CONE (Gr. Kwvos)  , in geometry, a surface generated by a line (the generator) which always passes through a fixed point (the vertex) and through the circumference of a fixed curve (the directrix) . The two sheets of the surface, on opposite sides of the vertex, are called the " nappes " of the cone . The solid formed between the vertex and a plane cutting the surface is also called a " cone "; this is contained by a conical surface and the plane of section, Euclid defines a " right cone " as the solid figure formed by the revolution of a right-angled triangle about one of the sides containing the right angle . The axis of the cone is the side about which the triangle revolves ; the circle traced by the other side containing the right angle is the "base"; the hypotenuse in any one of its positions is a generator or generating line ; and the intersection of the axis and a generator is termed the vertex . The Euclidean definition may be modified, so as to avoid the limits thereby placed on the figure, viz. the notion that the solid is between the vertex and the base . A general definition is as follows :—If two intersecting straight lines be given, and one of the lines is made to revolve about the other, which is fixed in such a manner that the angle between the lines is everywhere the same, then the surface (or solid) traced out by the moving line (or generator) is a cone, having the fixed line for axis, the point of intersection of the lines for vertex, and the angle between the lines for the semi-vertical angle of the cone . An " oblique cone " is the solid or surface traced out by a line which passes through a fixed point and through the circumference of a circle, the fixed point not being on the line through the centre of the circle perpendicular to its plane . A " quadric cone " is a cone having any conic for its base . The plane containing the vertex, centre of the base, and perpendicular to the base is called the principal section; and the section of a cone by a plane containing the vertex is a triangle if the solid be considered, and two intersecting lines if the surface be considered . The "subcontrary section " of an oblique cone is made by a plane not parallel to the base, but perpendicular to the principal section, and inclined to the generating lines in that section at the same angles as the base ; this section is a circle . The planes parallel to the base or subcontrary section are called " cyclic planes." The Greeks distinguished three types of right cones, named " acute," " right-angled " and " obtuse," according to the magnitude of the vertical angle; and Menaechmus showed that the sections of these cones by planes perpendicular to a generator were the ellipse, parabola and hyperbola respectively . Apollonius went further when he derived these curves by varying the inclination of the section of any right or oblique cone (see CONIC SECTION) .

It is to be noted that the Greeks investigated these curves in solido, and consequently the geometry of the cone received much

attention . The mensuration of the cone was established by Archimedes . He showed that the volume of the cone was one-third of that of the circumscribing cylinder, and that this was true for any type of cone . Therefore the volume is one-third of the product area of base X vertical height . The surface of a right circular cone is equal to one-half of the circumference of the base multiplied by the slant height of the cone . Analytically, the equation to a right cone formed by the revolution of the line y=mx about the axis of xis z=m(x2+y2) . Obviously every tangent plane passes through the vertex; this is the characteristic property of conical surfaces . Conical surfaces are also " developable " surfaces, i.e. the surface can be applied to a plane without wrinkling or rending . Connected with quadric cones is the interesting curve termed the " spheroconic," which is the curve of intersection of any quadric cone and a sphere having its centre at the vertex of the cone . References should be made to the articles GEOMETRY and SURFACE for further discussion; and to the bibliographies of these articles for sources where the subject can be further studied . The geometrical construction of the curves of intersection of the cone with other solids is given in treatises on descriptive solid geometry, e.g . T .

H . Eagles, Constructive Geometry .