How do you prove addition? A. By adding the columns downwards. Why should that prove it? A. It is putting together the same numbers that composed the first amount, only in a different order. How do you write down Decimal or Federal money for adding? A. Dollars under dollars, cents under cents, and mills under mills: After your numbers are written down, how do you proceed in adding? A. The same as in whole numbers. Why do you add the same? A. Because one in a superiour column, is equal to ten in the next inferiour column, the same as in whole numbers. In federal money what are the figures called at the left hand of the separatrix? A. Dollars.What are the two next at the right called? A. Cents. What is the next at the right called? A. Mills. If a denomination be wanting what do you write in its place? A. A Cipher or ciphers to fill the vacant places. Do ciphers at the right hand of a decimal alter its value? A. They do not. When you have the value of a half given, how do you find the value of the whole? A. Add the given sum to itself and the amount will be the value of the whole. If you have the value of one third given, how do you find the value of the whole? A. Set down. the value of the third three times and add; the sum will be the value of the whole.

SIMPLE SUBTRACTION,

Is taking a less number from a greater. The greater number is called Minuend, and the less the Subtrahend. The number produced from the operation of the work, is called the remainder or difference.

Subtraction being exactly the reverse of addition, your knowledge of that rule will be of great service in this; you have already learned by addition that 4 and 3 make 7; and now you will readily perceive that if either be taken away from 7, the other will remain. You also understand that addition is collecting numbers together: you must now learn that subtraction is taking them apart.

RULE.--Place units under units, tens under tens, &c. as in addition, with the greater number above. Draw a line under them.--Begin at the right hand, and subtract the units of the subtrahend from those of the minued, and write the difference directly below. Proceed in like manner with the remaining figures at the left. But if a figure in the subtrahend exceed that above it in the minuend, you must add 10 to the upper figure and from the amount take the figure below; remembering to add 1 to the next figure of the subtrahend; this is adding equals to both the given numbers, for the convenience of work; and adding equals to both numbers, their difference must ever remain the same.

Subtraction is denoted by a horizontal line; thus, 6-3÷3 signifies that the difference between 6 and 3, is 3.

PROOF. 1st. Method. Add the difference between the

given numbers to the subtrahend, and if the amount equal the minuend, the work is right. 2nd. Method. Or you may subtract the difference from the minuend, and if the remainder equal the subtrahend, the work is right.

SUBTRACTION TABLE.

2 from 2 leaves 04 from 4 leaves 06

2

∞∞∞∞ co co co.co/ 10.10 10 10 10 10 10 10 10 10 10

3

8

9;

3 from 3 leaves 015

3

3

3

པ་

67

3

11

8

1

18

9

1

19 from 9 leaves 0

11

12

دو

9

6 from 6 leaves 09

The student should be required mentaily to answer the following questions. If there be a class, let each student answer his question directly after reading it.

1. John having 9 cents, paid 6 cents for apples; how ma2 ny had he left?

2. George having 11 oranges, gave 4 to his mother ; how many had he left?

3. A man paid 15 dollars for a cow, and sold her for 20 : how much did he gain by the trade?

4. Charles bought a knife for 15 cents, but soon became sick of his bargain, and sold it for 12 cents; how much did he lose?

5. Charles gave Henry 24 cents, George gave him 20: how many did Charles give more than George? How many are left after taking 20 from 24?

6. Joseph has 16 peaches, Henry 4, and George 24; how many has George more than the other two?

7. Henry bought an English Reader for 30 cents, and at writing book for 20; but George offers for his writing book what his English Reader cost him; how much will Henry gain by selling?

8. Charles carried 24 eggs to market, and sold them for 8 oranges; how many more eggs had he than oranges? 1. Examples to be performed on the Slate.

From 4 6 Minuend.
Take 3 2 Subtrahend.

DEMONSTRATION.Commencing at the right hand as the rule directs

1 4 Remainder or difference. we say 2 from 6 leaves 4,

placing it directly under; because it is plain if 2 units be taken from 6 units, there will remain 4 units. Then we say 3 from 4 leaves 1, placing it directly under; for it is evident, if 3 tens be taken from 4 tens there will remain 1 ten; consequently we have 14 for the difference between 46 and 32.

PROOF. According to the 1st method.

46 DEMONSTRATION.-Nothing can be more plain, than the 32 proof of this rule. Because it is clear, that it can take no more than the difference between two numbers, to make the less e1 4 qual to the greater. Consequently when we add 14, the difference between 46 and 32, to 32 the less, we find the amount 46 to be 46, a number equal to the greater.

46

PROOF. According to the 2nd method.

DEMONSTRATION.-You already understand, that the dif32 ference between two uumbers when added to the less, gives a sum equal to the greater, and you will now readily perceive, 14 that if the difference between two numbers be taken from the

greater, it must leave a number equal to the less. Because it 3 2 can take no more than the difference between two numbers,

when taken from the greater, to diminish it, so that, it shall be equal to the less; for you perceive when we take 14 from the minuend, it is reduced to 32, a number equal to the subtrahend. Consequently subtraction may be made to prove itself.

NOTE. We might very properly have offered subtraction, as a proof of addition, had you been acquainted with the rule. It is a saying of the school boy, that it is a poor rule that will not work both ways. We have already proved subtraction by addition; then to make the rule good according to the test of the school boy, we will prove addition by subtraction. Let 46 and 32 be added, the amount is 78; now it is plain, if the numbers which compose this amount, be taken away, nothing will be left; thus, if from 78 you subtract 46, the first number, 32, will remain, and if 32, the other number, be taken away from 32 the remainder; nothing will remain. Now you may treasure up in your mind this fact; that if you subtract from the amount of any sum, the several numbers that compose the amount; you destroy it or reduce it to nothing.

2. From 4 2 Take 2 4

18

In this example, we find that the 4 units of the subtrahend cannot be taken from the 2 in the minuend; therefore we add 10 to the 2 which increases it to 12, then we say 4 from 12 leaves 8, placing it under. Then we add 1 to the 2, the next figure of the subtrahend which increases it to 3, we then say 3 from 4 leaves 1, placing it directly under, which leaves 18 for the difference of the 2 given numbers.

DEMONSTRATION.-The reason of this operation is plain when we recollect that the 1 ten which we ad to the subtrahend is equal to the 10 units which we add to the minuend; because 1 in a superiour column is equal to 10 in an inferiour column; and adding equal sums to two numbers, or subtracting equal sums, the difference between the two numbers must ever remain the same; thus,

NOTE.-Adding 10 to a figure in the minuend before we can subtract and then one to the next left hand figure of the subtrahend, is by some called borrowing.

3. Again, to show the principles of this rule in a different, though if possible, in a clearer light, we introduce the following example.

From 5 4
Take 2 6

(3)

DEMONSTRATION.-When no figure in the subtrahend, is greater than that directly above it in the minuend; the student finds no difficul28 difference. ty. And even when a figure in the subtrahend, is greater than that directly above it in the minnend, the difficulty vanishes, when he properly understands the local value of numbers. In this example our minuend (54) consists of 5 tens and 4 units; and our subtrahend (26) consists of 2 tens and 6 u

nits. Now to take the 6 units of the subtrahend from the 4 units of the minuend is impossible; yet it is evident that the minuend (54) is greater than the subtrahend (26;) therefore, we will resolve our minu end, 5 tens and 4 units, into 4 tens and 14 units, which can be evidently done. And now we can take the 6 units of the subtrahend from the 14 units of the minuend and have 8 units remain; we then take the 2 tens of the subtrahend from the 4 tens of the minuend and have 2 tens left, then our remainder or difference is 2 tens and 8 units, or 28, which is the same.

Although further examples may seem unnecessary to illustrate this rule; yet as the following example includes every variety in subtraction, it may not be thought amiss to introduce it here.

(4)

From 3 2 0 50 41
Take 78700 2

2 4 1 8 0 3 9 Rem.

We begin at the right hand as our rule directs; and since we cannot take 2 from 1, we add 10 to the 1, which makes 11, and then say, 2 from 11 leaves 9, placing it directly under. We then add 1 to the next left hand figure of the subtrahend (for reasons already given under the 2nd example) which is a cipher (0,) and say 1 from 4 leaves 3, placing it under. Next we say take a cipher (0) from a (0) and a cipher (0) remains. We next say take 7 from 5 we cannot, but according to our rule, we add 10 to the 5 and then say, 7 from 15 leaves 8, placing it directly under. Next, according to our rule, (which has been explained,) we add 1 to the 8, which makes 9 and then say, 9 from a cipher (0,) we cannot; but when we add 10 to the cipher (0,) we say 9 from 10 leaves 1, placing it under. We now add 1 to the 7 which makes 8, and say, 8 from 2 we cannot subtract; we then add 10 to the 2, which increases it to 12, we then say 8 from 12 leaves 4, placing it under. Lastly, we say 1 from 3 leaves 2, placing it under. We obtain this 1 in the vacant place of the subtrahend at the left hand, on account of our having added 10 to the next right hand figure of the minuend.

5. What is the difference between 478 and 320 ?

Ans. 158.

6. If 1 be taken from a 1000; what will remain ?

Ans. 999.

7. If one be subtracted from a million; what will remain ? Ans. 999,999.

S. What is the difference between one million and ten thousand? Ans 990,000.

9. At the census taken in 1810, the number of inhabitants in the six N. England States, was 1,471,973; and at the census taken in 1820, the number of inhabitants was 1,659,854;