To find the solid content of a prism, multiply the area of one end by the perpendicular height.

We have already said that, in calculating volume, whatever method is adopted amounts practically to multiplying three numbers together; the product of two of the numbers gives us the area of a base, and the product of this and the third number gives the solid content of the body. This is exactly the way in which we have treated the cube and parallelopiped, and it will be readily seen that the same rule holds good for the prism.

Let us suppose that the area of each triangular end of the prism A is exactly 1 square foot; then it follows that if the prism is 1 foot high its solidity is 1 cubic foot, and that, whatever it be, the number of cubic feet in the whole body is equal to the number of linear feet in the height. But whatever may be the area of each end, a certain height cut off from the prism will give a volume of 1 cubic foot, and the volume of the whole prism will be just as many cubic feet as the number of times this height is contained in the whole height of the body.

If the prism be oblique, the area of one end must be multiplied by the perpendicular height; for it is manifest that the volume decreases as the perpendicular height is increased, so that the more the sides slope the less is the solid content.

EXERCISES (L).

1. A triangular prism is 21 ft. long, and of its end one side is 14 ft. long, and the perpendicular let fall on it from the opposite angle is 1 ft. 3 in.; find the solid content.

2. An octagonal prism is 40 ft. long, and each side of its end is 10 ft.; find its volume.

3. The triangular ends of a prism have sides measuring 6, 8, and 10 ft., and the prism is 10 ft. long; find its volume.

4. Find the volume of a square prism, whose length is 41 ft., one side of its base 15 in.

and

5. The length of a hexagonal prism is 36 ft., and each side of its base 1 ft. 4 in.; find its solid content.

6. Each side of a pentagonal prism measures 10 ft., and its length is 50 ft.; find its volume.

7. What would be the cost of gilding the entire surface of a pentagonal prism whose length is 321⁄2 in., and each side of its end 61 in., at lid. per cubic inch.

8. A hexagonal prism is 50 ft. 6 in. long, and a straight line across the centre of its end from corner to corner is 30 in.; find its solidity.

9. The volume of a hexagonal prism is 93.532 cubic ft., and each side of its end measures 18 in.; find its length.

10. The volume of a triangular prism is 64.95 cubic ft., and each side of its end measures 30 in.; find its length.

11. A hexagonal prism is 21 ft. long, and contains 1651 564 cubic ft.; find the length of a side of its end.

12. A triangular prism, each side of which measures 14 ft. 10 in. is 10 ft. 3 in. high; find its volume.

LESSON XVI. THE CYLINDER.

The cylinder may be described as a circular prism, i.e., a prism with circular ends.

It has been already said that a circle is practically a regular polygon with an unlimited number of sides, and this being the case, we have found that the circle is treated in mensuration exactly as if it were a polygon. In the same way the cylinder is really a prism, and the volume or solidity of both is found in the same way, i.e., by multiplying the area of the base by the perpendicular height. Although, from the above definition and the figure, you will know what a cylinder is, it may be well to add the following more perfect definition:

A cylinder is a solid figure described by the revolution of a rectangle about one of its sides which remains fixed.

It is evident that the whole surface of a cylinder consists of the two circles forming its ends, and the rectangle forming the sides; the convex surface is really a rectangle whose height is the height of the cylinder and base the circumference of the circular end.

The volume of a ring may be easily found by the same rule as that finding the volume of a cylinder. If the ring (A) has a circular section, we may regard it as a cylinder

whose height is equal to the middle line, or mean circumference, of the ring.

If the ring be flat (B) its volume is equal to the difference of the volumes of the outer and inner cylinders of the same height.

EXERCISES (M).

1. A cylindrical rod of iron is 42 ft. long, and its diameter is 1 ft. 3 in.; find its volume.

2. What is the volume of a piece of the above rod 50 in. long? 3. Find the convex surface of a cylinder whose length is 20 ft., and radius of the base 2 ft.

4. What is the whole surface of a cylinder whose length is 20 ft. 4 in., and radius of base 5 ft. 3 in.?

5. A well is 15 ft. deep, and its diameter is 3 ft. 6 in.; how many cubic yards of earth were taken out in digging it.

6. A cylinder is 20 ft. 8 in. long, and the diameter of its base 11 ft.; find its volume.

7. A circular tunnel shaft is 90 ft. deep and 45 in. in diameter; what was the cost of sinking it at 14s. 6d. per cubic yard?

8. What is the volume of a cylinder 48 ft. long and 6 ft. 8 in. in circumference?

LESSON XVII.—THE PYRAMID.

A pyramid is a solid, the base of which is any plane figure, and its sides are triangles which meet in a point called the vertex of the pyramid.

A more precise definition which you will find prefixed, among others, to the eleventh book of Euclid, is that a pyramid is a solid figure contained by planes that are constituted betwixt one plane and one point above it in which they

meet.

According to the form of its base, a pyramid is either triangular, square, pentagonal, hexagonal, etc.

The axis of a pyramid is the line which is drawn from its vertex to the centre of its base.

A right pyramid is one the axis of which is perpendicular to the base.

The altitude (or perpendicular height) of a pyramid is the perpendicular distance de from the vertex to the plane of the base. In right, but not in oblique (or slanting), pyramids this corresponds to the axis of the pyramid.

The slant height of a pyramid is the distance fg of the vertex from the centre of one of the sides of the base.

B

The volume of a pyramid is equal to one third of the product of the area of its base and its altitude.

In other words, the volume of a pyramid is exactly onethird of the volume of the prism upon the same base. The correctness of this rule can best be demonstrated practically by a model, as illustrated by the accompanying figures.

Any triangular prism, whether right (as 1), or oblique (as 2), may be divided into three equal and similar pyramids. Thus, if the pyramid ABCD be cut away by a

plane passing through the points A, B, and D, there remains a pyramid with a quadrangular base, BDFE, and apex at A; this again can be divided into two triangular pyramids by a plane passing through A, E, and D, and all three pyramids will then be found to be equal and similar. But the area of the prism is the product of the base and the altitude, and therefore the area of the pyramid is one-third of the product.

The same illustration may be made to apply to a pyramid with a polygonal base, which can always be divided into a number of component triangular pyramids, by planes passing through the apex and the corners of the polygonal base.

Note. The surface of a pyramid is equal to the sum of the areas of the triangles, with or without that of the base, as the case may be.

EXERCISES (N).

1. The perpendicular height of a pyramid is 20 yds., and each side of its triangular base 6 ft.; find its solidity.

2. A pyramid with a pentagonal base, each side of which measures 4 ft., is 30 ft. high; find its volume.

3. Find the whole surface of a pyramid whose slant height, measured from the apex to the centre of one of the pentagonal sides of the base, is 10 ft., and each side of the base 20 in.

4. A hexagonal pyramid is 10 ft. high, and each side of the base 15 in., and it is of uniform density and weighs 340 lb. per cubic ft.; find its weight.

5. A square pyramid is 15 ft. 9 in. high, and each side of its base 2 ft. 6 in.; find its volume.

6. A triangular pyramid is 19 ft. high, and the sides of its base are 16 ft., 141 ft. 18 ft.; find its volume.

LESSON XVIII.-THE CONE.

A cone may be described as a circular pyramid, i.e., a pyramid with a circular base. A more precise definition however is

A cone is the solid figure described by the revolution of a right-angled triangle (ABC) about one of the sides (AB) containing the right angle, which side remains fixed (fig. I.)

If the side which remains fixed is equal to the other side con