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 weigh ing 8oz. 10pwt. 12gr.; the 2d, 9oz. 3pwt. 5gr.; the 3d, 8oz. 9pwt. 7gr. If the loss in refining be 5 pwt. 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 104 A. 2R. 37 P., the other, 87 A. 1R. 38P.: I reserve for myself 40A. 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. 3 fur.

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 15o. (360÷2415°). 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

1. How many hr. min. and sec. of time correspond to 18° 25' 30" of longitude? Ans. 1hr. 13 min. 42 sec.

ANALYSIS. By inspection of the Table, it is evident that,

Degrees (°) of longitude divided by 15, give hours of time: 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 Iligher 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. 4hr. 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? What do degrees, minutes, and seconds of longitude divided by 15 give? 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. 40sec. 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. 18 sec. A. M.

What the time east
Why earlier west?

REVIEW. 109. When is it noon at any place? or west of that place? Why is the time later east? 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 ivided? 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 cdd? Give examples.

DEF. 10. A multiple (dividend), of a number is the product arising from taking it a certain number of times: thus, 6 is a multiple of 2, because it is equal to 2 taken 3 times. Hence,

A multiple of a number can be divided by it without a remainder. Therefore, every multiple is a composite number.

DEF. 11. A factor of a number is a number that will exactly divide it thus, 4 is a factor of 8, 12, 16, &c.

REM. The terms, factor, divisor and measure, all mean the same thing. Every composite number being the product of two or more factors, each factor must exactly divide it, (Art. 37).

Hence, every factor of a number, is a divisor of that number.

DEF. 12. A prime factor of a number is a prime number that will exactly divide it: thus, 3 is a prime factor of 12; while 4 is a factor of 12, but not a prime factor.

Therefore, all the prime factors of a number, are all the prime numbers that will exactly divide it: thus, 1, 3, and 5, are all the prime factors of 15.

Every composite number is equal to the product of all its prime factors: thus, all the prime factors of 10 are 1, 2, and 5; 1X2X5=10.

DEF. 13. An ALIQUOT part of a number, is a number that will exactly divide it: thus, 1, 2, 3, 4, and 6, are aliquot parts of 12.

RESOLVING NUMBERS INTO PRIME FACTORS.

ART. 111. The smaller composite numbers may be resolved into their prime factors by inspection; thus,

6=2X3; 8=2X2X2; 9-3X3; 10=2x5.

In the case of large numbers, their factors are found by trial; that is, by dividing by cach of the prime numi

REVIEW.-110. REM. Are the even numbers prime or composite? Aro the cdd numbers? What is a divisor of a number? Give examples. When is one number divisible by another? Give examples.

110. What is a multiple? Give examples. A factor? Give examples. REM. What terms besides divisor are used in the same sense? Why is every factor a divisor? What is a prime factor? Give an example.