281 On Fractional Values fur the Heptagon and Circle. 



some such side exactly contained either in one circle or in 

 several circles ; and when they thought they had made the dis- 

 covery, if they wished for further accuracy they commenced their 

 bisection with it. It is a curious fact that the most simple frac- 



355 

 tional value of the circle =^5 is founded on a diameter which 

 110 



forms the hypothenuse of one of these triangles (15, 112, 113). 

 It represents an angle of about 7°'657875, of which 47' 195 . . . 

 constitute a circle. It was therefore of no use for the experi- 

 mental determination of the value of the circumference. Pro- 

 bably they did not attempt to measure the circumference itself, 

 not being able to do it with sufficient accuracy, but confined 

 themselves to measurement by inscribed polygons; this would 

 account for their not discovering the above fraction if really 

 they were not acquainted with it. But there was another 

 rational right-angled triangle, 100, 2499, 2501, with an angle 

 of about 2°*291526. The diameter being 2501, the chord would 

 be 100, of which 1571 would very nearly compose 20 circles, 

 the angle being contained about 157*10054 . . . times in a circle. 

 Now, 157 being greater than (2 7 =128), this side, compared 

 with the side of an inscribed square, would save them seven bi- 

 sections ; as compared with that of a circumscribed square, 6 bisec- 

 tions ; and about as many compared with those required in com- 

 mencing with the hexagon. Designating the angle of the above 

 triangle by A, its half by B, and its quarter by C, I find 



cosB= ^/ V + ^° sA =2500-499 950 199 570, 



sin C = \J Y- ^ ° sB = 25*006 244 728 82, 



sin C x 4 x ^l- = 7850-962 093 795 .. . 



Seeing how nearly this approached to 7857, and knowing that 

 the actual value must be somewhat greater, I suspect they deter- 



7857 

 mined their first value at \ then finding the odd unit in 



<vOUl 



the denominator inconvenient, they altered the numerator by 3, 



. . , ,. ' 3 , . 7854 3927 ™ . . 



m the rough proportion of y, making = . This is 



J- <voUU l^ou 



= 3*1416, a much better fraction than the one they had calcu- 

 lated ( = 3*14152987. . .), or the approximate value 



