NOTES.-I. Repeating a letter repeats its value. Thus, I denotes one; II, two; III, three; X, ten; XX, twenty, etc.

2. Placing a letter of less value before one of greater value, diminishes the value of the greater by that of the less; placing the less after the greater increases the value of the greater by that of the less. Thus, I denotes one, and V five; but IV is four, and VI six.

3. Placing a horizontal line over a letter increases its value a thousand times. Thus, I denotes a thousand; X, ten thousand; C, a hundred thousand; M, a million.

4. Four was formerly denoted by IIII; nine, by VIIII; forty, by XXXX; and ninety, by LXXXX.

Express the following numbers by letters:

Numeration is the art of reading numbers expressed by figures, letters, or other numeral characters. 26. How read numbers expressed by figures

Divide them into periods of three figures each, counting from the right.

Beginning at the left hand, read the periods in succession, and add the name to each, except the last.

27. Why is the name of the last period omitted?

Because the right hand period always denotes simple units, therefore its name need not be mentioned.

Repeat the Numeration Table, beginning with units.

NUMERATION TABLE.

The periods in the Table are thus read: 298 trillions, 570 billions, 923 millions, 874 thousand, two hundred and sixty-seven.

NOTE.—This method of reading numbers is commonly ascribed to the French, and is thence called the French Numeration. Others ascribe it to the Italians, and thence call it the Italia, Method.

Copy and read the following:

Express by figures, and read the following numbers: 1. Two hundred and five thousand, six hundred, and ninety-one.

2. Eight hundred and forty thousand, five hundred, and nine.

3. Two millions, four hundred thousand, and seventy. 4. Forty-five millions, sixty thousand, two hundred, and sixty.

5. Three hundred and ninety millions, four thousand, a seventy-two.

6 Six hundred millions, forty-eight thousand, and ten. 7. Five billions, six hundred and ten millions, and three hundred.

8. One hundred and forty billions, and thirty-five millions.

9. Forty-five millions, seven hundred and sixty thou

sand.

10. Three hundred and twenty-nine trillions, six hundred and thirty-seven billions, three hundred and forty millions, four hundred and nineteen thousand, two hundred and eighty-four.

NOTE.-Dictation exercises in reading and writing numbers should be continued, till the class is perfectly familiar with both.

ADDITION.

MENTAL EXERCISES.

TO TEACHERS.-The object of this Exercise is to teach beginners the process of adding two digits together. If young, let them illustrate the examples by counters or unit marks.

1. If you have 1 apple, and I give you 1 apple more, how many apples will you have?

"One apple and 1 apple more are 2 apples."

2. If you have 2 cents, and you find 1 more, how many cents will you have?

"Two cents and 1 cent more are 3 cents."

3. How many are 3 marbles and 2 marbles?

4. Show this by your fingers.

5. Sarah has 3 red roses, and 3 white ones: how many roses has she of both kinds? Show it.

6. If an orange costs 5 cents, and a lemon 4 cents, how much will both cost? Show it.

7. William earned 6 cents in the morning, and 4 in the afternoon: how much did he earn in both?

8. Sanford obtained 6 credit marks, and his sister 6: how many did both obtain? Show it.

9. If I gather 6 quarts of cherries, and buy 8 quarts, how many quarts shall I have? Show it.

10. Joseph picked 5 quarts of blackberries, and his brother 7 quarts: how many quarts did both pick? 11. If I pay 8 dollars for a barrel of flour, and 7 dollars for a ton of coal, what shall I pay for both?

12. A teacher received two bouquets, one containing 8 flowers and the other 10: how many flowers were there in both?

13. George picked 12 peaches from one tree, and 3 from another: how many did he pick from both?

1. Show by counters or unit marks how many 2

added to 3 will make.

2. Show how many 2 added to 4 will make.
3. Show how many 2 added to 5 will make.
4. Show how many 3 added to 4 will make.
5. Show how many 3 added to 5 will make.
6. Show how many 4 added to 5 will make, etc.

Particular care should be taken to see that beginners understand how the Addition Table is constructed; that they fully comprehend the results of adding two digits together, before they are required to commit them to memory.