Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/54

In China a work was published by Imperial order in 1713 which contained a chapter on the quadrature of the circle where the first 19 figures in the value of $\pi$ are given.

At the beginning of the eighteenth century, analytical methods were introduced into China by Tu Tê-mei (Pierre Jartoux) a French missionary; it is, however, not known how much of his work is original, or whether he borrowed the formulae he gave directly from European Mathematicians.

One of his series

$\pi =3\left(1+{\frac {1^{2}}{4{\,.\,}6}}+{\frac {1^{2}{\,.\,}3^{2}}{4{\,.\,}6{\,.\,}8{\,.\,}10}}+{\frac {1^{2}{\,.\,}3^{2}{\,.\,}5^{2}}{4{\,.\,}6{\,.\,}8{\,.\,}10{\,.\,}12{\,.\,}14}}+\ldots \right)$

was employed at the beginning of the nineteenth century by ChuHung for the calculation of $\pi$. By this means 25 correct figures were obtained.

Tsêng Chi-hung, who died in 1877, published values of $\pi$ and $1/\pi$ to 100 places. He is said to have obtained his value of $\pi$ in a month, by means of the formula

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

and Gregory's series.

In Japan, where a considerable school of Mathematics was developed in the eighteenth century, $\pi$ was calculated by Takebe in 1722 to 41 places, by employment of the regular 1024agon. It was calculated by Matsunaga in 1739 to 50 places by means of the same series as had been employed by Chu-Hung.

The rational values $\textstyle \pi ={\frac {5419351}{1725033}}$, $\textstyle \pi ={\frac {428224593349304}{136308121570117}}$, correct to 12 and 30 decimal places respectively, were given by Arima in 1766.

Kurushima Yoshita (died 1757) gave for $\pi ^{2}$ the approximate values 227/13, 10748/1089, 10975/1112, 98548/9885.

Tanyem Shǒkei published in 1728 the series

$\pi ^{2}=8\left(1+{\frac {1}{6}}+{\frac {1{\,.\,}4}{6{\,.\,}15}}+{\frac {1{\,.\,}4{\,.\,}9}{6{\,.\,}15{\,.\,}28}}+\ldots \right)$,
$\pi ^{2}=4\left(1+{\frac {2}{6}}+{\frac {2{\,.\,}8}{6{\,.\,}15}}+{\frac {2{\,.\,}8{\,.\,}18}{6{\,.\,}15{\,.\,}28}}+\ldots \right)$

due to Takebe, and ultimately to Jartoux.

The following series published in 1739 by Matsunaga may be mentioned:

$\pi =9\left(1+{\frac {1^{2}}{3{\,.\,}4}}+{\frac {1^{2}{\,.\,}2^{2}}{3{\,.\,}4{\,.\,}5{\,.\,}6}}+{\frac {1^{2}{\,.\,}2^{2}{\,.\,}3^{2}}{3{\,.\,}4{\,.\,}5{\,.\,}6{\,.\,}7{\,.\,}8}}+\ldots \right)$.