Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/46

The work of Descartes

The great Philosopher and Mathematician René Descartes (1596—1650), of immortal fame as the inventor of coordinate geometry, regarded the problem from a new point of view. A given straight line being taken as equal to the circumference of a circle he proposed to determine the diameter by the following construction:

Take $AB$ one quarter of the given straight line. On $AB$ describe the square $ABCD$; by a known process a point $C_{1}$ on $AC$ produced, can be so determined that the rectangle $\textstyle BC_{1}={\frac {1}{4}}ABCD$. Again $C_{2}$ can be so determined that rect. $\textstyle B_{1}C_{2}={\frac {1}{4}}BC_{1}$; and so on indefinitely. The diameter required is given by $AB_{\omega }$, where $B_{\omega }$ is the limit to which $B,B_{1},B_{2},\ldots$ converge. To see the reason of this, we can

shew that $AB$ is the diameter of the circle inscribed in $ABCD$, that $AB_{1}$ is the diameter of the circle circumscribed by the regular octagon having the same perimeter as the square; and generally that $AB_{n}$ is the diameter of the regular $2^{n+2}$-agon having the same perimeter as the square. To verify this, let

$x_{n}=AB_{n},x_{0}=AB;$

then by the construction,

$x_{n}(x_{n}-x_{n-1})={\frac {1}{4^{n}}}x_{0}^{2},$

and this is satisfied by $x_{n}={\frac {4x_{0}}{2^{n}}}\cot {\frac {\pi }{2^{n}}}$; thus

$\lim x_{n}={\frac {4x_{0}}{\pi }}=$ diameter of the circle.

This process was considered later by Schwab (Gergonne's Annales de Math. vol. vi), and is known as the process of isometers.

This method is equivalent to the use of the infinite series

${\frac {4}{\pi }}=\tan {\frac {\pi }{4}}+{\frac {1}{2}}\tan {\frac {\pi }{8}}+{\frac {1}{4}}\tan {\frac {\pi }{16}}+\ldots ,$

which is a particular case of the formula

${\frac {1}{x}}-\cot {x}={\frac {1}{2}}\tan {\frac {x}{2}}+{\frac {1}{4}}\tan {\frac {x}{4}}+{\frac {1}{8}}\tan {\frac {x}{8}}+\ldots ,$

due to Euler.