1911 Encyclopædia Britannica/Cone

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CONE (Gr. κῶνος), 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 “nappés” 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 ✕ 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 x is 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.