See also:

• CONE (Gr. Kwvos)

CONE (Gr. Kwvos)  , in See also:

• GEOMETRY

geometry, a See also:

• SURFACE

surface generated by a See also:

• LINE

line (the generator) which always passes through a fixed point (the vertex) and through the circumference of a fixed See also:

• CURVE (Lat. curvus, bent)

curve (the directrix) . The two sheets of the surface, on opposite sides of the vertex, are called the " nappes " of the See also:

• CONE (Gr. Kwvos)

cone . The solid formed between the vertex and a See also:

• PLANE

plane cutting the surface is also called a " cone "; this is contained by a conical surface and the plane of See also:

• SECTION
• SECTION (Lat. sectio, cutting, secare, to cut)

section, See also:

• EUCLID
• EUCLID [EucLEIDES]

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 See also:

• ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

angle . The See also:

• AXIS (Lat. for " axle ")

axis of the cone is the See also:

• SIDE
• SIDE (mod. Eski Adalia)

side about which the triangle revolves ; the circle traced by the other side containing the right angle is the "See also:

• BASE

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 See also:

• DEFINITION (Lat. definitio, from de-finire, to set limits to, describe)

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 See also:

• GENERAL
• GENERAL (Lat. generalis, of or relating to a genus, kind or class)

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-See also:

• VERTICAL (from Lat. vertex, highest point)

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 " See also:

• QUADRIC

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 See also:

• PRINCIPAL

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 See also:

• ELLIPSE (adapted from Gr. EXkei 'tc, a deficiency, iXXeirely, to fall behind)

ellipse, See also:

• PARABOLA

parabola and See also:

• HYPERBOLA

hyperbola respectively . See also:

• APOLLONIUS

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 See also:

• ATTENTION (from Lat. ad-tendo, await, expect; the condition of being " stretched " or " tense ")

attention . The See also:

• MENSURATION (Lat. mensura, a measure)

mensuration of the cone was established by See also:

• ARCHIMEDES (c. 287–212 B.C.)
• ARCHIMEDES, SCREW OF

Archimedes . He showed that the See also:

• VOLUME

volume of the cone was one-third of that of the circumscribing See also:

• CYLINDER (Gr. KvAwSpos, from KvXivaety, to roll)

cylinder, and that this was true for any type of cone . Therefore the volume is one-third of the product See also:

• AREA

area of base X vertical height . The surface of a right circular cone is equal to one-See also:

• HALF

half of the circumference of the base multiplied by the slant height of the cone . Analytically, the See also:

• EQUATION (from Lat. aequatio, aequare, to equalize)

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 See also:

• PROPERTY

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 See also:

• SPHERE (Gr. vclsa?pa, a ball or globe)

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 See also:

• BIBLIOGRAPHIES

bibliographies of these articles for See also:

• SOURCES

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 See also:

• TREATISES

treatises on descriptive solid geometry, e.g . T .

H . Eagles, Constructive Geometry .