5. What are the contents of a pyramid, the area of whose base is 540, and altitude 32?

6. A pyramid has a rectangular base, the sides of which are 50 and 24; the altitude of the pyramid is 36: what are its contents?

7. A pyramid with a square base, of which each side is 15 has an altitude of 24: what are its contents ?

Cone.

426. A CONE is a volume generated by the revolution of a right-angled triangle ABC, about the side CB. The point C is the vertex, and the line CB is called the axis, or altitude.

427. To find the volume or contents of a

cone.

Rule.-Multiply the area of the base by the altitude, and divide the product by 3; or, multiply the area of the base by one-third of the altitude (Bk. VIII., Prop. V.).

Examples.

1. Required the contents of a cone, the diameter of whose base is 6, and the altitude 11.

ANALYSIS.-We first square the diameter, and multiply it by .7854, which gives the area of the base. We next multiply by the altitude, and then divide the product by 3.

OPERATION.

62 = 36

11

36 × .7854 = 28.2744

3)311.0184

Ans. 103.6728

2. What are the contents of a cone, the diameter of whose base is 36, and the altitude 27?

3. What are the contents of a cone, the diameter of whose base is 35, and the altitude 27?

4. What are the contents of a cone, whose altitude is 27 t, and the diameter of the base 20 feet?

GAUGING.

428. CASK-GAUGING is the method of finding the number of gallons which a cask contains, by measuring the external dimensions of the cask.

429. Casks are divided into four varieties, according to the curvature of their sides. To which of the varieties any cask belongs, must be judged of by inspection.

1st Variety-least curvature.

2d Variety-least mean curvature.

3d Variety-greatest mean curvature.

4th Variety-greatest curvature.

8000

430. The first thing to be done is to find the mean diameter. To do this,

Rule.-Divide the head diameter by the bung diameter, and find the quotient in the first column of the following table, marked Qu. Then if the bung diameter be multiplied by the number on the same line with it, and in the column answering to the proper variety, the product will be the true mean diameter, or the diameter of a cylinder having the same altitude and the same contents with the cask proposed.

1. Supposing the diameters to be 32 and 24, it is required to find the mean diameter for each variety.

Dividing 24 by 32, we obtain .75; which being found in the column of quotients, opposite thereto stand the numbers,

2. The head diameter of a cask is 26 inches, and the bung diameter 3 feet 2 inches: what is the mean diameter, the cask being of the third variety?

3. The head diameter is 22 inches, the bung diameter 34

inches what is the mean diameter of a cask of the fourth variety?

:

431. Having found the mean diameter, we multiply the square of the mean diameter by the decimal .7854, and the product by the length; this will give the contents in cubic inches. Then, if we divide by 231, we have the contents in wine gallons (see Art. 475); or if we divide by 282, we have the contents in beer gallons (Art. 476).

ANALYSIS.-For wine measure, we multiply the length by the square of the mean diameter, then by the decimal .7854, and divide by 231.

OPERATION.

1 x d2 x 1854 =

231

1 x d x .0034.

If, then, we divide the decimal .7854 by 231, the quotient carried to four places of decimals is .0034; and this decimal, multiplied by the square of the mean diameter and by the length of the cask, will give the contents in wine gallons.

For similar reasons, the content is found in beer gallons by multiplying together the length, the square of the mean diameter, and the decimal .0028.

Hence, for gauging or measuring casks,

OPERATION.

1 x d2 x 7854

282

1 x d x .0028.

=

Rule.-Multiply the length by the square of the mean dr ameter; then multiply by 34 for wine, and by 28 for beer measuré, and point off in the product four decimal places. The product will then express gallons, and the decimals of a gallon.

1. How many wine gallons in a cask, whose bung diameter is 36 inches, head diameter 30 inches, and length 50 inches; the cask being of the first variety?

2. How many wine, and how many beer gallons in a cask whose length is 36 inches, bung diameter 35 inches, and head diameter 30 inches, it being of the first variety?

3. How many wine gallons in a cask of which the head diameter is 24 inches, bung diameter 36 inches, and length 3 feet 6 inches, the cask being of the second variety?

OF THE MECHANICAL POWERS.

432. There are six simple machines, which are called Mechanical powers. They are, the Lever, the Pulley, the Wheel and Axle, the Inclined Plane, the Wedge, and the Screw.

433. To understand the nature of a machine, four things must be considered.

1st. The power or force which acts. This consists in the efforts of men or horses, of weights, springs, steam, &c.:

2d. The resistance which is to be overcome by the power. This generally is a weight to be moved:

3d. The center of motion, called a fulcrum or prop. The prop or fulcrum is the point about which all the parts of the machine move :

4th. The respective velocities of the power and resistance.

434. A machine is said to be in equilibrium when the resistance exactly balances the power; in which case all the parts of the machine are at rest, or in uniform motion, and in the same direction.

Lever.

435. THE LEVER is a bar of wood or metal, which moves around the fulcrum. There are three kinds of levers.

1st. When the fulcrum is between the weight and the

ower:

2d. When the weight is between the power and the fulcrum :