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Machin's Formula

        

The hunt for PI began in Egypt and in Babylon about two thousand years before Christ. The Egyptians obtained the value (4/3)4=3.160493827 and the Babylonians the value 3 1/8=3.125 for PI.About the same time, the Indians used the square root of 10 for PI.
In the 3rd century B.C. Archimedes considered inscribed and circumscribed polygons of 96 sides and deduced that 3 + 10/71 < PI < 3 + 1/7 .
Various people calculated PI, including:

    Ptolemy (c.150 AD) 3.1416
    Tsu Ch`ung Chi (430-501 AD) 355/113
    Al`Khwarizmi (c.800) 14 places
    Vičte (1540-1603) 9 places
    Roomen (1561-1615) 17 places
    Van Ceulen (c.1600) 35 places
There was no theoretical progress involved in these improvements, only greater stamina in calculation.
The European Renaissance brought about in due course a whole new mathematical world. Among the first emergence of mathematical formulae for PI.One of the best-known was that of James Gregory (1638-1675):



Gregoru also showed the more general result:



from which the first series if we put x=1.
Archimedes`constant PI has many infinite series, infinite product, definite integral and continued fraction representations. It was proved to be irrational by Lambert and transcendenal by Lindemann. The first truly attractive formula for coputing decimal digit of PI was found by John Machin ( 1680-1752), in 1706.



Gauss, Stirling and others used the following formulae for PI/4:



The adventage of Machin`s formula is that the second term converes very rapidly and the first is nice for decimal arithmetic.
Here are the computations:
  All computations to 15 decimal places:
  
  arctan(1/5)                               arctan(1/239):
  1/5              = 0·200000000000000      1/239          = 0·004184100418410
  1/375            =-0·002666666666666      1/40955757     =-0·000000024416591
  1/15625          = 0·000064000000000      1/3899056325995= 0·000000000000256
  1/546875         =-0·000001828571428         
  1/17578125       = 0·000000056888889                          
  1/537109375      =-0·000000001861818                          
  1/15869140625    = 0·000000000063015
  1/457763671875   =-0·000000000002184
  1/12969970703125 = 0·000000000000077
  1/362396240234375=-0·000000000000002
  
  SUMMING:
  arctan(1/5)      = 0·1973955598498807  and arctan(1/239) = 0·004184076002074
  
  Putting these in the Machin's formula gives:
          Pi/4=  4xarctan(1/5 )        -   arctan( 1/239 )
   or     Pi  = 16xarctan(1/5 )        - 4xarctan( 1/239 )
              = 16x0·1973955598498807  - 4x0·004184076002074
              = 3·1415926535897922
  

Using this, Machin became the first individual to correctly compute 100 digits of PI.


John Machin was born 1689 in England. We know that he acted as a private tutor to Brook Taylor teaching him mathematics in 1701. He continued to correspond with Taylor for many years and this is a useful source for understanding his mathematical thinking. The two met in coffeehouses, a standard place where mathematical discussions were held during this period. We also know that Machin was friendly with Keill, who taught at Oxford, and with de Moivre who like Machin was a private tutor of mathematics at this time. In 1706 William Jones reports that Mr John Machin`s formula allows PI be calculated:- ... to above 100 places.

No indication is given in Jones's work, however, as to how Machin discovered his series expansion for p so when de Moivre wrote to Johann Bernoulli on 8 July 1706 telling him about Machin's series for PI he suggested that Johann Bernoulli might tell Jakob Hermann about Machin's unproved result. He did so and Hermann quickly discovered a proof that Machin's series converges to PI. He produced techniques that show other similar series also converge rapidly to PI and he wrote on 21 August 1706 to Leibniz giving details. Two years later, on 6 July 1708, de Moivre wrote again to Johann Bernoulli about Machin's series, on this occasion giving two different proofs that it converged to PI.
On 16 May 1713 Machin was appointed as Professor of Astronomy at Gresham College, London. He succeed Dr Torriano and went on to hold the chair until his death 38 years later. For nearly 30 of these years he acted as Secretary to the Royal Society, being appointed in 1718 and holding the post until 1747. John Machin died 9 June 1751 in London.

        

An Englishman named Shanks used Machin`s formula to calculate PI to 707 places, publishing the results of many years of labour in 1873.
Here is a summary of how the calculating of PI went:
    1699: Sharp used Gregory`s result to get 71 correct digits.
    1701: Machin used an improvement to get 100 digits and the following used his methods:
    1791: De Lagny found 112 correct digits.
    1789: Vega got 126 places and in 1794 got 136.
    1841: Rutherfor calculated 152 digits and in 1853 got 440.
    1873: Shanks calculated 707 places of which 527 were correct.
    1949: A computer was used to calculate PI to 2000 places.
    1973: Guilloud and Bouyer used a version of it to compute one million digits of Pi on a CDC 7600.
    1997: Y.Kanada calculated PI to over 51 billion digits.

Here, we shall give a proof for the John Machin`s formula:

Well-known trigonometrical identity that can be used to prove Machin`s formula is:



We shall put α = β



For α=2α



In general case



Here are valid identities for polynomials Pn(x) and Qn(x) :



Using former reccurency relations we get for n=4 the next polynomials:



Finally we have:



For tg( 4α - β ) we get



or:



If we introduce supstitations:



where



We have:



Now we get for t the next relation:



In case of x=1 and n=4 we get t=4/956=1/239 and arctg(1/1)=PI/4:
So, we have proved Machin`s formula:



Let us take n=1,2,3,4,5,... - we get series of Machin`s formulae:



We have just seen that there are infinitely many formulae for PI/4 using arcustanges function.

        

        

     

     

        

  2001-2005 Radoslav Jovanovic                 created:  January 2005.