274 PHILOSOPHICAL TRANSACTIONS. [aNNO 1738. 



writers, and therefore may be considered as confirmed by experience. But as 

 he has never seen any demonstration of it, he adds the following. 



Take the binomial (x -\- y), and expand it ; take also this other binomial 

 {x — yY, which also expand ; put {x + y)" = i, and {x — y)n = p ■ now it 

 is evident that, if the expanded binomials be united by addition, their sum will 

 give double the sum of the uneven terms of the first binomial ; but if the latter 

 be subtracted from the former, then the remainder will be double the sum of 

 the even terms of the same binomial ; hence it follows that —— is the sum of 



the uneven terms, and ^-^ the sum of the even terms. 



From "i+lPl+JP^ the square of the first sum, taken '"^P^ + PP ^ the 

 square of the last, the remainder will be ps = (x -\- y)' X {x — y)" = 

 (xx — yy)", the nth root of which is xx — yy. 



Carol. — By putting 1x =■ V a ■\- ^/b + V a — V b, and making Vaa — h 

 = TO, then expounding n successively by 1,2, 3, 4, 5, 6, &c. there will arise 

 the following equations : 



7th. OAx' — li2mx^ -f 56toV — 7m^x = a 

 &c. 



Now these equations are of the same form as the equations for cosines, 

 though they are things of a quite different nature. Thus, let r be the radius of 

 a circle, c the cosine of any given arc, and x the cosine of another arc, which 

 is to the former, as 1 to n. Then it will be, 



1st. X =: c 



2d. 20^ — r^ = re 



3d. Ao^ — 3r^x = r'^c 



4th. Bx* — BrV 4- r* = r^c 



5th. iQx'' — lOr'a? + 5r*x =. r*c 



6th. 32a?* — 48rV + 18rV — r" = r^c 



7th. QAkP — 112rV + 56rV — Tr^x — r^c 



&c. 



