SECTION I.

FUNDAMENTAL RULES OF ARITHMETIC.

THESE are four, ADDITION, SUBTRACTION, MULTIPLI

CATION, and DIVISION; they may be either simple or compound; simple, when the numbers are all of one sort or denomination; compound, when the numbers are of different denominations.

THEY are called, Principal or Fundamental Rules, because that all other rules and operations in arithmetic are nothing more than various uses and repetitions of these four rules.

THE object of every arithmetical operation, is, by certain given quantities which are known, to find out others which are unknown. 'This cannot be done but by changes effected on the given numbers; and as the only way in which numbers can be changed is either by increasing or by diminishing their quantities, and as there can be no increase or diminution of numbers but by one or the other of the above operations, it consequently follows, that these four rules embrace the WHOLE art of Arithmetic.

§. 1 Simple Addition.

SIMPLE ADDITION is the putting together of two or more numbers, of the same denomination, so as to make them one wkole or total number; as 3 dollars, 6 dollars, and 8 dollars added or put together, make 17 dollars.

RULE.

"WRITE the numbers to be added one under another, with units under " units, tens under tens, and so on. Draw a line under the lower number, then "add the right hand column; and if the sum be under ten, write it at the foot " of the column; but if it be ten, or an exact number of tens, write a cypher; and "if it be not an exact number of tens, write the excess above tens at the foot of the column; and for every ten the sum contains, carry one to the next column, "and add it in the same manner as the former. Proceed in like manner to "add the other columns carrying for the tens of each to the next, and marl"down the full sum of the left hand column.'

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PROOF.

RECKON the figures from the top downwards, and if the work be right, this amount will be equal to the first or, what is often practised, "cut off the 66 upper line of figures and find the amount of the rest; then if the amount "and upper line when added be equal to the sum total, the work is supposed "to be right."

EXAMPLES.

Thous.

Hund.
Tens.

Units.

∞ Thous.

Hund.

Tens.

w Units.

1. WHAT will be the amount of

dollars

dollars, and of 3 dollars, when added together?

dollars;

HERE are four sums given for addition; two of them contain units, tensy hundreds, thousands; another of them contains units, tens, hundreds; and a fourth contains units only. The first step, to prepare these sums for the op eration of addition, is to write them down, units under units, tens under tens, and so on, as in the following manner.

T. of Thou.

Thousands.
Hundreds,

Tens.

& Units.

THE four given sums for addition placed as the rule directs.

Answer, or, amount, 1

dollars.

Proof, 1 2 3 O

9.

Amount of the three lower lines,

To find the answer or amount of the sums given to be added, begin with the right hand column, and say, 3 to 1 is 4, and 3 is 7, and 2 is 9; which sum (9). being less than ten, set down directly under the column you added. Then proceeding to the next columu, say again; 5 to 4 is 9, and 1 is 10 being even ten, set down 0, and carry 1 to the next column, saying 1, which I carry to 6. is 7, and is nothing, but 6 is 13; which sum (13) is an excess of 3 over even ten; therefore, set down 3 and carry 1 for the ten to 8 in the next column, ing 1 to 8 is 9, and 3 is 12; this being the last column, set down the whole number, (12) placing the 2, or unit figure, directly under the column, and carrying the other figure, or the 1, forward to the next place on the left hand, or to that of Tens of Thousands, and the work is done.

say

Ir may now be required to know if the whole be right. To exhibit the method of proof let the upper line of figures be cut off as seen in the example. Then adding the three lower lines which remain, place the amount (8697) under the amount first obtained by the addition of all the sums, observing carefully that each figure fall directly under the column which produced it; then add this last amount to the upper line which you cut off; thus, 7 to 2 is 9; 9 to 1 is 10; carry 1 to 6 is 7 and 6 is 13; 1 which I carry again to 8 is 9 and 3 is 12, all which

Hund.

Units,

or Tens.

being set down in their proper places, and as seen in the example, compare the amount (12309) last obtained with the first amount (12309) and if they agree, as it is seen in this case they do, then the work is judged to be right.

NOTE. THE reason of carrying for ten in all simple numbers is evident from what has been taught in Notation. It is because 10 in an inferior column is just equal in value to 1 in a superior column. As if a man should be holding in his right hand half pistareens, and in his left hand, dollars. If you should take 10 half pistareens from his right hand, and put one dollar into his left hand, you would not rob the man of any of his money, because 1 of these pieces in his left hand is just equal in value to 10 of those in his right hand.

THE Scholar who has given proper attention to his rule, and the foregoing examples, will of himself be able to work the following --always remembering to carry one for every 10, and at the last column to set down the whole number.

9 7 1 6 1 1 1 1 6 8 1 3 3 9 1 3 7

SUPPLEMENT to Addition.

THE attentive Scholar who has understood, and still carries in his mind, what has already been taught him of Addition, will be able to answer his Instructor to the following

QUESTIONS.

1. WHAT is Simple Addition ?

2. How do you place numbers to be added?

3. WHERE do you begin the Addition ?

4. How is the sum or amount of each column to be set down?

5. WHAT do you observe in regard to setting down the sum of the last column? 6. WHr do you carry for 10 rather than any other number?

7. How is Addition proved?

8. OF what use is Addition ?

NOTE 1. SHOULD the Learner find any difficulty in giving an answer to the above questions, he is advised to turn back and consult his Rule, with its Illustrations.

NOTE. 2. In treating of the Rules of Arithmetic the Scholar,in all instances, is not particularly instructed in the use and application of them to the purposes of life. This is a point, however, to which his thoughts should be called; therefore it is made a question here. A consideration of the Rule and of the questions, which it involves, naturally suggests an answer. To consideration, therefore, let the Scholar apply himself. The mind acquires strength by exercise; instruction ought ever to be plain, but never so full as to preclude a necessity that the Scholar should in some degree exercise his own thoughts; it should be given in such a manner as to force him into some reflections of his

NOTE. THE Scholar who looks at greatness in his class will not be discouraged by a little difficulty which may at first occur in stating his question, but will apply himself the more closely to his Rule and to thinking, that if possible he may be able of himself to answer, what another may be obliged to have taught him by his Instructor.

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