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ARITHMETIC,

IN WHICH THE

PRINCIPLES OF OPERATING BY NUMBERS

ARE

ANALYTICALLY EXPLAINED,

AND

SYNTHETICALLY APPLIED;

THUS

COMBINING THE ADVANTAGES

TO BE DERIVED BOTH FROM

THE INDUCTIVE AND SYNTHETIC MODE

OF INSTRUCTING:

THE WHOLE

MADE FAMILIAR BY A GREAT VARIETY OF USEFUL AND INTER-
ESTING EXAMPLES, CALCULATED AT ONCE TO ENGAGE
THE PUPIL IN THE STUDY, AND TO GIVE HIM A
FULL KNOWLEDGE OF FIGURES IN THEIR
APPLICATION TO ALL THE PRAC-

TICAL PURPOSES OF

LIFE.

DESIGNED FOR THE USE OF

SCHOOLS AND ACADEMIES

IN THE UNITED STATES.

BY DANIEL ADAMS, M. D.
AUTHOR OF THE SCHOLAR'S ARITHMETIC, SCHOOL GEOGRAPHY, &C.

KEENE, N. FL.
PUBLISHED BY J & J. W. PRENTISS.

1830.

DISTRICT OF NEW-HAMPSHIRE.

District Clerk's Office. BE IT REMEMBERED, That on the eighteenth day of September, A. D. 1827, in the fifty-second year of the Independence of the United States of America, DANIEL ADAMS, of said district, has deposited in this office the title of a book, the right whereof he claims Es author, in the words following, to wit :

". ARITHMETIC, in which the Principles of operating by Numbers are analytically ex. plained, and synthetically applied ; thus combining the Advantages to be derived both from the inductive and synthetic Mode of instructing: the whole made familiar by a great Variety of useful and interesting Examples, calculated at once to engage the Pupil in the Study, and to give him a full Knowledge of Figures in their Application to all the practical Purposes of Life. Designed for the use of schools and Academies in the United States. By DANIEL Adams, M. D. Author of the Scholar's Arithmetic, School Geography, &c."

In conformity to the act of Congress of the United States, entitled, “ An Act for the encouragement of learning, by securing the copies of maps, charts, and books, to the authors and proprietors of such copies during the times therein mentioned ;" and also to an act, entitled, " An Act supplementary to an act for the encouragement of learning, by securing the copies of maps, charts, and books, to the authors and proprietors of such copies during the times therein mentioned ; and extending the benefits thereof to the arts of designing, engraving and etching historical and other prints."

CHARLES W. CUTTER,

Clerk of the District of Ner-Hampshire. A true copy

Attest, C. W. CUTTER, Clerk.

Stereotyped at the
Boston Type ana Stereotype Foundry.

,-1.4 50703.

PREFACE.

THERE are two methods of teaching,--the synthetic and the analytic. In the synthetic method, the pupil is first presented with a general view of the science he is studying, and afterwards with the particulars of which it consists. The analytic method reverses this order : the pupil is first presented with the particulars, from which he is led, by certain natural and easy gradations, to those views which are more general and comprehensive.

The Scholar's Arithmetic, published in 1801, is synthetic. If that is a fault of the work, it is a fault of the times in which it appeared. The analytic or inductive method of teaching, as now applied to elomentary instruction, is among the improvements of later years. Its introduction is ascribed to Pestalozzi, a distinguished teacher in Switzerland. It has been applied to arithmetic, with great ingenuity by Mr. COLBURN, in our own country.

The analytic is unquestionably the best method of acquiring know ledge; the synthetic is the best metlicd of recapitulating, or reviewing it. In a treatise designed for school education, both methods are useful. Such is the plan of the present undertaking, which the author, occupied as he is with other objects and pursuits, would willingly have forborne, but that, the demand for the Scholar's Arithmetic still continuing, an obligation, incurred by long-continued and extended patronage, did not allow him to decline the labour of a revisal, which should adapt it to the present more enlightened views of teaching this science in our schools. In doing this, however, it has been necessary to make it a new work.

In the execution of this design, an analysis of each rule is first given, containing a familiar explanation of its various principles; after which follows a synthesis of these principles, with questions in form of a supplement. Nothing is taught dogmatically ; no technical term is used till it has first been defined, nor any principle inculcated without a previous developement of its truth; and the pupil is made to understand the reason of each process as he proceeds.

The examples under each rule are mostly of a practical nature, beginning with those that are very easy, and gradually advancing to those more difficult, till one is introduced containing larger numbers, and which is not easily solved in the mind; then, in a plain, familiar manner, the pupil is shown how the solution may be facilitated by figures. In this way he is made to see at once their use and their application.

At the close of the fundamental rules, it has been thought advisable to collect into one clear view the distinguishing properties of those rules, and to give a number of examples involving one or more of them. These exercises will prepare the pupil more readily to understand the application of these to the succeeding rules; and, besides, will serve co interest him in the science, since he will find himself áble, by the application of a very few

principles, to solve many curious questions. The arrangement of the subjects is that, which to the author has appeared most natural, and may be seen by the Index. Fractions have received all that consideration which their importance demands. The principles of a rule called Practice are exhibited, but its detail of cases is omitted, as unnecessary since the adoption and general use of federal inoney. "The Rule of Three, or Proportion, is retained, and the solution of questions involving the principles of proportion, by analysis, is distinctly shown.

The articles Alligation, Arithmetical and Geometrical Progression, Annuities and Permutation, were prepared by Mr. Ira Young, a member of Dartmouth College, from whose knowledge of the subjoct, and experience in teaching, I have derived important aid in other parts of the work.

The numerical paragraphs are chiefly for the purpose of reference: these references tie pupil should not be allowed to neglect. His attention also ought to be particularly directed, by his instructer, to the illustration of each particular principle, from which general rules are deduced : for this purpose, recitations by classes ought to be instituted in every school where arithmetic is taught.

The supplements to the rules, and the geometrical demonstrations of the extraction of the square and cube roots, are the only traits of the old work preserved in the new.

DANIEL ADAMS. Mont Vernon, (N. H.) Sept. 29, 1827.

INDEX.

SIMPLE NUMBERS.
Numeration and Notation,
Addition,
Subtraction,
Multiplication,
Division,

Fractions arise from Division,
Miscellaneous Questions, involving the Principles of the preceding Rules,

Page

7
12
19
26
37
42
52

56
57
641

COMPOUND NUMBERS.
Different Denominations,
Federal Money,

to find the value of Articles sold by the 100, or 1000,

Bills of Goods sold,
Reduction,

Tables of Money, Weight, Measure, &c.
Addition of Compound Numbers,
Subtraction,
Multiplication and Division,

69-

89
93

.

FRACTIONS.
COMMON, or VULGAR. Their Notation,

101
Proper, Improper, &c.

102
To change an Improper Fraction to a Whole or Mixed Number,

103
a Mixed Number to an Improper Fraction,

104
To reduce a Fraction to its lowest Terms,

105
Greatest common Divisor, how found,

106
To divide a Fraction by a Whole Number; two ways,

107
To multiply a Fraction by a Whole Numher; two ways,

110
a Whole Number by a Fraction,

112
one Fraction by another,

113
General Rule for the Multiplication of Fractions,

114
To divide a Whole Number hy a Fraction,

115
one Fraction by another,

117
General Rule for the Division of Fractions,

118
Addition and Subtraction of Fractions,

119
Common Denominator, how found,

120
Least Common Multiple, how found,

121
Rule for the Addition and Subtraction of Fractions,

124
Reduction of Fractions,

124
DECIMAL. Their Notation,
Addition and Subtraction of Decimal Fractions,

135
Multiplication of Decimal Fractions,

137
Division of Decimal Fractions,

139
To reduce Vulgar to Decimal Fractions,

142
Reduction of Decimal Fractions,

145
To reduce Shillings, &c., to the Decimal of a Pound, by Inspection, 146

the three first Decimals of a Pound to Shillings, &c., by Inspection, 157

1.32

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