lar pyramid, whose altitude is 21 feet and whose bases are feet and 3 feet square? Ans. 343 cu. ft. 2. What are the contents of the frustum of a cone, whose hight is 12 feet and whose bases are 6 feet and 4 feet in diameter?

400. A SPHERE or GLOBE is a solid bounded by a curved surface, all parts of the surface being equally distant from a point within, called the center.

A diameter of the sphere is a line passing through the center, and limited in both directions by the surface.

401. PROBLEM 10. To find the surface of a sphere: RULE. Multiply the circumference by the diameter.

Ex. 1. What is the surface of a sphere, whose diameter is 100 inches? Ans. 31415.92 sq. in. 2. What is the surface of the earth, supposing it to be a sphere 8000 miles in diameter ?

3. What is the surface of the sun, supposing it a sphere whose diameter is 885680 miles?

402. PROBLEM 11. To find the contents of a sphere:

RULE 1. Multiply the surface of the sphere by one third of the radius.

RULE 2. Multiply the cube of the diameter by the decimal .523599; i. e. by of 3.141592.

Ex. 1. What are the contents of a sphere, whose diameter is 100 inches? Ans. 523598 cu. in. 2. What is the volume or solidity of the earth, supposing it a sphere whose diameter is 8000 miles?

3. What is the volume or solidity of the sun, supposing it a sphere whose diameter is 885680 miles?

400. What is a sphere? Its diameter? 401. Rule for finding the surface of a sphere? 402. Rule for finding the volume or solid contents of a sphere? Second rule?

MISCELLANEOUS EXAMPLES.

1. What number increased by of itself gives 20?

2. What number diminished by 43 gives 21?

3. The sum of two numbers is 54 and one of the numbers is

3 times the other; what are the numbers?

4. Three roods and ten rods are what part of an acre?

5. The difference between two numbers is 371⁄2 and the smaller number is 12; what is the larger?

6. What number multiplied by 33 gives 1000?

7. What number divided by 37 gives 64?

8. What is the greatest common divisor of 84 and 144? 9. What is the least common multiple of 72 and 364 ? 10. What is the interest of $756.64 for 8 m. 17 d.?

11. The difference between two numbers is 25, and the smaller number is 10; what is the larger? What the sum of the two numbers?

12. The difference of two numbers is 563492, and the larger number is 3642538; what is the smaller? What the sum of the two numbers? 1st Ans. 3079046.

13. How many bricks 8 inches long, 4 inches wide, and 2 inches thick, will be required to build a wall 20 feet long, 16 feet high, and 23 feet thick?

will

14. How many bricks whose dimensions are 8', 4', and 2', it take to build the walls of a house 40 ft. long, 28 ft. wide, and 22 ft. high, the walls to be 1ft. 6' thick, and no allowance made for doors and windows?

15. The salary of the President of the United States is $50000 per annum; what sum may he expend daily, and yet save $141560 in one term of office, viz. 4 years s? Ans. $40. 16. What number, multiplied by of itself, will produce 12? 17. What number, multiplied by of itself, will produce 27? 18. How many square feet of boards will it take to lay a floor 20 ft. long and 16 ft. wide?

19. How large a square floor can be laid with 676 square feet of boards?

20. The fore wheel of a carriage is 9 feet, and the hind wheel 10 feet in circumference; how many times will each turn round in running from Boston to Andover, 20 miles?

21. A rectangular piece of land, containing 60 acres, has its length to its breadth as 3 to 2, what are its length and breadth?

22. Bought a cask of molasses, containing 84 gallons, for $28; but 9 gallons having leaked out, at what price per gallon must I sell the remainder to gain $4.25 ? Ans. 43 cents.

23. If a pipe 6 inches in diameter will discharge a certain quantity of water in 4 hours, in what time will a 4-inch pipe discharge the same quantity? Ans. 9 hours.

24. In 12 gal. 3 qt. 1 pt. 2 gi., how many gills?
25. In 1846542 seconds how many weeks, days, etc.?
26. Resolve 25740 into its prime factors.

Ans. 2, 2, 3, 3, 5, 11, 13.

27. Reduce, 16, 1, and 27 to equivalent fractions having the least common denominator.

28. Reduce 3 s. 4 d. 2 qr. to the fraction of a pound.

29. Reduce of a pound to shillings and pence.

30. Add lb. oz. dwt. gr. together.

32. A colonel, arranging his men in a square battalion, found that he had 31 men remaining; but, increasing the rank and file by 1 soldier, he wanted 20 men to make up the square. Of how many men did his regiment consist? Ans. 656.

33. How shall I mark gloves that cost me 80 c. per pair so that I may discount 333 per cent. from the marked price and yet gain 25 per cent. on the cost? Ans. $1.50.

34. Suppose that in a shower the water falls to the depth of 2 inches, how many gallons will fall upon a township that is 6 miles square, each gallon containing 231 cubic inches?

35. How many bricks 8' long, 4′ wide, and 2' thick, will be required to build a house 32 ft. long, 24 ft. wide, and 20 ft. high, the walls being 1 ft. 4′ thick, the house having 2 doors, each 4 ft. * wide and 8 ft. high, and 21 windows, each 3 ft. wide and 6 ft. high, no allowance being made for the space occupied by the mortar?

36. What is the square root of the square root of 16 times 81?

37. If a horse travels 64 miles per hour, how many hours will it take him to travel as far as a rail car will run in 6 hours, the car running 223 miles per hour?

38. Light moves about 192500 miles per second and sound about 1090 feet per second; what is the ratio of the velocity of light to that of sound? Ans. 932477180.

39. What is the square root of 4 times the square of 8?

40. What is the cube of the square root of 25? 41. What is the cube root of the square of 8?

42. What is the square of the cube root of 8?

43. Two ships sail from the same port, one due north and the other due west, one at the rate of 6 miles and the other 8 miles per hour. Suppose the surface of the ocean to be plane, how far apart are the ships in 10 hours?

44. An army consists of 59049 men; how many shall be placed in rank and file to form them into a square?

45. What is the diameter of a circular pond which shall contain 36 times as much area as one 20 rods in diameter ?

46. What is the mean proportional between 16 and 64? 47. What is the third proportional to 3 and 30?

48. A ladder 41 feet long, will reach a window 40 feet high on one side of a street, and, without moving the foot, it will reach a window 9 feet high on the other side; how wide is the street? Ans. 49 ft.

49. What is the difference in the expense of fencing a circular 40-acre lot and one of the same area in a square form, the fence costing 50 c. per rod?

50. Sold to J. P. F. goods as follows: Jan. 18, 1862, on 6 m., 75 yd. of cloth,

at $4,

$300.

When shall he pay me the balance of the debt?

51. What is the side of a square equivalent in area to a rectangular field, which is 81 rods long and 49 rods wide?

52. Sent an invoice of goods to my agent in Liverpool which he sold for $25000; what sum can he invest for me, his commission for selling being 2 per cent. and for investing 1 per cent.? 53. A house worth $8000 is insured for its value; what is the premium at 3 of 1 per cent. ?

54. What is the amount of $325, at 6 per cent., compound interest, for 3 yr. 8 m. 12 d.?

55.

$1200.

Boston, May 12, 1870. For value received of A. B. I promise to pay him, or his order, one thousand two hundred dollars, on demand, with interest. CHARLES DANE. INDORSEMENTS: Aug. 18, 1870, $300; Dec. 18, 1870, $10; May 6, 1871, $16.50; June 24, 1871, $400; Dec. 24, 1871, $100; what was due Apr. 12, 1872?

56. A bushel measure is 18 inches in diameter and 8 inches deep; what are the dimensions of a similar measure that holds half a peck? Ans. 9 in. diameter; 4 in. deep.

57. Sold a lot of goods for $100 and thereby gained 25 per cent.; what per cent. should I have gained, had I sold them for $120?

58. A garden whose breadth is 5 rods, and whose length is 13 times its breadth, has a wall 3 feet thick and 4 feet high, around it, outside of the line; what was the cost of this wall at 31 c. per cubic foot?

59. What will be the cost of digging a ditch around the abovementioned garden, within and adjacent to the wall, 31⁄2 feet wide and 23 feet deep, at § of a cent per cubic foot?

60. What would be the cost of walling the above-mentioned garden, the central line of the wall to be on the bounding line, the wall to be 3 feet thick and 33 feet high and to cost 6 c. per cubic foot?

61. A hare has 45 rods the start of a hound, but the hound runs 12 rods while the hare runs 9; how many rods will the hare run before the hound overtakes him?

62. A hare has 32 rods the start of a hound, but the hare