QUADRATURE of the CONIC SECTIONS, &c. 305 



— [ - tan - s -+- - tan_ j -J- - tan - s ~\ — - tan —? s 

 \2 24 48 8 16 16 



tan j- \2 2 4 4 



2* tan— , i, nn A 



2* -r — tan - }. 



^ I 2" 2 7 



36. Let us now fuppofe the hyperbolic fector ACB to be di- 

 vided into 2" equal parts, by lines drawn from the centre to the 

 points 1, 2, 3, 4, .... 7 in the curve, and tangents to be drawn 

 at the extremities of the hyperbolic arch AB, and at the alter- 

 nate intermediate points of divifion 2, 4, 6, &c. fo as to form 

 the polygon AFF F" F'" BC. Then, by a known property of 

 the hyperbola, the triangles ACF, FC 2, 2 CF, F'C 4, . . . F" CB 

 are all equal, and as their number is 2", the whole polygon 

 bounded by the tangents, and by the ftraight lines AC, CB will 

 be equal to the triangle ACF taken 2" times. But the area of 



this triangle is - AC X AF = - tan ^ (becaufe AF = tan-^, 

 2 22 2 / 



therefore 2 n tan — exprefles twice the area of the polygon 

 AFF Y" F" BC. Let Qjlenote this area, then, fubftituting - Q^ 



2 



S 



for 2" tan — , and multiplying all the terms of the feries by 2, 



we have 



*=< 



2 / 1 ,i 1 .1 1 ? 1 ■■ 1 



( tan - s -f- - tan - s + - tan - s + - tan -> 



ms \ 22 44; 8 8 16 



+ -L t an±Y 



^ 2" 2V 



Now 



