 16 3.1415926535897932 Click for note 14 15 Otho 1573 6 3.1415929 Click for note 15 16 Viète 1593 9 3.1415926536 Click for note 16 17 Romanus 1593 15 3.141592653589793 Click for note 17 18 Van Ceulen 1596 20 3.14159265358979323846 Click for note 18 19 Van Ceulen 1596 35 3.1415926535897932384626433832795029 Click for note 19 20 Newton 1665 16 3.1415926535897932 Click for note 20 21 Sharp 1699 71 Click for note 21 22 Seki Kowa 1700 10 23 Kamata 1730 25 24 Machin 1706 100 Click for note 24 25 De Lagny 1719 127 Only 112 correct Click for note 25 26 Takebe 1723 41 Click for note 26 27 Matsunaga 1739 50 Click for note 27 28 von Vega 1794 140 Only 136 correct Click for note 28 29 Rutherford 1824 208 Only 152 correct Click for note 29 30 Strassnitzky, Dase 1844 200 Click for note 30 31 Clausen 1847 248 Click for note 31 32 Lehmann 1853 261 Click for note 32 33 Rutherford 1853 440 Click for note 33 34 Shanks 1874 707 Only 527 correct Click for note 34 35 Ferguson 1946 620 Click for note 35
General Remarks:
A. In early work it was not known that the ratio of the area of a circle to the square of its radius and the ratio of the circumference to the diameter are the same. Some early texts use different approximations for these two "different" constants. For example, in the Indian text the Sulba Sutras the ratio for the area is given as 3.088 while the ratio for the circumference is given as 3.2.

B. Euclid gives in the Elements XII Proposition 2:
Circles are to one another as the squares on their diameters.
He makes no attempt to calculate the ratio.

Computer calculations of π

General Remarks:

A. Calculating π to many decimal places was used as a test for new computers in the early days.

B. There is an algorithm by Bailey, Borwein and Plouffe, published in 1996, which allows the $n$th hexadecimal digit of π to be computed without the preceeding $n - 1$ digits.

C. Plouffe discovered a new algorithm to compute the $n$th digit of π in any base in 1997.

References (show)

1. D H Bailey, J M Borwein, P B Borwein, and S Plouffle, The quest for Pi, The Mathematical Intelligencer 19 (1997), 50-57.

Written by J J O'Connor and E F Robertson
Last Update September 2000