Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/53

due to Euler, and shewed that de Lagny's determination was correct with the exception of the 113th place, which should be 8 instead of 7.

Clausen calculated in 1847, 248 places of decimals by the use of Machin's formula and the formula

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

In 1841, 208 places, of which 152 are correct, were calculated by Rutherford by means of the formula

${\frac {\pi }{4}}=4\tan ^{-1}{\frac {1}{5}}-\tan ^{-1}{\frac {1}{70}}+\tan ^{-1}{\frac {1}{99}}$.

In 1844 an expert reckoner, Zacharias Dase, employed the formula

${\frac {\pi }{4}}=\tan ^{-1}{\frac {1}{2}}+\tan ^{-1}{\frac {1}{5}}+\tan ^{-1}{\frac {1}{8}}$,

supplied to him by Prof. Schultz, of Vienna, to calculate $\pi$ to 200 places of decimals, a feat which he performed in two months.

In 1853 Rutherford gave 440 places of decimals, and in the same year W. Shanks gave first 530 and then 607 places (Proc. R. S., 1853).

Richter, working independently, gave in 1853 and 1855, first 333, then 400 and finally 500 places.

Finally, W. Shanks, working with Machin's formula, gave (1873—74) 707 places of decimals.

Another series which has also been employed for the calculation of $\pi$ is the series

$\tan ^{-1}{t}={\frac {t}{1+t^{2}}}\left\{1+{\frac {2}{3}}{\frac {t^{2}}{1+t^{2}}}+{\frac {2{\,.\,}4}{3{\,.\,}5}}\left({\frac {t^{2}}{1+t^{2}}}\right)^{2}+{\frac {2{\,.\,}4{\,.\,}6}{3{\,.\,}5{\,.\,}7}}\left({\frac {t^{2}}{1+t^{2}}}\right)^{3}+\dots \right\}$.

This was given in the year 1755 by Euler, who, applying it in the formula

$\textstyle \pi =20\tan ^{-1}{\frac {1}{7}}+8\tan ^{-1}{\frac {3}{79}}$,

calculated $\pi$ to 20 places, in one hour as he states. The same series was also discovered independently by Ch. Hutton (Phil. Trans., 1776). It was later rediscovered by J. Thomson and by De Morgan.

An expression for $\pi$ given by Euler may here be noticed; taking the identity

$\tan ^{-1}{\frac {x}{2-x}}=2\int _{0}^{x}{\frac {dx}{4+x^{4}}}+2\int _{0}^{x}{\frac {xdx}{4+x^{4}}}+\int _{0}^{x}{\frac {x^{2}dx}{4+x^{4}}}$,

he developed the integrals in series, then put $\textstyle x={\frac {1}{2}}$, $\textstyle x={\frac {1}{4}}$, obtaining series for $\textstyle \tan ^{-1}{\frac {1}{3}}$, $\textstyle \tan ^{-1}{\frac {1}{7}}$, which he substituted in the formula

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