which is 0 units; the whole number is therefore four hundred and fifty-three thousand, two hundred and ten.

By examining this table, it will be seen, that a number containing but one figure can contain units only; two figures, tens and units; three figures, hundreds, tens, and units, &c.

To read any number, we commence on the right and say, units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, &c., until we have passed over all the figures in the number. Thus,

and taking all their values together, we read, five hundred and thirty-nine thousand, eight hundred and sixty-four.

Q. Repeat the numeration table. How is the value of any number determined? If the number 6 fall under thousands, what is its value ? If under tens of thousands? Millions? A number with but one figure contains what? Two figures? Three? How do you read any number? What is the figure on the right? The next? Read 124, 764. 2,004,761 76,030,147. 109,026,374.

EXAMPLES.

I. Write in words upon your slate the population of the six New England States, viz:

1. Maine,

501,793, 4. Massachusetts, 737,699, 2. New Hampshire, 284,574, 5. Rhode Island, 108,830, 3. Vermont, 291,948, 6. Connecticut, - 309,978.

Q. How many New England States? What are they? Which is the largest in population? Which the smallest? What is the population of Massachusetts in round numbers? Rhode Island, &c.? Is your state one of the New England States? What is its population?

II. Write in figures the population of the four Middle States, viz:

1. New York. Two millions, four hundred and twentyeight thousand, nine hundred and twenty-one.

2. New Jersey.-Three hundred and seventy-three thousand, three hundred and six.

3. Pennsylvania.- -One million, seven hundred and twenty-four thousand and thirty-three.

4. Delaware.-Seventy-eight thousand and eighty-five.

Q. How many Middle States? What are they? Which is the largest in population? The smallest? The population of New York in round numbers? Delaware? Is your state one of the Middle States? What is its population?

III. Read the population of the eight Southern States,

viz:

1. Maryland, 470,019, 5. Georgia, 2. Virginia, 1,239,797, 6. Alabama, - |

3. North Carolina, 753,419, 7. Mississippi, 4. South Carolina, 594,398, | 8. Louisiana,

Q. How many Southern States? What are they? Which is the largest in population? Which the smallest? Has North Carolina a greater or less number of inhabitants than South Carolina? Is your state one of the Southern States? What is its population?

IV. Write upon your slate the population of the eight Western States, in figures, viz:

1. Arkansas.-Ninety-seven thousand, five hundred and seventy-four.

2. Tennessee.-Eight hundred and twenty-nine thousand, two hundred and ten.

3. Kentucky.-Seven hundred and seventy-nine thousand, eight hundred and twenty-eight.

4. Ohio.-One million, five hundred and nineteen thousand, four hundred and sixty-seven.

5. Michigan.-Two hundred and twelve thousand, two hundred and sixty-seven.

6. Indiana.-Six hundred and eighty-five thousand, eight hundred and sixty-six.

7. Illinois.-Four hundred and seventy-six thousand, one hundred and eighty-three.

8. Missouri. Three hundred and eighty-three thousand, seven hundred and two.

Q. How many Western States? What are they? Which is the largest in population? The smallest? How many states altogether? Which is the largest in population of the twenty-six? Next? Which is the smallest? Is your state one of the Western States? What is its population?

V. Write in figures the population of the United States upon your slate, viz: Seventeen millions, sixty-three thousand, three hundred and fifty-three. Ans. 17,063,353.

OF THE ROMAN NUMERATION.

10. The above is the most common system of numeration, being used throughout the civilized portion of the world. It is called the Arabic method of numeration, as it was invented by the Arabs.

The Romans invented another system, which is sometimes used. They expressed numbers by the seven capitals of the alphabet, viz:

The other numbers were expressed by various combinations of these letters, after the following manner:

As often as any character is repeated, so many times is its value repeated. Thus II. is two, XX. is 2 tens, or twenty, CC. is two hundred, &c.

C.

M.

66

One million.

A less character before a greater diminishes its value as much as is denoted by the less. Thus, IV. is one from five, or four.

A less character after a greater increases its value as much as is denoted by the less. Thus, VI. is five and one, or six.

A bar

over a character increases it one thousand fold.

V. is five, V. is five thousand.

Q. What is the common method of numeration called? By whom invented? How extensively is it used? Is there any other method? What called? By whom invented? In what does it consist? How many letters are used? What do they express? How are the other numbers expressed? Write the various numbers from one to twenty upon your slate. From 20 to 50. From 50 to 1000. What effect is produced by repeating a character? Suppose a less character come before a greater? After a greater? What effect has a bar over a character?

ADDITION OF SIMPLE NUMBERS.

11. For the purposes of Arithmetic, numbers are divided into two kinds, viz: simple and compound.

Simple numbers are numbers which are composed of the same unit, whether the unit be dollars, pounds, yards, or miles. Thus,

4 dollars, 8 dollars, and 10 dollars, is a simple number. 15 pounds, 30 pounds, and 54 pounds, is a simple number. When the kind of unit is not expressed, the numbers are always simple numbers. Thus, 2, 3, 5, are simple numbers, and they may express so many pounds, or dollars, or other units.

Compound numbers are numbers which are composed of different units. Thus,

4 dollars, 5 cents, and two mills, is a compound number. 6 yards, 2 feet, and 3 inches, is a compound number.

Q. How are numbers divided in arithmetic? What are they? What are simple numbers? Compound? When the kind of unit is not expressed, what kind of numbers are denoted?

12. Addition, Subtraction, Multiplication, and Division, are the four fundamental or ground rules of arithmetic. Every question in arithmetic depends upon one or the other of these rules.

Addition consists in expressing by a single number the total value of several numbers.

Addition of simple numbers consists in expressing by a single number the total value of several simple numbers.

Thus, if John has two marbles, and James 3 marbles, they will have between them 5 marbles. Here, by a single number 5 we have expressed the number of marbles which John and James had between them.

Again-if Mary has 3 pins, Jane 4 pins, and Anna 5 pins, they will have altogether 12 pins.

The number which thus expresses the total value of several other numbers is called the sum. Thus 5 marbles is the sum of 2 marbles and 3 marbles; and 12 pins, the sum of 3 pins, 4 pins, and 5 pins.

Q. What are the four ground rules of arithmetic? Why are they important? In what does Addition consist? In what does addition of simple numbers consist? How do you explain this? What is the number called which expresses the total value of several numbers? What is the sum of 2 and 2? 2 and 3? 2 and 4? 2 and 5? 3 and 4? 3 and 5? 3 and 6? 4 and 5? 4 and 7? 4 and 9? 5 and 9?

13. There is no difficulty in adding numbers composed of a single figure. Before proceeding to the addition of larger numbers, every pupil should be able to add promptly those which contain but one figure.

To indicate that two numbers are to be added together, we use the sign +, which is called plus. Thus 2+3, denotes that 2 and 3 are to be added together, and is read 2 plus 3.

To indicate that two numbers are equal to each other we use the sign, which is called the sign of equality. Thus 2+3=5, denotes that 2 and 3 added together are equal to 5, and is read, 2 plus 3, equal to 5.