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Frustum of a cone.

the product will be the mean area between the two bases; lastly, multiply the mean area by the length of the frustum, and the product

will be the solid content.

8. What is the content of a stick of timber whose length is 40 feet, the diameter of the larger end 24 inches, and the smaller end 12 inches? Ans. 734 ft. nearly. PROBLEM 13.

To find the solidity of a sphere or globe.

RULE.-Multiply the cube of the diameter by .5236. 9. What is the solidity of a sphere or globe, whose axis or diameter is 12 inches?

PROBLEM 14.

Ans. 904.78+ inches.

[graphic]

To find the convex surface of a sphere or

globe.

RULE.-Multiply the diameter by the

circumference.

10. Required the superficial content of a globe whose diameter is 24 inches? Ans. 1809.55 + inches. To find how large a cube may be cut from any given sphere, or be inscribed in it.

RULE.-Square the diameter of the sphere, divide that product by 3, and extract the square root of the quotient for the answer.

1. How large a cube may be inscribed in a sphere 40 inches in diameter? Ans. 23.09 inches.

GUAGING.

The business of cask guaging is commonly performed by two instruments, namely, the guaging or sliding rule, and the guaging or diagonal rod.

Cask of the common form.

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diameter in inches by the length of the cask in inches. And, again, multiply this product by the mean diameter; deduct one-fifth of the sum so found for the roundness of the cask, and reduce the remainder to feet and inches, by the rule for measurement, which is correct if the content be required in solid feet, or, divide by 232 for ale gallons.

Example 1.-If a cask be in length 3 feet 9 inches, its head diameter 2 feet 6 inches, and its bung diameter 2 feet 10 inches, what are its solid contents in ale gallons? Length 3 ft 9 in.

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Multiply

45 length.

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2. Each side of the square base of a vessel is 40 in

ches, and its depth 10 inches. Required its contents in Ans. 56.7 gallons.

ale gallons?

8. The diameter of a cylindrical vessel is 32 inches, and its internal depth 45.5 inches. Required its content in ale gallons? Ans. 129.78 gals.

4. How many bushels of grain will a box contain that is 15 feet long, 5 feet wide, and 7 feet high?

Ans. 421.8 bushels.

ABSTRACT OF MECHANICS.

OF MATTER.

MATTER, is possessed of the following properties, viz: solidity, extension, divisibility, mobility, inertia, attraction and repulsion.

1. SOLIDITY is that property by which two bodies cannot occupy the same place at the same time. It is sometimes called the impenetrability of matter.

2. EXTENSION, like the solidity of matter, is proved by the impossibility of two bodies co-existing in the same place.

3. DIVISIBILITY, is that property by which bodies are capable of being divided into parts removable from each other.

4. MOBILITY expresses the capacity of matter to be moved from one position or part of space to another.

5. INERTIA, designates the passiveness of matter, which, if at rest, will forever remain in that state until compelled by some cause to move; and, on the contrary, if in motion, that motion will not cease, or abate, or change its direction unless the body be resisted.

1. SPACE is either absolute or relative.

2. ABSOLUTE SPACE, is merely extension, immoveable, illimitable and without parts, yet it is usually spoken of as if it had parts. Hence the expression:

RELATIVE SPACE, signifies that part of absolute space occupied by any body, as compared with any part occupied by another body.

ABSTRACT OF MECHANICS.

ATTRACTION.

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4. ATTRACTION denotes the property which bodies have to approach to each other.

There are five kinds of Attraction: 1, the attraction of cohesion; 2, of gravitation; 3, of electricity; 4, of magnetism; 5, of chemical attraction.

Cohesion is exerted only at very small distances, its strength varies in different kinds of matter, and is supposed to be the cause of the relative degrees of hardness of different bodies.

Capilliary Attraction is only a particular modification, or branch of the attraction of cohesion.

GRAVITATION decreases from the surface of the earth upwards, as the square of the distance increases; but from the surface of the earth downwards, it decreases only in a direct ratio to the distance from the centre.

REPULSION.

1. REPULSION is that property in bodies, whereby, if they are placed just beyond the sphere of each other's attraction of cohesion, they mutually fly from each other.

2. OIL refuses to mix with water, from the repulsion between the particles of the two substances; and from the same cause, a needle gently laid upon water will

swim.

MOTION.

Absolute Motion, is the actual motion that bodies have, considered independently of each other, and only with regard to the parts of space.

Relative motion, is the degree and direction of the motion of one body, when compared with that of another. Accelerated motion, is understood, when its velocity continually increases.

Retarded motion, when the velocity continually decreases, and the motion is said to be uniformly retarded. when it decreases equally in equal times.

Uniform motion is estimated by the time employed in

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moving over a certain space, or in other words, by the space moved over in a certain time.

CASE I.

To ascertain the velocity.

Divide the space run over by the time.

CASE II.

To ascertain the distance.

Multiply the velocity by the time.

In accelerated motion the space run over (or distance) is as the square of the time.

1. A body acted upon only by one force, will always move in a straight line.

2. Bodies acted upon by two single impulses, whether equal or unequal, will also describe a right line.

The Momentum of a body is the force with which it moves, and is in proportion to the weight or quantity of matter, multiplied into its velocity.

The action of bodies on each other are always equal, and exert in opposite directions; so that any body acting upon another, loses as much force as it communicates.

CENTRAL FORCES.

The central forces are its centrifugal and centripetal forces.

The centrifugal force is the tendency which bodies that revolve round a centre, have to fly from it in a tangent to the curve they move in, as a stone from a sling. 2. The centripetal force is that which prevents a body from flying off by impelling it towards the centre, as the attraction of gravitation.

CENTRE OF GRAVITY.

1. The centre of gravity is that point in a body about which all its parts exactly balance each other in every direction.

1. A vertical line passing through the centre of gravity of a body, is called the line of direction.

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