CONE

, a kind of round pyramid, or a solid body having a circle for its base, and its sides formed by right lines drawn from the circumference of the base to a point at top, being the vertex or apex of the cone.

Euclid desines a cone to be a solid figure, whose base is a circle, and is produced by the entire revolution of a right-angled triangle about its perpendicular leg, called the axis of the cone. If this leg, or axis, be greater than the base of the triangle, or radius of the circular base of the cone, then the cone is acute-angled, that is, the angle at its vertex is an acute angle; but if the axis be less than the radius of the base, it is an obtuseangled cone; and if they are equal, it is a rig^.ht-angled cone.

But Euclid's definition only extends to a right cone, that is, to a cone whose axis is perpendicular or at rightangles to its base; and not to oblique ones, in which the axis is oblique to the base, the general definition, or description of which may be this: If a line VA continually pass through the point V, turning upon that point as a joint, and the lower part of it be carried round the circumference ABC of a circle; then the space inclosed between that circle and the path of the line, is a cone. The circle ABC is the base of the cone; V is its vertex; and the line VD, from the vertex to the centre of the base, is the axis of the cone. Also the other part of the revolving line, produced above V, will describe another cone Vacb, called the opposite cone, and having the same common axis produced DVd, and vertex V.

Properties of the Cone.—1. The area or surface of every right cone, exclusive of its base, is equal to a triangle whose base is the periphery, and its height the slant side of the cone. Or, the curve superficies of a right cone, is to the area of its circular base, as the slant side is to the radius of the base. And therefore, the same curve surface of the cone is equal to the sector of a circle whose radius is the slant side, and its arch equal to the circumference of the base of the cone.

2. Every cone, whether right or oblique, is equal to one-third part of a cylinder of equal base and altitude; and therefore the solid content is found by multiplying the base by the altitude, and taking 1/3 of the product; and hence also all cones of the same or equal base and altitude, are equal.

3. Although the solidity of an oblique cone be obtained in the same manner with that of a right one, it is otherwise with regard to the surface, since this cannot be reduced to the measure of a sector of a circle, because all the lines drawn from the vertex to the base are not equal. See a Memoir on this subject, by M. Euler, in the Nouv. Mem. de Petersburgh vol. 1. Dr. Barrow has demonstrated, in his Lectiones Geometricæ, that the solidity of a cone with an elliptic base, forming part of a right cone, is equal to the product of its surface by a third part of one of the perpendiculars | drawn from the point in which the axis of the right cone intersects the ellipse; and that it is also equal to 1/3 of the height of the cone multiplied by the elliptic base: consequently that the perpendicular is to the height of the cone, as the elliptic base is to the curve surface. For the curve surface of all the oblique parts of a cone, see my Mensur. pa. 234 &c.

4. To find the Curve Surface of the Frustum of a Cone. Multiply half the sum of the circumferences of the two ends, by the slant side, or distance between these circumferences.

5. For the Solidity of the Frustum of a Cone, add into one sum the areas of the two ends and the mean proportional between them, multiply that sum by the perpendicular height, and 1/3 of the product will be the solidity. See also my Mensuration, pa. 189.

6. The Centre of Gravity of a cone is 3/4 of the axis distant from the vertex.

Cones of the Higher Kinds, are those whose bases are circles of the higher kinds; and are generated, like the common cone, by conceiving a line turning on a point or vertex on high, and revolving round the circle of the higher kind.

Cone of Rays, in Optics, includes all the several rays which fall from any point of a radiant object, on the surface of a glass.

Double Cone, or Spindle, in Mechanics, is a solid formed of two equal cones joined at their bases. If this be laid on the lower part of two rulers, making an angle with each other, and elevated in a certain degree above the horizontal plane, the cones will roll up towards the raised ends, and seem to ascend, though in reality its centre of gravity descends perpendicularly lower.