QUADRATURE oj the CONIC SECTIONS, &c. 305 



they are to. be deduced are in effect the fame as we have had 

 occafion to employ when treating of the circle, it will be pro- 

 per to ufe the fame form of reafoning, and the fame mode of 

 notation, in the one cafe as in the other. 



Therefore, in the equilateral hyperbola ABB', of which C 

 is the centre, (Plate IX. Fig. 1 .), and CA the femitranfverfe axis ; 

 let CB be drawn to any point B of the curve, and BD perpen- 

 dicular to CA ; then, in imitation of the notation commonly 

 ufed in the arithmetic of fines, which we have followed in the 

 former part of this paper, we fhall confider the co-ordinates 

 CD, DB, as functions of the hyperbolic fedor ACB. and put- 

 ting S to denote its area, we fhall denote the abfcifTa CD by 

 ab S, and the ordinate BD by ord S. 



Draw AE touching the curve at its vertex, and meeting CB 



DB 



in E ; then, from fimilar triangles, we have AE zr x CA ; 



CD 



therefore fuppofing the femitranfverfe axis AC to be unity, 

 AE zz . „ . Now this expreflion for the tangent correfpond- 



ing to a hyperbolic fedor S, being analogous to — % — , the ex- 



cof A 



preffion for the tangent of an angle A, we may fimilarly de- 

 note AE by the abbreviation tan S. In like manner, if CB' be 

 drawn to a point B' of the curve, bifeding the fedor ACB, and 

 meeting AE in E', and B'D be drawn perpendicular to C \ y 



then, as the fedor ACB' will be -S, it follows, that 



2 



CD' = ab I S, B'D' = ord - S, and AE' = tan - S : and fo on. 

 2 ,2 2 



35. FROM: 



