# 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 V*acb,* called the opposite cone, and having the
same common axis produced DV*d,* 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.