" To draw Tangents to the Parabola." If the point of contact C be given : (fig. 7) draw the ordinate C B, and produce the axis till A T be = A B ; then join T C, which will be the tangent. Or if the point be given in the axis produced : take A B = A T, and draw the ordinate B C, which will give C the point of contact ; to which draw the line T C as before. If D be any other point, neither in the curve nor in the axis produced, through which the tangent is to pass : draw 1) E G per pendicular to the axis, and take D H a mean proportional between I) E and D G, and draw IL C parallel to the axis, so shall C be the point of contact through which, and the given point D, the tangent D C T is to be drawn.
When the tangent is to make a given angle with the ordinate at the point of contact : take the absciss A I equal to half the parameter, or to double the focal dis and draw the ordinate I E : also draw A II to make with A I the angle H A I equal to the given angle ; then draw II C parallel to the axis, and it will cut the curve in C, the point of contact, where a line drawn to make the given angle with C B will Le the tangent re quired.
''1'o find the Area of a Parabola." Mul tiply the base E G by the perpendicular height A I, and 4 of the product will be the area of the space A E G A; because the parabolic space is 4 of its circumscrib ing parallelogram.
" To find the Length of the Curve A C," commencing at the vertex. Let y the ordinate B C, p = the parameter, 29 — , and s = 1 + q' ; then shall p x (q s + hyp. log. of g + 8) be the length of the curve A C.
PAnAnotA, Cartesian, is a curve of the second order, expressed by the equation d, containing four infinite legs, viz. two hyperbolic ones, M B nz, (Plate Parabola, fig. 8), (A E being the asymptote) tending con trary ways, and two parabolic legs B N, M N joining them, being the sixty-sixth species of lines of the third order, accord ing to Sir Isaac Newton, called by him a trident : it is made use of by Des Cartes, in the third book of his Geometry, for find ing the roots of equations of six dimensions by its intersections with a circle. Its most simple equation is x y = x3 + to, and the points through which it is to pass, may be easily found by means of a common para bola, whose absciss is a .ai-= b x + c, and an hyperbola, whose absciss is ---; for y will be equal to the sum or differ ence of the correspondent ordinates of this parabola and hyperbola.