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which is 1,080 times, with 25" remaining. Since 60' make a degree, in 1,080' there are as many degrees as there are times 60' in 1,080', which is 18 times. Hence, 64,825"-18° 25".

Reduce to higher denominations:

11. 46,824".

12. 32,295'.

13. 53,684".

14. 536,058".

15. How many signs in 972,000"?

16. How many quadrants in 8,100'?

17. A boy in going around a circular park on his bicycle traveled 10,368,000". How many times did he ride around the park?

18. If he traveled 6 mi., what was the length of a degree in feet?

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The names of the months and the number of days in

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1. The solar day is the time the earth takes to turn upon its axis so as to bring the sun to the same meridian again.

2. The time in which the earth makes one revolution around the sun is 365 days 5 hrs. 48 min. 50 sec., nearly. This is the

true year.

3. The common year thus loses 5 hrs. 48 min. 50 sec. (1 day, nearly), or one day, nearly, in four years. This defect is made up by adding one day to one year in four, thus making a year of 366 days. Such years are called leap years. But this adds about 44 min. 40 sec. too much. To correct this error, the closing year of each century is not counted as a leap year, excepting once in four centuries.

4. Every year that is exactly divisible by 4 is a leap year, except the centennial years, and every centennial year that is exactly divisible by 400 is a leap year.

5. The years are numbered from the beginning of the Christian era.

6. In most business transactions, 30 days are considered a month.

7. The civil day begins and ends at 12 o'clock midnight.

8. A. M. (ante-meridian) is used to denote the time between midnight and noon; M. (meridian) to denote noon; and P. M. (post-meridian) to denote the time between noon and midnight.

ORAL EXERCISES.

171. 1. How many seconds in 10 min. ? In 30 min. ? 2. How many minutes in 4 hrs.? In 540 sec.? 3. How many hours in 5 days? In 720 min.?

4.

5.

How many days in 13 wks.? In 960 hrs.?

Which of the following years were leap years: 1835, 1856, 1800, 1886, 1860, 1600, 1870, 1872, 1890?

6. What part of a minute are 20 sec.? 30 sec.?

7.

What part of an hour are 15 min.?

12 min.?

8. What part of a day are 9 hrs.? 20 hrs.?

9. How many years in a quarter of a century? In one half of a century?

10. How many days in Feb., 1882? 1896?

11.

How many hours in the month of June?

12. How many hours from 9.00 A. M. to 6.00 P. M.?

13. How long is the night when the sun sets at 7.40 and rises at 4.25?

14. How many days from Jan. 31 to May 30, 1894? From May 10 to Sept. 18? From Aug. 17 to Nov. 12?

15. What century is this? In what century was

1492?

WRITTEN EXERCISES.

172. 16. How many seconds in 365 days 5 hrs. 48 min. 50 sec.?

17. Reduce 5,683,762 min. to higher denominations. 18. How many minutes in the first three months of the year 1885?

19. Change 1,153,800 min. to higher denominations. How many seconds in the 19th century?

20.

21.

How many common years in 220,752,000 sec.?

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174. A stationers' table for quantities of paper:

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175. 1. Name some articles that are sold by the dozen.

By the gross.

2. How many pencils in a gross? In 3 score?

3. How many sheets of paper in a ream? ream? In a ream?

In a

4. What will 72 sheets of paper cost, at 20 cts. a quire?

5. How many dozens in a great gross?

6. How many quires in 5 rms.? In 96 sheets?

7. How many reams in 25 bun.? In 100 qu.?

8. How many are 3 score and 10 yrs.?

9. I bought 3 gross of pens at $1.40 a gross, and sold them at 2 cts. apiece. How many cents did I gain?

WRITTEN EXERCISES.

176. 10. What will 120 doz. of buttons cost, at 50 cts. a gross?

11. How many sheets of paper in 3 bales 3 bun. 1 rm. 16 qu. 20 sheets?

12. How many bales in 33,600 sheets?

13. How many weeks and days are there in 2 score and 7 yrs.?

14. How many rows of buttons, 12 in a row, are there in a great gross?

GENERAL RULE FOR ALL REDUCTIONS.

177. Definitions.

1. Reduction is changing the form of a number without changing its value. The form is changed by changing the unit of measurement. To preserve the value, of course, the number of units must be changed proportionally. When the change is made to a smaller unit, the process is called Reduction Descending; when it is made to a larger unit, the process is Reduction Ascending.

2. A ratio is the part which one number is of another; as, 3 is of 7-that is, is the ratio of 3 to 7.

3. A reciprocal of a number is that number, integral or fractional, by which it must be multiplied to give 1 for a product; thus, 3 and 4, and 9, and, are reciprocals.

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