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

441. From these 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.

1. The latitude of Washington is 38° 53' 39'' north, and that of Boston 42° 21' 27" north ; what is the difference of latitude ?

Ans. 3° 27' 48''. 2. The latitude of Philadelphia is 39° 56' 39'' north, and that of Montreal 45° 35' north ; what is the difference of latitude ?

Ans. 5° 38' 21". 3. The latitude of New York is 40° 24' 40" north, and of Cape Horn 55° 58' 30" south; what is the difference of latitude ?

Ans. 96° 23' 10". 4. The long. of Phila. is 75° 9' 5" west, of San Francisco 122° 26' 15" west; what is the difference? Ans. 47° 17'10".

5. The long. of San Francisco is 122° 26' 15'' west, of Pekin 118° east; what is the difference? Ans. 119° 33' 45'.

LONGITUDE AND TIME. 442. The earth revolves upon its axis from west to east once in 24 hours, which causes the sun to appear to revolve around the earth from east to west in the same time. Places east of a certain point have later time, those west of it earlier time, since the sun appears first to those on the east.

443. The circumference of a circle contains 360°, hence the sun appears to travel through 360° in 24 hours, and in 1 hour it travels 2 of 360° 15°; in 1 minute it travels og of 15° = 15'; and in 1 second it travels oo of 15' = 15''. Hence the following table :

TABLE OF LONGITUDE AND TIME.

15° of longitude
15' of
15" of

1 hour of time.
1 minute of time.
1 second of time.

CASE I.

OPERATION.

444. To find the difference of time of two places when their difference of longitude is given.

1. The difference of longitude between two places is 50° 45'; what is the difference of time?

SOLUTION.-Since 15° of longitude correspond to 1 h. of time, and 15' of longitude to 1 min. of time, it's

15)50° 45' of the number of degrees and minutes will equal the number of hours and minutes difference in time. Divid.

3 23 ing by 15 we have 3 h. 23 min. Hence the

Rule.—Divide the difference of longitude expressed in or" by 15; the result will be the difference of time in H. MIN. SEC.

2. The longitude of Philadelphia is 75° 9' 5" west, and that of New Orleans 90° west; what is the difference of time?

Ans. 59 min. 233 sec. 3. The longitude of Boston is 71° 3' 30" west, and that of San Francisco 122° 26' 15'' west; what is the time in Boston when it is 8 o'clock A. M. in San Francisco ?

Ans. 11 h. 25 min. 31 sec. A. M. 4. The longitude of Edinburgh is 3° 11' west, and that of Chicago 87° 44' 30" west; what change would it be necessary to make in our watches in coming from Edinburgh to Chicago ? Ans. Set back 5 h. 38 min. 14 sec.

5. The longitude of Dubuque is 90° 38' 30" west; what change must we make in our watches in coming from Dubuque to Philadelphia ? Ans. Set forward 1 h. 1 min. 57 sec.

6. The long. of Jerusalem is 35° 32' east; what time is it when it is 7 A. M. in Boston? Ans 2 h. 6 min. 22 sec. P. M.

7. St. Petersburg is in 30° 19' east longitude; what is the time there when it is 23 min. past 10 P. M. in Philadelphia ?

Ans. 5 h. 24 min. 52} sec. A. M. the day after.

CASE II. 445. To find the difference of longitude of two places when their difference of time is given.

1. The difference of time between two places is 3 h. 23 min. ; what is the difference of longitude ?

SOLUTION.—Since 1 h. of time corresponds to 15° of OPERATION. longitude, and I min. of time to 15' of longitude, 15 h. min. times the number of hours and minutes difference in time

3 23 will equal the number of degrees and minutes difference 15 in longitude. Multiplying by 15 we have 50° 45'. Hence the following

50° 45' Rule.- Multiply the difference of time expressed in H. MIN. SEC. by 15; the result will be the difference of longitude in O111

2. The difference of time between New York (long. 74° 3' W.) and Buffalo is 18 min. 48 sec.; required the longitude of Buffalo.

Ans. 78° 45' W. 3. The difference of time between Philadelphia (long. 750 9' 5" W.) and St. Louis is 1 h. 241} sec.; what is the longitude of St. Louis ?

Ans. 90° 15' 16'' W. 4. When it is noon at London (long. 9' 17'' W.) it is 7 h. 16 min. 23° sec. A. M. at Boston ; required the longitude of Boston.

Ans. 71° 3' 30'' W. 6. When it is 3 o'clock P. M. at Cambridge, England (long. 5' 21'' E.), it is 10 h. 45 min. 91 sec. A. M. at Cambridge, Mass. ; required the longitude of the latter place.

Ans. 71° 7' 21" W. 6. When it is 7} o'clock P. M. at Chicago (long. 87° 44' 30'' W.), it is 3 h. 43 min. 6 sec. A. M. at Jerusalem ; required the longitude of Jerusalem. Ans. 35° 32' E.

7. In going from Detroit (long. 82° 58' W.) to Baltimore, I found it necessary to set my watch forward 45 min. 32 sec.; what is the longitude of Baltimore?

Ans. 71° 35' W. 8. I left New Haven (long. 72° 55' 24'') at 111 o'clock A M. and when arriving in San Francisco I found it to be 9 P. M. by their time, while it was 12 h. 18 min. 3 sec. A. M. by my watch; required the longitude of San Francisco.

Ans. 122° 26' 15" W. 9. A captain of a vessel takes an observation and finds that by solar time it is 2 h. 25 min. 30 sec. past noon, but by his chronometer, set at Greenwich, it is 32 min. 42 sec. past 11 A. M.: what was his longitude ? Ans. 43° 12' E.

DENOMINATE FRACTIONS. 446. A Denominate Fraction is one in which the unit of the fraction is denominate; as, į lb., .36bu.

447. Denominate Fractions may be expressed either as common fractions or as decimals.

448. The Processes are Reduction, Addition, Subtraction, Multiplication, Division, and Relation.

REDUCTION OF DENOMINATE FRACTIONS.

449. Reduction of Denominate Fractions is the process of changing them from one denomination to another without altering their value.

450. There are two general cases, reduction descending and reduction ascending, which for convenience of treatment, are subdivided into several other cases.

REDUCTION DESCENDING.

OPERATION.

CASE I. 451. To reduce a common denominate fraction to a fraction of a lower denomination.

1. Reduce o of a shilling to farthings.

SOLUTION.—Since there are 12 d. in one shilling, 12 times the number of shillings equals the number

X1Xi= of pence; and since there are 4 farthings in one penny, 4 times the number of pence equals the number of farthings; hence of a shilling equals 4x12x 1 farthings, which, by cancelling and niultiplying, becomes of a farthing.

Rule.-Express the multiplication by the multipliers required, and reduce by cancellation.

Reduce
2. ^ of an oz. to the fraction of a gr.

Ans. Á 3. sy of a bu. to the fraction of a pint.

Ans. 4. Tóm of a gal. to the fraction of a gill.

Ans. 5. 179200U of a ton to the fraction of an ounce. Ans. 36. of a mile to the fraction of an inch.

Ans. [. 7. 55478000 of a com. yr. to seconds.

Ans. š.

Ans. Ž. 8. 7408967 of an A. to the fraction of a sq. in.

6. 500880

OPERATION.

CASE II. 452. To reduce a common denominate fraction to integers of a lower denomination.

1. What is the value of } of a pound Troy? SOLUTION.—There are 12 oz. in one pound, hence 12 times the number of pounds equals 7x12=84=9} oz. the number of ounces; 12 times ] equals $4,

*x20=0=6į pwt. or 9} oz.: there are 20 pwt. in one ounce, x 24=16 gr. therefore 20 times the number of oz. equals the number of pwt.; 20 times equals , or

OPERATION. 6į pwt., etc.

lb. oz. pwt. gr. SOLUTION 2D.— of a pound equals. f of 7

9)7 0 0 lb. and of 7 lb. we find by dividing is 9 oz.

0 6 pwt. 16 gr.

9 6 16 Rule 1.-Reduce the fraction until we reach an integer and a fraction of a lower denomination, set aside the integer and reduce the fraction as before, and thus continue as far as necessary.

Rule II.Regard the numerator as so many units of the given denomination, and divide by the denominator.

What is the value 2. Of şof a £?

Ans. 13 s. 4 d. 3. Of of a rod ?

Ans. 3 yd. 1 ft. 31 in. 4. Of ã of a bushel ?

Ans. 3 pk. 2 qt. 1} pt. 5. Of ã of a mile ?

Ans. 266 rd. 11 ft. 6. Of of a year ? Ans. 10 mo. 2 wk. 4 da. 16 h. 7. Of is of a ton ?

Ans. 2 cwt. 66 lb. 10ş oz. 8. Of g of an acre? Ans. 142 P. 6 sq. yd. 6 sq. ft. 72 s in.

2 sq.

OPERATION.

CASE III. 453. To reduce a denominate decimal to integers of lower denominations.

1. Reduce £.675 to integers of lower denomination.

SOLUTION.-There are 20 s. in £1, therefore 20 times the number of pounds equals the number of

.675 shillings; 20 times .675 equals 13 s. and .5 s.: there

20 are 12 d. in 1 shilling, therefore 12 times the number

13.500 of shillings equals the number of pence; 12 times .5

12 equals 6 d. Therefore, £.675 equals 13 s. 6 d.

6.000 Rule.- Reduce the decimal until we reach an integer and a decimal of a lower denomination, set aside the integer and

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