IN presenting this work to the public, the author makes no pretensions to having discovered any new spring by which to put the youthful mind into action, nor any new method of communicating a knowledge of Arithmetic. He has founded his work on the belief that labor and labor only, can insure success in any pursuit; and that labor should always be bestowed upon those objects which produce the greatest useful result.

In the selection and arrangement of matter, therefore, these rules that are of the most general use, have been presented first, and their exercises made extensive, that the pupil many early become familiar with their principles, and expert in their application.

The explanations accompanying the rules, are designed to facilitate the progress of private students, and to diminish the labor of teachers, especially in large schools, where they are unable to give to each pupil the necessary explanations.

The MENSURATION of Carpenters', Masons', Plasterers' and Pavers' work, &c., will be found an acceptable part of Arithmetic, to every man of business, and a practical knowledge of it will contribute much to the security and satisfaction of both workmen and employers, in estimating amounts of work. This has been introduced in consequence of numerous applications to the author to measure various kinds of work, and for instruction in particular rules of Mensuration.

The system of Book Keeping, is thought to be sufficient for all the purposes of farmers, mechanics and retailers, in that necessary

branch of a business education.

How far the author has succeeded in his attempts to compile a useful work, particularly adapted to the circumstances of the Western People, remains for them to judge, and for experience to determine.

NOTICE.

THE favorable reception of this treatise and the increasing demand for it, have induced the publishers to revise, enlarge, and otherwise improve the work. Such alterations and amendments have been made as the experience of the author and of other intelligent and successful teachers has suggested; it is therefore presumed, that the work will be found more useful, and consequently more acceptable than heretofore.

Numerous testimonials to the merits of the work, have been received; but its general adoption without any efforts to force its introduction, and its intrinsic worth, are our main reliance; we have therefore given it a thorough revision, and now submit the result of our labors to a discerning public.

February, 1841.

F MORGAN & Co.

ARITHMETIC.

ARITHMETIC is that part of MATHEMATICS which treats of numbers. It is both a science and an art;---the science explains the nature of numbers, and the principles upon which the rules are founded, while the art relates merely to the application of the various rules.

All the operations of arithmetic are conducted by means of FIVE fundamental rules, viz., Numeration, (which includes Notation,) Addition, Subtraction, Multiplication, and Division.

NUMERATION AND NOTATION.

Numeration is the art of representing figures or numbers by words; Notation is the art of representing numbers by characters called figures.

All numbers are represented by the following characters, which are called figures or digits.

0, 1, 2, 3, 4, 5, 6, 10%, 8, 9. nought, one, two, three, four, five, six, seven, eight, nine.

The one is often called a unit, it signifies a whole thing of a kind; two signifies two units or ones; three snifies three units or ones, &c.

The value which the figures have when standing. alone is called their simple value; but in order to denote numbers higher than 9, it is necessary to give them another value called a local value, which depends entirely on the order or place in which they stand. Thus, when we wish to write the number ten in figures, we do it by combining the characters already known, placing a 1 on the left hand of the 0, thus, 10, which is read ten. This 10 expresses ten of the units denoted by 1, but as it is only a single ten it is called a unit, and the 1 being written in the second order or second place from the right hand to express it, it is called a unit of the second order, the first place being called the place

of units, and the second, the place of tens ; ten units of the first order making one unit of the second order.

When units simply are named, units of the first order are always meant, when units of any other order are intended, the name of the order is always added.

The numbers between 10 and 20, between 20 and 30, between 30 and 40, &c. may easily be expressed by considering the tens and units of which they are composed. Thus, eleven being composed of one ten and one unit, is expressed thus, 11, twenty-three being composed of two tens, and three units, is expressed thus, 23. &c.

Sixteen being 1 ten and 6 units, is written thus, 16. Thirty-nine being 3 tens and 9 units, is written 39. Sixty-four being 6 tens and 4 units, is written 64. Ninety-five being 9 tens and 5 units, is written 95. Ten tens or one hundred forms a unit of the third order; it is expressed by placing a 1 in the third place, and filling the first and second places with cypher, thus, 100. Two hundred is expressed thus, 200 Three hundred thus, 300, Ezc.

With the orders of units, tens, and hundreds, all the numbers between one and one thousand may be readily expressed. For example, in the number four hundred and twenty-seven, there are 4 hundreds, 2 tens, and 7 units, that is, 4 units of the third order, 2 units of the second order, and 7 units of the first order.

4 2

Hence the number is written thus, In the number three hundred and five, there are 3 hundreds, no tens, and 5 units, or 3 units of the third,

none of the second, and five of the first order, hence the

Ten units of the order of hundreds, that is ten hundreds form a unit of the fourth order, called thousands, written thus, 1000.

In the same manner ten units of the fourth order form a unit of the fifth order, called tens of thousands.

The following may be regarded as the principles of Notation and Numeration.

1st. Ten units of the first or lowest order, make one unit of the second order; ten units of the second order, make one unit of the third order, and universally ten units of any order make a unit of the next higher order.

2d. All numbers are expressed by the nine digits, and the cypher, and this is effected by giving to the same figure different values according to the place it occupies. Thus, 4 in the first place is 4 units, in the second place 4 tens or forty, and so on. This tenfold increase of value by changing the place of the same figure is usually expressed by saying that figures increase from right to left in a tenfold proportion. names of the orders are to be learned from the

NUMERATION TABLE.

The

The orders are likewise divided into periods of six places each, according to the following table.