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Entered according to Act of Congress, in the year 1855, by

NOBLE HEATH, in the Clerk's Office of the District Court of the United States for the Eastern

District of Pennsylvania.




He who has, during the last forty years, been acquainted with the state of the science of Arithmetic, not only in the schools of this country, but in other parts, well knows that, at the commencement of that period, and for nearly half its duration, scarcely any thing better was found in the hands of the student than a miserable epitome or skeleton of the subject, presenting merely a few dogmas, followed by examples, without farther elucidation than the mechanical operation of one of these to serve as a model for the performance of all the others.

Divested almost entirely of scientific principles, Arithmetic was taught, in accordance with its title in the books then in use, as the art of computing by numbers; and a very clumsy, uninteresting, and wearisome art it was. Hence, the student, as might be expected, having no clue to the intricacies of the subject but memory, after ciphering through his Assistant,—as such a work was improperly called,--for the third, fourth, fifth, and even the sixth time, found himself almost as ignorant of the science as when he began.

Deeply felt and widely spread seem to have been the dissatisfaction, discouragement, and regret resulting from such palpable and enormous waste of the time and labour of the youth ; for, during the last twenty years, new publications on this science, of much less exceptionable character, have, in the schools of Germany, France, England, and America, succeeded and superseded each other with surprising rapidity.

It is interesting to trace in these works the extremely gradual, yet steady approach toward a full development of the science, which, apparently, no one has hitherto dared to attempt.

The student who is early accustomed to investigation, in tracing effects to their causė, 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.


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.


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 endeavoured 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.


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.


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.


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 sugtain 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 student; and 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.


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