EXERCISES. 688. Two pyramids, P, P', have square bases, and are such that the altitude of P equals twice the altitude of P', but the base edge of P is half as long as the base edge of P'. Find the ratio of their volumes. 689. Find the volume of a cube whose diagonal is √3.

690. A frustum of a pyramid has for its bases squares whose sides are respectively 0.6 m, 0.5 m; the altitude of the frustum is 0.9 m. Find the volume.

691. Given the volume v, and the bases b, b', of the frustum of a pyramid, to find a formula for (1) its altitude, (2) the altitude of the whole pyramid.

692. A granite monument is in the form of a frustum of a square pyramid, surmounted by a pyramid; the sides of the bases of the frustum are 1 m and 0.8 m, and the altitude of the frustum is 1.8 m; the altitude of the pyramidal top is 0.45 m. A cubic meter of water weighs a metric ton, and granite is 3 times as heavy as water. Find the weight of

the monument.

693. An excavation 1.5 m deep, rectangular at top and bottom, and in the form of a frustum of a pyramid, has its upper base 10 m wide and 16 m long, and the lower base 7.5 m wide. How many cubic meters of earth would it take to fill it to a depth of 0.75 m ?

694. In ex. 693, what is the total capacity of the excavation?

695. The volume of a cube is six times that of the regular octahedron formed by joining the centers of the faces of the cube.

696. Find the volume of a prismatoid of altitude 3.5 cm, the bases being rectangles whose corresponding dimensions are 3 cm by 2 cm, and 3.5 cm by 5 cm.

697. It is usual to find the volume of a pile of broken stones by taking the product of the altitude and the area of a transverse mid-section. Compare this with the Prismatoid Formula and find what relation it assumes between m and b + b'. Is this relation true in the case of a pyramid?

698. The volume of a pyramid equals the product of the altitude and a transverse section (parallel to the base) how far from the vertex?

699. A pyramid stands on a square base of edge 1 m; the lateral edge of the pyramid is also 1 m. Find the lateral area and volume.

700. An edge of a regular octahedron is 1 in. Find the volume. 701. In th. 14, cor., B' was supposed to decrease to 0; supposing, instead, that B' increases until it equals B, show that step 2 of the theorem gives the usual formula for the lateral area of a prism.

702. The Pyramid of Cheops was originally 480.75 ft. high, and 764 ft. square at the base. What was its volume ?

BOOK VIII.-THE CYLINDER, CONE, AND

SPHERE. SIMILAR SOLIDS.

DEFINITIONS. which is plane.

Section 1. The Cylinder.

A curved surface is a surface no part of

The number of kinds of curved surfaces is unlimited, just as the number of kinds of curves in a plane is unlimited. But as among plane curves the circumference is the best known, so there are certain curved surfaces which are better known than others, and these are treated in this book.

A cylindrical surface is a surface generated by a straight line, called the generatrix, which moves so as constantly to pass through a given curve, called the directrix, and to remain parallel to its original position.

A straight line in any position of the generatrix is called an element of the cylindrical surface.

If the directrix is a closed curve, the cylindrical surface incloses a space of unlimited length, called a cylindrical space.

A section of a cylindrical space, made by a plane cutting its elements, is called a transverse section. If it is perpenIdicular to the elements it is called a right section.

Β ́

B

One form of a cylindrical surface. ABCBD, the directrix; BB', an element; BCB', a portion of a cylindrical space.

may be a convex,

As a transverse section of a prismatic space concave, or cross polygon, so a transverse section of a cylindrical

space may be a curve of any shape if only its end-points meet. All theorems, if the signs are properly considered, will be seen to apply to each of the three forms of transverse section, corresponding to convex, concave, and cross polygons. The third is, however, too complex for treatment in elementary works.

The portion of a cylindrical space included between two parallel transverse sections is called a cylinder.

The terms bases and altitude of a cylinder will be understood, without further definition, from the corresponding definitions under the prism. The student should, throughout this section, notice the relation of cylindrical spaces to prismatic spaces.

A cylinder is considered as having the same directrix as its cylindrical space, and as having for elements the segments of the elements of the cylindrical surface included between its bases.

A cylinder is said to be right or oblique according as its elements are perpendicular or oblique to the bases.

If the base of a cylinder is a circle, the cylinder is said to be circular.

Theorem 1. Parallel transverse sections of a cylindrical space are congruent.

6. .. if L is placed on L' so that O falls on O' and OA

lies on O'A', A will fall on A'.

7. And B will fall on B', and C on C'.

Why?

8. Similarly, for every point of L there is a single corresponding point of L' on which it will fall.

9. .. the figures are congruent.

Def. congruence

COROLLARIES. 1. The bases of a cylinder are congruent.

2. The elements of a cylinder are equal. (Why ?)

Theorem 2. Cylinders cut from the same cylindrical space, and having equal elements, are equal.

4. .. solid CD' can be made to slide along in S, and coincide with solid AB' since they are equal in all their parts.

5. .. adding the common part A'D, AD = A'D'. Ax. 2

COROLLARY. The cylindrical surfaces of two cylinders cut from the same cylindrical space, and having equal elements, are equal.

DEFINITIONS. A conical surface is a surface generated by a straight line which moves so as constantly to pass through a given curve and contain a given point

called the vertex.

The terms generatrix, directrix, elements will be understood from Section 1.

The portions of the conical surface on opposite sides of the vertex are called the nappes, and are usually distinguished as upper and lower.

If the directrix is a closed curve, the conical surface incloses a double space, on opposite sides of the vertex, known as a conical space.

A section of a conical space made by a plane cutting all of its elements on the same side of the vertex is called a transverse section.

N'

A conical surface. DX, the directrix ; V, the vertex; N, N', the lower and upper nappes; V-DX, a cone, with base the closed figure DX.

The portion of a conical space included between the vertex and a transverse section is called a cone, the transverse section being called its base.

A cone is considered as having the same directrix and vertex as its conical space, and the segments of the elements between the vertex and base are called the elements of the cone.

The distance from the vertex of a cone to the plane of the base is called the altitude of the cone.

If the base of a cone is a circle, the cone is said to be circular. In that case, the line determined by the vertex and the center of the base is called the axis of the cone.

If