Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/40

The development of the theory of equations which later became of fundamental importance in relation to our problem was due to the Italian Mathematicians of the 16th century, Tartaglia (1506—1559), Cardano (1501—1576), and Ferrari (1522—1565).

The first to obtain a more exact value of $\pi$ than those hitherto known in Europe was Adriaen Anthonisz (1527—1607) who rediscovered the Chinese value $\textstyle \pi ={\frac {355}{113}}=3{\cdot }1415929\ldots$, which is correct to 6 decimal places. His son Adriaen who took the name of Metius (1571—1635), published this value in 1625, and explained that his father had obtained the approximations $\textstyle {\frac {333}{106}}<\pi <{\frac {377}{120}}$ by the method of Archimedes, and had then taken the mean of the numerators and denominators, thus obtaining his value.

The first explicit expression for $\pi$ by an infinite sequence of operations was obtained by Vieta (François Viète, 1540—1603). He proved that, if two regular polygons are inscribed in a circle, the first having half the number of sides of the second, then the area of the first is to that of the second as the supplementary chord of a side of the first polygon is to the diameter of the circle. Taking a square, an octagon, then polygons of 16, 32, … sides, he expressed the supplementary chord of the side of each, and thus obtained the ratio of the area of each polygon to that of the next. He found that, if the diameter be taken as unity, the area of the circle is

$2{\frac {1}{{\sqrt {\frac {1}{2}}}{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {\frac {1}{2}}}}}{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {{\frac {1}{2}}\ldots }}}}}}}}}$,

from which we obtain

${\frac {\pi }{2}}={\frac {1}{{\sqrt {\frac {1}{2}}}{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {\frac {1}{2}}}}}\ldots }}$.

It may be observed that this expression is obtainable from the formula

$\theta ={\frac {\sin {\theta }}{\cos {\frac {\theta }{2}}\cos {\frac {\theta }{4}}\cos {\frac {\theta }{8}}\ldots }}$ $(\theta <\pi )$

afterwards obtained by Euler, by taking $\theta ={\frac {\pi }{2}}$.

Applying the method of Archimedes, starting with a hexagon and proceeding to a polygon of 2¹⁶ . 6 sides, Vieta shewed that, if the diameter of the circle be 100000, the circumference is > 314159 26535/100000 and is < 3.14159 26537/100000; he thus obtained $\pi$ correct to 9 places of decimals.