A. Multiply the length by the breadth, and that product by the depth or thickness, and the last product will be the solid contents.

Q. What particular RULE is to be observed in the measurement of all superficies and solids?

A. In measuring superficies, if inches are multiplied by feet, or feet by inches, the product must be divided by 12, to bring it to square feet. In measuring solids, if feet are multiplied by inches, or inches by feet, the product must be divided by 144, to bring it to solid feet.

EXAMPLES.

1. What is the superficial area of a cubic box, whose sides are 4 feet, and how many solid feet does it contain ?

Operation. 4,5x4,5×6=1213 ft. superficial area.

Then 4,5x4,5×4,5=91,125, or 91g solid feet. 2. How many feet of boards will make a box, 6 ft. long, 4 ft. wide, and 3 ft. deep; and how many bushels will it contain, allowing a bushel measure to contain 2150 cubic inches?

5 It will require 108 feet of boards. Ans. It will contain 57 bu. 3 pk. 3+qts.

3. How many feet of boards will make a corn-bin, 24 feet long, 8 feet wide, and 4 feet deep, (without a top cover,) and how many solid feet will it contain; also, how many bushels will it contain, the bushel measure containing exactly 2150,4 inches? Ans. 448 feet of boards; 768 solid feet. 617,14 bush. 4. How many solid feet in a stick of timber 24 feet long, 16 inches one way, and 14 inches the other? Ans. 373 ft.

5. The grand Erie Canal is about 365 miles long, 30 feet wide, and 4 feet deep; allowing the excavation of it to cost, on an average, 12 cents the cubic yard, what was the expense of that portion of the labor, and what weight of water will it require to fill the whole length of the canal, to the depth of 3 feet, 6 inches, allowing a cubic foot of water to weigh 623 lbs. ? Ans. Expense of excavation, $1,070.6663. Weight of water, allowing 2000 lbs. to the ton, is 6,323,625 tons.

6. A man was hired to dig a cellar, 30 feet in length, 24 feet in breadth, and 6 feet deep, at the rate of 64 cents the cubic yard; how many solid feet did the excavation contain, and to what did his labor amount, at that rate? Ans. 4320 solid feet. $10 amount of labor.

7. There is a large pond, measuring two miles square; if et could be drawn from the surface, how many cubic feet

would be drawn off, and how many tons would it allowing 2000 pounds to the ton? Ans. 334,540,800 t. 10,454,400 tons weight.

ARTICLE IX. TO FIND THE NUMBER OF WINE GALLONS IN A ROUND OR CYLINDRICAL CISTERN.

Q. What is the RULE to find the number of wine gallons in a round, or cylindrical cistern.

A. Square the diameter (in inches) and multiply this product by the depth (in inches). Then multiply this product by 34, and from the right hand of the product, cut of four figures; then the figures standing at the left hand of the cut-off will show the number of wine gallons.

Q. When the cistern is smaller at the top, than it is at the bottom, what is the RULE ?

A. Multiply the greater diameter by the less, both in inches, and to this product, add of the square of the difference of the two diameters, then multiply this sum by the depth, and this last product by 34, and cut off 4 figures as before, and you will have the answer.

EXAMPLES.

1. How many wine gallons are contained in a cistern, 6 feet in diameter, and 8 feet deep?

Ft. in.

Operation. 6=72×72×96×34=1692,0576 or 1692 gallons.

8-96

2. If the bottom diameter of a cistern be 4 feet, and the top diameter 3 feet, and 5 feet deep, how many wine gallons will it contain? Ans. Diameters 4 ft.-48; then 48×36-1728 and 48-36-12 difference. Diameters 3 ft.-36; then 12X 12÷3+1728-1776; therefore, depth 5 ft. 60 ;=1776×60× 34-362,3040 or 362,3 gallons.

3. What is the difference in capacity of two cisterns, one measuring 48 inches in diameter at each end, and 70 inches deep, and the other 48 inches at the bottom, 28 inches at the top, and 70 inches deep? Ans. 196,7 gallons.

ARTICLE X. TO MEASURE A SPHERE OR GLOBE.

Q. What is a Sphere or Globe?

A. It is a round, solid body, whose surface is every where equally distant from a point within, called the centre, as a cannon ball, or any other round ball.

Q. How do you find the superficial area of a globe?

A. RULE 1.-Multiply the circumference of the globe by the diameter, and the product will be the area.

RULE 2.-Square the diameter, and multiply this

3,1416, and the product will be the area.

square

RULE 3.-Square the circumference, and multiply this square by the decimal,3183, and the product will be the area.

Q How do you find the solid contents of a globe?

A RULE 1.-Multiply the square of the diameter by the circumference, and divide this product by 6, and the quotient will be the solid contents.

RULE 2.-Multiply the superficial area by the diameter, and divide the product by 6, for the solidity.

RULE 3.-Cube the diameter, and multiply it by the decimal ,5236, or cube the circumference, and multiply it by the decimal,01688, for the solid contents.

EXAMPLES.

1. If a cannon ball be 9 inches in diameter, what is its cir. cumference, its superficial area, and solid contents? Ans. Circumference 28 in.; area 254,57 in.; contents 381,85 in.

2. What will be the solid contents of a ball, whose diameter is 7 inches, and circumference 22 inches? Ans. 179 in. 3. What is the diameter of a globe, whose solid contents are 905 inches?

Operation. 905÷,5236=1728 the cube of the diameter. Therefore, 31728-12 the answer.

4. The diameter of the earth is about 8000 miles, and its cir. cumference about 25000; how many square miles does its surface contain, and how many cubic miles in solidity? Ans. Supeficial area, 200,000,000. Solid contents, 266,666,666,666.

5. If the diameter of the moon be 2180 miles, what is its circumference, its superficial area in square miles, and solid contents in cubic miles; also, how many such globes would be equal in magnitude to the earth? Ans. Its circumference 68512 miles; superficial area 14,936,1142 miles; solid contents 5,426,788,1901 miles; it would require 49+ such globes.

ARTICLE XII. TO MEASURE ANY IRREGULAR BODY.

Q. By what RULE can you measure any irregular body, whose dimensions cannot be taken in the common way?

A. Put the irregular body into any regular vessel, either square or round, and put in water till the body is entirely covered. Measure from the top of the vessel to the water, and note the distance. Then take out the body, and measure again from the top of the vessel to the water; subtract the first measurement from the second, and the difference will ow the fall of the water. Next find the area of the vessel,

proper rule of its figure; multiply this area by the fall er, and you will have the solidity of the body required.

EXAMPLES.

1. An irregular body, being put into a vessel 18 inches long, and 12 inches broad, when taken out, the water sank 6 inches; what were the solid contents of the body?

Ans. 18x12×6=1296 inches. 2. An irregular body, being placed in a round tub, 18 inches in diameter, covered with water, on being taken out, it was found that the water had fallen 7 inches; what were the contents of the body in cubic inches?

Ans. 1908,522 inches.

MISCELLANEOUS QUESTIONS IN MENSURATION.

1. How many feet of boards will it take to enclose 1000 cubic feet of merchandize in boxes, measuring 4 feet in length, 2 feet in breadth, and 1 foot deep? Ans.. 3500 feet. 2. What is the difference in cubic feet, between a solid body of 12 feet cube, and two solid bodies of 6 feet cube? Ans. 1296 feet, difference.

3. How many feet of boards will it take to make 100 boxes, each 4 feet cube; and how many feet to make the same number of boxes, 8 feet long, 4 feet wide, and 2 feet deep; and what will be the difference in price at $1,75 cts. per hundred feet, for the boards?

{

For 100 cubic boxes 4 feet square, 9600 ft. of boards. Ans. For 100 boxes of 8,4 and 2 ft. 11200 ft. of boards. Difference in price of boards, 28 dollars.

4. Says a steady old farmer to a conceited young student, who was boasting of his great knowledge in mathematics and mensuration, can you tell how many solid inches are contained in that thorn bush? (pointing to a very gnarled and scragged bush near by.) No, says the student; to measure that would be impossible. Then, says the farmer, I will convince you that, with all your learning, I know more, at least about some things, than you do. Upon which he proceeded to cut up the bush in such a manner as to immerse it in a large water trough, measuring 8 feet in length, and two feet in width.. The bush being taken out, the water fell 7 inches. Now, says the farmer, I have reduced the bush to a solid of 8 feet long, 2 feet wide, and 77 inches thick; please to tell me the contents in inches yourself. Ans. 18144 in. or 10 solid ft.

5. A farmer borrowed of his neighbor, a solid stack of hay, measuring 20 feet cube. The farmer returned him two stacks, each measuring 10 feet cube, thinking he had paid him all. I wish you to inform me, what proportion of the debt, the farm er has paid? Ans. of the de

6. Suppose two large wheels, one of 14 feet in diameter, and the other 7 feet, were placed on the opposite ends of an axle, at the distance of 28 feet from each other, and put in motion upon a smooth plain; what would be the circumference and diameter of the circles, that each would describe, and what would be the area of each circle; also, how many times would each wheel turn in forming the circle? Ans. The larger circumference 352 ft. The diameter of the circle 112 ft. The smaller circumference 176 ft. And the shorter diameter 56 ft. Area of the larger circle 9856 ft. Area of the smaller circle 2464 ft. Each wheel will turn 8 times.

7. Six men bought a large grindstone 84 inches in diameter; how many inches of its diameter must each one grind off to get his proper share of the stone, if one first grind his share, and then the other, allowing 6 inches diameter for the eye, to be deducted?

RULE.-Square the diameter of the stone, and subtract the square of 6, for the eye, from the whole square. Then divide the remainder by the number of owners, and subtract the quotient from the dividend, and extract the square root of this last remainder, which shows the length of the diameter, after the first man has ground his share, which subtract from the whole diameter, and you will have the number of inches to be ground off, by the first owner. Thus continue the same process for each owner. Ans. A.'s diameter, 7,515. B.'s, 8,075. C.'s, 9,164. D.'s, 10,873. E.'s, 14,168. F.'s, 34,205, including the eye.

8. Allowing the national debt of England to be eight hundred million pounds sterling, and a cubic inch of silver to be worth 2 pounds sterling, and the whole, cast into a cubic block, what would be its cubic sides; also, if cast into a sphere or globe, what would be its diameter; if cast into a square bar, one foot square, what would be its length; and if coined into dollars, allowing 64 dollars to cover a square foot, what area of ground would it all cover? Ans. In a block, its cubed sides 61,4+feet. Its spherical diameter, 76,18 feet, very nearly. Square bar, 231481,481 feet, or 43,84+miles. Will cover an area of 1275,38+acres.

9. If a wheel, 6 feet in diameter, be placed on one end of an axle, and another wheel, 4 feet in diameter, be placed on the same axle at the distance of 12 ft. from the first, what length of axle must extend from the smaller wheel to enter an upright standard, 9 inches from the surface of the ground? Ans. 15 ft. 10. In the frustrum of a pyramid, the base diameter is 18 ft.,

diameter 14 ft., and its perpendicular height 28 ft.; what

height when carried to a complete cone? Ans. 126 ft.