NOTE. A pyramid is triangular, quadrangular, etc., according as its base is a triangle, quadrangle, etc.

A CONE is a solid, like a pyramid, except that its base is a circle. The altitude of the pyramid or cone is its perpendicular hight.

397. PROBLEM 8. To find the contents of a pyramid or of a cone:

RULE. Multiply the area of the base by one third of the altitude.

Ex. 1. What are the contents of a cone, whose base is 10 feet in diameter and whose altitude is 24 feet?

Ans. 628.3184 cu. ft.

2. What are the contents of a pyramid, whose altitude is 12 inches and whose base is a triangle, having its base 6 inches and its altitude 8 inches?

399. PROBLEM 9.

398. The FRUSTUM of a pyramid or cone is the part remaining after a portion next the vertex has been cut off by a plane parallel to the base. The two ends are called the upper and lower bases.

To find the contents of the frustum of a pyramid or cone:

RULE. Multiply the sum of the two bases, added to the mean proportional between the two bases, by one third of the altitude of the frustum.

Ex. 1. What are the contents of the frustum of a quadrangu

396. What is a Cone? Altitude of a pyramid or cone? 397. Rule for fiuding the solid contents? 399. What is the Frustum of a pyramid or cone? 399. Contents of a frustum, how found?

lar pyramid, whose altitude is 21 feet and whose bases are 5 feet and 3 feet square? Ans. 343cu. 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 centre.

A diameter of the sphere is a line passing through the centre, 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. 31415.92sq. 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. 5235983c. 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?

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 333 gives 1000?

7. What number divided by 371⁄2 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 8m. 17d.?

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 2 feet thick?

14. How many bricks whose dimensions are 8', 4', and 2′, will it take to build the walls of a house 40ft. long, 28ft. wide, and 22ft. 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 $25000 per annum; what sum may he expend daily, and yet save $41560 in one term of office, viz. 4 years? Ans. $40.

16. What number, multiplied by of itself, will produce 121? 17. What number, multiplied by 2 of itself, will produce 27? 18. How many square feet of boards will it take to lay a floor 20ft. long and 16ft. wide?

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

1

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 $1.25 ? 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 12gal. 3qt. 1pt. 2gi., 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,,, and to equivalent fractions having

the least common denominator.

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

29. Reduce of a pound to shillings and pence. 30. Add lb. oz. dwt. gr. together.

31. From 3lb take 13.

32. A colonel, arranging his men in a square battallion, 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 80c. 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 32ft. long, 243ft. wide, and 20ft. high, the walls being 1ft. 4' thick, the house having 2 doors, each 4ft. wide and 8ft. high, and 21 windows, each 3ft. wide and 6ft. 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 22 miles per hour?

38. Light moves about 192000 miles per second and sound about 1142 feet per second; what is the ratio of the velocity of light to that of sound? Ans. 887705##1.

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. 49ft.

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 50c. per rod?

50. Sold to J. P. F. goods as follows:

Jan. 18, 1862, on 6m.,

66

75yd. of cloth,

Mar. 12,

June 15, 66

at $4, $300. "331c.,

200.

" $8.

400.

When shall he pay me the balance of the debt?