An Elementary and Practical Arithmetic

of the Dividend (51)-Sign of Division (52)-Quotient when the dividend and
divisor are similar concrete numbers (53)-Quotient when the dividend and
divisor are dissimilar concrete numbers (54)-Remainder in division (55)-The
Remainder used to complete the Quotient (56)-Constant Quotient (57).

RULE VII. To Divide by a number not exceeding 12; or by such numbers with Os

annexed (58).

The figures of the dividend first taken in dividing-The first quotient

figure-How its proper local value is assigned to the quotient figure-The excess

belonging to any place in the dividend-Principle on which Os may be omitted

in the right of the divisor if the same number of figures be omitted in the right

of the dividend (57).

How the operation may be verified (59).

RULE VIII. To divide by any number exceeding 12, and containing two or more

SIGNIFICANT FIGURES (60).

How this Rule differs from Rule VII.

How the operation may be proved by Addition (61)—Sign of Aggregation (62)

-Two numbers found from their Sum and Difference (63).

CHAPTER III-PAGE 47.

COMPOSITE NUMBERS.-PRIME FACTORS.-COMMON MEASURE. COMMON

MULTIPLE.

COMPOSITE NUMBERS (64).

Decomposition of Numbers (65)-How a number is resolved in Division (66)

-How any number whatever may be resolved-Sign of Equality (67)--Con-

stant Product of several Factors (68)-Division by the Canceling of Factors

(69)-If the Product be divided by itself-How the cancellation of a number is

denoted.

RULE IX. To multiply by means of FACTORS (70).

RULE X. To divide by means of FACTORS (71).

A remainder after the first division-after the second division-A re-
mainder after the first division X the first divisor, what it produces-A remain-
der of the first quotient X the first divisor, what it produces-How these two
remainders are to be used to find the true remainder.

PRIME FACTORS.-PAGE 51

A prime number (72)-How every composite number may be resolved (73).

RULE XI. To resolve a COMPOSITE number into its prime factors (74).

COMMON MEASURE-PAGE 53.

One number a measure of another (75)-An even number-an odd number-A

common measure of two or more numbers (76)-The greatest common measure

of two or more numbers (77)-When two numbers are said to be prime to each

other.

RULE XII. To find the COMMON MEASURES, of two or more numbers (78)

The greatest common measure of the divisor and dividend (79)-The same

principle generalized for two or more numbers (80).

RULE XIII. To find the greatest common measure of two or more numbers (81).

Principle involved in this method of finding the greatest common mea-

sure of two numbers (79)—of three or more numbers (80).

COMMON MULTIPLE.-PAGE 57.

One number a multiple of another (82)-A common multiple of two or more

numbers (83)-The least common multiple of two or more numbers (84).

RULE XIV. To find the COMMON MULTIPLES of two or more numbers (85).

RULE XV. To find the least common multiple of two or more numbers (86).

What of the divisors and numbers in the lowest line-why their product

is the least common multiple.

CHAPTER IV.-PAGE 63

PRELIMINARY DEFINITIONS AND PRINCIPLES.-REDUCTION OF FRACTIONS.

FRACTIONS.

A Fraction (87)-One half, one third, two thirds, three fourths, &c.--How a
fraction is denoted (88)—What the upper number is called-the lower-What
each number shows-A proper fraction (89)—An improper fraction (90).

What every proper fraction expresses (91)-What every fraction, whether
proper or improper, is equal to (92)—Constant value of a fraction (93)—Princi-
ples on which that truth depends (92 and 57).

FRACTIONS REDUCED TO THEIR LOWEST TERMS.-PAGE 65.

Reduction in general, in what it consists (94)-A fraction reduced to lower

terms (95)--A fraction reduced to its lowest terms (96)—Of the numerator and

denominator when a fraction is in its lowest terms.

RULE XVI. To reduce a fraction to its LOWEST TERMS (97).

Principle on which the Rule depends (93)--Advantage of having a fraction

in its lowest terms.

FRACTIONS REDUCED TO A COMMON DENOMINATOR.-PAGE 67.

When two or more fractions are said to have a common denominator (98)—

How two or more fractions may be reduced, mentally, to a common denomina-

tor (99)-Two or more fractions reduced to their least common denominator (100).

RULE XVII. To reduce two or more fractions to a COMMON DENOMINATOR (101).

How the two terms of each fraction are operated on, in finding a com-

mon denominator-Principle on which the Rule depends (93). £

How two fractions, having different denominators, are most readily compared

with each other (102).

INTEGERS AND MIXED NUMBERS REDUCED TO IMPROPER FRACTIONS.-

PAGE 70.

An integer, or integral number, (103)-A mixed number (104)—An integer

under the form of an improper fraction (105)-What a mixed number may be

readily reduced to (106).

RULE XVIII. To reduce an integer or a mixed number, to an improper

fraction (107).

IMPROPER FRACTIONS REDUCED TO INTEGERS OR MIXED NUMBERS.-Page (72).

RULE XIX. To reduce an improper fraction to an integer, or a mixed number (108).

When the fraction formed of the divisor and remainder, will be in its lowest

terms-on what principle (79).

CHAPTER V.-PAGE 77.

ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION OF FRACTIONS.

ADDITION OF FRACTIONS (109).

By what means the sum of two or more fractions is found-How two or more

fractions may be added, mentally (110).

RULE XX. To add two or more fractions together (111).

How mixed numbers may be added.

SUBTRACTION OF FRACTIONS (112).-PAGE 81.

By what means the difference between two fractions is found-How one

fraction may be subtracted from another, mentally (113)-How a proper frac-

tion may be subtracted from an integer (114).

RULE XXI. To subtract one fraction from another (115).

How mixed numbers may be used in subtraction.

MULTIPLICATION OF FRACTIONS.-PAGE 85.

Multiplying by a fraction, in what it consists (116)-Compound fractions (117)

-Multiplying two or more fractions together, equivalent to what (118).

RULE XXII. To multiply a fraction by a fraction (119).

How an integer and a fraction are multiplied together-How a mixed number

may be used in multiplication.

Two principles involved in the Rule for multiplying a fraction by a

fraction.

Constant product of two fractions (120).

DIVISION OF FRACTIONS.-PAGE 89.

Dividing by a fraction, in what it consists (121)—The dividend as a product,

and the divisor as one of its factors (122)—Reciprocal of a fraction (123)-Re-

ciprocal of a mixed number-Complex or Mixed Fractions (124)-A complex

fraction, equal to what.

RULE XXIII. To divide a fraction by a fraction (125.)

How a fraction is divided by an integer-How an integer is divided by a
fraction-How a mixed number may be used in division-How a mixed number
may otherwise be divided by an integer.

Reason for dividing numerator by numerator, and denominator by

denominator-When the terms of the dividend cannot be divided by those of

the divisor without a remainder.

What part the quotient is of the dividend (126)—How compound and complex

fractions are prepared for addition, subtraction, &c. (127).

CHAPTER VI.-PAGE 97.

DECIMAL FRACTIONS.-DECIMAL OR FEDERAL MONEY.

A Decimal Fraction (128)—The simple term decimal-A vulgar fraction as

distinguished from a decimal (129)—Notation of Decimals (130)—The first two

figures on the right of the decimal point (131)—the first three-the first four-A

complex or mixed decimal (132)—What a vulgar fraction annexed to a decimal

denotes-Whether it is to be reckoned as in a separate place of decimals-

Scale of decimals (133)-The system of numbering by tens, how carried from

units.

RULE XXIV. To read a decimal fraction (134).

How to prevent an ambiguity which might arise in the enunciation of deci-

mals and mixed numbers.

Ciphers annexed to decimals (135)-Each 0 between the decimal point and the

first significant decimal figure (136).

RULE XXV. To denote tenths, hundredths, &c., by decimal fractions (137).

FEDERAL MONEY (138).-PAGE 101.

The units of Federal Money-Table of Federal Money-The only denomina-

tions in common use (139)-eagles, how expressed-dimes-cents-smaller

values.

RULE XXVI. For the decimal expression of federal money (140).

The first figure after the point in a decimal of a dollar (141)—the second-the

first two together-the third-the fourth.

RULE XXVII. To reduce a decimal to a vulgar fraction (142).

RULE XXVIII. To reduce a vulgar fraction to a decimal (143).

Principle on which the numerator may be divided by the denominator

(92)-Each 0 annexed to the numerator-Each decimal figure made in the quo-

tient-Effect of thus multiplying and dividing by the same number.

A complex reduced to a simple decimal (144)-Approximate decimals (145)—

The number of figures to which an approximate decimal need be carried

ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION OF DECIMALS AND

FEDERAL MONEY.-PAGE 105.

The objects of Addition, &c. (146)—Addition, &c., of Federal Money (147)-

Why the methods of adding, subtracting, &c., are the same for decimals and

integers (148).

RULE XXIX. For the addition of decimals (149).

RULE XXX. For the subtraction of decimals (150).

When the minuend has no decimal figures, or not so many as the subtrahend.

RULE XXXI. For the multiplication of decimals (151).

Integral Os in the right of the multiplier.

Why the product of two decimal fractions must contain just as many

decimal figures as both the factors.

RULE XXXII. For the division of decimals (152).

When the divisor has more decimal figures than the dividend, or is greater

than the dividend (regarding both as integers)-Ciphers annexed to the remain-

der-Integral Os in the right of the divisor.

Why the dividend must contain just as many decimal places as both the

divisor and quotient-Why the number of decimal places in the dividend can-
not be taken less than the number in the divisor-Why the Os are prefixed, in
supplying deficiencies in the quotient.

The quotient of a less integer divided by a greater, how represented (153)—

If the less integer be then divided by the greater, decimally.