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ing effects to their cause, the dictamen of rules to the principles on which they are founded, the mutual dependence and connection of principles, and the skilful arrangement and combination of elements, in order to produce a desired result, thereby acquires a readiness and justness of perception, a conciseness in reasoning, a power of comprehension, and a strength of judgment unattainable by other means.

Experience having for a long time strongly impressed the mind of the author of this work with the above views, he was induced, many years ago, to impose on himself the arduous, and to him, from various causes, extremely protracted and difficult task of preparing, for schools and private students a copious yet compact treatise of Arithmetic, containing, together with an ample number of practical examples, a bold, connected, logical, and lucid developement of the principles and rules of the entire science. How far, in the persent work, that object is accomplished, the intelligent reader must determine.

SUMMARY OF REASONS FOR UNDERTAKING THIS WORK.

1. The almost total absence of every thing like theory in many arithmetics, and the very great imperfection and deficiency in this respect in the best of those now in use.

2. The utter impossibility that a master, however capable, who presides over a numerous class of pupils in various stages of advancement in the science, can find time to give to each a lucid verbal explanation to make up for the deficiency of his book.

3. The inefficiency of such verbal explanation, even if it could be given, for want of sufficient reiteration: said reiteration irksome in the extreme, and calculated to irritate the teacher to the discouragement of his pupil.

4. The inability of teachers to give the requisite explanation ; for, without any disparagement to that self-devoted class of men, it is to be presumed that, taking the whole scope of the science, comparatively few are, impromptu, equal to the task.

BASIS OF THE WORK.

Whatever can be spoken can be written. Therefore, whatever could be said by the most enlightened teacher to his pupil, in explanation of every department of the science, the author has en

deavoured to say in the book; and thus the scholar, having it before him, can read it as often as he pleases, to his own edification and great relief of the teacher.

SCHEME.

The subject is divided into six books, each of which exhibits a different department of the science. In each of the first four books, the cardinal operations are thoroughly discussed: in the first in Integers; in the second in Vulgar Fractions; in the third in Decimals; and in the fourth in Compound Numbers. The fifth book exhibits those rules which are not cardinal: viz., Involution, Evolution, Proportion, Progression, &c. &c., including Logarithms. The sixth book is devoted to mercantile science; in some departments of which, the author is greatly indebted to Kelly's Universal Cambist.

METHOD.

For the sake of easy reference, as well as logical concatenation, the work is subdivided into Sections and Articles, which are regularly numbered. Also, those passages which involve the most important principles, or leading points of the argument, particularly in the first four books, are italicized, the more perfectly to impress the mind of the pupil, and thus induce the habit of reflection,

Questions may be put by the teacher, to elicit from the pupil the passages italicized, or answers in accordance with them, thus :-TEACHER. What is quantity ? SCHOLAR. A limited portion of any natural object.

This may be continued by the teacher throughout the work, which will be a good test of the progress of his pupil. The pupil, and he or she who is self-taught, may also pursue the same method for self-examination.

STYLE.

In presenting to the young, for daily contemplation, a subject of primary importance, much depends on the language in which that subject is invested. A style elevated, rich, clear, and concise, being at once most

appropriate and attractive, is best calculated to awaken and sug. tain a lively interest.

Prolixity wearies, puerility disgusts; both, therefore, as far as consistent with that most essential attribute, perspicuity, have been studiously avoided.

Brevity, wherever practicable, has been adopted, not only because it economizes time and expense, but because, by concentration, it lends force, and hence greatly facilitates comprehension.

ORIGINALITY.

The principle upon which this work is based, the plan upon which its greater divisions are arranged, and the method of conducting its details, both in calculation and language, combine to render it exceedingly unlike any of the popular works now

in use.

Much that is entirely new in calculation will be found in the work, especially in the first, fourth, and fifth books. The rule for finding the least common multiple of several numbers, the author first gave in the year 1826, at which time he was employed as teacher in the Senior Department of the High School of New York.

SCOPE.

To the many novel features already presented, we may add the explanation of the nature and use of Logarithms, accompanied by a Table of those numbers for practical application. This, then, is probably the only school treatise which embodies the whole science, and hence the only one by means of which the science can be perfectly learned without the aid of a teacher.

Having thus endeavoured, instead of a barren, uninteresting, mechanical abstract, or an unwieldy, incongruous compilation, to give the subject the form, consistent with its important nature, of a transcendently beautiful science, it is hoped that the work here presented will amply reward the industrious that, in its perusal, he will feel assured that he is, by a wholesome discipline, and consequent elevation of his mental power, preparing himself successfully to attempt the loftiest subjects to which human genius can aspire.

NOBLE HEATH.

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TABLE OF CONTENTS.

PAGE

BOOK II.

Of the Square

89

Cube and Parallelopiped.........

90

Powers of Numbers............

91

Prime Numbers ....

92-94

Properties of Numbers..

94-99

Prime Factors ......

99-100

Least Common Multiple ......

100-104

Greatest Common Measure..............

104-106

Ratio .................. .......................................

106-107

Reduction of Whole and Mixed Numbers..

108

Lowest Terms of a Fraction

109-110

Dissimilar Fractions reduced to their Least Com. Denom...... 110–112 .

Addition of Vulgar Fractions..

112-113

Addition of Mixed Numbers...

113–114

Subtraction of Fractions ....

114

Subtraction of Mixed Numbers.....

115-116

Multiplication of Fractions ...

116–119

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