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 Perial Units.

Quintillions. Quadrillions,

Trillions.

Billions.

Millions.

Thousands.

!

894

21 6 6 3 6

625

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, Nonil

糖

lions, 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 to Millions. They were read, Millions, Tens of Millions, Hundreds o Millions, Thousands of Millions, Tens of Thousands of Millions, Hundreds 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-nin housand, 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 hundred and forty-one thousand, four hundred and elever

Formation of Numbers.

20. ONE refers to any single thing, and has no reference to kind or quality. It is called an Abstract Unit.

ONE FOOT refers to a single foot, and is called a Denominate or Concrete Unit.

21. AN ABSTRACT NUMBER is one whose unit is abstract thus, three, four, six, &c., are abstract numbers.

22. A DENOMINATE or CONCRETE NUMBER is one whose unit is denominate or concrete; thus, three feet, four dollars, five pounds, &c., are denominate numbers.

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,

the 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?-21. What is an Abstract Number?22. What is a Denominate Number?-23. What is a Simple Number? When are two numbers of the same denomination? When of different denominations ?-24. What is a Compound Denominate Number?

make one unit of the next higher; twelve of the second, one of the third; and twenty of the third, one of the fourth.

When figures are written to express Avoirdupois weight, thus,

he language implies, that 16 units of the lowest denominatio make one unit of the next higher; 16 of the second, one o the third; 25 of the third, one of the fourth; 4 of the fourth, one of the fifth; and 20 of the fifth, one of the sixth. All the other compound denominate numbers are formed on the same principle: hence,

We pass from a lower to the next higher denomination by considering how many units of the lower make one unit of the next higher.

26. A SCALE is a series of numbers expressing the law of relation between the different units of any number. There are

two kinds of scales-Uniform and Varying.

A Uniform Scale is one in which the law of relation between the units, at any step of the scale, is the same.

A Varying Scale is one in which the law of relation between the units is different, at different steps of the scale.

The Units of a Scale, at any step, are denoted by the number of units of the lower denomination which make one unit of the next higher.

25. When several figures are written by the side of each other, what does the language imply?

In the English Currency, how many units of the lowest denomination make one of the next higher? How many of the second make one o the third? How many of the third, one of the fourth?

In Avoirdupois weight, how many units of the lowest denomination make one of the next higher? How many of the second, one of the third?

26. What is a Scale? How many kinds of scales are there? Name them. What is a Uniform Scale? What is a Varying Scale?