69 miles, (statute) or 60 miles, (geographical) S 1 degree on the earth, 360 degrees, the circumference of the earth. NOTE. This measure is used for lengths only, without regard to breadth. Distances, &c., are measured by it. Leagues are principally used in measuring distances at sea. Besides the above table, A A 4 inches make 1 hand; used in measuring the height of horses. 1 geometrical pace; used for distances. 1 fathom; used in measuring depths at sea. 1 line, used in measuring the lengths of pendulum 1 inch, rods, for time keepers. 6 points 12 lines 30. How many miles in 3,820 rods? A. 113, or 11 mls. 7 fur. 20 rods. 31. How many furlongs in 78,230 rods? A. 1,955 3, or 1,955 fur. 30 rods. 32. How many degrees in 12,472,098 ft. ? A. 34° 9 mls. 2 fur. 28 rds. 4 yds. 33. How many mls. fur. rods, &c., in 218,300,654 in. ? A. 3,445 mls. 3 fur. 8 rds. 3 yds. 2 in. 34. How many deg. in 27,999,666,333 b. c.? 35. How many hands high is a horse, that is 5 ft. in height? 38. How many ft. in 2,876 b. c. ? B B C E A figure which is 1 inch in length, and 1 inch in breadth, and has all its corners or angles equal, is called a square inch. If each side is a foot long, it is called a square foot, &c. The form of a square is familiar to every one. ABCD is a square. In measuring surfaces, such as the floor of a room, the size of a door, the size or surface of planks or boards, portions of land, &c., we generally express the amount of surface in square inches, square feet, square yards, square rods, square miles, &c. A B An ACRE, used often in measuring land, contains 160 square rods. A square yard is a square figure, one yard on each side. CD, represents a square yard. But a yard in length contains 3 feet; so that a square yard is a square figure, 3 feet long, and 3 feet wide. A figure 3 feet long and 1 foot wide, as FEG H, contains 3 square feet. Therefore, a figure 3 feet long, and 3 feet wide, (that is, a square yard,) contains 3 times as many, that is 9 square feet, H 1. In a figure 7 inches long and 1 inch wide, how many square inches? Ans. 7. 2. In a figure 7 inches long, and 7 inches wide, how many square inches? Ans. 49. 3. Then what is the difference between 7 square inches, and a figure 7 inches square? Ans. 42. 4. In a figure 12 inches long and 1 inch wide, how many square inches? 5. In a figure 12 inches long and 12 inches wide, how many square inches ? 6. Then what is the difference between 12 square inches and 12 inches square? 7. In a figure, 9 inches long, and 1 inch wide, how many square inches? 8. In a figure 9 inches long, and 9 inches wide, how many square inches? 9. Then, what is the difference between 9 square inches and 9 inches square? 10. How many square inches in a figure 3 inches long, and 1 inch wide ? 11. How many square inches in a figure 3 inches long, and 2 inches wide? 12. How many square inches in a figure 5 inches long, and 4 mches wide ? 13. How many square miles in a township of land, that is 7 miles long, and 6 miles wide? 14. How many square feet in a board, that is 12 inches long, and 3 feet wide? 39. How many square feet in a floor, that is 20 feet long, and 18 wide ? Ans. 360. 40. How many square inches in a figure 1 foot long, and 1 foot wide; that is, how many square inches in a square foot? Ans. 144 41. We have told you that 160 square rods make an acre. If a piece of land is 5 rods wide, how long must it be to contain an Ans. 32 acre ? 42. How many square rods in a square mile? Ans. 102,400 NOTE. First find how many rods will make a mile in length, and then multiply this number by itself. 43. If a piece of land is 16 rods wide, how long must it be to con tain a square mile? Ans. 6,400 44. If a piece of land is 25 rods wide, how long must it be to contain 75,075 square rods? SQUARE MEASURE. 144 square inches, (sq.in.) make 9 square feet 301 square yards, or 2721 square feet . 40 square rods 66 66 66 1 square foot, marked sq.ft. perch or pole, 66 sq.yd. 66 sq.rd. 66 3q. p. rood. 4 roods 640 acres, or 102,400 square rods, NOTE. In measuring land, a chain, called Gunter's chain, 4 rods in length, is used. It is divided into 100 links. Of course, 25 links make a rod, and 25 times 25 625 square links make a square rod. In 4 rods, there are 792 inches: Of course, 1 link is 7 inches. 99% As 100 links make a chain, it is easy to reduce any number of links to chains. For all the hundreds will be chains, and the lower orders links. If we point off, then, two figures from the right of any number of links, the remaining figures will be chains. This is like reducing cents to dollars. (§ xv.) A square chain contains 4 times 4-16 square rods. An acre contains 160 square rods. A square chain is, therefore of an acre. Therefore, square chains may be reduced to acres by pointing off one figure on the right. A square chain, likewise contains 100X100=10,000 square links. In the same manner, therefore, square links may be reduced to square chains by pointing off four figures on the right. And as these square chains may be further reduced to acres by pointing off one figure, it follows, that square links may be reduced to acres; by pointing off five figures.. EXAMPLES. 1. In 2,475 links, how many chains? In 13,892 square links, how many square chains ? 2. In 1,849 square chains, how many acres? In 81,943? In 27,854 ? 45. In 39,204 square inches, how many sq. rds.? 46. In 87,120 sq. ft. how many acres? Ans. 1 Ans. 2 47. In 201,283,876 sq. in. how many acres, sq. rds. sq. yds., &c.? Ans. 32 acres, 14 sq. rds. 8 sq. yds. 1 sq. ft. 28 sq. in. Ans. 9 48. In 921,600 sq. rds. how many sq. mls.? A solid or space which is 1 inch long, 1 inch wide, and 1 inch high, is called a solid, or cubic inch. The figure represents a cubé. If each side is a foot long, it is called a solid or cubic foot. In measuring solid things, such as timber, wood, stone, the magnitude of mountains, &c., and also, in measuring spaces which have the three dimen. sions, length, breadth, and height, depth or thickness; as the space in a room or vault; the size of a canal or ditch; or the quantity of room in the hold of a ship, or in a cellar; we express the amount of space in solid or cubic inches, feet, yards, rods, miles, &c. Wood and timber are also measured by CORDS and TONS, OF LOADS. A square yard, we have seen, contains 9 square feet. Now, upon each of these feet of surface, we can place a solid or cubic foot; so that a figure 3 feet long, 3 feet wide, and one foot high, contains 9 cubic feet. If we make the figure two feet high, it will contain twice as many cubic feet, that is, 18 cubic feet. If we make it three feet high, it will contain three times as many cubic feet, that is, 27 cubic feet. But if it be 3 feet high, it will be a yard high, and of course, the figure will be a cubic yard. Therefore, a figure three feet long, three feet wide, and three feet high, that is, a cubic yard, contains 27 cubic feet. 1. In a solid 7 inches long, 1 inch wide, and I inch high, how many solid inches? 2. In a solid 7 inches long, 7 inches wide, and 1 inch high, how inany solid inches? Ans. 49. 3. In a solid 7 inches long, 7 inches wide, and 7 inches high, how many solid inches? Ans. 343. 4. Then what is the difference between 7 cubic inches and 7 inches cube, or cubed. Ans. 336. 5. In a solid 9 inches long, 1 inch wide, and 1 inch high, how many solid inches? 6. In a solid 9 inches long, 9 inches wide, and 1 inch high, how many solid inches ? 7. What is the difference between 9 cubic inches, and 9 inches cube ? 49. How many solid inches in a cube, 31 feet on each side? 50. How many cubic feet in a cellar, 29 feet long, 18 feet wide, and 10 feet deep? 51. How many cubic yards in a ditch 279 feet long, 9 feet wide, and 3 feet deep? 52. A room is 11 feet high, 21 feet long, and 16 feet wide. How many cubic feet of space in it? 53. How many cubic inches in a solid, 1 foot long, 1 foot wide, and 1 foot high? SOLID OR CUBIC MEASURE. 1,728 cubic inches, (cu. in.) make 1 cubic foot, marked cu. ft. 27 cubic feet, 1 cubic yard, § 1 cubic rod, "perch or pole, 66 } 66 166 cubic yards, or 4,4921 cubic feet, 32,768,000 cubic rods, 40 feet of round timber, or 50 feet of hewn timber, 66 cu. yd. cu. rd..or cu.p. 1 cubic mile, cu. ml. 1 ton, or load. 66 { { 1 cord of wood. 128 solid feet, or a solid 8 feet long, 4 feet high, 4 feet wide, NOTE. A cord of wood contains what are called 8 FEET OF WOOD, or 8 CORD FEET. Then, as a cord of wood is 4 feet high, 4 feet wide, and eight feet long, a foot of wood, or a cord foot, is a solid 4 feet high, 4 feet wide, and one foot long.. A foot of wood, therefore, contains sixteen solid, or cubic feet. 54. How many cubic feet in a pile of wood, 23 feet long, 5 feet wide, and 3 feet high? How many feet of wood in the same pile ? How many cords? Ans. 345 cu. ft.-21 or 2 cords, 89 cu. ft. 86 55. In 468,374 cubic inches, how many cubic feet? Ans. 271,99 56. In 86,400 solid inches, how many tons of hewn timber? 57. In 221,184 solid inches how many cords? Ans. 1. TABLE OF PARTICULARS. 12 single things, make 1 dozen, marked 12 dozen or 144 single things, 1 gross Ans. 1. doz. gro. SCO. 12 gro., 144 doz., 1,728 single things, 1 great grò.grt.gro. 20 single things, 24 sheets of paper, 20 quires, 200 pounds 100 pounds, 66 1 quintal of fish. 21 stone, 66 1 pig of lead. 8 pigs of lead, 58. A man bought beef to the amount of $96,00, at 4 cents a pound. How many barrels did he buy? 59. How many grt. gro., gro., doz. and single things, in 689,432 single things? 60. How many reams, quires, and sheets, in 598,476 sheets of paper? Here we have been bringing lower denominations to higher. The pupil has been told that this is REDUCTION ASCENDING. He will perceive that we have proceeded according to the following rule. DIVIDE BY THE NUMBER REQUIRED OF THE DENOMINATION DIVIDED, to MAKE ONE OF THE NEXT HIGHER. DIVIDE THE QUOTIENT BY THE NUMBER REQUIRED TO MAKE ONE OF THE NEXT HIGHER STILL, AND SO ON, TILL YOU ARRIVE AT THE DENOMINATION REQUIRED. THE SEVERAL REMAINDERS BELONG TO THE ANSWER, AND ARE OF THE SAME. DENOMINATIONS AS THE DIVIDENDS FROM WHICH THEY ARE DERIVED. EXAMPLES TO EXERCISE REDUCTION. 61. How many gallons in 586 barrels ? NOTE. 31 gallons make a barrel. The pupil will readily multiply by 31, but he may not be able to do so with. But he should remember, that to multiply 586 by 3, we take 586 3 times; to multiply it by 2, we take it twice, or two times; to multiply it by one, we take it once, or one time; and, therefore, to multiply it by, we must take it a time; that is, we must take half of it, or divide it by 2. So, to multiply by, we must divide by 3, to multiply by, we must divide by 4, &e. As is twice as much as, in order to multiply by 3, we must divide by 3 and multiply by 2; or first multiply by 2, and then divide by 3, since the result will be the same in either case. So, likewise, to multiply by 3, we must divide by 4 and multiply by 3; or first multiply by 3, and then divide by 4, &c. Hence, to multiply a number by a fraction, MULTIPLY THE NUM BER BY THE NUMERATOR OF THE FRACTION, AND DIVIDE THE PRODUCT BY THE DENOMINATOR. The above example, then, is performed as follows ; 586X31=18,166, and 586X293. Then, 18, 166293-18,459, Ans. |