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The square made by the ten Arabic characters forms an Addition and Subtraction Table.

Beginning with the first line, thus, Zero and zero are zero; zero and 1 are 1; zero and 2 are 2; zero and 3 are 3; zero and 4 are 4; zero and 5 are 5; zero and 6 are 6, etc.

The second line, one and zero are one; 1 and 1 are 2: 1 and 2 are 3; 1 and 3 are 4; 1 and 4 are 5, etc.

2 and zero are 2; 2 and 1 are 3; 2 and 2 are 4; 2 and 3 are 5; 3 and 4 are 7, etc.

3 and 0 are 3; 3 and 1 are 4; 3 and 2 are 5; 3 and 3 are 6; 3 and 4 are 7, etc.

Continue this, taking the first figure of the 1st column and adding it to each successive figure in the first line; the adding of zero is only nominal, as it makes no increase.

It also becomes a subtraction table, the figures of the first column being the subtrahend, and those of the first line the remainders.

Take zero from 1 and 1 remains; 0 from 2, 2 remain, etc. It may be thus expressed: 101; 2 − 0 = 2; 3-03; 4-0 4; 505; which is read,

1 minus zero equals 1, etc.

Second line: take 1 from 2, 1 remains; 1 from 3, 2 remain; or, 211; 3-1=2; 4—1=3; 514, etc.

Third line; 3-21; 4-22; 5-2=3; 6-24; 7-25, etc.

In addition, we add two numbers at a time, never more, and in the first square we have the addition of every two units that can come together; so also in subtraction.

In the second square, the units correspond with the first square, and have an additional ten.

In the third square, the units again are repeated, and another additional ten.

As a column of tens, hundreds, and every higher or lower order is added and subtracted in the same way, the above table develops every principle of addition and subtraction.

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422; 22+ 6 = 28; 28 + 5 = 33; 33 + 3 = 36.

REM.-Although many numbers may be added together, in performing the operation only two at a time are added.

Add the following numbers jointly and separately;

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The sum of the column of units is 21; that is, 1 unit and 2 tens; the sum of the column of tens is 20; that is, 20 tens or 2 hundred; and the two sums united make

221; precisely the same as if the column of units is first added, and the units of the sum placed under the column. of units, and the tens added with the column of tens; and then the tens of the sum of the tens column placed under the column of tens, and the hundreds in place of hundreds.

COR. As the relation of each successive order is the same, hence for every ten of any order, the 1, or lefthand figure, belongs to the next order; and the process is the same in the addition of every column; that is, one is carried to the next column for every ten in the addition of each column.

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COR. 1.-The minuend is always equal to the sum of the subtrahend and remainder, and is therefore greater than either.

COR. 2.-Arithmetic is based upon the postulate contained in 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, which is addition, etc.; and the application of Axiom 6 (page 9) to this postulate proves the principle of subtraction; thus, 19 +6 25, then 25 - 6 must be equal to 19.

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REM. 1.-When the figure of the subtrahend is larger than the one above it of the same order of the minuend, 1 of the next order of the minuend must be united to the figure of the minuend and then the subtraction be performed; then in order to make up for this addition to the minuend, 1 must be added to the next order of the subtrahend, and then perform the subtraction; this process is called carrying and requires all the attention of the student.

REM. 2.-I prefer few examples, but these may be often repeated, and if thought necessary the teacher can give others in which the columns are longer.

REM. 3.-Each order may be regarded as units, and the sum may reach one, two, or more hundred of its order.

QUESTIONS.

1. When the sum of the column of units is 157, where do you place the 7, and what do you do with the 15?

2. When the sum of the column of tens and of the tens carried from the column of units is 246, what do you do with the 6, and what with the 24 ?

3. In what does the process of the other columns. differ from these?

REM. 1.-Let the class be thoroughly exercised in subtraction, especially in the matter of carrying from one order to another; one example is sufficient to impress it on the mind, as the solution may be repeated as often as is necessary.

REM. 2.-The teacher can add other questions and larger examples if he think proper.

PRACTICAL EXAMPLES.

1. A farmer has 7 cattle in one field and 8 in another; if he take 4 from the first field and put them in the second, how many will there then be in each field?

2. In one field there are 435 cattle and in another 657; if 320 be taken from the first field and put in the second field, how many cattle are then in each field?

REM.-In subtraction the larger number is the minuend and the smaller number the subtrahend.

3. A merchant bought dry goods to the amount of 1263 dollars, groceries for 734 dollars, hardware for 231 dollars, and notions for 137 dollars; what is the amount of his purchases?

4. A farmer sold a horse for 175 dollars, cows for 97 dollars, and sheep for 51 dollars; what was the amount of his sales?

5. A gentleman owns five farms; the first is worth 10600 dollars, the second 4970 dollars, the third 5000 dollars, the fourth 6500 dollars, and the fifth 8500 dollars; the amount of his indebtedness is 7984 dollars; if the whole is disposed of at the above rates and the debt paid, what sum will he then have?

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