184 PHILOSOPHICAL TRANSACTIONS. [aNNO 1738. 



multiple arch ; therefore if for cp be put t, for cs, /, according to the suppo- 

 sition; and if for pk be substituted ^-+4r X 4' + lr X^l + t^m x^ + &c. 



by cor. 2, lem. 2; and for nm 



,- ^ X 75 H i7- X r* ^=^ X -^^ + &c. according to cor. 1, 



1 3|, t 5,5 t 717 t ^ 



lem. 1, the area of the sector will appear in a series, as is above determined. 



But since the number n is greater than 1, and the given arch pn is less than 

 a semicircle, and consequently kl or ?/, the sine of the submultiple arch pk, is 

 less than the semidiamer cp or t\ it may thence be easily proved, that the series 

 will approximate to the just quantity of the area, ad libitum. 



Corol. 1. — Hence, if the number n be taken equal to \/5 ^.'^25 + ^> the 



sector Nsp will be = \npy + ^—^^"7^-^/ + **** + J^.y'' + &c. 



For the numerator of the coefficient of the third term in the series, that 



determines the area h, namely, 9; — n+~3| x f, 'S equal to 9^ — nn— I .nn — Q.f, 

 which, according to the above determination of the number n, will become 

 nothing; therefore, if for < — p be puty in the second term, and the value of 

 n be substituted for n in the third and fourth, the series for the area will appear 

 on reduction to be as is here laid down. 



Corol. 2. — Hence the area of the sector nsp may be always defined nearly by 

 the terms of a cubic equation. 



For the number n, as constructed in the former corollary, is always greater 

 than the square root of 10, and consequently ^ is always less than the sine of 



»' 

 one-third part of the given arch ; so that the fourth term „ y, with the sum 



of all the following terms of the series, can never be more than a small part of 



the whole sector. 



Corol. 3. — If K stand for 57,2957795 &c. degrees, or the number of degrees 



contained in an angle subtended by an arch of the same length with the radius 



of the circle, and m be the number of degrees in an angle which is to four 



right angles, as the area nsp to the area of the whole circle; then will m be 



= ^1X^ + nH-n:r^^LhI X %, nearly. 



For - X - will appear, by the construction, to be equal to the sector nsp. 

 Case 1. — If sp be greater than cp, then take an area h equal to the sum of 

 the terms in the following series: 



