From the same points of the cone, therefore, only one parabola can be drawn; all the other sections within these parallels being ellipses, and all without hyperbolas.
Properties of the Parabola. The square of an ordinate is equal to the rectangle of the abscissa, and four times the distance of the focus from the vertex.
The perpendicular on the tangent, from the focus, is a mean proportional between the distance from the vertex to the focus, and the distance of the focus from the point of contact.
All lines within the parabola, which are drawn parallel to the axis, are called diameters.
The parameter of any diameter is a right line, of such a nature that the product under the same, and the abscissa, are equal to the square of the semi-ordinate.
The squares of all ordinates to the same diameter, are to one another as their abscissas.
Cartesian Parabola, is a curve of the second order, expressed by the equation xy = ax³ + bx² + cx + d containing four infinite legs, being the 66th species of lines of the third order, according to sir Isaac Newton: and is made use of by Descartes, in the third book of his geometry, for finding the roots of equations of six dimensions by its intersections with a circle.
Diverging Parabola, a name given by sir Isaac Newton to five different lines of the third order, expressed by the equation yy = ax³ + bx² + cx + d.
PARABOLE, Fr. See [Parabola].
PARABOLOIDE, Fr. See Parabolic Conoid.