184. To find the solid contents of the walls of a cylindrical shaped structure; as round cisterns, wells, &c.

RULE. To the inner diameter of the cylinder, add the thickness of the wall; the sum will be the mean of the inside and outside diameters. Multiply this sum by 34, or by 3.1416; the product will be the mean circumference. For the solid content, Multiply this mean circumference by the height, and this product by the thickness.

185. To find the solid contents of the bottom, or foundation-work, of a cylindrical cistern.

RULE Multiply the diameter from outside to outside by 34 for the circumference: then multiply half the diameter by half the circumference, and this product by the thickness, for the cubical content.

NOTE. To find the capacity of a cistern, the top and bottom of which are not equal, see Art. 174.

186. To find the number of bricks it will take to build any wall, or other work, the solid contents of which are known.

A common brick is 8 inches long, 4 inches wide, and 2 inches thick; its solid content is therefore 64 cubic inches, or of a cubic foot. Hence the

RULE. Multiply the number of cubic feet by 27; the product will be the answer.

187. To find how many gallons a rectangular cistern will contain.

RULE. From the inside dimensions find the cubical contents in feet, (170,) and multiply the content thus found by 7. This will be the number nearly. Or, Find the content in cubic inches, and divide by 231.

188. To find how many gallons a cylindrical cistern will contain.

RULE. Multiply the inside diameter by itself, and this product by the height, the dimensions being taken in feet; then multiply the last product by 54.

189. To find the number of quarts a cylindrical vessel will hold.

RULE. Take the dimensions in inches; multiply the square of the diameter by twice the height, and divide the product by 147.

190. To find the number of bushels a rectangular bin will hold.

RULE. Take the dimensions in feet; multiply the length, width, and height together; then multiply this product by 45, and divide by 56. Or, Take the dimensions in inches; multiply the length, width, and height together; multiply this product by 5, and divide by 10752.

191. To find the inside dimensions of any box, cistern, &c., of a given capacity, the dimensions of which are to have a given proportion to each other.

RULE. Divide the capacity of the required box, &c., by the capacity of one whose dimensions are expressed by the numbers of the given proportion; and multiply each of these numbers by the cube root of the quotient; the several products will be the dimensions required.

1. Required the dimensions of a rectangular cistern which shall hold 3000 gallons, and whose length, width, and depth shall be as the numbers 5, 3, and 4.

SOLUTION.5X3 X4 X 7450 gallons, the capacity of a cistern whose dimensions are 5, 3 and 4 feet.

3000 ÷ 450

=

= length; the width; and 1.882 × 47.528

2. Required the inside dimensions of a round cistern to contain 10000 gallons, the diameter of which shall be to the height as 2 to 3.

gallons, the capacity

SOLUTION.-2 × 2 × 3 × 57 = 70 of a cistern whose dimensions are 2 and 3 feet. = 141.844: = 5.215. 5.215 × 2 = 10.43 ft. ter; 5.215 × 3 = 15.645 feet = the height.

10000 ÷ 701

the diame

NOTE. Of rectangular forms, the cube gives the greatest capacity from a given amount of materials, if the top is to be covered; if not, a square base is the best, with an indefinite height.

In cylindrical forms, the greatest capacity from a given amount of materials is obtained by making the diameter and height equal, if the top is to be covered; if not, a given amount of materials will enclose the greatest space when the diameter is one half the height.

192. To find the dimensions of any surface which shall enclose a given area, the dimensions of which are to have a given proportion to each other.

RULE. Divide the area of the required surface by the area of a surface whose dimensions are expressed by the numbers of the given proportion, and multiply each of these numbers by the square root of the quotient; the several products will be the dimensions required.

1X44 sq. rods

1. Required the dimensions of a rectangular field to contain 40 acres, and which shall be 4 times as long as it is wide. area of a surface whose dimensions are 1 and 4 rods. 6400÷4 : = 1600: = 40. 1 X 40=40 rods = the width; 4×40: = 160 rods = the length. 193. EXERCISES IN THE FOREGOING PROBLEMS. (182 to 192.)

1. How many bricks will it take to build the walls of a house 35 feet long, 25 feet wide, and 20 feet high, the perpendicular height of the ridge above the eaves being 10 feet, the walls 1 foot thick, and the gable ends 8 inches thick?

Solution. 120- 4116, the length of the walls. 116 X 20 X 1 =2320 cu. ft. in the walls. 25 X 10 X = 1663 cu. ft. in the gable ends. 2320+ 1663 = 24863 cu. ft., the solid content of the walls, including the gable ends. 2486 × 27=67,140, the number of bricks required.

2. How many bricks will it take to build a rectangular cistern, whose inside dimensions are to be 8 feet long, 5 feet wide, and 7 feet deep, the foundation and walls being 8 inches thick, and the top being left uncovered?

3. How many solid feet of masonry are there in a cylindrical cistern, the inside diameter of which is 5 feet, its depth 8 feet, and the walls 10 inches, and the foundation 8 inches thick, the top being left uncovered?

4. How many solid feet are there in the walls of a cellar, the inside dimensions of which are 38 feet long, 23 feet wide, and 5 feet high, the walls being 15 inches thick?

5. How many bricks are required to build a cylindrical cistern to contain 20,000 gallons, the diameter of which is to be two thirds of the depth, the walls and foundation being 8 inches, and the top 6 inches thick?

6. What must be the inside dimensions of a bin to hold

1000 bushels of wheat, the length, width, and height being as the numbers 4, 3, and 2?

7. How many feet of boards will it take to make the walls of the above bin, the boards being 14 inches thick, allowing nothing for waste; and what will they cost, at $18 per thousand feet, board measure? (180.)

QUESTIONS. What is a solid? a prism? a cube? a triangular prism? a square prism? a parallelopipedon? a pentagonal prism? a cylinder? a pyramid? the vertex of a pyramid? a cone? a segment of a solid a frustum? the axis of a solid? a sphere or globe? a hemisphere? the axis of a sphere? the height of a solid? the slant height of a pyramid? of a cone? a spheroid? a prolate spheroid? an oblate spheroid?

What is the rule for finding the area of the surface of a cube? for finding the solid content of a cube? for finding the surface of a prism, parallelopipedon, or cylinder? for finding the solid contents of a prism or cylinder? the rule for finding the surface of a pyramid or cone? the solid content of a pyramid or cone? the rule for finding the surface of a frustum of a pyramid or cone? to find the solid content of the frustum of a pyramid or cone? to find the surface of a wedge? to find the solid content of a wedge?

What is the rule to find the surface of a sphere, or of any segment or zone of it? to find the solidity of a sphere? to find the side of a cube, the content being given? What are similar solids? To what are the contents of similar solids proportional? How does a plane parallel to the base of a pyramid divide the lines it meets?

What is the unit measure for boards, plank, &c.? Give examples. What is the rule for measuring boards, plank, &c.? What is the rule for finding the side of the largest square stick that can be hewn from a round log whose diameter is given? Draw a circle and inscribe a square within it, and see if you can demonstrate the correctness of this rule. (165, quest. 22.)

What is the rule to find the solid contents of the walls of a rectangular building? to find the content of the gable ends? to find the solid contents of the walls of a cylindrical shaped structure? to find the solid contents of the foundation of a cylindrical cistern? to find the capacity of a cistern, the top and bottom of which are not equal? to find what number of bricks it will take to build any work, the solid contents of which are known? to find how many gallons a rectangular cistern will contain? a cylindrical cistern to find the number of quarts a cylindrical vessel will hold to find the number of bushels a rectangular bin will hold? to find the inside dimensions of any box, cistern, &c., the dimensions of which are to have a given proportion to each other? Of rectangular forms, what gives the greatest capacity from a given amount of materials? What is said of cylindrical forms? What is the rule to find the dimensions of any surface which shall enclose a given area?

SECTION XXI. - THE MECHANIC POWERS.

194. Machines are certain contrivances employed for the purpose of changing the direction of moving powers, of enabling them to produce any required velocity, or to overcome any required force.

All machines, however complicated, are formed by combining a few simpler machines, commonly called the "Mechanic Powers." They are, the Lever, the Wheel and Axle, the Pulley, the Inclined Plane, the Wedge, and the Screw.

In any machine, the force or original prime mover is called the Power. The resistance to be overcome by the power, through the intervention of the machine, is called the WEIGHT.

The great law of equilibrium in the Mechanic Powers, and which applies to all machines, whether simple or complex, is the following: "The power multiplied by the space through which it moves in a vertical direction, is equal to the weight multiplied by the space through which it moves in a vertical direction." Or,

The power is to the weight as the distance through which the weight moves is to the distance through which the power moves. Hence, (Art. 97, or 127,) any three of these terms being given, the fourth may easily be found.

In the practical application of the Mechanic Powers, a certain allowance must be made for friction. In some, this is much greater than in others. No account will be taken of friction in the following exercises, unless it is particularly mentioned.

195. THE LEVER.

The LEVER is a solid, unyielding rod or bar, working upon a fixed point, called its fulcrum or prop. In theory, it is an inflexible and imponderable line, supported upon one point, upon which it can turn. It is of three kinds.

No. 1.

No. 2.

No. 3.

In the first kind, the fulcrum is between the power and the weight, as in No. 1.

In the second, the weight is between the fulcrum and the power, as in No. 2.

In the third, the power is between the weight and the fulcrum, as in No. 3.