2. Reduce 71.m. 3wk. 5d. 19h. 53m. 45sec. to seconds.

Ans. 19252425 sec.

3. Reduce 53bush. 3pk. 7qt. 1pt. to pints.

4. Reduce 4c. yd. 24c. ft. 1695c. in. to inches.

5. Reduce 67t. 17cwt. 2qr. 23 lb. 14oz. 11dr. to drams.
6. Reduce 171b. 103. 73. 29. 19gr. to grains.

7. Reduce 273lb. 11oz. 19dwt. 21gr. to grains.
8. Reduce 87yd. 3qr. 2na. to nails.

9. Reduce 69m. 5fur. 37rd. to rods.

10. Reduce 5 yd. 2ft. 7in. 2b. c. to barleycorns.

11. Reduce 7m. 7fur. 9ch. 3rd. 21li. to links.

12. Reduce 37sq. m. 637a. 3r. 37sq.rd. to square rods. 13. Reduce 14c. 7c. ft. 15cu. ft. 1716 cu. in. to cubic inches. 14. Reduce 15gal. 2qt. 1pt. 3gi. to gills.

15. Reduce 14circ. 7s. 17° 57′ 14′′ to seconds.

16. Reduce 61.m. 2wk. 18h. 47 sec. to seconds.

17. Reduce 4m. 8ch. 2rd. to links.

18. Reduce 14t. 24lb. to ounces.

88. REDUCTION ASCENDING is performed by division; thus,

Ex. 1. To reduce 11415 farthings to pence, we divide the 11415 by 4, because there will be only one-fourth as many pence as farthings. Performing the division we obtain 2853d. and a remainder of 3qr. If we wish to reduce the 2853d. to shillings, we divide by 12, because there will be only one-twelfth as many shillings as pence, and obtain 237s. and a remainder of 9d. Again the 237s. may be reduced to pounds, by dividing by 20, giving 11 and a remainder of 17s. Thus we find that 11415qr. are equal to 11£ 17s. 9d. and 3qr. Like reasoning applies to all similar examples. Hence,

To reduce a number of a lower denomination to numbers of higher denominations,

RULE.-Divide the given number by the number it takes of that denomination to make one of the next higher; divide the quotient by the number it takes of THAT denomination to make

one of the NEXT higher, and so proceed till the number is brought to the denomination required. The last quotient, together with the several remainders (50) will be the answer.

89. Reduction Ascending and Reduction Descending prove each other.

Ex. 2. Reduce 19252425 seconds to numbers of higher denominations. Ans. 71. m. 3wk. 5d. 19h. 53m. 45sec.

OPERATION.

60) 1925 24 2.5

60) 3 2 0 8 7.3... 45sec.

24 8 X 3. 8) 5347...53m.

3) 668...3

7) 222...2...3 + 2 x 819h.

4) 31... 5d.

1. m. 7... 3 wk.

First divide by 60 (58, b) to reduce the seconds to minutes; then divide by 60 to reduce minutes to hours; then by 24, i. e. by 8 and 3 (56 and 57), to reduce hours to days; etc.

3. Reduce 3455 pints, dry measure, to higher denominations. 4. Reduce 229791 cubic inches to feet and yards.

5. In 34758123 drams, how many tons, cwt. etc.?

6. In 103199 grains, Apothecaries' weight, how many lbs. etc.? 7. In 1578237 grains Troy, how many lbs. etc.?

8. In 1406 nails how many yards?

9. Reduce 22317 rods to miles. 10. Reduce 635 barleycorns to yards. 11. Reduce 63996 links to miles.

12. Reduce 3890877 square rods to miles. 13. Reduce 3317748 cubic inches to cords.

14. How many gallons in 503 gills?

15. How many circumferences in 18964634 seconds? 16. How many lunar months in 15789647 seconds?

17. How many miles in 32850 links?

18. How many tons in 448384 ounces?

This subject will receive further attention in the sections on

§ 9.

SIGNS, DEFINITIONS AND GENERAL
PRINCIPLES.

90. The sign of inequality, > or <, signifies that the number at the opening of the sign is greater than that at the vertex; thus, 5 + 3 > 7, i. e. 5 and 3 are greater than 7. Again, 7 - 5 < 4, i. e. 7 minus 5 is less than 4.

91. Parenthesis, (), indicates that all the numbers within it are to be subjected to the same operation; thus, (8 + 4) × 3 241236; also, (156) 35-23.

NOTE.

- Without the parenthesis, the first example would stand thus: 8 + 4 X 3 = 8 +12= 20, i. e. the sign of multiplication would not So in the second example, if the parenthesis be removed, the sign of division will not affect the 15.

affect the 8.

(a) A vinculum, placed over several numbers, performs the same office as the parenthesis, and, in any example where their aid is needed, either may be used; thus,

(85) 3=8+5 × 3 = 24 + 15 = 13 × 3 = 39; also, ×

(5612) 456-12 414-3=44÷4=11.

NOTE. The numbers in parenthesis or under a vinculum may be taken separately, or they may first be united and then the result may be multiplied or divided, as in the above examples.

92. ALL numbers are even or odd.

(a) An even number is divisible by 2 without a remainder; as 2, 4, 12, etc.

(b) An odd number is not divisible by 2 without remainder; as, 1, 3, 7, 15, etc.

93. ALL numbers are concrete or abstract.

(a) A number that is applied to a particular object, is concrete; as, 6 books, 11 men, 25 horses, 4 bushels, etc.

(b) A number not applied to any particular object, is abstract; as, 6, 11, 25, 4, etc.

94. ALL numbers are prime or composite.

(a) A prime number can be divided by no whole number except itself and unity; as, 1, 2, 3, 5, 7, 11, 19, etc.

NOTE 1. divisible by 2. NOTE 2. - Two numbers are mutually prime when no whole number but one will divide each of them; thus, 7 and 11 are mutually prime; so, also, 9 and 16 are mutually prime, although neither 9 nor 16 is absolutely prime.

Two is the only even prime number, for all even numbers are

(b) A composite number (Art. 43) can be divided by other numbers besides itself and unity; as, 4 2 × 2, 6 = 2 × 3, 8 = 2 X 4 = 2 × 2 × 2, 15 = 3 X 5, etc.

NOTE 1. A composite number which is composed of any number of EQUAL factors is called a power, and the equal factors are called the roots of the power; thus, 9, which equals 3 × 3, is the second power or square of 3, and 3 is the second or square root of 9; 27, which equals 3 × 3 × 3, is the third power or cube of 3, and 3 is the third or cube root of 27; etc. NOTE 2. The power of a number is usually indicated by a figure, called an index or exponent, placed at the right and a little above the number; thus, the second power of 4 is written 42, which equals 4 × 4 = 16; the third power of 4 is 43, which equals 4 × 4 × 4 = 64; etc. NOTE 3. - A root is indicated by a fractional exponent or by the radical sign,; thus, the second or square root of 9 is written 9 or 9, either of which expressions is equal to 3, i. e. the square root of 9 is one of the two equal factors of 9; the third or cube root of 64 is 643 or 3/64, which equals 4, one of the three equal factors of 64; etc.

NOTE 4.

sign.

The third and higher roots require a figure over the radical

NOTE 5.- Every number is considered to be both the first power and the first root of itself.

95. ALL composite numbers are perfect or imperfect.

(a) A perfect number is equal to one half the sum of all its integral factors; thus, 28 is a perfect number, for the sum of its factors, 1+2+4+7+14 +28562 X 28.

NOTE 1.- All the perfect numbers known are 1, 6, 28, 496, 8128, 33550336, 8589869056, 137438691328 and 2305843008139952128.

NOTE 2.

All known perfect numbers, except 1, end with 6 or 28.

(b) An imperfect number is not equal to one half the sum of its factors; thus, 8 is an imperfect number, for the sum of its factors, 1+2+4+8=15 < 2. X 8.

96. ALL imperfect numbers are abundant or defective.

(a) An abundant number is one the sum of whose factors is greater than twice the number; thus, 12 is an abundant number, for the sum of its factors, 1+2+3+4+6+ 12 = 28 2 X 12.

(b) A defective number is one the sum of whose factors is less than twice the number; thus, 16 is a defective number, for 1+2+4+8+1631 < 2 X 16.

97. The factors of a number are those numbers whose continued product is the number; thus, 3 and 7 are the factors of 21; 3 and 6, or 3, 3 and 2 are the factors of 18; etc.

(a) The prime factors of a number are those prime numbers whose continued product is the number; thus, the prime factors of 24 are 2, 2, 2 and 3; the prime factors of 36 are 2, 2, 3 and 3; etc.

NOTE. - Since 1, as a factor, is useless, it is not here enumerated.

98. An aliquot part of a quantity is that which is contained in the quantity an exact number of times; thus 12 cents, 20 cents, 25 cents, 333 cents, etc. are aliquot parts of a dollar.

99. An aliquant part of a quantity is not contained in that quantity without remainder; thus, 12 cents, 23 cents, 75 cents, etc. are aliquant parts of a dollar.

100. A measure of any quantity is contained in that quantity a certain number of times without remainder; thus 3 is a measure of 6, and 8 of 24.

101. A multiple of any quantity contains that quantity a certain number of times without remainder; thus, 15 is a multiple of 5, and 21 of 3.