“ The definition of conics as sections of a cone was employed by the Greek geometers as the fundamental principle of their researches in this subject; but the subsequent development of geometrical methods has brought to light many other means for defining these curves. One definition, which is of especial value in the geometrical treatment of the conic sections (ellipse, parabola and hyperbola) in plano, is that a conic is the locus of a point whose distances from a fixed point (termed the focus) and a fixed line (the directrix) are in constant ratio. ”

“ When the cutting plane is inclined to the base of the cone at an angle less than that made by the sides of the cone, the latus rectum is greater than the intercept on the ordinate, and we obtain the ellipse; if the plane is inclined at an equal angle as the side, the latus rectum equals the intercept, and we obtain the parabola; if the inclination of the plane be greater than that of the side, we obtain the hyperbola. ”

“ Then the square of the ordinate intercepted between the diameter and the curve is equal to the rectangle contained by the portion of the diameter between the first vertex and the foot of the ordinate, and the segment of the ordinate intercepted between the diameter and the line joining the extremity of the latus rectum to the second vertex. ”