352 THREE GEOMETRICAL PROBLEMS [CH. XVI 



and 384 sides. The perimeter of the last is given as equal to 

 V9'8694, from which his result was obtained by approxima- 

 tion. 



Brahmagupta*, circ. 650, gave VlO, which is equal to 

 31622.... He is said to have obtained this value by inscribing 

 in a circle of unit diameter regular polygons of 12, 24, 48, and 

 96 sides, and calculating successively their perimeters, which 

 he found to be V9-65, Vim, V9-86, V9'87 respectively; and to 

 have assumed that as the number of sides is increased indefinitely 

 the perimeter would approximate to \/l0. 



Bhaskara, circ. 1150, gave two approximations. Onef — pos- 

 sibly copied from Arya-Bhata, but said to have been calculated 

 afresh by Archimedes's method from the perimeters of regular 

 polygons of 384 sides— is 3927/1250, which is equal to 31416: 

 the othei^ is 754/240, which is equal to 31416, but it is un- 

 certain whether this was not given only as an approximate 

 value. 



Among the Arabs the values 22/7, VlO, and 62832/20000 

 were given by Alkarismi§, circ. 830; and no doubt were derived 

 from Indian sources. He described the first as an approximate 

 value, the second as used by geometricians, and the third as 

 used by astronomers. 



In Chinese works the values 3, 22/7, 157/50 are said to 

 occur, probably the last two results were copied from the 

 Arabs. The Japanese|| approximations were closer. 



Returning to European mathematicians, we have the follow- 

 ing successive approximations to the value of ir: many of those 

 prior to the eighteenth century having been calculated origin- 

 ally with the view of demonstrating the incorrectness of some 

 alleged quadrature. 



* Algebra... from Brahmegupta and Bhascara, trans, by H. T. Colebrooke, 

 London, 1817, chap, xii, art. 40, p. 308. 



t Ibid., p. 87. 



t Ibid., p. 95. 



§ The Algebra of Mohammed ben Musa, ed. by F. Rosen, London, 1831, 

 pp. 71—72. 



|| Ou Japanese approximations and the methods used, see P. Harzer, 

 Transactions of the British Association for 1905, p. 325. 



