QUADRATURE of the CONIC SECTIONS, &c. 303 
they are to’ be deduced are in effe@ 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 fector ACB, and put- 
ting S to denote its area, we fhall: denote the abfcifla CD by 
ab S, and the ordinate BD by ord S. 
Draw AE touching the curve at its vertex, and meeting CB 
in E; then, from fimilar triangles, we have AE — DB x CA; 
CD 
therefore fuppofing the femitranfverfe axis AC to be unity, 
ordS © 
ab S°* 
AES Now this expreflion for the tangent correfpond- 
fin A 
cof A 
prefflion for the tangent of an angle A, we may fimilarly de-. 
note AE by the abbreviation tanS. In like manner, if CB’ be 
drawn to a point B’ of the curve, bifecting the fetor ACB, and 
meeting AE in E’, and B’D’ be drawn perpendicular to CA; 
ing to a hyperbolic fector S, being analogous to - , the ex- 
then, ‘as the feG@or ACB’ will be =S, it follows, that 
CD’ = ab ; S, B'D’= ord : S, and AE’= tan : S; and fo on. 
35. FRoM: 
