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ARC AND ANGLE MEASURES. 177. A Circle is a plane surface bounded by a curve, all points of which are equally distant from a point within called the center.

The circumference of a circle is its boundary line.

An arc of a circle is any part of the circumference; as A Cor D B.

The diameter of a circle is a straight line drawn through its center, and terminated by the circumference; as A B.

The radius of a circle is half its diameter; as A E or E D. 178. A Degree is to of any circumference.

179. An Angle whose vertex is the center of a circle is measured by the arc between its sides.

Thus, the arc A C measures the angle A EC.

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TABLE.
60 seconds (") are 1 minute, '.
60 minutes 6 i degree, .

360 degrees “ 1 circumference, C. 1. A degree of the earth's equator contains 69.16 miles, or about 694 miles.

2. A sign is an arc of 30°; as D B. A sextant is an arc of 60°; as DC. A quadrant is an arc of 90° ; as A C. A semi-circumference is an arc of 180°; as A C B. A right angle is an angle of 90°; as A E C.

NOTE. — The earth, by turning on its axis once in 24 hours, causes of of 360°, or 150, of longitude to pass under the sun, from east to west, in 1 hour's time.

ORAL EXERCISES. 55. How many degrees in 2 right angles ? In 4 right angles ?

56. How many minutes in 37°? Indo?

57. In 195' how many degrees ? In 360' ?

58. How many signs in a quadrant? Draw an angle of 45°; another of 75o.

59. How many degrees does the minute-hand of a clock move in 3 hours ? The hour-hand ?

60. How many degrees does the minute-hand of a watch move in 3 hours ?

61. How many degrees in half a meridian circle? How many miles in one such degree?

MISCELLANEOUS MEASURES.

180. Counting
12 ones are 1 dozen.
12 dozen “ 1 gross.
12 gross " 1 great gross.
20 ones « 1 score.

181. Paper.
24 sheets are l quire.
20 quires « 1 ream.
2 reams " i bundle.
5 bundles 4 i bale.

ORAL EXERCISES.

62. How many pencils in a gross ? How many score ?
63. What cost 15 doz. pens at $1 a gross ?
64. How many sheets in a ream ? In } ream?
65. What will 72 sheets cost at 15 cts. a quire ?

66. If I buy paper at 20 cents a quire, and sell for a cent a sheet, how much do I gain on a ream?

67. Name some articles that are sold by the gross ? By the dozen ?

68. How much will a great gross of tacks cost at 10 cents a paper ?

69. Bought 3 gross of pens at $ 1.50 a gross, and sold them for 2 cents each; how much did I gain ?

70. How many dozen pint bottles will be needed to hold 5 gal. 3 qt. of currant wine ?

71. If I use 1 ream of paper in 10 weeks, how many sheets do I use per day?

72. If you can count four score in a minute, how many can you count between 8.50 and 9.10, A. M. ?

73. How many gross of boxes at 29 each can be bought for $ 5.76 ?

74. If the sun requires 24 hours to make his apparent journey around the earth, how many degrees does he travel in an hour ?

75. A druggist buys a cask of liquor containing 30 gallons for $30, and sells it at $0.75 a pint. What does he gain ?

QUESTIONS.

161. What is a line ? 162. What are linear measures ? Recite the table of linear measures.

163. What is a surface ? A plane surface ? 164. What is an angle? 165. When are two lines perpendicular to each other ? 166. What is a rectangle ? 167. A square ? 168. For what are surface, or square measures used ? Give the table.

169. What is a solid, or volume ? 170. What is a cube? 171. What are cubic measures? Give the table.

172. For what are liquid measures used ? Give the tables. 173. What articles are measured by the bushel ?

174. For what are troy weights used ? Give the table. 175. For what are avoirdupois weights used ? Give the table. What is the long ton? Which is the heavier, a pound of gold or a pound of sugar ?

176. What is time ? Recite the table. When does the civil day begin and end ? Name the calendar months. When has February 29 days ?

177. What is a circle ? The circumference of a circle ? The diameter ? 178. What is a degree ? 179. How is an angle measured ? Give the table. What is a degree at the equator? What is a sign ? A right angle ?

180. Give the counting table. 181. Give the table for paper.

COMPOUND NUMBERS.

182. 1. How many inches in 3 feet 6 inches ?
2. How many feet and inches in 42 inches ?
3. How many quarts in 6 gallons 3 quarts ?
4. How many gallons and quarts in 27 quarts ?

183. A Denomination is the name of a unit of measure or weight.

184. A Denominate Number is a number composed of units of one or more denominations. Thus, 42 inches, 6 pounds 4 ounces, are denominate numbers.

185. A Simple Number is a number of a single kind or denomination. Thus,

2, $3, 5 months, 7 yards, are simple numbers. 186. A Compound Number is a number composed of two or more denominations of the same general kind. Thus,

3 ft. 6 in., 6 gal. 3 qt., are compound numbers.

REDUCTION.

To change Denominate Numbers to Smaller Denominations.

ORAL EXERCISES. 5. How many inches in 7 feet? In 5 feet 6 inches ? 6. How many gills in 5 gallons ? In 2 gal. 3 qt. ? 7. How many ounces in 3 lb. troy ? In 8 lb. 7 oz. ? 8. How many minutes in 4 h. 20 min. ? In 5 h. 15 min. ?

9. How many quires of paper in 10 reams ? How many sheets in 3 quires 16 sheets ?

10. How many days in 12 weeks 6 days? In 11 weeks 4 days ?

11. How many quarts in 3 pecks ? In 2 bu. 1 pk. ?

187. Reduction Descending is changing denominate numbers to smaller denominations.

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188. Rule for Reduction Descending. Multiply the largest denomination by the number of units it takes of the next smaller denomination to equal one of that larger, and to the product add the given number, if any, of the smaller denomination.

Multiply the sum in like manner, and so proceed until the given number is changed to units of the required denomina

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