In my previous article I talked about the history of Pi (Ï€): How various formulae have been evolved till date to reach the maximum accuracy of Ï€. In this article I will jot down the very same thing in nothing but a tabular format but more systematically. Below is the table.
Pre computer calculations of Ï€
Sl. No.  Mathematician  Date  Places  Comments  Notes 

1  Rhind papyrus  2000 BC  1  3.16045 (= 4(^{8}/_{9})^{2})  
2  Archimedes  250 BC  3  3.1418 (average of the bounds)  It is claimed that in a text which is now lost Archimedes gave better bounds whose average gives the value 3.141596 for Ï€, correct to seven places. 
3  Vitruvius  20 BC  1  3.125(= ^{25}/_{8})  Marcus Vitruvius Pollio was a Roman architect. He made his calculation of Ï€ by measuring the distance a wheel of a given diameter moved through one revolution. 
4  Chang Hong  130  1  3.1622(= âˆš10)  Chang Hong (78139) was an astrologer from China. He was one of the first to use this approximation. He deduced it from his value for the ratio of the volume of a cube to that of the inscribed sphere was ^{8}/_{5}. 
5  Ptolemy  150  3  3.14166  Ptolemy used a regular 360 polygon to approximate Ï€. He actually obtained the number 3 + ^{8}/_{60} + ^{30}/_{602} which written as a decimal is 3.1416666… 
6  Wang Fan  250  1  3.155555(= ^{142}/_{45})  Wang Fan (229267) was a Chinese astronomer. No method of arriving at his value of Ï€ is recorded, although it has been conjectured that he came up with the value ^{142}/_{45} “knowing that 3.14 was too low”. 
7  Liu Hui  263  5  3.14159  Liu Hui was a Chinese mathematician. He calculated Ï€ using an inscribed regular polygon with 192 sides. 
8  Zu Chongzhi  480  7  3.141592920(= ^{355}/_{113})  Zu Chongzhi produced a value of Ï€ which was not bettered for nearly one thousand years. In fact although he gives ^{355}/_{113} as a “very close” ratio, he actually calculated 3.1415926 < Ï€ < 3.1415927 which, taking an average, would give Ï€ correct to 8 places. 
9  Aryabhata  499  4  3.1416(= ^{62832}/_{20000})  Aryabhata made his approximation using an inscribed regular polygon with 384 sides. 
10  Brahmagupta  640  1  3.1622 (= âˆš10)  Brahmagupta criticises Aryabhata for his earlier, and much better, value of Ï€. He gave 3 as the approximate value for practical calculation and âˆš10 as the exact value. 
11  AlKhwarizmi  800  4  3.1416  Al’Khwarizmi states in his Algebra that the practical man uses ^{22}/_{7} for Ï€, the geometer uses âˆš10 for Ï€, while the astronomer uses 3.1416. 
12  Fibonacci  1220  3  3.141818  Fibonacci gave the approximation ^{864}/_{275} = 3.14181818… He used inscribed and circumscribed regular polygons of 96 sides, taking the mean, but made no reference to Archimedes. 
13  Madhava  1400  11  3.14159265359  Madhava was an Indian mathematician who discovered Gregory’s series long before Gregory and was the first to use a series to calculate Ï€. 
14  AlKashi  1430  14  3.141592653589 79 
Al’Kashi obtained his remarkably accurate value using a regular polygon with 3Ã—2^{28} = 805306368 sides. His aim was to calculate a value which was accurate enough to allow the calculation of the circumference of the universe to within a hair’s breadth. He takes the universe to be a sphere 600000 times the diameter of the earth. Note that in fact 39 places of Ï€ will compute the circumference of the universe (known in 2000) to an accuracy of the radius of the hydrogen atom. 
15  Otho  1573  6  3.1415929  Valentin Otho, a pupil of Rheticus, was a German engineer. He gave the approximation ^{355}/_{113}. 
16  ViÃ¨te  1593  9  3.1415926536  ViÃ¨te used Archimedes method with polygons of 6Ã—2^{16} = 393216 sides to obtain 3.1415926535 <Ï€ < 3.1415926537. He is also famed as the first to find an infinite series for Ï€. 
17  Romanus  1593  15  3.141592653589 793 
Romanus (or Adriaan van Roomen) used a regular polygon with 2^{30} = 1073741824 sides. He gaveÏ€ to 17 places but only 15 are correct. 
18  Van Ceulen  1596  20  3.141592653589 79323846 
Van Ceulen used inscribed and circumscribed polygons with 60 2^{33} = 515396075520 sides. He also used what in modern notation would be essentially 1 – cos A = 2sin^{2} A/2. He had discovered this result. 
19  Van Ceulen  1596  35  3.141592653589 7932384626433 832795029 
Van Ceulen used a polygon with 2^{62} = 4611686018427387904 sides to obtain Ï€ to 35 places of accuracy. He spent most of his life on this calculation which my computer now gives essentially instantly: 3.1415926535897932384626433832795029. Snell in 1621 showed that more accurate values of Ï€ could be obtained by a clever variation in the construction so that he needed only to use a polygon with 2^{30} sides to obtain the accuracy which Van Ceulen obtained by using ones with 2^{62 }sides. It was around this time that variants of Archimedes’ method stopped being used and future calculations used infinite series expansions. 
20  Newton  1665  16  3.141592653589 7932 
Isaac Newton used the series arcsin(x) = x +1/(2.3) x^{3} + 1.3/(2.4.5) x^{5} + 1.3.5/(2.4.6.7) x^{7} + … putting x = ^{1}/_{2 }to obtain a series for arcsin(^{1}/_{2}) = Ï€/6. This requires taking about 40 terms. 
21  Sharp  1699  71  Abraham Sharp (16511742), an English mathematician, calculated Ï€ at the suggestion of Halley. He used James Gregory’s series arctan(x) =x – x^{3}/3 + x^{5}/5 – … Putting x = 1/âˆš3 gives a series for arctan(1/âˆš3) = Ï€/6. To obtain Ï€ correct to 71 places he used nearly 300 terms of the series. 

22  Seki Kowa  1700  10  
23  Kamata  1730  25  
24  Machin  1706  100  John Machin used the formula Ï€/4 = 4 arctan(^{1}/_{5}) – arctan(^{1}/_{129}) and James Gregory’s series for arctan(x). 

25  De Lagny  1719  127  Only 112 correct  De Lagny used the same method as Sharp. 
26  Takebe  1723  41  Takebe Hikojiro Kenko (16641739) was a Japanese mathematician. 

27  Matsunaga  1739  50  Matsunaga Ryohitsu was a Japanese mathematician. He used the same method as Newton. This requires taking about 160 terms in the series. 

28  Von Vega  1794  140  Only 136 correct  Baron Georg von Vega used the relation Ï€/4 = 5 arctan(^{1}/_{7}) + 2 arctan (^{3}/_{79}), and James Gregory’s series for arctan(x). Euler had calculated Ï€ correct to 20 places in an hour in 1755 using this method. 
29  Rutherford  1824  208  Only 152 correct  William Rutherford published his result in the Transactions of the Royal Society in 1841. He used the relation Ï€/4 = 4 arctan(^{1}/_{5}) – arctan(^{1}/_{70}) + arctan(^{1}/_{99}) which Euler published in 1764. 
30  Strassnitzky, Dase  1844  200  Von Strassnitzky from Vienna gave the formula Ï€/4 = arctan(^{1}/_{2}) + arctan (^{1}/_{5}) + arctan(^{1}/_{8}) to Zacharias Dase, a calculating prodigy. He used took about two months to calculate Ï€ to 200 places using the formula. 

31  Clausen  1847  248  Thomas Clausen wanted to find the correct value of Ï€ to 200 places since he knew that Rutherford and Dase gave different values from the 153rd place. Clausen used Machin’s relation Ï€/4 = 4 arctan(^{1}/_{5}) – arctan(^{1}/_{129}). He discovered that Dase was correct and extended the number of correct places to 248. 

32  Lehmann  1853  261  W Lehmann from Potsdam used the formula Ï€/4 = arctan(^{1}/_{2}) + arctan(^{1}/_{3}) which was published by Charles Hutton in 1776. The formula Ï€/4 = 2 arctan(^{1}/_{3}) + arctan(^{1}/_{7}) was also given by Hutton in 1776 and Euler, independently, in 1779. 

33  Rutherford  1853  440  William Rutherford, having made an error in his published value of Ï€ in 1841, made another calculation. This time he used Machin’s formula. 

34  Shanks  1874  707  Only 527 correct  Shanks was a pupil of William Rutherford. He also used Machin’s formula. The first to suspect that Shanks’s work on Ï€ may not be correct was De Morgan who discovered that in an version with 607 places given by Shanks, there were too few occurrences of the digit 7. 
35  Ferguson  1946  620  D F Ferguson who worked at the Royal Naval College in England used the formula Ï€/4 = 3 arctan(^{1}/_{4}) + arctan(^{1}/_{20}) + arctan(^{1}/_{1985}) to compute Ï€. He worked on it from May 1944 until May 1945 by which time he had calculated 530 places and found that Shanks was wrong after place 528. He continued with his efforts and published 620 correct places of Ï€ in July 1946. Ferguson then continued with his calculations, but after this he used a desk calculator. This marks the point at which hand calculations of Ï€ ended and computer assisted calculations began. 
Computer calculations of Ï€
Sl. No.  Mathematician  Date  Decimal Places  Type of computer 

1  Ferguson  Jan 1947  710  Desk calculator 
2  Ferguson, Wrench  Sept 1947  808  Desk calculator 
3  Smith, Wrench  1949  1120  Desk calculator 
4  Reitwiesner et al.  1949  2037  ENIAC 
5  Nicholson, Jeenel  1954  3092  NORAC 
6  Felton  1957  7480  PEGASUS 
7  Genuys  Jan 1958  10000  IBM 704 
8  Felton  May 1958  10021  
9  Guilloud  1959  16167  IBM 704 
10  Shanks, Wrench  1961  100265  IBM 7090 
11  Guilloud, Filliatre  1966  250000  IBM 7030 
12  Guilloud, Dichampt  1967  500000  CDC 6600 
13  Guilloud, Bouyer  1973  1001250  CDC 7600 
14  Miyoshi, Kanada  1981  2000036  FACOM M200 
15  Guilloud  1982  2000050  
16  Tamura  1982  2097144  MELCOM 900II 
17  Tamura, Kanada  1982  4194288  HITACHI M280H 
18  Tamura, Kanada  1982  8388576  HITACHI M280H 
19  Kanada, Yoshino, Tamura  1982  16777206  HITACHI M280H 
20  Ushiro, Kanada  Oct 1983  10013395  HITACHI S810/20 
21  Gosper  Oct 1985  17526200  SYMBOLICS 3670 
22  Bailey  Jan 1986  29360111  CRAY2 
23  Kanada, Tamura  Sept 1986  33554414  HITACHI S810/20 
24  Kanada, Tamura  Oct 1986  67108839  HITACHI S810/20 
25  Kanada, Tamura, Kubo  Jan 1987  134217700  NEC SX 
26  Kanada, Tamura  Jan 1988  201326551  HITACHI S820/80 
27  Chudnovskys  May 1989  
28  Chudnovskys  June 1989  525229270  
29  Kanada, Tamura  July 1989  536870898  
30  Chudnovskys  Aug 1989  1011196691  
31  Kanada, Tamura  Nov 1989  1073741799  
32  Chudnovskys  Aug 1991  2260000000  
33  Chudnovskys  May 1994  4044000000  
34  Kanada, Tamura  June 1995  
35  Kanada  Aug 1995  4294967286  
36  Kanada  Oct 1995  6442450938  
37  Kanada, Takahashi  Aug 1997  51539600000  HITACHI SR2201 
38  Kanada, Takahashi  Sept 1999  206158430000  HITACHI SR8000 
39  Kanada et. al.  Nov 2002  1,241,100,000,000  HITACHI SR8000 
40  Takahashi et. al.  Apr 2009  2,576,980,377,524  T2K Open Supercomputer 
41  Fabrice Bellard  Dec 2009  2,699,999,990,000 

42  Shigeru Kondo  Aug 2010  5,000,000,000,000 

43  Shigeru Kondo  Oct 2011  10,000,000,000,050 

44  Shigeru Kondo  Dec 2013  12,100,000,000,050 

Various Formulas for Computing Ï€
 Wallis:
Ï€/2=(2.2.4.4.6.6.8.8. …)/(1.3.3.5.5.7.7.9. …)  Machin:
Ï€/4=4 arctan(1/5)arctan(1/239)  Ferguson:
Ï€/4 = 3 arctan(1/4)+arctan(1/20)+arctan(1/1985)  Euler:
Ï€/4 = 5 arctan(1/7)+2 arctan(3/79)  Euler:
Ï€^{2}/6=1/2^{2}+1/3^{2}+1/4^{2}+1/5^{2}+ …  Euler:
e^{iÏ€}+1=0  Borwein and Borwein:
1/Ï€ = 12Æ©[(1)^{n}(6n)!/(n!)^{3}(3n)!][(A+nB)/C^{n+1/2}];
where,
A=212175710912âˆš(61)+1657145277365;
B=13773980892672âˆš(61)+107578229802750;
C=[5280(236674+30303âˆš(61)]^{3}  Borwein, Bailey, and Plouffe:
Ï€ = Æ©[4/(8n+1)2/(8n+4)1/(8n+5)1/(8n+6)](16)^{n }
This formula enables one to calculate the nth digit of pi, in hexadecimal notation, without being forced to calculate the preceding n1 digits.
Some interesting things about Ï€
 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.
 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.
 Calculating Ï€ to many decimal places was used as a test for new computers in the early days.
 There is an algorithm by Bailey, Borwein and Plouffe, published in 1996, which allows the nth hexadecimal digit of Ï€ to be computed without the preceding n– 1 digits.
 Plouffe discovered a new algorithm to compute the nth digit of Ï€ in any base in 1997.
 As a postscript, here is a mnemonic for the decimal expansion of Ï€. Each successive digit is the number of letters in the corresponding word: 3.14159265358979323846264…
How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics. All of thy geometry, Herr Planck, is fairly hard…
Open questions about the number Ï€
 Does each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 each occur infinitely often in Ï€?
 Brouwer’s question: In the decimal expansion of Ï€, is there a place where a thousand consecutive digits are all zero?
 Is Ï€ simply normal to base 10? That is does every digit appear equally often in its decimal expansion in an asymptotic sense?
 Is Ï€ normal to base 10? That is does every block of digits of a given length appear equally often in its decimal expansion in an asymptotic sense?
 Is Ï€ normal ? That is does every block of digits of a given length appear equally often in the expansion in every base in an asymptotic sense? The concept was introduced by Borel in 1909.
 Another normal question! We know that Ï€ is not rational so there is no point from which the digits will repeat. However, if Ï€ is normal then the first million digits 314159265358979… will occur from some point. Even if Ï€ is not normal this might hold! Does it? If so from what point? Note: Up to 200 million the longest to appear is 31415926 and this appears twice.
Hope this was a good read for you. You think to share anything else, Do not forget to write here.. Have a good time till my next article. Bye Bye…
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Arindam Bose
Scientific History Blog Writer â€¢ Art enthusiast and Illustrator â€¢ Amateur Photographer â€¢ Biker and Hiker â€¢ Beer Enthusiast â€¢ Electrical Engineer â€¢ Chicago

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