160 PHILOSOPHICAL TRANSACTIONS. [aNNO 1738. 



let f, T, V be expounded as before; then, if any number n be taken, the arch 



of the circle will be 



1 I 



, . , 7*4- T|" r — t1 " 



always less than \ '— X «f, 



and greater than ^±^rjzliz3l — x wj . 



For if, by the preceding Corollaries, a regular rectilinear polygon be in- 

 scribed within, and another without, the arch a, each having half as many 

 sides as is expressed by the number n ; then will the former of these quantities 

 be the length of the bow of the circumscribed polygon, or the sum of all its 

 sides, which is always greater, and the latter will be the length of the bow of 

 the inscribed polygon, which is always less, than the arch of the circle : how 

 great soever the number n be taken. 



Carol. 1 . Hence the serieses for the rectification of the arch of a circle 

 may be derived. 



For by converting the binomials into the form of a series, that the fictitious 

 quantities, j , t, v may be destroyed ; it will appear, that no number n can be 

 taken so large as to make the inscribed polygon so great, or the circumscribed 

 so little, as the series 



"I" ~ ^ + 1» ~ ^ + ^*^" '" °"^ ^^^^' °^ ^^^ ^1"^^ 

 t' <* f 



t — T'i'h T7 — rrr -\- &c. in the other case. 



Therefore, since the quantity denoted by the sum of the terms, in either of 

 these serieses, is always greater than any inscribed polygon, and always less 

 than any circumscribed, it must therefore be equal to the arch of the circle. 



Carol. 2. If, in the first of the above serieses, the root \^rr — yy, be ex- 

 tracted, and substituted for z, there will arise the other series of Sir Isaac 

 Newton, for giving the arch from the sine; namely, 



y + & + 157 + "iS^ + ^^- °' otherwise, 



— V^ 1.2.3 ^ r^ ^ 1.2.3.4.5. ^ r* ^ 1.2.3.4.5.6.7. r" ^ ^^' 



Schol. In like manner, as the arches of the polygons serve to determine 

 the arch of the circle, so by comparing the areas of the circumscribed and in- 

 scribed polygons, -^nrT and ^n\z, the area of the sector of a circle may be 

 found. For if t, y and z be the tangent, sine and cosine of the arch a; then, 



by the second Lemma, the area of the circumscribed polygon 



^ 1^ 



will be found to be -j-nrj X ^+''1" — r— t|" _ ,^^.j,_ 



