16 MATHEMATICS. ° 
68), or oblique (fig.71), as this axis is perpendicular or oblique to the base. 
A right cylinder will manifestly be generated by the revolution of a 
rectangle about one of its sides. On the convex surface of the cylinder, 
innumerable straight lines may be drawn, parallel to each other and the 
axis. If a cylinder be intersected by a plane passing through the axis, the 
section will be a parallelogram (a rectangle in the right cylinder) ; if the 
plane be parallel to the base, the section will be a circle equal to the base ; 
if it have any other position, an ellipse will be formed. Every cylinder may 
be considered as a prism of an infinite number of sides; its volume, as in 
the prism, will evidently be obtained by multiplying the area of the base by 
the altitude. The convex surface of the right cylinder is equal to the area 
of a rectangle whose base is equal to the circumference of the base of the 
cylinder, and whose altitude is the altitude of the cylinder. The deter- 
mination of the convex surface of an oblique cylinder is very difficult. 
A cone (fig. 69) is bounded by a circle as base, and a convex surface 
running toa point. The latter is a simple curved surface, and is generated 
by the revolution of a line around the circumference of a circle, and fixed 
to a point not in the plane. A straight line from the vertex to the middle 
of the base, is called the axis of the cone, which is termed right or oblique, 
as this axis is perpendicular or oblique to the base. The ordinary right 
cone is produced by the revolution of a right angled triangle about one of 
the short sides. On the convex surface of the cone, from the vertex 
to the circumference of the base, innumerable straight lines may be drawn, 
which in the right cone are all equal to each other. Every cone may be 
considered as a pyramid of an infinite number of sides. Since, then, the 
pyramid is the third part of a prism of the same base and altitude, the 
cone will be the third part of a cylinder of the same base and altitude. 
When a cone is intersected by a plane we obtain, 1, a triangle, when the 
plane of intersection is parallel to the axis (isosceles, in right cones); 2, a 
circle, when the plane is parallel to the base; in any other position, one of 
the three curves, known as the conic sections, which are next in importance 
to the circle (ellipse, parabola, and hyperbola). When a cone has its 
upper part or vertex cut off by a plane parallel to the base, it is said to be 
truncated: this is equivalent to the sum of three cones, whose altitude is 
that of the truncated cone (or frustum), and which have for bases, the 
upper base of the frustum, the lower base, and a mean proportional 
between the two bases. The area of the convex surface of the right cone, 
is equal to that of a sector of a circle whose radius is the length of the side 
of the cone, and whose arc is equal to the circumference of the base 
The area of the convex surface of a truncated cone is equivalent to that 
of a rectangle whose altitude is the length of the side of the truncated 
cone, and whose base is equal to half the sum of the circumference of the 
two bases. 
A sphere is inclosed by a single curved surface, all of whose points are 
equally distant from a point within, called the centre. A straight line 
drawn from this centre to any point of the surface is called a radius; all 
radii of a sphere are equal. A diameter is a straight line passing through 
16 
