VOL. XL.} PHILOSOPHICAL TRANSACTIONS. Iftl 



and the area of the inscribed will appear to be 



But on the expansion of these binomials it will appear, that no number n 

 can be taken so large as to make the one so large, or the other so small, as 

 the area denoted by the series. 



Xr n\ t ~ h T-: — :r-s + «C' 



^ Zrr ' 5H 7r° 



So that this area being larger than any inscribed, and smaller than any cir- 

 cumscribed polygon, must be equal to the area of the sector. 



It may further be observed, that as the arch or area is found from the sine, 

 cosine, or tangent of the arch, by means of the limiting polygons, so may the 

 sine, cosine or tangent be found from the length of the arch, by the same 

 method. 



Thus, if A be the arch, whose tangent t, sine y, and cosine z, are to be 

 determined, then will the 



A - -i- X - + ^ X — - &c. 



1.2.3. '^ r^ ~ 1.2.3.4.5 r< 



Tangent T be = ■- .. , .4 



Cosine z = r - -jL X ^ + j^- X -^ - &c. 



For it may be made to appear, from the first Lemma, and its Corollaries, 

 that if in any of these theorems, as suppose in the first, the quantity a stand 

 for the bow of the circumscribed polygon, then will the quantity t, exhibited 

 by the theorem, be always larger; but if for the bow of the inscribed, always 

 less than the tangent of the arch, how great soever the number n be taken ; 

 and consequently, if a stand for the length of the arch itself, the quantity x 

 must be equal to the tangent ; and the like may be shown for the sine, and, 

 mutatis mutandis, for the cosine. 



These principles, from whence he has here derived the quadrature of the 

 circle, which is wanted in the solution of the problem in hand, happen to be, 

 on another account, absolutely requisite for the reduction of it to a manage- 

 able equation. But he has enlarged, more than was necessary to the problem 

 itself, on the uses of this sort of quadrature by the limiting polygons, because 

 it is one of that kind which requires no other knowledge but what depends on 

 the common properties of number and magnitude; and so may serve as an 

 instance to show, that no other is requisite for the establishment of principles 



