350 THREK GEOMETRICAL PROBLEMS [CH. XVI 



The ancient Egyptians* took 256/81 as the value of it, this 

 is equal to 3\L605... ; but the rougher approximation of 3 was 

 used by the Babylonians f and by the Jews J. It is not unlikely 

 that these numbers were obtained empirically. 



We come next to a long roll of Greek mathematicians who 

 attacked the problem. Whether the researches of the members 

 of the Ionian School, the Pythagoreans, Anaxagoras, Hippias, 

 Antipho, and Bryso led to numerical approximations for the 

 value of 7r is doubtful, and their investigations need not 

 detain us. The quadrature of certain lunes by Hippocrates of 

 Chios is ingenious and correct, but a value of w cannot be 

 thence deduced; and it seems likely that the later members 

 of the Athenian School concentrated their efforts on other 

 questions. 



It is probable that Euclid §, the illustrious founder of the 

 Alexandrian School, was aware that it was greater than 3 and 

 less than 4, but he did not state the result explicitly. 



The mathematical treatment of the subject began with 

 Archimedes, who proved that tr is less than 3f and greater 

 than 3f$, that is, it lies between 3\L428... and 31408.... He 

 established || this by inscribing in a circle and circumscribing 

 about it regular polygons of 96 sides, then determining by 

 geometry the perimeters of these polygons, and finally 

 assuming that the circumference of the circle was inter- 

 mediate between these perimeters : this leads to a result from 

 which he deduced the limits given above. This method 

 is equivalent to using the proposition sin 6 < 6 < tan 0, where 

 = -n-/96: the values of sin0 and tan# were deduced by 

 Archimedes from those of sin $w and tan §ir by repeated 

 bisections of the angle. With a polygon of n sides this 



* Ein mathetnatuches Handbuck der alten Aegypter {i.e. the Rhind papyrus), 

 by A. Eisenlohr, Leipzig, 1877, arts. 100—109, 117, 124. 



t Oppert, Journal AHatique, August, 1872, and October, 1874. 



X 1 Kings, oh. 7, ver. 23 ; 2 Chronioles, ch. 4, ver. 2. 



§ These results can be deduced from Euc. iv, IS, and it, 8: see also book hi, 

 prop. 16. 



|| ArchimedU Opera, KifcXov jttAynjiris, prop, m, ed. Torelli, Oxford, 1792, 

 pp. 205—216; ed. Heiberg, Leipzig, 1880, vol. i, pp. 263—271. 



