NOTE. If the divisor exceeds 12, and is a composite number, we may divide by its factors in succession, as in Art. 47.

10. Divide 69 A. 1R. 24P. by 16. (168×2, or 4 X 4.)

The true remainder is found in the same manner as in Division of Simple Numbers. (Art. 47).

11. 490 bu. 2 pk. 4qt.÷100. 12. 266 lb. 9oz. 10 dr. ÷ 50. 13. 339 lb. 7 oz. 9pwt. 18 gr.

OPERATION. A. R. P. 24

2)69 1

8)34 2 32

Ans. 4 1 14 Ans. 4bu. 3 pk. 5qt. Ans. 5lb. 5 oz. 5 dr.

14. 114 da. 22 hr. 45 min. 35 sec.

54.

Ans. 2da. 3hr. 5 min. 17 sec., and 17 sec. rem.

15. 45 T. 18 cwt. ÷ 17..

T. cwt.

OPERATION. 17)45 18 (2T. 14 cwt. Ans.

REVIEW.-106. Repeat the Rule for Division. REM. In both Simple and Compound Division, by what is each remainder multiplied? Of what denomination or order is each partial quotient figure?

ART. 107. PROMISCUOUS

EXAMPLES.

1. If from 1 lb of Ipecac there be taken, at one time 43 23 13gr., and at another 33 13 20 14gr., how much will be left? Ans. 43 33 20 13gr.

2. A silversmith has 3 pieces of silver, the first weighing 8oz. 10pwt. 12gr.; the 2d, 9oz. 3pwt. 5gr.; the 3d, 8oz. 9pwt. 7gr. If the loss in refining be 5pwt. 12gr., and the rest be made into 15 spoons of equal weight, what will each spoon weigh? Ans. 1oz. 14pwt. 12gr.

3. I have two farms, one 104A. 2R. 37P., the other, 87 A. 1R. 38P.: I reserve for myself 40 A. 1R., and divide the remainder equally among my 3 sons: required

the share of each.

Ans. 50 A. 2R. 25 P.

4. A boy residing 3fur. 25 rd. from school, attends twice a day: how far does he travel in 30 days? Ans. 54 mi. 3fur.

5. B loaned A money on the 27th of June, 1843, and A paid it February 3d, 1845; A then lent B a sum to be kept 5 times as long: how long is B to keep A's money? Ans. 8 yr.

6. A ship in latitude 35° 30′ north, sails 20° 35' south; then 14° 20′ north; then 25° 4′ 30′′ south; then 6° 19′ 20′′ north: what is now her latitude?

Ans. 10° 29′ 50′′ north.

ART. 108. LONGITUDE AND TIME.

Difference of Longitude and Time between different places.

The circumference of the earth, like other circles, is divided into 360°, (equal parts), called degrees of Longitude.

The sun appears to pass entirely round the earth (360°). once in 24 hours, one day; and in 1 hour, over 15°. (360÷24 15°). And,

As 15° equal 900' of a degree, and 1 hour equals 60 minutes of time, therefore, the sun passes in 1 min. of time over 15' of a degree. (900'÷60=15′). And,

As 15' equal 900" of a degree, and 1 min. of time equals 60 sec. of time, therefore, in 1 sec. of time the sun passes over 15" of a degree. (900"÷6015). Hence the

TABLE FOR COMPARING LONGITUDE AND TIME.
1 hour of time.

15° of longitude,
15' of longitude,= 1 min. of time.
15" of longitude, 1 sec.

1. How many hr. min. and sec.

to 18° 25′ 30′′ of longitude?

Ans.

of time.

of time correspond 1 hr. 13 min. 42 sec.

of time:

ANALYSIS. By inspection of the Table, it is evident that, Degrees (°) of longitude divided by 15, give hours Minutes () of longitude divided by 15, give minutes of time: Seconds (") of longitude divided by 15, give seconds of time. Hence, if 18° 25′ 30′′ of lon. be divided by 15, the quotient will be the time in hr. min. and sec. corresponding to that longitude.

To find the time corresponding to any difference of longitude:

Rule. Divide the longitude by 15, according to the Rule for Division of Compound Numbers, and mark the quotient hr. min. sec., instead of

:

Conversely To find the longitude corresponding to any difference of time:

Rule.-Multiply the time by 15, according to the Rule for the Multiplication of Compound Numbers, and mark the product • // instead of hr. min. sec.

For short methods of operation, see "Ray's Higher Arithmetic.”

2. The difference of longitude between two places is 30° what is their diff. of time?

Ans. 2hr.

3. The diff. of lon. between two places is 71° 4' what the diff. of time? Ans. 4 hr. 44 min. 16 sec.

REVIEW.-108. At what rate does the sun appear to move in a day? In an hour? In a minute? In a second? seconds of longitude divided by 15 give?

What do degrees, minutes, and Repeat the Rules.

4. The diff. of lon. between New York and Cincinnati is 10° 35': what the diff. of time? Ans. 42 min. 20 sec.

5. The diff. of time between Cincinnati and Philadelphia is 37 min. 20 sec.: what the diff. of lon.? Ans. 9° 20'.

6. The diff. of time between New York and St. Louis is 1 hr. 4 min. 56 sec.: what the diff. of lon.? Ans. 16° 14'. 7. The diff. of time between London and Washington is 5 hr. 8 min. 4sec.: what the diff. of lon.? Ans. 77° 1'.

DIFFERENCE IN TIME.

ART. 109. It is noon (12 o'clock), at any place when the sun is on the meridian of that place; and,

As the sun appears to travel from the east toward the west, when it is noon at any place, it is after noon east of that place, and before noon west of that place :

Hence, a place has later or earlier time than another, according as it is east or west of it. Therefore,

When the time at one place is given, the time at another, EAST of this, is found by ADDING their difference of time: Or, if WEST, by SUBTRACTING their difference of time.

8. When it is noon at Cincinnati, what is the time at Philadelphia? Ans. 37 min. 20 sec. past noon.

9. When it is 11 o'clock A. M. at New York, what is the time in lon. 30° east of New York? Ans. 1 P. M. 10. When 12 o'clock (noon) at Philadelphia, what is the time at Cincinnati? Ans. 11 hr. 22 min. 40 sec. A. M. 11. When it is 11 o'clock A. M. at New York, what is the time at St. Louis? Ans. 9hr. 55 min. 4sec. A. M.

12. Wheeling, Va., is in lon. 80° 42' west: the mouth of the Columbia river in lon. 124° west: when it is 1 o'clock, P. M., at Wheeling, what is the time at the mouth of Columbia river? Ans. 10 hr. 6 min. 48 sec. A. M.

REVIEW.-109. When is it noon at any place? What the time east or west of that place? Why is the time later east? Why earlier west? Having the time at one place, how find the time at another?

VIII. FACTORING.

ART. 110. DEFINITIONS.-1. An integer is a whole number; as, 1, 2, 3, &c.

DEF. 2. Whole numbers are divided into two classes; prime numbers, and composite numbers.

DEF. 3. A prime number can be exactly divided only by itself and unity, (1).

Thus, 1, 2, 3, 5, 7, 11, &c., are prime.

DEF. 4. A composite number (Art. 33) can be exactly divided by some other number besides itself and unity.

Thus, 4, 6, 8, 9, 10, &c., are composite.

DEF. 5. Two numbers are prime to each other when unity (1), is the only number that will exactly divide both. Thus, 4 and 5 are prime to each other.

REM. Two prime numbers are always necessarily prime to each other. Also, two composite numbers are sometimes prime to each other: thus, 4 and 9 are prime to each other.

DEF. 6. An even number can be divided by 2 without a remainder. Thus, 2, 4, 6, 8, &c., are even.

DEF. 7. An odd number can not be divided by 2 without a remainder. Thus, 1, 3, 5, 7, &c., are odd.

REM.-All even numbers except 2, are composite numbers, while odd numbers are partly prime and partly composite.

DEF. 8. A divisor of a number will exactly divide it; that is, without a remainder: thus, 2 is a divisor of 4; 5 of 10, &c.

A divisor of a number is a measure of that number.

DEF. 9. One number is divisible by another, when the former contains the latter without a remainder. Thus, 6 is divisible by 2.

REVIEW.-110. What is an integer? How are the whole numbers divided? What is a prime number? Give examples. What a composite? Give examples. When are two numbers prime to each other? Give examples. What is an even number? An odd? Give examples.