10. Three hundred six thousand, five hundred and two.

11. Nine hundred forty-six million, five hundred fourteen thousand, nine hundred and twenty-five.

12. Six billion, fifteen million, seven thousand, and four hundred. Ans. 6,015,007,400. 13. Five million, six hundred fifty-one thousand, four hundred and six.

14. Seventy-four million.

15. Sixty-three million, fourteen thousand, and seven hundred.

26. By the ENGLISH METHOD OF NUMERATION, the first period contains units, tens, hundreds, thousands, tens of thousands, and hundreds of thousands, and is called the period of units; the second period contains millions, tens of millions, hundreds of millions, thousands of millions, tens of thousands of millions, and hundreds of thousands of millions, and is called the period of millions; etc., as in the following

ENGLISH NUMERATION TABLE.

Hundreds of Thousands of Millions,
Tens of Thousands of Millions,

Thousands of Millions,

Hundreds of Millions,

Hundreds of Thousands,
Tens of Thousands,
Thousands,
∞ Hundreds,

Tens of Millions,
Millions,

4 3.

Tens,

ço Units,

26. By the English numeration what figures are in the first period? Second period' Third? Repeat the table.

27. The value of the figures in this table, is twenty-eight trillion, seven hundred sixty-nine thousand five hundred and forty billion, seven hundred six thousand four hundred and seventy-six million, one thousand eight hundred and forty-three.

28. The names of the figures and their values are the same in the two tables for the first nine places from the right, after which they are alike in value but different in name. A trillion by the English method is much more than by the French.

29. To numerate and read a number according to the English method:

RULE. 1. Beginning at the right, numerate and point off the number into periods of SIX figures each.

2. Beginning at the left, read each period separately, giving the name of each period except that of units.

31. To write numbers by the English method:

RULE. 1. Beginning at the left, write the figures belonging to the highest period.

2. Write the figures of each successive period in their order, filling each vacant place with a cipher.

EXERCISES IN ENGLISH NOTATION AND NUMERATION.

32. Write the following, and read by the English method: 1. Five units of the eighth order, six of the seventh, two of the fourth, and one of the third. Ans. 56,002100.

27. What number is expressed by the table? 28. Are the names of figures alike in the French and English tables? Their values, alike or unlike? 29. Rule for numerating and reading a number by the English method? 31. Rule for writing a number by the English method?

2. Nine units of the fourteenth order, two of the twelfth, three of the eleventh, six of the eighth, nine of the sixth, two of the fifth, and three of the fourth. Ans. 90,230060,923000.

3. Two units of the ninth order, six of the sixth, one of the fifth, two of the third, seven of the second, and five of the first. 33. Express the following numbers by the English Notation: 1. Seventy-two million, six hundred thirteen thousand four hundred and forty-six. Ans. 72,613446.

2. Five hundred seventeen billion, three hundred twenty-two thousand one hundred fourteen million, eight hundred forty-one thousand nine hundred and sixty-nine.

3. Two hundred and ten billion, and six thousand.

NOTE. These and other exercises will be varied and extended by the teacher as circumstances may dictate.

34. The ROMAN NOTATION, or that used by the ancient Romans, employs seven capital letters to express numbers, viz.: I, V, X, L,

C,

D,

M.

One, Five, Ten, Fifty, One hundred, Five hundred, One thousand. All other numbers may be expressed by combining and repeating these letters.

35. The Roman Notation is based on the following principles:

1st. When two or more letters of equal value are united, or when a letter of less value follows one of greater, the sum of their values is indicated; thus, XXX stands for 30, LXV for 65, CC for 200, MDCLXVII for 1667.

2d. When a letter of less value is placed before one of greater, the difference of their values is indicated; as, IX stands for 9, XL for 40, XC for 90.

3d. When a letter of less value stands between two of greater value, the less is to be taken from the sum of the other two; as, XIV stands for 14, XIX for 19, CXL for 140.

34. How many and what characters are employed in the Roman Notation! What is the value of each? 35. What is the first principle in Roman Notation! Second? Third?

4th. A letter with a line over it represents a number one thousand times as great as the same letter without the line; thus X stands for ten, but X stands for one thousand times ten, i. e. ten thousand; M stands for one thousand, but M for one thousand himes one thousand.

EXERCISES IN ROMAN NOTATION.

36. Express the following numbers by letters:

1. Twelve.

2. Eighteen.

Ans. XII. Ans. XVIII.

3. Twenty-nine.

4. Ninety-nine.

5. Two hundred and eighty-four.

6. One thousand four hundred and forty-six. 7. One thousand six hundred and forty-four.

8. The present year, A. D.

NOTE. The Roman notation is very inconvenient for Arithmetical operations, and the Roman numerals are now seldom used, except for numbering the pages of a preface, the divisions of a discourse, and the sections, chapters, and other divisions of a book.

35. What is the fourth principle in Roman Notation? 36. Are Roman nume rals much used in arithmetical operations? Why? For what are they used?

37. Besides the Arabic and the Roman figures, there are various marks used to indicate that certain operations are to be performed, such, e. g., as the sign of addition,+; the sign of subtraction, ; etc. These signs will be given, and their uses

explained when their aid is needed.

ADDITION.

38. ADDITION is the putting together of two or more numbers of the same kind, to find their sum or amount.

The sum or amount of two or more numbers is a number which contains the same number of units as the two or more numbers put together; thus, 7 is the sum of 3 and 4, because there are just as many units in 7 as in 3 and 4 put together; for a like reason 11 days is the sum of 2 days, 4 days, and 5 days.

Ex. 1. James has 4 marbles, John has 5, and Henry has 3; how many marbles have they all?

To solve this example, add the numbers 4, 5, and 3: thus, 4 and 5 are 9, and 3 are 12; therefore James, John, and Henry have 12 marbles, Ans.

2. How many are 3 and 6? 6 and 3? 2 and 5 and 7?,

39. A SIGN is a mark which indicates an operation to be performed, or which is used to shorten some expression.

40. The sign of dollars is written thus, $; e. g. $2 represents two dollars; $10, ten dollars, etc.

41. The sign of equality,, signifies that the quantities between which it stands are equal to each other; thus, $1100 cents, i. e. one dollar equals one hundred cents.

37. What characters are used in Arithmetic besides the Arabic and Roman Agures? For what?

38. What is Addition? Sum or amount? 39. A sign? 40. Make the sign of dollars on the black-board. 41. Make the sign of equality. What does it mean?