4l6 PHILOSOPHICAL TRANSACTIONS. [aNNO 1794. 



All which series are evidently of the 1st form of the 1st fluents, and therefore 

 their values may be expressed in the 2d form there given, or more neatly in the 

 Newtonian notation. In each of these series the value of 7j is 8 ; and the value 

 of m in the 1st series, is 1 ; in the 2d series, is 5 ; in the 3d series, is 3 ; in 

 the 4th series, is /. 



If now we take t ■=. v/-^, the tangent of 30°, which was chosen by Dr. Halley, 

 we shall have the arch of 30" 



( — \ — V • 1 -4- ^ _|_ —1 -|_ L-. 4_ ^ &f> 



._1_J -/3 ^ ' ~ 9.81 ~ 17.81' ~ 23.81' ' 33.81*' 



^ — ^ X • i -I- — ^ 4- — !— + -^ + — ^ &C 

 9^3 '^ • > T^ 13,81 ' 21. 8P ' 29-81* ~ 37.81''' 



f_L_ X • i -(- — I ^— A i— A ^— &c 



_ ) 3^/3 ^ ■ ^ ~ 11.81 ' 1Q.81' ~ 27.81' ' 35.81<' 



)_L_ V . ± 4- ^ J. ^ J_ 1 . 1 &c 



C 27 V3 ^ • 7 -r j5 81 -r 23.812 ~ 31.81' ~ 39.81*' 



Six times this quantity will be = the semi-circumference when radius is 1, 

 and = the whole circumference when the diameter is 1. If therefore we multi- 

 ply the last series by 6, and write ^\1 for —j^t and express their value in the 



form before given, we shall have the circumference of a circle whose dia- 

 meter is 1, 



f 81 V12 _ 8a 16b _ 24c 32d „ 



. , ) 80 9.80 * 17.80 ~ 2a. 80 "*" 33.80 ' ^' 



"T" j 81^12 _ 8a i6b _ 24c , 32d „ 



' 5.9. SO 13.80 ''" 21.80 ~ 29.8O "•" 37.80 ' 



/■81^12 8a , 16b 24.C , 32D . 

 V "— -4- — -I- &c 



__ 1 3.3.80 11.80 * 19.8O 27.8O ' 35.80' ' 



} 81^12 8a , i6b 24c , 32d „ 



»• 7.27 .80 15.80 ^ 23.80 31.80 ^ 39.80' 



All these new series, it is evident, converge somewhat swifter than by the 

 powers of 80. For in the first series, which has the slowest convergency, tlie 

 co-efficients — , — , —, &:c. are each of them less than 1 ; so that its conver- 

 gency is somewhat swifter than by the powers of 80. But another advantage of 

 these new series is, that the numerator and denominator of every term except the 

 first, in each of them, is divisible by 8 ; in consequence of which the arithmeti- 

 cal operation by them is much facilitated, the division by 80 being exchanged for 

 a division by 10, which is no more than removing the decimal point. These 

 series then, when the factors which are common to both numerators and deno- 

 minators are expunged, will stand as below, (each of which still converging some- 

 what quicker than by the powers of 80), and we shall have the circumference of 

 a circle whose diameter is 1, 



