28. Newspapers were first printed at Paris, in France, in 1631; how many years since, up to the year 1832? Ans. 201 years.

29. James is 19 years old, and William is 12; how much older is James than William? Ans. 7 years.

30. A merchant had in bank 4567 dollars, and drew out 3435 dollars; how much money had he remaining in bank? Ans. 1132 dollars.

31. Gunpowder was invented in the year 1302; how many years since, up to the year 1832? Ans. 530 years.

32. A man bought four barrels of flour for 24 dollars, and sold them for 28 dollars; how many dollars did he make by the sale? Ans. 4 dollars.

33. A lady went to a store with 26 dollars. She bought a dress for 12 dollars, and a shawl for 8 dollars; how many dollars had she left? Ans. 6 dollars.

Q. When a figure in the lower line, or subtrahend, is greater than the figure in the upper line, or minuend, how must it be substracted?

Á. Add ten to the figure in the upper line, or minuend, and then substract the figure in the lower line, or subtrahend, from the amount.

Q. What must you carry to the next figure in the lower line, for the ten which you borrowed and added to the figure in the upper line?

A. One must be added, or carried, to the next figure in the lower line.

Q. Why do you carry one to the next figure for the ten which you borrowed and added to the figure in the upper line?

A. Because ten in any one column is only equal to one in the next column at the left hand.

EXAMPLES

For Exercise on a Slate.

1. William had 183 dollars, and paid 69 dollars for a horse; how many dollars had he left? Ans. 114 dollars.

EXPLANATIONS.

183 minuend.

69 subtrahend.

Beginning with the 9, the first figure of the subtrahend, you will perceive that it is larger than the figure 3 above it in the minuend. 114 difference. You cannot take 9 from 3, therefore you must add ten to the 3 in the upper line, which increases it to thirteen, and then say, 9 from 13 leaves 4. You must set down the 4 under the 9, and carry one to the 6, for the ten which you borrowed, and added to the 3 in the upper line, and say, 7 from 8 leaves 1. You must set down the 1 under the 6, and as there is no figure in the place of hundreds in the subtrahend, and, consequently, nothing to be taken from the 1 in the minuend, you must set down the 1 in the place of hundreds; and thus you will find that he had 114 left, which is the difference between 183 and 69.

The principle of this operation will be perfectly plain to you if you remember that the 1 ten which you added, or carried, to the 6 in the subtrahend is equal to the ten units which you added to the 3 in the minuend; for 1 in a superiour column is equal

to 10 in an inferiour column; and it is also evident, that if you add equal sums to two numbers, or if you substract equal sums, the difference between the two numbers must always be the same.

2. A merchant bought a quantity of goods for 8143 dollars, and sold them for 6584 dollars; how many dollars did he lose by the sale? Ans. 1559 dollars.

EXPLANATIONS.

8143 minuend. 6584 subtrahend.

1559 difference.

Beginning with the 4 at the right hand in the subtrahend, you will say, 4 cannot be taken from 3; and, therefore, you must add ten to the 3 which makes it thirteen, and you will then say, 4 from 13 leaves 9, which you must set down under the 4; and proceeding to the next figure, you must say, 1, which you borrowed, added to 8 makes 9; and as 9 is greater than 4, adding ten as before, you must say, 9 from 14 leaves 5, which you must set down under the 8; and proceeding to the next figure, you must say, 1, which you borrowed, added to 5 makes 6; and as 6 is greater than 1, adding ten as before, you must say, 6 from 11 leaves 5, which you must set down under the 5; and proceeding to the fourth column, you must say, 1, which you borrowed, added to 6 makes 7, and 7 from 8 leaves 1; and you will then have 1559, the difference between 8143 dollars and 6584 dollars.

You will easily see that by this method the numbers are increased alike, because, in the above example, the 1 which you added to the 8 in the lower line being in the place of tens, is equal to the ten units which you added to the 3 in the upper line; the 1 which you added to the 5 in the lower line being in

the place of hundreds, is equal to the ten tens which you added to the figure 4 in the upper line; and, lastly, the 1 which you added to the 6 in the lower line being in the place of thousands, is equal to the ten hundreds which you added to the 1 in the upper line; and it is also evident, since the numbers added to each of the proposed numbers are equal, that in substracting the one of these equal numbers from the other, nothing will remain.

3. Take 678003 from 4306052.

EXPLANATIONS.

4306052 min.

678003 sub.

3628049 dif.

Beginning with the 3 at the right hand in the lower line; as the 3 is larger than the 2, you must add ten to the 2, which makes it twelve, and say, 3 from 12 leaves 9, which you must set down under the 3; and, proceeding, you must add 1 to the next figure, which is a cipher, and say, 1 from 5 leaves 4, which you must set down under the cipher; proceeding to the next cipher, you must say, cipher from cipher leaves nothing, and you must set down the cipher; and, proceeding to the figure 8, you will say, 8 cannot be taken from 6, and you must, therefore, add ten to the 6 which makes sixteen, and say, 8 from 16 leaves 8, which you must set down under the 8; and, proceeding, you must add 1 to the figure 7 which makes 8, and you will then say, 8 cannot be taken from cipher, and you must, therefore, add ten to it, and say, 8 from 10 leaves 2, which you must set down under the 7; and, proceeding, you must add 1 to the figure 6 which makes 7, and you will then say, 7 cannot be taken from 3, and you must, therefore, add ten to the 3, which makes thirteen, and say

7 from 13 leaves 6, which you must set down under the 6; and as there is no other figure in the subtrahend, you must say, 1, which you borrowed, taken from 4 leaves 3, which you must set down under the vacant place in the subtrahend, under the 4 which is in the minuend, and then the work is done.

Note.-TO TEACHERS. There is, perhaps, not an operation in the whole Arithmetick which is so imperfectly understood by the learner as that of "BORROWING" and "ADDING" ten to the figure in the upper line or minuend, when the figure in the lower line, or subtrahend, is greater than the one in the minuend. Great pains should be taken, therefore, that the learner be thoroughly acquainted with the principles of the operation. For this purpose let him pay particular attention to the preceding "EXPLANATIONS."