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Rule.-Beginning with the highest, multiply the units of each denomination by the number in the scale required to reduce it to the denomination next lower; add the units, if any, of such lower denomination, and so continue from the given to the required denomination.

351. To Reduce Time from Lower to Higher Denominations. EXAMPLE.—Reduce 112433138 seconds to years.

OPERATION. 60 ) 112433138 sec.

EXPLANATION.-Divide the given

seconds by 60, to reduce to minutes; 60 ) 1873885 min. + 38 sec.

the minutes thus obtained, by 60, to 24 ) 31231 hr. + 25 min.

reduce to hours; the hours by 24, to 30 ) 1301 da. + 7 hr.

reduce to days: the days by 30, to

reduce to months, and the months 12 ) 43 mo. + 11 da.

by 12, to reduce to years. + y mo. 112433138 sec. = 3 yr. 7 mo. 11 da. Y hr. 25 min. 38 sec.

3 yr.

Rule.- Divide the given units by that number in the scale which will reduce them to units of the next higher denomination, and so continue from the given to the required denomination. Any remainder obtained will be of the same denomination as the dividend from which it arises.

41 yr.

ADDITION OF TIME. 352. To Add Time.

Time expressions may be added as simple numbers, if only it be observed that the scale from the lowest to the higest order is 60, 60, 24, 30, and 12. The highest denomination in common use is the year. EXAMPLE.—Add 41 yr. 8 mo. 22 da. 19 hr. 27 min. 14 sec., and 5


6 mo. 11 da. 10 hr. 50 min. 56 sec. OPERATION.

EXPLAN A TION.—Arrange the 8 mo. 22 da 19 hr. 27 min.

14 sec.

numbers so that those of the same yr. 11 da. 10 hr. 50 min.

56 sec.

denomination stand in the same ver

tical line. Then begin with the 4da. 6 hr. 18 min.

10 sec.

lowest denomination, which is seconds, and add: 14 seconds plus 56 seconds equals 70 seconds, equals 1 minute plus 10 seconds; write the 10 underneath the column of seconds, and carry the 1 to the next column; 27 minutes plus 50 minutes equals 77 minutes, and 77 minutes plus 1 minute (to carry) equals 78 minutes, equals 1 hour plus 18 minutes; write and carry as before; 19 hours plus 10 hours equals 29 hours, and 29 hours plus 1 hour (to carry) equals 30 hours, equals 1 day plus 6 hours; 22 days plus 11 days equals 33 days, and 33 days plus 1 day (to carry) equals 34 days, equals 1 month plus 4 days; 8 months plus 6 months equals 14 months, and 14 months plus 1 month (to carry) equals 15 months, equals 1 year plus 3 months; 41 years plus 5 years equals 46 years, and 46 years plus 1 year (to carry) equals 47 years.

6 mo.

47 yr.

3 mo.

Rule.- Add as in abstract numbers, and reduce according to the table of Time.

SUBTRACTION OF TIME. 353. Difference in time is found in two ways:

1st. By counting the actual number of days from the given to the required date. Thus, the number of days between May 13 and September 7 is 117, counting 18 days left in May, 30 for June, 31 for July, 31 for August, and the 7 of September.

2d. By Compound Subtraction. Subtraction in either simple or compound numbers is really the same, except that in the latter a varying scale is employed. That is, it may, and usually does, involve a transformation in either case. This will always be required unless the several minuend terms, or orders are each equal to or greater than the corresponding subtrahend term.

8 yr. 5 yr.

4 mo.

2 yr.

354. To Find the Difference in Time by Compound Subtraction. EXAMPLE.—Subtract 5 yr. 4 mo. 21 da. fron 8 yr. 1 mo. 18 da. OPERATION.

EXPLANATION.-Write the numbers so that those of the 1 mo.

18 da. same denomination stand in the same column. Then begin 21 da.

with the lowest denomination to subtract. Since 21 days can

not be subtracted from 18 days, transform, or borrow one from 27 da.

the next denomination; 1 month = 30 days, and 18 days added = 48 days; 48 days — 21 days 27 days, which write underncath the column of days; the 1 month having been borrowed from the minuend, there are no months remaining from which to subtract the 4 months in the subtrahend, hence, borrow one from the next denomination; 12 months -- 4 months = 8 months, which write underneath the column of months; there now remains 7 years from which to subtract; 7 years — 5 years = 2 years, which write underneath the column of years. This completes the operation, giving a remainder of 2 years, 8 months, and 27 days.

8 mo.

Rule.-Subtract as in abstract numbers, observing the varying scale.


REMARK. - In the following examples, the difference in time should be found by compound subtraction, unless it be otherwise stated. 355. 1. Reduce 27051 seconds to minutes. 2. Reduce 83129 seconds to hours and minutes. 3. Reduce 610251 seconds to higher denominations. 4. How many years, months, days, hours, and minutes, in 749520360 seconds?

5. How many hours from half-past three o'clock P. M. Oct. 13, 1888, to noon on the fourth day of July, 1889?

6. A note entitled to 93 days' time was dated Oct. 13, 1888. Counting actual time, on what day should it be paid? 7. How many days between Nov. 3, 1890, and Mar. 1, 1900?

A mortgage dated July 2, 1888, was paid Sept. 14, 1891. How many days did it run?

9. How long does a note run if dated Sept. 22, 1887, and paid Aug. 31, 1888? 10. How much time will a man gain for labor in 60 years, by rising 45 minutes earlier each day, beginning Jan. 1, 1888.

11. How many more minutes in the eleven years before Jan. 1, 1890, than in the eleven years after that date ?

12. How many seconds of difference in the time of one solar year and 12 lunar months of 29 da. 12 hr. 44 min. and 3 sec. each ?


356. Circular Measure is used in surveying, navigation, astronomy, and geography; for reckoning latitude and longitude, determining location of places and vessels, and in computing differences of time.

357. Every circle, great or small, is divisible into four equal parts; these parts are called quadrants, and are divisible into ninety equal parts, each of which is called a degree; every circle, therefore, may be divided into 360 equal parts, called degrees.

REMARK.-The divisions into twelfths called signs, and into sixths called sextants, are in occasional use.

60 seconds (") = 1 minute ('). 30 degrees = 1 sign (S.)
60 minutes = 1 degree (°). 12 signs or 360° = 1 circle (C.)

descending, 12, 30, 60, 60; or, 360, 60, 60.
ascending, 60, 60, 30, 12; or, 60, 60, 360.

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REMARK.—Minutes of the earth's circumference are called nautical or geographic miles.



358. 1. Reduce 2154' to degrees.
2. Reduce 87406" to degrees, minutes, and seconds.
3. Reduce 330581" to higher denominations.
4. How many seconds in a circle?
5. How many minutes in 2 S. 21° 47'?
6. How many seconds in 1 S. 27° 8' 57"?
7. Reduce 8162 geographic miles to degrees.
8. How many geographic miles in the circumference of the earth?

9. By two different observations the position of a ship was shown to have changed 519 geographic miles. How much was her change in degrees and minutes?

LATITUDE, LONGITUDE, AND TIME, 359. Latitude is distance north or south from the equator. A place is said to be in north latitude if north of the equator; and to be in south latitude if south of the equator.

360. Longitude is distance east or west from any given starting point or meridian. A place is said to be in west longitude if west of the given meridian; and to be in east longitude if east of the given meridian.

361. Since every circle may be divided into 360 equal parts, or degrees, and the sun appears to pass from east to west around the earth, or through 360° of longitude, once in every 24 hours, it will pass through at of 360°, or 15° of longitude, in 1 hour; through 1° of longitude in 1 of 1 hour, or 4 minutes; and through 1' of longtitude ind of 4 minutes, or 4 seconds.



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360° of longitude = 24 hours or 1 day of time, da.
1 hour of time,

1° 56
4 minutes

1' “

4 seconds 66 REMARK.-Standard Time.- Previous to 1883 there were fifty-three different time standards in use by the railroads of the United States, and as these standards were based on the local time of the principal cities which served as the center of operations of the different roads, they were a constant source of annoyance and trouble, both to the railroads and to the traveling public. To obviate this difficulty the principal railroads of the United States and Canada adopted, in 1883, what is known as the “Standard Time System.” This system divides the United States and Canada into four sections or time-belts, each covering 15° of longitude, 71° of which are east and 71° west of the governing or standard meridian, and the time throughout each belt is the same as the astronomical or local time of the governing meridian of that belt. The governing meridians are the 75th, the 90th, the 105th and the 120th west of Greenwich, and as these meridians are just 15° apart, there is a difference in time of exactly one hour between any one of them and the one next on the east, or the one next on the west; the standard meridian next on the east being one hour faster, and the one next on the west one hour slower. The time of the 75th meridian, which is about 4 minutes slower than New York time and about 1 minute faster than Philadelphia time, is called “Eastern Time," and when it is astronomical noon on this meridian it is noon on every railroad clock from Portland, Me., to Buffalo and Pittsburg, and from Quebec to Charleston. The time of the 90th meridian, one hour slower than “ Eastern Time," and 9 minutes slower than Chicago time, is known as “Central Time," and all roads operated in the second belt are run by “Central Time.” The time of the 105th meridian, one hour slower than “ Central Time,” is distinguished as “ Mountain Time.” Time in the fourth belt, which is governed by the 120th meridian, and extends to the Pacific coast, is called “Pacific Time;" it is one hour behind “Mountain Time," two behind “ Central Time," and three behind “ Eastern Time.” The changes from one timestandard to another are made at the termini of roads, or at well-known points of departure, and where they are attended with the least inconvenience and danger. As this system has produced satisfactory results and has been adopted by most of the principal cities for local use, it is probable that the business of the whole country will, before many years, be regulated by standard railroad time.

362. To Find the Difference in Time, when the Difference in Longitude is given.

EXAMPLE.—If the difference in longitude of two places be 9° 15', what must be their difference in time?

OPERATION. EXPLANATION.–Since each minute of distance equals 4 seconds of 9° + 15'

time, 15 minutes of distance will equal 15 times 4 seconds, or.60 seconds,

which equals one minute of time. And since each degree of distance 4

equals 4 minutes of time, 9 degrees will equal 9 times 4 minutes, or 36 37 min. 0 sec. minutes; adding the one minute obtained above, gives 37 minutes as the

required result.

Rule.-Multiply the units of distance by 4, and reduce according to the table of Time.


REMARK.—Examples under this topic will be restricted to variations of solar time.

363. 1. Cincinnati is 84° 24', and San Francisco 122°, west longitude. What is their difference in time?

2. New York is 74° 1', and Halifax 63° 36', west longitude. Find their difference in time.

3. St. Petersburg is 30° 19' east, and St. Louis 90° 15' west longitude When it is noon at St. Petersburg, what is the time at St. Louis.

REMARK.—If one place be east and the other west of the given meridian, to find their difference in longitude, add their respective distances from the meridian taken.

4. The longitude of the City of Mexico is 99° 5', and that of Boston 71° 3', west longitude. Find their difference in time.

5. If on leaving London, 0° 0' of longitude, my watch, keeping correct time, indicates 46 minutes, 15 seconds, after 3 P. M., what time should it indicate on my arrival at Astoria, Oregon, 124° west, where it is then noon?

364. To Find the Difference of Longitude, when the Difference in Time is Given.

EXAMPLE.-The difference in time between two places is 2 hours, 19 minutes, and 48 seconds. What is their difference of longitude? OPERATION.

EXPLANATION.-2 hours, 19 minutes, and 2 hr. 19 min. 48 sec.

139 min. 48 sec.

48 seconds equal 139 minutes and 48 seconds;

since each 4 minutes of time equal 1 degree 4 ) 139 min. 48 sec.

of distance, 139 minutes and 48 seconds equal 34° + (3 min. 48 sec.)

34 degrees, with 3 minutes and 48 seconds, or 3 min. 48 sec. = 228 sec.

228 seconds, remainder; and since each 4 sec

onds of time equal l' of distance, 228 seconds 4 ) 228 sec.

equal 57' of distance. Therefore, if the dif57

ference in time between two points be 2 2 hr. 19 min. 48 sec. = 34° 57.

hours, 19 minutes, and 48 seconds, their dif. ference in longitude will be 34° 57'.

Rule.-Reduce the difference in time to minutes and seconds, and divide by 4; the quotient will be the difference of longitude, in degrees, minutes, and seconds.


365. 1. What is the difference in the longitude of New York and San Francisco, their difference of time being 3 hr. 11 min. 56 sec.

2. The longitude of Sitka is 135° 18' west. What is the longitude of the city of Jerusalem if, when it is 9 o'clock and 5 minutes A. M. at Sitka, it is 27 minutes and 4 seconds after 8 P. M. in Jerusalem ?

3. Find the difference in latitude of Chicago, situated 41° 54' north, and Valparaiso, 33° 4' south.

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