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WRITTEN EXERCISES. Express the following numbers by the Roman method : 1. Twenty-seven. 2. Seventy-seven. 3. Two hundred and one. 4. Six hundred and fifty-six. 5. One thousand seven hundred and seventy-six. 6. Four thousand seven hundred and fifty-seven. 7. 25007. 8. 206484.

Read the following numbers:

1. LXXVII. 2. MỐC. 3. MDCCCLXXVI. 4, MMMCCCXXXIII. 5. XVDCCXLIV. 6. clxxxviii. 7. xlix 8. xcix.

LUMBERMEN'S NOTATION. 53. Lumbermen in marking lumber employ a modifica tion of the Roman Method of Notation. The first four characters are like the Roman. The others are as follows:

À AI AI AI X X X X X
XXV XVI XVIII XIX X XII Xlll X

XV XVII XVII XVIII X4 X Y

16

18

19

20

23

24

26

26

28

30

40

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Note.-For a full discussion of Arithmetical Language, see the author's Philosophy of Arithmetic.

INTRODUCTION TO ADDITION.

MENTAL EXERCISES.

1. If theru are 7 boys in one class and 6 in another class, how mary boys in both classes ?

SOLUTION.-If there are 7 boys in one class and 6 in another class, in hoth classes there are 7 boys plus 6 boys, which are 13 boys.

2. In a garden there are 9 apple-trees and 10 pear-trees; how many trees in the garden?

3. Anna bought a bonnet for 15 dollars and a cloak for 24 dollars, how much did both cost her?

4. Mary gave 19 apples to one of her schoolmates and 15 to an. other; how many did she give to both ?

5. If a boy gave 75 cents for a pair of skates and 25 cents for a ball, what did they both cost him?

6. A lady gave 10 cents for needles, 14 cents for thread, and 15 cents for muslin; what did she give for all?

7. A farmer sold some wheat for 15 dollars, some corn for 25 dullars, and some oats for 30 dollars; what did he receive?

8. How many are 6 and 21? 9 and 22? 8 and 31 ? 10 und 171 4 and 1877 and 27? 9 and 287 12 and 11? 13 and 11?

9. How many are 6 and 17? 8 and 16? 9 and 189 8 and 19? 7 and 12? 10 and 21 ? 12 and 20? 14 and 187 11 and 21 ?

10. How many are 3 and 14? 4 and 17? 5 and 159 7 and 17? 6 and 16? 8 and 18? 9 and 197 11 and 21? 10 and 20?

11. Count by 2’s from 2 to 20; from 20 to 40; from 40 to 60; from 1 to 21; from 21 to 41; from 41 to 61.

12. Count by 3's from 3 to 21; from 21 to 42; from 1 to 22; from 22 to 43; from 2 to 23; from 23 to 44.

13. Count by 4’s from 4 to 40; from 40 to 80; from 1 to 25; from 25 to 41; from 2 to 30; from 30 to 50; from 3 to 35; from 35 to 55.

14. Count by 5's from 5 to 60; from 1 to 61; from 2 to 62; from 3 to 63; from 4 to 64.

15. Count by 6's, 7's, 8's, 9's, 10's, 11's, and 12's, as in the previous problems.

The uniting of two or more numbers into one sum is called Addi tion. The sign of addition is t, and is read plus.

Find the sum of 3+5+4 6+8+5 8+7+6 10+5+7 13+ 6+ 8 4+3+6 3+9+6 7+9+8 11+6+5 147 of 9 6+7+5 5+7+8 8+9+7 12+7+8 157 877 518+2 479+7 9-7-7+8 12+8+9 16 417 418

SECTION II.

FUNDAMENTAL OPERATIONS.

ADDITION.

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54. Addition is the process of finding the sum of two or more numbers.

55. The Sum of several numbers is a number which con. tains as many units as the numbers added.

56. The Sign of Addition is +, and is read plus. It denotes that the numbers between which it is placed are to be added.

57. The Sign of Equality is' =, and is read equals. Thus, 10=4+6 is read 10 equals 4 plus 6.

Notes.-1. The Sign of Addition consists of two short lines bisecting each other, the one in, and the other perpendicular to, the line of writing.

2. The symbol + was introduced by Stifelius, a German mathematician, in a work published in 1544.

PRINCIPLES.

1. The numbers added must be similar. 2. The sum is a number similar to the numbers added.

CASE I.

OPERATION.

58. To add when the sum of no column exceeds nine units of that column.

1. What is the sum of 34, 23, and 32?

SOLUTION.- We write the numbers so that figures of the same order stand in the same column, draw a line 34 beneath, and begin at the right to add. 2 units and 23 3 units are 5 units, and 4 units are 9 units, which we 32 write under the column of units; 3 tens and 2 tens are 89 5 tens, and 3 tens are 8 tens, which we write under the column of tens. Hence the sum of 34, 23, and 32 is 8 tens and 9 units, or eighty-nine. Hence the following

Rule.-I. Write the numbers to be added so that terms of the same order stand in the same column.

II. Begin at the right, add each column separately, and write each sum under the column which produced it

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12. What is the sum of $21.35, $15.02, and $50.617

Ans. $86.98. 18. What is the sum of $32.16, $21.50, and $24.23?

Ans. $77.89. 14. What is the sum of 2131236, 2314010, and 4520652?

Ans. 8965898. 15. A man gave his son $13.56 and his daughter $46.21; bow many dollars did he give to both? Ans. $59.77

16. Marie's book contains 3516 words, and Jennie's book 4271 words; how many words in both ? Ans. 7787.

17. John takes 3250 steps in going to school, and Henry takes 2518 steps; how many do both take? Ans. 5768.

18. If Mr. Martin rides 3164 miles in a year and walke 513 miles; how far does he travel in a year? Ans. 3677.

19. A house cost $4160, a barn $1702, and a store $2027; required the cost of all.

Ans. $7889. 20. A book cost $2.25, a kite $1.10 and a ball 62 cents; what was the cost of all ?

Ans. $3.97. 21. A little girl's pulse beats 105216 times in a day, and ner mother's pulse beats 93470 times; how many beats do both make in a day?

Ans. 198686. 22. Jobn bought a Higher Arithmetic for $1.38, a Peter son's Familiar Science for $1.60, and a Loomis's Algebra for $2.00; how much did he pay for all ? Ans. $4.98.

23. An army consisting of 24,360 men received at one time a reinforcement of 10,220, and at another time of 6400; of how many men did it then consist ? Ans. 39.980.

CASE II.

OPERATION,

59. To add when the sum of a column exceeds nino units of that column.

1. What is the sum of 475, 384, and 896 ? SOLUTION.-We write the numbers so that terms of the same order stand in the same column, and begin 475 at the right to add. 6 units and 4 units are 10 units, 384 and 5 units are 15 units, which equal 1 ten and 5 units; 896 we write the 5 units under the column of units, and

1755 sdd the 1 ten to the column of tens: 1 ten and 9 tens are 10 tens, and 8 tens are 18 tens, and 7 tens are 25 tens, which equal 2 hundreds and 5 tens; we write the 5 tens under the column of tens, and add the 2 hundreds to the column of hundreds : 2 hundreds and 8 hundreds are 10 bundreds, and 3 hundreds are 13 hundreds, and 4 hundreds are 17 hundreds, which equal 1 thousand and ? hundreds; we write the 7 hundreds under the column of hundreds, and place the 1 thousand on the left in the place of thousands. Hence, the sum of 475, 384, and 896 is 1755.

Rule.-I. Write the numbers to be added so that terms of the same order stand in the same column, and draw a line beneath

II. Begin at the right, add the terms of each column separately, and write the sum underneath, if less than ten.

III. When the sum of any column is len or more than ten, write the units figure only, and add the tens to the nex column.

IV. Write the entire sum of the last column. NOTE.-In adding dollars and cents, dollars must be written under do lars and cents under cents, so that the points may be in a vertical line.

Proof.—Begin at the top and add the columns downward and if the work is correct the two sums will be equal.

SECOND METHOD. Separate the number into two or more parts, add these parts, and then add the sum of these parts and if the wnia 18 correct the two results will be equal.

NOTES.-1. We write figures of the same order in the same column fui onvenience of adding, since only units of the same order can be dinactly added.

2. We begin at the right to add for convenience, so that when the sum of any column exceeds $ we may add the left hand term of such sum to the next column.

WRITTEN EXERCISES. %. What is the sum of 4326+7428+4675+3857+1231 ?

Ans. 21517

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