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

 The more quickly convergent series

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

and gave the general expression

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 $$\frac{x}{y} = 1$$,

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

which follows from the relation

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