50. A man, having three sons, gave 50 dollars to the oldest, 40 dollars to the second, and 30 dollars to the youngest: how many dollars did he give to the three?

17. The learner will perceive that the solution of each of the preceding examples, consists in finding a single number which will exactly express the value of the several given numbers united together.

18. The process of uniting two or more numbers together, so as to form one single number, is called ADDITION.

The answer, or the number thus found, is called the

sum or amount.

OBS. When the numbers to be added are all of the same denomination, as all dollars, or all pounds, &c., the operation is called Simple Addition.

19. Signs-Addition is often represented by the sign (+), which is called plus. It consists of two lines, one horizontal, the other perpendicular, forming a cross, and shows that the numbers between which it is placed, are to be added together. Thus the expression 6+8, signifies that 6 is to be added to 8. It is read, "6 plus 8," or "6 added to 8."

Note.-Plus is a Latin word, originally signifying "more," hence "added to."

20. The equality between two numbers, or sets of numbers, is expressed by two parallel lines (=), called the sign of equality. It shows that the numbers between which it is placed, are equal to each other. Thus the expression 5+3=8, denotes that 5 added to 3 are equal to 8. It is read, " 5 plus 3 equal 8," or "the sum of 5 plus 3 is equal to 8." So 7+5=8+4=12.

QUEST.-17. In what does the solution of the preceding examples consist? 18. What is addition? What is the answer called? Obs. When the numbers to be added are all of the same denomination, what is the operation called? 19. What is the sign of addition called? Of what does it consist? What does it show? Note. What is the meaning of the word plus? 20. How is the equality between two numbers represented? What does the sign of equality show? How is the expres sion 5+3=8, read? How, 7+5=8+4=12?

EXERCISES FOR THE SLATE.

21. Examples in which the numbers to be added are small, should be solved mentally; but when the numbers are large, the operation may be facilitated by setting them down upon a slate, or black-board. The manner of doing this will now be explained.

Овя. Pupils not unfrequently seem to infer, that when they take up the slate and pencil, they can lay aside thinking; that the hands are to solve the question without the aid of the intellect. Hence operations upon the slate are often a merely mechanical effort, as listless and mindless as the talking of a parrot, or the trudging of a dray-horse. This is a sad mistake. It is sure to render the study of arithmetic irksome, and to destroy the progress of the learner.

It is not the object in using the slate to supersede thinking and reasoning, but to assist the memory in retaining the numbers and the several steps of the operation, while the intellect is carrying on the process of thinking and reasoning.

The hands simply write down the figures or the result of the operation, but it is the mind, and the mind only, that performs the addition and all other arithmetical calculations, whether we use the slate or not. Hence, whoever wishes to become a proficient in arithmetic, must never allow his mind to become inactive when using his slate, nor pass a single solution without understanding the reason of the several steps.

Ex. 1. A man bought a pound of tea for 63 cents, and a pound of coffee for 24 cents: how much did he pay for

both?

Directions. -Write the numbers under each other, so that units may stand under units, tens under tens, and draw a line beneath them. Then, beginning at the right hand or units, add each column separately in the following manner: 4 units and 3 units are 7 units.

→ tens

units

Operation.

6 3 price of tea.
24
66 of coffee.
8 7 cts. pr. ce of both.

QUEST.-21. How should examples, in which the numbers to be added are small, be solved? When they are large, how may the operation be facilitated? Obs. Is the slate designed to supersede thinking and reasoning? What is its use? How are all arithmetical calculations performed? What direction is given to those who wish to become proficients in arithmetic?

Write the 7 in units' place, 2 tens and 6 tens are 8 tens. The amount is 87 cents.

under the column added. Write the 8 in tens' place.

Note. The learner will perceive that the operation upon the slate is essentially the same as the mental solution of the same question; (Art. 16. Ex. 41 ;) and that both give the same result.

2. A butcher purchased two droves of sheep, the first containing 436, and the second 243: how many sheep did both droves contain?

Write the numbers under each other, and proceed as before. Thus, 3 units and 6 units are 9 units; 4 tens and 3 tens are 7 tens; 2 hundreds and 4 hundreds are 6 hundreds. The amount is 679.

Operation.

436 First drove. 243 Second "

679 Ans.

22. It will be perceived, from these examples, that units are added to units, tens to tens, and hundreds to hundreds; that is, figures of the same order are added to each other. This is the only way numbers can be added. For, figures standing in different orders or columns, express different values; (Art. 8;) consequently, if united together in a single sum, the amount can neither be of one order nor another. Thus, 3 units and 3 tens will neither make six units, nor six tens, any more than 3 oranges and 3 apples will make 6 apples, or 6 oranges. In like manner it is plain that 4 tens and 4 hundreds will neither make 8 tens, nor 8 hundreds.

OBS. In writing numbers to be added, great care should be taken to place units under units, tens under tens, &c., in order to prevent mistakes which would otherwise be liable to occur from adding different orders to each other.

3. A man found two purses of money, one containing 425 dollars, the other 361 dollars: how many dollars did both purses contain?

QUEST. In the 1st example how do you write the numbers for addition? Which column do you add first? Which next? Note. Does the operation upon the slate differ from the mental solution of the same question? 22. Can figures standing in different orders be added to each other? Why not? Illustrate by an example. Obs. What is the object in writing units under units, &c. ?

4. What is the sum of 3261 and 5428?

5. What is the sum of 45436 and 12321?

6. What is the sum of 420261 and 231204? 7. What is the sum of 3021040 and 5630721? 8. What is the sum of 730043000 and 268900483? Write the following examples upon the slate, and find the sum of each:

23. When the sum of a column does not exceed 9, it must be written, as we have seen, under the column added. But when the sum of a column exceeds 9, it requires two or more figures to express it; (Art. 7;) consequently, it cannot all be written under the column added. What then must be done? We will now illustrate this case.

13. A man paid 98 dollars for a horse, and 65 dollars for a wagon: how much did he pay for both?

Operation.

98 price of horse. 65 66 of wagon.

163 Amount.

Hence

Directions. Write the numbers, and begin at the right hand, as before. Thus 5 units and 8 units are 13 units. Now 13 is 1 ten and 3 units, and requires two figures to express it; (Art. 7;) consequently it cannot be written under the column of units. we write the 3 units in the units' place, and reserving the 1 ten or left hand figure in the mind, add it with the tens in the next column. Thus 1 ten (which was reserved) and 6 tens are 7 tens, and 9 are 16 tens, which are equal to 1 hundred and 6 tens. Write the 6 tens under the col umn added, and the 1 hundred in the place of hundreds The amount is 163 dollars.

QUEST.-23. When the sum of a column does not exceed 9, where is it written? Can the whole sum be written under the column when it exceeds 9? Why not? In the 13th example, what is the sum of the units' column? How do you dispose of it? What do you do with the sum of the next column?

OBS. It will be perceived that the operation upon the slate is substantially the same as the mental solution of the same question. (Art. 16. Ex. 40.) In each case, we add the orders separately; in each, finding the sum of the units' column to be 13, or 1 ten and 3 units, we add the 1 ten to the number of tens which is contained in the example; and in each we obtain the same result.

14. A gentleman bought a span of horses for 645 dollars, a carriage for 467 dollars, and a set of harness for 158 dollars: how much did he give for the whole establishment?

Operation. 645 price of horses. 66 carriage.

467

158

66 harness.

1270 dollars. Ans.

Proceed as before. Thus 8 units and 7 units are 15 units, or we simply say, 8 and 7 are 15, and 5 are 20. Set the 0 under the column added, and, reserving the 2, add it with the next column. 2 (which was reserved) and 5 are 7, and 6 are 13, and 4 are 17. Set the 7 under the column added, and add the 1 with the next column. 1 (which was reserved) and 1 are 2, and 4 are 6, and 6 are 12. Set the 2 under the column added, and since there is no other column to be added, write the 1 in the next place on the left. The amount is 1270 dollars.

24. The process of reserving the tens or left hand figure, when the sum of a column exceeds 9, and adding it mentally to the next column, is called carrying tens.

25. When the sum of the column exceeds 9, set the units or right hand figure under the column added, and carry the tens or left hand figure to the next column. In adding the last column on the left, it will be noticed we set down the whole sum. This is done for the obvious reason that there are no figures in the next column to which the left hand figure can be added, and is in fact carrying it to the next order.

QUEST. Obs. Does the operation upon the slate differ essentially from the mental solution of the same example? In what respects do they coincide? 24. What is the process of reserving the tens and adding them to the next column, called? 25. When the sum of any columa exceeds 9, what is to be done with it? When the sum is 20, what do you set down, and what do you carry? If 27, what?