Rule.-Reduce both dividend and divisor to the lowest denomination mentioned in either, and then divide as in simple numbers.

Proof. The same as in division of simple numbers.

NOTE.-The division may also be made without reducing to the lowest denomination, and this will be shorter when the quotient is integral.

WRITTEN EXERCISES.

2. Divide £48 7 s. 4 d. by £6 11 d.

3. 69 bu. 3 pk. 6 qt. by 6 bu. 3 pk. 6 qt.

Ans. 8. Ans. 10.

4. If a man feeds his horse 1 pk. 6 qt. of oats a day, how long will 3 bu. 2 qt. last him? Ans. 7 days.

5. How many demijohns, each containing 2 gal. 3 qt. 1 pt., can be filled from a tank holding 71 gal. 3 qt. 1 pt. of wine? Ans. 25.

6. A drove of cattle ate 6 T. 15 cwt. 87 lb. of hay in a week; how long will 33 T. 19 cwt. 35 lb. last them?

Ans. 5 weeks.

DIFFERENCE BETWEEN DATES.

CASE I.

313. To find the difference of time between two dates.

1. Washington was born Feb. 22d, 1732, and died Dec. 14th, 1799; what was his age?

the

SOLUTION.-Dates are expressed in the number of year, the month, and the day; hence the date of his birth is 1732 yr. 2 mo. 22 da., and the date of his death is 1799 yr. 12 mo. 14 da.; and the difference of these two dates will equal his age, which we find to be 67 yr. 9 mo. 22 da.

OPERATION.

yr. mo. da, 1799 12 14 1732 2 22

67 9 22

Rule.- Write the number of the year, month, and day of the earlier date under the year, month, and day of the later date, and take the difference of the numbers.

NTE. In this method we reckon 30 days to the month; when greater accuracy is required, we reckon the actual number of days in each month. The exact time between two dates is found by the table, Art. 296.

WRITTEN EXERCISES.

2. What is the difference in time from Dec. 12th, 1850, to Jan. 5th, 1860 ? Ans. 9 yr. 23 da.

8. Milton was born Dec. 9th, 1608, and died Nov. 8th 1675; what was his age? Ans. 66 yr. 10 mo. 29 da. 4. Franklin was born Jan. 6th, 1706, and died April 17th, 1790; required his age. Ans. 84 yr. 3 mo. 11 da. 5. What is the exact time a note has to run, dated Dec 30th, 1862, and payable Jan. 16th, 1864? Ans. 1 yr. 17 da. 6. The Revolution was commenced the 19th of April, 1775, and terminated January 20th, 1783; how long did it continue? Ans. 7 yr. 9 mo. 1 da.

7. The Civil War began April 11, 1861, and closed April 9, 1865; how long did it last? Ans. 3 yr. 11 mo. 28 da. 8. How many years, months, and days, from your birthday to the present date?

SUPPLEMENTARY PROBLEMS,

To be omitted unless otherwise directed.

9. A man was born the 29th of Feb., 1824, and died the 3d of March, 1860; how many birthdays did he see, and what was his exact age? Ans. 10; Age, 36 yr. 4 da. 10. Two men were born on the first day of January, one in the year 1756, the other in 1822; both died 37 years after they were born, ipon the 1st of January; required the difference of their ages?

Ans. 1 day.

11. What is the exact time from 45 min. and 25 sec. past 12 o'clock noon, May 20, 1874, to 24 min. 36 sec. past 4 o'clock P. M., July 4, 1876 ? Ans. 2 yr. 1 mo, 15 da. 3 h. 39 min. 11 sec.

CASE II.

314. To find the day of the week upon which any given day of the month will fall, the day of the week of some other date being given.

NOTE.-A common year ends on the same day it begins, hence each year begins one day later than the preceding year. A year following leap year begins two days later.

1. If the 3d of April, 1876, occur on Monday, on what day of the week will the next 29th of August be?

SOLUTION. By the table we find the difference of time to be 148 days: dividing by 7, the number of days in a week, we have 21 weeks and 1 day; the 29th of August must therefore be one day after Monday or Tuesday.

OPERATION.

148÷7=21,+1

Rule.-Find the number of days between the two dates, reduce this number to weeks; the number of days remaining will be the number of days from the given day of the week to the required day.

NOTE.-In solving the problems after the 2d, the date in a previous problem may be assumed as the starting-point.

WRITTEN EXERCISES.

2. If the year 1776 began on Monday, on what day of the week was the 4th of July following? Ans. Thursday. 3. What day of the week was the surrender of Burgoyne, Oct. 17, 1777? Ans. Friday. 4. On what day of the week was Daniel Webster born, Jan. 18, 1782? Ans. Friday. 5. On what day of the week was the battle of Waterloo fought, June 18, 1815? Ans. Sunday. 6. On what day of the week was President Lincoln assas sinated, April 14, 1865? Ans. Friday. 7. On what day of the week was General Grant inaugurated the first time, March 4, 1869? Ans. Thursday.

SUPPLEMENTARY PROBLEMS.

To be omitted unless otherwise directed.

8. The Centennial opened on Wednesday, May 10, 1876; on what day will the next open on the same date?

Ans. Monday.

9. Suppose a letter should be published purporting to have been written at New Orleans on the day of the battle, and dated Wednesday, Jan. 8, 1815; would it be genuine?

10. Let the pupils now determine, from the above principles, the day of the week upon which they were born.

LATITUDE AND LONGITUDE.

315. The Latitude of a place is its distance from the equator, north or south. It is reckoned in degrees, minutes, and seconds, and cannot exceed 90°, or a quadrant.

316. The Longitude of a place is its distance, east or west, from a given meridian. It is reckoned in degrees, minutes, and seconds, and cannot exceed 180°, or a semicircumference.

NOTE. In adding two longitudes, if their sum exceed 180 degrees, it must be subtracted from 360 degrees for the correct difference of longitude.

1. From what is latitude reckoned? From what is longitude reckoned?

2. What is the greatest latitude a place may have? What is the greatest longitude a place may have?

3. What places have no latitude? What places have no longitude What place has neither latitude nor longitude?

4. What is the use of latitude and longitude? Has every place meridian of longitude?

5. What is the latitude of the equator? the latitude of the poles? the longitude of the poles?

317. From the above principles, to find the difference of latitude or longitude, we have the following rule:

Rule. When the latitudes or longitudes are both of the same name, subtract the less from the greater; when they are of different names, take their sum.

WRITTEN EXERCISES.

1. The latitude of Richmond, Va., is 37° 20' north, and of Savannah 32° 4′ 56′′ north; what is the difference of latitude? Ans. 5° 15' 4".

2. The latitude of Charleston, S. C., is 32° 46′ 33′′ north, and of Quito, 13' 27" south; what is the difference of latitude? Ans. 33°.

3. The longitude of Portland is 70° 13′ 34′′ W., and of Mobile, 88° 1′ 29′′ W.; what is the difference of longitude? Ans. 17° 47' 55". 4. The longitude of New Orleans is 90° W. and of Geneva 6° 9′5′′ E.; what is the difference of longitude?

Ans. 96° 9' 5".

and

5. The longitude of St. Paul, Minn., is 95° 4′ 55′′ W., of Berlin 13° 23′ 45′′ E.; what is the difference of longitude? Ans. 108° 28′ 40′′.

6. The longitude of Paris is 2° 20′ E., and of New York 14° 3′ W.; what is th› difference of longitude?

Ans. 76° 23'.

LONGITUDE AND TIME

318. The earth revolves on its axis from west to east once in 24 hours, and this causes the sun to appear to revolve around the earth from east to west in the same time. Places on the east of a certain point have later time, those on the west earlier time, since the sun appears to those on the east first.

319. The circumference of a circle contains 360°, hence the sun appears to travel through 360° in 24 hours, and in 1 hour it travels of 360°=15°; in 1 minute it travels of 15° 15′, and in 1 second it travels of 15′15′′. Hence the following table:

TABLE OF LONGITUDE AND TIME.

1. In what time does the earth revolve on its axis? What part of a revolution does it make in 12 hours? in 6 hours?

2. How many degrees of the earth's surface pass under the sun's rays in 24 h.? in 12 h.? in 4 h.?

3. How many degrees of longitude make a difference of 1 hour in time? 2 hours? 3 hours? 4 hours?

4. In what direction does the earth turn on its axis? In what di rection does the sun appear to move?

5. Does the sun appear first to places east or west of a given point? 6. When it is noon with us, is it earlier or later east of us? west of us?

7. When it is noon at Boston, what is the time 15° east of Boston? 15° west? 30° east? 30° west?

8. What difference in longitude makes a difference of 1 hour of time? of 1 minute? of 1 second?

9. What is the difference of longitude between two cities, if the difference of time is 1 hour? 1 h. 30 min.? 2 h.? 2 h. 45 min.?

10. If I start at New York and travel until my watch is 1 h. 30 min. too fast, in what direction and how far do I go?

11. If I start at Chicago and travel until my watch is 2h. 15 min Goo slow, how far and in what direction do I travel"