polyedral angles must be either triedral, tetraedral or pentaedral; for six angles of 60° each are equal to four right-angles, and cannot therefore form a convex polyedral angle.

Triedral Fig.153.

The regular tetraedron is contained by four equilateral triangles; and has four triedral angles. angles admit only of four faces in the body.

The regular octaedron is contained by eight equilateral triangles; and it has six tetraedral angles. The te- Fig.154.

traedral angle allows only eight faces in the body.

The regular icosaedron, contained by twenty equilateral triangles, will have twelves pentaedral angles. These Fig.155. are all the regular polyedrons which can be formed with triangular faces.

The regular hexaedron or cube is evidently the only Fig.156. regular polyedron which can be formed by square faces.

The regular dodecaedron contained by twelve pentag- Fig.157. onal faces, has twenty triedral angles.

These five bodies are the only regular polyedrons. They can each of them be inscribed in a sphere, and each may have a sphere inscribed in it; and, in each case, the inscribed and circumscribed sphere will have the same centre. The right cone whose angle at the apex is 60°, and the right cylinder whose height is equal to the diameter of its base, may each have an inscribed and circumscribed sphere whose centres are the same point. The right prism and pyramid, whose heights have the proper ratio to their other dimensions, and whose bases are regular polygons, may have inscribed and circumscribed spheres with a common centre.

Besides the five polyedrons mentioned above, the sphere is the only other regular body.

Practical Questions.

What is the volume of a prism whose height is 20 feet and whose base is a rectangle the contiguous sides of which are 3 feet and 8 feet?

Required the cubic feet in a stick of timber 60 feet long, 18 inches wide and 8 inches thick ?

What is the volume of a prism whose height is 7 yards, and whose base is a right-angled triangle one side of which is one foot and the other 30 inches?

Required the volume of a prism whose height is 12 feet and whose base is an equilateral triangle whose side is 6 feet.

A rectangular parallelopiped has its three contiguous edges, 4 feet, 6 feet, and 3 feet; what is it its volume?

An obelisk is composed of the frustum of a square pyramid, the side of whose inferior base is 12 feet, the side of the superior base 6 feet and the height 684 feet, this is surmounted by a pyramidal summit 11 feet high, making the entire height 70 feet; what is the entire volume of the obelisk ?

Suppose the above obelisk to be of stone, what would it cost at one dollar per foot of surface, the base of the obelisk not being included?

What is the area of a rectangular parallelopiped whose three contiguous edges are 3 feet, 4 feet, and 5 feet?

What is the area of a right pyramid, the base of which is a square of 6 feet on a side, and whose height is 4 feet?

A house 40 feet wide and 50 feet long, has a cellar under the whole of it, 8 feet deep below the sills; the bottom of the sills (which are 10 inches wide) is 2 feet above the natural surface of the ground; the inside of the cellar wall, which is 22 inches thick, coincides with the inside of the sills; the underpinning stones are 18 inchess deep and 8 inches thick; there are 6 cellar windows of 3 feet each in width and 18 inches high. How many cubic yards of earth were removed from the cellar? What did the cellar wall cost at $3 per cubic yard? How many cubic feet of stone in the underpinning; and what did it cost at 50 cents per foot of surface?

A ditch is to be dug 1 mile in length, 5 feet in width

and 4 feet in depth; what will it cost at 10 cents per cubic yard?

1

An embankment for a road is to be made across a meadow, of a mile in width; the road is to be 6 feet above the surface of the meadow, 30 feet wide at top, with banks sloping 2 feet in 3; what will this embankment cost at 12 cents per cubic yard?

A bridge is to be built of stone; its length 60 feet, its width 25 feet and its height 12 feet, with two semi circular arches of 8 feet radius and 9 feet high in the intrados; What will be the cost of this bridge, at $2 per cubic yard?

What is the area of a right cone the radius of whose base is 36 feet, and whose height is 48 feet? What is the volume of this cone?

What is the area of a cylinder whose height is 20 feet and the radius of whose base is 10 feet? What is its volume ?

What is the area of a sphere inscribed in the same cylinder; and what is its volume ?

What are the area and volume of a right cone inscribed in this cylinder ?

How many cubic feet of water will a hemispherical vessel contain, its radius being 10 feet?

How many cubic feet of water in a pond of 1000 acres, 10 feet deep? How many miles of canal will such a reservoir supply for three months, at 50 cubic feet per mile per minute; supposing one foot in depth of the reservoir to be lost by evaporation?

Supposing this reservoir is at one extremity of the canal to be supplied by it, what will be the velocity of the current in this part of the canal, with 5 feet of water and an average width of 40 feet?

A cubic block measures 6 feet on a side; the edge of a cubic block eight times as large ?

what is

Cleopatra's Needle, a celebrated pyramidal obelisk near Alexandria in Egypt, consists of a single piece of red granite; its height is 60 feet, and its inferior base is 7 feet square. Supposing its superior base to be 5 feet square, what will be its area and volume; and what its weight at 170 lbs per cubic foot ?

The pyramid of Cheops, the largest of the Egyptian pyramids, has a square base whose side is 693 feet; its height is 499 feet. How many acres of ground does it

cover; and what is the side of a cube of equal volume?

Suppose 1000 men to be employed at a time, in erecting this structure, and suppose 10 men, on an average, to complete 29 cubic feet of the work per day: How long would it be in building?

AN INTRODUCTION

TO

DESCRIPTIVE GEOMETRY.

SECTION I.-Preliminary Explanations.

1. IN the Second Part of the Elements of Geometry we discussed geometrical magnitudes embracing the three dimensions of space. But it must have been perceived by the learner, that, in the graphical representations of these bodies, there was much indefiniteness respecting the relative position and relative measure of their several parts. In the application of Geometry to the arts, we have frequent occasion, not only to represent the various forms and relative positions of bodies, but accurately to determine the various dimensions and other relations of their parts, and to deduce from these determinate relations, a series of results, the accuracy of which must depend entirely upon the fidelity of the graphical representation of the given magnitudes, and of the graphical construction of the consequent relations.

2. The object of Descriptive Geometry is to teach this accurate representation of bodies upon a plane, and this graphical solution of problems. A geometrical magnitude is represented upon a plane by a method called projection.

The simplest projection, and that in most frequent use, is called orthogonal or othographic projection. It consists in drawing straight lines from the points to be projected, perpendicular to the plane on which the projection is to be made, and which is called the plane of projection; the straight lines by which the projection is made are called projecting lines.

3. When the projecting lines are inclined to the plane of projection, the projection is called oblique. When the