Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/37

in his work The crowning of the system; and he describes this value as exact, in contrast with the inexact value 22/7. His commentator Gancea explains that this result was obtained by calculating the perimeters of polygons of 12, 24, 48, 96, 192, and 384 sides, by the use of the formula

$a_{2n}={\sqrt {2-{\sqrt {4-a_{n}^{2}}}}}$

connecting the sides of inscribed polygons of $2n$ and $n$ sides respectively, the radius being taken as unity. If the diameter is 100, the side of an inscribed 384agon is ${\sqrt {98694}}$ which leads to the above value^[1] given by Áryabhatta. Brahmagupta (born 598 A.D.) gave as the exact value $\pi ={\sqrt {10}}$. Hankel has suggested that this was obtained as the supposed limit (${\sqrt {1000}}$) of ${\sqrt {965}}$, ${\sqrt {981}}$, ${\sqrt {986}}$, ${\sqrt {987}}$ (diameter 10), the perimeters of polygons of 12, 24, 48, 96 sides, but this explanation is doubtful. It has also been suggested that it was obtained by the approximate formula

${\sqrt {a^{2}+x}}=a+{\frac {1}{2a+x}}$

which gives ${\sqrt {10}}=3+{\frac {1}{7}}$.

The work of the Chinese Mathematicians

The earliest Chinese Mathematicians, from the time of Chou-Kong who lived in the 12th century B.C., employed the approximation $\pi =3$. Some of those who used this approximation were mathematicians of considerable attainments in other respects.

According to the Sui-shu, or Records of the Sui dynasty, there were a large number of circle-squarers, who calculated the length of the-emmlar circumference, obtaining however divergent results.

Chang Hîng, who died in 139 A.D., gave the rule

${\frac {({\text{circumference}})^{2}}{({\text{perimeter of circumscribed square}})^{2}}}={\frac {5}{8}}$

which is equivalent to $\pi ={\sqrt {10}}$.

Wang Fau made the statement that if the circumference of a circle is 142 the diameter is 45; this is equivalent to $\pi =3{\cdot }1555\ldots$. No record has been found of the method by which this result was obtained.

1. See Colebrooke's Algebra with arithmetic and mensuration, from the Sanscrit of Brahmagupta and Bháskara, London, 1817.