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Entero, according to act of Congress, in the year 1834, by Frederick Emerson, in the Clerk's Office of the District Court of the District of Massachusetts


CAMBRIDGE, Oct. 31, 1834. To the Publishers of Emerson's Arithmetic.-Gentlemen - I have exam. wed the Third Part of Mr. Einerson's Arithmetic, with great pleasure. The perspicuity of its arrangeinent, and the clearness and brevity of its explanations, combined with its happy adaptation to the purposes of practical busimess are its great recoinmenilations. hope it will suon be introduced into ll our schools and take place of the ill-digested 'Treatises, to which our astructers have bitherto been compelled to resort.

Respectfully, BENJAMIN PEIRCE.
(Professor of Mathematics and Nat. Philo. Harvard University )

Boston, Nur. 10, 1834. Messrs. Russell, Odiorne, & Co.- I have carefully examined the Third Part of the North American Arithinelic, by Mr. Emerson; and am so well satisfied that it is the best treatise upon the subject, with which I am acquainted, that I have determined to introduce it as a text-book into my school Very respectfully, &c. yours, E. BAILEY.

[Principal of the Young Ladies' High School, Boston.) Erom the Boston Public Schoolmasters.

Boston, Nov. 16, 1834. We have considered it our duty to render ourselves acquainted with the more prominent systems of Ariihinatic, published for the use of schools, and to fix on some work villich-appears coʻunite the greatest advantages, and report the same to the School Committee of Boston, for adoption in the Public Schools. After the inost careful exc.rication, we have, without any hesitancy, come to the conclusion, that Einerson's North American Arithmetic [First, Second, and Third Parts] is the work best suited to the wants of all classes of scholars, and most convenient for the purposes of instruction, Accordingly, we have petitioned for the adoption of this work in the Public Schools.



ABNER FORBES. Orders of the Boston School Committee. At a Meeting of the School Committee, Nov. 18, 1834,

Ordered, That Emerson's North American Arithmetic, Second, and Third Parts, be substituted in the Writing Schools, for Colburn's First Lessons and Sequel.*

Ordered,' That the Arithmetics now in use be permitted to their present owners; but that whenever a scholar shall have occasion to purchase a new one, the North American Arithmetic shall be required.

Attest, S. F. M'CLEARY, Secretary • The First Part was already adopted, by a previous order.


The work new presented, is the last of a series of books, under the general title of The North AMERICAN ArithMETIC, and severally denominated Part First, Part Second, and Part Third.

Part First is a small book, designed for the use of children between five and eight years of age, and suited to the convenience of class-teaching in primary schools.

Part Second consists of a course of oral and written exercises united, embracing sufficient theory and practice of arithmetic for all the purposes of common business.

Part THIRD comprises a brief view of the elementary principles of arithmetic, and a full development of its higher operations. Although it is especially prepared to succeed the use of Part Second, it may be conveniently taken up by scholars, whose acquirements in arithmetic are considerably less than the exercises in Part Second are calculated to afford. While preparing this book, I have kept in prominent view, two classes of scholars;. viz. those wbo.are to prosecute a full course of mathérapical studies and inose who are to embark in cominèrce. In attempting to place arithmetic, as a science, before the scholar in that light, which shall prepare him for the proper requirements of college, I have found it convenient to draw a large portion of the ese ainples for illustration and practice; bom,mercantile transactions; and thus pure and mercantile'arithmetic are united. No attention has been spared, to render the mercantile information here presented, correct and adequate. Being convinced, that many of the statements relative to commerce, which appear in books of arithmetic, have been transmitted down from ancient publications, and are now erroneous, I have drawn new data from the counting-room, the insurance office, the custom-house, and the laws of the present times. The article on Foreign Exchange is comparatively extensive, and I hope it will be found to justify the confidence of merchants. Its statements correspond to those of the British Universal Cambist,' conformably with our value of foreign coins, as fixed by Act of Congress, in 1834.

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Although a knowledge of arithmetic may, in general, be well appreciated as a valuable acquisition, yet the effect produced on intellectual character, by the exercises necessary for acquiring that knowledge, is not always duly considered. In these exercises, the inental effort required in discovering the true relations of the data, tends to strengthen the power of comprehension, and leads to a habit of investigating; the certainty of the processes, and the indisputable correctness of the results, give clearness and activity of thought; and, in the systematic arrangement necessary to be observed in performing solutions, the mind is disciplined to order, and accustomed to that connected view of things, so indispensable to the formation of a sound judgment. These advantages, however, depend on the manner in which the science is taught; and they are gained, or lost, in proportion as the teaching is rational, or superficial.

Arithmetic, more than any other branch of learning, has suffered from the influence of circumstances. Being the vade-mecum of the shop-keeper, it has too often been viewed as the peculiar accomplishment of the accountant, and neglected by the classical student. The popular supposition, that a compendious treatise can be more easily mastered than a copious one, has led to the use of textbooks, which are deficient, both in elucidation and exercises. But these evils seem now.to be dissipating.–The elements of archmetic have becorde a subject of primary instruction; and teachers of higher schools, who have adopted an elevated course of südy, are no longer satisfied with books of indifferent character..

It has been my belief,:that: a treatise on arithmetic might be so constructed; tbat.the Hearder should find no means of proceeding in the exercises, without mastering the subject in his own mind, as he advances; and, that he should still be enabled to proceed through the entire course, without requiring any instruction from his tutor. Induced by this belief, I commenced preparing The North American Arithmetic about five years since; and the only apology I shall offer, for not earlier presenting its several Parts to the public, is the unwillingness that they should pass from my hands, while I could see opportunity for their improvement. Boston, October 1834.


A KEY to this work (for teachers only) is published separately





QUANTITY is that property of any thing which may be increased or diminished —it is magnitude or multitude. It is magnitude when presented in a mass or continuity; as, a quantity of water, a quantity of cloth. It is multitude when presented in the assemblage of several things; as, a quantity of pens, a quantity of bats. The idea of quantity is not, however, confined to visible objects ; it has reference to every thing that is susceptible of being more or less.

NUMBERS are the expressions of quantity. Their names are, One, Two, Three, Four, Five, Six, Seven. Eight, Nine, Ten, &c. In quantities of multitude, One expresses a Unit; that is, an entire, single thing; as one pen, one bat. Then each succeeding number expresses one unit more than the next preceding. In quantities of magnitude, a certain known quantity is first assumed as a measure, and considered the unit; as oue gallon, one yard. Then each succeeding number expresses a quantity equal to as many times the unit, as the number indicates. Hence, the value of any number depends upon the value of its unity.

When the unit is applied to any particular thing, it is called a concrete unit; and numbers consisting of concrete

units are called concrete numbers: for example, one dollar, two dollars. But when no particular thing is indicated by the unit, it is an abstract unit; and hence arise abstract numbers: for example, one and one make two.

Without the use of numbers, we cannot know precisely how much any quantity is, nor make any exact comparison of quantities. And it is by comparison only, that we value all quantities; since an object, viewed by itself, cannot be considered either great or small, much or little; it can be so only in its relation to some other object, that is smaller or greater.

ARITHMETIC treats of numbers: it demonstrates their various properties and relations; and hence it is called the Science of numbers. It also teaches the methods of computing by numbers; and hence it is called the Art of numbering.


NOTATION AND NUMERATION. Notation is the writing of numbers in numerical char. acters, and NUMERATION is the reading of them.

The method of denoting numbers first practised, was undoubtedly that of representing each unit by a separate mark. Various abbreviations of this method succeeded; such as the use of a single character to represent five, another to represent ten, &c.; but no method was found perfectly convenient, until the Arabic FIGURES or DIGITS, and decimal system now in use, were adopted. These figures are, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; denoting respectively, nothing, one unit, two units, three units, &c.

To denote numbers higher than 9, recourse is had to a law that assigns superior values to figures, according to the order in which they are placed. viz. Any figure placed to the left of another figure, expresses ten times the quantity that it would express if it occupied the place of the latter. Hence arise a succession of higher orders of units.

As an illustration of the above law, observe the differ. Int quantities which are expressed by the figure 1.

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