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numbers,* which he believes are more concise and convenient than any others now in use.

Questions on the nature of the principal rules of Arithmetic, are inserted in the work, the answers to which are contained in the rules to which the questions relate. These questions are calculated to lead the learner to examine those rules attentively, in order to find out the proper answers; and it will, perhaps, be well for teachers, in the general way, to see that their pupils are able to answer, correctly, the questions on each particular rule, before they allow them to proceed to the next.

The following work contains, besides the System of Arithmetic, Rules for the Mensuration of the different kinds of Superfices and Solids, with directions for applying them to various practical purposes; and also, Rules for solving a number of useful and interesting Mathematical, Philosophical, and Chronological Problems--all of which are illustrated by suitable examples: Likewise, a collection of curious and instructive miscellaneous questions for exercise, and several useful Tables. The great practical utility of the rules of Mensuration, and of the subsequent Prob-' lems, &c. will, it is presumed, be considered a sufficient reason for inserting them in the work, at the end of the treatise on Arithmetic.

In composing this work, the writer has taken the liberty to make use of some short extracts from other mathematical works; and he acknowledges himself indebted to other writers for some of the illustrations of the principles of Arithmetic, and for a number of the questions for exercise. Some

*The method of extracting the cube rooi, given in this work and also an accara e general method of fiuding the roots of compound algebraical pquations, were discovered by the author a number of years ago ; and subsequently, he and some of his friends made those meibods known to several mathematicians, who resided in diflerent parts of the U. States, Tbose rules, (or very similar ones) have since been published in this country, by several persons with a bom the anthor of this work bas had no acquaintance; and he has not yet been able satisfactorily to ascertain whether the rules were, or were nol, known to any mathematician prior to the discovery of them by bimself.

Teachers who think it advisable for their pupils to acquire a tolera. ble knowledge of Practical Arithmetic before they undertake to learn the Theory of the rules, can direct them to omil ik Theoretical Questions, and ihe demonstrations of the rules, until they bave made considerable pre. gress in the study of Arithmetic, and then to allend to them as a review.

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of those extracts are contained in so many different works that the original writers of them could not easily be ascertained; and the rest have, for the most part, been so disposed of, as to make it very inconvenient to refer the reader to the works from which they were taken; for which reasons such references are not generally inserted. The author has not aimed at novelty where he saw no prospect of making improvements; and he hopes the use which he has made of the labors of his predecessors will be considered pardonable by those who know how difficult it would be for any person, at the present day, to write a Treatise on Arithmetic without repeating much which has already been written by others. The practice of copying has indeed been general with almost all writers on Arithmetic, though few of them have candidly acknowledged it.

The author is aware that there is a number of works of this nature already in use, in different parts of this country. It does not appear, however, that any one of them is now, or is likely very soon to be, in general use in schools throughout the United States; and hence it may be infered that none of them have met the approbation of the public, generally. The author is not disposed to expatiate on the defects of those systems, nor on the merits of his own; being persuaded that enlightened persons will examine and judge for themselves, and that his work, as well as others, must stand or fall, according to its real merit or demerit.

In submitting the work to the public, the author cannot forbear expressing his gratitude for the liberal patronage already bestowed upon this edition. To those persons who have encouraged the publication of the work, the author tenders his sincere thanks; and he would request of them, and others, that indulgence towards its imperfections, which the nature of the work must necessarily require. Should the present edition prove in some respects imperfect, it is hoped that such improvements will hereafter be made in the work, as will render the subsequent editions more deserving of public patronage.

IRA WANZER. New-Fairfield, Conn., Oct. 1, 1831.







MATHEMATICS IN GENERAL. MATHEMATICS is the science which treats of all kinds of quantity whatever, that can be numbered or measured. By quantity, is meant any thing that will admit of increase or decrease; or that is capable of any sort of calculation or mensuration : such as numbers, lines, space, time, motion, weight.

Note.--In the Mathematics, a quantity is said to be simple, when its value or magnitude is expressed by one term or denomination only ; and compound, when its value is expressed by two or more terms.

Those parts of the Mathematics on which all the others are founded, are Arithmetic, Algebra, and Geometry:

ARITHMETIC is the science of numbers. Its aid is required to complete and apply the calculations in almost every other department of the mathematics.

ALGEBRA is a general method of computing by letters and other symbols. It is of extensive use in mathematical investigations.

GEOMETRY is that part of the mathematics whích treats of magnitude. By magnitude, in the appropriate sense of the term, is meant that species of quantity which is extended; that is, which has one or more of the three dimensions, length, breadth, and thickness.

Mathematics are either pure, or mixed. In pure or abstract mathematics, quantities are considered independently of any substances actually existing. But, in mixed mathematics, the relations of quantities are investigated, in connection with some of the properties of matter, or with reference to the common transactions of business. Thus, in Surveying, mathematical principles are applied to the measuring of land; and in Astronomy, to the motions of the heavenly bodies, &e.

Mathematics are also distinguished into Theoretical, or Speculative, and Practical, viz. Theoretical, when concerned in investigating and demonstrating the variotis properties and relations of quantities; and Practical, when applied to practice and real use concerning physical objects.



There are various characters or marks which are used in Arithmetic and Algebra, to denote the operations of Addition, Subtraction, &c.; the chief of which are the following:




Plus, † or more; the sign of Addition, or the
positive sign, signifying that the number to
which it is prefixed, § is to be added to some
other number. Thus, 2+5, denotes that 5 is
to be added to 2; and the expression is read thus,
2 plus 5, or, the sum of 2 and 5.
Minus,t or less; the sign of Subtraction, or
the negative sign, signifying that the number
to which it is prefixed

is to be subtracted from
some other number. Thus, 8–5, signifies that
5 is to be subtracted from 8, and is read thus,
8 minus 5, or the difference of 8 and 5.
Into, or with; the sign of Multiplication, sig-
nifying that the numbers between which it is
placed, are to be multiplied together. Thus,
8X2, signifies that 8 is to be multiplied by 2,
and is read, 8 into 2, or the product of 8 and 2.


* The preceding “ Introductory Observations” have been mostly ex. tracted from Hution's and Day's Courses of Mathematics.

Plus and minus are Latin words; the former signifying more,and the lalter less.

§ To prefix, is to place before : to annex, is to place after.

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