A chronology of pi

Pre computer calculations of π

	Mathematician	Date	Places	Comments	Notes
1	Rhind papyrus	2000 BC	1	3.16045 (= 4(⁸/₉)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 (= ²⁵/₈)	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 (= ¹⁴²/₄₅)	Click for note 6
7	Liu Hui	263	5	3.14159	Click for note 7
8,	Zu Chongzhi	480	7	3.141592920 (= ³⁵⁵/₁₁₃)	Click for note 8
9	Aryabhata	499	4	3.1416 (= ⁶²⁸³²/₂₀₀₀)	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	14	3.14159265358979	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

Mathematician	Date	Places	Type of computer
Ferguson	Jan 1947	710	Desk calculator
Ferguson, Wrench	Sept 1947	808	Desk calculator
Smith, Wrench	1949	1120	Desk calculator
Reitwiesner et al.	1949	2037	ENIAC
Nicholson, Jeenel	1954	3092	NORAC
Felton	1957	7480	PEGASUS
Genuys	Jan 1958	10000	IBM 704
Felton	May 1958	10021	PEGASUS
Guilloud	1959	16167	IBM 704
Shanks, Wrench	1961	100265	IBM 7090
Guilloud, Filliatre	1966	250000	IBM 7030
Guilloud, Dichampt	1967	500000	CDC 6600
Guilloud, Bouyer	1973	1001250	CDC 7600
Miyoshi, Kanada	1981	2000036	FACOM M-200
Guilloud	1982	2000050
Tamura	1982	2097144	MELCOM 900II
Tamura, Kanada	1982	4194288	HITACHI M-280H
Tamura, Kanada	1982	8388576	HITACHI M-280H
Kanada, Yoshino, Tamura	1982	16777206	HITACHI M-280H
Ushiro, Kanada	Oct 1983	10013395	HITACHI S-810/20
Gosper	Oct 1985	17526200	SYMBOLICS 3670
Bailey	Jan 1986	29360111	CRAY-2
Kanada, Tamura	Sept 1986	33554414	HITACHI S-810/20
Kanada, Tamura	Oct 1986	67108839	HITACHI S-810/20
Kanada, Tamura, Kubo	Jan 1987	134217700	NEC SX-2
Kanada, Tamura	Jan 1988	201326551	HITACHI S-820/80
Chudnovskys	May 1989	480000000
Chudnovskys	June 1989	525229270
Kanada, Tamura	July 1989	536870898
Chudnovskys	Aug 1989	1011196691
Kanada, Tamura	Nov 1989	1073741799
Chudnovskys	Aug 1991	2260000000
Chudnovskys	May 1994	4044000000
Kanada, Tamura	June 1995	3221225466
Kanada	Aug 1995	4294967286
Kanada	Oct 1995	6442450938
Kanada, Takahashi	Aug 1997	51539600000	HITACHI SR2201
Kanada, Takahashi	Sept 1999	206158430000	HITACHI SR8000

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 nth hexadecimal digit of π to be computed without the preceeding n- 1 digits.

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

Reference (One book/article)

Other Web sites:

Astroseti (A Spanish translation of this article)
Simon Fraser University (Tables of Computations)

Article by: J J O'Connor and E F Robertson

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JOC/EFR September 2000

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