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

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 given by Áryabhatta. Brahmagupta (born 598 ) 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

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

The earliest Chinese Mathematicians, from the time of Chou-Kong who lived in the 12th century, 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, gave the rule

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.