The constant The constant p

2  The classic period

This period began around 1650 and lasted up to 1980. During this period many analytic expressions for p were discovered and the number of digits computed became impressive.

It started in 1655 with Wallis infinite product, then in 1658, Lord Brouncker gave an infinite continued fraction for p. A few years later, Gregory (1671) and Newton (1676) founded the power series development for the arctan and arcsin functions respectively. In 1706, Machin was the first to pass the limit of 100 decimal places with an arctan formula. During the eighteenth century other arctan formulae were founded (Klingenstierna, Euler, Hutton, ...) and Vega gave, in 1796, 136 correct digits.

The nineteenth century was the time of huge hand calculation. Incredible human calculators evaluated p to more and more decimal places. Among them we find: Dahse, Lehmann, Clausen, Rutherford and finally William Shanks who gave in 1874 the greatest hand calculation of p with a publication of 707 computed digits.

The quest for more digits started again in 1945, with electronic calculators, when Ferguson found a mistake in Shanks calculation at the 528th place. Up to the late seventies arctan relations were used to increase the number of digits. Wrench, D. Shanks, Guilloud, Kanada, ... are some of the actors of this period.

2.1  Wallis formula

Using the well-known infinite product for sin(x):

sin(x) = x

1- x2
p2




1- x2
4p2




1- x2
9p2


with x = p/2 leads to

p
2
= 2.2
1.3
4.4
3.5
6.6
5.7
8.8
7.9
...

John Wallis (1613-1703) gave this beautiful relation in 1655 ([1]). He founded it by a difficult and complicated proof. Even if the convergence of the product is very slow, it opened the era of analytic methods.

2.2  Infinite series

2.2.1  The arctangent function

A major improvement was made in 1671 when James Gregory (1638-1675) discovered the series for the arctangent function:

arctanx =
x

0 
dt
1+t2
=

k = 0 
(-1)k x2k+1
2k+1
with x < 1

This discovery was the basis for many fast convergent algorithms. Setting in this series x = 1 gives the famous Leibniz-Gregory-Madhava formula:

p
4
= 1- 1
3
+ 1
5
- 1
7
+...

Unfortunately the convergence of this formula is extremely slow (logarithmic convergence), the first 1000 terms only give 2 digits, the first 500000 terms give 5 digits ... The series becomes powerful for small values of x. In 1699 Abraham Sharp (1651-1742) (England) used Gregory's series by taking x = 1/3:

p
6
= 1
3
(1- 1
3.3
+ 1
32.5
- 1
33.7
+...)

The first iterates are:

x1
=
3.(4641016151377545...)
x2
=
3.(0792014356780040...)
x5
=
3.14(26047456630846...)
x10
=
3.14159(05109380800...)

By computing x150, Sharp calculated p to 72 digits.

2.2.2  The arcsines function

In 1676, Isaac Newton gave the following series

arcsinx = x+ 1
2
x3
3
+ 1.3
2.4
x5
5
+...with -1 < x < 1,

jointly with the formula

p
6
= arcsin 1
2
,

it's easy to find a few digits for p. Newton gave 14 digits with this series.

2.3  Machin's formula

In 1706, John Machin (1680-1752), a professor of astronomy in London, founded the very famous formula which is known as Machin's formula:

p
4
= 4arctan 1
5
-arctan 1
239

Using this formula, he computed p up to 100 digits. This relation can easily be, nowadays, deduced from elementary considerations on complex arithmetic:

(5+i)4
239+i
= 2(1+i)      (*)

A complex number z = x+iy can also be represented by the polar form z = reiq (r = |z| is the modulus and q = arg(x+iy) = arctan(y/x) is the argument of z). We have

arg(z1z2)
=
arg(z1)+arg(z2)
arg(zm)
=
marg(z)

so taking arguments in both side of (*) yields

4arg(5+i)-arg(239+i) = arg(1+i) = arctan(1) = p
4
,

which, thanks to the relation arg(x+iy) = arctan(y/x) is the Machin formula.

2.4  Machin type formulae

After Machin's discovery, many formulae of this type were founded by famous or not mathematicians [3], [4],[8]. We give a list of the most famous relations:

p
4
=
8arctan 1
10
-arctan 1
239
-4arctan 1
515
    Klingenstierna 1730
(1)
p
4
=
5arctan 1
7
+2arctan 3
79
       Euler 1755
(2)
p
4
=
4arctan 1
5
-arctan 1
70
+arctan 1
99
       Euler 1764
(3)
p
4
=
arctan 1
2
+arctan 1
3
       Hutton1776      
(4)
p
4
=
2arctan 1
3
+arctan 1
7
       Hutton 1776
(5)
p
4
=
arctan 1
2
+arctan 1
5
+arctan 1
8
       Strassnitzky 1844
(6)
p
4
=
12arctan 1
18
+8arctan 1
57
-5arctan 1
239
       Gauss 1863
(7)
p
4
=
6arctan 1
8
+2arctan 1
57
+arctan 1
239
       Störmer 1896
(8)
p
4
=
44arctan 1
57
+7arctan 1
239
-12arctan 1
682
+24arctan 1
12943
(9)
p
4
=
5arctan 29
278
+7arctan 3
79
(10)
p
2
=
2arctan 1
2
+arctan 1
22
       Wetherfield 1996
(11)
p
6
=
2arctan 1
33
+arctan 1
43
      
(12)

Some of those relations may be founded using properties of the arctan function:

arctan 1
p
=
2arctan 1
2p
-arctan 1
4p3+3p
     (**)
arctan 1
p
=
arctan 1
p+q
+arctan q
p2+pq+1

For example, if we set p = 5 in (**), we may prove Klingenstierna's formula by replacing arctan1/5 by 2arctan1/10-arctan1/515 in Machin's formula.

Nowadays, the computer can be used to find many such formulae. In 1997, Hwang found a very efficient Machin like formula, this formula has 6 terms ([8]).

2.5  Variation on arctan formulae

2.5.1  Euler's calculation

In 1755, Euler used a new series for the arctan function:

arctanx = x
1+x2


1 + 2
3
y + 2·4
3·5
y2 + 2·4 ·6
3·5 ·7
y3 +

,        y = x2
1+x2
.
(13)

in conjunction with the formula

p
4
= 5arctan 1
7
+2arctan 3
79
.

The values x = 1/7 and x = 3/79 respectively give

y = 1
50
= 2
100
   and     y = 9
6250
= 122
105
.

These values made the computation convenient for Euler in decimal base. He computed 20 decimal places of p whithin one hour.

If we set x = 1 in Euler's series 13, we obtain

p
4
= arctan1 = 1
2


1+ 2
3
· 1
2
+ 2·4
3·5
· 1
22
+

giving

p = 2

1+

k = 1 
2k(k!)2
(2k+1)!


The convergence is not so bad

x5
=
3.12(15007215007215...)
x10
=
3.141(1060216013776...)
x15
=
3.1415(797881375958...)
x20
=
3.141592(2987403396...).

This series is the basic formulae of the very small p program presented in Tiny programs for constants computation.

2.5.2  Frisby formula

In 1872, Frisby used the same technique to compute 30 decimal places but this time, with Hutton's formula

p
4
= 2arctan 1
3
+arctan 1
7

leading to the nice sequence (for a computation in base 10)

p = 12
5


1+ 2
3
( 1
10
)+ 2.4
3.5
( 1
10
)2+...

+ 14
25


1+ 2
3
( 2
100
)+ 2.4
3.5
( 2
100
)2+...

2.6  Record of computation during the classical period

Until recent years such formulae were used to compute more and more digits of p.

1794 Baron Georg von Vega found 136 correct digits with relation (2)

1841 Rutherford used relation (3) to compute 152 correct digits

1844 Dahse gave 200 correct digits in a two months calculation with relation (6)

1853 Lehman reached 261 decimals with formula (5)

1874 Shanks William found 527 correct digits with Machin's formula. He published 707 digits but only the first 527 were correct. The error was detected, much latter in 1945, by Ferguson.

This terminates the hand calculation period. Here are some prophetic notes from Shanks in his report of his hand calculation of p [2]:

''Whether any other Mathematician will appear, possessing sufficient leisure, patience, and facility of computation, to calculate the value of p to a still greater extent, remains to be seen: all that the Author can say is, he takes leave of the subject for the present ...''

With the advent of digital computers the number of digits founded became impressive.

1949 John von Neumann and his team used the ENIAC to determine p (and also e) up to 2,000 digits. Machin's formula was used [5].

1958 Felton on the Pegasus computer found 10,000 digits. Machin's formula was used again.

1961 Shanks Daniel and John W. Wrench published the first 100,000 digits. The calculation was made on an IBM 7090 system. Formulae (7) and (8) were used, one for the computation and the other for the check [6].

1974 Guilloud and Bouyer reached 1,000,000 digits on a CDC7600. Formulae (7) and (8) were also used.

1983 Kanada and Ushiro gave 10,013,395 decimal places with Gauss formula (7) [7].

References

[1]
J. Wallis, Arithmetica infinitorum, sive nova methodus inquirendi in curvilineorum quadratum, aliaque difficiliora matheseos problemata, Oxford, (1655)

[2]
W. Shanks, Contributions to Mathematics Comprising Chiefly the Rectification of the Circle to 607 Places of Decimals, G. Bell, London, (1853)

[3]
C. Stömer, Sur l'application de la théorie des nombres entiers complexes à la solution en nombres rationnels x1,x2,...,xn,c1,c2,...,cn,k de l'équation c1arctg x1+c2 arctg x2+...+cn arctg xn = kp/4, Archiv for Mathematik og Naturvidenskab, (1896), vol. 19

[4]
D. H. Lehmer, On Arctangent Relations for p, The American Mathematical Monthly, (1938), p. 657-664

[5]
G. W. Reitwiesner , An ENIAC Determination of p and e to more than 2000 Decimal Places, Mathematical Tables and other Aids to Computation, (1950), vol. 4, p. 11-15

[6]
D. Shanks and J.W. Wrench, Jr., Calculation of p to 100,000 Decimals, Math. Comput., (1962), vol. 16, p. 76-79

[7]
Y. Kanada, Y. Tamura, S. Yoshino & Y. Ushiro, Calculation of p to 10,013,395 decimal places based on the Gauss-Legendre Algorithm and Gauss Arctangent relation, Computer Centre, University of Tokyo, (1983), Tech. Report 84-01

[8]
C. L. Hwang, More Machin-Type Identities, Math. Gaz., (1997), p. 120-121


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