Arctan Formulae for Computing π

Derivation of the Machin Formula


In 1706 John Machin, Professor of Astronomy in London, computed π to 100 digits using the formula (Formula 8):

π = 16·arctan(1/5) − 4·arctan(1/239)

This is not an arbitrary formula. It can be derived easily.

First, start with:

tan α = 1/5

for some angle α that we need not be concerned with since it will disappear from the final result. (For the curious, α ≈ 11° 18' 35.8").

Using the trig formula for double angles:

tan 2α  =    2·tan α      =        2/5        =    5 
1 − tan2 α 1 − 1/25 12

Applying the double angle formula once more:

tan 4α  =    2·tan 2α      =        5/6        =    120 
1 − tan2 1 − 25/144 119

This differs from 1 by only the small value 1/119, but tan π/4 = 1; therefore, 4α differs from π/4 by a small value so that tan (4α − π/4) is small. Indeed, it is just:

tan (4α − π/4)  =    tan 4α − tan π/4     =     tan 4α − 1     =    120/119 − 1    =      1   
tan 4α + tan π/4 tan 4α + 1 120/119 + 1239

Taking the arctan of both sides results in:

4α − π/4 = arctan (1/239)

Solving for π and substituting α = arctan(1/5) finally gives:

π = 16·arctan(1/5) − 4·arctan(1/239)