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188. The circumference of any circle is supposed to be divided into 360 equal parts, called Degrees, each degree into 60 Minutes, and each minute into 60 Seconds.

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189. A Semi-circumference is half a circumference, a Quadrant one fourth, and a Sextant one sixth. A Sign, used only in astronomy, equals 30°.

Fig. 3.

a

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190. An Angle is the opening between two lines which meet each other. The point of meeting is called the Vertex of the angle. The angle in the annexed figure may be read, "the angle a bc," or simply "the angle b." An

angle is measured by that part of the circumference of a circle included between its sides, the centre of the circle being at the vertex of the angle; thus,

g

Fig. 4. h

e

In fig. 4, the angle def is measured by the arc mn; that is, if the arc mn contains 70°, the angle def is an angle of 70°.

191. An angle which inclv.des f 90°, or of a circumference, is a Right Angle, the sides of which are said to be perpendicular to each other; in fig. 4, the angle geh is a right angle. An angle greater than a right angle is an Obtuse angle. An angle less than a right angle is an Acute angle; hed is an acute angle ged is an obtuse angle.

and

NOTE.

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As arcs are measurements of angles, the table for angular measure is the same as the table for circular measure.

192. EXAMPLES.

1. Reduce 148° 54′ 18" to seconds.

2. Reduce 354° 0' 16" to seconds.

Ans. 536058:

3. Reduce 53684" to numbers of higher denominations.

4. Reduce 359° 59′ 59′′ to seconds.
5. Reduce 1 quadrant to seconds.
6. How many seconds in 1 sextant?
7. How many minutes in a sign?

Ans. 14° 54' 44"

8. Reduce 35467" to numbers of higher denominations.

TIME MEASURE.

193. The length of an Astronomical or Sidereal Day is the time the earth takes to turn once upon its axis; the length of a Solar Day is the time the earth takes to turn so as to bring the sun to the same meridian again. The solar day is divided into 24 hours, each hour into 60 minutes, and each min, ute into 60 seconds.

The denominations of time are centuries, years, months, weeks, days, hours, minutes and seconds.

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The time which the earth takes to revolve around the

years.

But

194. sun is 365 d. 5 h. 48 m. 50 s. nearly. The common year (365 days) thus loses nearly one day in 4 years. Hence the leap year of 366 days was established, which occurs once in 4 this adds too much by about 11 m. a year, which in 100 years amounts to nearly 183 h. To balance this error, every 100th year is not regarded as a leap year. But this drops too much by a little more than 5 h., which in 4 centuries amounts to nearly 1 d. Hence every four-hundredth year is a leap year. This leaves an error which is less than 1 d. in 3600 years. Hence the

RULE for ascertaining when any year is a leap year. When the number denoting the year is divisible by 4, and not by 100, it is a leap year; and any year that is divisible by 400 is a leap year.

195. A year is divided into four seasons, of three calendar months each, and commences with January, the second winter month.

The succession of the seasons, quarters and months, and the number of days in each month, are shown by the following diagram :

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"Thirty days hath September,
April, June, and November;
All the rest have thirty-one,

Except February alone,

To which we twenty-eight assign,

Till leap year gives it twenty-nine."

Second

NOTE. In the following examples, common and leap years are understood unless the Julian is specified.

* Leap year, 29 d.

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1. Reduce 8 y. 3 w. 19 d. 7 h. to hours, allowing for 2 leap

years.

NOTE. 2 d. 3 w. 19 d. 42 d., ... =

8 c. y. 42 d. 7 h. to hours.

Ans. 71095 h.

the example may be stated, Reduce

2. Reduce 13 y. 8 w. 2 d. 3 h. 18 m. to minutes, allowing for 3 leap years.

3. Reduce 180739 hours to numbers of higher denominations. Ans. 20 y. 225 d. 19 h. 4. Reduce 5683762 minutes to numbers of higher denomina

tions.

5. How many minutes in the 1st century? Ans. 52594560 m. 6. How many hours in 10 y. 36 d., beginning with Jan. 1st, Ans. 88536 h.

1852?

7. How many seconds in the 3 summer months? 8. How many days from April 12th, 1831, to May 3d, 1832? NOTE. - From April 12, 1831, to April 12, 1832 366 days; to May 3, 21 days more. Ans. 387 days.

9. How many days from Jan. 1st, 1832, to Jan. 1st, 1863? 10. How many days from March 1st, 1850, to Jan. 1st, 1864? 11. How many seconds in 10 years, 36 minutes, allowing 3651 days to the year?

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EXAMPLES.

1. How many rows of buttons, 6 in a row, are there in a great gross of buttons?

2. In 3 score and 6 years how many days?

3. How many sheets of paper in 3 reams, 7 quires, 21 sheets? 4. How high must a doorway be for a horse to pass freely under that is 15 hands high?

5. How many loaves of bread can be made from a barrel of flour, allowing 124 oz. to the loaf?

6. If pork is worth $18.75 a bbl., what is it worth per lb.?

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