sect each other; and the point of intersection is the same for all these lines.

6. The diagonals of a rectangular parallelopiped are equal.

7. The square of the diagonal of a rectangular parallelopiped is equivalent to the sum of the squares of its length, breadth, and altitude.

8. A cube is the largest parallelopiped of the same extent of surface.

9. If a right prism is symmetrical to another, they are equal. 10. Within any regular polyedron there is a point equally distant from all the faces, and also from all the vertices.

11. Two regular polyedrons of the same number of faces are similar.

12. Any regular polyedron may be divided into as many regular and equal pyramids as it has faces.

13. Two different tetraedrons, and only two, may be formed with the same four triangular faces; and these two tetraedrons are symmetrical.

14. The area of the lower base of a frustum of a pyramid is five equare feet, of the upper base one and four-fifths square feet, and the altitude is two feet; required the volume.

CHAPTER XI.

SOLIDS OF REVOLUTION.

716. Of the infinite variety of forms there remain but three to be considered in this elementary work. These are formed or generated by the revolution of a plane figure about one of its lines as an axis. Figures formed in this way are called solids of revolution.

717. A CONE is a solid formed by the revolution of a right angled triangle about one of its legs as an axis. The other leg revolving describes a plane surface (521). This surface is also a circle, having for its radius the leg by which it is described. The hypotenuse describes a curved surface.

The plane surface of a cone is called its base. The opposite extremity of the axis is the vertex. The altitude is the distance from the vertex to the base, and the slant hight is the distance from the vertex to the circumference of the base.

718. A CYLINDER is a solid described by the revolution of a rectangle about one of its sides as an axis. As in the cone, the sides adjacent to the axis describe circles, while the opposite side describes a curved surface.

The plane surfaces of a cylinder are called its bases,

and the perpendicular distance between them is its altitude.

These figures are strictly a regular cone and a regular cylinder, yet but one word is used to denote the figures defined, since other cones and cylinders are not usually discussed in Elementary Geometry. The sphere, which is described by the revolution of a semicircle about the diameter, will be considered separately.

719. As the curved surfaces of the cone and of the cylinder are generated by the motion of a straight line, it follows that each of these surfaces is straight in one direction.

A straight line from the vertex of the cone to the circumference of the base, must lie wholly in the surface. So a straight line, perpendicular to the base of a cylinder at its circumference, must lie wholly in the surface. For, in each case, these positions had been occupied by the generating lines.

One surface is tangent to another when it meets, but being produced does not cut it. The place of contact of a plane with a conical or cylindrical surface, must be a straight line; since, from any point of one of those surfaces, it is straight in one direction.

CONIC SECTIONS.

720. Every point of the line which describes the curved surface of a cone, or of a cylinder, moves in a plane parallel to the base (565). Therefore, if a cone or a cylinder be cut by a plane parallel to the base, the section is a circle.

If we conceive a cone to be cut by a plane, the curve formed by the intersection will be different according to the position of the cutting plane. There are three dif

ferent modes in which it is possible for the intersection to take place. The curves thus formed are the ellipse, parabola, and hyperbola.

These Conic Sections are not usually considered in Elementary Geometry, as their properties can be better investigated by the application of algebra.

CONES.

721. A cone is said to be inscribed in a pyramid, when their bases lie in one plane, and the sides of the pyramid are tangent to the curved surface of the cone. The pyramid is said to be circumscribed about the cone.

A cone is said to be circumscribed about a pyramid, when their bases lie in one plane, and the lateral edges of the pyramid lie in the curved surface of the cone. Then the pyramid is inscribed in the cone.

722. Theorem.-A cone is the limit of the pyramids which can be circumscribed about it; also of the pyramids which can be inscribed in it.

Let ABCDE be any pyramid circumscribed about a

alternate side of the second polygon coinciding with a side of the first. This second polygon may be the base of a pyramid, having its vertex at A. Since the sides of its bases are tangent to the base of the cone, every

side of the pyramid is tangent to the curved surface of the cone. Thus the second pyramid is circumscribed about the cone, but is itself within the first pyramid.

By increasing the number of sides of the pyramid, it can be made to approximate to the cone within less than any appreciable difference. Then, as the base of the cone is the limit of the bases of the pyramids, the cone itself is also the limit of the pyramids.

Again, let a polygon be inscribed in the base of the cone. Then, straight lines joining its vertices with the vertex of the cone form the lateral edges of an inscribed pyramid. The number of sides of the base of the pyramid, and of the pyramid also, may be increased at will. It is evident, therefore, that the cone is the limit of pyramids, either circumscribed or inscribed.

723. Corollary. The area of the curved surface of a cone is equal to one-half the product of the slant hight by the circumference of the base (660). Also, it is equal to the product of the slant hight by the circumference of a section midway between the vertex and the base (666).

724. Corollary. The area of the entire surface of a cone is equal to half of the product of the circumfer ence of the base by the sum of the slant hight and the radius of the base (499).

725. Corollary. The volume of a cone is equal to one-third of the product of the base by the altitude.

726. The frustum of a cone is defined in the same way as the frustum of a pyramid.

727. Corollary.-The area of the curved surface of the frustum of a cone is equal to half the product of its slant hight by the sum of the circumferences of its bases. (664). Also, it is equal to the product of its slant