25. What is the content of a mast 57 ft. high, and the girths at its ends 63 and 38 inches?

26. What are the contents of a

square stick of timber 25 feet long, the sides of the ends being 18 and 13 inches?

27. What is the weight of a square

stone pillar, 12 feet high, each side of whose base is 4 feet, and of the top 3 feet, allowing 12 cubic feet to weigh a ton, 2000 pounds?

175. TO FIND THE SURFACE AND SOLID CONTENT OF A WEDGE.

RULE FOR THE SURFACE. Find the areas of the rectangle, the two parallelograms or trapezoids, and the two triangles of which its surface consists, and add them together.

RULE FOR THE SOLID CONTENT. To twice the length of the base add the length of the edge, and multiply the sum by the breadth of the base, and by one sixth of the perpendicular from the edge upon the base; the product will be the content.

28. Required the superficial and the solid contents of a wedge of which the sides of the base or "head" are 36 and 9 inches, the edge 44 inches, and the perpendicular height 22 inches.

9

-

=

√(4})2 + (22)2 = 22.456, slant height of the sides. 36 X the area of the rectangle; 22 × 9= the two triangles, and (3644) × 22.456 the two trapezoids. Hence, 324+ 1981796.482318.48 sq. in. in the whole surface. (3 ft.+3 ft. +3 ft. 8 in.) × 9 in. × 22 × in. 7", solid content.

= = 2 c. ft. 2

29. How many solid feet in a wedge, of which the height is 25 inches, the edge 28 inches, and the sides of the base 34 and 10 inches?

176. TO FIND THE SURFACE OF A SPHERE, OR OF ANY SEGMENT

OR ZONE OF IT.

RULE. Multiply the circumference of the sphere by the axis, or by the part of it corresponding to the zone or segment required; the product will be the surface.

20

30. What is the surface of a globe whose axis is 15 feet?

31. If the diameter of the earth is 8000 miles, how many square miles of surface does it contain?

II. 32. The part of the earth's axis corresponding to each of the frigid zones is 327.19 miles, and to each temperate zone 2053.467 miles, and to the torrid zone 3150.677 miles. What is the surface of each zone the earth's diameter being 7912 miles.

177. TO FIND THE SOLID CONTENT OF A SPHERE.

RULE. Multiply the cube of the axis by .5236.

33. What is the solidity of a sphere, of which the diameter is 16 inches?

34. What is the solidity of the moon, supposing her to be a perfect sphere, the axis being 2180 miles?

178. TO FIND THE SIDE OF A CUBE, EQUAL IN CONTENT TO ANY

GIVEN SOLID.

RULE. The cube root of the cubical content given is the side of a cube of equal solidity.

35. The diameter of a globe is 3 feet; what is the side of a cube of equal solidity?

36. A chest is 4 ft. 7 in. long, 2 ft. 3 in. broad, and 1 ft. 9 in. deep. Required the side of a cube of equal solidity.

37. What must be the side of a cubical cistern that shall hold 5000 gallons of water? (59.)

38. A cylindrical vessel is 15 inches in diameter, and 18 inches deep; what would be the side of a cubical vessel of equal capacity?

39. The side of a cubical vessel is 12; what must be the side of a cubical vessel that will contain 3 times as much? 40. The side of a cubical vessel is 24; required the side of another vessel that will contain only as much as the first.

41. What must be the side of a cubical bin which will contain 1000 bushels of corn? (58.)

179. SIMILAR SOLIDS.

42. There is a cubical box, each of whose inside dimensions is 1 inch. What is the content of the cube? What is the content of the largest globe that can be included within the box? Ans., .5236 of a cubic inch.

43. Suppose each side of the box to be 2 inches; what would be its capacity? How many times as large as the other? How times as large would be the globe that it would include?

44. Suppose each side of the box to be 3 inches. Ask and answer the same questions.

45. What is the content of a block of marble, if its length, width, and thickness are 5, 4, and 3 feet respectively? What, if 10, 8, and 6 feet? What, if 15, 12, and 9 feet? What, if 21, 2, and 11 feet?

NOTE 1. Similar solids are those whose corresponding sides or dimensions are proportional. Thus the different solids mentioned in example 45 are similar.

NOTE 2. The contents of similar solids are in proportion to the cubes of their corresponding dimensions; that is, The content of one solid is to the content of another similar solid, as the cube of any dimension of the former is to the cube of the like dimension of the latter.

46. Can you illustrate this truth by the above and similar examples?

47. Can you show from the above examples why the solidity of a sphere is obtained by multiplying the cube of the diameter by .5236?

NOTE 3. If a plane pass through a cone or pyramid parallel to its base, it divides the lines it meets proportionally; the small cone or pyramid cut off by it is, therefore, similar to the whole cone or pyramid.

48. If a ball 20 inches in diameter weighs 555 pounds, what is the diameter of one of the same metal that weighs 15 pounds?

49. If a vessel, one of whose sides is 2 feet, will contain 37.63 gallons, what will another similar vessel hold, whose corresponding side is 15 feet?

50. If a tree whose diameter is 2 feet at the base contains 3 cords of wood, how much wood will there be in a tree of the same shape, the diameter of which is 3 feet?

51. If an ox whose girt is 7 feet weighs 1000 pounds, how much will an ox of the same form weigh, whose girt is 6 ft. 6 in.? What should be the girt of an ox of the same form, which weighs 1500 pounds?

52. A square pyramid of wood, 12 feet long, and each side of whose base is 18 inches, is to be balanced upon a pivot passing through it. How far from the base must the pivot be placed to balance it?✨

53. A cone is 15 feet high; how many feet of its top must be taken off to remove one half of it? one third of it? two thirds?

54. What part of the cone will a plane 5 feet from the base cut off?

55. If a stack of hay 12 feet high weighs 4 tons, how much will a similar stack weigh, that is 15 feet high?

56. If a common brick 8 inches long weighs 4 pounds, how much will a brick of similar shape weigh, that is 12 inches long?

57. If a man 6 feet high weighs 200 pounds, what will a giant of similar form and of equal solidity weigh, that is 81 feet high?

58. A cubic foot of lead weighs 11352 ounces; how much will a leaden ball 2 inches in diameter weigh? 4 inches? 6 inches? 8 inches? 12 inches?

59. How large a ball of lead will weigh 709 ounces?

709 11352

NOTE. The ball will contain = of a cubic foot. ✔.5236 answer. Or, .5236: 13: the cube of the answer; that is, by Note 2, the solid content of a ball 1 foot in diameter, is to the solid content of the required ball, as the cube of the diameter of the first ball, is to the cube of the diameter of the second.

180. MENSURATION OF BOARDS AND TIMBER.

The unit of measure for boards, plank, joists, beams, &c., is the square foot; they are usually surveyed by board measure, the board being estimated at one inch thick. Thus, a board

10 ft. long, and 1 feet wide, contains 15 square feet, if it is 1 inch thick; if it is 1 inch thick, it contains 22 square feet; if 2 inches thick, 30 square feet. Round timber is sometimes measured by the ton, and sometimes by board measure.

RULE FOR MEASURING BOARDS, PLANK, JOIST, BEAMS, &c. Multiply the length in feet by the width in inches, and this product by the depth, or thickness, in inches, and divide the last product by 12; the quotient will be the number of square feet.

1. How many square feet in a board 23 feet long, 17 inches wide, and 1 inch thick? in. thick? in. thick? 14 in. thick?

2. How many square feet in a joist 30 feet long, 6 in. wide, and 3 in. thick? 51⁄2 in. wide, and 21⁄2 in. thick?

To find the solid contents of any rectangular stick of timber that does not taper, see Art. 170; if it does taper, see Art. 174.

181. To find the side of the largest square stick of timber that can be hewn or sawn from a round log, whose diameter is given.

RULE. Multiply the diameter of the smaller end by .7071. (Art. 165, quest. 22.)

SECTION XX.- PROBLEMS IN MENSURATION.

From the principles and illustrations detailed in the two preceding sections, we deduce the following useful and practical problems.

182. To find the solid contents of the walls of a rectangular cistern or building, of any given dimensions.

RULE. From the OUTSIDE perimeter of the walls, (p. 214, def. 32,) subtract four times the thickness of the walls; the remainder will be the length of the walls. Then multiply this length by the height, and this product by the thickness.

183. To find the content of the gable ends.

RULE. Multiply the breadth of the house by the perpendicular height of the ridge above the eaves; the product will be the area of both gable ends. For the solid contents, Multiply this arca by the thickness.