ART. 349. To find the surface of a sphere.

RULE. – Multiply the diameter by the circumference, and the product will be the surface.

1. What is the convex surface of a globe, whose diameter is 20 inches?

2. If the diameter of the earth is 8000 miles, what is its convex surface?

ART. 350. To find the solidity of a sphere.

RULE.-Multiply the cube of the diameter by .523598, and the product is the solidity.

1. What is the solidity of a sphere, whose diameter is 20 inches?

2. If the diameter of a globe or sphere is 5 feet, how many cubic feet does it contain?

ART. 351. 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 the 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 10 inches in diameter?

2. What is the side of a cube that may be cut from a sphere 30 inches in diameter ?

THE SPHEROID.

ART. 352. A SPHEROID is a solid, generated by the revolution of an ellipse about one of its diam

eters.

If the ellipse revolves about its longer or transverse diameter, the spheroid is prolate or oblong; if about its shorter or conjugate diameter, the spheroid is oblate or flattened.

ART. 353. To find the solidity of a spheroid.

RULE.-1. Multiply the square of the shorter axis by the longer axis, and this product by .523598, if the spheroid is prolate, and the product will be its solidity.

QUESTIONS. - Art. 349. What is the rule for finding the surface of a sphere? Art. 350. What is the rule for finding the solidity of a sphere? Art. 351. What is the rule for finding how large a cube can be cut from a given sphere?-Art. 352. What is a spheroid? What is a prolate spheroid? What an oblate spheroid? - Art. 353. What is the rule for finding the solidity of a spheroid?

2. If it is oblate, multiply the square of the longer axis by the shorter axis, and this product by .523598; the last product will be the solidity. 1. What is the solidity of a prolate spheroid, whose transverse axis is 30 feet, and the conjugate axis 20 feet?

2. What is the solidity of an oblate spheroid, whose axes are 30 and 10 feet?

§ XLVI.

MENSURATION OF LUMBER AND

TIMBER.

ART. 354. All rectangular and square lumber and timber, as planks, joists, beams, &c., are usually surveyed by board measure, the board being considered to be one inch in thickness. Round timber is sometimes measured by the ton, and sometimes by board measure.

ART. 355. To find the contents of a board.

RULE. Multiply the length of the board, taken in feet, by its breadth taken in inches, and divide this product by 12; the quotient is the contents in square feet.

1. What are the contents of a board 18 inches wide and 16 feet long?

2. What are the contents of a board 24 feet long and 30 inches wide?

ART. 356. To find the contents of joists, beams, &c.

RULE.- Multiply the depth, taken in inches, by the thickness, and this product by the length, in feet; divide the last product by 12, and the quotient is the contents in feet.

1. What are the solid contents of a joist 4 inches wide, three inches thick, and 12 feet long?

2. What are the contents of a square stick of timber 25 feet long and ten inches thick?

ART. 357. To find the contents of round timber.

RULE. Multiply the length of the stick, taken in feet, by the square of one fourth the girt, taken in inches; divide this product by 144, and the quotient is the contents in cubic feet.

QUESTIONS. Art. 354. By what measure are planks, joists, &c., usually surveyed? What is the usual thickness of a board? How is round timber measured? - Art. 355. What is the rule for finding the contents of a board? Art. 356. What is the rule for finding the contents of joists, &c.?

NOTE 1.

The girt is usually taken about one third the distance from the larger to the smaller end.

NOTE 2. A ton of timber, estimated by this method, contains 50% cubic feet.

1. How many cubic feet of timber in a stick, whose length is 50 feet and whose girt is 60 inches?

2. What are the contents of a stick whose length is 30 feet and girt 30 inches?

§ XLVII. MISCELLANEOUS QUESTIONS.

1. WHAT is the difference between 7 pence and 10 cents?

2. What number is that, to which if be added, the sum will be 7?

3. What number is that, from which if 34 be taken, the remainder will be 41 ?

4. What number is that, to which if 32 be added, and the sum divided by 5%, the quotient will be 5?

5. From of a mile take of a furlong.

6. From 7 acres take of a rood.

7. John Swift can travel 7 miles in of an hour, but Thomas Slow can travel only 5 miles in of an hour. Both started from Danvers at the same time for Boston, the distance being 12 miles. How much sooner will Swift arrive in Boston than

Slow?

8. If of a ton cost $ 49, what will lcwt. cost?

9. How many bricks, 8 inches long, 4 inches wide, and 2 inches thick, will it take to build a wall 40 feet long, 20 feet high, and 2 feet thick?

10. How many bricks will it take to build the walls of a house, which is 80 feet long, 40 feet wide, and 25 feet high, the wall to be 12 inches thick; the brick being of the same dimensions as in the last question?

11. How many tiles, 8 inches square, will cover a floor 18 feet long, and 12 feet wide?

12. If it cost $ 18.25 to carry 11cwt. 3qr. 19lb. 46 miles,

how much must be paid for carrying 83cwt. 2qr. 11lb. 96 miles?

13. A merchant sold a piece of cloth for $24, and thereby lost 25 per cent.; what would he have gained, had he sold it for $34?

14. Bought a hogshead of molasses, containing 120 gallons, for $30; but 20 gallons having leaked out, for what must I sell the remainder per gallon to gain $10?

15. How many acres are there in a piece of land 117 rods long, and 1123 rods wide?

16. Bought a quantity of goods for $128.25, and, having kept them on hand 6 months, for what must I sell them to gain 6 per cent.?

17. If 27 bushels of potatoes cost $8.75, what must be paid for 36 bushels ?

18. How many bushels of oats, at 50 cents per bushel, must I give Moses Webster for 93 bushels of corn, at $1.25 per

bushel?

19. How many bushels of salt, at $1.30 per bushel, must be given in exchange for 75 bushels of wheat, at $1.25 per bushel?

20. If a sportsman spends of his time in smoking, & in "gunning," 2 hours per day in loafing, and 6 hours in eating, drinking, and sleeping, how much remains for useful purposes?

21. If a lady spend of her time in sleep, in making calls, at her toilet, in reading novels, and 2 hours each day in receiving visits, how large a portion of her time will remain for improving her mind, and for domestic employments?

22. What will a piece of land, 73 rods long, and 5 rods wide, come to at $ 25.75 per acre?

23. If 5% ells English cost $15.16, what will 712 yards

cost?

24. If a staff 4 feet long cast a shadow 5 feet, what is the height of a steeple whose shadow is 150 feet?

25. Borrowed of James Day $150 for six months; afterwards I lent him $ 100; how long shall he keep it to compensate him for the sum he lent me ?

26. A certain town is taxed $6045.50; the valuation of

the town is $293275.00; there are 150 polls in the town, which are taxed $1.20 each. What is the tax on a dollar, and what does A pay, who has 4 polls, and whose property is valued at $3675 ?

27. What is the value of 97 pigs of lead, each weighing 2cwt. 3qr. 11lb., at 3£. 17s. 9d. per cwt.?

28. What is the interest of $17.86, from Feb. 9, 1840, to Oct. 29, 1842, at 74 per cent. ?

29. What is the interest of $97.87, from Jan. 7, 1840, to Sept. 25, 1842, at 9 per cent. ?

30. Required the superficial surface of the largest cube that can be inscribed in a sphere 30 inches in diameter ?

31. $1000. Salem, N. H., Oct. 29, 1836. For value received, I promise to pay Luther Emerson, Jr., or order, on demand, one thousand dollars with interest. Emerson Luther.

Attest, Adams Ayer.

On this note are the following endorsements:

Jan. 1, 1837, was received

$ 125.00,

$316.00,

$ 417.00,

$100.00,

$50.00.

What is due, at compound interest, Oct. 29, 1842 ?

Ans. $53.79.

32. D. Sanborn's garden is 234 rods long, and 134 rods wide, and is surrounded by a good fence 74 feet high. Now if he shall make a walk around his garden within the fence 7 feet wide, how much will remain for cultivation ?

Ans. 1A. 3R. 7p. 85181ft.

33. J. Ladd's garden is 100 feet long and 80 feet wide; he wishes to enclose it with a ditch 4 feet wide; how deep must it be dug, that the soil taken from it may raise the surface one foot?

34. How many yards of paper that is 30 inches wide, will it require to cover the walls of a room that is 15 feet long, 11 feet wide, and 73 feet high?

35. Charles Carleton has agreed to plaster the above room at 10 cents per square yard; what will be his bill?