CH. XVI] THREE GEOMETRICAL PROBLEMS 351 



process gives a value of ir correcb to at least the integral part 

 of (21ogw — T19) places of decimals. The result given by 

 Archimedes is correct to two places of decimals. His analysis 

 leads to the conclusion that the perimeters of these polygons 

 for a circle whose diameter is 4970 feet would lie between 

 15610 feet and 15620 feet— actually it is about 15613 feet 

 9 inches. 



Apollonius discussed these results, but his criticisms have 

 been lost. 



Hero of Alexandria gave* the value 3, but he quoted f the 

 result 22/7 : possibly the former number was intended only for 

 rough approximations. 



The only other Greek approximation that I need mention 

 is that given by Ptolemy J, who asserted that ir = 3° 8' 30". 

 This is equivalent to taking ir = 3 + & + jf-jfe = 3 T W « 31416. 



The Roman surveyors seem to have used 3, or sometimes 4, 

 for rough calculations. For closer approximations they often 

 employed 3£ instead of 3|, since the fractions then introduced 

 are more convenient in duodecimal arithmetic. On the other 

 hand Gerbert§ recommended the use of 22/7. 



Before coming to the medieval and modern European mathe- 

 maticians it. may be convenient to note the results arrived at in 

 India and the East. 



Baudhayana|| took 49/16 as the value of ir. 



Arya-Bhatal, circ. 530, gave 62832/20000, which is equal 

 to 3-1416. He showed that, if a is the side of a regular 

 polygon of n sides inscribed in a circle of unit diameter, and if 

 b is- the side of a regular inscribed polygon of 2ra sides, then 

 Ja _ ^ _ ^ (i _ a s)i From the side of an inscribed hexagon, he • 

 found successively the sides of polygons of 12, 24, 48, 96, 192, 



* Mensurae, ed. Hultsoh, Berlin, 1864, p. 188. 



+ Geometria, ed. Hultsoh, Berlin, 1864, pp. 115, 136. 



% Almagest, bk. vi, chap. 7; ed. Halma, vol. i, p. 421. 



§ (Euvres de Gerbert, ed. Olleris, Clermont, 1867, p. 453. 



|| The Sulvasutras by G. Thibaut, Asiatic Society of Bengal, 1875, arts. 



26—28. 



V Lecons de calcul d'Aryabhata, by L. Bodet in the Journal Asiatique, 1879, 



series 7, vol. xm, pp. 10, 21. 



