Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/52

The more quickly convergent series

$\sin ^{-1}{x}=x+{\frac {1}{2}}{\frac {x^{3}}{3}}+{\frac {1{\,.\,}3}{2{\,.\,}4}}{\frac {x^{5}}{5}}+\ldots$,

discovered by Newton, is troublesome for purposes of calculation, owing to the form of the coefficients. By taking $\textstyle x={\frac {1}{2}}$, Newton himself calculated $\pi$ to 14 places of decimals.

Euler and others occupied themselves in deducing from Gregory's series formulae by which $\pi$ could be calculated by means of rapidly converging series.

Euler, in 1737, employed special cases of the formula

$\tan ^{-1}{\frac {1}{p}}=\tan ^{-1}{\frac {1}{p+q}}+\tan ^{-1}{\frac {q}{p^{2}+pq+1}}$,

and gave the general expression

$\tan ^{-1}{\frac {x}{y}}=\tan ^{-1}{\frac {ax-y}{ay+x}}+\tan ^{-1}{\frac {b-a}{ab+1}}+\tan ^{-1}{\frac {c-b}{cb+1}}+\ldots$,

from which more such formulae could be obtained. As an example, we have, if $a,b,c,\ldots$ are taken to be the uneven numbers, and ${\frac {x}{y}}=1$,

${\frac {\pi }{4}}=\tan ^{-1}{\frac {1}{2}}+\tan ^{-1}{\frac {1}{2{\,.\,}4}}+\tan ^{-1}{\frac {1}{2{\,.\,}9}}+\ldots$.

In the year 1706, Machin (1680—1752), Professor of Astronomy in London, employed the series

${\frac {\pi }{4}}=4\left({\frac {1}{5}}-{\frac {1}{3{\,.\,}5^{3}}}+{\frac {1}{5{\,.\,}5^{5}}}-{\frac {1}{7{\,.\,}5^{7}}}+\ldots \right)$${}-\left({\frac {1}{239}}-{\frac {1}{3{\,.\,}239^{3}}}+{\frac {1}{5{\,.\,}239^{5}}}-{\frac {1}{7{\,.\,}239^{7}}}+\ldots \right)$,

which follows from the relation

${\frac {\pi }{4}}=4\tan ^{-1}{\frac {1}{5}}-\tan ^{-1}{\frac {1}{239}}$,

to calculate $\pi$ to 100 places of decimals. This is a very convenient expression, because in the first series ${\frac {1}{5}},{\frac {1}{5^{3}}},\ldots$ can be replaced by ${\frac {4}{100}}$, ${\frac {64}{1000000}}$, &c., and the second series is very rapidly convergent.

In 1719, de Lagny (1660—1734), of Paris, determined in two different ways the value of $\pi$ up to 127 decimal places. Vega (1754—1802) calculated $\pi$ to 140 places, by means of the formulae

${\frac {\pi }{4}}=5\tan ^{-1}{\frac {1}{7}}+\tan ^{-1}{\frac {3}{79}}=2\tan ^{-1}{\frac {1}{3}}+\tan ^{-1}{\frac {1}{7}}$,