5. Write 9 units of the 3d order, with 6 of the 2d, and 1

of the 1st.

6. Write 0 units of the 2d order, 8 of the 1st, with 4 of the 3d, and 7 of the 4th.

of the 6th order, 7 of the 4th, 9 of the

and 1 of the 1st.

7. Write 8 units 5th, 0 of the 3d, 2 of the 2d, 8. Write 8 units of the 8th order, 6 of the 7th, 0 of the 1st, 3 of the 2d, 4 of the 3d, 9 of the 4th, 0 of the 6th, and 2 of the 5th.

9. Write 4 units of the 10th order, 8 of the 7th, 3 of the 9th, 2 of the 8th, 0 of the 6th, 3 of the 1st, 6 of the 2d, 0 of the 3d, 1 of the 4th, and 2 of the 5th.

10. Write 3 units of the 2d order, 2 of the 1st, 9 of the 3d, 0 of the 4th, 9 of the 9th, 6 of the 8th, 7 of the 7th, 0 of the 6th, and 4 of the 5th.

11. Write 3 units of the 11th order, 0 of the 10th, 8 of the 4th, 0 of the 5th, 2 of the 6th, 0 of the 7th, 3 of the 8th, 4 of the 9th, 1 of the 3d, 2 of the 2d, and 3 of the 1st. 12. Write 3 units of the 12th order, 6 of the 11th, 3 of the 8th, 7 of the 6th, 2 of the 4th, and 1 of the 2d.

13. Write 5 units of the 13th order, 8 of the 12th, 0 of the 9th, 6 of the 7th, 8 of the 3d, and 12 of the 1st.

14. Write 7 units of the 14th order, 5 of the 13th, 6 of the 12th, 5 of the 10th, 7 of the 8th, 9 of the 6th, 5 of the 4th, and 8 of the 1st.

15. Write 9 units of the 15th order, 4 of the 13th, 8 of the 9th, 2 of the 6th, 7 of the 3d, and 2 of the 2d.

16. Write 6 units of the 16th order, 9 of the 12th, 7 of the 9th, 4 of the 7th, 0 of the 6th, 8 of the 4th, 9 of the 5th, and 2 of the 2d.

17. Write 8 units of the 20th order, 5 of the 18th, 6 of the 13th, 4 of the 11th, 9 of the 9th, 1 of the 17th, 4 of the 5th, and 9 of the 3d.

10th order, 5 of the 8th, 9 of

18. Write 6 units of the the 3th, 0 of the 4th, and 1 of the 1st

19. Write 9 units of the 18th order, and then diminish the figure of each order by 1 till you come to and include 0; then increase the figure of each order by 1, till you reach the first order; and then read each order.

Numeration Table

7th Period. 6th Period. 5th Period. 4th Period. 3d Period. 2d Period. 1st Period, Trillions.

Quintillions. Quadrillions.

Billions.

Millions.

Thousands.

Units.

NOTES.-1. Numbers expressed by more than three figures are writ ten and read by periods, as shown in the above table.

2. Each period always contains three figures, except the left-hand period, which may contain one, two, or three figures.

3. The unit of the first, or right-hand period, is 1; of the second period, 1 thousand; of the third, 1 million; of the fourth, 1 billion; and so on, for periods, still to the left.

4. To Quintillions succeed Sextillions, Septillions, Octillions, Nonillions, Decillions, Undecillions, Duodecillions, &c.

5. The pupils should be required to commit, thoroughly, the names of the periods, so as to repeat them in their regular order from left to right, as well as from right to left.

6. Formerly, in the English Notation, six places were given t Milliona. They were read, Millions, Tens of Millions. Hundreds of Millions, Thousands of Millions, Tens of Thousands of Millions, Hundred, of Thousands of Millions. This method produced great irregularity in the Notation, as it gave three places to the units of the first two periods (viz.: units and thousands), and six places to the next denomination. The French method, which gives three places to the unit of each period, is fully adopted in this country, and must soon become universal.

Notation and Numeration.

Rule for Notation.

I. Begin at the left hand and write each period in order, as if it were a period of units:

II When the number, in any period except the left-hand period, can be expressed by less than three figures, prefix one or two ciphers; and when a vacant period occurs, fill it with ciphers.

Rule for Numeration.

I. Separate the number into periods of three figures each, beginning at the right hand:

II. Name the unit of each figure, beginning at the right: III. Then, beginning at the left hand, read each period as if it stood alone, naming its unit.

Examples for Practice.

Express the following numbers in figures. 1. Six hundred and twenty-one.

2. Five thousand seven hundred and two.

3. Eight thousand and one.

4. Ten thousand four hundred and six.

5. Sixty-five thousand and twenty-nine.

6. Forty millions two hundred and forty-one.
7. Fifty-nine millions three hundred and ten.

8. Eleven thousand eleven hundred and eleven.

9. Three hundred millions one thousand and six.

10. Sixty-nine billions three millions and two hundred.

Let the pupil point off and read the following numbers; then write them in words:

Let each of the above examples, after being written on the blackboard, be analyzed as a class exercise; thus

1. In how many ways may the number 97 be read ?

1st. The common way, ninety-seven.

2d. We may read, 9 tens, and 7 units.

2. In how many ways may 326 be read?

1st. By the common way, three hundred and twenty-six. 2d. Three hundred, 2 tens, and 6 units.

3d. Thirty-two tens, and six units.

3. In how many ways may the number 5302 be read? 1st. Five thousand three hundred and two.

2d. Five thousand, three hundred, 0 tens, and 2 units. 3d. Fifty-three hundred, 0 tens, and 2 units.

4th. Five hundred and thirty tens, and 2 units.

4. In 65042, how many ten thousands? How many thousands? How many hundreds? How many tens? How many units?

5. In 742604, how many hundred thousands?

How many

ten thousands? How many thousands? How many hundreds? How many tens? How many units?

Let the pupil express the following in figures:

32. Forty-seven quadrillions, sixty-nine billions, four hundred and sixty-five thousand, two hundred and seven.

33. Eight hundred quintillions, four hundred and twenty-nine millions, six thousand and nine.

34. Ninety-five sextillions, eighty-nine millions, eighty-nine thousand, three hundred and six.

35. Six quintillions, four hundred and fifty-one billions, sixty five millions, forty-seven thousand, one hundred and four.

36. Nine hundred and ninety-nine billions, sixty-five millions, eight bundred and forty-one thousand, four hundred and eleven.

Formation of Numbers.

20. ONE refers to a single thing, and has no reference to kind or quality. It is called an Abstract Unit: hence, AN ABSTRACT NUMBER is one whose unit is abstract.

21. A DENOMINATE NUMBER is one whose unit is named, or denominated: thus, three feet, four dollars, five pounds are denominate numbers.

22. A CONCRETE NUMBER is a denominate number which carries with it the idea of matter; as six pounds,

one ton.

23. A SIMPLE NUMBER is a single unit, or a single collection of units, either abstract or denominate.

Two numbers are of the same denomination when they have the same unit; and of different denominations when they have different units.

24. A COMPOUND DENOMINATE NUMBER is one expressed by two or more different units; as, 1 yard 2 feet 6 inches.

Laws of the Units and Scales.

25. We have seen that when figures are written by the side of each other, thus,

678904,

the language implies that ten units, of any place, make one unit of the place next to the left.

When figures are written to express English Currency, thus,

he language implies, that four units of the lowest denomination

20. To what does one refer? What is it called? To what does one foot refer? What is it called? What is an Abstract Number?21. What is a Denominate Number?-22. What is a Concrete Number?-23. What is a Simple Number? When are two numbers of the same denomination? When of different denominations ?-24. What 1s & Compound Denominate Number?