Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/51

depending upon the expression for ${\frac {\pi }{8}}$ the area of a semi-circle of

diameter 1 as the definite integral $\int _{0}^{1}{\sqrt {x-x^{2}}}dx$.. The expression has the advantage over that of Vieta that the operations required by it are all rational ones.

Lord Brouncker (1620—1684), the first President of the Royal Society, communicated without proof to Wallis the expression

${\frac {4}{\pi }}=1+{\frac {1}{2+}}{\frac {9}{2+}}{\frac {25}{2+}}{\frac {49}{2+}}$,

a proof of which was given by Wallis in his Arithmetica Infinitorum. It was afterwards shewn by Euler that Wallis' formula could be obtained from the development of the sine and cosine in infinite products, and that Brouncker's expression is a particular case of much more general theorems.

The calculation of π by series

The expression from which most of the practical methods of calculating $\pi$ have been obtained is the series which, as we now write it, is given by

$\textstyle \tan ^{-1}{x}=x-{\frac {1}{3}}x^{3}+{\frac {1}{3}}x^{5}-\ldots$ $(-1\leq x\leq 1)$.

This series was discovered by Gregory (1670) and afterwards indepen- dently by Leibnitz (1673). In Gregory's time the series was written as

$a=t-{\frac {t}{3r^{2}}}+{\frac {t}{5r^{4}}}-\ldots$

where $a$, $t$, $r$ denote the length of an arc, the length of a tangent at one extremity of the arc, and the radius of the circle ; the definition of the tangent as a ratio had not yet been introduced. The particular case

${\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-\ldots$

is known as Leibnitz's series ; he discovered it in 1674 and published it in 1682, with investigations relating to the representation of $\pi$, in his work "De vera proportione circuli ad quadratum circumscriptum in numeris rationalibus." The series was, however, known previously to Newton and Gregory.

By substituting the values ${\frac {\pi }{6}}$, ${\frac {\pi }{8}}$, ${\frac {\pi }{10}}$, ${\frac {\pi }{12}}$ in Gregory's series, the calculation of $\pi$ up to 72 places was carried out by Abraham Sharp under instructions from Halley (Sherwin's Mathematical Tables, 1705, 1706).