DIAMETER AND CIRCUMFERENCE OF A CHICLE. 



The first column represents the successive value of P„ in which four times each 

 number, minus twice the preceding, equals the following one ; and bence it i> easily 

 continued. 



In the fifth place, if we take o = —n, then sin cj = - - , and (4 ) gives a = 2 — fa" In 



6 2 *J ' 



this case a is again not a rational integral number. But in the same manner as in 



the preceding case, we obtain from (5) 



(10) 



p 1 p 1-2 1.2.3 



1+ 2.3 - + 3. 4. 5^ + 4.5.67V 4 



in which 



P x = 3 , 



P 8 =3(2-J3), 



and generally after the second 



Pi = 4P,_! — P,_, . 

 By means of this relation, the first two being known, the others are readily obtained, 

 and with tbem the value of the whole expression, as follows: — 



3.000000000 x 

 .803847577 x 



.215390308 x 



57713655 x 



15464312 x 



4143593 x 



1110060 x 



1 



1 



273 



1.2 



37T75 



1.2.3 



4.5.6.7 



1.2.3.4 



576777879 



1.2.3.4.5 



6.7.8.9.10.11 



1.2.3.4.5.6 



= 3.00000000C 

 = 133974596 



= 7179677 



= 412240 



= 24547 



= 1495 



92 



7.8.9.10.11.12.13 



7i: = 3.14159265 



In this case, in the first column representing tlie successive values of P., four 

 times each number, minus tbe preceding, equals tbe following one ; hence the law 

 of continuance is very simple and convenient. 



By continued bisection of tbe arcs, series of any degree of eonvergency may be 

 obtained, but the preceding are the only ones found in which a is a rational integral 

 number, or in which the successive values of P ( may be obtained by a simple re- 

 curring series. 



Each term of the series in (5) is equal to the preceding multiplied into } } .' 



in which i is the exponent of a, and consequently (i + 1) the number of the term 

 in the series. Hence this is the expression of the ratio of eonvergency, and when 

 the value of i is considerable, differs but little from \ a, which is the limiting ratio 

 of eonvergency. The first of the preceding series, on account of the large values 



