GENERAL FORMULAS.

If we denote the volume of any prism by V, its base by B, and its altitude by H, we shall have (P. XIV.),

If we denote the volume of any pyramid by V, its base by B, and its altitude by H, we have (P. XVII.),

If we denote the volume of the frustum of any pyramid by V, its lower base by B, its upper base by b, and its altitude by H, we shall have (P. XVIII., C.),

A REGULAR POLYEDRON is one whose faces are all equal regular polygons, and whose polyedral angles are equal, each to each.

There are five regular polyedrons, namely:

1. The TETRAEDRON, or regular pyramid-a polyedron bounded by four equal equilateral triangles.

2. The HEXAEDRON, or cube-a polyedron bounded by six equal squares.

3. The OCTAEDRON-a polyedron bounded by eight equal equilateral triangles.

4. The DODECAEDRON-a polyedron bounded by twelve equal and regular pentagons.

5. The ICOSAEDRON-a polyedron bounded by twenty equal equilateral triangles.

In the Tetraedron, the triangles are grouped about the polyedral angles in sets of three, in the Octaedron they are grouped in sets of four, and in the Icosaedron they are grouped in sets of five. Now, a greater number of equilateral triangles can not be grouped so as to form a salient polyedral angle; for, if they could, the sum of the plane angles formed by the edges would be equal to, or greater than, four right angles, which is impossible (B. VI., P. XX.).

In the Hexaedron, the squares are grouped about the polyedral angles in sets of three. Now, a greater number of squares can not be grouped so as to form a salient polyedral angle; for the same reason as before.

In the Dodecaedron, the regular pentagons are grouped about the polyedral angles in sets of three, and for the same reason as before, they can not be grouped in any greater number so as to form a salient polyedral angle.

Furthermore, no other regular polygons can be grouped so as to form a salient polyedral angle; therefore, Only five regular polyedrons can be formed.

TETRAEDRON

OCTAEDRON

ICOSAEDRON

HEXAEDRON

DODECAEDRON

EXERCISES.

1. What is the convex surface of a right prism whose altitude is 20 feet and whose base is a pentagon each side of which is 15 feet?

2. The altitude of a pyramid is 10 feet and the area of its base 25 square feet; find the area of a section made by a plane 6 feet from the vertex and parallel to the base.

3. Find the convex surface of a right triangular pyramid, each side of the base being 4 feet and the slant height 12 feet.

4. A right pyramid whose altitude is 8 feet and whose base is a square each side of which is 4 feet, is cut by a plane parallel to the base and 2 feet from the vertex; required the convex surface of the frustum included between the base and the cutting plane.

5. The three concurrent edges of a rectangular parallelopipedon are 4, 6, and 8 feet; find the length of the diagonal.

6. Of two rectangular parallelopipedons having equal bases, the altitude of the first is 12 feet and its volume is 275 cubic feet; the altitude of the second is 8 feetfind its volume.

7. Two rectangular parallelopipedons having equal altitudes are respectively 80 and 45 cubic feet in volume, and the area of the base of the first is 12 square feet; find the base of the second and the altitude of both.

8. Find the volume of a triangular prism whose base is an equilateral triangle of which the altitude is 3 feet, the altitude of the prism being 8 feet.

9. The volumes of two pyramids having equal altitudes are respectively 60 and 115 cubic yards and the base of the smaller is 8 square yards; find the base of the larger.

10. Given a pyramid whose volume is 512 cubic feet and altitude 8 feet; find the volume of a similar pyramid whose altitude is 12 feet, and find also the area of the base of each.

11. Find the volume of the frustum of a right triangular pyramid with each side of the lower base 6 feet and each side of the upper base 4 feet, the altitude being 5 feet.

12. Find the volume of the pyramid of which the frustum given in the last example is a frustum.

[Find the radii of the inscribed circles of the upper and lower bases (B. IV., P. VI., C. 2); then the altitude of the pyramid, slant height, and the two radii form two similar triangles from which the altitude may be found.]

13. Given two similar prisms; the base of the first contains 30 square yards and its altitude is 8 yards; the altitude of the second prism is 6 yards-find its volume and the area of its base.

14. A pyramid, whose base is a regular pentagon of which the apothem is 3.5 feet, contains 129 cubic feet; find the volume of a similar pyramid, the apothem of

whose base is 4 feet.

15. Show that the four diagonals of a parallelopipedon bisect each other in a common point.

16. Show that the two lines joining the points of the opposite faces of a parallelopipedon, in which the diagonals of those faces intersect, bisect each other at the point in which the diagonals of the parallelopipedon intersect. 17. Show that two regular polyedrons of the same kind are similar.

18. Show that the drons are to each

homologous edges

surfaces of any two similar polyeother as the squares of any two

BOOK VIII.

THE CYLINDER, THE CONE, AND THE SPHERE.

DEFINITIONS.

P

A

H

1. A CYLINDER is a volume which may be generated by a rectangle revolving about one of its sides as an axis. Thus, if the rectangle ABCD be turned about the side AB, as an axis, it will generate the cylinder FGCQ-P. The fixed line AB is called the axis of the cylinder; the curved surface generated by the side CD, opposite the axis, is called the convex surface of the cylinder; the equal circles FGCQ, and EHDP, generated by the remaining sides BC and AD, are called bases of the cylinder; and the perpendicular distance between the planes of the bases is called the altitude of the cylinder.

M

B

The line DC, which generates the convex surface, is, in any position, called an element of the surface; the elements are all perpendicular to the planes of the bases, and any one of them is equal to the altitude of the cylinder. Any line of the generating rectangle ABCD, as which is perpendicular to the axis, will generate a circle IK, whose plane is perpendicular to the axis, and which is equal to either base: hence, any section of a cylinder by a plane perpendicular to the axis, is a circle equal to either base. Any section, FCDE, made by a plane through the axis, is a rectangle double the generating rectangle,

K