History topic: A chronology of pi


Pre computer calculations of π

MathematicianDatePlacesComments
1Rhind papyrus2000 BC13.16045 (= 4(8/9)2)

2Archimedes250 BC33.1418 (average of the bounds)

3Vitruvius20 BC13.125 (= 25/8)

4Chang Hong13013.1622 (= √10)

5Ptolemy15033.14166

6Wang Fan25013.155555 (= 142/45)

7Liu Hui26353.14159

8,Zu Chongzhi48073.141592920 (= 355/113)

9Aryabhata49943.1416 (= 62832/20000)

10Brahmagupta64013.1622 (= √10)

11Al-Khwarizmi80043.1416

12Fibonacci122033.141818

13Madhava1400113.14159265359

14Al-Kashi1430143.14159265358979

15Otho157363.1415929

16Viète159393.1415926536

17Romanus1593153.141592653589793

18Van Ceulen1596203.14159265358979323846

19Van Ceulen1596353.1415926535897932384626433832795029

20Newton1665163.1415926535897932

21Sharp169971

22Seki Kowa170010
23Kamata173025
24Machin1706100

25De Lagny1719127Only 112 correct

26Takebe172341

27Matsunaga173950

28von Vega1794140Only 136 correct

29Rutherford1824208Only 152 correct

30Strassnitzky, Dase1844200

31Clausen1847248

32Lehmann1853261

33Rutherford1853440

34Shanks1874707Only 527 correct

35Ferguson1946620


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

MathematicianDatePlacesType of computer

FergusonJan 1947710Desk calculator
Ferguson, WrenchSept 1947808Desk calculator
Smith, Wrench19491120Desk calculator
Reitwiesner et al.19492037ENIAC
Nicholson, Jeenel19543092NORAC
Felton19577480PEGASUS
GenuysJan 195810000IBM 704
FeltonMay 195810021PEGASUS
Guilloud195916167IBM 704
Shanks, Wrench1961100265IBM 7090
Guilloud, Filliatre1966250000IBM 7030
Guilloud, Dichampt1967500000CDC 6600
Guilloud, Bouyer19731001250CDC 7600
Miyoshi, Kanada19812000036FACOM M-200
Guilloud19822000050
Tamura19822097144MELCOM 900II
Tamura, Kanada19824194288HITACHI M-280H
Tamura, Kanada19828388576HITACHI M-280H
Kanada, Yoshino, Tamura198216777206HITACHI M-280H
Ushiro, KanadaOct 198310013395HITACHI S-810/20
GosperOct 198517526200SYMBOLICS 3670
BaileyJan 198629360111CRAY-2
Kanada, TamuraSept 198633554414HITACHI S-810/20
Kanada, TamuraOct 198667108839HITACHI S-810/20
Kanada, Tamura, KuboJan 1987134217700NEC SX-2
Kanada, TamuraJan 1988201326551HITACHI S-820/80
ChudnovskysMay 1989480000000
ChudnovskysJune 1989525229270
Kanada, TamuraJuly 1989536870898
ChudnovskysAug 19891011196691
Kanada, TamuraNov 19891073741799
ChudnovskysAug 19912260000000
ChudnovskysMay 19944044000000
Kanada, TamuraJune 19953221225466
KanadaAug 19954294967286
KanadaOct 19956442450938
Kanada, TakahashiAug 199751539600000HITACHI SR2201
Kanada, TakahashiSept 1999206158430000HITACHI 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.

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


September 2000

MacTutor History of Mathematics
[http://www-history.mcs.st-andrews.ac.uk/HistTopics/Pi_chronology.html]