The foregoing table may be read thus, 2 and 0 are 2; 2 and 1 are 3; 2 and 2 are 4; &c.

To enable the learner to acquire accuracy and dispatch in addition, it is proper to train him to add in the following manner, till he can do it with facility.

Since 4 and 5 are 9, 14 and 5 are 19; 24 and 5 are 29; 34 and 5 are 39; &c.—the right hand figure of each amount being 9.

Since 5 and 5 are 10, 15 and 5 are 20; 25 and 5 are 30; 35 and 5 are 40; &c.

Since 7 and 6 are 13, 17 and 6 are 23; 27 and 6 are 33; 37 and 6 are 43; &c.

In this way pupils may soon learn to put numbers together readily, so as not to hesitate, or stop to count, whilst adding up a column of figures.

When only a few very small numbers are to be put together, the addition may be readily performed in the mind,*

Very young pupils may be taught the art of putting numbers together, by the following method: Let the pupil take one or more cents, (or

The addition of large

without writing down the numbers.
numbers may be performed by the following

RULE,

For the Addition of Simple or Whole Numbers.

1. Write the numbers which are to be added together, one under another, so that units shall stand under units, tens under tens, &c. and draw a line under the lowest number, to separate the numbers from their sum, when it shall be found.

2. Begin at the bottom of the right hand (or units) column of figures, and add together all the figures in the column: If the amount does not exceed 9, (or one figure,) set it down under that column, and then proceed to the next column; but if the amount exceeds 9, set down only the right hand (or units) figure of it; and the number expressed by the other figure, or figures, (which will be the number of tens in the amount,f) carry to the next column, or column of tens.

3. Then, add up the figures in the second column, (or column of tens,) together with the number, if any, carried from the first column; and, if the amount is only one figure, set it down under the second column; but if it is more than one figure, set down only the right hand figure, (which will denote tens,) and carry the rest, (which will denote hundreds,) to the third column, or column of hundreds.

4. Proceed in the same manner through all the remaining

beans, &c.) into each hand, and tell how many cents each hand contains; then request him to tell the number of cents in both hands, &c.

A similar method may be practised in Subtraction; viz. request the learner to tell how many more cents one hand contains than the other,&c. Familiar methods of instruction, similar to the above, may sometimes be practised to advantage, in various other parts of Arithmetic.

In any expression of a whole number, consisting of several figures, the figures exclusive of the one at the right hand, express the number of tens, and the right hand figure the remaining units, or ones, contained in the given number: the former figures being (as will hereafter be seen) the quotient, and the other figure, the remainder, which would be found by dividing the given number by 10. Therefore, if in adding together several numbers, we set down the right hand figure of the sum of each column of figures, and carry to the next column the number expressed by the other figure or figures of the sum, we carry from each column the number of tens contained in the sum, and set down the ones which remain over.

columns of figures, and set down the whole amount of the last column.'

Note. In writing down the numbers, it will be convenient to begin on the right hand, and proceed towards the left, setting down the units, or right hand figure of each number, first, the tens next, and so on.

To prove Simple Addition.

Begin at the top of the right hand column of figures, and perform the addition of this, and of all the other columns downwards, proceeding in other respects according to the foregoing Rule; and, if the work be right, the total sum, thus found, will be the same as that found by adding the columns upwards.-There are several other methods of proving addition, but this is the most convenient one.

EXAMPLES.

1. Find the sum of the numbers 4262, 7683, and 91. Operation.

Thousands.
Hundreds.

Tens.

Units.

Explanation. First, I write down the given numbers, one under another, so that units stand under units, tens under tens, &c.; then I draw a line under them, and add as follows:

I begin at the bottom of the right hand 4 2 6 2 column, or column of units, and say 1 76 8 3 and 3 are 4, and 2 are 6. So, the a91 mount of the first column is 6, which

Sum, 120 3

Proof, 1 2 0 3 6

being only one figure, I set it down un6 der that column, and proceed to the next. Then, proceeding to the bottom of the second column, or column of tens, I add thus, 9 and 8 are 17, and 6 are 23. this amount exceeds 9, I set down only the right hand figure, 3, and carry the other figure, 2, to the next column. Then, proceeding to the third column,

As

* This Rule is founded on the known axiom, "the whole of any quantity is equal to the sum of all its parts." The method of placing the given numbers, and carrying for the tens contained in the sum of the figures in each place, is evident from the nature of notation; for any other dispo. sition of the given numbers would alter their value; and carrying one for every ten, from an inferior to the next superior column, is evidently

or column of hundreds, I say, 2 carried to 6 makes 8, and 2 are 10: I set down 0 under the third column, and carry 1 to the next. Then, 1 carried to 7 makes 8, and 4 are 12; which being the amount of the last column, I set down the whole of it, and the work is done. So, the total sum, or answer, is 12036.

To prove the work, I add all the columns downwards, thus: I begin at the top of the right hand column, and say, 2 and 3 are 5, and 1 makes 6. I set down this amount below the first column, and then I proceed to the top of the second column, and add thus; 6 and 8 are 14, and 9 are 23. I set down 3 below this column, and carry 2 to the next. Then, 2 carried to 2, makes 4, and 6 are 10: I set down 0, and carry 1 to the next column. Then, I carried to 4 makes 5, and 7 are 12, which I set down. So the total sum is 12036, the same as before; and therefore I conclude the work is right.

Note. As young beginners are apt to make mistakes in carrying from one column to another, as well as in adding up the columns, it will be well for them to set down the sum of each column, placing the several sums one under another, in regular order, as in the next following example. These sums the learner should retain on his slate until he has finished the work; and then if it should contain any errors, le may the more easily correct them when he comes to prove the work.

2. What is the sum of the numbers 3106, 812, 48, 500, and 2019?

3106
812

4 8
500

2019

Answer, 6 48 5

25 Amount of the units, or first column.
8 Do. of the tens, or 2d do.
14 Do. of the hundreds, or 3d do.

Proof,

6485

right; because one unit in the latter case is equal to ten units in the former; that is, ten in the column of units are equal to one in the colum of tens, and ten in the column of tens, to one in the column of hundreds, &.c.